F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 1
CHAPTER 8 (with Fabio Ravagnani)
WALRAS, INTERTEMPORAL EQUILIBRIUM, TEMPORARY EQUILIBRIUM
(first draft 9 november 2005; Part IV added July 2012)
Contents:
Part A: Walras 1
Part B: Intertemporal general equilibrium 18
Part C: Overlapping generations 74
Part D (with Fabio Ravagnani): Temporary equilibrium 99
Appendix 1: Example of intertemporal equilibrium without UERRSP 151
Appendix 2: Mandler's proof of tâtonnement stability under WA 157
Appendix 3: Discounted and undiscounted prices 161
Appendix 4: Lucas and Sargent on learning rational expectations 167
PART I : WALRAS
8.A.1. In chapter 7 it was pointed out that a number of marginalist economists, and with particular clarity Wicksell, saw problems with the specification of the capital endowment of long-period general equilibria as an amount of exchange value.
Probably because of Wicksell’s doubts a pupil of Wicksell, Erik Lindahl, and an
Austrian economist familiar with Wicksell’s writings, Friedrich von Hayek, in the years around 1930 came to an outright rejection of the notion of a single factor of production
‘capital’, measured as an amount of exchange value[ 1 ]. But, like Wicksell, they could not imagine forces determining distribution other than the marginalist interplay of supply and demand functions for factors of production, so they proposed, not an alternative theory of distribution, but a reformulation of the marginal approach believed capable of overcoming the problem. They believed that the fundamental insights of the marginal approach could be maintained while treating each capital good as a distinct factor of production with its own endowment and its own explicit role in production functions – the same specification of the capital endowment as in Walras. Under their influence, a few years later John Hicks adopted the same perspective and helped to spread it by presenting it in 1939 in a very influential book, Value and Capital (usually quoted in 2nd edition, 1946). The notions of general equilibrium developed by these authors are nowadays often called 'Walrasian' because they have in common with Walras the
5).
1
More details on this issue are available in Milgate (1982), Gehrke (2003), Petri (2004, ch.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 2 specification of the equilibrium's capital endowment as a given vector ; but they are otherwise very different from Walras's equilibrium, and the differences are instructive on the roots of some difficulties of the modern versions of general equilibrium theory.
Therefore we must study, however briefly, Walras's own theory.
8.A.2. In his
Eléments d’Economie Politique Pure
(1st ed. 1876-7, 4th ed.
1900[ 2 ]) Walras studies first the general equilibrium of the pure exchange economy; then the one of the ‘acapitalistic’ economy (he calls it the production economy) where there is no production of capital goods (Walras assumes that there are capital goods among the factors, but they are assumed eternal and hence are formally identical to lands), and there is no interest rate, as in chapter 5 here; and then the one of the economy ‘with capitalization’, that is, with production of capital goods, amortization, and a rate of interest. The first two models are similar to the ones studied here in chapters 4 and 5; on the number of firms, Walras is fully in agreement with the argument advanced in chapter
5 here, that the number of firms cannot be taken as given, one must assume industries with CRS. The model ‘with capitalization’ is the one we need to discuss.
Walras aims, as much as Jevons or J. B. Clark or Böhm-Bawerk or Wicksell, at determining a long-period equilibrium. This is shown:
(i) by his characterization of equilibrium as the average situation around which the market oscillates (he likens it to the normal level of water of a lake, always different from the actual level because of waves), and toward which it tends to return via timeconsuming adjustment processes[ 3 ];
(ii) by his assumption of URRSP (§7.7) − uniform rate of return on the supply
2
A fifth edition, published posthumously in 1920, contains no relevant novelty relative to the
4th edition.
3
“It never happens in the real world that the selling price of any given product is absolutely equal to the cost of the productive services that enter into that product, or that the effective demand and supply of services or products are absolutely equal. Yet equilibrium is the normal state, in the sense that it is the state towards which things spontaneously tend under a régime of free competition in exchange and in production.” (Walras 1954, pp. 224-5) And even more clearly: "Such is the continuous market, which is perpetually tending towards equilibrium without ever actually attaining it. ...Viewed in this way, the market is like a lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it. But whereas there are days when the surface of a lake is almost smooth, there never is a day when the effective demand for products and services equals their effective supply and when the selling price of products equals the cost of the productive services used in making them. The diversion of productive services from enterprises that are losing money to profitable enterprises takes place in various ways, the most important being through credit operations, but at best these ways are slow
.” (Walras 1954, p. 380, italics added). This passage is present from the first (1874) to the last (5th, also called 4th définitive) posthumous edition (1926).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 3 price (equal to minimum average cost) of capital goods −, and by his traditional description of what would bring such a result about: changes in the relative endowments of the several capital goods[
4
];
(iii) by his neglect of the changes that equilibrium relative prices may be undergoing over time( 5 );
(iv) by his description of the tâtonnement[ 6 ], in the first three editions of his treatise, as involving actual productions and exchanges of disequilibrium quantities;
(v) by the absence of any consideration of fixed factors of firms – all factors are freely variable and earn the same rental in equilibrium; for durable capital goods, which are not easily transferable across firms once installed, this can only result from nonreplacement in the industries where they earn less, and production and installation in the
4
The following passage is present in all editions of Walras’s treatise: “Capital goods proper are artificial capital goods; they are products and their prices are subject to the law of cost of production. If their selling price is greater than their cost of production, the quantity produced will increase and their selling price will fall; if their selling price is lower than their cost of production the quantity produced will diminish and their selling price will rise. In equilibrium their selling price and their cost of production are equal." (Walras, 1954, p. 271). Walras’s
“selling price” is what nowadays we would call demand price , the value obtained by capitalization of expected future net rentals, i.e. (v i
–m i
P
Ki
)/r in the symbols of ch. 7, which,
Walras says, must be equal to P
Ki
in equilibrium; our equations (E’) of ch. 7 are therefore
Walras’s own (with the sole difference that, for simplicity, we have neglected the insurance charges which Walras subtracts from capital rentals in addition to amortization charges in order to obtain the net rentals, and which may be interpreted as a way to take account of risk). In this quotation Walras admits that the equality between “selling price” and “cost of production” of capital goods, equivalent to uniformity of rates of return on supply price, is brought about by changes in the endowments of capital goods that change their scarcity.
5
As explained in the previous footnote, Walras’s system of equations includes equations equivalent to our equations (E’) of ch. 7, where the prices of capital goods are implicitly assumed constant through time, as shown by the fact that the rate of return on the purchase of a newly produced durable capital good does not include any appreciation or depreciation of the capital good due to changes in relative prices over time. Also, Walras determines the purchase price of land (§236: 1954, p. 270) as its rental divided by the rate of interest, the capitalization formula for perpetual constant yields and a constant rate of discount.
6
Up to the 3rd edition of his
Eléments
, Walras describes the process of trial-and-error through which the economy tends toward equilibrium as a tâtonnement (French for 'blind groping') including the actual production and sale of disequilibrium quantities of products at disequilibrium prices; he envisages this tâtonnement as proceeding in successive ‘rounds’; in each round, at provisionally fixed factor rentals, a certain quantity is produced of each product, and the price necessary to sell it all (a sort of very-short-period partial-equilibrium price) stimulates an increased production in the next round if it is higher than average cost, a decrease in production if lower than average cost. When quantities produced render prices equal to average costs, the resulting excess demands for factors determine changes in factor rentals; the rounds of produced quantities start again at the new factor rentals, and the thing is repeated until factor demands adjust to supplies at product prices equal to average costs.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 4 industries where they earn more, that is, from long-period adjustments.
But Walras, alone among the founders of the marginal approach, does not treat capital as a single factor of variable 'form'; his equilibrium is based on given endowments of the several capital good. He seems not to have initially realized that there was a contradiction between his notion of equilibrium and this specification of the capital endowment[ 7 ], nor that his own admission, that the URRSP condition would be brought about by adjustments of the quantities of capital goods (see (ii) above), implied that he should have left the equilibrium composition of capital to be determined endogenously. With the 4th edition there appears a beginning of awareness of problems, evidenced by three important novelties relative to the 3rd edition[ 8 ].
The first novelty is that Walras no longer assumes actual productions and exchanges at each ‘round’ of the tâtonnement, but only the signing of provisional promises of purchase and of sale and delivery, that he calls 'bons' (translated as ‘tickets’ or ‘pledges’); a ‘bon’ is only valid if it comes out that the prices assumed in it are equilibrium prices; otherwise it is scrapped and the agents who signed it are free to propose new contracts. The tâtonnement must then be conceived as a series of ‘rounds’ in each one of which provisional ‘bons’ are signed, but only to be scrapped if it comes out that equality between supply and demand has not been reached on all markets[ 9 ]; in
7
In the lines quoted in footnote 2 above, Walras admits that the adjustments toward equilibrium are “slow”, thus leaving ample time for changes in the endowments of the several capital goods during the adjustments toward equilibrium. It is still an unsolved puzzle why he was unable to realize that this implied that he had no right to take the endowments of the several capital goods as among the equilibrium’s data, rather than endogenously determined. Possibly he was influenced by an initial mistaken conception of circulating capital as analytically eliminable, so that capital could be treated as consisting of durable capital goods only; this may have induced him to think he could treat the quantities of capital goods as nearly as persistent as, say, the quantity of labour. This would explain how in the equilibrium equations of the production economy he felt he could include, among the given factor supplies, given quantities of capital goods.
8
No clear explanation of the origin of these novelties can be gleaned from Walras’s 4 th edition nor from his other writings or correspondence. But a repeated stress on the assumption that newly produced capital goods only enter production in a period subsequent to the one for which equilibrium is determined suggests that the origin may have been the extension, in a 1899 paper, of the general equilibrium equations to include given inventories of goods already produced and waiting to be sold, including inventories – that is, endowments – of circulating capital goods; reflection on how to treat inventories may have helped Walras to realize that disequilibrium productions would alter them, and that the same problem in fact arose for all capital goods.
9 Walras’s description of the tâtonnement is rather different from the modern one, but we cannot stop on this. In modern general equilibrium literature, it is assumed that a special agent,
→
4

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 5 the next round of the tâtonnement new ‘bons’ are signed, obviously at higher prices than in the previous round for the goods and factor services whose excess demand had been positive in the previous round, at lower prices in the opposite case. The ‘bons’ become binding only when equilibrium is reached. Thus now during disequilibrium the economy is, as it were, congealed; there is no exchange and no production. Walras clearly intended to prevent the endowments of capital goods, and hence the equilibrium itself, from being altered by disequilibrium adjustments; but his way to avoid this problem undermines the significance of the equilibrium thus reached, because in real economies adjustments are not instantaneous, and will include changes in the amounts of capital goods, hence if the economy tends to some equilibrium it cannot be the one defined on the basis of the initial capital endowments, and the usefulness of the latter equilibrium as indication of the tendency of the economy becomes therefore unclear[ 10 ].
8.A.3. In order to grasp the second and third novelty, one must understand
Walras’ system of equations. Walras assumes product prices equal to average costs, and quantities produced of consumption goods equal to demands at those prices; he also assumes equality between supply and demand for services of all factors, including capital goods. No difference here from the long-period equilibrium of ch. 7. He differs in the treatment of capital, investment, and savings. The main difference is the absence of the notion of capital as a single factor of variable ‘form’ (necessarily measured as an amount of exchange value). Walras considers each capital good as a separate factor, with its given endowment. This also implies that he has no notion of a demand curve for value capital, from which one might derive an investment function decreasing in the rate of interest; so his way of arguing the stability of the savings-investment market is necessarily different from the one described in ch. 7.
To describe the resulting system of general equilibrium equations we need not add to the equations written down in chapter 7, §§7.6-7.8. Walras’ system corresponds to equations (A’), (A’-K), (B’), (B’-K), (C’), (D’), (E’), (H’), but with the difference that the variables X
1
,...,X h
– the endowments of the several capital goods – are now given quantities, i.e. they are data of the equilibrium; correspondingly, the total value the auctioneer , exists and, at each round, proposes a price for each good and service, collects the intentions of agents at those prices, checks whether the ‘bons’ realize equilibrium or not, and if not, declares the ‘bons’ not valid, and proposes new prices. This fairy-tale picture ensures that, at each moment of the tâtonnement, there is a unique price for each good or service.
10 Walras seems not to realize this difficulty; the passages quoted in footnotes 2 and 3 are maintained in the 4th and 5th editions.
5

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 6 endowment of capital K* is no longer a datum and has no role to play, it can be eliminated from the aggregate savings function S(p,v,r,K*) that appears in equation (H’); this savings function results now from choices of consumers whose given endowments include physical amounts of the several capital goods[ 11 ]. Equation (K') of §7.7 does not appear among the equations of the Walrasian system, so K* disappears completely from the system of equations once it is eliminated from the savings function. Another difference is that the first h of equations (D’) must be re-written as inequalities like the other n–h ones, since it is now possible that the demand for the endowment of a capital good be less than the endowment even when the rental of that capital good falls to zero.
The meaning of this difference in the list of equations will become clearer if we first consider the model of a simpler economy, that will allow a direct comparison of a long-period and a Walrasian determination of equilibrium. The economy is so simple that we can afford to make explicit the cost minimization conditions and the determination of consumer demand, that so far we assumed already solved and therefore not needing explicit listing (see §5.26).
Suppose an economy where a single consumption good, c, which is the numéraire, is produced in yearly production cycles by labour, land, and two circulating capital goods, according to a differentiable CRS production function q c
=F(L c
, T c
, K
1
, K
2
). The capital goods do not need the index c in F(·) because they are only used in the production of the consumption good. The two capital goods are produced in yearly production cycles by labour and land alone, according to differentiable CRS production functions q k1
=G(L k1
,T k1
), q k2
=H(L k2
,T k2
). Factor rentals, paid at the end of the year, are v
L
for labour, v
T
for land, v
1
and v
2
( gross rentals) for the two capital goods. The prices of the two capital goods are p
1
and p
2
; r is the rate of interest. For simplicity assume rigid supplies, equal to the endowments, of labour L* and of land T*. The endowments of the two capital goods, X
1
and X
2
, are endogenously determined variables in long-period equilibrium, while they are given (and then indicated as X
1
*, X
2
*) in Walras. Cost minimization requires factor rentals equal to marginal revenue products (partial derivatives are indicated as F
L
, F
T
, etc.); now these conditions will be explicitly indicated.
Let us first see the determination of the long-period equilibrium. For that, we know that we need a given supply of value capital K*, the treatment of the two capital endowments as variables, and the condition of a uniform rate of return on supply price,
11
Walras takes the capital endowments of each consumer as given; for simplicity I assume again that we can disregard how the total endowment of each capital good is distributed among consumers.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 7 equal to the rate of interest. The static assumption (no net savings) implies that the demand for the consumption good equals the economy’s net income. Accordingly, the long-period equilibrium equations are the following (for simplicity I neglect the possibility of inequalities).
1 q c
= F(L
2 v
L
= F
L
3 v
T
= F
T
4 v
1
= F x1 c
,T
[5 v
2
= F x2
] c
,K
1
,K
2
) 16 Q c
= q c
17 q k1
= D k1
18 q k2
= D k2
19 L d
=L c
+L k1
+L k2
20 T d =T c
+T k1
+T k2
21 L d =L*
22 T d =T*
6 1[=p c
]=(v
L
L c
+v
T
T c
+v
1
K
1
+v
2
K
2
)/c
7 q k1
= G(L k1
,T k1
)
8 v
[9 v
10 p
11 q
12 v
L
1
L
[13 v
= p
T k2
T
= p
= p
1
= (v
= p
1
2
G
L
2
H
L
G
L
= H(L
L
H
T
T
] k1 k2
+v
]
,T
T k2
T
) k1
) / q k1
23 K
1
= X
1
24 K
2
= X
2
25 (v
1
−p
1
)/p
1
= r
26 (v
2
−p
2
)/p
2
= r
27 K
1
= D k1
28 K
2
= D k2
29 p
1
X
1
+p
2
X
2
= K*. 14 p
2
= (v
L
L k2
+v
T
T k2
) / q k2
15 Q c
= v
L
L*+v
T
T*+rK*
Equations 1-6 determine p c
=MinAC c
for the consumption good; one of these equations is not independent of the other ones (this is why one of them is in square brackets) owing to Euler’s Theorem (see §5.26). So they correspond to equations (A’) with the difference that they specify the conditions for the determination of MinAC c
.
Equations 7-14 do the same for the two capital goods, and correspond to equations (A-
K’); two of these equations are not independent. Equations 15 and 16 correspond to equations (B’), the first one specifies the demand for the consumption good, equal to the economy’s net income; the second one equates supply and demand for the consumption good. Equations 17 and 18 establish the equality between the production of new capital goods and the demand for them, and correspond to equations (B-K’). Equations 19 and
20 determine the demand for labour and for land, so they correspond to equations (C’); the demands for the existing capital goods do not need further equations determining them because they coincide with the quantities K
1
and K
2
used in the production of the consumption good. Equations 21−24 determine the equality between supply and demand for physically specified factors, and correspond to equations (D’). Equations 25 and 26 specify the uniformity of rates of return on supply price and their equality to the rate of
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 8 interest, and correspond to equations (E)[ 12 ]. Equations 27 and 28 determine the quantities produced of new capital goods as the ones needed to replace the used-up capital goods, and correspond to equations (F’). Equation 29 is the form taken in this economy by equation (K’), the condition of equality between endowment and demand for capital the single value factor of variable ‘form’.
We know that three of these equations are not independent owing to Euler’s
Theorem. One more equation is not independent because of Walras’ Law: one of the conditions of demand=supply can be derived from the other ones. So the independent equations are 25. The variables to be determined are 25 too: q c
, L c
, T c
, K
1
, K
2
, v
L
, v
T
, v
1
, v
2
, q k1
, L k1
, T k1
, p
1
, q k2
, L k2
, T k2
, p
2
, Q c
, D k1
, D k2
, L d , T d , X
1
, X
2
, r.
Let us now see how Walras would have formulated the equilibrium equations of this economy. He would have taken the initial endowments of the capital goods as given :
X
1
* and X
2
* replace X
1
and X
2
and are now data, not variables to be determined. This means one variable less than the number of independent equations (equation 29 disappears as no longer necessary), and if there were h>2 capital goods, it would mean h−1 fewer variables than equations. Hence a clear overdetermination, if one keeps assuming a well-defined composition of investment. The important thing here is not the absence of net savings, but whether the composition of gross investment must obey some pre-defined criterion, for example that the growth of the endowments of capital goods must maintain all of them equally profitable also in the future, which can generally be reasonably approximated by having them all increase at the same speed: then one might assume positive net savings, for example that the demand for the consumption good Q c
is less than net income depending on a given propensity s to net savings, with a modification of equation 15:
15’ Q c
= (1-s)(v
L
L*+v
T
T*+rK*), and that the remainder of net income goes to demand new capital goods so as to cause their endowments to increase all in the same proportion: this would mean a modification of equations 27 and 28 into, for example, equations
27’ q k1
/ q k2
= X
1
/ X
2
28’ p
1
(q k1
−K
1
)+p
2
(q k2
−K
2
) = v
L
L*+v
T
T*+rK* − Q c
.
12 If one supposes the factor employments per unit of output (the technical coefficients) to have been determined for all three goods, then by replacing v
1
with (1+r)p
1
and v
2
with (1+r)p
2 one can write the price equations as establishing that for each good the price must equal the wage and land rent cost, plus the purchase cost of the capital goods multiplied by (1+r), obtaining a system of equations equivalent to Sraffa’s (except that here only the consumption good uses the capital goods).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 9
This reformulation makes it explicit that there are two issues involved in the determination of investment, its composition (here determined by equation 27’) and its amount (here determined by the condition that the value of net investment must equal the value of net savings, equation 28’)[ 13 ]. To take as given the endowments of the two capital goods still causes one less variable than equations, and with h>2 capital goods it would cause h−1 fewer variables than equations (h−1, and not h, because equation 29 disappears as now useless). Hence a clearly overdetermined system. Walras’ way out is not to make any assumption as to the composition of investment. He does assume positive net savings, hence for this economy he would modify equation 15 into something like equation 15’ and he would add an equation imposing the equality of net savings and net investment like our equation 28’, but he would not add any equation corresponding to 27’. What then according to him would determine the composition of investment is explained below.
8.A.4. This simple economy hopefully has made the difference between a longperiod equilibrium and Walras’ equilibrium clear. Now it should be easier to grasp the meaning of the more general list of equations of a Walrasian general equilibrium, derived from the equations of chapter 7. The equations raise no problem of interpretation for the factor markets and the markets of consumption goods. Consumers, on the basis of their given endowments and of prices, formulate consumption decisions and gross saving decisions that aggregate to a total represented by the savings function S(p,v,r); net savings are assumed by Walras to be positive; equation (H’) imposes that savings translate into investment (Say's Law). But, again, this system of equations includes no equation determining the composition of investment. Relative to the long-period system of equations (A’, A-K’, B’, B-K’, C’, D’, E’, F’, K’), Walras drops h variables because he takes as given the h endowments of capital goods, and he has h fewer equations: he drops equation (K’) which is now unnecessary (and the datum K* in it), he drops the h equations (F’) that determine amount and composition of gross investment[ 14 ], and he adds one equation that determines the total amount of gross investment by establishing its equality with gross savings, equation (H’) (but without the variable K* in it). So we
13
The two equations 27 and 28 too embody a condition of equality between net savings
(zero) and net investment (zero), and a condition establishing the composition of gross investment: they are indeed equivalent to the following two equations: p
1
(q k1
−K
1
)+p
2
(q k2
−K
2
)=0; q k1
/q k2
=K
1
/K
2
.
14
Equations (F’) can be seen as embodying one condition of equality between gross savings and gross investment, and h−1 conditions establishing the composition of gross investment.
Reformulate them to make this explicit, as an easy Exercise .
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 10 are back to as many equations as variables, but the determination of the composition of investment has been dropped without being replaced by anything explicit.
Walras’s idea is that the composition of investment is determined by equations
(E’), that is, by the condition of URRSP, because if a capital good yields a higher-thanaverage rate of return on its supply price, in the tâtonnement with provisional 'bons' the amount of it that firms intend to produce will increase because savers increase their demand for it[ 15 ], and this – Walras argues – reduces the rate of return on the purchase of that capital good, bringing it back toward equality with the other rates of return, and conversely for rates of return below the average. In the first three editions this idea that the URRSP condition embodied in equations (E’) is achieved by changes in the composition of investment, and therefore that its achievement also determines the composition of investment, rests essentially on the idea that increased production of a new capital good reduces the rental of that type of capital good because it makes it less scarce[ 16 ]; but this is in contradiction with the treatment of the capital endowments as given, part of the data of equilibrium; Walras confusedly reasons as if the production of new capital goods could be considered to alter the endowments of capital goods.
With the 4th edition Walras, evidently having realized the mistake, radically changes the argument (this is the second novelty ), dropping all mention of a negative effect of an increased production of a new capital good on its rental, and relying only on the argument[ 17 ] that the increased production of the capital good will cause an increase in the rentals of the factors used in relatively higher proportion in its production, thus bringing about an increase in its supply price and hence a reduction of the rate of return on supply price yielded by an unchanged rental[ 18 ].
8.A.5. But this effect is generally unable to guarantee the reaching of URRSP. If for example a capital good’s given endowment is so abundant that its marginal product is
15
This is, assuming that consumers invest their savings into the direct purchase of capital goods to be then rented out to firms. But it would make no difference to assume that it is firms that purchase the new capital goods, with savings borrowed from the savers.
16 That is, Walras illegitimately treats the production of new capital goods as altering the scarcity of existing capital goods (it is this scarcity that determines capital rentals), without realizing that this is in harmony with his persuasion that it is changes in the endowments of capital goods that render their demand prices equal to their costs of production (see the passage quoted in footnote 3), but it is in contradiction with his taking those endowments as given.
17
Already mentioned in the previous editions, but in a subordinate role.
18 This is an application of the role of changes in demand composition on relative factor rentals in the marginal approach, studied in §3.6.5 and §5.28.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 11 zero, then that capital good’s gross rental will be zero, implying a negative net rental and a negative rate of return on supply price, rendering impossible to satisfy equations (E’)
(or both equations 25 and 26) if a positive rate of return can be obtained on other capital goods (as Walras obviously assumes). And even if no gross rental becomes zero, still there is no guarantee that equations (E’) can be satisfied, because changes in the composition of investment can easily have a very limited effect on the relative costs of production of capital goods (factor rentals are in all likelihood not much affected by changes in the quantity produced of a single capital good). Thus the tâtonnement, in reallocating savings toward the new capital goods offering a higher rate of return, might easily bring to zero the amounts produced of many capital goods, concentrating investment on very few capital goods, even only one[ 19 ].
Walras realizes it, and we have the third novelty : he does not modify his system of equations but in an unobtrusive passage first introduced in that edition he writes[ 20 ]:
If we suppose that old fixed capital goods proper of the types (K), (K’),(K”), (K”’) ... are already found in the economy in quantities Q
K
, Q
K'
, Q
K''
... respectively [...] it is not at all certain that the amount of savings E will be adequate for the manufacture of new fixed capital goods proper in just such quantities as will satisfy the last l equations of the above system. In an economy like the one we have imagined, which establishes its economic equilibrium ab ovo, it is probable that there would be no equality of rates of net income. Nor would such an equality be likely to exist in an economy which had just been disrupted by a war, a revolution or a business crisis. All we could be sure of, under these circumstances, is: (1) that the utility of new capital goods would be maximized if the first new capital goods to be manufactured were those yielding the highest rate of net income, and (2) that this is precisely the order in which new capital goods would be manufactured under a system of free competition. On the other hand, in an economy in normal operation which has only to maintain itself in equilibrium, we may suppose the last l equations to be satisfied (Walras, 1954, p. 308: §267 [of fourth and fifth editions]; for the French original cf. Walras, 1988, pp. 430–1).
19 A further reason (not seen by Walras) why this might happen is that a capital good can be self-intensive , i.e. it can utilize itself as a factor of production in a higher-than-average proportion: then an increase in its production raises its rental, with a clear danger of instability.
20
In the quotation, ‘rate of net income’ is Walras’s term for rate of return on supply price; the
‘last l equations’ are the equations corresponding to our equations (E’); E is Walras’s symbol for gross savings.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 12
Here Walras admits that the amount of savings may easily not be ‘adequate’ for changes in the composition of investment to bring about URRSP. He thus admits that his system of equations is generally devoid of solutions. This amounts to admitting a contradiction between the given endowments of the several capital goods and the assumption of URRSP; the latter condition, he now admits, will only hold “in an economy in normal operation” (that is, an economy where the proportions among endowments of capital goods have already adjusted so as to yield a URRSP).
Walras does not consider this problem as undermining his entire theory; a new sentence, introduced in the 4 th
edition several pages before the one just quoted, and actually incomprehensible at that point of the exposition, is located at the end of a discussion of how the equality of ‘rates of net income’ is reached, and specifies that that equality “will be satisfied for those equations which survive after the elimination of those new capital goods which it is not worth while to produce” (Walras, 1954, p. 294;
§258 of fourth and fifth editions; translation modified on the basis of Walras, 1988, p.
401)[ 21 ]. Thus Walras has realized that, in the adjustment process described above that alters the composition of investment, the quantity demanded of a new capital good may decrease even to zero without raising its rate of return on supply price to the same level as for other capital goods; but he thinks it sufficient to let this reduction of the number of new capital goods produced in positive amount continue, until only those new capital goods are produced for which a URRSP obtains[ 22 ].
21
The idea that there may be capital goods “which it is not worth while to produce” is incomprehensible at that point because it has never appeared up to that point, and the passage is not accompanied by any explanation. This suggests that the changes introduced in the fourth edition were rushed.
22
However, this solution may well entail the implausible result that only one capital good is produced. Furthermore, Walras forgets that some capital goods may have zero endowments in the period considered but may be worth producing (for example, capital goods just invented); the rentals of these capital goods are not determined by his equations, which are therefore unable to indicate whether it is convenient to produce them. This again reveals the long-period nature of Walras’s original notion of equilibrium, where this problem would not arise because the composition of capital has already adjusted. It is opportune to clarify that the important thing is whether the URRSP condition is satisfied relative to capital goods in use : even in a normal position there will be capital goods that it is not worthwhile to produce because it is not convenient to use them; but the equilibrium endowments of these capital goods will be zero; capital endowments will be positive only for the capital goods that it is convenient to utilize, so the URRSP condition is satisfied for all capital goods that appear with positive quantities in the economy’s equilibrium factor endowments . In Walras’s equilibrium some capital goods, that in a long-period position would be utilized and produced, may be in use but not produced because their initial endowments are excessive.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 13
8.A.6. Walras does not seem to have realized the implications of his admission.
First, he seems unable to realize that his modified ‘equilibrium’ does not have the persistence needed for the role of centre of gravitation of time-consuming adjustments that he continues to assign to it[ 23 ]; already the need to assume an adjustment on ‘bons’ should have awakened him to the contradiction; the quick changes in relative capital endowments implied by the non-production of some capital goods should have made the thing even more obvious.
Second, he continues to assume that all units of a factor earn the same rental, which outside a long-period position is highly unlikely for durable capital goods, which are not easily transferable from one to another employment, or for lands with fixed plants built on them: transferability to other uses is essential for competition to bring about the same rental for all units of the same factor; indeed the absence in Walras of a fixed factors-variable factors distinction in the production functions of firms renders evident that Walras considers all factors as variable, a long-period assumption.
Third, he does not realize he can no longer neglect price changes over time (see
(iii) in §8.A.2): a composition of investment possibly radically different from the one that would maintain the composition of capital unchanged can cause very quick changes of the scarcity of the several capital goods, hence of their rentals, hence of all costs and prices, and investors will be conscious of this fact; therefore Walras has no right to assume that investment decisions are based on the expectation of future rentals of durable capital goods equal to the current ones; but he continues to assume it, as if his equilibrium were still determining very persistent relative prices. This problem in particular explains the forms taken by modern general equilibrium theory, see §8.B.1.
8.A.7. Walras on investment . Walras does not conceive capital as a single factor of variable 'form'; he seems unable even to grasp the presence of that conception in other economists of the period. He never entertains the idea that the endowments of the several capital goods might be endogenous variables, rather than data of the equilibrium. But then, how does he obtain the adjustment of aggregate investment to aggregate savings? I have argued in ch. 7 that it was from the view of the investment function as a reducedscale copy of the demand function for the stock of capital (the single factor of variable
‘form’) that the thesis could be derived that investment, the flow, was a decreasing function of the rate of interest, and that therefore the rate of interest was capable of
23
The 4 th
and 5 th
editions of the
Eléments
still contain the passages quoted in fn. 2 above.
13

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 14 bringing investment into equality with full-employment savings, a necessary condition for the full employment of all factor supplies. Walras's equilibrium does have the full employment of factor supplies, so it too needs equilibration between full-employment savings and investment; how is it achieved?
Walras synthesizes the answer in the Preface to the 4th edition of the
Eléments
.
There, for simplicity disregarding depreciation and insurance, he writes his equation corresponding to our equation (H’) as follows (1954, p. 46):
D k p k

D k ' p k '

D k " p k " i

...

F e
( p t
...
p p
...
i )
On the right-hand side of this equality, F e
(∙)
is Walras’s way to indicate the aggregate savings function.On the left-hand side, D k
, D k'
, ...
, are the quantities of newly produced capital goods of type k, k’, …
“determined by the condition of equality between their selling price and their cost of production”; p k
, p k’ , ...
, are “the prices of the services of the capital goods ... determined by the theories of exchange and production”[ 24 ]; their capitalization via division by the rate of interest i means that p k
/i, p k’
/i , etc., are the demand prices (selling prices) of new capital goods[ 25 ], so the fraction on the left-hand side is the aggregate value of the production (and sale) of new capital goods evaluated at supply prices (costs of production) equal to their demand prices, so it is aggregate investment. (Note the treatment of rentals as if unchanging for the infinite future.) The adjustment will happen, Walras argues, because “The left-hand side of the above equation constitutes the supply of new capital goods in terms of numéraire, and is manifestly a decreasing function of i ... Equality between the two sides of the equation is achieved through an increase or decrease in the price of new capital goods brought about by a fall or a rise in i
” (Walras 1954, p. 46).
What Walras has in mind is therefore the following: if there is excess supply of savings the rate of interest decreases, and this raises the demand price of new capital goods because their unchanged rentals are capitalized at a lower interest rate ; furthermore, this rise stimulates an increased production of new capital goods; for both reasons, the value of the production of new capital goods (investment) rises, tending toward equality with savings.
24 The symbols p k
, p k’
, ...
, refer to gross rentals in Walras’s treatise, but in this Preface they indicate net rentals because here Walras for simplicity is assuming zero depreciation, that is, eternal capital goods.
25
And of the already existing durable capital goods of equivalent efficiency and durability
(Walras assumes radioactive depreciation/efficiency loss, so if the loss of efficiency of a capital good is 10% a year, 11 one-year-old units of it are equivalent to 10 new units of it).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 15
No substitution between capital and labour is going on in this adjustment process; it might seem that capital-labour substitution is not necessary to argue that investment tends to adjust to savings. But Walras is mistaken. He has no right to keep factor rentals, and among them the rentals earned by new capital goods, unchanged as the rate of interest varies.
Walras’ reasoning might be reconstructed as follows. The rentals of capital goods are determined by their scarcity, which depends on their endowments; the amounts produced of new capital goods can only affect these rentals through their affecting the composition of demand for products which in turn influences the demand for factors, but this influence will generally be weak, so as a first approximation one can take the rentals earned by existing capital goods as given; the production of new capital goods takes time to alter the scarcity of the several capital goods, so investors can reasonably assume the new capital goods they buy will earn roughly the same rentals as existing capital goods.
If one assumes differentiable production functions, it is the marginal products of the several capital goods that can be taken as roughly given in this reasoning.
Now, in this reasoning the marginal products that one takes as roughly given can only be the marginal products corresponding to full factor utilization. The mistake lies in assuming the full employment of all factors independently of factor rentals, i.e. independently of income distribution. The point can be more easily grasped by referring to the neoclassical one-good economy where corn is produced by labour and corn-capital
(seed), according to a differentiable production function G=F(L,K). Given the supplies
(assumed rigid) of labour L* and of corn-capital K*, the full-employment marginal products are of course given; but if income distribution is not the equilibrium one, firms will find it convenient to adopt a capital-labour proportion different from K*/L*, and precisely such a factor proportion as will render marginal products equal to the given factor rentals[ 26 ], and therefore different from the full-employment marginal products; with some unemployment of at least one factor (which is precisely what, according to the marginal approach, will cause factor rentals to change in the direction required for them to tend to the full-employment marginal products and thus to make the full employment of all factors possible).
The thing can be put somewhat intuitively as follows: a lower rate of interest will
26
The assumption, made by Walras too, that competition brings product prices to equal costs of production, that is, the absence of pure profits, implies that marginal products will be not only proportional to factor rentals (the condition for cost minimization), but actually equal to factor rentals. In a more complex economy one needs only refer to marginal revenue products instead of marginal products.
15

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 16 induce the owners of capital goods to be content with lower net rentals earned by their capital goods. If production methods in an industry are given and the money purchase cost of capital goods and the money rentals of other factors are given, the result of competition will be a decrease of the product money price since it must cover lower rentals of the capital goods. The decrease of product prices[ 27 ] increases the real rentals earned by labour and land: the lower interest rate changes income distribution in favour of factors other than capital.
So if marginal products exist, they must be treated as determined by income distribution, when income distribution is treated as a parameter that one changes in order to study the stability of equilibrium; if technical coefficients are given, income distribution determines the ratio of product prices to the money rentals of factors other than capital goods, and the net rentals of capital goods adapt so as to yield just the rate of interest over their values. Either way, the rentals of capital goods adapt to the rate of interest, so as to yield just that rate of return on the investment consisting in the purchase of capital goods. So Walras should have admitted that when the rate of interest decreases, the rentals p k
, p k’
, ...
, do not stay unchanged but decrease too, causing no general increase in the demand price of capital goods, and offering therefore no incentive to an increase of investment[ 28 ].
Walras is making here a mistake that will often reappear in later authors, the mistake of treating factor rentals as unchanged when the rate of interest changes, and of relying on the (extra)profits or losses thus appearing, to argue for a generalized tendency of firms to expand or contract production. Any such profits or losses can only be transitory, due to the time required for competition to bring prices to equal the new average cost; the re-establishment of prices equal to normal average costs causes the disappearance of any reason for the increase of investment. No such mistake is made in
27
The decrease of product prices reduces also the purchase costs of new capital goods, reinforcing the tendency of money prices of consumption goods to decrease. If there is also change of production methods, that too reinforces the fall in product prices because it further reduces cost of production.
28
Nor can one defend Walras by suggesting that he might have had in mind that when the rate of interest decreases the reduction of product money prices does not happen, and firms expand because of the positive extraprofits (profits in the neoclassical sense) created by the reduction in costs due to the lower interest rate; as noted by Garegnani (1979, p. 66, footnote ‡), since the approach “assumes that labour and natural resources always tend to be fully utilised, the tendency to increase the scale of production in the aggregate could only lead to changes in the real remuneration of labour and natural resources”, which (this time by raising money wages and money rents) will reduce the rate of return on capital back to equality with the rate of interest, destroying any incentive to increase investment due to positive extraprofits.
16

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 17 the marginalist theory of investment, explained in chapter 7; this theory assumes equality
(neglecting risk for simplicity) between rate of interest and rate of return—hence zero extraprofits—both before and after the change in the rate of interest, and obtains the change in the demand for capital, and hence in investment, from the changes, induced by the changed rate of interest, in consumer choices and in the capital-labour ratios adopted by firms. Unfortunately, this theory rests upon an indefensible conception of capital.
8.A.8. In conclusion, the very high esteem that nowadays Walras enjoys among neoclassical theorists needs reconsideration: he was less clear than the other founders of the supply-and-demand approach on the logical requirements of the (long-period) notion of equilibrium that he, no less than they, was trying to determine; he did not realize the incompatibility between that notion, and given endoments of the several capital goods.
The admission, implicit in the recourse to the unrealistic device of ‘bons’, that realistic adjustments would alter the equilibrium’s data, highlights a problem: since adjustments are not instantaneous, even adjustments working in the direction needed by the neoclassical approach[ 29 ] would not take the economy to the original equilibrium, and one does not know the distance between the original equilibrium and the actual market outcomes. Even if this problem could be neglected, the equilibrium equations remain unsatisfactory because still based on the idea that the prices this equilibrium determines are persistent, which they cannot be. And the justification of the adaptation of investment to savings is unacceptable because it neglects that changes of the interest rate change rentals.
These difficulties had to be faced by the marginalist theorists who in the 1930s resume Walras’s treatment of the capital endowment as a given vector. Their attempts at a solution gave birth to the contemporary, neo-Walrasian theory of general equilibrium that will be studied in the remainder of this chapter.
29 That is, that when a factor is not fully employed and its rental decreases, this would tend to raise the demand for the factor if the data of equilibrium did not change. Of course one can question this presumption too, one example being the criticism of Walras’ arguments supporting the stability of the savings-investment market.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 18
PART B: NEO-WALRASIAN GENERAL EQUILIBRIA
8.B.1. When in the 1930s Lindahl, Hayek and Hicks propose to do without the conception of capital as a single ‘fund’ and hence to return to Walras’s treatment of the capital endowment as a given vector, they are confronted with Walras’s shortcomings.
Differently from Walras, these authors are clear that they are abandoning the attempt to determine long-period equilibria, and that their equilibria are very-shortperiod ones; so they know that a neglect of the changes that relative prices may be undergoing over time is not legitimate, and the agents' equilibrium decisions must be reformulated to take awareness of this fact into account. Two possible solutions are explored.
The first one, discussed in the present Part B, and then in Part C for the versions with overlapping generations, is that of intertemporal equilibria , where future prices are determined simultaneously with relative current prices, through an assumption either of existence, already in the period when equilibrium is established, of markets for all future goods, or of perfect foresight. A market where one buys or sells a ‘future’ (a promise of future delivery of a good) is called a futures market. The second solution, to be discussed in Part D, is that of temporary equilibria (without perfect foresight), where for most goods only spot markets exist, and agents take their decisions in the initial (‘current’) period on the basis of expectations of future prices, expectations that will generally differ among agents and therefore will turn out to have been mistaken for at least some agents.
8.B.2. The notion of intertemporal equilibrium was born as a reinterpretation of the exchange-and-production acapitalistic model without a rate of interest (described in chs. 5 and 6); its variables and equations were reinterpreted as referring to dated commodities. Its first mathematical formulations had then to cover only a finite number of periods because the acapitalistic model has a finite number of commodities.
Nowadays, frontier research is on models extending into the infinite future, but we study first the finite-horizon model.
It is possible to study equilibrium over a continuous time interval, but the interpretation is awkward since transactions are not continuous flows in real economies; so let us restrict ourselves to the versions where time is divided into short periods (Hicks called them ‘weeks’); each period t starts at instant/date t in the flow of time, and ends infinitesimally before instant/date t+1 ; thus the number that identifies a period coincides
18

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 19 with its initial date/instant[ 30 ].
There is an initial date 0, where equilibrium is reached for all exchanges and productions to be carried out not only in that period but also for all subsequent periods up to a terminal date and period T, where the economy ends; the periods go from the initial one, period 0, to period T called the horizon of the economy; markets are assumed to exist at the initial date for all goods, present and future, over all periods up to the horizon. The markets for immediate delivery (that is, for goods of the initial period 0) are called spot markets; the markets for goods to be delivered in the future are called markets in futures , or futures markets. A ‘future’ is a contract for delivery of a good at a future date. All contracts for delivery of goods, both current and future, are signed at the initial date. Prices are now prices to be paid at date 0 for current and future goods.
The reaching of equilibrium relies on the auctioneer, to be imagined as some institution that announces prices, collects intentions of supply and demand[ 31 ], and then changes prices depending on excess demands. At the initial date the auctioneer cries out prices for all present and future goods. The relative prices thus proposed must be interpreted as follows. Distinguish prices by two indices, the second one indicating the date of delivery and the first one indicating the type of good or service; then p it
/p js indicates the quantity of good j which at the initial date one must promise one will deliver at date s in order to obtain in exchange a promise of delivery of one unit of good i at date t. Equilibrium is reached at date 0, and then agents need no further opening of markets, they must only carry out the promises they signed. Iit is implicitly assumed that it is certain that contracts will be honoured; this is called the no default assumption. For the moment we neglect uncertainty; the state of the world in future periods, for example the weather, is assumed known or not affecting decisions taken at date 0.
Endowments too are distinguished according to the date in which they start being available (which is always the beginning of a period); endowments of factors available at the beginning of period t supply services during that period; for example among the endowments there are the endowment of labour of date 0 (that is, of labour services deliverable during period 0), the endowment of labour of date 1, and so on. The
30 Consider time as continuous, starting at a certain instant called date zero, and divided into periods of unit length, that go from date 0 to date 1, from date 1 to date 2, and so on. One must decide whether the period starting at date 0 and ending infinitesimally before date 1 is to be called period zero, or period 1. In common language it would be usually called the first period.
In economic literature, it is usually called period zero, a convenient convention that identifies the numbering of periods with the number that indicates their initial instant or date.
31 The auctioneer must also be supposed to check that consumers respect their budget constraint and firms respect their technological constraints.
19

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 20 endowments at date 0 are called initial endowments. Among the initial endowments there are endowments of capital goods produced in the periods before date 0, and inventories of consumption goods that have just come out of their production process and have just been delivered to consumers; but we will mostly neglect these inventories in what follows). The capital goods extant at date 0 are treated like exhaustible natural resources, how they were produced is irrelevant; now they are simply given quantities of nonproduced factors (in the sense of not produced inside the equilibrium’s periods
).
Thus, like in Walras (and differently from long-period equilibria), the equilibrium's data include a given initial endowment of each capital good. Equilibrium prices therefore depend on the accidents that determined the given initial composition of the capital endowment( 32 ).
From the point of view of the Marshallian classification of equilibria as veryshort-period, short-period, long-period, or secular depending on which supplies of produced goods are given and which are determined endogenously by the equilibrium, this type of equilibrium is a very-short-period equilibrium because at date 0 there are given supplies of producible goods, for example if fish is on offer its supply is given, as in Marshall's very-short-period partial equilibrium of a fish market[ 33 ].
8.B.3. In the next chapter a further reinterpretation of the same model will be presented, that intends to enlarge the same formal system of equations to cover uncertainty too. It consists of admitting that promises of delivery can be conditional on the realization of a certain state of nature, in which case the goods traded are contingent commodities , i.e. promises of delivery of commodities contingent on the realization of certain events (an example is “you will receive 10000 dollars if you have an accident during the trip you start tomorrow”); then the price one is ready to pay for one of these commodities will also reflect the probability one attributes to the corresponding state of nature. Each future period is subdivided into a partition of possible states of nature for that period, and if one wants to be certain to obtain a commodity at date t, then one buys a promise of its delivery for each one of the possible states of nature for that date. The purchase of a contingent commodity can often be seen as a form of insurance. The
32
This dependence will be particularly strong in the first period or periods; in subsequent periods the composition of capital will tend to adapt to the demand for capital goods at their supply prices.
33 Warning: recently the short-period/long-period terminology has been sometimes used as referring to how many periods are covered by the equilibrium; then intertemporal equilibria are called long-period equilibria, a terminology rejected here because it obfuscates the traditional meaning of long-period equilibrium, which is analytically very important.
20

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 21 general equilibrium model with complete markets in contingent commodities over a finite number of dates is what is normally meant by Arrow-Debreu model. But in this chapter the same name will be applied to the model without uncertainty about future states of nature, and therefore without contingent commodities. This model is sometimes called Arrow-Debreu-McKenzie model[ 34 ].
What is generally argued is that, if there are markets for all goods and services for all dates from the initial one to the horizon date T, then the intertemporal equilibrium is formally identical to the equilibrium of the acapitalistic economy without capital; only a reinterpretation is argued to be necessary: prices now indicate the prices to be paid at the initial date for (promises of delivery of) present and future commodities; utility functions are defined on present and future consumptions; production functions are defined on inputs and outputs of different dates. Let us examine this formal similarity.
THE CONSUMER
The utility function
8.B.4. The utility function of a consumer is now defined on dated commodities.
The consumer decides everything at the initial date, date zero, so the relevant thing is her utility function at that date. Indicate with Q jt
the quantity of good j consumed at date t
(or, equivalently, in period t), and with p jt
the price, to be paid at date zero , of a promise of delivery of one unit of good j at date t; or more concisely, the price of one unit of good (j,t), where the two indices designate the good, and its date of delivery. (Unless indispensable, in what follows we do not use an index to indicate which consumer we are talking of; otherwise we could write Q jt h for the amount of good (j,t) consumed by consumer h.) The endowment of the consumer consists of a vector ω of vectors ω t
of endowments of dated commodities ω jt
; the endowments in ω
0
need some clarification, only at that date besides amounts of (services of) the several types of land or of labour they can include capital goods or inventories of consumption goods; in subsequent periods, endowments can only include (services of) types of land or of labour: even durable capital goods or inventories surviving from the initial endowments cannot be endowments , because the approach views them as different goods, ‘produced’, via jointproduction processes (for capital goods) or via storage processes, and therefore resulting from acts of production and hence bought by the consumer if one distinguishes the
34 Actually there is a difference between the Arrow-Debreu formalization and the McKenzie one: the former authors assume a given number of (potential) firms, possibly with decreasing returns to scale; McKenzie assumes CRS industries owing to free entry, the procedure we argued in ch. 5 to be the more acceptable one.
21

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 22 consumer from the entrepreneur/producer even when the agent is the same.
The consumer's formal utility maximization problem is identical to the one for the acapitalistic economy, with the sole difference that consumption goods and endowments have two indices instead of one (it would however be possible, if one so wanted, to renumber goods progressively so as to assign a single index number to each one, and thus reach complete formal identity with the acapitalistic economy): maximize u(Q
10
,..,Q n0
,Q
11
,..,Q n1
,...,Q nt
,..,Q nT
) under the constraint Σ j,t p jt
Q jt
≤Σ j,t p jt
ω jt
.
Now it is better to drop the assumption that we can distinguish factors from consumption goods, because many consumption goods (e.g. sugar) are also inputs to other goods, so let us simply assume n types of goods and services. Neglecting corner solutions, marginal rates of substitution will be made equal to price ratios. Formally the consumer’s UMP is identical to the one in the acapitalistic economy. Only the interpretation is different: the marginal rates of substitution are now possibly intertemporal, like relative prices.
In model building, specific convenient forms of the intertemporal utility function are often chosen. The most widely assumed utility function is the sum of the
‘subutilities’ derived from consumption in each period according to a sub-utility function, that may or may not be period-specific, and may or may not be discounted. Let
Q t
stand for the vector (Q
1t
,Q
2t
,...,Q nt
); sub-utility u t
(Q t
) is also called a felicity function ; the utility function is
U(Q
0
, ... , Q
T
) =
 T
0
δ t u t
(Q t
).
The δ, positive and less than 1, is a constant discount factor that reflects the assumption that consumers prefer present to future consumption[ 35 ]. It is also possible to assume no discounting. It is usually assumed that the felicity function is time invariant.
Of course this utility function is very restrictive, its temporal separability neglects the frequent dependence of the marginal utility of a good in a period on what was consumed in previous periods, and there is no reason why δ should be constant.
35
Empirical evidence suggests that a more appropriate theory of how consumers actually consider consumption at future dates is hyperbolic discounting : the discount factor from t to date zero is not δ t
<1 but 1/(1+kt) with k>0, which decreases slower than δ t
and implies time inconsistency , that is, when a consumer is at time 0, the marginal rate of substitution between consumption at date 2 and consumption at date 3 is different from the one when the consumer reaches date 1 ( Exercise : prove it). This may occasion regrets. The discount factor δ t avoids time inconsistency. But in this textbook we will not further discuss these issues.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 23
In macroeconomic models with a single consumption good c the following felicity function is often assumed: u t
(c t
) =
1 c t
1
 
 
, for σ>0 and ≠1; u t
(c t
) = ln c t
for σ=1.
The resulting U(·) has constant elasticity of substitution[ 36 ] between consumption at any two consecutive dates, equal to 1/σ. When the analysis admits uncertainty, this utility function can be used to describe attitudes toward risk and then σ is the coefficient of relative risk aversion , a notion to be explained in ch. 9; for this reason U(·) with the above felicity function is called the constant relative risk aversion (CRRA) utility function.
Another additive utility function frequently used, again in models with a single consumption good, is the constant absolute risk aversion (CARA) utility function, which has the following felicity function: u t
(c t
) =

1
αe
αc t
.
This function too is discussed in ch. 9.
Time preference, own rate of interest, discounted prices
8.B.5. The marginal rate of substitution between the same good at two different dates can be expressed as including a rate of time preference for that good. I only consider one-period rates of time preference, but rates of time preference for intervals greater than one period might be derived without difficulty. Consider an indifference curve with good (i,t) measured in abscissa and good (i, t+1) in ordinate, and express its slope as –(1+ ϱ it
); then ϱ it
is the (one-period) marginal rate of time preference for good
(i,t). Thus, suppose that at the margin the consumer accepts to give up one (small) unit of good i at date t only in exchange for at least 1.2 units of good i one date later; then the slope of his indifference curve at that point is –1.2 and his marginal rate of time preference for good i between dates t and t+1 is ϱ it
=20%.
All prices are prices quoted and to be paid at date 0 for purchases, agreed at date
0, of present or future goods. As we are not introducing money, the price p it
of good (i,t) indicates the amount of numéraire to be paid to obtain one unit of the good. The relative price p it
/p js
indicates the amount of good (j,s) to be given up to obtain one unit of good
(i,t). The relative price p it
/p i,t+1
of the same physical good at two consecutive dates can be seen as indicating an own rate of interest r it
defined as
36 See §4.11.3.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 24
[8.1] p it
/p i,t+1
= 1+ r it
.
If the own rate of interest, say, of gold between periods 0 and 1 is 0.05, that is, 5%, one can purchase 1 unit of gold at date 0 by paying with a promise of delivery of 1.05 units of gold at date 1: that is, one can borrow gold at date zero at a physical 5% rate of interest, in gold.
Own rates of interest must not be confused with money rates of interest nor with real rates of interest. The issue is discussed below in §8.B.8.
When considering the choice between the same good at consecutive dates, the consumer will be at the tangency between indifference curve and budget line if p it
/p i,t+1
=
1+ ϱ it
. Thus in equilibrium, for interior solutions of the consumer problem, it is r it
= ϱ it
.
The similarity with the discounted or present value of future sums of money suggests an interpretation of the prices established at date zero for future goods in such an equilibrium as present values , or discounted prices . Suppose gold of date zero, good
(1,0), is the general numéraire, hence p
10
=1; consider p
11
, and the associated own rate of interest r
10
=1/p
11
– 1; define the undiscounted price of good (i,t), to be indicated as π it
, as the amount of the same numéraire good (gold) but of date t to be given in exchange for one unit of good (i,t)[ 37 ]; then the undiscounted price of gold is 1 for all dates; and p
11 can be seen as the present value, or discounted price, at date 0 of gold of date 1. For example, p
11
=0.91 if r
10
=10%. Then if p i1
=1.82 and therefore π i1
=p i1
/p
11
=2, one can say that if good (i,1) were paid spot , that is at date 1, with gold of that same date, it would cost 2 units of date-1 gold; then p i1
can be seen as the undiscounted price π i1
=2 discounted to date zero with the own rate of interest of the numéraire, 10%.
The above explains why the prices in the equations of intertemporal equilibrium are called discounted prices. Formally, their role in consumer choices is the same as that of the prices of acapitalistic equilibria; only the interpretation changes.
(An Appendix at the end of the chapter is suggested if you still find discounted and undiscounted prices a bit confusing.)
PRODUCTION
8.B.6. Now let us see how the treatment of production in intertemporal equilibria can be formulated so as to make it formally identical to the one of the acapitalistic model
(A-B-C-D) of chapter 5. Because I want to keep things simple I avoid joint production; in the discussion below all production processes last one period (inputs applied in period
37 If one admits money and suppose money consists of gold, then π it
is the amount of money to pay for a unit of good (i.t) if the payment is spot, that is, made at date t.
24

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 25 t produce their output at date t+1) and all capital goods are circulating capital goods which disappear in a single production cycle. The formalization of production functions is the same as in ch. 5 with the sole added condition that inputs must precede outputs by one period.
The usual assumption that factor rentals are paid after the factor services have been utilized suggests that the rentals of factors utilized in period t (whose endowments have date t as time index) have date t+1 as time index; thus, rental v it
will refer to factor i utilized during period t–1, because, if payments consisted of delivery of an actual good, say corn, after a labourer has worked during period t–1, this delivery would be effected at date t. But our picture is one in which all prices are discounted prices paid at date 0; then labour utilized in period t is paid a discounted wage at date zero. For circulating capital goods, this implies that the discounted purchase price of a capital good bought at date t–1, and the discounted gross rental the same capital good earns at date t (which, in undiscounted terms, would be the purchase price plus interest), coincide in equilibrium.
Still, to make the connection easier between intertemporal prices and the picture of prices paid spot, the general rule adopted here is that rental v it
is the rental paid at t for factor i utilized in period t–1.
It is opportune to distinguish the factor services utilized by firms in the intertemporal economy into two categories:
– services of ‘original’ or ‘nonproduced’ factors, of two types: (i) services of labour(s) or land(s) distinguished by date of delivery of the service: thus the services supplied by a certain land are treated as so many different factor services according to the period in which they are supplied; (ii) services of the capital goods already existing at date 0, which are treated as original factors, analogous to natural exhaustible resources, because how and why they were produced is no longer important;
– services of capital goods produced inside the equilibrium.
The endowments of factors (or factor services) in the first category are data that economic choices cannot modify[ 38 ]; they have the same role as the given endowments of labour and land of the acapitalistic equilibrium. They are the factors that appear in the reinterpreted equations (C) and (D).
The services of the second category, supplied by capital goods produced inside
38
Actually how many children to have is largely a choice, so labour supply should be considered endogenous to an extent, and the more so the longer the time span covered by the intertemporal equilibrium. But the influence of economic variables on population growth is best studied separately, so here labour endowment will be assumed given in each period (not necessarily constant).
25

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 26 the intertemporal equilibrium, can be made to disappear from the equilibrium equations by imagining all production of ‘final’ (i.e. consumption) goods to be performed by vertically integrated firms that produce internally, as intermediate non-marketed inputs, all the intermediate capital goods needed for their final output, and purchase only services of nonproduced factors (including date-0 endowments of capital goods). I give an example.
Suppose that the production of bread at date 3 requires flour and labour of date 2; that the production of flour of date 2 requires corn and labour of date 1; and that the production of corn of date 1 requires corn and labour of date 0 (for simplicity I neglect land). Labour of all dates, and corn of date 0, are 'original' factors. If we imagine the firm producing bread of date 3 to be vertically integrated, then the production function of bread of date 3 will have as inputs only corn of date 0 and labour of dates 0, 1 and 2.
Date-1 corn and date-2 flour need not be considered explicitly, because in a perfectly competitive economy efficiency in the vertically integrated firm will entail exactly the same production methods in each stage of production, and the same average cost, as if each stage were performed by a separate firm and the intermediate inputs were sold by the firms producing them to the firms utilizing them.
This for a simple reason. Assume given rentals of the ‘original’ factors, i.e. of corn of date 0, and of labour of the various dates. Then in equilibrium a profitmaximizing firm producing corn of date 1 must sell it at a price equal to its minimum average cost, MinAC. Hence the firm producing flour of date 2 and buying corn of date
1 has all input costs perfectly determined, and (assuming the same efficiency in all firms
– remember that competition eliminates less efficient firms) if it were vertically integrated the cost of producing internally the corn of date 1 would be the same, since the firm could do no better than minimize that average cost and get to an average cost equal to the price at which it buys corn. Therefore the MinAC of flour is the same whether produced by a vertically integrated firm or not; but then, for the same reason, the MinAC of bread of date 3 is the same whether produced by a vertically integrated firm that starts with corn of date zero, or by a firm that buys flour from other firms[ 39 ].
The above reasoning is general, but I confirm it with a formal reasoning applied to flour, assuming differentiable production functions, and admitting land use too. Now there are further goods besides those considered in the acapitalistic economy: the produced capital goods, ‘intermediate’ goods if they are produced and utilized by vertically integrated firms. Let their quantities too be indicated by q jt
, where the date t
39 There is here an implicit assumption that contracts of sales and purchases do not entail specific costs of contracting, that would be eliminated if firms were vertically integrated.
26

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 27 indicates their date of emergence as outputs, and also their date of purchase and employment as inputs – the use of the same symbol q jt
for all produced goods allows us to consider such goods as beans or sugar which can be both consumption goods and capital goods. (Therefore n is now the number of all goods, consumption or capital goods). The quantities used of ‘original’ factors are indicated as x it
.
Suppose then that labour is factor 1, land is factor 2, corn is good 3, iron is good
4, flour is good 5; assume the non-integrated production function of flour at date 2 is: q
5,2
= f(x
1,1
, x
2,1
, q
3,1
, q
4,1
) where x
1,1
, x
2,1
are labour and land utilized during period 1 (and paid at date 2), while q
3,1 and q
4,1
are corn and iron bought at the beginning of period 1. Assume that the production functions of these two capital goods are
[8.5] q
3,1
= g(x g1,0
, x g2,0
, x g3,0
) and q
4,1
= h(x h1,0
,x h2,0
,x h3,0
), where index g or h indicates the industry in which the input is employed; inputs 1 and 2 are labour and land; input 3 is corn, which being of date 0 is now classed as ‘original’ factor because date 0 is the initial date, its endowment at date 0 is among the data of the equilibrium, and therefore symbol x is used to indicate its quantity. The quantity produced of q
5,2
in the vertically integrated firm can then be represented as
[8.6] q
5,2
= f(x
1,1
, x
2,1
, g(x g1,0
, x g2,0
, x g3,0
), h(x h1,0
,x h2,0
,x h3,0
)) =
= F(x
1,1
, x
2,1
, x
1,0
, x
2,0,
x
3,0
).
The passage from f(·) to F(·), where intermediate goods have disappeared and no distinction appears between the two uses of factors x
1,0
, x
2,0
and x
3,0
, is made possible by the requirement of efficiency in the vertically integrated firm, which imposes that the total quantities employed of factors x
1,0
, x
2,0
and x
3,0
be efficiently allocated among the production processes g and h so as to have (assuming no corner solutions) the same indirect marginal product of flour in the two uses, e.g., for labour of date zero x
1,0
:
[8.7] (
 f/
 q
3,1
)

(
 g/
 x g1,0
)=(
 f/
 q
4,1
)

(
 h/
 x h1,0
) and analogously for the other two factors. This allows one to derive F(·) from f(·). Now we can note that, in competitive equilibrium, [8.7] is verified even when production is not vertically integrated and q
3,1
and q
4,1
are bought by the firm producing q
5,2
. This is because the tangency between isocost and isoquant in the production of q
5,2
requires (in terms of discounted rentals and prices):
[8.8] (
 f/
 q
3,1
)/(
 f/
 q
4,1
) = v
3,2
/v
4,2
= p
3,1
/p
4,1
= (
 h/
 x h1,0
)/(
 g/
 x g1,0
) where v i,t+1
is the discounted rental of capital good q it
, and p it
its discounted purchase price; in equilibrium they coincide, as already noted, because for a circulating capital good the rental, discounted to one period earlier, is the demand price; the last equality derives from the fact that the optimal employment of factor x
1,0
in firms producing q
3,1 and q
4,1
requires the equality between its factor rental v
1,1
and its value marginal product:
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 28
[8.9] p
3,1
·(  g/
 x g1,0
)=p
4,1
·(  h/
 x h1,0
)=v
1,1
.
Once the rentals of the ‘original’ factors of the several relevant dates are given, the vertically integrated firm will therefore adopt the same production processes as the non-integrated firms, and its production function can be formulated with only ‘original’ factors among the inputs.
8.B.7. Once intermediate capital goods are made to disappear inside the production functions of vertically integrated firms (a procedure that alters nothing in the equilibrium conditions), in the equations of the intertemporal equilibrium the only outputs are consumption goods, and the only inputs are the services of ‘original’ factors whose endowments are data: the formal analogy becomes perfect with the acapitalistic economy of equations (A-B-C-D) of ch. 5. Having established this result, one is free to avoid performing the vertical integration and to let the intermediate capital goods appear explicitly with their costs of production and their market prices, as in §6.35 (where it is found more convenient to use the netput notation); the reader is invited to reconstruct, as an Exercise , how equations (A-B-C-D) of ch. 5 should be then modified, so as to arrive at the very concise redefinition of the equilibrium price vector in §6.35. Of course the formal analogy with the acapitalistic economy is still there.
But the possibility to reinterpret the equations of an intertemporal equilibrium over a finite number of periods as equations of an acapitalistic model must not hide a fundamental difference . The acapitalistic model with only labour(s) and land(s) as factors has data, relative to factor endowments, that are sufficiently persistent and independent of disequilibrium events as to allow conceiving the equilibrium as unchanged during a repetition of disequilibrium productions and exchanges, and capable therefore of having the role of centre of gravitation for market prices and quantities (if stability can be argued), and hence the role of indicator of their averages and, with its changes, of their trends. This is no longer true for the intertemporal reinterpretation, because now the data include given endowments of each capital good, and these lack persistence, because altered by disequilibrium production decisions; therefore the equilibrium lacks the persistence necessary to view it as the position around and toward which market prices and quantities gravitate. Its capacity to indicate the average and the trend of market prices and quantities is thereby questioned. This very important problem, already noticed when discussing Walras because due to the Walrasian specification of the capital endowment as a given vector, will be further explored below, in §8.B.15, after pointing out some other aspects of intertemporal equilibria, relevant to this issue.
TWO MEANINGS OF UNIFORM RATE OF RETURN, WITH AN EXAMPLE
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 29
8.B.8. In an intertemporal equilibrium, relative prices of goods of the same date do not, in general, remain the same from one date to another; in symbols, in general
[8.10] p it
/p jt

p i,t+1
/p j,t+1
.
The reason is easy to understand: the arbitrary initial composition of the endowment of capital goods will be in general quickly altered. As a result, factor rentals and costs of production of products will be changing over time.
Since p it
/p jt
 p i,t+1
/p j,t+1
implies p it
/p i,t+1
 p jt
/p j,t+1
, the change over time in relative prices implies that in general own rates of interest are not equal ; remembering that p j,t+1
/p jt
= 1/(1+ r jt
), we obtain that p it
/p jt
<p i,t+1
/p j,t+1
(good i rising in price relative to good j from t to t+1) implies p it
/p i,t+1
<p jt
/p j,t+1
i.e. (1+r it
)<(1+r jt
); that is, the own rate of interest is lower, in equilibrium, for the good whose relative price is increasing . The reason is that there must be no arbitrage opportunity: it must be equally convenient to obtain a good (i,t+1) by lending some amount of good (i,t) or by exchanging good (i,t) with good
(j,t), lending the latter against good (j,t+1), and then exchanging it against good (i,t+1).
So suppose that at date t good (i,t) and good (j,t) have the same value, 1 for both, and that r it
=10% (lending one unit of good (i,t) one obtains 1.1 units of good (i,t+1)); and suppose that from t to t+1 good i rises 20% in price relative to good j, i.e. p i,t+1
/p j,t+1
=1.2; then it must be r jt
=32%, because at date t+1 one needs 1.32 units of good j to buy 1.1 units of good i. In equilibrium all possible indirect exchanges must bring to the same result. (We are assuming there are no transaction costs.)
When there are differences in own rates of interest, then the real rate of interest depends on the choice of numéraire. The adjective ‘real’ is to distinguish it from the money rate of interest. The money rate of interest indicates how much more money one obtains by lending money for one period; the real rate of interest indicates how much more purchasing power (in terms of some basket of goods) one obtains by lending purchasing power for one period; if relative prices are changing from one period to the next, the real rate of interest depends on the basket of goods chosen to measure purchasing power. Thus assume that, from one period to the next, the money price of corn remains unchanged at 100, while the money price of iron rises from 100 to 120, and that the money rate of interest is 20%. Lending the 100 money units corresponding to the value, in the first period, of 1 unit of corn or of one unit of iron yields 120 units of money next period, which means 20% more purchasing power in terms of corn, but 0% more purchasing power in terms of iron. So if one chooses corn as the reference good, the real rate of interest is 20%; if one chooses iron, it is 0%; if one chooses a basket that includes both corn and iron, the real rate of interest will depend on the proportion in which they enter the basket. In intertemporal equilibria too, the real rate of interest measures the change in purchasing power in terms of the chosen numéraire. Once a
29

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 30 numéraire is chosen, the real rate of interest one obtains on a loan is the own rate of interest of the numéraire, whatever the good one is actually lending and the good one is obtaining in exchange the following period. Thus in the example in the previous paragraph, where the own rates of interest are r it
=10% and r jt
=32% because good i appreciates 20% relative to good j, in period t+1 the value of a unit of good j in terms of good i of the same period is 1/1.2=0.833; if good i is the numéraire, all values are expressed in good i; then a loan of one unit of good (j,t) obtains 1.32 units of good (j,t+1) but it means lending a good of value equal to 1 unit of the numéraire good to obtain, one period later, a quantity of good j of value (1.32×0.833) = 1.1 units of the numéraire good
(of that period), i.e. 10% more purchasing power; the real rate of interest is 10%. If we take good j as numéraire, then a loan of one unit of good i against 1.1 units of good i one period later means, in undiscounted value terms, lending a good of value equal to 1 unit of the numéraire good to obtain, one period later, a quantity of good i of value equal to
(1.1×1.2)=1.32 units of the numéraire good (of that period), i.e. 32% more; the real rate of interest is 32%.
Don't confuse this real rate of interest on loans with own rates of interest. The latter are physical percentages, defined for a loan of a unit of a good in exchange for an amount of the same good one period later. The real rate of interest is in terms of purchasing power, or of exchange value if you like.
One important aspect of the above considerations must be kept in mind. The differences in equilibrium own rates of interest does not mean that loans in different goods are differently convenient. In the previous example, to lend good i against 10% more of it the next period, or to lend good j against 32% more of it the next period, are equally convenient, because the difference in own rates of interest reflects and compensates the change in the relative price of the two goods from one period to the next. Therefore, in equilibrium, to invest one’s savings into a firm that produces good i by using good i as input, or into a firm that produces good j by using good j as input, is equally convenient even if the first firm obtains 10% more output than the input it used, while the second firm obtains 32% more. The measure of the rate of return depends on the choice of numéraire, and therefore it is arbitrary within limits; but this does not disturb the equal convenience of all employments of savings in an intertemporal equilibrium, and this finds expression in the equality of real rates of interest once one has chosen a numéraire. One says then that in an intertemporal equilibrium there is an effective uniformity of rates of return on all investments. The numerical expression of this uniform effective rate of return , UERR, depends on the choice of numéraire.
8.B.9. But then, is there no difference between an intertemporal equilibrium and
30

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 31 long-period analysis? The distinguishing characteristic of long-period analysis was said to be, the need to leave the composition of capital to be determined endogenously, otherwise a uniform rate of profit, or uniform rate of return in all investments, was not obtainable. But haven’t we obtained this uniformity here too, in spite of the given initial composition of the capital endowment?
No. The uniform rate of profits of long-period analyses (classical as well as traditional marginalist) is not simply a UERR but a UERRSP, uniform effective rate of return on supply price , i.e. on the costs of production of capital goods[ 40 ]. This means an additional condition, relative to the simple UERR that holds in an intertemporal equilibrium. The additional condition is: the value of capital goods, obtained by capitalization of their future rentals, i.e. their demand price (the maximum price at which one is ready to buy them), must be equal to their supply price or cost of production.
In a neo-Walrasian equilibrium, where the initial endowments of capital goods are given, in the first period and usually also for some subsequent periods, in other words as long as the endowments of some capital goods remain largely exogenous, this additional condition does not generally obtain[ 41 ]. If for example the given initial endowment of a durable capital good is so abundant that its marginal product and thus its rental is zero and is expected to remain zero for some periods, then since the cost of producing a new capital good of that type is positive, the demand price for that capital good in the first period(s) will be inferior to the cost of producing it[ 42 ], and there will be no production
40
I mean by URRSP a uniform rate of return on supply price in a situation of constant relative prices ; the UERRSP condition admits changes of relative prices over time, therefore it is more general and it is the one to be considered to ascertain whether a position is a normal (or long-period) one, in a situation in which relative prices are not constant.
41
Here I admit durable capital to render the argument more intuitive.
42 (This footnote touches upon advanced issues and may be skipped.) Three additional observations can be useful. First, if production is instantaneous (a continuous flow), then the supply price of capital goods can be defined already at the initial moment of the time interval covered by the equilibrium, but a different formalization of the equilibrium is then required, in continuous time rather than in separate periods. Second, if, as assumed in the text, production takes time and is assumed to be in separate cycles, a 'supply price' of the initial endowments of capital goods can be determined only if these capital goods can be consumed, and therefore using them as capital goods requires paying a price – to prevent their use as consumption goods
– which can be called a supply price; a UERRSP condition might still be defined (although it would not be on cost of production), and it will not hold if some of these endowments of capital goods are not utilized in production in period 0, and go entirey to consumption. Third, the holding or not of UERRSP as revealed by whether all capital goods are produced requires defining the rate of return as on cost of production, and therefore it requires that cost of production can be determined, which can be done only for capital goods produced inside the
→
31

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 32 of that capital good. In this case there isn't an effective uniform rate of return on supply price on all the capital goods employed. An analogous situation of demand price inferior to supply price will obtain for the durable capital goods which have been made obsolete by an innovation just prior to when the equilibrium is established; a new type of capital good can supply the same services but it has a lower cost of production; only the new type will be produced; for the old type of durable capital goods the demand price has fallen, and become inferior to the cost of production; demand for them at a price equal to cost of production is zero, and they are not produced; but the already existing ones are there, and will keep being utilized as long as their quasi-rent remains positive. Even for circulating capital goods it is possible that UERRSP does not obtain, if the initial endowment of a non-perishable circulating capital good (e.g. bricks) is so abundant that the economy uses only part of it in the first period, and if the rest, transferred to subsequent periods through storage, is sufficient for those periods’ needs: then the good is not produced for some periods, UERRSP does not obtain. The traditional expression
“uniform rate of profits” as a distinguishing characteristic of long-period positions must be intended as on supply price . This uniform rate of profits, or of return, does need that the existing quantities of capital goods be not given but rather adapted to the demand for them.
The uniform effective rate of return on investment in an intertemporal equilibrium is on the contrary on the demand prices of capital goods, which, for the capital goods already existing at the initial date, can be lower or higher than their costs of production.
8.B.10. The demand price of a capital good at date 0 can also be higher than its cost of production, because since date 0 is at the beginning of the first period, there is no time to produce more of that capital good; and the same can be true for some subsequent dates if the time required to produce that capital good is several periods. From the equilibrium, hence from date 1 onwards. It is then possible to imagine a case in which the initial composition of capital is far from 'adjusted', so that one would not consider the resulting equilibrium a long-period position, and yet UERRSP holds: the case where initial endowments of capital goods (all circulating) are in very different proportions from those that will be reached subsequently, and nonetheless, owing to a high substitutability in production functions, all initial endowments are fully utilized, all rentals are positive, and all capital goods are produced in period 0 and subsequently. However, this case appears so implausible that one can neglect it: capital goods are generally to be combined in fixed proportions with one another and with labour, and can generally be transferred to subsequent periods, so that the inappropriateness of the initial composition of the capital endowment will generally show up in the impossibility fully to utilize some endowments, and/or in the fact that some capital good in positive endowment is not produced, only initially or even for ever.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 33 moment when it is possible to obtain it by production decided within the equilibrium’s periods, the demand price of a capital good cannot be higher than its cost of production; it can however be lower, if the capital good is durable (or non-perishable) and present in a very abundant endowment that makes its rentals lower than required for its present value to be at least equal to its cost of production; then the capital good will not be produced.
Therefore in an intertemporal equilibrium there is an equal convenience of investment in all possible employments of savings (i.e. there is an equality of effective rates of return), but this does not mean that it is a long-period equilibrium; a long-period equilibrium is one in which there is equality of rates of return on supply price (= cost of production), while in intertemporal equilibria the equal rates of return are on demand price. Long-period positions, and long-period equilibria as one type of long-period positions, are positions in which there has been time to adjust the composition of capital to the composition of demand so that all capital goods required by the dominant technology are produced; which means that, once one leaves out the capital goods that would be utilized only by methods of production that are not convenient at the ruling income distribution, in a long-period position all the dominant-technique capital goods are produced, their demand price equals their supply price. In neo-Walrasian
(intertemporal or temporary) equilibria it is generally the case that some dominanttechnique capital goods are not produced in the initial period or periods, owing to the arbitrarily given initial composition of the capital endowment.
Appendix 1 presents an example that illustrates this.
RADNER SEQUENTIAL EQUILIBRIA (without uncertainty)
8.B.11. The reinterpretability of the formal structure of the acapitalistic model as applying to intertemporal equilibria requires a finite number of goods, hence a finite number of periods, beyond which the economy does not continue – it ends with the last period. There cannot be further life of the economy after the last period; otherwise one would have to admit savings and loans in the last period, and expectations on prices etc. after that last period; the introduction of expectations would destroy the formal analogy with the acapitalistic model[ 43 ].
This creates a difficulty. Since it would be ridiculous to assume that agents expect the economy to end in only a few years' time, then the last period must be far into the future. But then the equilibrium must extend to future periods where demands and
43
This will become clearer when we come to temporary equilibria in Part D of this chapter.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 34 supplies will also come from yet-to-be-born consumers, so the assumption of complete markets in futures requires that yet-to-be-born consumers are already present at the initial date to exchange promises of delivery for the dates in which they will demand goods and will supply factor services. Which is impossible.
So it is impossible to assume complete futures markets. The usual way to surmount this difficulty is a re-interpretation of the equilibrium – formally unchanged – as describing, not an economy in which all futures markets actually exist at the initial date for all periods up to the terminal date T, but rather an economy where, at each date, there are spot markets for goods of that date plus a limited number of futures markets, for example, only for gold; then it is possible to transfer purchasing power across periods via purchase and sale of bonds that promise to deliver gold at a future date; at each date the gold bonds coming due are honoured, and markets re-open for spot transactions of goods of that date, and possibly for new futures contracts in gold. The reinterpretation is completed by adding a fundamental assumption: that at the initial date there is perfect foresight of the spot prices and of the own rates of interest of gold that will rule at each subsequent date up to date T.
The equilibrium is then called sequential , because it consists of a sequence of
(correctly forecasted) one-period equilibria. In such a sequential economy, let gold be the only good for which at date 0 there are complete futures markets; and let us suppose that at date 0 a consumer formulates a consumption plan as follows: for all dates except date t and date t+h, there is equality between her income of the date (i.e. the value of the endowments of that date) and the value of that date’s consumption bundle; for date t the discounted value of planned consumption is less than the discounted value of that period’s income (i.e. the consumer plans to save in that period), and for date t+h the consumer plans to dissave (spend more than that period’s income) for an equal discounted value. This consumer wants to transfer purchasing power from date t to date t+h. She can do it by selling (at date 0) promises (bonds) of delivery of a quantity A of gold at date t whose discounted value equals that of her date-t intended savings, and by buying bonds that promise delivery at date t+1 of a quantity B of gold for the same discounted value. When date t arrives, having correctly predicted date-t prices the consumer finds she has an excess of date-t income over her planned purchases, that allows her to buy the quantity A of gold on the gold spot market and honour the bonds she had sold at date 0, and when date t+1 arrives she receives the quantity B of gold and by selling it she obtains the desired excess of purchasing power over that date’s income.
In this way she can attain the same consumption plan as she would be able to attain if there were complete futures markets and she could directly use at date 0 her excess of income over consumption of date t to buy consumption goods of date t+1.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 35
The bonds might even be promises to deliver (fiat) money if existence of the latter is admitted (see §8.B.13 for problems on this account): then perfect foresight must mean correct foresight of future money prices too, and then there is no difference in the capacity to transfer wealth across dates relative to bonds that promise delivery of physical commodities, ‘real’ bonds. An Arrow (financial) security for date t is a bond that promises delivery of one unit of money at date t and only at that date[ 44 ]. To purchase an Arrow security for date t is to lend money (purchasing power) that will be repaid (plus interest) at date t. With correctly forecasted money prices, complete markets in Arrow securities will allow consumers to reach the same consumption plans as with complete futures markets.
The same result can be achieved even if at each date of the sequential economy only one-period Arrow financial securities , i.e. bonds that promise one unit of money at the next date, are available. The sole difference is that a consumer, who wants to transfer purchasing power from period t to period t+h with h>1, will wait for date t, will then buy one-period Arrow securities, at date t+1 will use the repayment of the securities to buy new Arrow securities, and will then renew the securities in subsequent periods up to period t+h-1, finally spending their value at date t+h. Perfect foresight means that her plans will not be frustrated by spot money prices at t, t+1,... different from the ones she expected to rule when at date 0 she formed her consumption plan for the entire sequence of periods.
In the sequential economy thus described, futures markets are not complete but it is possible to achieve the same consumption and production plans as in the economy with complete futures markets. The resulting notion of sequential competitive equilibrium with perfect foresight (a sequence of one-period equilibria) is called an equilibrium of plans, prices and price expectations or also a Radner equilibrium (from
Roy Radner, who first rigorously formalized the notion), and it can be rigorously proved that in it agents make the same consumption (and production) choices as with complete futures markets at the initial date. This equivalence between Arrow-Debreu equilibria and Radner equilibria[ 45 ] allows the re-interpretation of intertemporal equilibria as
44 In an economy with uncertainty, an Arrow security is a promise to deliver one unit of money at date t if and only if some specified state of nature (ascertainable only when date t arrives) occurs. See chapter 9, §9.B.6.
45
The intuition behind the equivalence result appears so clear that I do not find it necessary to provide a more formal proof. A fairly simple formal treatment is in M. Blad, H. Keiding,
Microeconomics: Institutions, Equilibrium, Optimality (North-Holland, 1990), ch. 12; as usual, a more advanced one can be found in Mas-Colell et al. (1995). Actually the equivalence proved by Roy Radner (1967, 1972) was for economies with uncertainty and contingent commodities,
→
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 36 sequential equilibria with perfect foresight.
8.B.12. The reinterpretation of the acapitalistic equilibrium as an intertemporal equilibrium is generally presented as unproblematical, but in fact many elements of traditional analysis, motivated by the aim to determine long-period equilibria, become problematical in the new framework.
One example, discussed in §6.38, is the difficulties with the assumption that production adjusts to demand at cost-covering prices. The argument will not be repeated here, the reader is invited to re-read that paragraph if she does not remember it.
Another example is the legitimacy of the assumption that all resources are owned by consumers. In real market economies, firms own capital goods, lands, mines, money, possibly credit titles and shares of other firms, and at any given moment a high number of firms will be holding (i.e. owning) inventories of not yet sold products. Therefore the treatment of the competitive firm as simply a production function plus the aim of profit maximization may seem rather distant from reality. The issue arises already in longperiod equilibria. But as long as the aim of the analysis was the determination of longperiod equilibria, most of these ownerships could be seen as transitory states required by profit maximization, and negligible in the same way one neglects to specify the details of the production process that remain internal to the firm. Thus, the long-period equilibrium of the labour-land economy of ch. 5 was only interested in determining the normal flow of consumption goods sold by firms and the normal flow of labour and land services utilized by firms, so (especially since one is leaving money, bonds and speculation out of the analysis) it was of no interest that at any given moment there would be inventories of consumption goods on the shelves of shops. However, for land owned by firms the solution could not be this one; but I am not aware of discussions of the issue. It would seem that it was implicitly considered (and not unreasonably) that it was perfectly equivalent, for the determination of equilibrium, to assume the owner of the land to be not the firm, but the owners of the firm, who loan it to the firm. So for long-period equilibria the assumption that all resources are owned by consumers seems acceptable.
But the assumption that all resources are owned by consumers at date 0 of an intertemporal equilibrium is much more far-fetched[ 46 ]. The capital goods produced in the past and owned by firms at date 0, perhaps incorporated into fixed plants, as well as the finished goods just produced and not yet sold, now must appear explicitly among the and the reader will find in §9.B.6 a sketch of a proof of this equivalence; but the intuition is the same.
46
The issue arises for temporary equilibria as well.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 37 given initial resources; how can one assume that they are owned by consumers? Again, I am not aware of discussions of the issue; in all treatises and textbooks I know, the assumption that all initial endowments are with consumers is just made, with no attempt to justify it. It would seem that the traditional procedure was imported into neo-
Walrasian equilibrium theory without asking how legitimate it was in the new framework. Since we must study the theory as currently presented, we too must make the assumption, but keeping a doubt in our minds as to whether it is really possible to justify it without changing anything in the formalization of the agents’ problems. (It would seem necessary that at each moment all goods legally owned by a firm should be considered to be in fact owned by the owners of the firm and loaned by them to the firm.
But this would pose constraints on what each consumer can do with the resources she/he owns, since there must exist some pre-existing contractual arrangement with the firm that the consumer must respect, and furthermore if ownership is shared, a single owner cannot independently decide what to do with her share of the firm’s resources. So, it would seem, consumer theory would need a different formalization.)
Another example is inheritance. There is an issue here that was left aside in
§8.B.4. The longer the horizon, the more it becomes inevitable to admit that some consumers will die before date T[ 47 ]. This could be neglected in the determination of long-period equilibria as what was relevant there was the current data, altered by the death of some consumers (and the arrival of other consumers to adulthood) with a slowness analogous to that of changes of total population, and therefore negligible. But intertemporal equilibria must admit and determine bequests . (Note that bequests can be seen as including the resources allotted to education of children.) Then the endowments of some consumers at some future date are not given, they depend on decisions of
‘earlier’ consumers. This is a clear analytical difference from the acapitalistic economy, where endowments of consumers are all given. A frequently adopted solution consists of simply neglecting the issue, assuming that the consumers—the households—are the same from the first to the last period, and trying to justify this by interpreting each consumer as representing a 'dynasty' as long lived as the economy: the utility function of each consumer embodies the utility that the descendants will derive from the consumption plan decided at date 0 for all periods. This maintains the formal similarity with the acapitalistic economy. Of course this means that the first decision maker in each dynasty correctly forecasts the tastes of descendants, which might be seen as just one implication of the assumption of correct foresight of future states of nature. But a
47 As in the equilibria of this chapter there is no uncertainty, consumers know when they will die.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 38 problem arises owing to the fact that children have a mother and a father, and therefore do not belong to a single dynasty. This makes the (archaic) notion of a 'dynasty' impossible to defend. For the determination of bequest decisions, the implication is that, since one’s child will often form a household with another person and share living standards with him/her, the living standard that one’s child will enjoy will also depend on the bequest to the spouse; then the bequest decision of one bequeathing household may depend on how much the inheriting household is going to inherit from the other bequeathing household. What in game theory is called a chicken game (also called hawkdove game ) arises: if I let the parents of my son-in-law know that I am going to leave a very small bequest to my daughter, I may induce them to increase their bequest. No price taking is available here to reach a well-determined decision. If the horizon is very long, with the income of offsprings of offsprings entering the utility function of the first-period consumers, the problem is compounded. The assumption of a finite number of consumers/households/dynasties present from the first to the last period appears therefore unacceptable; either one assumes finitely-lived households, that is,
‘overlapping generations’, or one ought to admit that one is neglecting consumer heterogeneity and the problem of bequests, in order to concentrate on other issues, and it will be more honest, then, to assume a single infinitely-lived ‘representative’ household, which is what will be mostly assumed in this Part. Overlapping generations raise new issues that will be discussed in Part C.
MONEY
8.B.13. We pass to study some criticisms that have been moved to the notion of intertemporal equilibrium.
We start with a difficulty with admitting fiat money in the model as defined so far, that is, with a finite horizon. The equilibria illustrated in chapters 3 to 7 included no treatment of money (in the sense of a commodity universally accepted as means of payment and desired for that use, not for an intrinsic utility of the physical substance of which it is made); but this was not due to an assumption that money was absent in the economy. Traditional neoclassical authors took it as obvious that the economy whose relative prices and produced quantities gravitate toward a (long-period) equilibrium uses money, and argued that such gravitation went together with a gravitation of the money price level and of the distribution of money among agents toward an equilibrium. The normal price level was determined by the quantity theory of money, expressed by a
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 39
Fisherine equation MV=PT or a Cambridge equation M=kPY[ 48 ]. The 'real' part of the overall equilibrium, that is the normal quantities and normal relative prices determined by the equations we have studied, implicitly assumed an equilibrium price level and an equilibrium distribution of money balances among agents. This is why no constraint deriving from money holdings appeared in the decisions of consumers or firms described by the equilibrium equations: not because money was assumed not present, but because money holdings were assumed to have adapted to the needs of the agent (Petri 2004,
Appendix 5A1 pp. 166-186). This is the so-called 'Neoclassical Dichotomy': long-period equilibrium quantities and relative prices are independent of the quantity of money, which only determines money prices.
This dichotomy is no longer defensible in very-short-period analyses. The given initial endowments of capital goods oblige one to conceive the equilibrium as established at a precise moment. Then the endowment of money of each economic agent must also be considered given, it must be the amount of money the agent has at that moment. Then, if money is essential for transactions (i.e. if money buys goods and goods buy money, but goods do not buy goods), money holdings are an additional influence on the agents' decisions. This influence was of course admitted by traditional authors, it is clear for example in the passage by Wicksell quoted in §3.3.8, which the reader is invited to reread. Wicksell's reasoning also illustrates the disequilibrium effects of total money balances appropriate to the price level but not distributed among agents in proportion to their needs/desires. The individuals who have greater money balances than they need will tend to get rid of the excess by spending more, while the individuals in the opposite situation will behave in the way indicated by Wicksell, and the result will be a redistribution of average money balances[ 49 ] that will tend toward an equilibrium, simultaneously with the tendency of the real choices of agents toward their normal values. In equilibrium the average real money balances of each agent are not data, they are endogenously determined as the ones appropriate to the carrying through of the normal transactions of the agent (and also appropriate to the agent’s desire for
48
The reader of course will be already acquainted with these theories, anyway I remember that M is the given quantity of money, V its velocity (the average number of times each unit of money changes hands in a period, because used for payments), P an index of the price level, T an index of the 'volume' of real transactions effected in the time period, k the average desired demand for money (i.e. holding of money) as a proportion of nominal income, Y real income,
PY nominal income; M, V, and T in the Fisherine equation. or M, k and Y in the Cambridge approach, are taken as given, and P is the variable determined by the other three.
49 Average money balances, because each agent's money balances will be changing continually as she sells and buys goods.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 40 precautionary real balances, if one wishes to admit them); the long-period 'real' choices of an agent are therefore independent of her money balances because the latter are determined by the long-period 'real' choices once the price level is given, and the price level is such as to make the total needed money balances equal to the money supply.
But when the equilibrium is a very-short-period one so that the money balances of each agent must be assumed given, then these balances do influence decisions. For example, if one assumes that a period is so short that it is not possible to utilize the money obtained from a sale within a period for a purchase within the same period, then in each period an agent can only purchase up to the value of the money with which she starts the period. The agent's decisions will be influenced also by her need for money balances in subsequent periods; for example an agent may decide to anticipate the sale of a good, relative to the date she would have otherwise chosen, as the only way to have enough cash for certain desired purchases.
Any attempt to take these cash-in-advance constraints into account in intertemporal equilibria will encounter a danger of indeterminateness: how a given initial money balance constrains an agent's decisions depends on the agent's time structure of transactions (which may depend on accidents, for example on how quickly she finds purchasers for her net supplies), on her possibility to postpone payments even only by a few hours, etcetera. Any assumption aimed at surmounting this indeterminateness would be largely arbitrary. This indeterminateness arises in the determination of the price level too.
If instead one conceives exchanges as not requiring money as an indispensable intermediary−which is the natural conception if one postulates the auctioneer−then it becomes impossible to justify a positive value of fiat money. If the money good has no intrinsic utility, then in the last period T no one will want to remain with a positive amount of it since there is no subsequent period in which to utilize it for purchases; as a result, in period T everybody will try to exchange the money she owns against goods, but no one will accept it, and the value of money will fall to zero; but then the value of money will also fall to zero in period T–1, because all agents, knowing that money will have value zero the next period, will want to get rid of all their money before the last period arrives, but again no one will accept it; then the same fall of the value of money to zero will happen in the preceding period, and so on, and money will have value zero in all periods including the initial one. This argument assumes that exchanges directly of goods against goods are possible, in other words, it assumes the existence of the auctioneer, evidently charged with the additional task of acting as a clearing house, that is, of collecting all net supplies of each good and then distributing them to the net demanders, at the same time keeping accounts of the values of the supplies and demands
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 41 of each agent to ensure that the budget constraints are not violated. (Such a task of the auctioneer is implicitly supposed by the absence of any other mechanism determining to which demander the supply of a good is assigned.)
We conclude that the Arrow-Debreu equilibrium has great difficulty with making room for (fiat) money, because it faces an unhappy dilemma between admitting money as indispensable for exchanges (which causes indeterminateness), and considering money as not indispensable (because of the auctioneer), which causes a zero value of money. This conclusion holds for the sequential reformulation à la Radner too: if money is indispensable for transactions and these follow one another, then the constraint due to money balances and to the accidents of transaction sequences causes indeterminateness; if money is not indispensable, then perfect foresight causes its expected value to be zero in the last period and then in all periods.
One thing we learn from this is that fiat money has positive value because it will keep having positive value in the future; its existence requires a never-ending economy.
But even the extension of equilibrium to an infinite future (which will be discussed later) does not avoid the other difficulty, the indeterminateness of transactions if money holdings are a constraint on possible transactions. This difficulty did not arise in traditional analyses aiming at determining long-period positions, because the latter were to indicate only averages, and not the precise sequence of transactions moment by moment, as on the contrary a neo-Walrasian intertemporal equilibrium with cash-inadvance constraints would have to determine owing to its very-short-period nature.
EXISTENCE, UNIQUENESS, STABILITY
8.B.14. Once vertical integration of firms is performed, and neglecting the difficulties pointed out in §8.B.12, the equilibrium conditions of the finite-horizon intertemporal equilibrium model are the same equations (A),(B),(C),(D) as for the economy of ch. 5; therefore the conclusion of the analysis of ch. 5, that everything depends on factor rentals, and that once these are given all prices and quantities of produced goods as well as all supplies and demands for factors are determined[ 50 ], remains valid for the intertemporal economy with a finite number of periods, with the sole need to refer the term ‘factors’ only to the ‘original’ factors: labour(s) and land(s) of the several periods, and initial capital goods endowments (and inventories of unsold consumption goods: but below we will neglect these inventories).
It is then generally argued that, since the equations are the same, the conclusions
50 Of course, under an assumption that optimal production methods at the given factor prices are uniquely determined.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 42 on existence, uniqueness and tâtonnement stability must be the same too. Thus, it is argued, for existence of an equilibrium what is necessary is the continuity of excess demands and some assumption to surmount the minimum-income and the survival problems; at least among differentiable economies, regular economies are the generic case and thus the number of equilibria is generically finite; the Sonnenschein-Mantel-
Debreu results remain valid; quasi-uniqueness is guaranteed by the assumption that the consumers’ aggregate excess demands obey the Weak Axiom (of Revealed Preferences) in the Aggregate, WAA; if the issue of stability is not emptied by the assumption of perfect foresight (see below), tâtonnement stability is guaranteed by the same assumption, as long as one assumes a ‘factor tâtonnement’, that is, that at each stage of the tâtonnement productions adapt to quantities demanded as assumed in equations (B), and that the same adaptation of productions to quantities demanded happens for the intermediate goods that do not explicitly appear in the formalization owing to the treatment of firms as vertically integrated. In this way the only markets on which there can be disequilibrium are the markets for ‘original’ factors. Hence the name ‘factor tâtonnement’ (due to Mandler, 2005), whose stability under WAA was proved in
Appendix 3 of ch. 6; its reinterpretation for an intertemporal model is explained in
Appendix 2 of the present chapter.
However, the analogy has limitations. There are important differences relative to the acapitalistic model of chapters 5 and 6, that question the applicability or the significance of the above results.
CAPACITY TO GIVE INDICATIONS ON ACTUAL PATHS: THREE NEW
PROBLEMS DUE TO THE VECTORIAL CAPITAL ENDOWMENT
8.B.15. A very important difference concerns the doubtful capacity of intertemporal equilibria to give indications on the behaviour of actual economies, because of their specification of the capital endowment as a given vector which makes them very-short-period equilibria. The main reference here is pp. 49-58 of P. Garegnani,
“Quantity of capital”, in John Eatwell, Murray Milgate, Peter Newman, eds.,
The New
Palgrave: Capital Theory (1990, Macmillan) a paper whose reading is strongly recommended.
A first problem was already briefly mentioned at the end of §8.B.7 but deserves extended discussion. The impermanence problem (Petri 2004, ch. 2) arises because of the lack of persistence of the capital endowment thus specified – an aspect that intertemporal equilibria share with temporary equilibria. This lack of persistence is implicitly admitted, in the usual presentations of modern general equilibrium theory, when in order to study how equilibrium is reached the assumption is made of the fairy-
42

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 43 tale auctioneer, who congeals the economy until, through a tâtonnement that mysteriously takes no time, equilibrium is achieved. If this patently unrealistic assumption is not made, we have the problem admitted by a highly esteemed neoclassical microeconomist:
"In a real economy, however, trading, as well as production and consumption, goes on out of equilibrium ... in the course of convergence to equilibrium (assuming that occurs), endowments change. In turn this changes the set of equilibria. Put more succinctly, the set of equilibria is path dependent ... [This path dependence] makes the calculation of equilibria corresponding to the initial state of the system essentially irrelevant" (Franklin M. Fisher, 1983, p. 14).
Let us make the problem more explicit. We are at date 0, the moment when the equilibrium’s data are observed to which a yet-to-be-found equilibrium path corresponds
(let us assume, for the sake of argument, uniqueness of equilibrium). There is no auctioneer; disequilibrium adjustments take time, during which time production and exchanges go on. Before the economy is able sufficiently to correct or compensate disequilibrium decisions, the proportions between capital endowments may have significantly changed, because there is no reason why the ongoing production of new capital goods should maintain their endowments unchanged, since the initial composition of capital is arbitrary and firms will generally wish to alter it in the search for the highest rate of return. The original equilibrium cannot be trusted to give a good indication of the behaviour of the economy in period 0, because adjustments in reality are far from instantaneous[ 51 , 52 ]. But then at date 1 the capital endowments will be different from the
51
Think for example of the adjustments required by a disequilibrium on the labour market, due e.g. to immigration: changes in wages take time, because many wage contracts last several months, even years; the wage changes will entail changes of production methods, and of demands for consumption goods, that in turn will change the demands for all factors, requiring adjustments also on the markets for natural resources, with changes in their rentals that will again cause changes in production methods and in demand for consumption goods, and so on. It seems clear that there will be plenty of time for the quantities in existence of circulating capital goods, and even of many durable capital goods, to change considerably. (The quantity of tractors of a certain type, for example, may require at least a couple of years to decrease considerably through non-replacement, but much less to increase considerably through increased production.) In the versions of general equilibrium theory that assume a given number of firms and the possibility of positive profits in equilibrium – versions criticized in §5.25 – another element lacking persistence is the shares owned by each consumer.
52 As Garegnani has put it: “To the extent to which this is true, a purely methodological reason, quite independent of the content of the theory , arises for questioning the capacity of
→
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 44 ones predicted by the equilibrium path for that date, so the economy would not be able to reach the original equilibrium predicted for period 1 even if at date 1 adjustments were instantaneous; the equilibrium path has been altered; furthermore, at date 1 too there will be disequilibrium decisions and productions, and the danger arises of a further deviation from the original equilibrium path, and of a cumulation of deviations as time proceeds[ 53 ]. Therefore the intertemporal equilibrium path corresponding to the initial capital endowments can be trusted to give a good indication neither of the initial behaviour of the economy, nor of the path over a sequence of periods, for reasons independent of whether the tâtonnement is stable or not, and depending simply on the fact that realistic adjustments shift the equilibrium itself, in directions that the theory is unable to indicate. Franklin Fisher (1983; 2003) seems to be the only neoclassical author to have openly admitted the problem and to have looked for assumptions that might surmount it while admitting disequilibrium productions in an economy with capital goods ; unfortunately he is not able to avoid an indeterminate path; the interested reader is invited to consult Petri (2004, pp. 67-71) for further details.
Therefore the trustworthiness of intertemporal equilibria as indicators of the path of actual economies would require that a convincing theory of the actual path were able these equilibria to offer sufficient guidance to the behaviour of the economy. On the one hand, the levels of the prices and of the distributive variables determined in these equilibria clearly cannot correspond to the actual levels of those variables, as they are at any one moment of time
(or over a period of time short enough for no appreciable changes to occur in the composition of the capital stock). This will be so because of those ‘fitful and short-lived causes’ (Marshall) that will keep the economy out of these equilibria, just as they kept it out of the traditional normal positions, at any one particular moment of time. On the other hand, these same equilibrium levels cannot correspond to any average of the observable levels taken over a period of time long enough for those ‘fitful and short-lived causes’ to efface one another’s influence through the repetition of the activities. This is so because, over such a period, considerable changes are bound to occur in the composition of capital, and therefore in the equilibrium itself.” (Garegnani
1990, pp. 49-50; initial italics added).
53
Disequilibrium decisions depend on a myriad of accidental causes and therefore their details contain random elements. Now, it suffices to think of the unpredictable evolution of random walks to realize that a cumulation of random ‘deviations’ in a certain direction is perfectly possible. So the equilibrium path is uninformative on the actual path even assuming the ‘deviations’ are purely random. In fact, ‘deviations’ need not be random but will have precise causes and direction the moment one proposes a theory of the actual path. But given the unpredictability of the precise disequilibrium path of prices and quantities, a theory of the actual path can only aim at determining averages and trends; the neoclassical approach does have a tradition of analyses of this type, but they all rely on capital the single factor of variable ‘form’, criticized in ch. 7. So a rejection of neo-Walrasian equilibrium theory obliges one to turn to some totally different approach.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 45 to show that the equilibrium path yields a good approximation to it. But with modern general equilibrium theory the traditional way to sketch a theory of the actual path in spite of the impossibility to specify the accidental events of disequilibrium – by arguing that, through error correction or compensation, the actual path gravitates around and toward a persistent equilibrium defined independenty of the accidents of disequilibrium – is lost; the fact that disequilibrium decisions cause the neo-Walrasian equilibrium itself to change, and in ways that equilibrium theory does not indicate, renders the tendency of the actual economy impossible to establish (Petri 1999, pp. 24, 50). F. M. Fisher has written: “If disequilibrium effects are in fact unimportant we need to prove that they are.
If such effects are important, then the way in which we tend to think about the theory of value needs to be revised.” (Fisher, 1983, p. 217). The Great Crisis of the 1930s would seem to suffice as proof that ‘disequilibrium effects’ are important.
These observations throw light on an issue that may have troubled thoughtful students: why do modern discussions of the stability of general equilibrium accept the patently unrealistic auctioneer-guided tâtonnement as their frame of reference? The answer is, because the moment the presence of capital goods among the factor endowments is admitted, it is indispensable to prevent the implementation of disequilibrium decisions, which would otherwise cause unpredictable changes of those data of equilibrium: the changes would render all discussion of the stability of the original equilibrium meaningless. But, as Franklin Fisher’s observation implies, the tâtonnement stability of the original equilibrium is ‘essentially irrelevant’ anyway, because the economy will not tend to that equilibrium, independently of whether it is tâtonnement stable or not[ 54 ].
8.B.16. A second problem derives from the need to admit an awareness of decision makers, in the initial period, that equilibrium prices cannot be expected to remain unchanged or nearly unchanged; the likely rapid change of the endowments of some capital goods will entail quick changes of their prices over time. The price-change problem that thus arises is the need to determine how this awareness influences decisions. In Part D we will see the form the price-change problem takes in temporary
54
The way to deal with this difficulty in modern literature seems to consist, increasingly, of avoiding it by omitting all mention of stability issues, and mentioning only existence and determinateness (that is, local uniqueness) of equilibria as the issues requiring exploration. Thus there is no mention of the tâtonnement or more generally of stability in, for example, Aliprantis,
Brown and Burkinshaw (1990), Ellickson (1993), Magill and Quinzii (1996, Theory of
Incomplete Markets Volume 1 , The MIT Press), Florenzano (2003), or Kreps (2013,
Microeconomic Foundations I , Princeton University Press).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 46 equilibria. For the determination of intertemporal equilibria, one must assume either complete futures markets (with no default, otherwise agents would mostly prefer spot transactions), or Radner sequential equilibria, hence perfect price foresight. An immediate criticism is then, that to arrive at defining the equilibrium the theory must assume the presence of elements with no correspondence with the real world : either complete futures markets, or perfect price foresight. Equilibrium cannot be even defined unless one assumes the world to be radically different from what it is! One more reason to view the connection between the equilibrium path, and actual paths, as totally unclear.
The absence of complete futures markets in the actual world is evident: there are only very few markets in futures in real economies, and for only a few years ahead at most[ 55 ].
The alternative assumption of perfect foresight (self-fulfilling expectations) in a sequential economy deserves more extended discussion in view of its widespread utilization. It is possible to point out analytical and logical difficulties of this assumption that go beyond a generic accusation of lack of realism.
First, in order for agents to reach the same forecasts and for these forecasts to be self-fulfilling, (i) everybody must agree on the theory of how the economy functions, and furthermore (ii) this theory must be correct. But there are wide disagreements among economists, as the present textbook shows, on which is the correct theory[ 56 ].
Second, equilibrium need not be unique; then expectations can be self-fulfilling only if everybody agrees on which of the several possible equilibria will be realized, and it is unclear what might ensure such unanimity.
Third, the idea behind the assumption of perfect foresight is that people are able to calculate what future equilibrium prices will be and, being price takers, find it optimal to behave according to that prediction, and the resulting choices do indeed produce that equilibrium; but correct calculation of which prices will be equilibrium prices need not determine optimal decisions uniquely (those prices may be necessary but not sufficient for equilibrium); in particular, this is the case with constant-returns-to-scale firms or free entry; in this case, as shown in §6.38, correct foresight of the equilibrium output price leaves firms’ output decisions indeterminate, with no guarantee that they will sum up to the equilibrium aggregate quantity demanded at that price. We noted the need, to
55 Some reasons will be discussed in chapter 9. Here it will suffice to remember that it is impossible at the initial date to stipulate contracts with consumers or firms not yet born.
56 Actually, more than sharing the correct theory is required: people must also be able correctly to determine future equilibrium prices (see under Third).
46

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 47 surmount this problem, for the auctioneer to become a command planner[ 57 ], which would mean the economy is no longer a market economy; but without complete futures markets the planner-auctioneer cannot operate; then the correct determination of future equilibrium prices plus a decision to consider them as the prices that will indeed rule in the future would not result in the production of the equilibrium quantities, but then those prices cannot be expected to rule, so decisions will be other, reached in some other way
– how, general equilibrium theory does not tell us.
Fourth, the future cannot be predicted more than extremely vaguely and tentatively, if for no other reason, because too many things cannot logically be predicted.
The perfect foresight assumption excludes all true novelties , in particular new scientific discoveries or theories, new technological inventions: these, if predictable, would not need to be discovered in the future because known already at date zero[ 58 ].
Fifth, it is unclear how to reconcile the perfect foresight assumption with the secular tradition of concern over the stability of equilibrium. Disequilibrium processes, motivated by the fact that equilibrium is not known in advance and must be found, are not easily reconciled with an assumption of perfect foresight: a tâtonnement, for example, would have to be conceived as operating only on current prices (remember that the economy is sequential) but with perfect foresight of future prices: but what could this mean? That agents have perfect foresight of the future consequences of any vector of current prices proposed by the auctioneer, even when these are not equilibrium prices?
No one has dared propose such a notion of perfect foresight. Indeed, no tâtonnement is ever proposed for the sequential Radner model ; I am not aware of discussions of how the tâtonnement idea, originally proposed for the equilibrium of the acapitalistic economy, might be transferred to this type of economy. One might even suggest that an assumption of perfect foresight implies that there is no need to find the sequential equilibrium through some kind of disequilibrium trial-and-error adjustment: perfect foresight, it might be argued, implies that even for date zero the equilibrium is known before the
57 This problem does not appear in the discussion of Radner equilibria because in the latter only the existence of a set of mutually compatible equilibrium decisions is discussed, and one such set would exist; the problem is that there is no reason why it should be hit upon.
58
It is worthwhile to point out the different approach of the long-period method to this issue.
Unpredictable novelties do not undermine the long-period method because the latter only aims at determining what the economy will tend to gravitate towards after the novelty has happened.
Even a stream of novelties does not undermine the analysis, since the long-period method does not lose usefulness only because the economy is always, so to speak, chasing a moving longperiod position without ever reaching it: the direction of change of the averages would still be correctly indicated by the direction of change of the long-period prices.
47

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 48 markets open, so the moment the markets open the correct equilibrium prices on the current markets are immediately established. This is because, if agents are capable correctly to predict future prices, why should they not be able correctly to predict current equilibrium prices? Indeed, if at date 0 agents can predict date 1 equilibrium prices, then at date -1 they must have been able correctly to predict date 0 prices, and when date 0 arrives they cannot have forgotten their predictions. But this only makes the question even more evident, how can perfect foresight have been acquired?
The answer is implicit in the following representative lines:
Although it is capable of describing a richer set of institutions and behaviour than is the Arrow-Debreu model, the perfect foresight approach ... is contrary to the spirit of much of competitive market theory in that it postulates that individual traders must be able to forecast, in some sense, the equilibrium prices that will prevail.... [this] seems to require of the traders a capacity for imagination and computation far beyond what is realistic.... An equilibrium of plans and price expectations might be appropriate as a conceptualization of the ideal goal of indicative planning, or of a long-run steady state toward which the economy might tend in a stationary environment. (Majumdar & Radner, 2008, p. 444; based on Radner, 1982, p. 942[
59
].)
It is admitted here that, as part of a descriptive theory, the perfect foresight assumption is legitimate only for the determination of situations where relative prices have no reason to change , and where therefore past prices are an excellent guide to future prices. If the economy tends to such a situation, then a mechanism of expectation correction must be part of the forces ensuring this tendency, and expectations become correct through trial and error correction . However, a situation of unchanging relative prices requires an endogenously determined composition of capital: therefore, Radner is implicitly admitting here that the sequential reinterpretation of Arrow-Debreu equilibria proposed by him is vitiated by an internal contradiction if intended to apply to generic
Arrow-Debreu equilibria, because the perfect foresight assumption is incompatible with
59 Grandmont (1982, pp. 879-880) writes extremely similar lines (quoted at the end of this chapter), and concludes that one should rather develop the temporary equilibrium approach.
Also interesting is the following admission: “Perfect foresight models are not designed to deliver descriptive accuracy ... It may well be that this abstraction undermines the value of any lessons drawn” (Mandler 2005: 487). But then, are there general equilibrium models designed to deliver descriptive accuracy? Certainly the complete-markets assumption is no improvement.
Also, especially in mainstream macroeconomic theory, perfect-foresight models are used as descriptive models.
48

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 49 an arbitrarily given vectorial capital endowment[ 60 ]. Furthermore, if the sole situation in which a perfect-foresight equilibrium might reasonably describe the actual behaviour of an economy is a ‘long-run steady state ... in a stationary environment’, then the question, whether the economy tends to a situation for which one might assume perfect foresight, would require a theory not based on perfect foresight, that intertemporal equilibrium theory does not provide.
In conclusion, we have here a second reason why the equilibrium path cannot be trusted to give an acceptable representation of the actual path.
8.B.17. A third problem of intertemporal equilibria due to the vectorial capital endowment is the lack of sufficient factor substitutability. This is the substitutability problem . Different production methods generally require different capital goods, not the same capital goods in different proportions; and capital goods of a certain type generally require rigid proportions to labour and to other capital goods; so only the treatment of capital as a single factor of variable 'form' can give plausibility to the assumption of extensive variability of factor proportions (§5.5). The absence of substitutability connected with a given vector of capital endowments undermines the indirect factor substitution mechanism too, because industries cannot expand beyond the limits imposed by the availability of the needed capital goods. In the initial period the demand for labour will be very rigid, with the risk of implausible levels of the equilibrium real wage; furthermore, the fixed coefficients together with the arbitrary capital endowments will generate high numbers of capital goods in excess of demand, with their rentals risking a fall to zero. A theory with such consequences lacks plausibility.
60
Similar considerations apply to an admission by Frank Hahn: “equilibria which cannot be reached from historically given initial conditions by an acceptable process of learning should, I contend, be ruled out. What that means is that the equilibrium definition should include the requirement of reachability... All this may be summed up by saying that economic theory should deal with equilibria which are stable under some acceptable process.” (Hahn, “History and
Economic Theory” 1991, pp. 70-71, emphasis added). Ariel Dvoskin, after quoting these seldom noted lines, rightly comments that “any ‘acceptable process’ of adjustment must allow for the implementation of actual , i.e. disequilibrium, activities and therefore, the position to which the adjustment converges cannot be defined and considered ‘stable’ on the basis of a given vector of capital endowments known before the adjustment has been completed”, therefore Hahn should have admitted “that the neo-Walrasian treatment of capital is simply incompatible with the
‘requirement of reachability’ ” (Dvoskin 2014: “An unpleasant dilemma for contemporary general equilibrium theory”, The European Journal of the History of Economic Thought , DOI:
10.1080/09672567.2014.881898, pp. 22-23 of online advance version).
49

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 50
We grasp here one important reason for the reliance of older marginalist economists (apart from Walras and his few pupils) on the conception of capital as a factor capable of changing 'form'. Dennis Robertson, in (1931, p. 227), explained that when nine workers dig with nine spades, in order to determine a plausible marginal product of a tenth worker one must allow the nine spades to become ten smaller spades having the same total value, or perhaps nine smaller spades plus a bucket with which the tenth worker brings refreshment to the other nine. The same conception of capital-labour substitution as needing changes in the 'form' of capital emerges in the passage from
Hicks's Theory of Wages quoted in ch. 7, footnote 35??. Hicks admits the little variability of labour employment in the very short period also in Value and Capital , where he states that in the first ‘week’ (his term for the short period over which a temporary equilibrium is established) and in ‘weeks’ in the near future the level of output of most firms will be dictated by the amount of intermediate goods ('work-in-progress') already in the pipeline, and therefore “The additional output which can be produced in the current week, or planned for weeks in the near future, will usually be quite small” (1946, p. 206), and for the same reason the variation in inputs can only be very small ( ibid ., p. 211)[ 61 ].
These admissions by Hicks have an important implication: in the first period(s) of an intertemporal equilibrium a real wage ensuring equality between supply and demand for labour might easily be so low that workers would prefer to turn to looting and revolts, or conversely so high as to absorb nearly the entire product (because of difficulties with the full employment of the other factors). As I have written elsewhere:
... short-period analyses, and even more neo-Walrasian analyses, cannot aim at autonomously determining the real wage as an equilibrium real wage. In order to avoid implausible results (such as a high probability of a zero or near-zero wage, or enormous changes of the real wage from one "week" to the next), the marginalist theorist must admit that the real wages are not so flexible as to try and bring into equality supply and short-period demand for labour; the short-period excess demand for labour can only be admitted to govern the direction of a gradual movement of wages from an initial level, at the beginning of Hicks's "week", which must be taken as exogenously given. Only long-period analysis, based on the conception of capital as a single factor capable of changing 'form', can plausibly try to explain the (trend or average) level of the real wage in terms of supply and demand, because only the demand for labour derived from the schedule of the long-period marginal product of labour (determined by allowing the given 'capital' to take the 'form' best appropriate to the various levels of labour
61
In a later article Hicks repeats that within a single ‘week’ “The actual outputs of products, and probably also the actual input of labour, would be largely predetermined” (Hicks 1980-81, p. 55).
50

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 51 employment) can be argued not to be so inelastic as to yield implausible results. (Petri
1991, pp. 271-272).
In Value and Capital Hicks admits precisely that wages are generally sticky and change only slowly in the direction indicated by the excess demand for labour; thus he admits that in general for at least some periods the economy cannot be assumed to have equilibrium factor rentals: these would be too implausible and incapable of assuring an orderly economic activity. This is a third reason why the equilibrium path cannot be trusted to give correct indications on the actual path.
As a digression, but on a very important issue, let me note that in this way Hicks admits that other elements besides excess demand enter the determination of wages, elements (e.g. the role of trade unions, feelings of class solidarity, or the status considerations that may induce each category of workers to reject wage reductions, fearing a loss of status relative to other categories[ 62 ]) capable of slowing down wage decreases in situations of unemployment. Now, once these elements are admitted, one gets very close to the theory of wages of classical authors, because it is unclear why these social forces could not be capable of completely preventing real wages from decreasing even in situations of unemployment, thus being the basis for a theory of wages alternative to the marginalist one. There would remain, as a marginalist element, the decreasing demand curve for labour, which, once the real wage were given, would determine labour employment; but Hicks’s rejection of the notion of a short-period marginal product of labour implies that the decreasing demand curve for labour cannot be based on a given vectorial endowment of capital goods; and the alternative of a given quantity of capital the single value factor is indefensible, as argued in ch. 7; so serious doubts can be advanced on the possibility of giving a logically consistent foundation to the decreasing demand curve for labour.
8.B.18. The price-change problem, the impermanence problem and the substitutability problem arise in intertemporal equilibria because of the abandonment of the specification of the capital endowment as that of a single factor of variable 'form'.
The latter specification of the capital endowment (neglecting its illegitimacy) avoided these problems. The impermanence problem did not arise, because the quantity of capital, being only slowly altered by net savings, was as persistent as the endowment of labour, and could therefore be included among the data determining the equilibrium as legitimately as the endowment of labour; the persistence of the resulting (long-period)
62
These were stressed by Keynes as a very important cause of rigidity of money wages.
51

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 52 equilibrium allowed neglecting the very slow changes that equilibrium prices might undergo as a result of the endogenous gradual change of the data of equilibrium owing to population growth or net savings[
63
], so the price-change problem did not arise; and the possibility to change the 'form' of capital, by allowing changes in the types of capital goods, avoided or reduced the substitutability problem.
63 Technical progress, on the other hand, would be treated through comparative statics, as determining new equilibria, toward which the economy would then gravitate.
52

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 53
THE SAVINGS-INVESTMENT PROBLEM
8.B.19. Another problem of intertemporal equilibria, the savings-investment problem , differently from the problems discussed in §§8.B.15-18, was already present in long-period equilibria, but it takes now a new form.
In order for a neoclassical equilibrium to be established in an economy with capital goods, in each period all resource supplies not demanded for production of consumption goods must be demanded for production of new capital goods; this production, in order to be decided, needs that there be a demand for it; and this requires that investment be equal to full-employment savings. The presence of savings and of investment in an intertemporal equilibrium may be hidden by the representation of production of consumption goods as effected by vertically integrated firms, but it is there nonetheless. So the theory must have some argument to justify the adjustment of investment to full-employment savings in each period.
In the analyses based on capital conceived as a single value factor, the adjustment of investment to savings was based on the assumed capacity of the rate of interest to adjust firms’ demands for capital to its supply, and therefore also to adjust changes in the demand for capital (net investment) to changes in its supply (net savings). The basis for the stability of the adjustment process, as explained in ch. 7, was the decreasing demand curve for capital the value factor, that we know now to be an unwarranted assumption because undermined by reverse capital deepening[
64
].
In the theory of intertemporal equilibria, this argument no longer appears explicitly (but see §8.B.21); the adaptation of investment to full-employment savings is simply assumed
, as made necessary by the tâtonnement.
A premise is necessary. Neoclassical theorists will generally admit that complete futures markets at the initial date are a wildly unrealistic assumption, only a first step toward the notion of sequential equilibrium with perfect foresight. But when they discuss the tâtonnement stability of general equilibrium, neoclassical theorists seem to take it for granted that the discussion applies to intertemporal equilibria too, neglecting the difficulties (§8.B.16) with reconciling perfect foresight with a need to find the
64 A new argument, advanced by Petri (2013, 2015), contends that this traditional neoclassical reasoning requires that the full employment of labour be assumed rather than derived from the analysis, and that if it is not assumed to start with, then even the traditional marginalist conception of capital-labour substitution does not suffice to obtain a tendency toward the full employment of labour. If the argument will resist criticism, it will be presented in a next edition of this textbook.
53

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 54 equilibrium by some form of groping[ 65 ]. So one must presume that they are assuming complete futures markets, not perfect foresight; and I will follow this interpretation here, because stability issues cannot be disregarded.
Given the impermanence problem, if one does not want to fall into the indeterminacy of the economy’s development encountered by Franklin Fisher (1983) the discussion of stability must be based on the auctioneer and recontracting. Then, since free entry and/or firm-level constant returns to scale must be admitted, once nonproduced factor rentals are given one must assume zero-profit product prices, and, to avoid the indeterminacy of the production decisions of firms, there seems to be little alternative to assuming the planner-auctioneer or some equivalent trick ensuring the adaptation of quantities produced to the quantities demanded at those prices. In this way one arrives at the ‘factor tâtonnement’ (§6.38), explicitly presented by Michael Mandler
(2005) as applicable also to finite-horizon intertemporal equilibria[ 66 ]. Now, the factor tâtonnement contains an assumption that implies that, in intertemporal economies (for the moment, over a finite number of periods), all problems in the determination of investment disappear.
This is because the quantities demanded are derived, directly or indirectly, from consumer choices based on incomes corresponding to the full employment of all factor supplies , independently of whether there is or not a demand for them. At each round of the tâtonnement[ 67
], (discounted) factor rentals of nonproduced factors are announced, minimum-cost prices of all produced goods are derived and announced, and consumers have all they need to determine their demands and supplies on the basis of the assumption that their factor supplies will find purchasers. The demand for each consumption good is determined, and from them plus the choice of cost-minimizing production methods all the direct and indirect demands for inputs are derived, both for
65
To the best of my knowledge, none of the advanced textbooks that do discuss the stability of general equilibrium contains a warning that the discussion (summarized here in ch. 6) of the stability conditions of tâtonnement in terms of gross substitutes, weak axiom in the aggregate,
Lyapunov function etc., does not apply to intertemporal equilibria or to their reinterpretation as
Radner sequential equilibria with perfect foresight.
66 Mandler’s paper is directed at refuting Garegnani’s (2000) thesis that the uniqueness and stability of intertemporal equilibria must be doubted owing to capital-theoretic problems, even if one assumes no problems coming from income effects and heterogeneous consumers. See footnote 63?? below.
67 The tâtonnement is generally formalized as a dynamical process in continuous time, but to help intuition it is best to imagine it as consisting – as in Walras – of a succession of ‘rounds’ in each one of which a vector of prices is proposed by the auctioneer and ‘bons’ are collected declaring the agents’ intentions.
54

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 55 nonproduced inputs (labour and land of the several dates, initial endowments of capital goods) and for capital goods to be produced inside the equilibrium[ 68 ]. Because of the finite number of periods covered by the equilibrium and of the assumption that in the last period there is demand only for consumption goods, all produced capital goods are used for productions for which there is a market inside the equilibrium. By assumption, the demands for capital goods to be produced are met by equal production decisions, so there never is disequilibrium on the market of an ‘intermediate’ good; in fact all production decisions are to order , because decided on the basis of known demands; and therefore all investment decisions too – decisions to buy produced capital goods – are to order , because those capital goods will produce goods for which sale is guaranteed.
There is never a decision to buy a capital good except based on a certainty as to what that decision will earn. If savings is defined, for each period, as in national accounting usual conventions, as the value of the income produced in that period (that is, the value of the inputs utilized in that period) that is not employed for the purchase of consumption goods produced in that period, then, since this value equals the value of the capital goods produced in that period, and all goods produced are sold, necessarily savings equals the value of the capital goods produced and sold in the period, that is, savings equals investment.
Of course, equilibrium in all markets for produced goods does not mean general equilibrium: unless the rentals of nonproduced factors are the equilibrium ones, there will be disequilibrium on their markets[ 69 ]. But, differently from what can happen in
68 Intuition is helped by assuming no joint production; then for each consumption good one can write a vertically integrated production function in terms of nonproduced factors only, and through cost minimization one can derive the demands for nonproduced factors, but these technical choices implicitly determine also the demands for the ‘intermediate’ goods that, if produced and sold, correspond to investments.
69
A different definition of savings and investment has been adopted in Garegnani (2000,
2003), who considers a two-dates economy (hence, a single production cycle) where the endowments at date 0 include commodities that can be consumed or used as capital goods, and
(gross) savings at date 0 are defined as the value of the portion of these endowments that is not consumed and is offered to firms, while investment is defined as the value of the demand for these goods that comes from firms. Then disequilibria on the markets for the services of
‘original’ factors can mean inequality between savings and investment so defined. In the simple economy that produces corn with labour and seed-corn in yearly cycles, Garegnani would ask us to consider the economy just after a harvest, therefore with a given endowment of corn bought by consumers with the payments to the factors that produced it, an endowment whose allocation to consumption or to production has not been decided yet, and he would call savings the supply of seed-corn out of that corn endowment, and investment the demand for it. I prefer to remain with the traditional use of the terms savings and investment as referring to quantities and
→
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 56
Keynesian models, there cannot be inequalities between aggregate demand and aggregate supply; by construction, to excess demand on some markets of nonproduced factors there will correspond excess supply on other ones; Walras’ Law holds, because the aggregate value of consumer demands equals by assumption the value of their factor supplies. There cannot be a problem of insufficient aggregate demand.
This tâtonnement need not be stable; but, since it is formally identical to the one for the acapitalistic economy presented in ch. 6, Appendix 3, it is stable if consumer excess demand satisfies WAA[ 70 ]. Instability can only arise from violations of WAA due to heterogeneous consumers, not from instability of savings-investment markets: by assumption these are in equilibrium at all stages of the tâtonnement.
8.B.20. But there is no right to assume that the incomes on which consumers can count in formulating their demands for consumption goods are the incomes corresponding to the value of their intended supplies of factors[ 71 ], if one wants the tâtonnement to try and mimic, however remotely, the functioning of markets.
Consider the simplest neoclassical CRS production economy, where labour and land produce a single output, vegetables, according to a standard neoclassical diffeerentiable production function, and are paid their marginal products, and there are no savings: every period the income earned by factors is spent entirely on the vegetables produced in that period. Suppose land is fully employed, and the real wage is higher than the equilibrium level; then there is labour unemployment, and the unemployed workers have no income, hence cannot demand the product; the realistic assumption that only employed factors earn an income and can demand the product implies that in this economy there will be disequilibrium only on the labour market. This is in fact the way to make sense, in the neoclassical approach, of the effect of a real wage kept above its equilibrium level by trade unions or by law.
Having grasped the reasonableness of the assumption that only employed factors earn an income and can demand produced goods, let us now assume that, in the same incomes produced during the period under consideration. One difficulty with Garegnani’s terminology is that in the neo-Walrasian framework there is no analytical basis to distinguish, at date zero, the endowment of corn from the endowment of labour services or land services (these too might be partly directly ‘consumed’, as leisure and parks).
70
See Appendix 1 to this chapter for an illustration of the reinterpretation required to apply the analysis of ch. 6, Appendix 3, to the intertemporal economy.
71
According to a very different criticism advanced by Garegnani (2000, 2003, 2005a,
2005b), even such an assumption does not eliminate the savings-investment problem. An intuitive presentation of the argument is supplied in Appendix 5.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 57 economy, each consumer supplies one unit of only one factor, either labour, or land.
Now suppose relative factor rentals are the equilibrium ones, but firms produce one half the full-employment output. Half the supply of labour and half the supply of land remain unemployed. But again there is no disequilibrium on the output market: earnings and hence the demand for the product still equal the value of the product. This example makes it still more evident that consumer incomes cannot be assumed given before firms decide factor employments. It is firms’ decisions that determine consumer incomes and hence consumer demands for produced goods. Therefore there is no right to take consumer incomes as given (and equal to full-employment incomes) and to derive production decisions (and hence decisions as to purchases of capital goods) from them, if one wants to understand disequilibrium processes.
The moment this is accepted, the announcement of prices does not suffice to determine consumer demands; at least part of firms' decisions as to productions and hence factor demands must be determined first[ 72 ], and will determine what portion of factor supplies finds purchasers and therefore what income consumers have at their disposal for consumption purchases or savings. Investment decisions are necessarily at least partly autonomous.
The issue can be grasped by considering an example. Assume a three-dates intertemporal economy where only one good, corn, is produced by labour and corncapital. Production cycles take one period; the economy starts at t=0 and ends at T=2, so there is production in periods 0 and 1. At date 0 the economy starts with an initial endowment of corn C
0
produced the previous period; for simplicity assume a rigid propensity to gross savings s, so sC
0
is offered as corn-capital for period-0 production, and (1-s)C
0
is eaten and disappears. So the economy can be viewed as starting with endowments of corn-capital sC
0
and of labour L
0
. Gross output at date 1 is C
1
=F(K
0
,N
0
), at date 2 it is C
2
=F(K
1
,N
1
); K t
, N t
are respectively the corn-capital utilized by firms, and labour employment, that need not coincide with the respective supplies. The supplies of corn-capital at the two relevant dates are sC
0
and sC
1
; labour supplies, assumed rigid for simplicity, are L
0
and L
1
. The production function is a standard differentiable one with constant returns to scale and strictly convex isoquants; profits must be zero, so the undiscounted real wages w
1
, w
2
(quantities of corn paid at the end respectively of period
0 and of period 1) univocally determine the corresponding undiscounted gross rentals
72 At least part, because once a firm decides to employ certain factors, the consumption demands from the incomes of those factors can be determined, and one can determine the demands for inputs deriving from those consumption demands as well as from the demand for those, of the factors demanded by the first firm, that are capital goods.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 58 earned by capital ρ
1
(w
1
), ρ
2
(w
2
), which of course imply rates of return r
1
=ρ
1
-1, r
2
=ρ
2
-1.
(Use of undiscounted prices makes intuition easier in this case.)
In the standard tâtonnement, at each round the auctioneer must call only w
1
and w
2
; outputs C
1
and C
2
adjust[ 73 ] to consumer demands determined by incomes equal to the value of their factor supplies, and to a capital demand K
1
derived from date-2 consumer demand. Demands are, in undiscounted terms, the right-hand sides of these supply-equals-demand equations:
(*) C
1
= (1-s)(w
1
L
0
+ρ
1 sC
0
) + K
1
(**) C
2
= w
2
L
1
+ ρ
2 s(w
1
L
0
+ρ
1 sC
0
) .
K
1
, investment at date 1, is determined as the amount of corn-capital of date 1 needed to produce the quantity demanded of C
2
with the optimal factor proportions determined by w
2
, and the same holds for the other demands for factors:
K
0
= F
K
-1 (C
1
,w
1
)
N
0
= F
L
-1 (C
1
,w
1
)
(***) K
1
= F
K
-1 (C
2
,w
2
)
N
1
= F
L
-1
(C
2
,w
2
).
Once w
1
and w
2
are announced by the auctioneer, C
1
, C
2
, K
1
are determined, and then disequilibria on factor markets can be determined. The demand for C
2
is the equilibrium amount independently of w
1
and w
2
because of the assumption of given factor supplies and given gross saving propensity out of the income of each period, plus the product exhaustion theorem that guarantees that for each period w t
and ρ t
are so tied to each other that the period’s undiscounted total income is independent of income distribution.
Therefore w
1
L
0
+ρ
1 sC
0
, the first period’s total income on which consumers count, is independent of income distribution and equal to the date 1 output obtained if date-0 inputs L
0
and sC
0
are fully utilized, let us indicate it as C
1
*; for the same reason, w
2
L
1
+
ρ
2 sC
1
* is independent of income distribution. For the same reason the demand for consumption at date 1 is independent of income distribution and equal to the equilibrium amount. On the other hand the total demand for C
1
need not equal the full-employment supply of it, because investment might not be equal to full-employment savings. If K
1
< s(w
1
L
0
+ρ
1 sC
0
), then C
1
<F(sC
0
,L
0
) and there may be excess supply on both date-0 factor markets. But since the amount of corn of date 2 that firms intend to produce is given and equal to the equilibrium amount, an excess demand for date 1 labour will correspond to the less-than-equilibrium demand for corn-capital to produce C
2
, so w
2
rises and ρ
2 decreases, hence K
1
rises until it becomes equal to s(w
1
L
0
+ρ
1 sC
0
), that is, demand for C
1
73
We are assuming the planner-auctioneer.
58

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 59 becomes equal to its full-employment supply, and then if one date-0 factor is in excess demand the other factor is in excess supply, and w
1
changes in the direction required to bring complete equilibrium about. Stability is guaranteed.
If instead one assumes that consumer income corresponds to the value of factor supplies that find purchasers , then consumption demand at dates 1 and 2 is no longer determined once w
1
and w
2
are given. Equations (***) continue to determine demands for factors once C
1
and C
2
are given, but now factor demands determine consumer income. Therefore (still assuming for simplicity a given gross propensity to save s out of
C
0
and C
1
) the supply=demand conditions for output at dates 1 and 2 are the following
(where the demands for C
1
and C
2
are the right-hand sides):
(*') C
1
= F(K
0
,N
0
) = (1-s)(w
1
N
0
+ρ
1
K
0
) + K
1
(**') C
2
= F(K
1
,N
1
) = w
2
N
1
+ρ
2
K
1
.
Here K
0
,N
0
, K
1
,N
1
are the quantities of corn and labour supplies that find purchasers. The second equation leaves the possibility open that at date 1 the income distributed (in corn) to factor owners amounts to w
1
N
0
+ρ
1
K
0
, of this income the fraction s is saved and offered for loans, but only the amount K
1
finds purchasers, the rest (if a positive amount) goes wasted, analogously to excess supplies of labour services (the same holds for date-0 excess savings). It is also possible that demand for a factor at date
0 or at date 1 exceeds supply; since we are assuming a (non-standard) tâtonnement, we can assume that, in such a case, the auctioneer calculates herself what the incomes of consumers would be if factor supplies equalled demands (on our assumptions on consumer preferences this is easy), and derives the excess demands so as to proceed to determine the wages to be proposed in the next round. (This tâtonnement does not pretend to be fully realistic, it aims at being no more than an initial unveiling of the complications that the standard tâtonnement hides under the carpet.)
The main implication of this slightly more realistic tâtonnement is that demand for C
2
is equal to C
2
whatever its level, so C
2
is indeterminate . Once the amount produced of C
2
is given, w
2
determines K
1
and N
1
, and then the fixed propensity to save determines C
1 if one assumes that the planner-auctioneer ensures equality between supply and demand for corn-capital at date 1, K
1
= s(w
1
N
0
+ρ
1
K
0
) = sC
1
, thus also determining demands for factors of date zero once w
1
is given. But there is no way round the indeterminacy of C
2
. A theory of investment, rather than simply an assumption of investment adjusting to independently determined savings, seems indispensable in order to surmount this indeterminacy. Whether one can assume a tendency toward the full employment of resources will depend on the theory of investment that will be found
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 60 most convincing[ 74 ]. But then one cannot help wondering whether it is worthwhile to try and imagine a theory of investment for a world of complete futures markets and virtual adjustments, rather than for a more realistic picture of the economy.
8.B.21. Even assuming that consumers count on incomes corresponding to their intended factor supplies, the need to consider investment decisions as something different from productions to order would be evident, in intertemporal equilibria over a finite number of periods, if one assumed (i) that the economy continues beyond the last period T considered by the equilibrium, (ii) at the initial date there aren’t complete markets, or perfect foresight, covering the economy’s evolution beyond date T. Then savings and investment would be present in period T too, but the equilibrium between them would be an open problem: investment in the last period could not be to order.
General equilibrium theorists admit that one should assume (i), but recoil from assuming
(ii). The assumption that the economy truly ends at a certain date is generally admitted to be only a first approximation that must be abandoned at some point; but the problem of justifying the equality between full-employment savings and investment is dealt with by refusing to admit that investment might not be to order: that is, by assuming equilibrium over the infinite future. Therefore we pass to this topic.
EQUILIBRIUM OVER THE INFINITE FUTURE
8.B.22. The formalization of infinite-horizon intertemporal equilibria with a representative consumer will not be discussed in this book. Topology would be required for a rigorous analysis. But it seems possible to convey an intuitive understanding of the main messages of these models. These can be glimpsed already from the model of a neoclassical economy, presented in ch. 3, where a single product, corn, is produced with the use of labour and corn-capital (seed) as inputs.
Let us remember the model. Land is overabundant and hence free. Production is in yearly cycles; competition imposes the same production function to all firms, with
CRS at least at the level of the industry (owing to free entry), that is, in this model, of the entire economy. Technical knowledge is given. Labour supply L is given, and fully employed because wages are flexible. One passes from the gross output production function G = G(K,L) to the net output production function C=F(K,L) by subtracting capital depreciation; for simplicity let us assume circulating capital, then
74
In order to justify a tendency toward the full employment of factors, recourse to the argument presented in §3.3.8 and based on the role of money holdings is made impossible by the difficulty intertemporal equilibria encounter in making room for fiat money (§8.B.12).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 61
F(K,L)=G(K,L)–K and F(·) too has CRS. Income distribution is determined by the fullemployment net marginal products of labour and capital derived from F(·). What we must do is derive some implications of the model for growth theory.
A common simplifying assumption in neoclassical growth theory is the so-called
Inada conditions on marginal products: it is assumed that as the amount of a factor increases while the other factor is fixed and positive, the marginal product of the first factor remains indefinitely positive, tending aymptotically to zero; and the marginal product of the first factor tends to +∞ as its amount tends to zero. But always positive marginal products would be too unrealistic if assumed for the net output production function: it is hard to accept that, however great the amount of capital already employed with a given amount of labour, one more unit of corn-capital not only always increases output, but always increases it by more than one unit. When assumed to apply only to the gross output production function, the Inada conditions do not prevent the marginal net product of capital from becoming negative, as shown in Fig. 8.??, where G
K
satisfies the
Inada conditions and yet F
K
becomes negative for K>K*. intl 1, char12
K
gross output
K
K* K + net output
Fig. 8.B.1. Gross and net output as functions of corn-capital in the corn-labour economy with a fixed labour supply.
Assume that initial K is less than K*, and that tastes are such that every period a positive fraction s<1 of net output is saved, and re-invested: investment is, neoclassically, determined by full-employment savings. Let K t
be the corn-capital stock with which the economy is endowed at the beginning of period t. The economy reaches the full utilization of both capital and labour each period, with income distribution determined by the marginal products of labour and capital; so the qualitative paths of capital, output, and income distribution are easy to deduce. K grows from one period to the next; since C t
=F(K t
,L), then K t+1
=K t
+sC t
, therefore K grows, and as long as the net
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 62 marginal product of capital is positive, net output grows too but by a smaller percentage than K owing to the decreasing marginal product of K[ 75 ]; therefore net savings sC t
are less than required for an unchanged growth rate of K, and the rate of growth of K slows down gradually. As K grows, the real wage rises, and the rate of interest decreases. Since the net marginal product of capital, and hence the rate of interest, becomes zero when
K=K*, and then negative, sooner or later the assumption of a fixed s becomes implausible: one can plausibly assume that the community will realize the little or even negative effect of further net savings, and that the propensity to net savings s will decrease. For example one can assume that preferences are such that aggregate s is an increasing function of the rate of interest, that is, that s decreases as the rate of interest decreases; assuming that s becomes zero for a positive interest rate r*, we obtain that output growth slows down gradually, tending asymptotically to stop as the net marginal product of K, and hence the rate of interest, approaches r*. The economy tends to a stationary state with K<K*.
Now assume that labour supply is not given, but grows at a given rate n>0, and go back to assuming a given average propensity to net savings s. In a given period t, output is C t
=F(K t
,L t
), net savings is sC t
, and there results a certain K t+1
, that is, a certain growth rate of K t
, call it g
Kt
=sC t
/K t
. As long as g
K
>n, K/L grows at rate g k
–n; the marginal product of K decreases, hence C grows slower than K, and this reduces g
K
, which tends asymptotically toward n; conversely, if g
K
<n, K/L decreases, therefore C grows at a higher rate than K, so g
K
rises, again tending toward n. When g
K
=n, K/L is constant, the economy is on a steady growth path. If s is an increasing function of the rate of interest, the tendency of g
K
to approach n is faster, but the tendency toward a steady growth at rate n is still there; but the change of s is no longer necessary for a steady state to be reached.
Solow’s 1956 growth model assumes that pretty much the same can be assumed also for a complex modern economy, where capital is a multitude of heterogeneous capital goods. Fully accepting the traditional marginalist/neoclassical conception of capital as a single value factor, therefore measured in the same units as national output or income Y, Solow’s model assumes that one can represent the entire economy’s production possibilities via an aggregate CRS net production function Y=F(K,L).
Production is now described as a continuous process, so Y is a flow per unit of time and the analysis is in continuous time. Labour supply is given by
75
If an increase of K by 1% causes C to increase by 1% or more, and the marginal product of labour is positive, then an increase of both K and L by 1% causes C to increase by more than
1%, implying increasing returns to scale against the assumption of CRS.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 63
L(t)=L
0 e nt .
Assuming again a given average net saving propensity s, we have dK/dt = sY.
Let y=Y/L and k=K/L. Because of CRS, Y/L=F(K/L,1) or y=F(k,1) which we can represent as y=f(k), net output per unit of labour as a function of capital per unit of labour, an increasing strictly concave function. Note that since dK/dt = sY, net savings per unit of labour are sf(k) = sY/L = (sY/K)·(K/L) = [(dK/dt)/K]·k = g
K k.
By definition (dk/dt)/k=g
K
-n, and multiplying both sides by k one obtains dk/dt = g
K k-nk = sf(k)-nk.
Representing graphically sf(k) and nk as functions of k, one obtains Figure 8.B.2[ 76 ].
In Fig. 8.B.2 the line nk has been drawn for two possible values of n. For a sufficiently high initial net marginal product of capital[ 77 ], sf(k) is initially above nk and crosses it from above. It is clear that k grows if it is less than the value at the crossing point, and decreases in the opposite case, thus tending to the value k^ that corresponds to steady growth. It is perfectly possible that this k^ be greater than k*, it is what happens in Fig. 8.B.2 for n=n
2
. In this case, clearly, consumption per unit of labour is not maximized in the steady state: a reduction of s would shift downwards proportionally the entire sf(k) curve, shifting k^ to the left and thus ensuring both a higher net output per unit of labour, and smaller net savings per unit of labour. In fact, steady-state consumption per unit of labour keeps increasing for a while even if s is decreased beyond the level that causes k^=k*. intl 1 char 12
y n
1 k
n
2 k
sf(k)
O k
1
^ k* k
2
^ k
76 Strangely, Solow’s 1956 article does not consider at all the possibility, indeed plausibility, that sf(k) may start decreasing beyond a certain value of k.
77 The Inada conditions imply that f(k) has infinite slope at k=0; a less extreme assumption will do, what is necessary is that the slope of f(k) at k=0 be greater than n/s.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 64
Fig. 8.B.2. Two possible steady-growth values of k in Solow’s model, depending on n; the second one is a clear case of overaccumulation.
A word of caution: in the real world, ownership of capital is unequally distributed, some people live on the interest earned by their capital without working, and the economy is possibly very far from a steady state (which, in this model, is approached very slowly, on a time scale of several decades), so the relevance of maximizing steadystate average consumption per unit of labour (which is not the same thing as per person) is unclear. But if you are interested in it, then the problem to be solved is max s,k
(1-s)f(k) subject to sf(k)=nk.
Straightforward application of the method of Lagrange multipliers yields the
Lagrangian function (1-s)f(k)+λ[sf(k)-nk]; set its partial derivatives with respect to s, k,
λ, equal to zero and you will obtain (check this result as an
Exercise ) f ’(k) = n, the marginal net product of capital – that is, the interest rate, in this neoclassical economy – must equal the steady-state growth rate; this is the so-called Golden Rule of
Accumulation . We can see from Fig. 8.??? or Fig. 8.?? that this implies that the k^ that maximizes steady-state consumption per unit of labour is less than k*; indeed f(k) is horizontal at k*, that is, f ’(k*)=0<n, while ‘optimal’ k^ requires f(k) to have slope n.
This result implies f ’(k^)·k^=sf(k^); that is, assuming the rate of interest equals the marginal net product of capital, net savings must equal the income from capital ownership , as when all wages are consumed and all income from capital ownership is invested[ 78 ].
The Solow growth model is generally complicated with the introduction of technical progress: the latter is usually assumed to be of the labour-augmenting type, that is, such that the production function instead of having the form F(K,L) has the form
F(K,M) where M=τ(t)L (with τ>1 and growing with time) is augmented labour , that is, the increase in labour productivity caused by technical progress is represented as causing each unit of labour to count as if it were more than one unit of labour, and more and more so as time passes. If labour grows at rate n and τ grows at rate μ, then augmented labour supply grows at rate n+μ. The basic idea remains the same: the reinvestment of savings with a given net savings propensity will cause the capital-labour ratio and the
78
If n=0, f’(k^)=0 implies k^=k* and the fact that the rate of interest is zero implies zero net savings, as required by the zero growth rate. Thus, the Golden Rule of Accumulation applies to the case n=0 too.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 65 corresponding income distribution variables to tend toward constant levels[ 79 ], associated with a rate of growth equal to the growth rate of augmented labour supply[ 80 ].
Now assume that the same economy is controlled by a benevolent planner who knows the utility function of the infinitely-lived representative consumer, who is both worker and capital owner; suppose this planner wants to find the accumulation path that maximizes the welfare of the representative consumer. The planner can do it by opportunely choosing what fraction of net output to save each period, and thus choosing a time path of consumption (consumption per person, if population is growing) from the initial period for the infinite future. The task is not necessarily easy, as the utility function may be such that utility over an infinite consumption path is infinite, in which case maximization may make no sense. But supposing a criterion exists that allows deciding which one, of any two alternative consumption paths, is preferred by the representative consumer, the planner must choose the feasible consumption path that the representative consumer prefers. A consumption path is feasible if the consumption assumed for date t is technically achievable, on the basis of given consumption at previous and at following dates, and of the economy’s technology and factor availability.
An optimal consumption path must be technically efficient, that is, such that it is not possible to increase consumption at some date without decreasing consumption at some other date; efficiency can be shown often to imply that the same path might be achieved as an infinite-horizon intertemporal equilibrium.
Even in the very simple case, the so-called Ramsey model, that makes the same assumptions about production as the Solow model (a single good produced by the economy, utilizable as consumption good or as capital good), and even with a utility function that remains finite[ 81 ], maximization requires the use of optimal control or of dynamic programming , mathematical techniques not presupposed in this textbook; but, if
79
However, now a constant wage per unit of M will mean an increasing wage per unit of actual labour.
80
Endogenous growth theory further complicates the model by making the rate of technical progress depend on the resources dedicated to ‘producing’ technical progress, or even directly on the growth rate of output (on the basis of Smithian ideas about the effects of increasing
‘division of labour’, or of learning by doing). But the study of the determinants of technical progress and of their implications is heavily conditioned by the basic framework adopted, so this book prefers to concentrate first on the choice of the framework, classical-Keynesian, or neoclassical (Petri, 2003b).
81 The usual assumption is that (if formalization is in discrete time) utility is additive, the sum of countably infinite discounted period utilities (or felicities
) ∑ t
δ t u(c t
) for t=0,1,..., where δ<1 is the discount factor, and the felicity function is sufficiently concave (e.g. logarithmic) to ensure convergence of the sum.
65

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 66 certain assumptions are made, the maximization can be performed, and the outcome of the maximization can be decentralized , that is, it can be supported as an infinite-horizon intertemporal equilibrium. Under assumptions similar to those of the Solow growth model about labour supply and technical progress, the optimal time path of the economy can often be shown to converge to a steady growth path. The difference from the Solow growth path is, in the end, small: in both cases growth is supply-determined, and changes in the capital-labour ratio alter income distribution in the way indicated by neoclassical theory; the main difference is the endogenously determined propensity to save, that changes along the time path. But for practical purposes this difference is of little relevance: how the propensity to save changes is anyway largely arbitrary, depending on the assumed form of the utility function of the representative consumer; also, in a real economy there is no representative consumer (and no ‘dynasties’), people decide on how much to save as individuals, unaware of the overall consequences of their decisions, not to speak of the consequences some decades hence. When these weaknesses of the model are taken into account, the main surviving messages are, that market economies tend to the full employment of resources; that income distribution is determined by the marginal products of labour and capital, which will change if capital grows at a rate different from the rate of growth of (augmented) labour; that growth is supply-determined (investment adapts to full-employment savings) and hence a faster growth rate requires more savings and less consumption; all messages derivable already from Solow’s model.
Fundamentally the same messages emerge from the extension of the model to accommodate heterogeneous consumption and capital goods. Each period, the available factors (with the capital endowment now a heterogeneous vector) imply a production possibility frontier, PPF, that – because of the full employment of resources – entails a tradeoff between more consumption that period, or more capital goods (and hence, it is argued, a PPF shifted outwards) the next period. The need to choose also the composition of consumption and the composition of investment every period does not introduce relevant additional problems for the benevolent planner (except the need to assume even more computational capacity); and again – under certain assumptions – it is possible to show that the outcome of the maximization can be supported as an infinitehorizon intertemporal equilibrium[ 82 ].
82
Assuming, of course, the planner-auctioneer in order to surmount the indeterminacy of supply decisions at equilibrium prices. A discussion of the problems with existence, determinateness, uniqueness and stability of infinite-horizon models would require the technical details of the formalization, that will not be discussed in this book. On uniqueness, let it suffice to say that the assumption of a representative consumer ensures uniqueness in these models too,
→
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 67
8.B.23. It is the latter type of disaggregated intertemporal equilibrium over the infinite future (with the addition of stochastic elements) that is considered the rigorous microfoundation of current mainstream DSGE (Dynamic Stochastic General
Equilibrium) macroeconomic models, or more generally, DGE models (the acronym proposed by Wickens, 2008, to include the models without stochastic elements, e.g. the
Ramsey model): “it is now widely agreed that macroeconomic analysis should employ models with coherent intertemporal general-equilibrium foundations” (Woodford, 2009, p. 269). The premise of these models is therefore that infinite-horizon intertemporal general equilibrium theory is a robust starting point for a descriptive theory. But the extension of the equilibrium to cover an infinite number of future periods does not eliminate the impermanence problem, the price-change problem, the substitutability problem and the savings-investment problem.
The substitutability problem still arises for the first periods of the equilibrium.
On the impermanence problem, it can be noted that the farther into the future the equilibrium path extends, the greater the potential cumulative deviations from it. Japan from 1990 to 2000 had a growth rate around 0.5% a year, after several decades of a growth rate approximating 5%, which we can assume here – for illustrative purposes only – to be the growth rate corresponding to full-employment savings; had Japan grown even only at 3.5% a year for those ten years, in 2000 its output would have been 34% greater than it was, and its stock of capital goods correspondingly greater – an enormous difference.
On the savings-investment problem, the dependence of consumer incomes on firms’ decisions holds over infinite horizons too. To view this fact clearly, imagine a full-employment equilibrium over the infinite future, in an economy with many types of identical consumers, each type supplying only one nonproduced factor, one unit per consumer. Now imagine that, at the same prices, firms produce of each good one half of but with heterogeneous ‘dynasties’ the Sonnenschein-Mantel-Debreu results obtain. Stability is simply not studied: the reasons can be supposed to be that a tâtonnement on an infinite number of markets seems impossible to conceive, and that anyway the need for perfect foresight renders the issue of stability difficult even to conceive, as explained in §8.B.16; but one can only guess, because there is a disconcerting absence of discussions of the legitimacy of assuming instantaneous continuous equilibration, as if existence were all that is needed for the equilibrium to have relevance. Existence and determinateness do raise some additional problems in the infinite-horizon case relative to the finite-periods case; but the negative conclusion that, as I will have to conclude, must be reached anyway on the usefulness of these models discourages me from imposing on the reader a discussion of these additional issues.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 68 the equilibrium quantity and employ one half of the equilibrium inputs, leaving one half of factor supplies unemployed. One half of each type of consumers is unemployed and with no income. Product markets are in equilibrium; investment equals savings. If firms decided to produce more, factor unemployment would decrease in spite of no change in relative prices. The equality of savings and investment at equilibrium prices does not determine production levels, because production decisions of firms determine incomes and hence savings, which are therefore determined by investment decisions. The general equilibrium requires an additional assumption of full employment of resources. But then different assumptions, for example the Keynesian approach that postulates some autonomous expenditure plus the multiplier, appear to be no more arbitrary, and with more support from the empirical evidence.
On the price-change problem, a correct foresight assumption becomes the less credible, the farther into the future the periods to which it is applied, in view of the logical unpredictability of novelties if for no other reason. Essentially the same issue can be viewed from a different perspective. The need to calculate the equilibrium prices for an infinity of future periods imposes impossible computational burdens on agents, unless the calculation can be effected on the basis of a finite system of mathematical relations
(e.g. a finite system of differential equations whose solution determines the entire time path of the variables)[ 83 ]. This means that the ‘laws of time change of the economic variables’ must be finite in number, hence, essentially, unchanged at least from a certain period onwards. And since politics and institutions influence the economy, one must assume complete predictability of historical developments in these fields too. If one recoils from such an assumption, then one must accept that the finite system of mathematical relations that specify the ‘laws of time change of the economic variables’ cannot be correct over the infinite future. But then, what can the infinite-horizon intertemporal equilibrium aim to indicate?
8.B.24. So the question posed at the end of §8.B.18 acquires, if possible, even greater urgency: why should we believe that intertemporal equilibrium models depict, albeit only approximately, the fundamental tendencies of income distribution,
83
No mathematical result can require the actual execution of an infinite number of operations. To grasp the point, think e.g. of proofs based on mathematical induction, that conclude that a certain property applies to all natural numbers because if it applies to n then it applies to n+1, and it applies to 1. In spite of the ‘all’ in the statement proved, a proof of this kind does not imply the actual performance of an infinite number of operations; it states that, whenever it is needed to prove that the property holds for the natural number n, then the proof can be achieved in a finite number of steps.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 69 employment and growth in actual economies?
It is not easy to find an explicit answer to this question; but an implicit answer appears supplied by some writings of Robert Lucas and Thomas Sargent, the economists who started post-monetarist neoclassical macroeconomics (Real Business Cycle models,
Dynamic Stochastic General Equilibrium models, recent New Keynesian or New
Neoclassical models). Lucas and Sargent have been fundamental in the spread of the
Rational Expectations Hypothesis , which, in the absence of stochastic variability of the determinants of future economic variables, amounts to perfect foresight; if there is stochastic variability (around a trend), the Rational Expectations Hypothesis corresponds to the assumption that the true probability distributions, together with the structure of the model and the trends of the variables under study, are known, so that expectations are correct on average [ 84 ]. Now, both Lucas and Sargent argue that rational expectations and the connected optimal decision rules are what economic agents will converge to, given time, if the economic environment is sufficiently constant as to allow rational expectations to be learned [ 85 ]. This means accepting that, if some event (e.g. an unexpected technical progress, a change of tastes) has recently changed the probability distributions, one cannot expect the Rational Expectations Hypothesis to be satisfied; correct expectations cannot be assumed except for situations in which learning has been completed. Therefore equilibria including a Rational Expectations assumption can only aim at determining the average behaviour, the trend, of economies where expectational mistakes in fact do happen, but are considered to cause only minor alterations of the trend relative to the one with Rational Expectations. Now, in his famous ‘islands’ model
84 The REH does not, by itself, require the model in which it is introduced to be a neoclassical model, but in fact the two things have so far been always coupled.
85 "The economic interpretation of this assumption of rational expectations is that agents have operated for some time in a situation like the current one and have therefore built up experience about the probability distribution which affects them. For this to have meaning, these distributions must remain stable through time." (Lucas, 1974: 190). Later in the same article
Lucas repeats that the probability distributions of the variables under examination "are learned by processing observed frequencies in some sensible fashion… which has the property that the
‘true’ distributions become ‘known’ after enough time has passed." (Lucas, 1974: 204). And
Sargent: “rational expectations models impute much more knowledge to the agents within the model (who use the equilibrium probability distributions in evaluating their Euler equations) than is possessed by an econometrician, who faces estimation and inference problems that the agents in the model have somehow solved. ... Rational expectations is an equilibrium concept that at best describes how such a system might eventually behave if the system will ever settle down to a situation in which all of the agents have solved their ‘scientific problems’ ” (Sargent
1993, p. 23).
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Lucas (1972) assumes market clearing even for the situations in which information is incomplete and therefore some expectations are not correct; but again, the implicit admission that expectations are not continuously correct implies that market clearing too can only refer to average outcomes, not to what really goes on moment by moment. For example, if correct expectations must be learned, then after any unexpected novelty a seller may well misjudge for some time whether a drop in the demand for her product is only a stochastic irregularity or indicates a permanent demand shift, and thus may produce a disequilibrium quantity of the good for a number of periods. To assume correct expectations, in order to determine the equilibrium of the market in which this seller operates, can only mean that one is determining the situation the market converges to or oscillates around, given time: the traditional centre of gravitation! And in (1986)
Lucas does indeed admit that the assumption of continuous market clearing is only a simplification intended to depict the trend of markets in which there is no auctioneer but the time-consuming adaptive behaviour of agents in the setting of prices converges to the equilibrium price[ 86 ]. Therefore the equilibria of the models formulated by Lucas and
Sargent are intended to determine the normal or average or trend situation of economies where disequilibria are in fact present, but can be neglected because markets gravitate around and towards the REH equilibrium. The irrelevance of mistakes, and in particular of the errors and corrections in the determination of the composition of capital, implies that what is taken as given in these equilibria as capital endowment is not the vector of capital goods, but capital conceived as a single amount, of endogenously determined composition: the traditional marginalist conception of capital. One understands better, then, why Lucas and Sargent always use models where capital is a single factor, as in the
Solow-Ramsey models, in spite of referring to Arrow-Debreu as the rigorous notion of equilibrium. The equilibria of their models are long-period equilibria as much as
86
For example: “decision rules are continuously under review and revision ... We use economic theory to calculate how certain variations in the situation are predicted to affect behavior, but these calculations obviously do not reflect or usefully model the adaptive process by which subjects have themselves arrived at the decision rules they use. Technically, I think of economics as studying decision rules that are steady states of some adaptive process, decision rules that are found to work over a range of situations and hence are no longer revised appreciably as more experience accumulates” (Lucas, 1986, pp. S401-402). As noticed by
Dvoskin (2014, §6), time-consuming adaptive processes imply that adjustment of prices to their equilibrium levels cannot be instantaneous. In a widely cited early survey of the rational expectations literature, the admission is clear that continuosly clearing markets is only a way to indicate the average: “In the other economic markets [i.e. other than labour markets] mistaken plans are undertaken and expectations falsified, but the markets are more or less continuously cleared” (Kantor, 1979, p. 1435, emphasis added).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 71
Solow’s[ 87 ], centers of gravitation of economies actually all the time in disequilibrium, where capital is treated like a single factor homogeneous with output because the belief is still present that capital can be treated like a single factor of variable ‘form’, necessarily measured therefore in the same units as output: as an amount of value.
Subsequent New Classical authors discuss the meaning of their equilibria even less, but they are great admirers of Lucas, and their models clearly are concerned only with averages and trends (the vast use of detrending would suffice to prove it); so it is safe to conclude that they accept Lucas’ views. We discover then that the continuousequilibrium macro models, where nowadays the neoclassical approach is most clearly applied, aim at describing only the trend that the economy is presumed to gravitate around owing to time-consuming disequilibrium adjustments; where this presumption derives from a continuing faith in the traditional neoclassical conception of capital as a single factor . Behind the smokescreen of the reference to Arrow-Debreu and of the assumption of continuous equilibrium in the models utilized, little has changed relative to the traditional method of admitting that the economy is actually always in disequilibrium and that the theoretical model can only aim at describing the trend which time-consuming stabilizing adjustment mechanisms acting in disequilibrium tend to reestablish when the economy diverges from it[ 88 ]. But these can only be the adjustment mechanisms postulated by the traditional versions of the marginal approach, the ones relying on capital the single value factor: above all, labour demand increasing if real wages decrease owing to unemployment; and investment increasing if the real interest
87 The quantity of the single capital good of Solow’s model is persistent enough to allow for time-consuming disequilibrium adjustments, so the so-called ‘momentary equilibrium’ of
Solow’s model has no need for the auctioneer or any other kind of instantaneous adjustment in order for the economy to gravitate towards it. The time scale over which the tendency towards it can be assumed to operate can well be years. So it is in fact a long-period equilibrium, a centre of gravitation of time-consuming adjustments. Because of this, it is not so illegitimate to modify the model by making the propensity to save depend on expectations as to future income and income distribution, expectations that have plenty of time to be corrected, and in this way one passes, without much loss of credibility, to Ramsey-type descriptive models, whose ‘momentary equilibria’ can again be seen as centres of gravitation of time-consuming adjustments. Of course the models can be interpreted as applying to real economies only if one believes that the behaviour of economies with heterogeneous capital is correctly grasped by the treatment of capital as a single factor homogeneous with output and substitutable for labour in the same way as land.
88 However, relative to pre-Keynesian economic theory, modern mainstream macroeconomics differs in that it generally assumes that labourers are on their supply curve all along the business cycle, thus admitting less extensive economy-wide disequilibria than in pre-
Keynesian business cycle theory.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 72 rate decreases. Therefore as I have written elsewhere, the people, who attribute to disaggregated intertemporal equilibria the role of indicators of the tendencies of actual economies in spite of the inability of these equilibria to say anything on the distance between equilibrium path and the behaviour of economies not continually perfectly in equilibrium, must be holding a more or less conscious belief that the undeniable occurrence, in actual economies, of disequilibrium and time-consuming adjustments does not destroy the traditional neoclassical theses as to the trend the economy follows, a trend which is believed to be reasonably approximated by the intertemporal equilibrium path. Only an idiot would deny that in actual economies there is no auctioneer and no complete futures markets, but rather time-consuming trial-and-error adjustments, mistakes, disequilibria, imperfect foresight; so [these] theorists must believe that actual economies are not all the time in equilibrium, there is in fact continuous error-correction, discovery of novelties, discrepancies between supply and demand on the several markets, but there are persistent forces that cause these disequilibria to be sufficiently corrected or compensated so that the trend the economy follows is not too far from the path described by their continuous-equilibrium models. The intertemporal equilibrium is then to be understood as only an indication of the qualitative properties of the average trajectory of the actual economy, which is never in equilibrium. Behind the reference to modern neo-Walrasian intertemporal general equilibrium theory as the microfoundation of the macro models there must therefore be a much more traditional and less absurd position than the belief that the economy is actually continually perfectly in intertemporal equilibrium in all markets: namely, a belief that the assumption of continuous equilibrium does not do excessive violence to the description of actual economic behaviour, because the tendency toward full employment, toward equality between supply and demand on the several markets, and toward income distribution determined by marginal products, does exist in reality although it is far from instantaneous, and it causes the behaviour of the economy to be not too far from what it would be with continuous equilibrium. But then the reference to disaggregated intertemporal equilibrium with perfect foresight as the ‘rigorous’ microfoundation of the models is only a smokescreen, behind it there must be and there can only be a belief in the time-consuming disequilibrium adjustment mechanisms on whose basis the marginal approach was born and accepted... Mechanisms ultimately based on the conception of capital as a single factor of variable ‘form’. (Petri 2015 p.??)
In other words, the belief in the conception of capital as a single factor, and in the connected traditional adjustment mechanisms, justifies the belief in the gravitation of real economies around a trend like the one described by Solow-Ramsey models. The central gravitational mechanisms are, first, the tendency of firms to employ more labour if the real wage decreases, and to employ more capital per unit of labour in new plants
(i.e. to invest more) if the rate of interest decreases; and second, the tendency of the real
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 73 wage and of the rate of interest to respond to excess demands on the respective markets.
From these a tendency is derived toward the full employment of resources; this tendency is taken to imply that the economy's evolution over longer periods is not too far from the continuous full-employment one. Then the similarity between this trend, and the path generated by disaggregated intertemporal equilibrium, authorizes the belief that, although of course a real economy cannot follow exactly the latter path, still it follows a roughly similar trend, and therefore disaggregated intertemporal equilibrium paths do give indications on the general qualitative character of the trend that the economy follows.
Therefore the usual characterization of the aggregative neoclassical growth models used in mainstream macro literature as simplified versions of the ‘rigorous’ disaggregated infinite-horizon intertemporal equilibrium models, and deriving their legitimacy from the latter ones, appears to be the opposite of the truth . The neoclassical analyses based on the capacity of traditional capital-labour substitution to cause the economy to gravitate, through time-consuming adjustments [ 89 ], toward full employment are the real microfoundation of the claimed validity of intertemporal equilibrium theory as a positive theory, not the reverse. Without a faith in those analyses the implausible assumptions needed by neo-Walrasian equilibria would make it impossible to attribute descriptive relevance to these equilibria.
We can now reconsider what was stated in §8.B.15, namely, that a theory of the actual path would be needed to assess whether the intertemporal equilibrium path gives an acceptable approximation to the behaviour of real economies. What emerges here is that behind the continuing faith in the descriptive validity of intertemporal equilibria there is an implicit theory of the actual path, and it is the traditional neoclassical theory based on long-period equilibria, because the faith persists in capital the single factor and in traditional capital-labour substitution, in spite of the criticisms explained in ch. 7. To put it in an expressive although imprecise way[ 90 ], it is Solow’s model that, if accepted, allows assigning some descriptive value to Arrow-Debreu, not the opposite.
89
The legitimacy of the assumption of continuous equilibrium for a Solow-type model does not derive from intertemporal equilibrium theory, where this assumption is an unfortunate necessity, but from the long-period nature of the equilibrium of a Solow-type model, that only aims at describing a trend around which the economy gravitates, and could be considered little affected by disequilibrium actions (see the previous footnote), if the conception of capital as a single factor were legitimate.
90
Imprecise, because it does not stress that capital in Solow’s model is not really a physically homogeneous good but is the traditional single factor of variable ‘form’, embodied in the heterogeneous capital goods, necessarily an amount of exchange value.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 74
8.B.25. Two important implications follow.
First, the great investment of intellectual energies, in recent decades, into the rigorous formalization of intertemporal equilibrium over an infinity of periods has added next to nothing, in terms of reasonable predictions, to what the simple one-good growth model of Solow (1956) indicated. Since disaggregated intertemporal equilibria cannot aim at being more than approximate qualitative indicators of the actual aggregate path, they contain only one potentially significant extension relative to Solow's model: the abandonment of the rigid propensity to save, replaced by the assumption that the aggregate propensity to save depends to some extent on the present rate of interest and its expected evolution over time. But as noted in §8.B.22, even this extension is of doubtful significance: even the sign of the dependence of savings on the rate of interest cannot be known in advance, it can only be arbitrarily assumed [ 91 ]. And even if this functional dependence could be empirically ascertained and assumed unchanging over decades (another highly debatable assumption), what difference would it make to the qualitative characteristics of the trend, relative to Solow’s model? The growth path would remain a supply-determined one, with real wage and rate of interest slowly changing owing to technical progress and to the difference between the rates of growth of labour supply and of capital; one would only be able to add, to the predictions of
Solow’s model, a rather irrelevant prediction of a slow change of the propensity to save as a percentage of GDP.
Second, the existence and determinacy of infinite-horizon equilibrium paths, even if solidly demonstrable, would not add to the credibility of Solow-like paths as approximate indicators of actual paths; the credibility goes the other way if at all, and rests on the credibility of the traditional marginalist time-consuming adjustment mechanisms. To plunge students as soon as possible into the study of optimal control, dynamic programming, or topology, so as to master infinite-horizon equilibria, has then the effect of taking their minds off the question, whether satisfactory arguments exist for believing that these equilibria do indicate, however approximately, actual paths; this question, if asked, would bring them to understand that the really important questions concern the credibility of the traditional time-consuming neoclassical adjustment
91
The assumption, that the rate of interest and its evolution over time is determined by utility maximization over the infinite future, produces results that depend on the arbitrary choice of the form of the utility function, and on the arbitrary assumption that this form will not change even many years into the future, and therefore appears to be more an excuse for display of impressive mathematics than a real increase in the realism and reliability of results.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 75 mechanisms. Hence the importance of the contents of ch. 7. To which one can add the empirical evidence: the numerous historical episodes, of nations suffering high rates of unemployment and undergoing slow or negative growth for many years, strongly suggest that it is empirically false that continuous full resource employment can be taken to approximate, however roughly, the actual behaviour of market economies.
8.B.26. Before closing this Part, a warning on the use of the term ‘stability’ in the recent literature on intertemporal equilibria over the infinite future. Nowadays, when the question is posed whether an infinite-horizon intertemporal equilibrium (or sequential
Radner equilibrium) is stable , what is meant is not whether disequilibrium adjustments would converge to it, but the totally different issue whether the equilibrium path converges to a steady state: an issue of doubtful relevance, unless it is previously established that the equilibrium path is a good indicator of the actual path of the economy. The modern neoclassical literature motivated by macro applications proceeds as if this good-indicator property could be taken for granted, with no explicit supporting argument[
92
]. The implicit argument has been pointed out and criticized in §8.B.21.
92 "We will repeatedly exploit these classical connections between competitive equilibria and
Pareto optima as a device for proving the existence of equilibria in market economies and for characterizing them. That is, we will solve planning problems, not for the normative purpose of prescribing outcomes, but for the positive purpose of predicting market outcomes from a given set of preferences and technology." (Stokey and Lucas with Prescott, 1989, p. 31). The book from which this quotation is taken contains no discussion of whether the infinite-horizon intertemporal competitive equilibria it examines can really be considered good predictors of market outcomes.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 76
PART C : OVERLAPPING GENERATIONS
8.C.1. We pass now to an introduction to the infinite-horizon equilibrium models that dispense with the assumption of infinitely-lived consumers and assume instead overlapping generations (OLG). These models too suffer from the impermanence, pricechange, substitutability, and savings-investment problems, and for the same reasons; so those criticisms will not be repeated. The exposition will concentrate on the differences from the infinitely-lived consumer models. The presentation of OLG equilibria will also afford the opportunity to present the notion of core .
Overlapping-generations (OLG) models assume consumers who live for a finite number of periods, and whose lives partially overlap with the lives of other consumers.
This is realistic, but in equilibria over an infinity of periods it means that there is a numerable infinity not only of goods but also of consumers. It has emerged that this combination causes new problems in the theory of general equilibrium.
The simplest model of OLG general equilibrium assumes that each consumer lives two periods; at the beginning of each period, that is, at each date, there are the young, who will also live the next period, and the old, who know that the period just starting is their last period. For simplicity, in each generation all agents are assumed identical, hence aggregable into a representative consumer.
Suppose that in this economy population is stationary, and in each period income is only produced by the young: a single perishable consumption good is gradually produced by labour as the sole input, and it is consumed the moment it is produced.
Preferences are such that labour supply per person is given.
Suppose an initial social arrangement where the young work and consume, while the old, who are unable to work, survive on theft or by scavenging rubbish dumps, or simply die of hunger. No way of transferring income over time exists. People are selfish, the young do not care about the old. The equilibrium of this OLG economy consists of a separate extremely simple equilibrium for each period, where the young produce on the basis of their given labour supply, and consume what they produce, while the old lead a horrible existence at the margin of society. This equilibrium is not Pareto efficient.
Suppose at a certain point (date 0) a religious leader proposes a social pact: each young generation will pass to the old generation of the same period half its production. By kind concession of the gods, all future generations are present when this proposal is made, and each generation can accept or reject it. Suppose that preferences (assumed to be the same for all generations) are such that the young prefer splitting the income over the two periods to enjoying the whole of it when young and then having a miserable old age; but if they are going to receive nothing when old, then the young want as much income
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 77 when young as possible. If the pact is accepted, the young of date 0 pass half their income to the old of date 0, and when old (date 1) they receive from the young of date 1 half their production. By assumption, they prefer it to the previous situation. The old of date 0 also prefer it, and all subsequent generations too. So the pact is accepted and a
Pareto improvement is obtained. (Who guarantees, and how, that the pact will be respected in the future is a problem we will not discuss.)
Note that this is only possible because of the infinity of future periods. If the economy ends at a last period T, the young of that period reject the pact because they have no incentive to pass half their income to the old of that period, they want to enjoy as much as they can the single period they are going to live; then the young of period T–
1 have no incentive to pass half of their income to the old of period T–1 because they will receive nothing when old; and so on backwards: the pact is rejected by all generations (except the old of period 0). Assuming a very high but finite number of periods makes no difference: the real difference is made by the assumption of an infinite number of periods, i.e. of a never-ending economy. (Or at least, of the possibility that there will always be still a few more periods: certainty of indefinite continuation may not be necessary, it may be enough that there be always a positive probability that the economy continues. We do not stop on this.)
8.C.2. Let us now study slightly more complex OLG economies. We will see that some unexpected possibilities arise, e.g. of a continuum of equilibria[ 93 ].
Time is divided into periods. Period t goes from date t to date t+1. Initially we concentrate on a stationary economy that not only will go on forever but has also forever existed, that is, t can take any integer value in the open interval (

,+

). All individuals in each generation are identical (in preferences and endowments) so we aggregate them into a single individual: each generation is an individual who lives two periods.
Generation t is the generation that is young in period t. In each period t there are two generations, the young (first life period of generation t), and the old (second life period of generation t-1).
Let us initially consider a pure exchange economy. Assume there is a single perishable consumption good , and each generation, both the young and the old, starts each period with an endowment ('manna from heaven') of this good. Each generation is only interested in her consumption over the two periods of her existence. Assume a logarithmic utility function (equivalent to a Cobb-Douglas) for each generation t:
93
We will take the existence of OLG general equilibria for granted.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 78
U t (...0,x t t
,x t t+1
,0...) = a t ln x t t
+ (1 – a t
) ln x t t+1 where x t j
is the consumption of generation t at time j. These utility functions are defined over all vectors x = (...,x
-1
,x
0
,x
1
,...)

L
+
, where L
+
stands for the space of vectors of nonnegative real numbers with infinite countable elements. Marginal utilities are a t
/x t t
and
(1-a t
)/x t t+1
.
Each generation has a positive endowment only in the two periods of its life. We indicate the endowment of the generation that is young at t as
 t

(
 t t
,
 t t+1
).

(...;
 t-1 t-
1
,
 t-1 t
;
 t t
,
 t t+1
;
 t+1 t+1
,
 t+1 t+2
;...) is the infinite-dimensional vector of all endowments, that includes two elements for each period. The overall endowment of the good at the beginning of period t is
 t-1 t
+
 t t
. There are no bequests[ 94 ].
Let us define an allocation as an infinite-dimensional vector with two elements per period x=(...;x t-1 t
,x t t
;x t t+1
,x t+1 t+1
;...) that specifies the consumption of each generation in each period, and therefore also the consumption of each generation over the two periods of its life. An allocation is admissible if in every period t total consumption equals total endowment: x t-1 t
+x t t
=
 t-1 t
+
 t t
.
Prices p t
are discounted prices of the single consumption good, for example in terms of some date chosen as date 0 where p
0
=1. An equilibrium is defined by a vector of discounted prices p

L
++
(strictly positive because preferences are monotonic) and by an admissible allocation x such that for each generation at the given prices utility is maximized (under a hypothesis of price-taking) i.e. that with p t
 t t
+p t+1
 t t+1
<

,
 t: x t

(x t t
,x t t+1
)

ArgMax

U t
(x t t
,x t t+1
)
 p t
(x t t
-
 t t
)+p t+1
(x t t+1
-
 t t+1
)=0

.
The budget constraint is an equality because of non-satiation.
If p is an equilibrium price vector then k p is also an equilibrium price vector with k a positive scalar; the normalization
 t p t
=1 is not convenient because there is an infinite number of prices; a different approach is to define q t
=p t+1
/p t
,
 t

Z.
Since prices p t
are discounted prices, it is (1+r t
)p t+1
=p t
, thus q t
=1/(1+r t
) is the discount factor from period t+1 to period t (if an amount of the consumption good at date
1 costs C in undiscounted terms, its value at date 0 is Cq t
), or the relative price of the consumption good of date t+1 in terms of the consumption good of date t. This
94
The assumption of no bequests is the usual one in OLG models, but it obscures the important role of inherited wealth in capitalist economies. The rich pass their wealth to their children, consuming only a small part of it when old; the middle class does largely the same at least for the property of houses, and for education expenses. Inheritance plays a fundamental role in the maintenance of social structure.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 79 normalization makes it possible to redefine the equilibrium as a sequence of relative prices q=(...,q
-1
,q
0
,q
+1
,...) and an admissible allocation x such that x t

(x t t
,x t t+1
) satisfies x t

ArgMax

U t (x)
 x t t
+q t x t t+1
=
 t t
+q t
 t t+1
 where the new expression for the budget constraint is obtained by dividing by p t
both sides of the original budget constraint. The maximization of the utility of generation t, once its endowments are given, depends only on q t
.
Let z t t
(q t
)
 x t t
-
 t t
be the excess demand of generation t at time t, i.e. of the young at time t, and let z t t+1
(q t
)
 x t t+1
-
 t t+1
be the analogous excess demand of the same generation when old, i.e. at time t+1. With this notation we can re-write the equilibrium condition as:
[1] z t-1 t
(q t-1
)+z t t
(q t
)=0,
 t
Let us study the case of a stationary economy, with the same preferences and endowments for all generations and the same number of identical agents in each generation. The parameter a t
=a is the same in all utility functions, and the same goes for
ω t
. We study the stationary solution, so we suppose that t goes from –

to +

.
Let us set
 t-1 t
+
 t t
= 1 for simplicity, and again for simplicity let us indicate the endowment when young as

=
 t t
,
 t, and therefore the endowment when old with 1-

.
Let us also assume

> a

1/2.
Thus we are assuming that the young's endowment is greater than the old's endowment and furthermore that in the utility function consumption when young is given greater weight, or at most the same weight, than consumption when old. Suppose each generation can transfer income across its two periods. The maximization problem of generation t is
MAX xt, xt+1
[a ln x t
+ (1-a) ln x t+1
] subject to x t
+q t x t+1
=

+q t
(1-

).
We can re-write the budget constraint as x t
=-q t x t+1
+

+q t
(1-

) and use it to eliminate x t
from the objective function:
MAX xt+1
u t = a ln [

+q t
(1-

)-q t x t+1
] + (1-a) ln x t+1
.

The sole variable is x t+1
. Given the shape of Cobb-Douglas indifference curves, the maximum is internal, so we impose:
 u/
 x t+1
= –q t a/[

+q t
(1-

)-q t x t+1
] + (1-a)/x t+1
= 0 i.e.
-q t a/x t
+ (1-a)/x t+1
= 0 .
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 80
This yields the form taken in this example by the so-called Keynes-Ramsey Rule ( 95 ):
[2] aq t
/x t
= (1-a)/x t+1
.
The marginal utility of one more small unit of consumption tomorrow must just offset the marginal disutility of the decrease in consumption to-day, made necessary by the given rate of exchange q t
between consumption to-day and tomorrow. Now using jointly the budget constraint and equation [2] we eliminate x t+1
and obtain:
[3] x t
= a[

+(1-

)q t
].
Thus consumption when young is a fraction a

1/2 of the present value of the total endowment.
Let us look for the equilibria of this stationary economy.
First of all there is an autarchic equilibrium in which each generation consumes its endowments, with no exchanges nor intertemporal reallocations of consumption. This equilibrium is defined by the allocation x=

and by a vector of constant relative prices q=(...,q*,q*,q*,...) determined as follows. Because the equilibrium is autarchic it is x t t
=
 and x t t+1
=1-

; by substitution in equation [2] we obtain q t a/

= (1-a)/(1-

) and therefore q t
= q* = [(1-a)/(1-

)]/(a/

)>1.
When q t
=q*, it is x t t
=

from equation [3], and therefore x t t+1
=1-

from the budget constraint, so there is equilibrium. Note that q*>1 (which follows from the assumption that

>a) implies a negative interest rate. The interest rate must be negative enough to induce each agent to consume each period that period's endowment: owing to the assumed form of the utility function and the assumption

>a

1/2, the marginal utility of consuming one's endowment when old, (1-a)/(1-

)>1, is greater than the marginal utility of consuming one's endowment when young, a/

<1: now, q t
is the relative price of the consumption when old with respect to consumption when young, and it must be equal to the ratio between the two marginal utilities.
8.C.3. But the autarchic equilibrium is not the only stationary equilibrium.
Another equilibrium is the golden rule equilibrium (so called because Pareto-efficient,
95
This rule takes a different form in different models, but it always expresses the need to balance, at the margin, the advantage of more consumption to-day with the disadvantage of the decrease in consumption in subsequent periods made inevitable by the assumption of full utilization of resources and therefore of a trade-off between consumption and savings in each period (in models with production, more consumption to-day means less capital tomorrow). In economies where this constraint does not hold because production is constrained by insufficient aggregate demand and not by the availability of resources, the rule does not apply.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 81 as we will see), with equilibrium relative price vector q=(...,1,1,1,...), hence zero interest rate, and consumptions (a,1-a) for each generation. The marginal utility of consumption when young and when old is the same, 1; the allocation is admissible. It is an equilibrium because for q t
=1 the optimal choice of each generation is x t
/x t+1
=a/(1-a) which, because of the budget constraint, imposes x t
=a, x t+1
=1-a. (We might also show that if population grows at rate g, then there is a golden rule equilibrium with interest rate equal to g, but we omit this demonstration.)
We now show that the autarchic equilibrium is Pareto-inferior to the golden-rule equilibrium.
Proof . We have seen that in the autarchic equilibrium the marginal utility of consumption when young is less than the marginal utility of consumption when old, so for each generation a diminution of consumption when young, and increase by the same amount when old, increases utility until it brings the two marginal utilities into equality.
Thus each generation is better off in the golden-rule than in the autarchic equilibrium. █
We have thus shown that the autarchic equilibrium is not Pareto efficient. We have confirmed what had been shown informally in §8.C.1: the First Welfare Theorem does not hold for OLG equilibria .( 96 )
We prove now that the golden rule equilibrium is Pareto efficient, by proving that, starting from it, any increase in the welfare of a generation implies that the utility of some other generation decreases.
Proof . Assume, without loss of generality, that starting from the golden rule equilibrium the utility of generation 0 is increased via an increase of its consumption when old, x 0
1
. (At least one of its two consumptions must increase if its utility is to increase.) This implies a decrease of x 1
1
, and, in order to maintain the utility of generation 1 at least unchanged, the latter's consumption when old, x 1
2
, must increase by a greater amount than the decrease of its consumption when young: this is because the starting situation is one of equal marginal utilities, i.e. of maximum utility achievable from a total two-period consumption, and then any reallocation of a given quantity of consumption from one period to the other decreases utility. So it must be x 1
2
>x 0
1
; for the same reason, it must be x 2
3
>x 1
2
, and so on; and the increase x t+1 t+2
-x t t+1
must itself increase with t, because, owing to decreasing marginal utility, even successive equal decreases of consumption when young would require greater and greater increases of consumption when old in order to leave utility unaffected, while here the decreases of consumption when young are themselves increasing as t increases; so at a certain point
96 On the contrary, it does hold for infinite-horizon economies with a finite number of infinitely-lived consumers, cf. e.g. Mas-Colell et al. (1995, pp. 766-68).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 82 x t t+1
becomes greater than 1 and therefore impossible. If the utility of generation 0 is increased by increasing its consumption when young, the same reasoning applies in the opposite direction. We have thus proved that it is inevitable that some generation's utility decreases if some other generation's utility increases starting from the golden rule equilibrium. █
These are not the only equilibria of this economy; there are others, which we will now study as part of the study of the core in OLG models.
OLG models and the core
8.C.4. Another difference between the usual equilibrium models and OLG models concerns the so-called core of the economy, a notion that has given rise to a very complex literature but will be discussed here in simple terms only.
The framework for this notion is an economy described in the marginalist way: there is a given number of individuals, each one with given endowments, and there is a given production possibility set; these determine a set of feasible allocations reachable via exchange and production.
The core of the economy is a subset of this set of feasible allocations; an allocation is in the core if no subset of individuals in the economy (including the set that includes all the individuals in the economy) can, by using only their endowments (plus production), reach an allocation restricted to this subset, that makes at least one individual in the subset strictly better off and no one worse off than in the initial allocation; with divisible goods and continuous non-satiable utility functions, this is equivalent to saying that it is possible to make everybody in the subset strictly better off.
For example, an allocation in a pure-exchange economy with three consumers is in the core if the following conditions are simultaneously satisfied: 1) no consumer can be better off counting only on her own endowments; 2) no couple of two consumers can, with the sole endowments of the couple, reach an allocation that makes both of them better off; 3) no Pareto-superior economy-wide feasible allocation exists.
The last condition shows that only Pareto-efficient allocations can be in the core; but not all Pareto-efficient allocations are in the core. For example, in a standard twoconsumers pure-exchange economy represented by an Edgeworth box with smooth strictly convex indifference curves and an interior Pareto set (except of course at the two endpoints), the core is the contract curve , the portion of the Pareto set included in the
‘lens’ formed by the two indifference curves that pass through the endowment point.
This is because to be in the core an allocation must be Pareto efficient, but a Pareto-
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 83 efficient allocation outside the contract curve does not satisfy condition 1.
In the theory of the core[ 97 ] the subsets of individuals are called coalitions , and if one of these coalitions can with its sole resources make all its members better off than in an initial allocation, then it is said that the coalition can improve on the initial allocation, or also that it blocks the initial allocation (what this can mean is discussed below). The core is the set of allocations that are not blocked by any coalition .
In Arrow-Debreu-McKenzie economies, if an equilibrium exists, the core is not empty, because equilibrium allocations are in the core . I prove this last claim for exchange economies.
Proof . By contradiction. Consider an exchange economy with n individuals where individual i has endowment vector ω i . Let x*={x* 1 ,...,x* n } be an equilibrium allocation at prices p*. Let S stand for the set that collects the indices of members of a coalition.
Suppose a coalition S can improve on x*, that is, there is a feasible sub-allocation x such that for each i

S yields x i
≻ x* i , and such that Σ i

S x i = Σ i

S
ω i , which implies
Σ i

S p*x i = Σ i

S p*ω i .
By the theory of revealed preference, it must be p*x i
>p*x* i =p*ω i
for all i in S, otherwise at the equilibrium prices p* consumer i would prefer to use her equilibrium income to purchase x i and not x* i . This implies
Σ i

S p*x i > Σ i

S p*ω i that contradicts the previous equality. So the assumption that there exists a coalition that can block x* brings to a contradiction, hence there cannot be such a coalition, and x* must be in the core. █
The fundamental result of core theory, which I only enunciate very intuitively and will not prove, is that, as consumers become more and more numerous and ‘smaller and smaller’ relative to the entire economy, the core shrinks, tending in the limit to coincide with the general equilibrium allocations. ‘Smaller and smaller’ is generally formalized as follows: in the framework of an atemporal pure-exchange general equilibrium, starting from a given number of agents with given endowments, one assumes a ‘replication’ of each agent (endowments included), the same number of times for all agents; therefore the m-th replica economy includes m agents identical to each i-th agent in the starting situation; in this way as the replication proceeds each agent becomes smaller and smaller relative to the size of the markets where he/she operates. In the limit, as agents tend to become infinitesimal relative to the market, competitive equilibria tend to become the sole non-blocked allocations. This is called the core equivalence theorem or also the
97 As applied to economic allocations. Core theory is broader than this, but here the term will be used in this more restricted sense.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 84 core convergence theorem .
To understand why the theorem is found interesting, one must start from Francis
Ysidro Edgeworth (1845-1926), from whom the idea of core is derived. Against some opinions expressed by Stanley Jevons, Edgeworth pointed out that repeated barter between two persons does not bring to a univocal result, because any point on the contract curve can be reached, depending on the succession of barters and on the barter abilities of the individuals. All one can say is that if, as long as mutually more favourable exchange opportunities exist, they are discovered and a further barter is proposed, then barter will stop only when some point on the contract curve is reached. But, Edgeworth went on to argue, this indeterminacy is reduced when the number of exchangers increases, at least as long as one admits that ‘recontracting’ is possible, that is, that
‘contracts’ are only tentative promises of exchange that can be cancelled when one of the partners can at the same time stipulate a more favourable contract with other exchangers.
To prove this claim Edgeworth introduced indifference curves and an argument that, in more modern terms, can be formulated as follows. Suppose two goods, x and y, and, initially, two individuals with strictly convex indifference curves, A and B. Assume given endowments and draw the corresponding Edgeworth box, assuming that A’s and
B’s indifference curves through the endowment point Ω are not tangent but form a ‘lens’ and therefore the contract curve is a curve and not a single point, as in Fig. 8.C.1.
Suppose the repeated barter produces the allocation indicated by point α, with B extracting all possible advantage while A has the same utility level as on her own. Once that allocation is reached, if the only person with whom A can barter is B, no recontracting will be accepted by B since he would be worse off if A is better off.
But now assume two more consumers appear, A
2
who is identical to A, and B
2 who is identical to B. Measure A
2
’s allocation from the same origin as A, and B
2
’s from the same origin as B. A
2
has the same initial endowment as A, but A now disposes of allocation α, and can exchange with A
2
: their total availabilities of x and y allow A’s allocation to move downwards from α, along the segment joining α and Ω, any distance up to reaching the mid-point β, if A
2
’s allocation moves upwards from Ω the same distance on the same segment. Again, the result of this bargaining cannot be known in advance, perhaps A stops before β because she reaches a tangent indifference curve, perhaps β is reached. Suppose the latter: now B is at α, A and A
2
are at β, and B
2
is at Ω.
But now B
2
can offer A to recontract, that is, to undo all previous contracts and to bargain with him, and to reach some point on the contract curve where A is at least no worse than before recontracting, that is, no worse than at point γ where her indifference curve through β crosses the contract curve. Let us suppose again that A gains nothing, and ends at γ. This pushes B and A
2
back to the initial endowment Ω, but by assumption
84

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 85 recontracting is allowed, so they cannot prevent this; but they can contract among them, and A
2
will not accept a point on the contract curve worse than the point obtained by A, otherwise she will propose to B
2
a better contract. Furthermore, if the mid-point of the segment from γ to Ω touches an indifference curve of A (and hence of A
2
) higher than the one through γ, A
2
(or A) can recontract and go back to Ω, and since A (or A
2
) is at γ the two can contract to reach that mid-point; then the counterproposals of B and B
2
will be accepted only if A and A
2
obtain a point on the contract curve yielding them no less utility; repeating the reasoning (always under the assumption that B or B
2
extracts all the advantage from contracts with A or A
2
) we conclude that recontracting will go on until a point δ on the contract curve is reached by both couples, such that A’s (and A
2
’s) indifference curve through it goes also through the mid-point η of the segment that connects δ to Ω, and therefore A and A
2
can no longer increase their utility by one of them going back to Ω and then the two contracting to reach the same allocation. This means that any ‘double’ allocation in the portion of the contract curve from α to δ (this last point excluded) cannot be part of the core. By a symmetrical reasoning, assuming now that it is A and A
2
that are able to extract all possible advantage from bargaining with B or B
2
, an analogous portion at the other end of the contract curve is shown to be out of the core. The core has shrinked.
Edgeworth showed that further restrictions of the indeterminacy are operated by the appearance of further ‘replica’ competitors, the core shrinking more and more as the number of replicas increases, until it comes to coincide with (in the example shown) the only equilibrium allocation. He also argued that the analogous process also happens with more goods as long as multilateral barter exchange is admitted (everyone exchanging with everyone else at once).
O
B
or O
B2
contract curve with only A and B
γ δ
α η
▪β
▪Ω
O
A
or O
A2
Fig. 8.C.1.
He concluded that, differently from what Jevons had done, the ‘law of one price’
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 86
(absence of arbitrage) and price taking need not be assumed , they can be derived from a description of competition as repeated bargaining and recontracting among very numerous price-making competitors, a process that produces price taking at the same time as it produces a tendency toward an equilibrium allocation.
The modern theory of the core reformulates the same idea in terms of ‘improving coalitions’, profiting from the fact that all the successive stages of bargaining and recontracting in Edgeworth’s process can be seen as the formation of coalitions that
‘block’ the previously reached allocation by actually implementing another allocation that is an improvement for the members of the coalition. For example the passage from the allocation with A and B at α, and A
2
and B
2
at Ω, to the allocation with B at α, A and
A
2
at β, and B
2
at Ω, can be seen as B, A and A
2
forming a coalition that changes the allocation of A and A
2
; the subsequent contract between B
2
and A that reaches point γ is a coalition that ‘blocks’ the previous allocation. The theory of the core generalizes this insight by asking us to imagine agents engaged in a ‘market game’ in which agents do not use prices, “simply wander around and make tentative arrangements to trade with each other” (Varian 1992 p. 388), and these tentative arrangements, called ‘coalitions’, compete with one another, until no better arrangements can be found: a core allocation has been reached.
What remains undiscussed, in core theory, is whether it is plausible that, when an allocation not in the core is proposed, agents will discover that there is a better arrangement that they can reach, and will be able to contact the other persons required by the ‘improving coalition’, and to co-ordinate with them, even when the coalition is large.
As Varian admits, the idea that the core will be reached “places great informational requirements on the agents—the people in the dissatisfied coalition have to be able to find each other. Furthermore, it is assumed that there are no costs to forming coalitions so that, even if only very small gains can be made by forming coalitions, they will nevertheless be formed” (Varian 1992 p. 388). Now, it seems clear that, if the formation of the coalition requires the co-ordination of a very great number of people (and remember, sometimes only a coalition including all individuals in the economy can block a non-core allocation), then generally it is an impossible task, equivalent to perfectly efficient central planning. Note also the need for coalitions to include yet-to-beborn consumers, if the economy is intertemporal , plainly an impossibility. And yet some modern authors seem blind to such considerations. In a recent contribution it has been written: “it would be surprising to find the economy settling on an allocation outside the core, since that would indicate there is a coalition which could have made each of its members better off, using only its own resources, but for some reason has failed to coalesce and do so” (Anderson,
New Palgrave 2008, “ core convergence”, vol. 2 p. 238).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 87
But a feeling of surprise would be justified only if the reasons preventing the coalition from forming were implausible – and the author makes no attempt at all to argue it. Nor would he have found it an easy task, in the face of the unemployment and crises exhibited by real economies.
The fascination of Edgeworth's suggestion that the more numerous the pricemaking individuals competing in the same market, the more the indefiniteness due to price making is reduced by the presence of competitors, so that in the end agents behave as price takers at the prices bringing about equilibrium, largely comes from its resemblance with the traditional picture of how concretely on a single product market the price tends toward the level that makes demand equal to supply, a picture indeed based on pricemaking agents: a seller unable to sell all her supply proposes a lower price to the buyers she is able to contact, then some buyers turn to her, then in order not to lose their customers other sellers lower their prices... In this picture, prices are potentially specific to each seller-buyer couple, the participants are price-makers, and yet they are obliged by the counterproposals of the other participants finally to converge onto a single price, the one that equilibrates the market. This picture[
98
] too can be described as a process of repeated two-goods barters (but with money as one of the goods), and of proposals of two-person ‘coalitions’, one buyer and one seller, that can ‘block’ other two-person ‘coalitions’ (that is, sales) by proposing a mutually more convenient exchange of the good against money; but one very important aspect of this picture is that the ease of realization of what actions are convenient makes the more convenient
‘coalitions’ indeed form when these exist, as the result of clear individual incentives to act in that direction, incentives deriving from inequality between supply and demand, or from arbitrage opportunities due to lack of price uniformity or non-correspondence of direct and indirect exchange ratios . (The elimination of arbitrage opportunities is of course a clear reason for the formation of two-person or three-person ‘improving coalitions’, so clear indeed, that Jevons took it as obvious that arbitrage opportunities would quickly disappear.) That allocations in the core will be reached requires that
98
Note how this realistic picture implies that, except in organized auction markets, adjustments may take considerable time: the seller must realize she is selling less than expected, which takes time; then she must let buyers know her new price, the buyers must decide to turn to her, the other sellers must realize what is happening and must react, all this again taking time; and if changes of production flows are also involved, the passage of time is even more obvious.
Which is why this picture has been abandoned in modern mainstream textbooks in favour of the auctioneer: not because it was defective, but because of the new need, created by the shift to neo-Walrasian general equilibria with their data lacking persistence, to exclude time-consuming adjustments.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 88 whenever a ‘blocking coalition’ exists, there be clear signals that induce individuals to form that coalition, and furthermore that the formation and dissolution of coalitions converges to a final result, and core theory does not prove that there will be such signals nor that there will be convergence. Supply-and-demand analyses of markets, realistically construed as in the above picture, do provide such signals (when the market equilibrium is stable), and in fact signals that require only local action, just one agent proposing a different contract to another agent, or a firm deciding to produce a different quantity, not the co-ordination of large coalitions. A ‘market game’ of coalitions is too vague and implausible an alternative to such analyses, and therefore it is unclear what it can add to them.
Furthermore, the core equivalence theorem suffers from a problem similar to the unreality of the auctioneer-guided tâtonnement, the moment one tries to give some more concreteness to the process bringing to the core. Elimination of ‘blocked’ allocations by repeated bargaining along Edgeworthian lines requires recontracting, that is, a treatment of all ‘contracts’—differently from the usual meaning of this word—as only tentative promises that can be cancelled, allowing agents to start all over again with their initial endowments. If exchanges irreversibly alter endowments, then repeated barter need not reach the core. This is shown by the following example from Malinvaud (1969, pp. 139-
140). Assume an exchange economy with two goods, 1 and 2, and three agents A,B,C.
The initial endowments are ω
A
=(0,2), ω
B
=(1,1), ω
C
=(1,1). The utility function is the same for the three agents, U(x
1
,x
2
)=x
1 x
2
. In a first barter exchange B gives A 1/4 units of good 1 and receives from A 3/2 units of good 2; now the endowments are ω'
A
=(1/4, 1/2),
ω'
B
=(3/4, 5/2), ω
C
=(1,1); this improves the utility of A from 0 to 1/8 while the utility of
B passes from 1 to 15/8. In a second barter exchange C gives B 1/4 units of good 1 and receives from B 1/2 units of good 2; B’s utility passes from 15/8 to 2, C’s utility passes from 1 to 9/8, because now the allocation is ω'
A
=(1/4, 1/2), ω"
B
=(1,2), ω"
C
=(3/4, 3/2).
All consumers now have the same marginal rate of substitution, 2, so this allocation is
Pareto-efficient, and once it is reached it is sustainable as a no-exchange equilibrium at relative price p
1
/p
2
=2. But it is not in the core, because it is blocked by the coalition formed by A and C, who with their endowments can reach the following allocation: x
A
=(1/4, 1), x
B
=(3/4, 2), superior for both consumers to the allocation reached by the two assumed barter exchanges.
So recontracting cannot be dispensed with; but it requires us to visualize the repeated bargaining as an ex-ante comparison of possible allocations with no actual exchange of endowments, hence instantaneous and fairy-tale as much as the tâtonnement with ‘bons’ and the auctioneer; furthermore, for intertemporal economies it requires complete futures markets, an impossibility. Alternatively, the repeated bargaining must
88

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 89 be interpreted as referring to a repetition of markets that at each round start with the same data (as in the ‘forest economy’ used in §6.27 to try and give a non-fairy-tale interpretation of the tâtonnement), an interpretation incompatible with the data of neo-
Walrasian equilibria, in particular with intertemporal equilibria. The same dilemma would seem to arise for any vaguer interpretation of the core as resulting from an unspecified ‘market game’ of coalitions[ 99 ].
In conclusion, there being no reason, based on core theory, to believe that complex economies tend to the core (and reach it so fast that the tendency is compatible with the given vectorial capital endowment), and since the core notion itself can be defined for intertemporal economies only under the unsustainable assumption of complete futures markets, what remains of Edgeworth’s argument seems to be only that economic agents should be considered price-makers , and competition should be viewed not as passive price-taking but as active rivalry, the active interest of agents to exploit all advantageous opportunities causing a tendency toward a common price for all the units exchanged of a good, and the elimination of arbitrage opportunities. But this is the realistic notion of competition found not only in Marshall, but also in Smith or Marx, and in the latter authors it does not imply the neoclassical tendencies; so to accept it leaves the question open as to the overall working of a competitive economy.
8.C.6. The evaluation of the core equivalence theorem presented here is not the one of most neoclassical theorists; many consider the core equivalence theorem an argument supporting the view that the economy will likely tend to a general equilibrium allocation. It is then useful to know that this argument cannot be advanced for OLG economies, because the core equivalence theorem does not hold in OLG economies; in these the core may be empty in spite of the existence of competitive equilibria. This of
99
Another claim frequently advanced for the core is that “While the notion of Walrasian equilibrium is based entirely on the institution of trading via prices, and assumes that individuals take prices as given, the definition of the core is completely institution free” (Anderson cit. p.
238). This is not true: the notion of core is far from institution-free, not only because it assumes private property, but also because it excludes those institutions that would allow the formation of coalitions capable of achieving objectives different from the one of reaching an allocation with the sole endowments belonging to the members of the coalition. In the real world, coalitions form in order to struggle, to exercise pressure or violence, to increase bargaining power; these ideas are absent from the notion of core, and with little justification, because if the capacity to co-ordinate and form binding agreements (the basis of the theory of cooperative games of which the notion of core is part) is assumed, then the same capacity may well help the birth of those other more realistic coalitions, and the ‘competition’ among coalitions will then take a totally different form – e.g. class struggle.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 90 course implies that in OLG economies not all equilibria need be in the core .
One example is supplied by the OLG stationary pure-exchange economy we have discussed. In it, besides the two stationary equilibria already discussed, there is a continuum of other equilibria; we will see in the next paragraph that for each q
0 satisfying 1<q
0
<q* (that is, strictly included in between the two q's corresponding to the two stationary equilibria) there exists a sequence of (non-constant) equilibrium prices q=(...,q
-1
,q
0
,q
1
,...); there exists, in other words, a continuum of equilibria indexed on the value assigned to q
0
. We will see that in these equilibria q t
 q* for t

+

, and q t

1 for t
 − 
. I assert, omitting the proof, that these equilibria, plus the two stationary equilibria that we already know, are all the equilibria of this economy.
Now that we know all the equilibria, we prove that none of them is in the core .
Let us start with perhaps the most unexpected result, concerning the Paretoefficient golden rule equilibrium. In it, since x t t
=a<

, each generation consumes when young less than its endowment when young, ω. Let us choose any t
0
and let us consider the coalition (with an infinite number of members) formed by the generations with index t
 t
0
. This coalition can make all its members better off than in the golden rule equilibrium, by exploiting the fact that, if it separates itself from the rest of the economy, it need not give a part of ω t0 t0 to the old of generation t
0
-1. The utility of generation t
0
is increased by increasing x t0 t0
until it equals

: this makes it possible to decrease somewhat that generation's consumption when old and still leave her utility higher than in the golden rule equilibrium; the consequent increase in the consumption of generation t
0
+1 when young makes it possible to reduce this generation's consumption when old and still assure her a higher level of consumption than in the golden rule equilibrium; by doing the same for all subsequent generations, the utility of all subsequent generations can be increased too, although by a smaller and smaller amount (tending asymptotically to zero, but remaining eternally positive) as t tends to +

. Thus the golden rule equilibrium can be blocked.
Let us now consider the autarchic equilibrium. The proof given earlier that this equilibrium is Pareto inferior to the golden rule equilibrium implies that any coalition including the generations from any t
0
onwards can improve the wellbeing of all its members by redistributing its resources so as to consume a when young and 1-a when old; this leaves an amount ω-a in period t
0
that can be given as a gift to the old generation of that period, so if t
0
= 0 all generations obtain a utility increase. Let us finally consider any one of the equilibria with 1<q
0
<q*. In this equilibrium too, like in the autarchic equilibrium, the marginal utility of consumption when young is less than when old, and therefore a coalition including all generations from any t
0
onwards can improve the wellbeing of all its members by redistributing its consumption so as to
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 91 achieve the golden rule consumptions. Thus for all equilibrium allocations there exist blocking coalitions. The core is empty. █
A continuum of equilibria in OLG models
8.C.7. In the course of the proof that, for the OLG model we have been using, the core is empty, I have stated without proof that the model has a continuum of equilibria.
Now I prove it.
Let us construct the excess demand functions. Let z t t
stand for the excess demand of the young, z t t+1
the excess demand of the old. It is: z t t
= x t
-

= a[

+(1-

)q t
]-
 z t t+1
= x t+1
– (1-

) =

(1-a)/q t
– a(1-

).
Equation [1] can be rewritten as
[4]

(1-a)/q t-1
– a(1-

) + a[

+(1-

)q t
] -

= 0.
From this equation we can derive the equilibrium value of q t
as a function of q t-1
, or the equilibrium value of q t-1
as a function of q t
, obtaining two non-linear first order difference equations each one of which allows us to obtain all future or earlier equilibrium q t
's once any one equilibrium q t
is assigned:
[5] q t
=

( 1 a ( 1

 a

)
)
+ 1 - a

( 1
( 1

 a

) q t
)

1
= q* + 1 - q*/q t-1
[6] q t-1
= a ( 1
 
)
 

( 1
 a )
( 1
 a )
 a ( 1
 
) q t
.
These equations (which are the same equation written in two different ways) imply that q t
=q t-1
in two cases: when q t
=1 and when q t
=q*=

( 1 a ( 1

 a

)
)
>1. This shows that the autarchic and the golden rule equilibria are the sole stationary equilibria.
Let us now prove that for each q
0
such that 1<q
0
<q*, there exists a non-stationary equilibrium in which q t
 q* for t

+

, and q t

1 for t

-

. Equations [5] and [6] show that an infinite sequence
 q t

with t=-

,...,+

can be obtained for any positive q
0
; it remains to check whether this sequence is compatible with equilibrium (i.e. is nonnegative). Equation [5] shows that the derivative of q t
with respect to q t-1
, that is q*/(q t-
1)
2 , is positive, therefore, since q t
=q t-1
for q t
=1 and for q t
=q*>1, we obtain that, as we increase q t-1
from 1 to q*, q t
increases too and cannot become greater than q*; in other words, for 1<q t-1
<q* it is also 1<q t
<q*, and vice-versa. Furthermore this derivative q*/(q t-1)
2
is greater than 1 for q t-1
sufficiently close to 1: this result, given that q t
=q t-1 when q t-1
=1, implies that, as q t-1
increases starting from q t-1
=q t
=1, q t
increases faster than q t-1
, so it is q t
>q t-1
, at least for q t-1
sufficiently close to 1, and this means that, in an open
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 92 right neighbourhood of 1, q t
increases as t increases. On the contrary q*/(q t-1)
2 <1 for q t-1 sufficiently close to q*, therefore as q t-1
decreases starting from q t-1
=q t
=q*, q t
decreases slower than q t-1
so again we obtain q t
>q t-1
, hence q t
increases with t also in an open left neighbourhood of q*. This also proves that for 1<q t-1
<q* it is always 1<q t-1
<q t
<q*, because this is true in a right neighbourhood of 1 and it remains true as long as q*/(q t-1)
2

1, and when the increase of q t-1
causes q*/(q t-1)
2 <1 this derivative no longer changes sign, so if as q t-1
increases it became q t
<q t-1
before q t-1
reaches q*, it would be impossible that q t
=q t-1 when q t-1
=q*. On the other hand as t increases or decreases from t=0, q t cannot go outside the interval (1,q*) if q
0
is internal to this interval, because we have seen that for 1<q t-1
<q* it is also 1<q t
<q* and vice-versa; therefore q t
 q* for t

+

, and q t

1 for t

-

. Hence q t
is always non-negative if 1<q
0
<q*. (We omit the proof that if q
0
is outside this interval then there is no equilibrium. The proof is based on showing that in this case q t
diverges as one gets farther away from t=0, and therefore in one of the two directions it finally becomes negative.)
We have thus proved that for any q
0
satisfying 1<q
0
<q*, there exists a sequence
 q t

of equilibrium prices that covers the infinite past and future. So we can arbitrarily choose q
0
within that interval: there is a continuum of equilibria( 100 ). █
8.C.8. The reader might object that the assumption that, differently from what is assumed in Arrow-Debreu equilibria, the equilibrium does not have a beginning (the model is a two-way infinity model because t extends to infinity in both directions) is legitimate when one wants to study stationary equilibria, but otherwise it is debatable; it might be argued that a difference between OLG and finite-horizon economies on the possibility of a continuum of equilibria, in order to be really demonstrated, should be proved for equilibria sharing the fact that they are established at a given initial date: so the OLG model should be a one-way infinity model, with t extending from 0 to +

.
However, the possibility of a continuum of equilibria has been demonstrated also when t goes from 0 to +

. The reason for this possibility can be first explained intuitively. It may happen that the equilibrium choices relative to period t depend on the choices in period t+1, which in turn depend on the choices in period t+2, and so on. For
100
Since we are determining equilibria without a beginning, the choice of t=0 is arbitrary and therefore equilibria which have the same price at two different dates must be considered the same equilibrium; but this fact does not eliminate the existence of a continuum of equilibria, because infinitesimal variations of q
0
entail infinitesimal variations of all terms of the price sequence, whose terms differ by finite amounts; in other words, given a certain q
0
’ with the associated sequence (...,q
-1
’,q
0
’,q
1
’,...), every q
0
” in the interval (q
-1
’,q
0
’) determines a different equilibrium.
92

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 93 example, assume that there is a single consumption-capital good produced by capital and labour (as in the Solow model, except that here time is divided into discrete periods). In each period labour and indestructible capital produce the output which can be allocated to consumption or to investment (i.e. to increasing the capital of the next period), according to a standard differentiable production function, and factors receive their marginal products. The choice of generation t about how much labour to supply in period t depends on the income they will earn on their savings in period t+1, which depends on the marginal product of capital in period t+1, which depends on the supply of labour of generation t+1; the latter supply of labour will in turn depend, for the same reason, on the supply of labour in period t+2; and so on. Then if the functional dependence is invertible one can assume a certain labour supply in period 0, and derive what the labour supply must be in period 1, in order that the assumed period-0 labour supply be an equilibrium choice; the period-1 labour supply thus determined, if feasible, will in turn require, in order to be an equilibrium choice, a certain labour supply in period 2; and so one can recursively determine the equilibrium choices in periods 3, 4, .... from the assumption of a certain equilibrium choice in period zero. By altering the period-0 choice one can alter the path that makes that initial choice an equilibrium choice. It may happen that by going forward into the future one discovers that the path becomes unfeasible (e.g. some quantity or price becomes negative), and this means that the initial choice is incompatible with an equilibrium over infinite periods; but it might happen that the recursively determined equilibrium path remains feasible for ever, in which case one has found an equilibrium. This is for example the case if the path asymptotically approaches a steady state (finding paths that asymptotically approach a steady state is the usual trick in the examples produced so far). If the asymptotic tendency to the steady state exists for a continuum of period-0 choices, then the economy has a continuum of equilibria, equilibrium is indeterminate in a strong sense.
8.C.9. I present two simple examples. The first example (Geanakoplos and
Polemarchakis, 1991, pp. 1943-4; I correct a misprint in their presentation) requires first to consider a two -way infinity OLG exchange economy similar to the one already studied in that it is stationary in its data, there is a single (perishable) consumption good, each generation (a single consumer) t lives two periods and consumes (x t t
,x t t+1
); now the endowments of each generation are 1 when young and ω when old, and the utility of generation t is u t = x t t
+ (1/
 ) δ 
-1 (x t t+1
)

, 0<

<1.
The budget constraint is x t t
+q t x t t+1
=
 t t
+q t
 t t+1
. The maximization of utility requires (for simplicity we omit the superscript t indicating that we are referring to generation t): x t+1
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 94
= q t
1/(

-1) /δ, x t
= 1 + q t
ω – q t

/(

-1) /δ .
Therefore the excess demand functions of generation t are z t t
= x t
– 1 = q t
ω – q t

/(

-1) /δ , z t t+1
= x t+1
– ω = q t
1/(

-1) /δ – ω .
They only depend on q t
. Equilibrium requires z t t
(q t
)+z t-1 t
(q t-1
)=0, t=-

,...,+

.
Therefore equilibrium relative prices are determined by the non-linear difference equation q t
ω – q t

/(

-1) /δ + q t-1
1/(

-1) /δ – ω = 0 . Let us assume ω = 0 : this difference equation simplifies to q t-1
= q t

, or q t
= q t-1
1/

.
This means that for any assigned q
0
>0 and different from 1, we can determine an equilibrium sequence q
-1
= q
0 that is to say q t
= q
0
(

–t
)

, q
-2
= q
0
(

2
) , ... , q
-t
= q
0
(
 t
) , and q
1
= q
0
1/

, ... , q t
= q
0
(1/
 t
)
. (For later reference let us note that we can express the same
; result as q t
= q
1
(

1–t
) .) This two-way infinity OLG economy has therefore a onedimensional continuum of equilibria( 101 ).
We build now a one -way infinity OLG exchange economy formally equivalent to the one just illustrated. Time now extends infinitely into the future but not into the past, t=1,2, ...,+

. Two commodities are available each period, and two representative individuals are born each period and live two periods. Individual (h,t) where h=1,2 is individual h young in period t. Commodity (s,t) where s=1,2 is commodity s available in period t. The consumptions of individual (h,t) are (x h,t
1,t
, x h,t
2,t
) when young and (x h,t
1,t+1
, x h,t
2,t+1
) when old, where the two superscripts indicate the individual, and the two subscripts indicate the commodity and the period in which it is consumed. The utility functions and the endowments of the two individuals in each generation are different
(but the same for all generations). Individual 1 only cares for commodity 1, individual 2 only cares for commodity 2; the utility function of individual (1,t) is u(1,t) = x 1,t
1,t
+ (1/
 ) δ 
-1 (x 1,t
1,t+1
)

, 0<

<1, and her endowment is (1,0) in period t and (ω,0) in period t+1, and zero in all other periods. The utility function of (2,t) is u(2,t) = (1/
 ) δ 
-1 (x 2,t
2,t
)

+ x 2,t
2,t+1
, and her endowment is (0,ω) in period t and (0,1) in period t+1. In addition, an individual
(3,1) is assumed to be alive in the sole period 1 with utility u(3,1) = (1/
 ) δ 
-1
(x
3,1
1,1
)

+ x 3,1
2,1
and endowment (ω,1) in the sole period 1.
This economy has equilibria that are equivalent to those of the previous two-way
101
This continuum (cf. footnote 43?? above) only corresponds to different equilibria for values of q
0
between, say,

1 and n

. Since equilibrium extends indefinitely in both time directions, which period is considered period 0 is arbitrary, and therefore the equilibria with q
0
=n and q
0
=n

are the same equilibrium with a different naming of the time periods, period 0 in one is called period 1 in the other.
94

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 95 infinity economy: it suffices to identify commodity (1,t) with the single commodity of period t of the two-way infinity economy, and commodity (2,t) with the single commodity of period (1-t), and analogously individual (1,t) with generation t, and individual (2,t) with generation (1-t), of the two-way infinity economy, for t going from
1 to +

. Let us prove it for the case ω=0. For brevity we indicate commodities in the one-way infinity economy as (s,t) and in the previous two-way infinity economy as simply (t), and analogously individuals in the one-way infinity economy as (h,t) and in the two-way infinity economy as simply (t). We let q
1
= p
2
/p
1
> 0 be assigned in the twoway infinity economy. Since we identify commodity (1,t) with commodity (t), we obtain p
1,2
/p
1,1
=q
1
and, if the equilibria are to be equivalent, it must be in equilibrium p
1,t+1
/p
1,t
= q t
= q
1
(

1–t
) . And since we identify commodity (2,t) with commodity (1-t), we obtain p
1,1
/p
2,1
= q
0
= q
1

and p
2,1
/p
2,2
= p
0
/p
-1
= q
-1
= q
1
(

2
) and in general p
2,t
/p
2,t+1
=p
1-t
/p
1-(t+1)
=q
t
=q
1
(

1+t
) . Thus individual (1,t), t

1, has the same utility function and faces the same relative prices as generation t and makes the same choices; therefore the markets for commodity (1,t) are in equilibrium for t

2. Individual (2,t) has the same utility function as generation (1-t) but with the place of the two goods inverted, but it also faces relative prices which are the same ones faced by generation (1-t) but inverted, so it makes the same choices but with good t+1 in place of good t and vice-versa; therefore the markets for commodity (2,t) are in equilibrium for t

2. There remain the markets for commodities (1,1) and (2,1). The excess demand for good (1,1) of individual (1,1) is – q t

/(

-1) /δ, the excess demand for good (2,1) of individual (2,1) is q t
1/(

-1) /δ. It is left to the reader to check that the excess supplies of individual (3,1) exactly balance these excess demands. Therefore this economy has as many equilibria as the two-way infinity economy we started with( 102 ); it has a continuum of equilibria. (In this example one sees the necessity of at least one-period-ahead perfect foresight for non-autarchic OLG equilibria too: each young generation must correctly predict what it will obtain the next period, which depends on the choices of a not-yet existing generation.)
However, in this example in order to make it possible for individuals to transfer part of their income from when they are young to when they are old it is necessary to assume that there is some kind of financial institution that in each period borrows from the young and uses the amount of consumption good thus obtained to pay back its debt to the old of the same period: the equilibrium would be unreachable if borrowing and
102 The example was presented also as a way to illustrate a general result, that for each twoway infinity OLG economy there is a formally equivalent one-way infinity OLG economy, cf.
Geanakoplos and Polemarchakis, "Overlapping generations", 1991, cit., p. 1944 and Lemma 1, p. 1907.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 96 lending had to occur directly between agents, because the young could only lend to the old of the same period, who, being dead the next period, would be unable to pay back their debt.
Therefore I proceed now to illustrate a second example (Mas-Colell et al., 1995, pp. 770-776), where the financial intermediary is not necessary.
8.C.10. Let us consider a one-way infinity OLG economy where each generation consists of a single type of individuals who live two periods and, taken together, are endowed with one unit of labour when young. Each generation is equally numerous and can be treated as a single individual. The economy is also endowed with a given amount of indestructible land, which in each period belongs to the old of that period for the reason that we now explain. In each period labour and land produce a single consumption good via a standard CRS production function which is the same in every period. The full-employment production of the consumption good is 1 unit. The good comes out at the end of the period and is immediately and instantly consumed; labour and land are paid (in kind, we may assume) their marginal products, which because of
CRS exhaust the product; the young consume part of their labour income and use the remainder – their savings – to buy the land from the old at the end of the period, who immediately consume all their income; in this way the old add, to what they earn as marginal product of land, what they earn by selling their land to the young. If we indicate with w the labour income, with ε=1−w the land income, and with c yt
the consumption of the young at the end of period t-1 (remember our convention that period t goes from date t to date t+1, therefore the end of period t-1 is date t, and generation t-1 consumes at dates t and t+1), then the consumption of the old at the end of period t-1, to be indicated as c ωt
, is equal to the marginal product of land, ε, plus the savings of the young at the end of period t-1, w−c yt
=1−ε−c yt
. (In c yt
and c ωt the subscripts y and ω are mnemonic for 'young' and 'old'.) By assumption ε and w are constant.
Let
 t
be the price of land at the end of period t-1, and p t
the price of the consumption good (all prices are to be interpreted as discounted prices; remember that the equilibrium is reached simultaneously for all periods from the beginning). Our description of the functioning of the economy implies that in equilibrium c yt p t
+
 t
= (1-
ε)p t
, which says that what the young consume, plus what they pay to purchase the land from the old, must equal their labour income; and c ω,t+1 p t+1
= εp t+1
+
 t+1
. The sole decision in each period is the young's decision as to how much to save at the end of the period.
The budget constraint of generation t-1 (for t≥1) is
c yt p t
+
 t
+ c
ω,t+1 p t+1
= (1-ε) p t
+ εp t+1
+
 t+1 but in equilibrium it is
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 97
 t
= εp t+1
+
 t+1 because in the absence of arbitrage opportunities the discounted price of an asset must equal the sum of the discounted prices of the future earnings it will earn. Therefore we can write the budget constraint as c yt p t
+ c ω,t+1 p t+1
= (1-ε)p t
.
This is the standard intertemporal budget constraint for a consumer who must choose between consumption to-day and consumption tomorrow, with endowment (1-ε) to-day and zero tomorrow. Once we are given a utility function, we can derive the choice curve of the consumer as the locus of tangencies of the budget lines that go through (1-ε, 0) with parametric slope –p t
/p t+1
, with the indifference curves between c yt
and c
ω,t+1
. This choice curve yields c yt
and c ω,t+1 once we are given p t
/p t+1
, but it can also be used to derive c
ω,t+1
and p t
/p t+1
once c yt
is given.
Since choices only depend on relative prices, let us put p
1
=1. In period zero land belongs to the old of generation (-1) and their consumption c
ω1
is residually determined once c y1
is given, because c y1
+c ω1 =1. Suppose we pick c y1
and hence c ω1 arbitrarily and ask whether an equilibrium path exists corresponding to this initial choice of generation
0. In order for the given c y1
to be chosen, p
1
/p
2
must be such as to make it an optimal choice; if such a p
2
exists, then c
ω2
is also determined as the one corresponding to the given c y1
on the choice curve. Consider now two possible shapes of the choice curve, shown in Figures 8. 10 and 8.11. In each of these Figures, besides the choice curve, there is also represented a negatively sloped 45° line with intercepts (1,1) on the axes, which represents the points satisfying c ωt +c yt
=1. interl. 15pt
c
ωt
choice curve c
ωt
+1
c ω1
1-γ c
ω3
c
ω1 c
ω2
c ω2
unfeasible
γ c y1
c y2
1-ε +1 c yt
c y1
γ
c y2
1-ε c yt
Fig. 8.10 Fig. 8.11
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 98
Let us initially examine the simpler case of Fig. 8.10 where c
ω2
is univocally determined. Then c y2
is univocally determined too, as the one corresponding to the given c
ω2
on the 45° line; then the choice curve determines c
ω3
and from this, the 45° line determines c y3
; recursively, we can therefore determine c y,t+1
from c yt
. This determination in Fig. 8.10 shows that, unless c y1
= γ such that c
ω1
= 1-γ corresponds to the intersection of the choice curve with the downward-sloping 45° line with intercepts equal to 1 on the axes, the sequence {c yt
} inevitably sooner or later becomes unfeasible
(the reader is invited to check it on the Figure for c ω1 >1-γ); this shows that there is only one equilibrium path, the steady state with c yt
= γ.
Figure 8.11 differs from Fig. 8.10 in that the slope of the choice curve is positive and less than 1 when it crosses the 45° line. The result is that, in a neighbourhood of c y
=γ, the recursion yields a sequence {c yt
} that converges to γ. This means that there is an interval of values of c
ω1
within which an equilibrium path exists for each c
ω1
. There is a continuum of equilibria. (In this interval, it might appear that the recursion is not well determined because a given c yt
can be optimal for two different values of p t
/p t+1
; but one of these generates a sequence that eventually becomes unfeasible and therefore cannot be an equilibrium. The reader is invited to explore what happens if the choice curve crosses the 45° line with a positive slope greater than 1.)
The continuum of equilibria of Fig. 8.11 is robust, in the sense that it does not disappear with small modifications of the utility function and hence of the choice curve.
Other examples of robust indeterminacy in OLG models have been produced; in one of them, indeterminacy derives from the existence of many consumption goods and from assuming that the composition of demand when young and when old is determined simultaneously: then relative prices at date 1 depend on relative prices at date 2, because the composition of the demand of the young generation of period 0 also depends on relative prices at date 2; but relative prices at date 2 depend among other things on the composition of the demand of the young generation of period 1, which depends on relative prices at date 3, and so on. The degrees of freedom in this case are n-1 if there are n different consumption goods. (Some models produce indeterminacy because fiat money is introduced as the store-of-value asset allowing intertemporal transfers of purchasing power, but these models are unpersuasive, because in them nothing determines the price level apart from the self-fulfilling expectations of agents: therefore the indeterminacy is due to the neglect of all the reasons traditionally adduced to explain
98

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 99 why the price level cannot be totally independent of the quantity of money.)[ 103 ] (No worked-out example of indeterminacy has so far been produced, based on labour supply in period t depending on labour supply of period t+1 in a Solow-type model, along the lines suggested in §8.C.8; the attempt to produce such an example could be a useful exercise for a research student interested in this topic.)
There is therefore a possibility of strong indeterminacy of OLG equilibria (the possibility, not of measure zero in the space of parameters, of a continuum of equilibria).
(There has been debate on the root cause of this unexpected result, and no agreed conclusion has been reached. Certainly the possibility of indeterminacy does not depend on a neo-Walrasian treatment of the initial endowment of capital goods: indeterminacy can arise in models without capital, or in models with only one capital good homogeneous with the product. Also, strong indeterminacy does not arise in infinitehorizon models with a finite number of infinitely-lived consumers[ 104 ]; it requires the existence of an infinity of consumers with limited life spans.)
It would seem therefore that the inner logic of the notion of intertemporal equilibrium – once it is admitted that a finite life of the economy, and infinitely-lived consumers, are implausible assumptions – has brought the theory into a blind alley. One can easily obtain a continuum of equilibria, based on self-fulfilling expectations. Of course when many self-fulfilling expectations are possible, it is unclear why one rather that another one should be held, so it becomes totally unclear why the several agents should share the same expectations, perfect foresight becomes even more difficult to assume, and the relevance of the equilibria determined by the model becomes even more doubtful. A highly respected neoclassical theoretician has felt obliged to admit: "It seems that one must now conclude that the forces of supply and demand are not sufficient to determine the rate of interest in the overlapping generations model” (J. Geanakoplos,
"Overlapping generations model of general equilibrium", New Palgrave Dictionary of
Economics, I edition, p. 768).
8.C.11. We can conclude that Arrow, Debreu, Koopmans, Malinvaud and other economists were wrong when confidently arguing that a re-interpretation of the
103
For more on OLG models, cf. J. Geanakoplos, H. Polemarchakis, “Overlapping generations”, in Handbook of Mathematical Economics , vol. 4, 1991, pp. 1943-5; G. Calvo, ”On the indeterminacy of interest rates and wages with perfect foresight”,
Journal of Economic
Theory , 1978; J. Geanakoplos, “Overlapping generations models of general equilibrium”, The
New Palgrave Dictionary of Economics , I ed., 1987; Mas-Colell et al., 1995, cit., pp. 769-777;
Heijdra and Van der Ploeg, ??, chs. 16 and 17.
104
This will not be proved here; cf. the references in the previous footnote.
99

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 100 acapitalistic general equilibrium model of production and exchange, in terms of a neo-
Walrasian intertemporal equilibrium with a vectorial endowment of capital goods, could accommodate capital goods within that model. The reinterpretation causes disasters: it makes the equilibrium a very-short-period one needing instantaneous adjustments, therefore incompatible with the role of center of gravitation of realistic, time-consuming adjustments, therefore silent on the distance between equilibrium path and actual path; it needs an absurd assumption of complete futures markets or perfect foresight; it suffers from insufficient substitutability; it assumes the equality of investment and fullemployment savings but does not explain what realistic processes might bring it about; once a finite horizon is admitted to be unacceptable, and the need for the OLG structure is admitted, it can generate a continuum of equilibria.
Is there some other way to defend the supply-and-demand approach to value and distribution? The alternative of temporary equilibria deserves examination.
100

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 101
Fabio Petri and Fabio Ravagnani
PART D : TEMPORARY EQUILIBRIA [ 105 ]
Part D, Section I: An informal presentation of some problems
8.D.1. This Part describes the alternative, within the neo-Walrasian approach, to complete futures markets or perfect foresight: the temporary general equilibrium (TGE) approach, where equilibrium is reached only for one period, on the basis of expectations of future prices that may turn out to be mistaken and may differ from agent to agent. To minimize complications, in the formal models to be presented these expectations will be mostly assumed to be point expectations, that is, the agent is assumed to be certain as to what price will occur; this allows us to avoid introducing probabilistic considerations almost completely; for the same reason we assume certainty about future states of nature, so we avoid having to consider contingent commodities.
This Part consists of five Sections. Section I is an informal presentation of the temporary equilibrium approach and of some of its difficulties. Section II presents a temporary equilibrium model of a pure exchange economy. Section III introduces production. Section IV summarizes the attempts to introduce money into temporary equilibria. Section V draws conclusions not only on temporary equilibria but also on the entire chapter, with special emphasis on the labour demand curve and on the investment function.
The direction advocated by Lindahl and by Hicks, when they decided that the concept of long-period equilibrium had to be abandoned, was precisely the study of temporary general equilibria (and of their sequences); intertemporal equilibria were judged too unreal. But Lindahl’s proposal of the temporary equilibrium approach in 1929 had little influence for many years; it was translated into English only in 1939. In that same year Hicks’s
Value and Capital was published and owing to the reputation of the author it was widely read, and had great influence. In that book equilibrium is established for one ‘week’[ 106 ], and the agents’ supply and demand decisions for that
105
The formal analyses of Sections II to IV of this Part are by Ravagnani (cf. Ravagnani
(2010)), except for §8.D.11 and §8.D.12 written by Petri. The remainder of this Part draws on observations of both authors.
106 Hicks defines the ‘week’ as a period short enough for the assumption of constant prices during the period to be legitimate: this leaves the length of the ‘week’ rather indeterminate. The choice of the term ‘week’ is anyway significant in indicating that the duration of the period over which the temporary equilibrium is established cannot be long. Hicks also assumes that
→
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 102
‘week’ are based on expectations of the prices at which it will be possible to buy or sell in subsequent ‘weeks’. Hicks’s discussion is only verbal; no formalization is supplied of the new notion of equilibrium in spite of the fact that the formal analogy with the acapitalistic equilibrium model no longer holds owing to the presence of expectations.
One must wait the end of the 1960s for the first attempts to go beyond the Arrow-
Debreu-McKenzie model in the direction of formal temporary equilibrium theory. The motivation adduced for these attempts is the same as in Lindahl and Hicks: the dissatisfaction with the assumption of perfect foresight[ 107 ].
The basic features of temporary equilibrium theory can be summarized as follows.
As in the Arrow-Debreu model, time is divided into a sequence of periods. With a view to the realistic representation of trading processes it is assumed that spot markets for commodities are active in every period, and there aren’t complete futures markets nor perfect foresight; there is only a possibility of transferring purchasing power to the future via the hoarding of money, or via the purchase of securities promising future delivery of a restricted set of commodities or of money. Within this framework, the theory focuses on the behaviour of agents in the initial period, stresses the dependence of agents’ choices on their individual expectations as regards future prices, and discusses the existence of general equilibrium on current markets. No assumption of unanimity is placed on the expectations held by agents at the beginning of the first period. Temporary equilibrium theory is thus ready to acknowledge that economic agents have limited predictive capabilities and may for this reason base their choices on erroneous expectations[
108
].
Research in the field of temporary equilibrium theory attracted many distinguished scholars during the 1970s but was gradually abandoned in the subsequent decade. The work carried out in the field has since fallen into a sort of oblivion, as attested by the fact that temporary general equilibrium models are not even mentioned in recent advanced textbooks. It is, however, our belief that knowledge of the basics of this area of research can be of use for a correct appraisal of the current situation in general equilibrium for the ‘week’ is reached on the Monday of the ‘week’, i.e. at the beginning. Thus
Hicks's temporary equilibrium includes in fact some futures markets, although only for a few
'days'; these markets are complete for the 'week', so no need arises to form expectations on the prices for the remaining days of the ‘week’, and anyway the ‘week’ is defined as that period of time over which prices can be assumed not to change.
107 In a later footnotes I quote the lines by Grandmont mentioned in footnote 58??.
108
Radner sequential equilibria might be seen as consisting of sequences of temporary equilibria with perfect foresight of future prices. But it is generally preferred to restrict the term
‘temporary equilibria’ to the case without perfect foresight.
102

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 103 equilibrium analysis. In accordance with this conviction, we shall endeavour to provide an accessible exposition of temporary equilibrium theory and highlight the analytical problems that emerged in the literature in the 1970s and were no doubt largely responsible for disenchantment with the approach. Several other problems arise, akin to the ones pointed out in §§8.B.15-20 for intertemporal equilibria, some of them admitted by Hicks himself (Petri 1991); some consciousness of these additional difficulties, although less often openly admitted, has no doubt contributed to the oblivion in which the temporary equilibrium approach has fallen.
In this Section a number of the problems of the temporary equilibrium approach will be described informally.
8.D.2. A first group of difficulties can be called of formalization : they arise in deciding how to model the behaviour of agents, and in ensuring the existence of an equilibrium. The problems of this type that have received most attention in the specialist literature will be discussed with the help of formal models in Sections II to IV. Here we point out some additional difficulties of formalization, also occasionally admitted in the literature but given less attention.
A first difficulty concerns the possible non-existence of equilibrium owing to discontinuities of excess demands caused by bankruptcies. In the models to be presented in Sections II to IV, in order to minimize complications the existence of initial debts is neglected, but obviously in actual economies in any period there are agents who in previous periods contracted debts which must now be honoured. These debts may have been incurred on the basis of expectations that turn out to have been wrong: so – assuming the equilibrium on current markets is reached by an auctioneer-guided tâtonnement – it may well happen that, depending on the prices announced by the auctioneer for the current markets and the connected expectations of future prices, some agents find it impossible to honour their debts, and are obliged to declare bankruptcy.
This fact creates doubts on the possibility of determining well-defined temporary equilibria, given the ample room for discretionary decisions on whether to declare bankruptcy, and the slowness with which, very often, bankruptcies produce definite results in real economies (lengthy legal controversies often arise). If this problem is set aside by assuming definite conditions in which bankruptcy will be declared, and definite consequences following from the declaration, one meets the problem of discontinuities of excess demands. Assumptions sufficient to highlight the problem are that bankruptcy is declared for any current price array that, in the opinion of the indebted agent , makes it impossible for the agent to honour her debts and survive, and that if bankruptcy is declared the agent's assets become the property of her creditors in proportion to their
103

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 104 credits. There will be then for each indebted agent ‘knife-edge’ price arrays, such that if any one price by changing infinitesimally reduces infinitesimally the agent’s income, bankruptcy is declared; at these knife-edge prices the agent esteems that the present value of her assets is just sufficient to repay her debts; but her creditors, because of the non-uniformity of expectations, may value those assets differently[ 109 ], so the infinitesimal price change that causes the agent to declare bankruptcy may cause the creditors to undergo a discontinuous jump in their wealth because they pass from owning a credit title promising a certain quantity of goods or of money, to owning assets to which they may not attribute the same value. This discontinuous jump in their wealth causes a discontinuity in their excess demands, and we know that discontinuities of excess demands can cause the non-existence of equilibrium.
Other difficulties arise with price taking, and with the number of firms. We saw in chs. 5 and 6 that price taking can be justified (without recourse to the implausible assumption of infinitesimal consumers and firms) as due to free entry, that will compel firms (that are in fact price-makers) to resign themselves to product prices barely covering average cost. But this justification requires free entry to have the time to produce its effects on supply and hence on price. Now, in temporary equilibria, in many industries firms purchase factor services in the current period in order to start production processes that will be completed in future periods, so the reaching of equilibrium on the current markets does not establish equilibrium on the markets of these products, and free entry cannot be adjusted consequently; therefore how many firms enter these industries, and in order to produce how much, will depend on accidental elements, on the unknowable expectations of managers who need not expect a future price for their product equal to average cost , and are justified in this because there is no mechanism ensuring it, since neither future demand nor future supply is known for sure (the auctioneer only announces current prices, managers do know know how much the other managers decide to produce). The fairy tale of the auctioneer is unable to surmount this problem: price taking can be justified via the auctioneer assumption for the current markets, not for production decisions that will yield revenue in markets that will only open up in subsequent periods. In such a situation, how is one to determine the investment decision of a constant-returns-to-scale firm whose manager expects for its future product a price higher than average cost? (We saw in ch. 5 that many economists have strong doubts on the existence of relevant scale diseconomies setting a limit to firm size.) The firm will necessarily have to take into account the probable influence of its
109 In intertemporal equilibria with perfect foresight all expectations coincide and this possibility cannot arise.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 105 supply upon the product’s future price, as well as the influence of the imperfectly known entry and supply decisions of competitors. It seems impossible here to avoid an indeterminateness of theoretical predictions; the traditional long-period method avoided this problem by concentrating on trends resulting from tentative decisions and their corrections , a method incompatible, however, with the very-short-period nature of temporary equilibria. This indeterminateness is avoided in the temporary equilibrium literature by postulating price taking relative to future expected prices too, and by assuming (in the not many papers where production is admitted) a given number of firms with decreasing returns to scale; but none of these assumptions appears acceptable. The indeterminateness of many production decisions in temporary equilibrium theory makes equilibrium on current factor markets indeterminate too.
Indeterminateness as to how to formalize agents’ decisions arises on many other issues too (for example, to whom credit will be conceded and why; whether an indebted firm will declare bankruptcy or look for further loans; whether a consumer will purchase a consumer durable in the current period or postpone the purchase to next period) and appears inevitable; it could be surmounted only if one could know all the myriad elements determining the details of the day-by-day behaviour of consumers, entrepreneurs, bank employees and so forth. Such a detailed knowledge appears impossible[ 110 ]. Traditional economic theorizing had more realistic ambitions: it aimed at determining normal averages resulting from giving time to the more persistent tendencies to emerge and dominate over the vagaries and accidents of day-by-day decisions. For example, the rate of birth of new firms in an industry did not need to be specified beyond the statement that it would be favoured by an average price greater than average cost and discouraged in the opposite case: the competitive interaction of firms over longer periods would correct or compensate an excessive or insufficient entry into an industry, and thus it authorized the conclusion that on average the price would be close to average cost, without needing the impossible specification of the details of the process.
8.D.3. A second group of difficulties, that can be called of predictive capacity , concerns the right to presume that temporary equilibria describe with sufficient approximation actual economic behaviour: these difficulties include our old friends the
110 The need for such knowledge is rendered less evident in the case of intertemporal equilibria only owing to the artificial assumptions of (i) complete futures markets (or perfect foresight), (ii) no default on obligations, and (iii) adjustments organized by the auctioneer and thus only virtual.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 106 price-change problem, the impermanence problem, the substitutability problem, and the savings-investment problem. The form taken by some of them is now different[ 111 ].
The price-change problem is dealt with by admitting non-coordinated expectation functions that, for each agent, determine future expected prices as functions of currently observed prices. This is clearly more realistic than assuming perfect foresight, but it introduces its own problems. Expectations are subjective, hence mostly unobservable; thus one must make assumptions about them which cannot but be largely arbitrary, with a resulting arbitrariness of the equilibrium itself. Therefore in temporary equilibrium theory the price-change problem takes the form of an indefiniteness problem : the temporary equilibrium corresponding to a given set of observable data is indefinite until unobservable expectations are specified by the theorist, but it seems impossible to specify them without arbitrariness. Thus for comparative statics or predictions the approach is unable to go beyond a list of possible results depending on the largely arbitrary assumptions about expectations. For example, if the rate of interest is determined by the equilibrium between savings and investment, the optimism or pessimism of investors becomes a fundamental determinant and one can only list the possible results depending on their expectations[ 112 ].
111
An excellent reading on the difficulties with attributing explanatory relevance to temporary equilibrium prices is Roberto Ciccone, “Classical and neoclassical short-run prices. A comparative analysis of their intended empirical content”, in G. Mongiovi and F. Petri, eds,
Value, Distribution and Capital. Essays in honour of Pierangelo Garegnani , Routledge, 1999, pp. 69-92.
112
The problem and the way it was avoided in traditional theorizing have been summarized as follows:
“In traditional theory, not only the long-period position, but also the gravitation towards it were generally explained without introducing price expectations. It may be objected that a particular treatment of price expectations was in fact always implied: that, for example, Smith's argument about the tendency of the market price to fall when it exceed the natural price implies that producers should expect the high market price to last long enough for them to reap the extra profit by acting now in order to produce more of the commodity later. But the important point of Smith's procedure is precisely that this effect upon the minds of people of a market price exceeding the natural price appeared to be so inescapable as to permit proceeding directly to its objective consequence, increased production. This would seem to be the procedure to be aimed at with respect to
'expectations' in the theory of value: to relate them uniquely to objective phenomena, so as to bypass them and relate the facts explaining the expectations directly to the actions of the individuals. The procedure which Hicks (1946) adopted, by which unobservable quantities, the expected prices, are introduced as independent variables, runs the risk of depriving the theory of any definite results....The values which the unknowns assume in a
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 107
Furthermore, since a temporary equilibrium determines prices and quantities only for a short period of time (Hicks called it a ‘week’), any explanation or prediction of the path of the economy over several months or years must be based on determining a sequence of temporary equilibria, which – even assuming uniqueness of each temporary equilibrium – requires assumptions as to how expectations will evolve from one 'week' to the next, that cannot but be, again, largely arbitrary.
Coming now to the impermanence problem , this is made even worse by the presence of expectation functions. The insufficient persistence of the endowments of the several capital goods is present in temporary equilibria as well, but there is now one further group of data lacking persistence: the form of the expectation functions. With respect to these data not even the fairy-tale fiction of the auctioneer-guided tâtonnement suffices to exclude their changes during disequilibrium . The reason is that since expectations can be mistaken, it can happen that during the tâtonnement the auctioneer proposes exactly the prices that, in the previous period, an agent had expected to be the equilibrium prices for the current period, and that these prices come out not to be equilibrium prices; the agent thereby discovers that her expectations had been wrong, and is stimulated to alter the way she forms her expectations: the way expected prices depend on current prices must therefore be admitted to be susceptible to change during the tâtonnement
. And the change may well be drastic: it might consist of a switch to a completely different, rival theory of how the economy works[
113
]. situation which is fully defined in its objective data can be made to vary almost indefinitely by varying the hypotheses about expected prices.” (Garegnani, 1976, p. 39).
Actually Hicks in 1936, in a review of Keynes’s General Theory that had just been published, criticized Keynes precisely because of the danger of inconclusive results due to the use of expectations: “The method [of including among the data exogenously given expectations] is thus an admirable one for analysing the impact effect of disturbing causes, but it is less reliable for analysing the further effects....it is probable that the change in actual production during the first period will influence the expectations ruling at the end of that period; and there is no means of telling what that influence will be. The more we go into the future, the greater this source of error, so that there is a danger, when it is applied to long periods, of the whole method petering out.” (Hicks, 1936: 87) It cannot cause great surprise, then, that Hicks, after adopting temporary equilibria in Value and Capital (1939) in spite of these misgivings, should have later come to recant and reject temporary equilibria as a fruitful conception (Petri, 1991).
113 This impermanence problem caused by the presence of expectation functions among the equilibrium’s data should not be confused with the indefiniteness problem. For the initial temporary equilibrium the indefiniteness problem consists of the fact that one does not know what to assume about current expectations; the aspect of the impermanence problem connected with expectations consists of the fact that, even if one could ascertain the expectation functions of the agents at the moment when the disequilibrium adjustment toward equilibrium starts, they
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 108
Thus anyone who wanted to study the stability of the single temporary equilibrium via a tâtonnement process, in order to avoid the impermanence problem would have to make the additional debatable assumption that expectation functions remain unchanged during the tâtonnement. The indefiniteness problem would anyway show up in the fact that the results would depend on the assumptions on expectation functions, as shown for example by the cobweb model of §6.26. As it turns out, the formal literature on temporary equilibrium theory has produced no attempt to study the stability of the single temporary general equilibrium, thus implicitly admitting the difficulties of such a study[ 114 ].
On the substitutability problem we can be brief. It is obviously present in temporary equilibria too: Hicks’s admissions mentioned in §8.B.17 were actually formulated with reference to the ‘week’ of his temporary equilibrium; the observations I advance there on the theory of wages are therefore applicable to temporary equilibria too.
8.D.4. Let us come to the savings-investment problem . In this literature, the adjustment of investment to full-employment savings is part of the definition of temporary equilibrium; whether there are reasons to accept such an adjustment is not discussed. (The previous observations on the indeterminateness of investment and of entry decisions should suffice to raise doubts.) The considerations advanced in §8.B.24 seem applicable here too: that investment adjusts to full-employment savings is not proved by temporary equilibrium theory, the arguments in its support are the long-period would not remain unaltered during the disequilibrium adjustment even if this consisted of an auctioneer-guided tâtonnement. For the evolution of expectation functions from one temporary equilibrium to the next, the indefiniteness problem consists of the fact that, even if we could ascertain the expectation functions that have determined the temporary equilibrium of one period, we would not know what they will have become at the beginning of the next period’s tâtonnement; the impermanence problem consists of the fact that anyway the expectation functions at the beginning of the next period’s tâtonnement may change during that tâtonnement; so there are two reasons why the expectation functions determining the next period’s temporary equilibrium are unknowable.
114
Whether the single temporary equilibrium is stable – the issue discussed in the text – is a different issue from the one of ‘stability’ of sequences of temporary equilibria, by which it is meant whether a sequence of temporary general equilibria converges to a repetitive situation qualifiable as a long-run equilibrium. The original project of the theorists who embarked on the study of temporary equilibrium was to arrive at studying this ‘stability’, but the difficulties of formalization of temporary equilibrium plus the indefiniteness problem have prevented reaching any significant result.
108

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 109 arguments explained in ch. 7 and relying on capital the single value factor; evidently, since the adjustment was considered valid on average, to assume the adjustment operative even in the very short period was believed to produce results not too much at odds with actual economic paths. The issue requires radical reconsideration, given that its traditional foundation was rejected in ch. 7. However, the present chapter is already long, so it has seemed best to present in a subsequent chapter (ch. 13) what we must add, on investment theory, to the considerations of ch. 7.
Part D, Section II: More formally: An introductory pure-exchange model
8.D.6. Let us now proceed to a formal examination of some simple models of temporary equilibrium so as to grasp the formal problems more directly responsible for the abandonment of the temporary equilibrium approach. We first describe a pureexchange model; then we extend it to the case of economies with production. Next we discuss the way money has been introduced in temporary equilibria; some concluding remarks follow.
We begin our formal exposition of temporary equilibrium theory by focusing attention on the simplest analytical case. Consider a pure-exchange economy with H price-taking households (indexed by h = 1, …, H ) and, in each period, N

2 nonstorable consumption goods[ 115 ] (indexed by n = 1, …, N ); assume this economy is active for only two periods of time, period 1 (the present) and period 2 (the future).
??cambia tutte le date a 0 e 1? Assume that, at the beginning of period 1, there are N distinct spot markets for the different consumption goods and a futures market for good
1 of date 2, i.e. a market on which households can trade (against immediate payment) promises of delivery of physical units of good 1 (or of equivalent purchasing power[
116
]) at the beginning of the next period. Only the N spot markets for commodities are open in period 2.
Now assume that, at the beginning of period 1, each household observes the prices quoted on the N +1 current markets and forms definite expectations of the future relative prices of commodities in terms of good 1 of date 2. Under these circumstances the generic household h will calculate that, by appropriately trading on the single forward market in existence, it can purchase or sell commodities for future delivery as
115
Non-perishable goods would introduce the complication that it becomes possible to
‘produce’ goods of period 2 by storage; now we want to exclude any kind of production.
116
Cf. §8.21.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 110 freely as in the presence of a complete system of futures markets. To clarify this point, consider any of the N -1 commodities different from good 1, say ‘grapes’, and assume household h expects that a unit of ‘grapes’ will exchange in period 2 for three units of good 1. Then the household will calculate that, if it wishes to purchase in the present a unit of grapes for future delivery, it can do so by buying forward[ 117 ] three units of good
1 in the anticipation of exchanging those units in period 2 for the desired unit of grapes.
Similarly the household will calculate that if it wishes to sell in the present a promise of future delivery of one unit of grapes, it can do so by selling forward three units of good 1 in the anticipation of surrendering a unit of grapes in period 2 against three units of good
1 and then using those units of good 1 to honour its forward sale.
The above example shows that, for a household endowed with definite expectations of future relative prices, trading on the single forward market open in period 1 allows transferring purchasing power across time, that is, saving or dissaving.
By buying promises of future delivery of units of good 1 the household in fact performs a loan, transfers to period 2 some present purchasing power, i.e. saves; and by selling forward units of good 1 the household can capitalize in the present the expected purchasing power (in terms of good 1) of any commodity, or commodity bundle, that it plans to surrender in period 2, i.e. obtains a loan, dissaves. In order to highlight this aspect, from now on we shall refer to the single futures market in existence as a market for one-period bonds specified in terms of good 1, where the unit bond is defined as a promise to deliver one physical unit of good 1 at the beginning of the second period.
Given that the bond market allows intertemporal transfers of purchasing power, it is reasonable to assume that households will simultaneously plan both their present and their future consumption at the beginning of period 1. We shall accordingly assume that at the initial date, given the current and the expected prices, each household trades commodities for present consumption and bonds so as to attain the most preferred consumption stream over periods 1 and 2. By definition, a state of the economy in which all households trade in this way, and individual trades are such that all the N+ 1 current markets clear, is a temporary equilibrium of the exchange economy for period 1.
8.D.7. Let us now provide a more precise description of households’ behaviour in the first period. It will be assumed that good 1 is each period’s numeraire for the
117 To avoid linguistic contortions, we will speak of ‘buying or selling forward’ as synonymous with buying or selling a promise of future delivery against immediate payment (in real economies forward contracts usually stipulate payment upon delivery ), and we will occasionally use ‘forward market’ as synonymous with ‘futures market’.
110

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 111 determination of relative spot prices, and good 1 of date 1 is the numéraire for the determination of discounted prices.
Let us first deal with the characteristics of the H households operating in the economy at the beginning of period 1. Let a two-period consumption stream of the generic household h be denoted by vector h x
12
= ( x
1 h vector x t h = ( x
1 t h , … , h x
Nt
, x
2 h ), with x
1 h

) denotes a consumption bundle for period
0 t
,
( t x h
2

0, where sub-
= 1, 2). Assume that the set of admissible consumption streams, or two-period consumption set , of the generic household is X h
12
=
 2

N . Assume also that, at the initial date, the generic household h knows with certainty both its current commodity endowments

1 h = (
 h
11
, …,
 h
N 1
) and its future endowments

2 h = (
 h
12
, …,  h
N 2
). In order to keep the model as simple as possible, we assume that the households’ endowments are strictly positive and do not include credits nor debts[ 118 ], and that its preferences can be represented by a well-behaved utility function:
Assumption 9.1. (a) The generic household h has a preference ordering over twoperiod consumption streams in X h
12 that can be represented by a continuous, strictly increasing and strictly quasi-concave utility function U h = U h ( x
1 h , x
2 h );
(b)

1 h

0,
 h
2

0 for each h .
Coming to the prices that guide households’ choices, let the price system ruling in period 1 in terms of good 1 be denoted by vector p = ( p
1
, q
1
), where sub-vector p
1
=(p
11
,...,p
N1
)
  N
 scalar
, in which p
11
=1, refers to the N spot markets for commodities, and q
1

0 is the price of a unit bond, that is, the discounted price of good 1 of period
2, which implies an own rate of interest r
11
determined by q
1
=1/(1+ r
11
). In p ij
the first index denotes the good, the second the period or date. As regards the future spot prices expected at the initial date, we assume that price expectations are subjective, and therefore likely to differ among agents, and ‘certain’, in the sense that each household believes that a definite price system will hold in the future with probability 1. Let us denote by p
2 h = ( h p
12
, … , p
N h
2
) the future relative spot prices in terms of good 1 as
118 Since the economy cannot have miraculously come into existence at date 1, it has a past, and there will have been sales of bonds in the previous period, which means that some households (the ones who sold bonds) have debts to be repaid at date 1, while other households
(the ones who bought bonds) have credits, and the budget constraint should make room for this.
The problem then arises that, since past expectations may have been mistaken, debtors may have overestimated the value of their period-1 endowments and may find it impossible to repay their debts, as mentioned in §8.D.2. For simplicity we neglect this issue and assume no debts inherited from the past.
111

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 112 expected by household h . The first element of this vector is h p
12
= 1, independent of expectations because numéraire. In general, expected prices will depend on both the prices observed in the past and those currently observed; for the moment, however, let us assume that price forecasts do not depend on current prices ( fixed expectations). Let us further assume that expected prices are strictly positive.
Assumption 9.2. (a) The system of future prices expected by the generic household h, p
2 h , is given at the initial date independently of current prices;
(b) p
2 h

0 for each h .
Let us now examine the behaviour of the generic household at the opening of markets in period 1. One way to understand its budget constraint is by assuming that the household in a first stage issues a quantity of bonds b
1 h corresponding to the maximum it expects to be able to repay in the future and adds the purchasing power thus obtained to the value of her period-1 endowments; in this way the household has available for expenditures the entire capitalized value of its incomes, i.e. its wealth; then in a second stage (always at the beginning of period 1) the household uses this total purchasing power to buy period-1 goods, and to buy bonds for a quantity b
1 h ≥0 (it cannot sell further bonds because its expected income in period 2 is already just sufficient to repay b
1 h and we exclude planned bankruptcy). Since a bond entitles to the future delivery of one unit of numéraire
, the number of bonds supplied by household h in the first stage coincides with the total spot value of the household’s future endowments as forecasted by the household itself .
Assumption 9.3. At the beginning of period 1 the generic household h in a first stage supplies a quantity of bonds b
1 h such that b
1 h = p
2 h

2 h .
By issuing bonds in accordance with Ass. 9.3, given the price q
1
of these bonds, the household obtains purchasing power equal to the discounted expected value of its future endowments. Since the current receipts from this operation are equal to ( q
1 b
1 h ) units of numeraire, the total wealth that the household can spend in the second stage in period 1 on goods for present consumption and bonds is W
1 h = p
1

1 h + q
1 b
1 h . The first period budget constraint of household h can therefore be written p
1 x
1 h + q
1 b
1 h = p
1

1 h + q
1 b
1 h (1) where b
1 h denotes the non-negative quantity of bonds demanded by the household in the
112

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 113 second stage, and the right-hand side indicates the value at date 1 of the household’s endowments, where the second term is the discounted or present value of the future endowment according to the household’s own expectations of future relative prices and to the market’s price at date 1 of good 1 of date 2. On the other hand, the household is aware that in period 2 it will have to surrender its entire endowment

2 h in order to honour the bonds b
1 h issued in the first stage, and therefore calculates that the purchasing power it will be able to spend in the future for its own consumption is entirely determined by the income from the bonds purchased in the second stage. The ( expected) second period budget constraint of household h thus reads: p
2 h x
2 h = b
1 h (2)
We see therefore that in (1) the term q
1 b
1 h on the left-hand side is the discounted or present value of the possible future consumption; the budget constraint (1)-(2) states therefore that the present value of present and future consumption must equal the present
(expected) value of present and future endowments . The description of agents’ behaviour at the beginning of period 1 can be completed by assuming that each household chooses its current consumption of goods, current demand for bonds and planned future consumption so as to attain the most preferred two-period consumption stream subject to budget constraints (1)-(2). Formally we can say that the choice of the generic household h at given current prices p and expected prices p
2 h is a solution to the following problem:
[ I ] Maximize U h ( x
1 h , x
2 h ) with respect to x
1 h

0, b
1 h

0, x
2 h

0 subject to constraints (1)-(2) and Ass. 9.3.
Let a solution to problem [ I ] be denoted by the triple ( x
1 h * , b
1 h * , x
2 h * ). It is clear that only the first two components will manifest themselves on current markets in the form of demand for commodities to be consumed in the present and demand for bonds, while planned consumption x
2 h * will remain, as it were, in the household’s mind. It can therefore be stated that a solution to problem [ I ] identifies the corresponding optimal action a h * = ( x
1 h * , b
1 h * ) taken by the generic household on period-1 markets after the action of Ass. 9.3. (So the complete action on period-1 markets should actually be indicated by a triplet: ( b
1 h , x
1 h * , b
1 h * ).)
By substituting for b
1 h and b
1 h in constraint (1) according to constraint (2) and
Ass. 9.3, one gets the equation p
1 x
1 h + q
1 p
2 h x
2 h = p
1

1 h + q
1 p
2 h
 h
(3)
2
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 114 which by adopting the convention q h = q
1 emerge in the next paragraph: p
2 h can be written in a way whose interest will p
1 x
1 h + q h x
2 h = p
1

1 h + q h
 h (3’)
2
Equations (3) or (3’
) express, as already indicated, that the present value of present and future consumption must equal the present expected value of present and future endowments.
Note that in these equations the variables b
1 h
and b
1 h do not appear. On the other hand, if household h determines a most preferred consumption stream h * x
12
= ( x
1 h * , x
2 h * ) by solving the problem
[ II ] Maximize U h ( x
1 h , x
2 h ) with respect to x
1 h

0, x h
2

0 subject to constraint (3) or (3’). the solution ( x
1 h , x
2 h ) is certainly also a solution, for these variables, of problem [I][
119
].
It emerges then that problem [I] can be solved in two consecutive steps. In the first step the household determines the optimal two-period consumption stream by solving problem [II], in the second step the household determines through the constraint (2) under Ass. 9.3 the quantity of bonds that in its opinion must be purchased (after selling b
1 h ) in order to finance planned future consumption, i.e. b
1 h * such that b
1 h * = p
2 h x
2 h * ; this second step, if one assumes that the household does not actually first of all sell the amount b
1 h of bonds but only reasons as if it had, concretely means that the household determies and exercises on the market its net or excess demand for bonds, to be indicated as z b h * = b
1 h * – b
1 h = p
2 h ( x
2 h * – 
2 h ) , whose present value q
1 z b h * is the amount of savings
(positive or negative) the household decides to perform in period 1. Then the optimal action of household h on current markets is the couplet α h* =( x
1 h , z b h * ) .
8.D.8. Let us examine the first step more closely. Constraint (3) or (3’) in problem
[II] can be interpreted as the single budget constraint that household h faces when choosing its consumption stream at the initial date: recall that, in the presence of a bond
119 Exercise: prove it, by proving that (assuming U(∙) differentiable, and interior solutions) the consumption variables are determined in problem [I] by the equalities between marginal rates of substitution and ‘price ratios’ p it
/q
1 p jt’
, plus the budget constraints reduced to (3), without a need simultaneously to determine b
1 h
, which is determined only afterwards, from the budget constraint (2) under Ass. 9.3.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 115 market, household h considers that there is no impediment to its purchase of the quantity

1 h * of bonds, which is not therefore something constraining its consumption choices, it is only an action necessary to implement those choices. Recall also that we have seen that the recourse to purchasing or selling bonds allows the household to trade goods for future delivery as if it faced a complete system of futures markets[ 120 ], at the future prices it expects. Let us then consider the budget constraint (3) in the form (3’). In the opinion of the household, the N components of the vector q h appearing in (3’) are precisely the prices at which it would be possible to trade in period 1 [promises of delivery of] future goods if complete futures markets existed, i.e. they are the ‘present or discounted prices’ of future goods in the household’s mind, given the discount rate q
1 supplied by the market. To see this it suffices to examine the first two elements of vector q h . By the convention adopted, the first element is q
1 h = q
1 h p
12
= q
1
, i.e. the ‘present or discounted price’ of a unit of good 1 for future delivery as actually quoted on the current bond market. The second element is q
2 h = q
1 h p
22
, where h p
22
is the future price for commodity 2 as expected by household h . By arguing like in the example that opens
§8.D.6, we see that q h
2
is the ‘present price’ at which household h esteems it can trade a unit of commodity 2 of period 2, because it is the price that the household should pay in the present for buying forward h p
22
units of good 1 given its expectation that h p
22
is the quantity of good 1 that will allow the purchase of one unit of good 2 in period 2.
(Reformulate the reasoning for the case of a forward sale of good 2.)
Because of this interpretability of q h as a vector of esteemed or perceived
‘present prices’ for commodities to be delivered in period 2, the constraint in problem
[II] is strictly analogous to the intertemporal budget constraint household h would face at the initial date in this economy if there were complete futures markets; the only difference is that it is the intertemporal budget constraint perceived by household h , owing to its expectation of future relative prices. It can therefore be concluded that, in the first step of the procedure indicated at the end of §8.D.7, the choice of the two-period consumption stream on the part of the generic household h at current prices p = ( p
1
, q
1
) and ‘fixed’ expected prices p
2 h is formally equivalent to standard consumer choice under complete forward markets at prices p
’ = ( p
1
, q h ), where q h = q
1 p
2 h .
In view of the abovementioned formal equivalence, and of Ass. 9.1 and Ass. 9.2, the following results are immediate: (a) the first step of the procedure univocally determines the consumption stream h * x
12
= ( x
1 h * , x
2 h * ) chosen by the generic household h at any p such that p
  N



1 ; (b) each component of h * x
12
changes continuously with p =
120 This is the same principle behind the equivalence between Arrow-Debreu equilibria and
Radner equilibria, mentioned in §8.18.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 116
( p
1
, q
1
) as the latter ranges in
 N
 

1 . In other words, both the current and the planned future demands for consumption goods are continuous functions of period-1 prices, provided that the latter remain strictly positive. In what follows we shall respectively denote those individual demand functions for commodities by x
1 h ( p ) and x
2 h ( p ), h = 1,
…,
H . A third result follows immediately, (c) the household’s demand for bonds, to be indicated as b
1 h ( p ), as well as its excess demand for bonds z b h (p) , are continuous functions of strictly positive current prices. In view of the foregoing discussion it can be finally concluded that, under the stipulated assumptions, the optimal action taken by the generic household on period-1 markets is well-defined and continuous for p
  N
 

1 .
8.D.9. We have now what we need for the formal definition of temporary equilibrium for the exchange economy under examination. Let us restrict our analysis to strictly positive first-period prices, we can then define the excess demand functions for commodities of the generic household h as z
1 h ( p ) = household’s excess demand function for bonds as x
1 h z b h
( p )–

1 h
( p ) =
, and we have defined the b
1 h ( p )– b
1 h (where b
1 h is a given parameter in view of Assumptions 9.3 and 9.2(a)). Summation over the H households then yields the corresponding aggregate excess demand functions z
1
( p ) and z b
( p ), that are obviously continuous.
121 A temporary equilibrium for period 1 is finally defined as a system of current prices actions
 a
1 ( p
* ), … , a
H ( p
* p
*
  N
 

1 and a corresponding set of optimal
)

on the part of the H households, with a h ( p
* )
  N
 

1 for each h , such that the N +1 market clearing conditions z
1
( p ) = 0, z b
( p ) = 0 are simultaneously fulfilled. (To drop Ass. 9.3 and to define the optimal actions as α h*
would make no difference since the several b
1 h are given parameters.)
It can be proved that temporary equilibrium of the exchange economy exists under assumptions 9.1-9.3. Moreover, the existence of temporary equilibrium is preserved if it is assumed that individual expectations depend continuously on the current prices, i.e. if a continuous expectation function
 h such that p
2 h =
 h (p) is introduced for each h . We shall refrain from substantiating these assertions, as the introductory model examined here is a particular specification of the temporary equilibrium model put forward by Arrow and Hahn (1971: Ch. 6), to which readers are referred for existence proofs. (See footnote 26?? for the relationship between the introductory model and the Arrow-Hahn model.) We shall instead focus in the remainder of this section on the analytical scope of the introductory model, which has taken us quite comfortably from the Arrow-Debreu world with complete forward markets to the
121 It is a simple Exercise to prove that the aggregate excess demand functions defined in the text are homogeneous of degree zero in p and satisfy Walras’s Law.
116

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 117 more realistic environment of temporary equilibrium theory. It will be argued that the model contains a hidden problem and is not really robust.
Problems with the introductory model
8.D.10. The introductory model assumes that a single futures market is open in period 1 together with the spot markets for the N period-1 consumption goods. However, there is no reason why there should be a futures market for only one good (in real economies there are markets for many different types of bonds and futures markets for many financial, and some real, assets); nor does the notion of temporary equilibrium imply anything like it, it only postulates that futures markets are incomplete and there isn’t perfect foresight; for the two-periods exchange economy we are discussing, we have a temporary equilibrium if the number of futures markets is lower than N. It is therefore natural to ask whether the model is susceptible of generalisation to economies with a larger set of futures markets. As we shall now see, unfortunately even a slight increase in the number of futures markets in existence has serious consequences for temporary equilibrium analysis.
Let us modify the introductory model by assuming that N ≥3 consumption goods are traded in the economy and that two distinct futures markets are open at the initial date, say the futures market for good 1 and the futures market for good 2. (We could equivalently state that two distinct bond markets are open, one for bonds promising a unit of good 1 of period 2, and the other for bonds promising a unit of good 2 of period
2.) This change in market structure necessitates some adjustment of the formal description of the economy. To begin with, current prices in terms of good 1 will be denoted by vector vector q = ( q
1
, q
2 p = ( p
1
, q ), where sub-vector
), with q
  2
 p
1
  N

refers to spot markets and sub-
, to forward markets (or bond prices). It will also be convenient to denote the quantities of goods that the generic household h trades on forward markets by the vector b h = ( b
1 h , b
2 h ), where by assumption b i h >0 denotes a quantity of good i demanded and b i h <0 a quantity of good i supplied ( i = 1, 2) by the household (i = 1, 2). In this notation, the first period budget constraint of the generic household h reads: p
1 x
1 h + q b h = p
1

1 h (4 ). and the household’s expected budget constraint for period 2 can be written p
2 h x
2 h = p
2 h

2 h + p
2 h ' b h (5 )
117

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 118 where p
2 h
 

N

denotes the future prices in terms of good 1 anticipated by the household, and p
2 h ' = (1, h p
22
) is the vector whose components coincide with the first two components of p
2 h .
Once these adjustments have been introduced, the economy can be described along the same lines as in the introductory model. We accordingly assume that given the current and expected prices, the generic household h chooses its current consumption, current trading on forward markets and planned future consumption at the initial date so as to maximise the utility function U h ( x
1 h , x
2 h ) subject to budget constraints (2.4)-(2.5).
Provided that it is well-defined, this choice in turn identifies the optimal action a h * =
( x
1 h * , b
1 h * , b
2 h * ) taken by the household on period 1 markets, where a h*
  N

2 . Within this framework, a temporary equilibrium of the modified exchange economy is finally defined as a system of current prices and a corresponding set of optimal actions on the part of the H households such that the N+2 current markets are simultaneously cleared.[ 122 ]
For this modified exchange economy a striking result is that a temporary equilibrium will generally not exist under Assumptions 9.1-9.2. As we shall see presently, the reason for this negative result is the possibility of speculative behaviour, that can cause the household’s demands and supplies to become unbounded.
The following example illustrates the nature of the problem. Assume that the price system ruling on futures markets at the beginning of period 1 is q = ( q
1
, q
2
) such that q
2
/ q
1
= 2, while household h expects (with certainty) that the future price of good 2 will be h p
22
=3. Under these circumstances the household will have a strong incentive to trade on forward markets for speculative purposes. Suppose the household buys forward a unit of good 2 and simultaneously sells forward two units of good 1: under the postulated price conditions, the total cost of implementing that scheme would be zero; but the household calculates that in period 2 it will be able to exchange the unit of good 2 that it will then receive for three units of good 1, that is one more than the two units of good 1 that it has committed itself to deliver. The household will conclude that by carrying out the trading scheme under consideration it can increase its future wealth at no cost or, to use a concise technical expression, that trading on forward markets provides an opportunity for profitable arbitrage . Now recall that, by assumption, the household is price-taker, and is never satiated in both present and future consumption: therefore it will tend to increase without limit the quantity of good 2 for future delivery demanded in the
122 Under the notation adopted in the text, the market clearing condition on the forward market for good i is
 h b i h *
= 0 ( i = 1,2).
118

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 119 present and financed by selling forward good 1. This means, however, that at the assumed current and expected prices the household’s optimal action cannot be determined and a temporary equilibrium cannot therefore exist.
The reader can check that the foregoing argument applies to all situations in which the current prices and the prices expected by the generic household h are such that
( q
2
/ q
1
)< h p
22
. By adapting that argument it can also be seen that, should the ruling and the expected prices be such that ( q
2
/ q
1
)> h p
22
, household h would tend to increase without limit the quantity of good 1 for future delivery demanded in the present and financed by selling forward good 2; as a consequence the household’s optimal action cannot be determined in this case either, and for this reason ( q
2
/ q
1
)> h p
22
too is incompatible with the existence of a temporary equilibrium. A system of period-1 prices can support a temporary equilibrium of the modified exchange economy only if at those prices profitable arbitrage is considered impossible by all households, i.e. only if the non arbitrage condition ( q
2
/ q
1
) = h p
22
holds for each h : which implies that all households have exactly the same expectation of the future price of good 2. But since perfect foresight is not assumed, one cannot expect a system of current prices generally to exist at which the required coincidence of forecasts could be fulfilled. This is quite obvious in the case of ‘fixed’ expectations, i.e. expectations independent of current prices: in this case it suffices that just two households disagree concerning the future price of good 2 to prevent the existence of a temporary equilibrium. But even assuming the expected prices to be continuous functions of current prices, it is perfectly possible that, for two or more households, the individual expectation functions be such that they generate different expected prices of good 2 of period 2 at any system of current prices, thereby preventing the existence of a temporary equilibrium; and even if some system of current prices existed rendering ( q
2
/ q
1
) = h p
22
for each h, it would be a fluke if it were also capable of ensuring equilibrium on the current markets.
8.D.11. The problem that (perceived) arbitrage opportunities create for the existence of temporary equilibrium was pointed out by Green (1973) within the context of a pure exchange economy close to that examined in this section. Green pointed out that the problem is reduced when expectation functions are ‘probabilistic’, i.e. take the form of probability distributions of future prices. However, Green made it clear that the problem is not completely ruled out under that formulation of price forecasts, because, in order for an equilibrium to exist, unlimited arbitrage operations on forward markets must be prevented, and this requires an assumption that there be an ‘overlap’ of individual expectations. A simple example will illustrate the meaning of that ‘overlap’, and will also highlight a debatable aspect of the equilibrium whose existence is proved under that
119

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 120 assumption.
Assume again the two-goods, two-periods economy with possibility of forward trading of both goods. Assume two price-taking households, A and B[ 123 ]; each one has an estimate, of the possible relative price of good 2 of period 2 in terms of good 1 of that period, which consists of a probability distribution over an interval. For simplicity, assume that A has a uniform probability distribution over an interval ( A
 p
22
,
A
 p
22
) which is independent of period-1 prices, while B has an analogous uniform probability distribution independent of prices, over the interval ( B
 p
22
,
B
 p
22
) generally different from
A’s interval. (Note that these are expected relative spot prices, i.e. h p
12
=1.) Those intervals are the supports of the probability distributions and will be indicated as supp(A), supp(B). The expected value of p
22
for household h , to be indicated as hE p
22
, is then the middle point of the relevant interval. Assume furthermore, again for simplicity, that both A and B have such endowments and preferences that they decide to perform no saving nor dissaving, i.e. p
1 x
1 h = p
1

1 h for h =A,B; hence q
1 b
1 h  q
2 b
2 h 
0 , that is, if a household speculates, it buys forward one good and sells forward the other good for the same value, balanced speculative trading . Remember that b i
> 0 indicates a purchase forward, b i
< 0 indicates a sale forward; under our assumption, b
1 h
, b
2 h have opposite sign.
When period 2 arrives, and actual spot prices p
12
, p
22
become known to a household, its budget constraint is p
12 x
12
+p
22 x
22
=p
12 b
1
+p
22 b
2
+ p
12
ω
12
+p
22
ω
22
, or rather, normalizing prices so that p
12
=1: x
12
+p
22 x
22
=b
1
+p
22 b
2
+
ω
12
+p
22
ω
22
, where for simplicity the household index is omitted. If b
1
is negative, that is, if the household sold good 1 forward, then in period 2 it must give
–b
1
units of good 1 to others, and if
–b
1
>ω
1
the household must buy some good 1 in order to be able to fulfill its contractual obligation; it is then possible that b
1
+p
22 b
2
+
ω
12
+p
22
ω
22
<0, which makes it impossible to satisfy the budget constraint. Green argues that this possibility must be excluded in order for the household’s utility maximization problem to be well defined; actually, it may be possible to render the utility maximization problem well-defined all the same, by stipulating that in that case the household goes bankrupt and there is a legal provision establishing what happens in this case, e.g. that x
12
=x
22
=0 ; but the literature has preferred to assume that no household ever wants to run the risk of going bankrupt even with very small probability, and then for it to be possible that an equilibrium exists one obtains the same necessary constraints as Green on the household’s speculative
123 Price taking can be justified by assuming that there are in fact 10000 huseholds identical to A and 10000 identical to B.
120

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 121 forward trades and on supp(A) and supp(B). To save on symbols, let
Q≡q
2
/q
1
; then for each household b
2
= –b
1
/Q, b
1
= –b
2
Q .
The result reached by Green, translated to our simple example, is that a necessary condition for an equilibrium to exist (under the stated assumption of no risk of bankruptcy) is that the intersection of supp(A) and supp(B) must be an interval of positive length (i.e. there must be an ‘overlap’ and it must not consist of a single point).
The reason is the following. If Q is not internal to supp(A), then A feels certain that it can speculate with no risk of bankruptcy; this is because there may be a subjective risk of bankruptcy only if it is esteemed that it can happen that b
1
+p
22 b
2
<0 , which, remembering the connection between b
1
and b
2
under our assumptions, becomes b
1
(1– p
22
/Q)<0 ; but if Q
≤ p
A

22
< AE p
22
, A finds it convenient to speculate by selling good 1 forward and buying good 2 forward, and therefore b
1
<0
; and given Q and A’s expectations, A is certain that (1–p
22
/Q)≤0
and therefore is certain that b
1
(1–p
22
/Q)≥0
; the same result b
1
(1–p
22
/Q)≥0
is reached if Q > A p
22

> AE p
22
(in which case the speculation requires b
1
>0). So unless Q is internal to supp(A), A finds no reason not to engage in unlimited speculation, similarly to the case of subjective certainty (point expectation) on the future value of p
22
; no equilibrium can exist. The same convenience of unlimited speculation applies to B if Q is not internal to supp(B). A necessary condition for an equilibrium to exist is therefore that there exists some Q internal to both supp(A) and supp(B): this requires that the two intervals overlap for some positive length. For an economy with several households and several forward markets, the equivalent condition is that the intersection of the supports of the price probability distributions of the households must have a non-empty interior. Therefore, in an exchange economy with many households and many goods tradable forward, it suffices that for at least one forward good (other than the numéraire one) there are two households whose probability distributions have supports that do not ‘overlap’ (i.e. that there is no price interval to which both households attribute a nonzero probability of containing the future spot price of that good), and a temporary equilibrium will not exist. The risk that no equilibrium exists appears considerable.
8.D.12. Green is able to prove that, under the stated ‘overlap’ condition plus the no-risk-of-bankruptcy assumption plus various (sometimes debatable) assumptions of continuity[ 124 ], an equilibrium does indeed exist[ 125 ]. But a perplexing aspect of this
124 E.g. for each household h, supp(h) must be a continuous correspondence of period-1 prices. This excludes the possibility that a small change in present prices may induce some agent
→
121

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 122 equilibrium seems to have escaped recognition: the equilibrium may require some household to engage in speculative trading from which it expects to gain nothing , undergoing a waste of time and energy that it seems rather implausible to assume the household will accept to bear. The reason is that equilibrium on the forward markets may require that Q equals a household’s expected value of p
22
. In such a case, assuming for example the household is A, it is Q = AE p
22
, hence A expects no profit from any extent of balanced speculative trading. To give an idea of why A may be nonetheless required to engage in some speculative trading for equilibrium to exist, we study the implications of the no-risk-of-bankruptcy assumption on the amount of speculative balanced trading a household undertakes.
Without loss of generality consider household A. Suppose initially p
A

22
< Q < AE p
22
; the household finds it convenient to buy good 2 forward and to sell good 1 forward, hence b
1
<0, b
2
>0, and the no-risk-of-bankruptcy condition is b
1
+p
22 b
2
+
ω
12
+p
22
ω
22
≥0 for all p
22
 supp(A); using b
1
=-Qb
2
and with reference to the least favourable case ( p
22
= A
 p
22
) this constraint becomes b
2
≤ ( ω
12
+ω
22
As Q tends to
A
 p
22
)/(Q– A p
22

). p
A

22
, the right-hand side tends to +∞; as Q tends to AE p
22
, the righthand side tends to a finite positive value that we may indicate as D A+ . Now suppose
AE p
22
< Q < p
A

22
; the household finds it convenient b
1
>0, b
2
<0, and the no-risk-ofbankruptcy condition with reference to the least favourable case ( p
22
= A p
22

) becomes b
2
≥ –(
ω
12
+ω
22
A p
22

)/( A p
22
 –
Q ).
The right-hand side tends to –∞ as Q tends to A
 p
22
, and tends to a finite negative value D A– as Q tends to AE p
22
. We can conclude that the household’s demand for b
2
as a function of Q has the shape of Fig. 9.??: it has no discontinuities because it is assumed that the household is ready to undertake no-profit speculative arbitrage at Q= AE p
22 within the vertical segment from D
A+
to D
A–
, where therefore its demand is multivalued; from the extremes of this segment two branches of hyperbola start, tending to positive infinity as Q decreases, and to negative infinity as Q increases. If the other household has a different probability distribution, its function b
2
(Q) will have the same type of shape, but the vertical segment will be of different length, and located at a different value of Q.
One can then have a situation like that of Fig. 8.D.1, where the only equilibrium on the b
2
market is at Q= AE p
22
and requires household A to locate itself on its vertical segment at to expect some event, e.g. a state intervention, causing a discontinuous change in expected prices.
125 Green (1973) refers to other papers for important passages in his proof; cf. Grandmont
(1982, p. 892) for a more complete proof.
122

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 123 a value of b
2
different from zero (point ■ in Fig. 8.D.1), while the household has no incentive to do so. And an equilibrium, which in order to be established requires an agent to choose one particular element of a set among whose elements the agent is indifferent, cannot be presumed to be ever established if no reason can be found why the agent should be induced to choose that particular element. (Note if one were to assume even minimal transaction costs, all points of the vertical segments different from zero would disappear, and in the case depicted the discontinuity would cause the non-existence of equilibrium.) intl. 19 pt b
2 b
2
A
( Q ) b
2
B
( Q )
D A+
A p
22

B p
22

AE p
22
BE p
22
A p
22

B p
22

Q
D A– ▀
Fig. 8.D.1
Part D, Section III: Extension to the case of economies with production
8.D.13. We shall now see how the introductory model with a single forward market can be modified so as to transform it into a model of exchange and production.
This will allow pointing out some of the issues that arise in attempts to introduce production into the framework of temporary equilibrium analysis.
123

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 124
The first step toward the proposed extension consists of introducing the following basic changes in the model. To begin with, we assume that the N commodities traded in the economy include not only consumption goods but also goods and services susceptible of being used as production inputs. Second, we assume that a given number
F of firms (indexed by f = 1, … , F ) are active in the economy[ 126 ]. Third, as in the
Arrow-Debreu model, we assume that the ownership of each firm is divided among households at the initial date in accordance with a given allocation of ‘ownership shares’. The last basic change to be made is closely related to the third: differently from the Arrow-Debreu model, we allow for the possibility to trade shares. It will be shown below that households are generally willing to trade their shares of ownership in firms within a temporary equilibrium framework. We therefore assume that F distinct markets for the shares in the different firms are active in period 1 in addition to the N spot markets for commodities and the single market for bonds specified in terms of good 1.
Having thus altered the structure of the economy, we shall now go on to analyse the behaviour of agents in period 1. As before it is assumed that the consumption good listed as ‘good 1’ is the numéraire and that agents have fixed subjectively certain price expectations.
We assume that the production processes available to firms develop in cycles, i.e. that inputs are employed at the beginning of period 1 and the corresponding outputs emerge at the beginning of period 2. A two-period production plan of the generic firm f will accordingly be denoted by the netput vector y
1 f f y
12
= ( y
1 f , y
2 f ), where the sub-vector
  N

denotes first period inputs (negtive numbers) and sub-vector y
2 f
  N

the associated future outputs. The set of production plans that are technically feasible for firm f (the production set of the firm for short) will be denoted by Y
12 f .
Due to the cyclical nature of production, the economy is endowed at the beginning of period 1 with given stocks of commodities derived from the activity of firms in the previous period. We assume that these stocks are part of the initial endowments of households, hence firms do not own endowments and must finance their current input expenditure entirely by issuing bonds. We finally assume that each firm is run by a manager who is responsible for selecting the two-period production plan. Under
126
In earlier chapters we have argued that not even the assumption that the tendency toward equilibrium operates through an auctioneer-guided tâtonnement justifies taking the number of firms as given, because firms are legal entities, distinct from plants (which are tradeable), and new firms might be created during the tâtonnement. But the literature on temporary equilibria has followed the Arrow-Debreu assumption of a given number of firms and given property shares of households in each firm, and our intention here is to explain the results reached by this literature.
124

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 125 these assumptions, the formation of production decisions can be described as follows.
Again for simplicity we assume certain and fixed expectations of future prices. At the beginning of period 1, the manager of the generic firm f is certain that the price vector p
2 f = ( p
12 f , …, p f
N 2
)

0 (which we normalize by setting f p
12
= 1) will obtain on future spot markets. Given these expected prices, the manager observes the prices p =
( p
1
, q
1
) quoted on current markets and assesses the profitability of the alternative plans in Y
12 f . In evaluating a hypothetical plan f y
12
= ( y
1 f , y
2 f ), the manager realises that the firm would have to issue a quantity of bonds b f such that q
1 b f = –( p
1 y
1 f ) in order to finance its current input expenditure and would accordingly have to repay b f = –
(1/ q
1
)( p
1 y
1 f ) units of numéraire at the beginning of period 2. At the same time, the manager anticipates that the plan would yield future receipts equal to ( p
2 f y
2 f ) units of numéraire. According to the manager’s subjective expectations, the hypothetical plan under consideration would therefore yield profits equal to

2 f = [ p
2 f y
2 f + (1/ q
1
)( p
1 y
1 f )] in period 2. It is convenient to introduce an alternative formulation of these expected profits. Given that a quantity

2
of the numéraire good for future delivery can be traded in the present on the bond market at the total price

1 profits expected by the manager is

1 f = q
1
[ p
2 f y
2 f
= (
+ (1/ q
1 q
1
)(

2 p
1
), the present value of the y
1 f )]. By adopting the convention q f = q
1 p
2 f , the present value of expected profits can be expressed in the equivalent form
 vector
1 f = ( q f y
2 f + p
1 y
1 f ), where, it should be noted, the components of q f are precisely the ‘present prices’ of commodities for future delivery as calculated by the manager of firm f . We assume that the manager of the generic firm chooses the production plan so as to maximise the present value of expected profits. In order for the analysis to continue, we assume (in accordance with this literature) that this maximization results in a finite dimension of the firm and hence in a finite maximum profit, i.e. that there are decreasing returns to scale at least beyond a certain scale of production.
Assumption 9.4. Given the expected prices p
2 f and the current prices p , the manager of the generic firm f chooses a production plan that maximises the ‘profit function’ 
1 f ( y f
12
) = ( q f y
2 f + p
1 y
1 f ), where q f = q
1 p
2 f , subject to y
12 f

Y
12 f . The production plan is always finite.
Under Ass. 9.4, the choice of the production plan at current prices p = ( p
1
, q
1
) and fixed expected prices p
2 f is formally equivalent complete forward markets at prices p
” = ( p
1
, q f
to standard producer choice under
) and might be analysed in the same way. If we now use y
12 f * = ( y
1 f * , y
2 f * ) to denote the plan chosen by the manager of the firm f in accordance with Ass. 9.4, the manager’s choice identifies not only the firm’s
125

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 126 current demand for inputs but also the current supply of bonds, where the latter is given by b f * = –(1/ q
1
)( p
1 y
1 f * ), and therefore determines the optimal action a f * = ( y
1 f * , b f * ) taken by the firm on period 1 markets.
Now let us go on to examine the household sector. As previously assumed, households are endowed at the initial date with given ‘shares of ownership’ in the different firms. We shall denote the share endowment of the generic household h by the vector
 h = (

1 h , …, 
F h ) and assume that
 
0 for all h,
 h
 f h = 1 for all f . It should be clear from the description of firms’ behaviour that the possession of an ownership share in a firm throughout period 1 entitles the holder to the same proportion of the profits accruing to the firm at the beginning of period 2. It should be noted, however, that when the firms’ plans are announced at the initial date, households will estimate the associated receipts according to their individual expectations and will thus typically form different opinions concerning the amount of profit to be earned by holding shares in any given firm. In the presence of those different opinions, it is natural to assume that households will find it advantageous to trade shares on the corresponding F markets in existence. Taking this aspect of the economy into account, we shall now examine the behaviour of households on the markets at the beginning of period 1 after the announcement of the production plans selected by managers.
As regards trading on share markets, we shall drastically simplify our analysis by assuming that the shares of each firm are automatically transferred to the household (or group of households) expecting the highest amount of profit from the firm’s plan, at a price exactly equal to the present value of those expected profits according to that household.[ 127 ] This assumption can be formally stated as follows. Define the present
127
This assumption is introduced for the sake of simplicity, but it might be acceptable in certain cases. Let us examine the demand for the shares of the generic firm f on the part of the generic household h at different prices. To begin with, let us assume that the price for the whole of firm f
’s shares coincides with ( q
1
 hf
2
), i.e. with the present value of the amount of future profits
 hf
2
that h expects from the plan announced by the firm. It is readily ascertained that in
( these circumstances, the question of whether to purchase the whole of firm f ’s shares or invest q
1
 hf
2
) units of numéraire on the bond market will be a matter of indifference to h . It follows that in the event of the price for 100% of the firm’s shares being higher than ( q
1

2 hf ), the household’s demand for shares in firm f would be zero, as it would prefer to invest its savings in bonds; in the event of the price for the whole of the firm’s shares being lower than ( q
1
 hf
2
), household h would have an incentive to buy the firm outright and finance the purchase by borrowing on the bond market, because in the household’s opinion the operation would ensure a positive profit in period 2 at no cost. Having established these preliminary results, let us assume for simplicity that there are only three households in the economy ( h = 1, 2, 3) and that
→
126

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 127 value of the profits that household

1 hf = q
1
[ p
2 h y
2 f * + (1/ q
1
)( p
1 h expects from the plan f y
12
* announced by firm f as y
1 f * )] and consider the equivalent formulation

1 hf =
( q h y
2 f * + p
1 y
1 f * ), where vector q h = q
1 p
2 h denotes the ‘present prices’ of commodities for future delivery as calculated by household h . Then denote by v f the current price for the whole of firm f
’s shares, or market value of the firm for short. Finally, denote by
 f h the share in firm f that household h owns after the announcement of production plans and the transferral of firm shares to the households that value them most. The following assumption is then made:
Assumption 9.5. (a) The market value of the generic firm f in period 1 is v f = Max h

1 hf = Max h
( q h y
2 f * + p
1 y
1 f * );
(b) for all h and all f,
 f h

0 ;
(c) for all h and all f ,
 f h > 0 if and only if

1 hf
(d) for all f,
 h
 h = f
 h
 h = 1. f
= v f ;
As regards the market for bonds, we assume as before that each household initially issues bonds so as to capitalise its expected future wealth, which is given in the present context by the expected value of future endowments plus the household’s expected profits from its shares of firms. individual price expectations are such that  1 f
2
  2
2 f
  3
2 f . In those circumstances, it can be argued (a) that the equilibrium price for the whole of firm f
’s shares cannot be higher than
( q
1
 1 f
2
) and (b) that the equilibrium price cannot be lower than ( q
1
 2
2 f ), since at a price lower than ( q
1
 1 f
2
) at least two households would be interested in purchasing the whole of firm f
’s shares and an aggregate excess demand would accordingly appear on the market for those shares. It can thus be concluded that the equilibrium price for 100% of the firm’s shares must lie in the interval [( q
1
 1 f
2
), ( q
1
 2
2 f )], the interval between the highest and the next-to-highest estimate by some household of the present value of expected profits. This in turn means that the assumption introduced in the text concerning the price for the shares of the generic firm can be justified in practice when the difference between
 1 f
2
and
 2
2 f is very small, and can be fully justified when two or more households share the most optimistic expectation as regards the firm’s profits (i.e. in the particular case in which  1
2 f = 
2
2 f
  3
2 f ). Thus that assumption requires that for each firm the two more ‘optimistic’ households have rather similar expectations. Note that if two or more households share the same most optimistic expectation about a firm’s profits, the way the firm’s shares are divided among these households does not affect their budget constraint because any division will result in the same (subjective) wealth for each household.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 128
Assumption 9.6.
At the beginning of period 1 the generic household h issues a quantity of bonds b
1 h such that q
1 b
1 h = q h
 h +
2
 f
 f h ( q h y
2 f * + p
1 y
1 f * ). is W
1 h
Under Ass. 9.6 the wealth at the disposal of household h after this issue of bonds
= p
1

1 h +
 f
 f h v f + q
1 b
1 h , but part of this wealth must be used to pay for the share transfers carried out in accordance with Ass. 9.5; only the remainder can be spent on purchase of commodities for current consumption and on purchase of a quantity b
1 h of bonds. The first period budget constraint of household h is therefore given by the equation p
1 x
1 h + q
1 b
1 h = p
1

1 h +
 f
 f h v f + q
1 b
1 h –
 f
 f h v f (6) which, by substituting for q
1 b
1 h according to Ass. 9.6 and taking Ass. 9.5 into account, can be written[ 128 ] p
1 x
1 h + q
1 b
1 h = p
1

1 h + q h

2 h +
 f
 f h v f (7)
This reformulation shows that what determines spendable income, besides the observed and expected prices and the endowments of goods, is the initial endowments of shares[
129
], not the endowments after the transferral of shares to the households that value them most. On the other hand, the household anticipates that in period 2 it will have to surrender both its commodity endowments and its share of firms’ profits in order
128 p
1
By substituting for

1 h +
 f
 f h v f q
1
+ q h  h
2 b
1 h the right-hand side of equation (6) becomes
+
 f
 f h ( q h y
2 f * + p
1 y
1 f * ).
As
 f h is strictly positive if

1 hf = ( q h y
2 f * + p
1 y
1 f * ) = v f and must otherwise be zero (Ass. 9.5
(b)-(c)), the right-hand side can be rewritten in the equivalent form p
1

1 h +
 f
 f h v f + q h 
2 h +
 f
 h f v f .
Once the right-hand side of equation (6) is reformulated in this way, elimination of the total expenditure for shares
 f
 f h v f from both sides yields equation (7).
129
The assumption of given initial shares in spite of the admission of trading in shares can only be acceptable if either the equilibrium prices are struck at the first attempt, or the tradings in shares during the disequilibrium adjustments are provisional, the adjustments being a tâtonnement where tradings in shares too are done through ‘bons’ that are cancelled once the auctioneer announces that the prices were not equilibrium prices and the tâtonnement must continue. Of course any realistic adjustment will on the contrary alter share ownerships during disequilibrium, so we have here one more element of impermanence of the data of equilibrium.
128

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 129 to repay the bonds issued in accordance with Ass. 9.6. The household’s (expected) budget constraint for period 2 is therefore p
2 h x
2 h = b
1 h (8)
Comparison of budget constraints (7)–(8) and budget constraints (1)–(2) of §
8.D.?? shows that once the firms’ plans have been announced and share transfers have taken place, households are fundamentally in the same position as in the introductory pure-exchange economy. We therefore assume that in these circumstances, the generic household h will choose its current consumption, current demand for bonds and planned future consumption so as to maximise the utility function U h ( x
1 h , x
2 h ) subject to constraints (7)–(8). As in the introductory model, this choice will in turn determine the optimal action a h * = ( x
1 h * , b
1 h * ) taken by the household on period 1 markets.
The description of agents’ behaviour at given current prices and fixed price expectations is now complete. Given that share markets are ‘automatically cleared’ in view of Ass. 9.5, a temporary equilibrium of exchange and production can be accordingly defined as a system of current prices, a corresponding set of F optimal actions on the part of firms, and a corresponding set of H optimal actions on the part of households such that the N spot markets and the market for bonds are simultaneously cleared in period 1.
The model outlined in this paragraph corresponds essentially to the temporary equilibrium model with production put forward by Arrow and Hahn (1971: Ch. 6)[
130
].
130 Our repeated reference to the contribution of these authors calls for some clarification as regards the link that can be established between the temporary equilibrium model with production of Arrow and Hahn (1971), and the models presented in this paragraph, and in
§§8.D.6 to 8.D.9.
To start with the model outlined in this paragraph, even though all the assumptions concerning the behaviour of agents are either borrowed from the Arrow-Hahn model or compatible with it, there are two differences in the formulation adopted. As readers can check, in the model of Arrow and Hahn prices are expressed in terms of a fictitious currency of account
(‘bancors’) and the unit bond is defined as a promise to pay a unit of that currency in period 2.
However, these differences are immaterial. To see why, consider a version of the Arrow-Hahn model in which all agents expect that the future price of good 1 in terms of ‘bancors’ will be equal to 1. In these circumstances, which are fully compatible with Arrow and Hahn’s formal treatment of expectations, the market for bonds specified in ‘bancors’ becomes the same thing as a market for bonds specified in terms of good 1.
As a result, the version of the Arrow-Hahn model under consideration coincides with the extended model outlined in this section except for the numéraire adopted. Given that the behaviour of agents in Arrow and Hahn’s contribution is
→
129

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 130
As regards the existence of temporary equilibrium we can therefore take advantage of the results obtained by those authors, who prove in this connection that temporary equilibrium of exchange and production exists under standard assumptions on preferences and productions sets. They also show that this result holds not only in the case of fixed expectations but also under the assumption that individual price expectations are continuous functions of current prices. Having thus briefly dealt with the question of existence, we shall now go on to a closer examination of the assumptions concerning production decisions and borrowing decisions made in the extended model. It will be argued that they are more problematic than they may appear.
Discussion of the extended model [
131
].
8.D.14. Let us first examine the production decisions of firms. In the intertemporal model of general equilibrium, the existence of complete forward markets for commodities permits a simple treatment of production decisions within firms. Let us consider, within that model, the position of the households holding ownership shares in a generic firm at the initial date. On the one hand, each household is interested in receiving the highest amount of profit from the firm, as any increase in profit would correspondingly increase the household’s initial wealth and therefore improve the household’s consumption opportunities. On the other hand, the profitability of the alternative production plans that are feasible for the firm can be assessed objectively on the basis of the prices observable on the current system of spot and forward markets. It follows from these considerations that the households sharing the ownership of a generic firm at the initial date will unanimously approve the choice of a production plan that independent of the numéraire measuring current prices, we can safely modify that version by taking good 1 as numéraire.
Having thus established that the model with production presented in this paragraph is simply a version of the Arrow-Hahn model, we shall now show that further specification of that version makes it possible to obtain precisely the pure-exchange model of §§8.D.6 to 8.D.9. Assume that there is only one firm in the economy ( F =1) and that its production set is Y
1
12
=   2

N . The last part of the assumption states that the only processes the firm can operate are free disposal processes, through which any good available in any of the two periods is instantaneously destroyed by using no other input than the good itself. Under this particular specification of the productive sector, which is compatible with Arrow and Hahn’s formal model despite its ad hoc nature, the single firm in existence will remain totally inactive in period 1 at every non-negative vector of current prices. As a result, the particular ‘production economy’ under consideration coincides in fact with the pure-exchange economy of §§8.D.6 to 8.D.9.
131
This part is based on Ravagnani (1989, 2000).
130

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 131 maximises profits calculated at the currently observed prices[ 132 ][ 133 ].
By contrast, the treatment of production decisions encounters considerable complications in a temporary equilibrium framework. In order to discuss the main issues that arise, let us return to the Arrow-Hahn model as presented in §8.D.13 and focus on the position of households at the initial date. Jointly considered, budget constraints (7)-
(8) show that the utility a household can plan to obtain by trading on current markets increases with the value of its period-1 wealth, which depends partly on the value of the household’s initial endowment of shares. This means that any household holding an initial share in the generic firm f will favour the choice of the production plan that receives the highest market valuation for the firm’s shares, i.e. the choice of the plan that maximises the market value v f of the firm . But according to Ass. 9.4 the manager of the firm will select the plan to which he individually attaches the greatest present value, so he does not try in general to act in the interest of the firm’s initial owners. An unsatisfactory feature of the model is therefore that the criterion of choice attributed to managers has no clear rationale. We shall now show that this shortcoming is not easily remedied, as it is a symptom of an authentic analytical problem.
Suppose for the sake of argument that the manager of the generic firm, in an effort to serve the interests of the initial owners, forms a definite opinion as regards the production plan that will generate the highest market value of the firm and then announces that he intends to implement precisely this project. However, the manager’s opinion is necessarily subjective: the firm’s initial owners may happen to have a different opinion and wish to alter the manager’s decision (perhaps by replacing the manager). But the initial owners may well have conflicting opinions as regards which plan will ensure maximisation of the firm’s market value; in these circumstances, no production plan could be unanimously approved by the initial owners and a sort of social choice problem would therefore arise within the constituency of the firm’s owners.
While this problem could be tackled in principle by assuming that some institutional rule leading to a definite production decision is at work within the firm, the fact that a variety of such rules can be conceived (e.g. different voting schemes) makes it hard to see how
132 Of course this argument presupposes that the owners of the generic firm are ‘price takers’, i.e. they believe that current prices are not appreciably altered by changes in the firm’s production plan.
133
Note that, in the Arrow-Debreu approach where share ownership is relevant owing to the possibility of positive firm ‘profits’, complete futures markets or shared perfect foresight imply that households have no interest in trading shares at the equilibrium prices, because everybody agrees on the value of shares at those prices, so buying a share would not change the wealth of a consumer.
131

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 132 that assumption should be precisely specified.
On the other hand, it is possible to adopt a pragmatic attitude and argue that the assumption that managers choose production plans according to their own evaluation of future receipts provides a realistic representation of where control over firms actually resides (see, for example, Bliss, 1976: 194–195). ‘Manager’ might also be interpreted
(and can be so interpreted in the remainder of this paragraph) as indicating whatever expectations profile emerges as the decisions-determining one from the owners’ expectations and preferences plus the institutional rule that governs the firm’s decisional process. This attitude may explain why that assumption has been commonly adopted in temporary equilibrium models with production. However, as discussion of a further shortcoming of the Arrow-Hahn model will presently show, the assumption of production plans autonomously chosen by managers (or by shareholder assemblies) is hardly tenable in a temporary equilibrium framework.
The aspect we shall now discuss concerns the financing of the production plans selected by managers in accordance with their personal expectations of future receipts.
As shown above, Arrow and Hahn assume that firms finance those plans by selling bonds on a single market where all bonds are traded at the same price and are therefore treated as perfect substitutes, who issued a particular bond being implicitly assumed to be irrelevant. It is highly doubtful, however, that rational households would be generally willing to treat bonds as perfect substitutes. A simple example will clarify this point.
Consider an economy with only two firms and assume that the manager of each firm selects a plan that maximises the present value of profits calculated on the basis of his individual price expectations. Then assume that when the manager of firm 1 announces the chosen plan (from which he expects non-negative profits), all the other agents in the economy expect that the future price of planned output will be so low as to generate negative profits for the firm in period 2. Finally, assume that all households expect positive profits from the plan announced by firm 2. In such circumstances, the entire ownership of firm 1 would be transferred to the firm’s manager when the markets open at the initial date. Moreover, the following situation would occur on the bond market. Except for the optimistic manager of firm 1, all households in the economy would calculate that firm 1 is going to issue bonds that cannot be repaid out of the firm’s future receipts – and since they do not know whether the future wealth of the firm’s new owner will be sufficient to guarantee repayment, those households would have to regard the bonds floated by firm 1 as risky assets. At the same time, they would regard the bonds issued by firm 2 as perfectly safe. The announcement of the production plans independently chosen by managers would thus signal to households that in the overall supply of bonds risky assets may coexist with others whose repayment is beyond doubt.
132

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 133
In this situation it is unreasonable to suppose, as the Arrow-Hahn model implicitly does, that households may be disposed to purchase bonds on a single, ‘anonymous’ market where risky securities cannot be distinguished from safe ones.
134
In order to avoid the abovementioned shortcoming, the model would have to be reformulated so as to enable potential lenders to identify the agents issuing bonds and to learn how they plan to repay their debts. This could be done by introducing a separate
134
Problematic situations such as the one described in the text may also arise if the Arrow-
Hahn model is modified by assuming that managers endeavour to select production plans that maximise the market value of their respective firms. For example , consider an economy with two firms, A and B, that can produce two different qualities of wine by employing grape must as the only input. Assume that each firm can produce any combination of wines by operating two independent processes defined by the production functions y
12
= (
 y
1
)
1 / 2 for wine of type 1 and y
22
= 2 (
 y
1
)
1 / 2 for wine of type 2, where y
1
(a negative number) denotes the quantity of must employed and y i 2
the output of wine of type i ( i = 1, 2). Assume further that there are four households in the economy characterised by the following fixed expectations. Household 1, which includes only the manager of firm A, expects that the price for wine of type 1 will be p
ˆ
12
> 0 and that the price for wine of type 2 will be zero. Household 2 has the same expectations as household 1. Household 3, which includes only the manager of firm B, expects that the price for wine of type 1 will be zero and that the price for wine of type 2 will be p
22
= ½ ˆ
12
. Finally, household 4 has the same expectations of household 3. Now recall that in the Arrow-Hahn model, the market value of each firm coincides with the present value of the profits that the most optimistic household (or group of households) expects from the firm’s plan (Ass. 9.6). Taking this assumption into account, we can readily see that at any given positive price for grape must, there are always two distinct production plans that ensure maximisation of the market value of the generic firm in the economy under consideration. The first involves producing only wine of type 1 in the quantity that maximises the present value of profits calculated at the positive price expected for that wine by households 1 and 2. The second involves producing only wine of type
2 in the quantity that maximises the present value of profits calculated at the positive price expected for that wine by households 3 and 4. Having established this point, assume that managers seek to maximise the market value of their respective firms and that if two or more plans ensuring this result are identified, each manager will choose the one that he thinks will yield the highest amount of profits (reasonable behaviour). Finally, assume for the sake of argument that both managers can correctly predict how individual households will evaluate any feasible production plan. Under these assumptions, each manager will be able to identify the pair of plans that ensure maximisation of the market value of his firm when markets open in period 1. Moreover, the manager of firm A will choose and announce the plan that involves producing only wine of type 1, while the manager of firm B will opt for and announce the plan that involves producing only wine of type 2. On the other hand, every household will calculate that one of the announced plans will yield positive profits while the other is bound to bring about losses. The announcement of production plans will thus signal to households that risky bonds may coexist with safe ones in the overall supply.
133

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 134 market for the bonds issued by each individual agent, but then the hypothesis that managers autonomously select production plans could hardly be retained. For example, suppose that the manager of the generic firm f selects a definite plan y
12 f with the intention of covering the input cost through the sale of a sufficient quantity of bonds at price p b f . When the plan is announced, households value future output according to their own price expectations (as well as the future wealth of the firm’s owners, if the latter are legally responsible for the firm’s debt) in order to assess the amount that can be paid back to lenders, and thus form an opinion about the rate of return that can actually be obtained on the bonds supplied by firm f . If this largely subjective rate of return proves to be lower than that expected on the bonds of some other firm, however, households will not buy firm f
’s securities. It would then be impossible to implement the plan chosen by the manager, and the theory would have to explain how the original project is to be revised.
8.D.15. Discussion of the Arrow-Hahn model thus shows that in the presence of subjective price expectations it is not reasonable to assume that managers or more generally firms can raise funds freely on capital markets. One is thus induced to explore different assumptions as to how firms finance their plans; e.g., as done by Grandmont and Laroque (1976), one may assume that firms own some wealth (some resources) and finance their input expenditure out of that wealth[
135
]; but this is unduly restrictive, in real economies firms do borrow; even when they own plants or mines or land they
135 This does not eliminate the possibility of conflict between manager and owners. Assume that the manager of the generic firm f , guided by his personal evaluation of future receipts, chooses a production plan that involves using the whole of the firm’s initial wealth to finance input expenditure. Assume further that when the manager’s decision is announced, all the households in the economy (except for the manager’s) anticipate that the firm’s planned output will have negligible value in the future. In these circumstances, it is reasonable to imagine that the current price for the whole of the firm’s ownership shares would be very close to zero.
Assume that this is indeed the case and consider the position of the initial owners of firm f .
Apart from the negligible price they could receive from the sale of their shares in the firm, these owners would calculate that the manager’s decision requires them to give up some of their potential period 1 wealth (corresponding to the value of the firm’s commodity endowment) in order to finance a project that they regard as a sheer waste of resources. At the same time, each owner would calculate that he would be better off if the firm were instructed to close down, as then he could regain his share of the firm’s initial wealth and improve his consumption opportunities. Even though there may be disagreements concerning the ‘optimal’ plan to put into operation, all the initial owners would thus prefer the firm not to engage in production, and in the presence of this unanimously preferred option it is paradoxical to suppose that they would passively agree to finance the manager’s project.
134

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 135 usually have obtained that property against indebtedness, so their net wealth is often not considerable and to restrict their plans to what can be financed with that net wealth would restrict their choices unrealistically. Or one may try to introduce collateral, but then the specification of equilibrium comes to depend on the specific assumptions as to what can be used as collateral, and how much collateral is needed, assumptions that cannot but be to some extent arbitrary, and furthermore do not avoid the loss of anonymity, with the unpleasant consequence that price taking can no longer be assumed in credit markets.
The considerations put forward thus far indicate that temporary equilibrium theory should admit that managers’ decisions are subject to the ultimate judgement of savers, who may refuse to supply the required funds and thus force revision of the original projects. In this situation it would appear more appropriate to assume that managers, when selecting production plans, take into account the opinion of the agents who provide funds to the productive sector. However, this assumption gives rise to a new problem, because in order to develop a plausible notion of temporary equilibrium the theory would have to explain how managers can succeed in correctly interpreting the private opinions of the potential financiers of firms.
Having started to see the problem with the assumption of indifference of lenders between potential borrowers, we can notice that the problem exists also for lending to consumers. The assumption that shares end up to the household who values them most is only acceptable if the household can finance the purchase by issuing bonds, as assumed in the formulation of budget constraint (6). But bond purchasers may well not share the optimism of the issuer: and indeed in the real world we do not expect the entire ownership of General Motors to go to the individual – perhaps a lunatic – who is most optimistic about its future prospects. Thus one must admit limits to what individuals can borrow, and again one has difficulty with determining these limits without arbitrariness and with arriving at a definite theory of how shares will end up being allocated.
We find here some examples of the need of the approach (mentioned in §8.D.2), due to its very-short-period nature, to specify behaviours that in real economies depend on a myriad different elements, with a consequent indeterminateness (that arbitrary assumptions can reduce only at the cost of excluding many other plausible behaviours).
Part D, Section IV: Temporary equilibrium in economies with ‘money’
8.D.16. The models discussed in the previous sections fail to capture one aspect of real-world trading processes, namely the fact that economic agents wish to keep stocks
135

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 136 of a special good – money – that has no intrinsic value and is used essentially in exchange against physical goods. It should be noted, however, that much of the research carried out by temporary equilibrium theorists had the precise aim of incorporating money into modern general equilibrium analysis. In this section we shall therefore illustrate some basic results emerging from that specific application of temporary equilibrium theory.
136 This will be done through reference to a simple model drawn from
Grandmont (1983), whose basic features are summarised below.
The model considers an exchange economy in which spot markets are active in each period, no forward market exists, and agents can transfer wealth from one period to the next only by holding a particular asset, ‘money’, which is available in the system in a constant amount. By assumption, the existing stock of this asset is made up entirely of outside or legal or base money (i.e. not created against debt) and can therefore be seen as part of the households’ net wealth. The stuff of which money is made has no intrinsic utility, so money is ‘fiat money’. Since there are no forward markets i.e. no bonds, money cannot be borrowed. The model is exclusively concerned with the store-of-value function of the asset (the possibility of using stored money as a way to transfer purchasing power across periods) and does not consider the other services performed by money in real-world economies, e.g. as a medium of exchange: the medium-of-exchange function of money does not appear in the model, in the sense that it poses no constraint on the agents’ ability to transact. The behaviour of agents is analysed under the condition of strictly positive monetary prices, and money is chosen as numéraire. It should be noted that this choice of numéraire is incompatible with states of the economy characterised by aggregate excess supply of money, as the exchange value of money in terms of any commodity would be zero in such circumstances. The main issue addressed by the model is therefore whether a temporary equilibrium for period 1 exists in which households are willing to hold the whole stock of money in circulation (if only the storeof-value function of money is considered).
Let us then go on to develop a detailed formal exposition in order to try and answer this question.
8.D.17. As in Section 2, we shall refer to an economy with H households and N
136 For an extensive treatment of the monetary issues addressed by temporary equilibrium theorists – which include the validity of the quantity theory of money, the possibility of monetary authorities to manipulate the interest rate, and the existence of a ‘liquidity trap’ – the reader is referred to Grandmont (1983).
136

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 137 non-storable consumption goods that is active for two periods of time.
137 At the beginning of period 1, the generic household h has both a commodity endowment vector

1 h and an endowment of money m h stemming from its past saving decisions, and knows that its future commodity endowment will be

2 h . The household observes the monetary prices p
1
  N
 
quoted on current spot markets and expects the system of monetary prices p
2 h to obtain in period 2. (For the sake of economy of notation we continue to denote prices as in the previous Sections even though they are now expressed in money.) Unlike the arguments developed in Sections 2 and 3, we do not regard the vector p
2 h as fixed but assume that expected prices depend on current prices. To be more precise, we assume that p
2 h =
 h ( p
1
), where the expectation function
 h can include past prices among its parameters. Finally we introduce the following assumption concerning the characteristics and expectations of households:
Assumption 9.7. (a) The generic household h has a preference ordering over two-period h consumption streams in the set X
12
=
 2

N that can be represented by the continuous, strictly increasing and strictly quasi-concave utility function U h
( x
1 h
, x
2 h
) ;
(b ) for all h ,

1 h 
0,
 
2
0 ;
(c) m h

0 for all h ,
Σ h m h = M

0;
(d) for all h , the expectation function
 h is continuous and such that
 h
( every p
1
  N
 
.
p
1
)
  N
 
for
Note that by postulating that households expect strictly positive but finite monetary prices for period 2, part (d) rules out one case in which there is no reason to transfer money to that period, namely the case in which households are certain that future commodity prices in terms of money will be infinite (i.e. that money will have no exchange value in period 2); it also rules out the case in which households think that the future money prices of all commodities will be zero, a case of more difficult interpretability (can one use money to buy commodities in this case?).
Let us now examine the behaviour of households at the beginning of period 1.
Given the ruling prices and the associated expected prices, the generic household h must choose a most preferred two-period consumption stream out of those it believes it can attain in view both of the value of its commodity endowments and of the possibility of
137
The analysis that follows can be readily extended to economies in which markets are active for more than two periods and households formulate their plans accordingly (cf.
Grandmont, 1983, Ch. 1).
137

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 138 transferring money to period 2. It can be stated in formal terms that at any given price system p
1
  N
 
, the generic household h must solve the following problem:

4.I

Maximise U h ( x
1 h , x
2 h ) with respect to x
1 h

0, m
1 h

0, x
2 h

0, subject to the current and expected budget constraints: p
1
 h x
1 h
(
+ p
1
) m
1 h x
2 h
= p
1
=
 h

1 h +
( p
1
) m h

2 h
(4.1)
+ m
1 h (4.2) where the choice variable m
1 h , the ‘demand for money’ as this term is used in this literature, denotes the amount of money with which the household wants to end the period in order to carry it over to period 2 to finance future consumption[ 138 ]. Note that m
1 h must be non-negative because, by assumption, the household cannot borrow money in period 1. Note also that the household does not plan to demand money in period 2, as it is aware that economic activity is going to cease at the end of that period[ 139 ].
138 This definition of ‘demand for money’, the amount of money one wants to end the period with , is very different from the traditional meaning of ‘demand for money’ (e.g. in the
Cambridge money equation) that refers to the average holding of money balances over a certain stretch of time desired by an agent in order to satisfy the transaction, precautionary and speculative motives for holding money. Consider a household which starts the period with zero money, sells labour services at the beginning of the period and thus obtains an amount of money m , spends it at a constant rate over the period, and ends the period with zero money, and is satisfied with the situation. The traditional demand for money of this household is m /2, its demand for money as defined in the text is zero. Cf. Petri (2004, ch. 5, Appendix 1).
139 The lack of incentives to demand money in period 2 creates a problem which we can neglect but it is opportune to remember, in fact the same problem with money as in the Arrow-
Debreu model, with expectations determining the same outcome as complete futures markets. If the generic household realizes that in period 2 all the other agents too have no reason to demand money (i.e. no reason to end the period holding a positive amount of money), it cannot reasonably expect money to have a positive exchange value in period 2 as stated by Ass. 9.7(d).
The same problem arises when the two-period model put forward in the text is extended to economies that are active for a higher but finite number of periods. We already know the argument, which needs only to be reformulated in terms of plausible formation of expectations.
Assume that economic activity comes to an end in an arbitrarily given period T > 2 and that all agents are aware of that future event. Then any non-stupid household operating in the economy at the beginning of period 1 will conclude that no agent will want to be left with money balances at the end of the terminal period T and that, for this reason, money will have no exchange value in that period. If the household assumes that the other households are not stupid, she will conclude that at the beginning of period T -1 all agents in the economy will realise that money is going to be worthless in the terminal period, and therefore will want to end period T -1 with zero money balances, causing the price of money to fall to zero in period T -1 too, and then in all previous periods for the same reason – including the initial period. In order to avoid this
→
138

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 139
Discussion of the solution to problem

4.I

will be facilitated by focusing on budget constraints (4.1)–(4.2). To begin with, it should be noted that the outcome of the problem remains the same if those constraints are modified by replacing the equality signs with inequality signs, since U h is strictly increasing. We can therefore consider the modified budget constraints p
1
 h x
1 h
(
+ p
1
) m
1 h x
2 h
 p
1
  h

1 h
( p
1
+
)
 m h
2 h + m
1 h
On adding up the modified constraints and eliminating m
1 h , it becomes clear that the consumption stream chosen by the household must fulfil the inequality p
1 x
1 h +
 h ( p
1
) x
2 h
 p
1

1 h +
 h ( p
1
)

2 h + m h (4.1’) which we shall call the intertemporal budget constraint of household h . On the other hand, we know that m
1 h must be non-negative, i.e. that the household cannot borrow money in period 1. This means that the consumption stream chosen by the household must also fulfil the inequality p
1 x
1 h
 p
1

1 h + m h (4.2’) which we shall call the liquidity constraint of household h [
140
]. It can be stated in the problem the temporary equilibrium model with ‘money’ should be modified by assuming that economic activity extends indefinitely over time. Within that context, the fact that human life has limited duration could be taken into account by assuming that two generations of households co-exist in the economy in every period of time, an ‘older’ generation initially endowed with the whole money stock and a ‘younger’ generation demanding money in the belief that the new younger generation will do the same in the subsequent period (cf., for example, Grandmont and
Laroque, 1973). The structure of the temporary equilibrium model with ‘money’ would become more complex, since the maximisation problem attributed to the younger generation should be distinguished from that attributed to the older one. However, there is no need to introduce this complex construction for our purposes, as the conditions ensuring the existence of temporary monetary equilibrium would remain essentially the same as those emerging from the simple model examined in this section.
140
Constraints (4.1’)-(4.2’) can be interpreted in economic terms as follows. Assume for the moment that household h is not only able to transfer money balances to period 2 but also to borrow money at no interest in period 1 within the limit set by its expected future wealth. It is easy to ascertain that in these circumstances, the household will only be subject to the intertemporal budget constraint (4.1’). Since we are assuming that the amount of money the
→
139

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 140 light of these considerations that problem

4.I

can be solved in two steps. In the first, the generic household determines its optimal consumption stream h * x
12
= ( x
1 h * , x
2 h * ) by solving the problem

4.II

Maximise U h ( x
1 h , x
2 h ) with respect to subject to constraints
(4.1’)–(4.2’) x
1 h

0, x h
2

0,
In the second, it determines its optimal ‘demand for money’ m
1 h * through the condition m
1 h * = p
1

1 h + m h – p
1 x
1 h * .
Let us focus on problem

4.II

and define the opportunity set of household h as the set of two-period consumption streams in X h
12
=
 2

N that fulfil both the constraints
(4.1’)-(4.2’). It is easily proved that this set is compact and convex under the assumption that both the current and the expected prices are strictly positive[ 141 ]. Given that U h is strictly quasi-concave, it follows from the properties of the opportunity set that problem

4.II
 uniquely determines the consumption stream h x
12
* h at any given p
1
  N
 
= ( x
1 h * , x
2 h * ) chosen by household
. In these circumstances, the amount of money demanded by the household is itself uniquely determined in the second step of the procedure. We therefore conclude that both the household’s current consumption demand x
1 h * and its money demand m
1 h * can be represented as functions of (strictly positive) vectors of current prices, which we shall denote by optimal action a h * = ( x
1 h * , m
1 h * x
1 h ( p
1
) and m
1 h ( p
1
) respectively. In this notation, the
) taken by household h at any given p
1
  N
 
is univocally identified by the function a h ( p
1
) = ( x
1 h ( p
1
), m
1 h ( p
1
)).
Let us now consider the first period excess demand function of the generic household h , defined as z
1 h ( p
1
) = x
1 h ( p
1
)–

1 h , and the household’s money demand function m
1 h ( p
1
). It can be proved that they are both continuous functions (Grandmont,
1983: App. B, p. 165). It should also be noted that since the household’s optimal choice must fulfil budget constraint (4.1), the equality p
1 z
1 h ( p
1
) + m
1 h ( p
1
) = m h necessarily holds at every strictly positive vector of current prices. It follows from this last consideration that first period aggregate excess demands inclusive of the excess demand for end-of-period-1 money balances satisfy what can be called
Walras’s Law with money : household can actually borrow is zero, however, the liquidity constraint (4.2’) on current consumption expenditure must also be introduced.
141 Key: under the assumption mentioned in the text, the opportunity set is the intersection of two convex and compact sets.
140

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 141 p
1
Σ h z
1 h ( p
1
) +
Σ h m
1 h ( p
1
) =
Σ h m h = M for every p
1
  N
 
(4.3)
Given the above formal description of the behaviour of households, a temporary monetary equilibrium of the exchange economy for period 1 can be finally defined as a system of monetary prices p
1
*
  N
 
, and a corresponding set of optimal actions on the part of households, such that the following market-clearing conditions are simultaneously satisfied:
Σ h z
1 h ( p
1
* ) = 0 ,
Σ h m
1 h ( p
1
* ) =
Σ h m h = M (4.4)
The second equality states that, in the aggregate, consumers must wish to end the first period with the same total amount of money they started with: if some consumer plans to spend in the first period more than his first-period income by using some of his initial money holding, some other consumer must wish to end the period with more money than he started with by spending less than his first-period income.
8.D.18. Let us now address the question of the existence of temporary monetary equilibrium. Existence is not guaranteed under Ass. 9.7. To clarify this point, let us focus on the simplified case of an exchange economy with a single consumption good ( N = 1).
In this case, the opportunity set of the generic household h at an arbitrary strictly positive price p
1
for the consumption good can be represented as in Figure 8.D.2.
141

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 142 x h
2
 h
2

 m h h
( p
1
)

2 h

 h h * x
12


1 h

1 h  m h p
1 x h
1
Figure 8.D.2
By examining constraints (4.1’)-(4.2’) taken for
N =1, it is easy to ascertain that the line going through points

and

, whose slope is p
1
/
 h ( p
1
), represents the intertemporal budget constraint , while the vertical half-line going through

represents the liquidity constraint . The curve is the highest indifference curve with a point in common with the budget set; the optimal choice of household h therefore corresponds to point h * x
12
, which identifies both the household’s current excess demand for the consumption good and the household’s demand for money balances. We shall now use this graphic device to analyse how the optimal choice of the generic household changes as the current price of the consumption good changes from p
1
. We shall only deal with a rise in the price, as the analysis that follows is easily adapted to the case of a fall.
Let us first assume that the household has unit elastic price expectations , i.e. that
 h (
 p
1
) =
  h ( p
1
) for every positive value of p
1
and every positive number

. In this case, an increase in the current price of the consumption good from p
1
to
 p
1
,

, causes both the intertemporal budget line and the liquidity line to move to the left,
  without however altering the slope of the former. What basically happens is that the price rise proportionately reduces the purchasing power of the household’s money endowment while leaving the ‘relative price’ of present consumption in terms of future
142

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 143 consumption unchanged at the initial level p
1
/
 h ( p
1
). The rise in the current price thus generates a real balance effect that will in turn normally reduce the current demand for the consumption good (Figure 8.D.3). x h
2

2 h 
'
 x
1 h

1 h
Figure 8.D.3
Let us now assume instead that the household’s expectations are not unit elastic, i.e. that
 h (
 p
1
)
   h ( p
1
) for every p
1
> 0 and every

> 0. In these circumstances, the change in the opportunity set generated by an increase in the current price from
 p
1 p
1
,
   , can be broken down into two ‘successive’ changes. The first is the shift to
to the left of both the intertemporal budget line and the liquidity line that would take place if expectations were unit elastic. This is precisely the real balance effect mentioned above. The second is the rotation of the intertemporal budget line around the new point
 ’ due to the fact that the ‘relative price’ of present consumption in terms of future consumption must now change with respect to its initial level. This change in the relative price will further affect the household’s choice by giving rise to a rotation effect, which can be decomposed into an intertemporal substitution effect and an intertemporal income effect whose sum is of uncertain sign. If one assumes that the substitution effect is stronger than the income effect (assuming both goods normal), when the elasticity of expectations is lower than 1 (
 h (
 p
1
)
   h ( p
1
)) and therefore the intertemporal budget line rotates upward , the intertemporal substitution effect reduces present
143

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 144 consumption, thereby reinforcing the real balance effect (Figure 8.D.4); when the elasticity of expectations is higher than 1, the budget line rotates downwards and the intertemporal substitution effect acts in the opposite direction to the real balance effect and the overall effect of the price rise on the household’s choice cannot be assessed a priori . x
2 h

2 h

2 h
 ' 

1 h x h
1
Figure 8.D.4
In the light of the above analysis, we can now discuss the existence of temporary monetary equilibrium for the one-commodity exchange economy. We shall show first of all that existence is not guaranteed when expectations ‘depend too much’ on the currently observed price. This will be done by means of two examples taken from
Grandmont (1983: 22-24), in which it is implicitly assumed that m h

0 holds for all h .
Example 1. A preliminary remark is necessary. Recall that the preferences of households can be represented by strictly increasing and strictly quasi-concave utility functions. It should thus be clear from Figure 8.D.5 that at any given demand of the generic household h p
1
> 0, the current consumption
will exceed the endowment

1 h if and only if the
‘relative price’ p
1
/
 h ( p
1
) is lower than the household’s marginal rate of
144

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 145 substitution[ 142 ] evaluated at point

= (

1 h ,

2 h + m h /
 h ( p
1
)). x h
2
 h
2

 m h h
( p
1
)

2 h

 h


1 h h x
1
Figure 8.D.5
Having established this preliminary result, let us assume that the households’ utility functions can be written w ( x
1 h ) +
 h w ( x
2 h ), h = 1, …, H , where w ( .) is strictly concave and differentiable and 0 <
 h < 1 for all h . Let us further assume that the expectation function of household h is such that the following condition holds:
 h p
1
( p
1
)

 w
' h
( w
'

1 h
(

)
2 h
) for every p
1
> 0 (4.5) where the term on the right-hand side of the inequality is the household’s marginal rate of substitution evaluated at the endowment point
 h = (

1 h ,

2 h ). Given that the
142 In this Section for simplicity we intend by ‘marginal rate of substitution’ its absolute value.
145

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 146 marginal rate of substitution increases as we move upward along the vertical half-line with origin (

1 h , 0) because w ( .) is strictly concave, it becomes clear that at any positive value of the current price, the household will be precisely in the position depicted in
Figure 8.D.5, and will therefore manifest an excess demand for the consumption good in period 1. If we finally assume that expectation functions are such that condition (4.5) holds for all h , it is clear that at every p
1
> 0 there will be an aggregate excess demand on the current commodity market, which will be accompanied in accordance with
Walras’s law with money
by a corresponding aggregate excess supply of money. This means that no temporary equilibrium exists for period 1, in which households are willing to hold the whole stock of money in circulation: assuming that the excess demand for the period-1 good causes an indefinite tendency of its money price to increase, it does not matter how high the price level becomes, in the aggregate consumers always wish to end the period with a smaller holding of money than they started with. In particular, the phenomenon described will occur when expectation functions are unit elastic and such that, for all h
, the (constant) ‘relative price’ p
1
/
 h ( p
1
) fulfils condition (4.5).
Example 2. Let us assume that preferences are such that, for all h , the value of the marginal rate of substitution along the vertical half-line with origin (

1 h , 0) is bounded above by a strictly positive number v h . Let us further assume that expectations are such that the following condition holds for all h :
 h p
1
( p
1
)
 v h for every p
1
> 0 (4.6)
Under these assumptions, each household will be in a situation opposite to the one shown in Figure 8.D.5 at any given p
1
> 0. As a result, z
1 h ( p
1
)

0, m
1 h ( p
1
)
 m h will hold for all h at every p
1
> 0 and temporary monetary equilibrium will not exist: however low the money price of the period-1 good, consumers still wish to end the period with more money than they started with. In particular, this phenomenon will occur when expectations are unit elastic and such that for all h
, the (constant) ‘relative price’ p
1
/
 h ( p
1
) fulfils condition (4.6).
Similar examples can be constructed for economies with a variety of consumption goods (cf. Grandmont, 1983: 25). The simple examples put forward are, however, sufficient to develop the relevant economic considerations. To begin with, let us consider
Example 1 on the hypothesis of unit elastic expectations. As we have seen, at an arbitrarily chosen p
1

0 there is aggregate excess demand on the period 1 commodity
146

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 147 market. We also know that an increase in the current price from p
1
would generate a real balance effect that is likely to reduce that excess but cannot eliminate it completely.
Similarly, on the hypothesis of unit elastic expectations, Example 2 shows that the real balance effect resulting from a fall in the current price may be not strong enough to compensate fully for an initial excess supply on the current commodity market. A negative conclusion therefore emerges from the temporary equilibrium model with
‘money’ as regards the effectiveness of the real balance effect as a mechanism capable of regulating the market. This negative conclusion attracted a great deal of attention when the original version of the model was published (Grandmont, 1974), as its was commonly held among neoclassical economists at the time that the real balance effect would normally ensure market clearing in economies endowed with outside money (cf., for example, Patinkin, 1965).
8.D.19. It should further be noted that conditions (4.5) and (4.6) in the examples presented are constraints on the variability of the ‘relative price’ p
1
/
 h ( p
1
) and therefore impose limits on the strength of the intertemporal substitution effects that can be generated by changes in p
1
. It may accordingly be conjectured that the introduction of restrictions on expectations capable of ensuring high variability of this ‘relative price’ could allow the intertemporal substitution effects engendered by changes in p
1
to become strong enough to reinforce the real balance effect and eliminate disequilibrium on current markets. The following argument indicates that this is a reasonable conjecture.
Assume that in the single-commodity exchange economy with ‘money’ there is a household h whose expectations are ‘insensitive’ to large changes in that two numbers

> 0,

> 0 exist such that
   h ( p
1
)
 
for every p
1
, in the sense p
1
> 0 ( bounded expectations). Under this assumption, it can be stated (a) that for p
1
large enough an aggregate excess supply appears on the period 1 commodity market and (b) that for p
1 low enough an aggregate excess demand appears on that market. By continuity, a price p
1
* > 0 should then exist that leads to equilibrium on the current commodity market and, in view of
Walras’s Law with money
, on the money market as well.
In the above argument, statement (a) can be justified on the grounds that when p
1 rises indefinitely, point

of the ‘insensitive’ household’s opportunity set tends to the endowment point
 h and the slope of the intertemporal budget line rises with no limit.
As a result, the household’s planned demand for future consumption tends to infinity together with the household’s current demand for money. Since by assumption the money demand of every household is bounded below by zero, this means that as p
1 progressively increases, an aggregate excess demand for money must eventually appear
147

F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 148 in period 1 with a corresponding aggregate excess supply on the current commodity market. As regards statement (b), note that when p
1
falls progressively towards zero, point

in the insensitive household’s opportunity set shifts indefinitely to the right, while the slope of the intertemporal budget line tends to zero, thus giving rise to a strong substitution effect in favour of current consumption. As a result, the insensitive household’s demand for present consumption tends to infinity. Since the current consumption demand of the generic household h is bounded below by –

1 h , this means that with the progressive fall in p
1
, an aggregate excess demand must eventually appear on the current commodity market.
The above heuristic argument indicating that the assumption of bounded expectations ensures the existence of temporary monetary equilibrium can be rigorously formulated and generalised to exchange economies with any finite number of goods. The following theorem has indeed been proved for N

1:
Theorem. Let Assumption 9.7 hold in the exchange economy with ‘money’. Assume further that there is at least one household h , with m h

0, whose expectations are bounded in the sense that two vectors
  N
 
,
  N
 
exist such that

  h ( p
1
)
 
for every p
1
  N
 
. Then a temporary monetary equilibrium exists.
Proof: cf. Grandmont (1983: Appendix B).
The assumption of bounded expectations has been generalised to the case of monetary economies in which agents’ predictions take the form of probability distributions over future prices (cf., for example, Grandmont, 1974). But is this really a plausible assumption? As noted, it postulates that price expectations are ‘rigid’ with respect to large variations in current prices and in particular that the price expected for any commodity remains practically unchanged when the current price keeps rising
(falling) beyond a sufficiently high (low) level. It is quite doubtful, however, that expectations would normally display this property. As an expert in the field has pointed out, ‘  p
 rice forecasts are indeed somewhat volatile, and are presumably quite sensitive to the level of current prices’ (Grandmont, 1983: 26). On the other hand, the examples presented in this section show that temporary monetary equilibrium may not exist under such circumstances. The conclusion that has been drawn is that ‘the existence of a
 temporary

equilibrium in which money has positive value is somewhat problematic’
(Grandmont, 1983: 27).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 149
Part D, Section V: Conclusions on the marginal/neoclassical approach, with special emphasis on the labour demand curve and on the investment function.
8.D.20. The studies in the field of temporary equilibrium theory carried out in the
1970s and early 1980s endeavoured to overcome the limitations of the Arrow-Debreu model by remaining inside the neo-Walrasian approach but focusing analysis on economies in which futures markets are limited in number or non-existent, trade takes place sequentially over time, and there isn’t perfect foresight. As we have seen, the models put forward in that period examine the behaviour of economic agents in an arbitrarily chosen ‘initial period’, stress the dependence of agents’ decisions on their subjective price expectations, and analyse the conditions ensuring the existence of general equilibrium on current markets. According to the scholars active in the field, the analysis concerning this isolated period was to be the first step of a more extensive research programme, whose ultimate goal was to model the evolution of the economy as a sequence of temporary equilibria without perfect foresight (Grandmont, 1977: 542–
543; 1989: 299).
However, the simplified exposition presented in this Part indicates that the very first steps of the programme pursued by temporary equilibrium theorists gave rise to serious problems. In particular, our presentation has shown that a major source of difficulties for the treatment of temporary equilibrium in a single market period is precisely the central role the approach must attribute to the subjective price expectations of economic agents. This point has been first illustrated with reference to economies with a numéraire commodity: Section II focused on the case of pure exchange economies and showed that substantial difficulties arise in the determination of households’ behaviour if individual expectations are not sufficiently uniform; Section III addressed the case of economies with production and argued that the divergence of individual expectations creates additional difficulties in the treatment both of the formation of production decisions within firms, and of the financing of production plans (and more generally of demands for loans).
8.D.21. Section IV went on to consider monetary economies and showed that a temporary equilibrium may not exist in those economies if expectations about the future level of money prices are rather sensitive to the level of current prices. However, differently from the cases studied in Section II and III, it is unclear whether the nonexistence of a monetary temporary equilibrium is due to the role of expectations, or to the way the functions of money are formalized in this model. The present textbook does
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 150 not intend to discuss money in any detail, but the quotation from Wicksell in chapter 3,
§3.3.8, permits some simple considerations. Let us take the first example studied in
§8.D.18. This example applies specific hypotheses on utility to the idea of the model that the demand for end-period money balances is determined exclusively by the desire to redistribute consumption across periods; what the example shows is that the wealth effect associated with changes in the purchasing power of the initial money holdings may have so little an influence on the desired redistribution of consumption, as not to correct an initial excess supply of money. But suppose money holdings are also desired because money has a medium-of-exchange function (because goods exchange only against money – no barter –, hence many transactions of goods against money must be carried out by each agent in each period, and the less money one has, the more difficult it is to perform the desired transactions); then the higher the price level expected for period
2 (i.e. the higher the price level observed in period 1), the more people will want to end the first period with a greater amount of money than they started with, in order to have fewer transaction difficulties in period 2, and this will tend to reduce demand for commodities in period 1; while the higher the price level in period 1, the more the agents have transaction difficulties already in that period, which also directly contributes to a decrease of demand for commodities in period 1. This creates a tendency for the demand for commodities to decrease and for the demand for end-period money balances to increase in period 1 as the price level rises, independently of any desire to transfer consumption across periods. Thus, it would seem that the possible non-existence of monetary equilibrium in this model is caused by the unjustified neglect of this very plausible effect of changes in the price level.
But why this neglect? One can only advance plausible conjectures. In all likelihood an important reason was the difficulty with admitting a medium-of-exchange function of money in the picture, dominant by the end of the 1960s, of general equilibrium as established by the auctioneer: in that picture, no money is needed during adjustments, and once equilibrium is reached, people must simply honour their promises of delivery. The abandonment of that picture in favour of a more realistic description of trading processes where money had an indispensable-medium-of-exchange function would have made it difficult to determine the very-short-period behaviour of each agent univocally (differently from long-period analyses, the approach cannot stay content with averages coming out of a multitude of diverse behaviours that are left unanalyzed in their individual, partly accidental, determinants: averaging requires that the analysis not be a very-short-period one): if exchanges can only be of goods against money, then at each moment the money balance held by an agent is a constraint on what the agent can purchase, and the agent’s maximization problem must take this constraint into account,
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 151 with a resulting difficulty with univocally determining the agent’s optimal choices because nothing can be specified without arbitrariness about the possible temporal succession of purchases and sales among which the agent can choose. It would seem, therefore, that the admission of a medium-of-exchange function of money would have perhaps reduced the problems highlighted in Section IV but would have introduced other problems.
8.D.22. The analytical difficulties highlighted in Sections II to IV are the ones that emerged more explicitly in the formal studies of temporary equilibria, and they largely explain why research in the field of temporary equilibrium theory was abandoned about thirty years ago[ 143 ]. Thus they afford some insight into why general equilibrium theorists have since chosen to study economies with sequential trade only under the assumption of correct (or self-fulfilling ) price expectations, for example along the lines indicated by Radner (1972). This assumption of 'perfect foresight' is normally taken for granted in advanced textbooks nowadays (cf., for example, Mas-Colell et al ., 1995: 696).
Basic knowledge of temporary equilibrium models makes us aware of the fact that modern general equilibrium theory can hardly dispense with that assumption.
And yet, the attempt to develop the temporary equilibrium programme was motivated by dissatisfaction precisely with the correct-price-expectations assumption.
The reasons for that dissatisfaction have not disappeared: Radner has repeated in 2008 the same considerations adduced in 1982 by Grandmont as justification for the temporary equilibrium research programme[
144
]; these reasons, one must conclude, are nowadays neglected only because otherwise the entire neo-Walrasian approach would have to be recognized as unsatisfactory.
In conclusion, the marginal/neoclassical/supply-and-demand approach to value and distribution had to abandon the method of long-period positions because unable to
143
Dissatisfaction with some of the other problems of the approach mentioned in §8.D.3 most probably played a role as well, but it is not easily documented, except for Hicks who already by
1965 had to all effects completely rejected the temporary equilibrium approach, see Petri (1991).
144 It is opportune to quote now the lines from Grandmont (1982) mentioned in footnote
58??: “In practice, economic agents have only a very imperfect knowledge of economic laws, and moreover their computing abilities are so limited that they cannot correctly forecast future economic events ... The perfect foresight approach is very useful as a tool for indicative planning or for the description of stationary states, where it seems natural to assume that agents correctly forecast their future environment. It can be employed also in order to check the validity of an economic proposition when agents do not make mistakes. However, it is surely an improper tool for the description of actual economies” (pp. 879-880).
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 152 defend the conception of capital as somehow a single factor of variable 'form', needed to formulate a notion of long-period general equilibrium; but the consequent shift to veryshort-period neo-Walrasian equilibria has compelled the approach to take refuge in notions of intertemporal equilibrium that cannot be seen as attractors for the actual path because when the latter differs from the former, it alters the equilibrium path itself and one no longer knows where the economy will drift. The attempt to minimize this problem, through an implicit faith (§8.B.24) that the traditional marginalist adjustment mechanisms continue to be operative in spite of the rejection of the conception of capital as a single factor, is undermined by the correct analysis of long-period choices: there is no guarantee of a negative interest-elasticity of investment, nor of a downward-sloping demand curve for labour. (More on these two issues will be said in ch. 13.)
Chapter 9 will conclude that these problems are in no way decreased by the introduction of contingent commodities to deal with (environmental) uncertainty; the final conclusion will be negative on the plausibility of the entire supply-and-demand, or marginal, or neoclassical approach to income distribution, employment, and growth.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 153 nel file MicroecsCriticCh8Ch9AppendicesScrap altre appendici
APPENDIX 1: EXAMPLE OF INTERTEMPORAL EQUILIBRIUM WITHOUT
UERRSP
The example is based on the assumption of discrete time, with production in discrete one-year cycles and prices only defined at the beginning of years. (The reader is invited to try and develop on her own an example in continuous time; it is suggested that she should then opt for a representation of production as employing factors that produce the products instantaneously, in a continuous flow, and with all capital goods durable, depreciating – for simplicity – radioactively.)
Assume an economy with two industries. The first industry produces corn by employing seed-corn, labour and tractors. The second industry produces tractors by employing labour alone. Both productive processes take one year. Time is divided into years, productive processes are started at the beginning of a year and the product comes out at the end. Seed-corn is a circulating capital good. Tractors are a durable capital good which lasts two production processes, with constant efficiency. Corn is the numéraire good (I will sometimes change numéraire in the course of the presentation of the example, always corn but of a date depending on analytical convenience).
I proceed to consider the sole aspects that interest us of a hypothetical intertemporal equilibrium over 5 years. The years are divided by dates, with prices defined at the dates; goods are distinguished according to the date at which they are available. Goods are exchanged at the dates and therefore are distinguished by the date at which they become available. The first date is, as usual, date 0, and year 0 goes from date 0 to date 1; year 1 goes from date 1 to date 2, and so on. The last year, year 4, starts at date 4 and with it the economy reaches its end: date 4 is the last one for which prices and exchanges are defined, after that date consumers simply consume what they got and die (this is the well-known consequence of the absurdity of assuming a ‘finite horizon’).
Production processes of year j produce their products at date j+1.
A picture illustrates all this. year 0 1 2 3 4
┼──────┼──────┼──────┼──────┼────── end date 0 1 2 3 4
Equilibrium is established at date 0 simultaneously for all dates. At date 0 there are given endowments of labour, corn, new tractors and one-year-old tractors. As we will
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 154 see, in equilibrium there is production of corn in all first four years, but production of tractors only in year 1. In the last year, year 4, there is no production because by the end of that year everybody will be dead, alas, so production would be useless.
The assumptions on the production side are that the production of tractors requires only labour (one unit of labour produces 20 new tractors), and that the production function of corn is a Cobb-Douglas that yields a net product of corn Y as a function of labour L, seed-corn C and tractors T (it does not matter whether the tractors are new or one-year-old, their efficiency is the same) as follows:
Y = T
1/4 ·C 1/4 ·L 1/2
.
On the consumer side there is no need, for our purposes, to specify utility functions or demand functions: as the Sonnenschein-Mantel-Debreu results show, with opportune assumptions as to differences in utility functions and distribution of endowments one can obtain any demand function, so I will directly assume certain equilibrium aggregate consumer choices. At date 0, out of the initial endowment of corn
(that need not be specified) consumers save 16 units of corn that is offered to firms as seed-corn; labour supply is 16 units; the endowment of tractors, whose services are entirely offered to firms, is 256 tractors; 160 of these are new. These factors are all employed in the production of date-1 corn, so the net product of date-1 corn Y(1) is:
Y(1) = 256 1/4 ·16 1/4 ·16 1/2 = 32
The gross production of corn is therefore 32+16=48; in addition, 160 one-year old tractors are produced as a joint product. The other 96 two-year old tractors are useless and thrown away at no cost. (From now on 'old tractors' means one-year-old tractors.)
The corn marginal product of tractors is
∂Y(1)/∂T = (1/4)·256–3/4·16 1/4 ·16 1/2 = 1/32 = 0.03125.
No production of tractors is performed in year 0, so at date 1 the endowment of tractors is 160 old tractors. The reason why tractors are not produced in year 0 is that their supply price at date 1 is higher than the date-1 demand price for new tractors. The supply price of a date-1 new tractor is 1/20 of the date-1 wage; the latter in undiscounted terms is equal to the corn marginal product of date-0 labour; this is determined by the data we have specified as:
∂Y(1)/∂L = (1/2)·256 1/4 ·16 1/4 ·16 –1/2
= 1 .
Therefore, if I choose as numéraire date-1 corn, the supply price of date-1 new tractors at the given equilibrium prices is 1/20=0.05. For production to be convenient, this supply price must be not higher than the demand price (i.e. the maximum price that a purchaser would be ready to pay), equal to the value, discounted to date 1, of the two corn marginal products that a new tractor earns (or would be able to earn if produced) at dates 2 and 3. This demand price can well be less than the supply price. This will be the
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 155 case if, for example, the factor employments and products in years 1 and 2 are (‘*’ means ‘together with’ and −−> means ‘produce’): in year 1:
160 old tractors * 10 seed-corn * 10 labour −−> 30 date-2 corn (gross);
8 labour −−> 160 date-2 new tractors. in year 2:
160 new tractors * 10 seed-corn * 10 labour −−> 30 date-3 corn (gross) plus 160 date-3 old tractors. (No production of new tractors in year 2; for the reason, see below.)
Because of the freedom with which we can fix consumer choices, the above can be assumed to be the equilibrium productions of years 1 and 2 (and therefore the equilibrium outputs at dates 2 and 3). The proportions in which factors are employed are the same as in year 1, so owing to constant returns to scale the corn marginal product of tractors is again 1/32 both at date 2 and at date 3. To find the demand price of a date-1 new tractor, we must discount these earnings to date 1. I have chosen corn as numéraire, so the rate of discount must be the own interest rate of corn, which is equal to the net marginal product of corn because that is the real rate of interest that firms will be ready to pay for the employment of seed-corn. The net marginal product of corn is the same in all three years:
∂Y/∂C=(1/4)·256 1/4 ·16 –3/4 ·16 1/2 =1/2.
Thus the one-year own interest rate of corn in between date 0 and date 3 is constant at 50%. The demand price for a date-1 new tractor is therefore:
1/32 1/32
––––––– + –––––––– = 0.0347 .
(1+50%) (1+50%)2
It is less than the supply price 0.05. Therefore it is not convenient to produce date-
1 new tractors. Date-1 savings are employed in the direct purchase of corn to be then lent as seed-corn to firms, or lent to firms or to other consumers: either way they earn a cornrate of interest equal to 50%. If one invested one's savings into the purchase of a date-1 new tractor, paying it 0.05, one would earn a rate of return less than 50% because in order to obtain 50% one would have to pay it 0.0347: the rate of return on supply price on the purchase of new tractors would be less than 50%. The price of date-1 old tractors in terms of date-1 corn is determined by arbitrage: it must be such that the rental earned
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 156 by such a tractor at date 2, i.e. 1/32 units of date-2 corn, guarantees a rate of return equal to 50% on this demand price, which is then obtained by discounting 1/32 at the 50% rate: it is 0.020833. Thus in year 1, i.e. for investments starting at date 1, there isn’t a uniform rate of return on supply price, because the year-1 rate of return on seed-corn is 50% while the one on new tractors would be less than 50%; this lower rate of return does not appear explicitly in the equilibrium because there are no new tractors at date 1 and hence it is impossible to invest in their purchase. (Note: new tractors are not produced in year 0 because it is impossible to reach the same rate of return on supply price for them as for corn at date 1 , because of the too low rental they would obtain at date 2 . The decision not to produce them, taken at date 0, depends on prices at two subsequent dates.)
In this equilibrium there is, on the contrary, a uniform rate of return on supply price in year 2: at date 2, the supply price and the demand price of tractors are equal, thus in year 1 there is production of new tractors; these are sold at date 2 and used for the production of date-3 and date-4 corn. At date 2 there are no old tractors, so through a sufficiently low supply the marginal product of new tractors can be such as to render their demand price not less than (and in equilibrium equal to) their supply price. The endogenous determination of their supply makes it possible to obtain the equality of rates of return on supply price for seed-corn and for tractors, the equality that could not be obtained at date 1 because then there already was a given, and large, endowment of tractors, albeit old ones, which rendered the marginal product of tractors too low.
The above is all we need in order to understand why a uniform rate of return on supply price is something different from an equal convenience of all realized investments, and why it may not (and generally will not) exist in the first periods of an intertemporal equilibrium. But for completeness let us now check that the numerical example is acceptable. There are many details of the equilibrium that we have not specified. Some need not be specified, e.g. the consumers’ utility functions or endowments; labour supply, for example, can be assumed rigid, or affected by labour migrations. We only need to check that we have not inadvertently incurred into negative quantities or prices on the markets that we have not explicitly analyzed.
For simplicity it was assumed that factor proportions in the production of date-2 and of date-3 corn were the same as in previous years, hence the marginal product of tractors was again 1/32 and the own interest rate of corn again 50%; therefore, since the supply price of date-2 new tractors, if we now choose as numéraire date-2 corn, is again
0.05, in order for their demand price to equal the supply price the marginal product of tractors in the production process of date-4 corn must be either higher than previously, or discounted at a lower discount rate, or both. For simplicity let us assume that from date 3 to date 4 the corn own rate of interest is still 50%; then we can derive the date-4
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 157 marginal product X of tractors that will permit the equality between demand price and supply price from the following equation:
1/32 X
––––––– + –––––––– = 0.05 .
(1+50%) (1+50%)2
The solution is X = 0.065625: in the production of Y(4), tractors must have a higher marginal product than in the three previous years. We have assumed that production employs 160 tractors and that the net marginal product of seed-corn is 0.5; these data plus the condition that the marginal product of tractors equals X suffice to determine the employments of labour and of seed-corn in the production of Y(4); they can be calculated from the system of two equations:
∂Y(4)/∂T = (1/4)·160 –3/4 ·C 1/4 ·L 1/2 =0.065625
∂Y(4)/∂C=(1/4)·160 1/4 ·C –3/4 ·L 1/2 =1/2.
The result (obtained from Maple) is L=30.43, C=21, and hence Y(4)=42.
The value of date-3 old tractors in terms of date-3 corn is their marginal product disconted to date 3, hence 0.065624/1.5=0.04375. This is different from 0.020833, value of date-1 old tractors in terms of date-1 corn. Own rates of return for tractors and for corn are different.
Let us finally check why in the years after year 1 there is no production of new tractors. Their production in year 2 would yield date-3 new tractors which would be employed for only one year, and their marginal product for a single year (equal to
0.065625) discounted to date 3 (yielding, as we have seen, 0.04375) would not cover the cost of production, equal to one twentieth of the undiscounted date-3 marginal product of labour and therefore equal (in terms of date-3 corn) to 0.05. Production of new tractors during year 3 would obviously make no sense because they would come out at date 4 when there no longer is any production activity.
Now, it is true that at date 3 there is again a non-uniform rate of return on supply price (investment in the purchase of new tractors is less convenient than in the purchase of seed-corn); but the resulting zero production of tractors in year 2 is not due to an insufficiently adjusted composition of the supply of capital goods in previous periods; rather, it reflects a change in what we might call technical conditions: it reflects the fact that, owing to the impending doom, date-3 new tractors can earn a rental for only one period. Similar phenomena might happen for more realistic causes in long-period analyses too, e.g. the gradual exhaustion of a natural resource might make certain capital goods no longer convenient from a certain point on (at that date, one would observe a
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 158 non-uniform rate of return, but the reason is that those capital goods no longer belong to the dominant long-period technique, thus the condition of a uniform rate of return on supply price is not violated, as explained presently). It is not here that the very-shortperiod nature of neo-Walrasian equilibria emerges as the cause of the non-uniform rate of return on supply price. This is confirmed by the fact that the finite horizon is not a necessary element of neo-Walrasian equilibria. The non-uniform rate of return on supply price due to the finite horizon is therefore irrelevant for a comparison with long-period equilibria. The important difference emerges in the fact that tractors are not produced in year 0, owing to their excessive endowment.
This example shows what it means to say that an intertemporal equilibrium, although it guarantees the same effective rate of return on all investments[ 145 ], is not a long-period equilibrium. A long-period position (the general notion, of which neoclassical long-period equilibria are one instance) aims at endogenously determining the composition of capital. If in a long-period position a capital good is not produced, the reason is not that its endowment is excessive, it is that it is not convenient to produce it because it is not required by the long-period cost-minimizing technique; in a fully adjusted long-period position that capital good will not be present at all.
Two concluding observations on aspects of reality that were not included in the example. First, the example was based on a durable capital good. As already noted, even a circulating capital good may fail to be produced in year 0 if it is not perishable (i.e. if it can be stored) and if its endowment at date 0 is so abundant that it is not entirely used in the first year and what is left of it and hence available at date 1 is still so much as to make it not convenient to produce it in year 0. Again, the reason for its non-production in period 0 is that its demand price at date 1 (that is, the rental it would earn at date 2, discounted to date 1) is lower than its supply price at date 1. But a formalization of this case would require the specification of storage possibilities besides production functions, rendering the example less simple.
Second, in this example, where production is assumed to happen in separate yearly production cycles, supply prices can only be defined at the end of the first year; supply prices of the initial endowments of capital goods cannot be defined. But in real
145
In the present example the possibility of changes in the numerical expression of rates of return depending on the numéraire chosen is not highlighted because unnecessary for the purposes of the example and also made difficult by the fact that, except in one year, only corn is produced. However, the interested student will notice that the value of date-3 old tractors relative to date-3 corn is different from the value of date-1 old tractors relative to date-1 corn, and from that she can infer that the two-period own rate of interest of old tractors between dates
1 and 3 is different from the corn own rate; still, investment in either is equally convenient.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 159 economies, production for most capital goods takes a very short time: in order to obtain the same situation as for dates 0 and 1 of this example, the production period should be taken as short as a few days or even shorter; since generally costs of production are not going to change appreciably in a few days, it will be generally legitimate, for practical purposes, to attribute to the initial endowment too of a capital good a supply price, the one corresponding to the minimum cost of production at the initial factor rentals. In order to stress this point, one might have built an example on the assumption of continuous-flow instantaneous production for all products. With such an assumption, the supply prices of capital goods are well defined from the initial moment. However, all capital goods have then to be treated as durable capital goods; it would be very hard to find room for the notion of circulating capital goods. Also, because the analysis must be in continuous time, integrals become necessary to determine demand prices. The analytical complications would be considerable.
********************************************************
APPENDIX 2: TÂTONNEMENT STABILITY
The simplified version, presented as Appendix 3 of ch. 6, of Mandler's (2005) proof of stability of the factor tâtonnement under WAA, for equilibria of constantreturns-to-scale acapitalistic production economies, can be reinterpreted as applying to finite-horizon intertemporal economies with capital goods and vertically integrated firms. The details of the reinterpretation are given here.
First, the equilibrium cannot be interpreted as a perfect-foresight sequential
(Radner) equilibrium; one must assume complete futures markets, for the reason indicated at the end of §8.B.16. Also, the formal analogy with the acapitalistic economy requires that the intertemporal equilibrium extends over a finite number of periods.
Second, all goods and factor services are dated, and the prices that appear in the theorem and its proof are all discounted prices. The m non-produced factors are what in this chapter have been also called ‘original’ factors: initial inventories of consumption goods or of circulating capital goods (the exclusion of joint production means there are only circulating capital goods) produced before the date when the equilibrium's periods start; and types of land and types of labour, in which case they are treated as different factors when in different periods, with the same endowment in the different periods if nothing is altering it, but with a possibly different ‘price’ (of their services, i.e. rental) for each period.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 160
Third, the economy now produces also capital goods, utilized as inputs in subsequent production, but it is assumed that firms can be treated as if vertically integrated, so the production functions leave the production and utilization of intermediate goods implicit: the inputs appearing in the production functions are only
‘original’ (nonproduced) inputs. This means that, if a good is both a consumption good and a capital good (e.g. sugar), the quantity of the good produced in a certain period may well be more than the demand for it that appears in the equations, because the latter demand is only the one coming from consumers, while there may be also production of the good as an input to subsequent production; the latter production remains hidden in the vertically integrated production functions. If these capital goods produced inside the equilibrium are in fact priced and sold because firms are not vertically integrated, still it is unnecessary to make their markets appear explicitly because by assumption quantities produced are always adapted to quantities demanded, so these markets are always in equilibrium, and therefore are irrelevant for the tâtonnement, that needs to act only on nonproduced factor ‘prices’ (rentals).
Fourth, the indexing of factors and of consumption goods by a single index must be understood as a way to renumber factors’ and goods’ double indices as indicated in
§8.B.4: for example, one may list first all factors of date 0, then all factors of date 1, and so on, and number them progressively with a single index.
Fifth, production functions obviously obey some restrictions as to the dating of inputs and outputs: inputs must be applied before the output comes out. The assumption made in this chapter is that all production processes take one period, starting at the beginning of a period and ending at the end of the period. Thus outputs coming out at date t are produced by inputs used in period t-1.
Sixth, the t that appears in the continuous-time dynamics of the tâtonnement does not refer to the dates or periods over which the equilibrium extends, it refers to the imaginary time along which the tâtonnement develops, a time which is assumed to take no time in the actual economy.
It is worth recalling that Mandler’s proof of stability, since it rests on WAA, requires that the observation by Hildenbrand and Grodal (§6.36) does not apply: there must not be two or more ‘original’ factors which yield no direct utility to consumers and whose endowments are therefore entirely supplied whatever their rentals. Now that the
‘original’ factors include the initial endowments of capital goods, it seems rather difficult to exclude such a case.
*************************************************************
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 161
APPENDIX 3. DISCOUNTED AND UNDISCOUNTED PRICES( 146 )
A full understanding of the connection between undiscounted money prices, undiscounted relative prices in terms of a numeraire, and the discounted prices of intertemporal equilibria, is not easy to reach. The following Appendix may help.
Consider two goods, 1 and 2, and two dates, 0 and 1. Let P j t be the money or nominal (undiscounted) price of good j at date t. Let r j
be the own rate of interest of commodity j for loans of that commodity from date 0 to date 1, i.e. the excess over 1 unit of commodity j the lender obtains at date 1 if she lends 1 unit of commodity j at date zero. Let m j
be the money rate of interest implicit in a loan of a commodity against itself yielding an own rate of interest r j
. If a commodity's money price at date zero is 100 and at date 1 it is 120, and if a loan of that commodity yields an own rate of interest of 20%, it must mean that if at date zero one lends one unit of that commodity (therefore for a money value of 100), at date 1 she obtains 1.2 units of that commodity, of money value
144: the money rate of interest on that loan is 44%. Therefore the following relationship holds, where q j
0 is the amount of the commodity lent at date 0:
[A8.3.1] P j
0 q j
0 (1+m j
) = P j
1 [(1+r j
)q j
0 ].
In words: the money value of the amount of the commodity lent, plus the money rate of interest on it, must equal the money value of the amount of the commodity obtained one period later. Eliminating q j
0 one obtains
[A8.3.2] (1+m j
)P j
0
= P j
1
(1+r j
) .
If one defines the rate of inflation of commodity j as φ j
=(P j
1 –P j
0 )/P j
0 , then the above equation can be re-written in the well-known form
[A8.3.3] (1+r j
)=(1+m j
)/(1+φ j
) .
In the absence of arbitrage, it is not guaranteed that money loans will be at the same money rate of interest for all commodities. If the money rate of interest is the same
(and neglecting differences in risk), then arbitrage has no further work to do; a difference in own rates of interest simply compensates for the different evolution of relative prices over time, and it is equally convenient a) to lend a commodity against itself, or b) to exchange it with a second commodity in the first period, lend the second commodity against itself, and then in the second period exchange the second commodity for the first one. Thus the following must hold:
146
In preparing this Appendix I found the first pages of an unpublished 1997 manuscript by
P. Garegnani (a long comment on Hahn (1982), whose contents partially reappear in Garegnani
(2000, 2003)) very useful.
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 162
[A8.3.4] (1+r
1
)/(1+r
2
) = (1+φ
2
)/(1+φ
1
) = (P
2
1 /P
2
0 )/(P
1
1 /P
1
0 ) = (P
2
1 /P
1
1 )/(P
2
0 /P
1
0 ).
For example, if the numbers given above refer to commodity 1 while the money price of commodity 2 increases by 44% from date 0 to date 1, then (1+φ
2
) / (1+φ
1
) = 1.2 and therefore r
2
=0 is the own rate of return on commodity 2 that makes loans in either commodity equally convenient. This equal convenience emerges numerically in a uniform monetary rate of interest, if one looks at the money rate of interest, or in a uniform real rate of interest if one chooses a commodity (or commodity bundle) as numéraire for both periods and therefore assigns undiscounted price 1 to that commodity in both periods, cf. §8.A.14; P. Garegnani has proposed to call such a situation one with unform effective rate of return, the word ‘effective’ intending to stress that the change of relative prices over time does not disturb the equal convenience of all loans, whatever the good lent and the goo in terms of which the repayment is specified. The numerical value of this uniform effective rate of interest depends on the choice of numéraire.
This equality of effective rates of interest does not necessarily obtain in the real world; it will be the task of arbitrageurs to bring it about. Therefore the equality sign in equation [A8.3.4] does not derive from definitions but from the assumption that arbitrage opportunities have disappeared. This task of arbitrageurs is made less clear in the definition of own interest rates on the basis of discounted prices p j t , the prices in terms of which intertemporal general equilibrium is determined. By definition,
[A8.3.5] (1+r j t
)=p j t
/p j t+1
.
Therefore by definition
[A8.3.6] (1+r
1
) / (1+r
2
) = (p
2
1
/p
2
0
) / (p
1
1
/p
1
0
) = (p
2
1
/p
1
1
)/(p
2
0
/p
1
0
).
This equation is identical to equation 4 of this Appendix, except for the fact that here the prices are discounted prices. But if exchange rates between contemporary goods are given, it cannot make any difference whether prices are discounted or not; risk apart, the discount rate cannot but be the same; so relative prices must be the same, whether discounted or undiscounted. In other words, it must be
[A8.3.7] P i t
/P j t
= p i t
/p j t
.
Therefore equations 4 and 6 are in fact identical; but in equation 6 the equality sign derives from the definition of own rate of return. The point I am trying to clarify here is that when one considers discounted intertemporal prices, arbitrageurs are implicitly assumed to have already completed their task of making exchange rates between goods mutually compatible, in the sense that if the relative price (the exchange rate) of good 1 in terms of good 2 is

and if the relative price of good 2 in terms of good
3 is

, then the relative price of good 1 in terms of good 3 is
 · 
. The exchange rate between a good to-day and the same good tomorrow is simply 1 + the own rate of interest of that good, and therefore the arbitrage that renders the rates of interest
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 163 effectively equal is submerged, so to speak, into the more general arbitrage that renders indirect exchange just as convenient as direct exchange.
If we set p j
0 =P j
0 , all j (a natural assumption, since prices of the initial date do not need discounting), then we can express the relationship between discounted and undiscounted price of a good as follows. Since
[A8.3.8] P j
0 = P j
1 (1+r j
)/(1+m j
) (from equation 2 of this Appendix) we obtain
[A8.3.9] p j
1 =p j
0 /(1+r j
)=P j
0 /(1+r j
)= P j
1 /(1+m j
) .
Let us now consider the difference between the representation of costs of production in terms of discounted and of undiscounted prices. Suppose that both goods 1 and 2 are produced by labour and by themselves (as inputs they are circulating capital goods, and production lasts one period), according to fixed-coefficients methods: a
11
 a
21

L
1

1 unit of good 1 a
12
 a
22

L
2

1 unit of good 2.
Suppose that inputs 1 and 2 must be bought by firms, therefore paid at date 0, while wages are paid in arrears, therefore paid at date 1. We indicate the undiscounted money wage rate as W 1 , the discounted wage rate as w. Let the money rate of interest m be uniform. In undiscounted terms the conditions "price = historical cost plus rate of interest" are
[A8.3.10] (P
1
0 a
11
+P
2
0 a
21
)(1+m) + W
1
L
1
= P
1
1
[A8.3.11] (P
1
0 a
12
+P
2
0 a
22
)(1+m) + W 1 L
2
= P
2
1 .
In discounted prices the same conditions are (where we may assume p
1
0
=P
1
0
, p
2
0
=P
2
0
):
[A8.3.12] p
1
0 a
11
+p
2
0 a
21
+wL
1
= p
1
1
[A8.3.13] p
1
0 a
12
+p
2
0 a
22
+wL
2
= p
2
1 .
(Notice the formal similarity with the conditions price=cost of the acapitalistic production economy.) The uniform effective rate of return depends on the numeraire adopted. Suppose we choose good 1 as numeraire for the undiscounted prices, that is, we divide the undiscounted prices of date 0 by P
1
0
and the undiscounted prices of date 1 by
P
1
1 and we indicate these new prices as
 j t . So

1
0 =1,

2
0 =P
2
0 /P
1
0 =p
2
0 /p
1
0 by equation
[A8.3.7],

1
1 =1,

2
1 =P
2
1 /P
1
1 =p
2
1 /p
1
1 . (Remember that
 j t is the relative price of good j of date t in terms of the numeraire good of that same date. We see now that it makes no difference whether this relative price is derived from undiscounted or from discounted prices.) Let us furthermore indicate with ω=W 1 /P
1
1 the real wage rate in terms of good 1 of date 1. If we divide both sides of equation [A8.3.10] by P
1
0
, remembering that
P
1
1 =(1+m)P
1
0 /(1+r
1
) and therefore W 1 /P
1
0 =ω(1+m)/(1+r
1
), we obtain the expression
(a
11
+

2
0 a
21
)(1+m) + ωL
1
(1+m)/(1+r
1
) = (1+m)/(1+r
1
), that is, simplifying and multiplying by (1+r
1
):
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 164
[A8.3.14] (a
11
+

2
0 a
21
)(1+r
1
) + ωL
1
= 1.
This expression shows that if one re-writes equation 10 in real terms with good 1 as numeraire, then the rate of interest is r
1
, the own rate of interest of good 1. The same obtains with equation 11; the derivation is left to the reader as an exercise, but we write down the result:
[A8.3.15] (a
12
+

2
0 a
22
)(1+r
1
) + ωL
2
=

2
1 .
Equations 14 and 15 are the form taken by the condition "price = historical cost plus rate of interest" when prices are expressed as relative undiscounted prices in terms of a numeraire, as in the usual equations of prices of production, except that relative prices of same-date goods are not assumed to remain unchanged over time.
It is easy to show that equations 14 and 15 are respectively equivalent to equations 12 and 13. I show it for equations 14 and 12. Divide both sides of equation 12 by p
1
0 ; multiply both sides by (1+r
1
); notice that equation [11.2] in the main text implies that w/p
1
0 =ω/(1+r
1
); you obtain equation 14.
Had we chosen good 2 as numeraire, then the rate of interest would have turned out to be r
2
, and the proof is again left to the reader as an Exercise .
Exercise : What is the relationship between W 1 and w? Check that the relationship derived from equation [A8.3.9] coincides with the relationship derived from the definition of ω and equation [A8.3.2].
Let us now notice that equation 4 and the definition of
 j t
imply
[A8.3.16]

2
1 /

2
0 = (1+r
1
)/(1+r
2
) and therefore

2
0 =

2
1 (1+r
2
)/(1+r
1
).
We can therefore re-write equations 14 and 15 as
[A8.3.17] (1+r
1
)a
11
+(1+r
2
)

2
1 a
21
+ ωL
1
= 1.
[A8.3.18] (1+r
1
)a
12
+(1+r
2
)

2
1 a
22
+ ωL
2
=

2
1
.
These equations express the condition "price = cost plus rate of interest" not on the historical costs but on the replacement costs of the capital goods. The prices are now the same on the left-hand and on the right-hand side of the equations, like in the usual equations of prices of production. The fact that relative prices of same-date goods have not remained constant over time shows up now in the two different rates of interest, which however, as we know, indicate in fact a uniform effective rate of return.
Garegnani concludes:
"Thus we can state that in each industry where in equilibrium production is carried out – i.e. where ... an inequality sign does not replace the equality sign in the
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 165 price equations – there will be a uniform rate of return on the prices of all the capital goods employed there, whether those prices are conceived as historical prices or as replacement prices."
**************************************************************
APPENDIX 5 Garegnani (2000, 2003, 2005) on intertemporal equilibria
Assume a single consumption good produced by heterogeneous circulating capital and labour, in an intertemporal economy with three dates 0, 1 and 2, and hence production during periods 0 and 1. Demand for date-2 consumption goods requires production of capital goods during period 0, to be sold at date 1 to the firms that, with them and date-1 labour, will produce the quantity demanded of date-2 consumption good. Suppose the auctioneer has just achieved equilibrium when she is informed of a change in tastes causing a decrease of date-1 consumption demand and increase of date-1 savings (and hence an increase of date-2 consumption demand, because now we are not questioning the usual way to determine consumers’ incomes and demands). At the old equilibrium prices, the excess demand for date-2 consumption good causes an excess demand for date-1 labour and date-1 capital goods, but labour is fully employed, so the only way to satisfy the increased demand for date-2 consumption is by coupling date-1 labour with different capital goods, capable of raising the output from the given amount of date-1 labour, and whose production is made possible by the date-0 factors left free by the reduced demand for date-1 consumption. Garegnani has argued that the required change in date-1 production methods is implicitly supposed by neoclassical theory to be accomplished by the decrease, due to the excess demand for date-2 consumption, of the interest rate implicit in the intertemporal discounted prices of the consumption good; that is, technical choice is supposed to behave essentially in the same way as in the one-good economy producing corn with corn and labour; the ‘greater quantity’ of capital produced in period 0 is supposed to absorb the period-0 resources no longer employed to produce date-1 consumption. But, Garegnani has argued, for reasons similar to those giving rise to reverse capital deepening in long-period analyses, in the presence of heterogeneous capital the lower interest rates may cause a change in production methods different from the required one, hence the possibility arises that the increased savings be not absorbed, causing instabilities analogous to the ones arising in long-period analysis owing to reverse capital deepening. An assumption of productions determined by demands would only transfer the problem to the markets of date-0 factors: the assumed initial excess of
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F Petri Microeconomics for the critical mind ch 8 intertemporal 17/04/2020 p. 166 date-1 savings becomes an initial insufficient demand for date-0 factors if production of date-1 capital goods is assumed adapted to the demand for them, because the latter has increased but not as much as savings (the increased value of demand for date-2 consumption includes also the value of an increased demand for date-1 labour); if then the reduction of the interest rates does not cause the ‘right’ change in production methods, the excess supply of date-0 factors does not disappear, and the price of some of them may go to zero[ 147 ]. The debate on this argument has been inconclusive so far. The latest contributions are Garegnani (2005, 2005b, 2012) , Mandler (2005b), Schefold
(2008), Petri (2011b). Garegnani’s argument, if correct, would imply that the vertically integrated intertemporal production functions with only ‘original’ factors as inputs do not behave like standard production functions; so far the issue has not been explored.
147
Mandler’s ‘factor tâtonnement’ excludes such an outcome because of the assumption of strictly monotonic preferences, which implies that all factor services, including the ones of capital goods, yield positive utility to the representative consumer. The implications, for
Mandler’s stability proof, of the Hildenbrand-Grodal observation on WAA (§6.36) have not been clarified yet. Garegnani has repeatedly insisted that not only instability, but also a nonnegligible likelihood of equilibria with zero wages or zero other incomes would undermine the plausibility of the marginal/neoclassical approach to income distribution.
166