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SOME INTRODUCTORY NOTES ON THE THEORY OF GROWTH
by
Giacomo Costa,
Università di Pisa,
e-mail: <costag@specon.unipi.it>
[Last revision: March 2004]
I reproduce here some (but not all) of the observations and derivations I presented in class (Siena,
March 11-13, 2003) some that I omitted having run short of time, and intersperse the exposition
with exercises, references and other bits of information about old neoclassical growth theory. Our
reference book will be R. Solow, Growth theory, an exposition, Second edition , Oxford Univ.
Press, 2000, whose origins are in Warwick (for the first part) and in Siena (for the more exciting
second part dealing with endogenous growth: see his 1992 Siena Lectures!) I will refer to this book
as “Solow”. We will be mainly concerned with the first chapter of the Second Part, a chapter that is
devoted to a careful if synthetic review of “old” neo-classical theory”. Starred sections in the
following are optional, or for reference purposes only.
1. The Keynesian precursors of growth theory: Domar, Harrod, and Kaldor.
1.1.Domar: the dual role of investment: on the one hand, investment contributes to
(current) aggregate demand and therefore helps to stabilize the economy; on the other
hand, it adds to productive capacity, thus making future investment less desirable! In
particular, a constant level of investment is not maintainable: for, through the multiplier,
it implies a constant level of output, thus making the increasing capital stock more and
more redundant! Only if investment grows over time at a specific rate will the additional
capacity that is continuously being added not be made superfluous.
The idea in its simplest formulation: Let there be a “normal”, or “desired”, capital output
ratio C, let s be the average and marginal savings propensity.
Supply side: Yc = K/C = I/C (this the dangerously useful, capacity enhancing side of
I);
demand side: Yd = (1/s)I (this is the short run contribution of I to aggregate demand
and equilibrium output);
equality of supply and demand requires Yc = Yd , i.e., I/I = s/C (= Y/Y.)
No mechanism inherent in a market economy, according to Domar, can be expected to
make investment and income grow at this rate. What if Yd > Yc ? The pressure on
capacity would make firms to wish to invest more, but that would mean, at a faster rate
than s/C. If the inequality wee reversed, then the desired proportional rate of investment
would be smaller than s/C. This is the famous instability proposition, present in one form
or another (but never in a clear, definite form) in the writings of both Domar and Harrod.
The Domar quotation (in Evsey Domar, “Expansion and employment”, A.E.R. 1947) from Lewis
Carroll, Through the looking glass (the sequel to Alice in Wonderland):
“A slow sort of country,” said the Queen. ”Now here, you see, it takes all the running you can do, to
keep in the same place. If you want to get somewhere else, you must run at least twice as fast as
that.”
1.2. Harrod: his concept of a warranted rate of growth is similar to Domar’s equilibrium
rate of growth. He indicates the desired capital-output ratio by Cr to emphasise its
dependence on r, the rate of interest. He introduced the idea of Harrod neutral technical
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progress, that should leave the competitive distributive shares constant over time at a
constant rate of interest, as can happen iff technical progress is labour increasing.
Assuming a constant rate of labour-increasing technical progress, he was able to
introduce the very important concept of a natural rate of growth, the analogous for growth
theory of full employment output. Does the warranted rate of growth equal the natural
rate of growth? Perhaps there is a (level of the) rate of interest at which they are. But if
the usual Keynesian monetary causes block or limit the variations of the rate of interest,
the two will not be equal. What will then happen to the actual rate of growth? It is by no
means equal to either the warranted, or the natural rate of growth. Indeed, it is likely to be
prey to an instability principle: in particular, if the warranted rate of growth were to be
larger than the natural rate of growth (too much savings) then the economy is doomed to
grow very little (at less than the natural rate of growth) or not at all. So, Harrod attempted
to provide a wide-ranging extension of the Keynesian ideas to the long run. Some of the
mechanics of his model (but not the crucial monetary ingredients) are discussed in the
Swan 1956 paper.
1.3. Harrod-Domar. It is interesting to compare their works. The main ideas are similar,
but there are differences. Harrod is more systematic, and only he produced a clear
distinction between warranted and natural rate of growth. He was able to do so since he
had an analysis of technical progress. He provided such an analysis because he was able
to rely on the neoclassical theory of production and distribution! See the important papers
by J. Robinson, “The classification of inventions”, REStud 1938, and by H. Uzawa
“Neutral inventions and the stability of growth equilibrium”, REStud 1961 (both
reprinted in J. Stiglitz and H. Uzawa, Readings in the modern theory of economic
growth, The MIT Press, 1969, one of the best sourcebooks on exogenous growth theory.)
1.4. Nicholas Kaldor. What are the facts of growth? A tentative but very influential
answer was provided in the mid-fifties by Kaldor. See the discussion of Kaldor’s stylized
facts in Solow, pp. 2-7, pp.181-2. Certain trends can be identified considering the
historical evolution of the main industrialized economies: the rate of profit not falling, the
real wage rate rising , productivity rising, the capital-output ratio more or less constant,
the capital to labour ratio rising; differences in the rates of growth experienced by the
same economy during its history and among different economies at the same time.
Of course it is the task of growth students to search for and eventually to provide new
facts, at least new “stylized facts” in the style of Kaldor. Has the recent, huge literature on
the “convergence” problem succeeded in this? In his Siena Lectures Solow had some
interesting comments on the new literature, but he left them out of his 2000 book.
2. The production function
2.1. Intensive form. Associated to a constant returns to scale (c.r.s) production function
(1)
Y = F(K, AN),
F(K , AN) = F(K, AN), all , K, AN,
is the so called production function in intensive form,
(1’)
y  Y/AN = F(K/AN, 1)  f(k),
where by definition k  K/AN. We can re-write (1’) as
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F(K, AN) = ANF(K/AN, 1),
an equality which is an identity in K, AN, not an equation. The equality is therefore preserved under
derivation with respect to K:
FK(K, AN) = AN F1(K/AN, 1). d(K/AN)/dK = AN F1(K/AN, 1).(1/AN) = f’(k),
where the last equality reflects the fact that by (1’) the partial derivative of F(K/AN, 1) with respect
to its first argument, F1(K/AN, 1) is f’(k) by the very definition of f(k). Similarly, it can be proved
(an easy exercise!) that FKK = f’’(k).
2.2.Euler and the regime of returns . Euler’s theorem applied to 1rst degree homogeneous
production functions gives
(E0)
FK(K, AN).K + FAN(K, AN).AN = F(K, AN) = Y,
i.e., competitive factor rewards exhaust output: there are no net profits (or losses) from production
when output and factor markets are competitive.
More generally, if F is hom. of degree h, so that
(E1) F(K, AN) = hF(K, AN),
then differentiating with respect to 
FK(K, AN).K + FAN(K, AN).AN = h h-1 F(K, AN),
and setting  = 1 (as we are free to do since these equalities are identities in K, AN, and )
(E2)
FK(K, AN).K + FAN(K, AN).AN = hF(K, AN).
This says that under the specific form of increasing returns to scale represented by (E1), with h > 1,
rewarding factors at the marginal products (i.e., competitive factor rewarding) is impossible, for it
would require over-exhaustion of output.
Notice that we have not proved all of Euler’s theorem: in fact, the hard part of the theorem (which
affirms the equivalence of E1 and E2) lies in going from (E2) to (E1)! This is so important for
economists that we will do it.
Since (E2) is an identity in K, AN, we can write it as
FK(K, AN). K + FAN(K, AN). AN = hF(K, AN).
Now (this is the trick invented by Euler) divide both sides by the positive quantity (h+1)
(E2’) (1/h) [FK(K, AN).K + FAN(K, AN).AN] = (h/(h+1))F(K, AN).
Now consider the ratio F(K, AN)/h. Its derivative with respect to  is but the l.h.s minus the
r.h.s. of (E2’). Therefore, its is zero for all positive . But then the ratio F(K, AN)/h is
independent of  and can be evaluated at  = 1:
F(K, AN)/h = a constant = F(K, AN)1h = F(K, AN),
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and finally
F(K, AN) = hF(K, AN).
Another finding by Euler is that if F (K, AN) is hom. of degree h, its first partial derivatives are
hom. of degree h-1.
Exercise: would you be able to show it? (It is easy).
Exercise: solve the partial differential equation xfx(x, y) – yfy(x, y) = 0.
2.3 The relation between K/Y and Y/AN. Notice that from 1rst degree homogeneity of F in its two
arguments we get from (1)
1 = F(K/Y, AN/Y),
and from this (and the fact that F is increasing in both its arguments) it is easy to deduce that Y/AN,
the productivity of effective labour, is an increasing function of the capital coefficient, K/Y:
Y/AN = (K/Y), ’ > 0.
The economic meaning of this is that, given the level of technology, with more capital actual and
effective labour are both more productive. Notice that growth theorists use the existence of the 
function all the time, to them it is an obvious feature of production conditions (1).
Exercise: Give a more precise argument by using the implicit function theorem (more precisely
called “Dini’s theorem” in Italy) and compute ’.
Can you express the competitive capital share and the competitive wage bill in the intensive
notation?
2.4. Increasing returns to scale
Definition: a production function F exhibits increasing returns to scale if for all  > 0,
(E3) F(K, AN) > F(K, AN).
If F satisfies (E1), with h > 1, then it is i.r.s. If it is i.r.s. (i.e., if it satisfy (E3)), does it have to
satisfy (E1)? Hint: look at the Swan production function (SWN0) in section 4 below.
Are there any scientific laws that dictate a specific regime of returns? No. Are there economic laws
to this effect? No. See the highly interesting methodological paper on the “laws” of production by
Karl Menger (son of Carl Menger, the famous Austrian economist), Selected papers in logic and
foundations, didactics, economics, Dordrecht, 1979. Then why are we assuming c.r.s. all the time?
Well, not all the time: see most theory of endogenous growth! Apart from that, the reason is
compatibility with the assumption of perfect competition, which is lost if we assume i.r.s.
2.5. The factor price frontier. Let r = q/p be the real rental rate of a unit of capital (also called the
rate of profit on capital), w/p the real wage of a unit of effective labour. Competitive factor
rewarding requires
(E4) r = FK = f’(k),
the rate of interest (and of profit) equals the marginal product of capital. It also requires
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(E5) w/p = FAN = f(k) – f’(k)k,
where the last equality is just a transcription of (E0). Eqs (E4), (E5) give the famous factor-price
frontier in parametric form, k being the parameter. An easy exercise is to show that they imply
dr/d(w/p) < 0.
Exercise: notice that w/p is here the relative price of a unit of effective labour, not of labour.
Suppose that A = A(t) is an exponential, and consider a situation where k is constant over time.
Then what happens to the real wage rate, i.e. to the relative price of one unit of labour?
3. The type of technical progress:
3.1. Economic grammar. Why should technical progress be (as it usually is assumed) labouraugmenting? In other words, why should the equality
F(K(t), N(t), t) = F(K(t), A(t)N(t))
necessarily hold for all t?
Surprising and perhaps disappointing answer: for no economic reasons at all; rather, it is the only
type of technical progress arithmetically consistent with steady state growth. See Solow, pp.109113. He provides a different and a little more laborious, but equally instructive proof in the first part
of the book, pp. 32-34. However, it can be noticed that his argument is not a complete proof: for he
presupposes that technical progress should take a factor augmenting form, then shows that capital
augmentation must be zero (or, the production function must be Cobb-Douglas). But labour
augmenting is not the only form that technical progress may take.
Not so easy Exercise: would you be able to provide a (real, or more general) proof?
Easier Exercise: use his method of proof to show that when technical progress takes the form
F(K, N, t) = A(t)G(K, N),
then steady state growth is only possible if G(K, L) is a Cobb-Douglas.
Exercise. Labour augmenting technical progress is also referred to as “Harrod-neutral”. Show that if
the production function is labour augmenting, the marginal product of capital depends only on the
K/Y ratio (and not directly on time). Very easy. The converse is also true: if the marginal product of
capital depends only on the K/Y ratio, then technical progress is labour augmenting. This converse
proposition however is not easy to prove.
3.2. Hicks neutral technical progress. Iff
(1’’) F(K(t), N(t), t) = A(t)F(K(t), N(t)), with of course A(t) an increasing function of time,
we have Hicks-neutral technical progress. In his 1956 paper, Swan assumed Hicks-neutral technical
progress. (Check that his equation (1’) comes from our (1’’) above.) But he also assumed a CobbDouglas production function, and it is easy to show that (Exercise!) under a Cobb-Douglas
production function Hicks-neutral technical progress, and any kind of factor augmenting technical
progress as well, can be re-described as labour-augmenting.
Exercise: verify that under Hicks neutrality the ratio of marginal products of the two factors remains
unchanged at a constant capital to labour ratio.
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4. Returns to scale and endogenous growth: the Swan Proposition
4.1. The Swan Proposition. Does the assumption of increasing returns to scale per se provide the
bridge between exogenous and endogenous growth theory? Does it make impossible a steady state
rate of growth (i.e., where Y and K grow at the same rate) determined exclusively by technological
and demographic parameters? Not necessarily, argues Solow, and he offers a specific functional
form. However, as he himself argues (p. 113) in commenting his chosen example, the following
(SWN0) Y = F(K, ANq), with F 1rst deg. hom. in its two arguments,
is the only (and therefore, the most general) type of production function compatible with steady
state growth. This is what says the so called (by me at least…) Swan Proposition, to be found in T.
W. Swan, “Golden ages and production functions”, pp.203-218 of A. Sen, Growth economics;
selected readings, Penguin, London, 1970. Notice that the Swan Proposition answers at once to two
distinct questions: the type of technical progress and the specific regimes of returns to scale that are
consistent with steady state growth. Remember that while a c.r.s. production function is first degree
homogeneous in factor quantities, an increasing return to scale production need not be
homogeneous (of some degree higher than 1) in factor quantities.
Exercise: is the function (SWN0) homogenous (of some degree…) in factor quantities?
Exercise: compute the rate of steady state grow for (SWN0): (the answer is in Solow!)
Exercise: take q>1 in (SWN0). Is this function homogeneous of some degree (higher than 1) in
factor quantities? Would there be over-exhaustion of product under competitive factor
remuneration?
4.2. A Swan model. Let’s take an example of the Swan production function above, the
Cobb-Douglas special case
Y = K (AN) , + 1, 0 <  <1,
and let us denote, following Solow, by g the rate of growth of output. By logarithmic
differentiation with respect to time we find
g = .(rate of growth of K) + (a + n),
the Swan growth equation. In a steady state, the rates of growth of output and of the
reproducible factors of production are equal (this the definition of a steady state!)
(SWN1)
g = .g + (a+n) ;
(1-)g = (a+n),
if + =1, (constant returns to scale),
g = a+n,
the steady state rate of growth equals the natural rate of growth. In particular, the steady
state rate of growth is independent of the savings ratio. These are, of course, the main
results of the “old” neo-classical theory. (They are an equivalent statement to Solow’s
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“conclusion number one”). If +>1 (increasing returns to scale) we obtain from
(SWN1)
g = (a+n)/(1-) > (a+n),
the rate of growth of output is larger than the natural rate of growth. Still, g is determined
only by technological and demographic factors, and therefore, by Solow’s own
definitions, this is still exogenous growth. The condition g > (a+n) is necessary, but not
sufficient, for endogenous growth.
Exercise: in section VI of his 1956 article “A contribution to the theory of economic
growth”, Solow considered the steady state growth of capital and output for an economy
with Hicks-neutral technical progress. He quickly moved to the Cobb-Douglas production
function
Y = egtKa(Loent)b, a+b = 1,
solved the corresponding growth equation
dK/dt = s egtKa(Loent)b
and found
K(t) = [Ko –(bs/(nb+g))Lob + (bs/(nb+g))Lobe(nb+g)t]1/b
and argued that “…in the long run the capital stock increases at the relative rate
(n+g/b)…the eventual rate of increase of real output is (n+ag/b)…the capital coefficient
K/Y eventually grows at the rate n+(g/b)-n-(ag/b) = g.”
Yet is easy to show (do so!) that if K = Koeft, and dK/dt = sY(t), with s constant over
time, then Y(t) is bound to grow at the same rate f as K…Did Robert Solow, the foremost
growth theorist, make an arithmetical error? (Remember: anybody can make an
arithmetical error…) Use the appropriate version of the Swan growth equation above to
give your answer.
5. Essentiality of labour
Definition. Given a production function y = f(x1, x2,…,xi,…,xn), factor i is essential if f((x1, x2,…,
xi-1, 0 , xi+1,…xn) = 0 identically in x1, x2,…, xi-1, xi+1,…xn.
Example: under a constant returns to scale Cobb-Douglas production function, both capital and
labour are essential.
(Mini-)Theorem: If the production function (1) is continuous and constant returns to
scale, and if labour is essential, then
Lim k-> f(k)/k = 0.
Proof:
F(1, AN) = ANF(1/AN, 1) = F(1/AN, 1)/(1/AN) = f(x)/x, x = 1/AN.
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Lim AN->0F(1, AN) = F(1, 0) = 0 = lim x-> f(x)/x .
(Mini-)corollary: Under the same hypotheses, lim x-> f’(x) = 0.
Exercise: prove the mini-corollary.
Question: Is capital essential under the production function
Y = AN(1- e –aK/AN)? Use this function to discuss the following question: does the
essentiality of capital imply some (positive) lower bound for f’(0)?
See E. Burmeister and R. Dobell, Mathematical theories of economic growth, Chapter 2,
on all this.
Exercise: consider the production function Y = Na(i (ki1-a), 0<a<1, and the summation
index i runs from 1 to n, n being the number of capital goods employed with labour to
produce the final good. (i) Check that this is c.r.s in its n+1 factors. (ii) Check that labour
is essential, and that no capital good is essential. (iii) Yet, if all capital goods are
withdrawn from the production process, no output will be forthcoming. So the n capital
goods, as members of a proper subset of factors, are collectively essential: at least one of
them must be available in positive quantity for production to be positive. Find a
production function where no factor is (individually) essential, and a (proper) subset of
the factors are collectively essential.
Find a production function where the first m factors, and the last n-m factors are two
proper non-empty subsets of collectively essential factors of production.
6. Solow’s “general point about endogenous growth models” (pp. 144-7)
Start from a constant returns to scale production function Y = F(K, AN), and find its Swan equation
(SLW1)
g = sFK - (n+a),
where  = KFK/Y, the competitive share of profits. I think the argument as it stands cannot be
supported, still the main idea that Solow wanted to put across is right. We can rewrite (SLW1) as
(SLW2)
g-(n+a) = [sY/K – (a+n)],
which says that the growth rate of output will be higher than the natural rate if the growth rate of
capital is higher than the natural rate. This is, or is an implication of, the Swan equation. But in
steady state, g = sY/K, and (SLW2) implies that this can only be true if  = 1. Thus, if the
production function is such that there is an upper bound on KFK/Y lower than 1 (as in the CobbDouglass case), then there can be no s.s. rate of growth higher than the natural rate of growth. Other
constant returns production functions, however, are not so restrictive, and the least upper bound on
the competitive share of profits is 1.
Then growth at a s.s. rate higher than the natural one will be possible, at least asymptotically.
Example: following Solow, p. 146, take F(K, AN) = K + K (AN)1-,  > 0, 0<<1:
make sure you understand and are able to show that this is a c.r.s. production function, with
essential capital, inessential labour. Check that
FK =  + K-1 (AN)1-  ,
KFK/Y = [K+ K (AN)1-]/[K+ K (AN)1- ]  1,
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lim K-> (KFK/Y) = 1.
Then there can be a s.s. rate of growth higher than the natural rate, at least asymptotically.
Can you find constant returns to scale production function such that FK is not bounded away from
zero, and still the competitive share of profits tends to 1 for increasing K?
Exercise: study the possibility of a s.s. rate of growth higher than the natural one assuming a C.E.S
production function (see Solow 1956 and in greater detail Burmeister and Dobell, Mathematical
theories of economic growth, 1970, p.59-60, exercises 10-14: they are very instructive exercises to
do.)
7. The intertemporal maximization problem
The intertemporal utility functional is the integral, from t = 0 to t = , of the current total utility
flow u(c(t))N(t) discounted at t = 0 using the social discount factor e(-t),  > 0.


0
e   .t u(c(t ))N (t )dt
The utility integral should be maximized subject to the initial conditions K = K(0), AN = A(0)N(0),
and to the basic constraint
(0)
C + S = C + I = Y = cN + dK/dt,
and another boundary condition to be discussed later (the “Transversality condition”.)
The intertemporal utility functional is a sort of intergenerational welfare function. It raises many
issues, most of which we cannot even begin to discuss.
Why should  be positive rather than zero, as Ramsey wanted it to be? (To him a positive discount
rate was just “a weakness of imagination” unjustifiable on ethical grounds.1)
Why should the integrand be e-t u(c(t))N(t) rather than e-t u(c(t))? Notice that while Jeremy
Bentham and the Pope (and unlikely but not impossible alliance) would both favour the former, any
individualist thinker, utilitarian or not, would certainly (as I would) prefer the second. It is a good
exercise to carefully observe what difference it makes to the solution to change from one to the
other.
What are the ordinal properties of a utility functional as the above? Impatience;
independence; stationarity; homogeneity. See for example Hicks, Capital and Growth, cap.
XXI. Hicks gives an interesting discussion of the properties of intertemporal utility
functions in the light of a paper, fundamental, but really tough, by T. Koopmans,
“Stationary ordinal utility and impatience”, Econometrica, 1960. A more recent, and not
too hard to follow, discussion of the same problems is in Mas-Collel, Whinston e Green,
Microeconomic Theory, 1995, pp. 733-736.
Another friend of Keynes’, Roy Harrod, thought that pure time preference was “a human infirmity, probably stronger
in primitive than in civilized man.” See his Towards a dynamic economics, p. 37. Of course one may disagree with
Harrod’s sententious puritanism. Keynes, for example, thought that civilized man had gone too far in deferring
enjoyment!
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Solow discusses both the general method to solve this problem and some properties of the solution
to a special case of our problem, which results when we assume a Cobb-Douglas production
function, and a current utility function of the type
u(c(t)) = (c(t)1- -1)/(1-), where  > 0.
The derivative of this utility function with respect to its argument c(t) is u’(c(t)) = c(t)- .
It is easy to check that u’’< 0, diminishing marginal utility.
The elasticity of u’(c) with respect to c is simply  >0. This is not the only interpretation
of , however: see Solow, p. 114. Let me explain by taking an inevitably round-about
way. In general, given the utility function U = U(c(1), c(2)), and letting MRS =
dc(2)/dc(1) be its associated marginal rate of substitution, the elasticity of substitution of
U is defined as
[MRS/(c(2)/c(1))]d(c(2)/c(1))/dMRS = dlog(c(2)/c(1))/dlogMRS,
i.e. the elasticity of the c(2)/c(1) ratio with respect to the MRS.
In particular, if U(c(1), c(2)) = [(c(1)1- -1)/(1-)] + [(c(2)1- -1)/(1-)],
the elasticity of substitution turns out to be = (1/)! (Check this.)
However our utility functional is not a mere sum of temporary utilities, it is a discounted
sum of them! We then need to consider the elasticity of substitution between c(t) e c(t+h),
and take its limit as h->0. It turns out that this limit is, again, 1/ .( See Blanchard e
Fischer, Lectures on Macroeconomics, The M.I.T. Press, 1989, pp. 40-41, where
unfortunately the second last formula of p. 40 is clearly wrong, and, unfortunately for the
mathematically lazy, the limit operation is not carried out.
8. A recipe for the solution
I will repeat here the general method indicated by Solow to solve intertemporal maximization
problems:
1) Form the so-called current value Hamiltonian
H = u(c)N + p[F(K, AN)- Nc],
a sort of current national income in utility terms: u(c(t))N(t) is the current utility of total
consumption, and, since p(t) is the utility price of a unit of commodity devoted at time t
to augmenting the capital stock), p(F – cN) is the utility of investment.
H must be maximized with respect to c(t), for all t. In an interior maximum,
(2)
p(t) = u’(c(t)) = c(t)-,
the marginal value of investment (its utility price p(t)) should equal the marginal utility of
consumption.
2) Introduce the so-called “co-state equation”,
(3)
dp/dt = p( - FK),
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this is called “Fisher equation” by Solow, perhaps “Ramsey-Fisher” would be more
appropriate. It is somewhat similar to the Fisher equation which relates the money and the
real interest rates and the rate of inflation, but, unlike it, it is an optimizing condition, not
a definitional identity. I will give an euristic derivation of this equation in the next
section. For the moment, accept it on faith.
3) The third component of the “recipe” for intertemporal maximization is the famous
“transversality condition” already known to the founders of the Calculus of Variations. In
our special context, it prescribes that
(TR) lim t-> e -t p(t)K(t) = 0.
Let’s try to interpret it economically. It prevents an unnecessary, useless accumulation of
capital in the far future. Suppose that the planning horizon is [0, T] instead of [0, ].
What would be the use of a positive capital stock a t = T? Its value would necessary be
zero, p(T) = 0. Therefore, either the mistake of ending up with a positive K(T) has been
made and p(T) = 0, or it has not, and therefore K(T) = 0. In either case, the current value
of the capital stock at t = T, p(T)K(T), equals zero, and therefore also its discounted
value would be zero,
(TRF)
e -T p(T)K(T) = 0.
Our transversality condition (TR) is simply the limit version of (TRF).
Equations (1), (2), (3) are a system in c(t), p(t), K(t) that can be reduced to a system in
two differential equations with unknows p(t) e K(t). The obvious initial conditions
(I)
K(0) = Ko, A(0)N(0) =AoNo,
but we need another boundary condition, and that is our (TR).
9. The co-state equation: an euristic derivation
(F) dp/dt = p( - FK),
This is also called “the Fisher equation” by Solow, and it should not be confused with
the Fisher formula giving the relation between the real and the monetary rates of interest.
The latter is merely definitional, it says, if not what the real rate of interest is, at least how
it should be computed. Solow’s “Fisher equation” is a necessary condition for the
maximization of intertemporal utility, and may well be violated. Perhaps it would be
better to call it “the Fisher-Ramsey equation”, since Ramsey discussed it more clearly,
perhaps, than Fisher. We can rewrite it as
(F-R) du’/u’dt =  - FK ,
and we will try to derive it, as Ramsey suggested, by an euristic economic reasoning
rather than from some mathematical theory. (Eq. (F-R) is actually eq. (9) in Ramsey’s “A
mathematical theory of saving”, E.J. December 1928, reprinted in Stiglitz e Uzawa,
12
Readings in the modern theory of economic growth, MIT Press 1969; in A. Sen, Growth
economics; and quite possibly in other collections too.
If at time t I withdraw from consumption 1 unit of the good, invest it and disinvest it at
time t+h in order to enjoy the benefits of my additional saving, my utility loss at t is
(almost by definition of marginal utility) u’(c(t)).1; my utility gain at t+h will be u’(c(t+h)
times the additional quantity of the good that will render itself available for me at that
time, i.e., (1+hFK) units of the good. Notice that FK is a quantity per unit of time, and
needs therefore to be multiplied by the time interval, h, during which the additional unit
of the good is applied to the production process. My utility increase at time t+h will
therefore be u’(c(t+h)). (1+hFK). In terms of utility at time t, this will be, using simple
rather than compound discounting, [1/(1+h)]. u’(c(t+h)).(1+hFK). Thus for the c(t) path
which maximizes intertemporal utility,
®
u’(c(t)) = [1/(1+h)]. u’(c(t+h)).(1+hFK),
and this will have to hold for all t and all h. We can then take the limit for h->0, and find
(F-R); or at least, so argues Ramsey, who passes on. Let us see how it can be done. We
can harmlessly rewrite ® as
u’(c(t)) +hu’(c(t)) = u’(c(t+h)) + u’(c(t+h))hFK ,
or
u’(c(t))- u’(c(t+h))FK = [u’(c(t+h))-u’(c(t)]/h,
where we recognize an incremental ratio on the R.H.S. We can then take limits for h->0
and obtain
( - FK)u’(c(t)) = du’(c(t))/dt,
as desired.
Exercise: Show that the error made in approximating compound with simple
capitalization (or discounting) is of second order of smallness with respect to the relevant
interval, h.
A slightly different, but equally instructive, euristic derivation of our co-state equation
can be found in Solow, pp. 76-77. Beware, however, of the two misprints in the key
formula in the middle of p. 77. Solow, pp.77-84, has also a very good discussion of the
economic implications and interpretation of this equation. It is highly recommended that
you study his “reader-friendly” discussion.
Exercise: explain in words the difference in the “variation” from the reference path of
capital accumulation considered by Ramsey and by Solow.
10. The “Golden rule”
This is an ethical rule of reciprocity, of Biblical origin (Jesus of Nazareth: “Do unto
others as you would have others do unto you”, applied to an intergenerational problem.
What is the K/AN ratio that gives the highest per capita consumption to each generation?
We want to compare different balanced growth paths, and look for that path, for which
13
the ratio C/N (which is increasing at the rate ) is the highest. Moreover, we wish to find
which I/Y = S/Y ratio, i.e., which value of s, will be necessary to support that path.
It is an arithmetical fact that
dk/dt = k[dK/Kdt - dA/Adt- dN/Ndt],
I repeat, a mere arithmetical truism. By our assumption that A(t) and N(t) are exponential
at constant rates  and ,
(S*)
dk/dt = k[dK/Kdt - - ].
Since we will consider only balanced growth paths where the factor quantity ratio K/AN
= k is unvarying,
d(K/AN)/dt = 0,
or, by the above equation,
dK/dt = I = (+)K,
an equality which says that the purpose of investment is merely to provide new capital to
the additional effective labour force so as to leave the K/AN ratio unchanged. By the
national income identity (0),
C/NA = Y/NA – I/NA = f(k) – [(+)K]/NA = f(k) - (+)k,
Our problem is to maximize the function C/N = A[f(k) - (+)k] with respect to k, but
since A is independent of k, this is equivalent to maximizing C/AN = f(k) - (+)k. For
an interior maximum,
f’(k) = +,
a condition that (remembering that f’(k) = FK,) can be verbally stated as follows: the rate
of profit (equal to the marginal product of capital) should equal the natural rate of growth.
But then
C/NA = f(k) – f’(k)k,
I/NA = f’(k)k,
I/Y = (I/NA) (NA/Y) = f’(k)k/f(k) = K FK/Y,
in words, the investment share in output, s, should equal the elasticity of output with
respect to capital. With a Cobb-Douglas production function,
(1C-D)
Y = F(K, NA) = K(AN)1-,
it can be easily checked that that elasticity equal . Then we get
s = ,
14
as a formulation of the Golden Rule, as is not indicated, but pre-supposed by Solow on p.
120 and 121.
Question: can the golden rule be extended to an economy with increasing returns to
scale?
In is very instructive discussion of the golden rule, Solow, p. 27, argues that “steady-state
consumption per head must fall with an increase in the rate of population growth.” Can
you obtain this result arithmetically, and then interpret it economically?
11. The Golden Rule in the simplest overlapping generations economy *.
The household sector is not described explicitly in the above construction. An interesting
way to do it is to introduce and endless chain of overlapping generations, each of which
lives two periods. In the first period of their lives, individuals work, in the second, they
are retired. Of course they consume in both periods. All people in the same generation are
equal, and all generations are equal. Suppose there is no technical progress, so that A(t) =
a constant that may be set equal to 1. The per unit rate of growth of the population is
(again) n. Then we can look for the production-consumption configuration that gives any
given generation the highest utility, under the constraint that no other generation should
receive an intertemporal consumption bundle of smaller utility. At time t, the output
produced by the people who at t are young, net of “maintenance investment”, is divided
between them and the people who at t are old (and are in retirement). Let U(c1, c2) be the
intertemporal utility function of each individual of each generation. What is the optimal
stationary pattern of capital intensity K/N and intergenerational transfers? However we
may decide to apportion the cake between the two groups, it had better be as large as
possible. (1) C/N is maximized if the condition FK = n is satified: the Golden rule.
The resource availability constraint becomes
(2) f(k) – nk = C/N = c1N/N + [c2(N/(1+n)]/N = c1 + c2/(1+n),
(3) U(c1, c2) should be maximized under it: this is the “second part2 of the famous “Twopart Golden Rule”.
Notice that the budget constraint (2) is cross-section, i.e., it refers to the distribution of
consumption between the two groups of people alive at time t; while the maximand (3)
goes forward in time, it refers to the intertemporal consumption bundle of the people who
at time t are young. Still, in a steady state they will get, at t+1, no more, no less than the
people who are old now, at time t. So the formulation of the problem is right. Thus, we
have found that the Golden Rule gives a proper steady state welfare optimum. Is it
supported by a competitive equilibrium? It would if r = f’(k) = n were part of a competitive
equilibrium for this Solow-Samuelson-Diamond economy. But it can be easily proved not
to be so. See the seminal, and not too difficult article by Peter Diamond “National debt in a
neo-classical growth model”, A.E.R., Dec. 1965.
Exercise: can the Diamond model be extended to allow for labour-augmenting technical
progress?
12. Solow’s 1956 (“behavioristic” ) growth model.
15
12.1. Solow’s growth equation. See R. Solow, “A contribution to the theory of economic
growth”, Q.J.E. 1956, reprinted in the collections on growth edited by Amatya Sen, by
Stiglitz e Uzawa, and in Peter Newman (ed.) Readings in Mathematical Economics, vol.
II, The Johns Hopkins Press, Baltimore, 1968 (This volume also contains the 1956 Swan
paper). See also the very instructive exposition given by Solow in chapter 2 (Part one) of
his Growth theory, an exposition, 2nd edition.
The arithmetical identity (S*) makes up almost 50% of the famous Solow model.
Inserting the “behavioristic” savings equation S = sY, it can be written as
dk/dt = k[sY/K - - ]
and inserting now the production function (1’)
(S) dk/dt = k[sNAf(k)/K - - ] = sf(k) – ( +)k,
the basic differential equation of Solow 1956. Here is its phase diagram
For ease of drawing I have temporarily reverted to the earlier Solow notation, but now I
will swich back to Greek again. By (S), on the left of k = k° dk/dt is positive, so k(t) is
increasing over time; on the right, dk/dt is negative, so k(t) is decreasing over time. This
is an illustration of famous Solow-Swan stability proposition (Solow’s “conclusion
number two”). Of course the diagram presupposes the existence of a positive k°. There
are two conditions sufficient, taken together, to guarantee this. One is the condition that,
at the origin, the slope of the sf(k) curve be greater than the slope of the straight line
(a+n)k (or (+)k). This condition, sf’(0) > ( +), can be given an interesting economic
interpretation, analogous to the famous Hawkins-Simon conditions. Since these were
illustrated and discussed in the classroom, the interpretation is left to the students as an
exercise. If f’(0) = , the first of the two well-known “Inada conditions”, then certainly
the condition above is satisfied. But making sure that sf(k) starts off from the origin
above the (+)k line is not enough. For, even though it is concave, it might never turn
down sufficiently to cross the straight line! The marginal product of capital might be
16
bounded from below by (+). The second Inada condition says that lim k f’(k) = 0,
the marginal product of capital should eventually approach zero. Under this condition, it
can be proved that the sf(k) line will cross the straight line from above. Do it as an
exercise. It can be proved that lim k f’(k) = 0 will necessarily happen if labour is
essential. Remember the mini-theorem. As another easy exercise, argue that k° > 0, if it
exists, is unique.
12.2 Conditional convergence in the Solow model. It can the seen from the graph of the
Solow model that for all k in the interval (k*, k°), the rate of accumulation, and hence the
rate of growth of output, is a decreasing function of k. A lower capital stock per unit of
effective labour is associated to a higher rate of growth. This may be taken to be a partial
answer to the momentous question: why do growth rates differ? This type of convergence
(late-comers in the growth process grow faster) is called “conditional” because it applies
to economies with the same production function and other growth parameters, except the
capital to effective labour ratio.
12.3 Solow –Swan and Harrod. Starting off from any K, AN, the system converges to the balanced
growth equilibrium characterized by a constant K/AN ratio. This implies that K and AN grow at
the same rate, (a+n) (or (( +)). Moreover, in the balanced growth equilibrium we have s/[K/Y] =
( +), the warranted rate of growth equals the natural rate of growth. Thus in the Solow model
both of Harrod’s contentions, the instability of the warranted growth path and the inequality of the
warranted and natural rates of growth, are upset. The question whether the Harrod model was
essentially different from Solow-Swan’s is controversial. It was not, according to Solow and
Swan. See in particular the striking quotations from Harrod given by Swan in his 1956 Economic
Record article. However, Harrod’s instability principle is based on the possibility of aggregate
demand exceeding or falling short of normal or desidered capacity, a possibility that is ruled out
by both Swan and Solow. For they show that under competition factor prices can vary at all times
to keep firms happy with the K/Y, or K/AN ratios that are actually prevailing! Solow does show
how the monetary factors could be introduced in his model in the last section of his paper. A new
look at this old debate was taken taken by Hukukane Nikaido, see his two papers "Factor
Substitution and Harrod's Knife-Edge", 1975, ZfN, and "Harrodian Pathology of Neoclassical
Growth: the irrelevance of smooth factor substitution", 1980, ZfN.
Solow and endogenous growth. Suppose the sf(k) line does not cross from above the
investment line. Then there is no value of the K/AN ratio that can act as a resting place
for the system. This does not, however, mean that there can be no steady state, i.e., a
situation where Y and K grow at the same rate. This is indeed what happens when there
are i.r.s., or endogenous growth.
Exercise: show that, under production function (1), constancy over time of the K/AN
ratio implies constancy over time of the K/Y ratio.
13. Characterization of the solution to the optimization problem: the “ modified
golden rule”
The modified golden rule describes the values that the key macroeconomic ratios (in
particular, I/Y) take up in the steady state associated to the solution of the intertemporal
maximization problem.
Let’s consider first of all Solow’s approach. He wants to solve the system formed by (0),
(1), (2), and (3), given the initial conditions (I) and the transversality condition (TR).
Moreover, he adopts the Cobb-Douglas specification (1C-D) of the production function
(1). He decides to solve the system with respect to K(t) e p(t). He conjectures (or rather,
17
already knows) that K(t) and p(t) tend to become exponential functions growing at
asymptotically constant rates, and sets out to look for these asymptotic, “ultimate”
proportional growth rates. He exploits knowledge not available to his readers that the
optimizing path approaches a steady state path, and is content to find the nature of this
steady state. Having found the asymptotic rates of growth of K(t) and p(t), he then looks
for the steady state rate of growth of c(t). Finally, he computes the asymptotic steady state
value of I/Y and compares it to its “golden rule” value. The outcome of this comparison
is that this value of I/Y is lower than its golden age value. This is the modified golden
rule.
Solow’s computations on pp. 116-118 are rather heavy, and moreover he does not explain
very clearly his procedure. Only by going over it repeatedly can one see his main thrust.
It is possible, however, to carry out the analysis in such a way as to make the
computations a little lighter, and at the same time prepare the stage for the phase diagram
which usually provides the context for a discussion of the solution to the intertemporal
maximization problem. This phase diagram is portrayed for example in the paper by R.
Dorfman “An economic interpretation of optimal control theory”, American Economic
Review, 1969, pp.817-831, and in the Blanchard-Fischer Lectures on Macroeconomics
textbook (p. 46) , and no doubt in many other publications. In the two source just quoted
technical progress is not considered, but, as we will see, a reinterpretation of the labour
force as the “effective labour force” is all that is needed to adapt the existing phase
diagrams.
The idea is simple: it is best to solve the system with respect to k and c*, an artificial
variable equal by definition to C/AN = c/A, consumption per unit of augmented (or
effective) labour. Of course individual utility depends on c(t), not on c*(t)! But, thanks to
the special form of our instantaneous utility function, this difficulty is easily by-passed, as
we will see at once.
We can re-write the Solow identity (S*), in the light of (0), (1’), as
(4) dk(t)/dk = f(k) – c* - (+)k.
Secondly, we can write (2) in the form
p(t) = [A(t)c*(t)]-,
from which we can easily obtain
dp/pdt = -[ +dc*/c*dt].
Combinig this with (3), we find the interesting
- FK = -[ +dc*/c*dt],
which can be put in the more revealing form
(5) dc*/c*dt = c*(t)[f’(k) - - ]/.
Now (4), (5) are a system of two autonomous differential equations in the two unknow
functions, c*(t) and k(t). This system can be studied qualitatively by drawing its phase
diagram. One can immediately notice that the steady state of the system satisfies
18
f’(k) =  + ,
a formula which is arrived at, albeit in a somewhat more tortuous way, by Solow, p.117.
From the transversality condition, Solow obtains
 > + - ,
and combining the last two, it is easy to establish the modified golden rule, for
f’(k) =  +  > (+ - ) +  = +.
It follows from this (since f’’(k) > 0) that the steady state K/AN ratio is lower than the
golden rule K/AN ratio. But then I/Y will also be lower: the modified golden rule!
14. The empirical success of neo-classical growth theory.
Solow was not content with armchair theorizing. He wanted to check –and to measure.
He did so in another seminal paper of his, “Technical change and the aggregate
production function”, Review of Economics and Statistics, vol. 39, 1957, pp. 312-20.
Here is technical ingenuity and his scientific integrity are at an unrivalled peak. His
purpose was to find a simple econometric technique that should allow him to separate
“shifts of the aggregate production function from movements along it”. He studied the
U.S.A. economy from 1909 to 1949. The main finding was that “Gross output per man
hour doubled over the interval, with 87.5 per cent of the increase attributable to technical
change and the remaining 12.5 to increased use of capital”.87.5 per cent of the increase
attributable to an exogenous factor! Disastrous for the theory! See on this the comments
by Nelson and Winter in chapter 8 of their An evolutionary theory of technical change,
Harvard University Press, 1982. It was this impasse –if not debacle- that set the stage for
the theory of endogenous growth.
15. The main papers and three main source-books on old neo-classical growth
theory*
The main papers: F. P. Ramsey 1928, Solow 1956 and 1957, Swan 1956 and 1963, J.
Robinson 1938, H. Uzawa 1961, K. Arrow 1962.
Ramsey pioneered the optimizing approach to growth. Solow and Swan 56 are
substantive; Solow 57 is empirical: Swan 63 methodological: Arrow 62 is highly
innovative, and already contains the engine of endogenous growth.
The main collections of papers:
1. Peter Newman (ed.), Readings in Mathematical Economics, vol. II (Capital and
Growth), The Johns Hopkins Press, 1968.
2. A. Sen (ed.), Growth economics; selected readings, Penguin, London, 1970.
3. J. Stiglitz and H. Uzawa (eds), Readings in the modern theory of economic growth,
The MIT Press, 1969.
Ramsey 1928 is present in 2 and 3,
Solow 56 in 1, 2, and 3;
Solow 1957 in 2,
Swan 56 in 1, 2, and 3;
19
Swan 63 in 2;
Robinson 38 and Uzawa 61 in 3;
Arrow 62 in 1 and 3.
16. Required readings for the exam: the first chapter of the Second Part of Solow 2000;
Domar ’47; either Solow 56 or Swan ’56. You should be fully acquainted with Solow’s
“six conclusions”. The exercises given above (and below) give an idea of what the exam
questions might be like. There will also be some questions more of the essay type.
17. Some (more) questions and problems.
1. The F-R can be written in the form
(F-R’)  = [dp(t)/pdt] + FK ,
and Solow argues (I quote from memory) that “if it were not satisfied at a given instant, by
some intertemporal substitutions total utility could be increased”. Suppose the l.h.s. of (F-R’)
exceeded the r.h.s. What would be the appropriate “intertemporal substitution?
2. Under what assumptions at the micro level is a regime of increasing returns at the macro
level compatible with competitive markets and prices?
3. Find the  function associated to a Cobb-Douglas production function.
4. Suppose two economies are equal in all respect, excepts in the intertemporal welfare
integrand. In economy A, it is as above. In economy B, the object is to maximize the
discounted value not of total current utility, but of current individual utility. Can you
compare the growth paths?
5. Consider the following statement by Aghion and Howitt: “Euler’s Theorem tells us
that with increasing returns not all factors can be paid their marginal products”. Are
increasing returns production functions necessarily homogeneous? In which way can we
apply Euler’s Theorem to increasing returns to scale production functions? Do you agree
with Aghion and Howitt?
6. Do the data discussed by Solow in Chapter 1 of his book support the idea of a “natural
rate of growth”?
7. On p. 147 of his 2000 book Solow writes that “there is endogenous growth whenever
the growth of output exceeds (a+n).” Is this right, or a minor slip? And if the latter, how
should it be rectified?