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Fabio Petri
Adv Micro
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Microeconomics for the critical mind
CHAPTER 5.
november 2015
THE THEORY OF THE FIRM, PARTIAL EQUILIBRIA, PERFECT
COMPETITION, AND THE ATEMPORAL (ACAPITALISTIC) GENERAL
EQUILIBRIUM WITH PRODUCTION
5.1. The present chapter extends the marginalist theory of competitive general equilibrium to
include production without capital goods and without a rate of interest; capital goods and rate of
interest raise special problems in the marginal/neoclassical approach and will be discussed in
Chapters 7, 8 and 9. However, in order to make our treatment of production decisions sufficiently
general and thus also useful for subsequent chapters, we admit the presence of capital goods (that is,
inputs to production that are produced goods) when discussing the single firm and the single
industry; we only leave them out when we come, in the third part of the chapter, to discuss the
general equilibrium of ‘atemporal’ production and exchange – as this kind of general equilibrium is
called (explanation for this terminology must wait for Chapters 7 and 8).
We start with the necessary notions about the theory of production and of price-taking firms.
Price-taking behaviour, which we have already assumed in the study of consumers, means the
economic agent treats prices as given parameters in her maximizations; hence a buyer (respectively,
a seller) believes that the price at which she can purchase (respectively sell) additional units of a
good is the same as the price of the previous units; therefore for a firm, revenue from sales or
expenditure on inputs are linear functions of the quantity sold or bought. A discussion of when the
price-taking assumption is legitimate, and of its connection with the notion of competition, is
provided in Part III of the chapter.
The firms we study in this chapter produce undifferentiated goods: each product is produced
by many firms and is so standardized (and unaccompanied by marketing expenses) that consumers
are indifferent as between the several producers. As a result, the only problem of the price-taking
firm is how much to produce and with what combination of inputs.
PART I
PRODUCTION POSSIBILITIES SETS AND PRODUCTION FUNCTIONS
5.2.1. The purpose of this chapter is to make more precise the study of ch. 3 of how the
marginal approach determines the competitive general equilibrium of an economy where a variety
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of consumption goods is produced without need for capital anticipations, i.e. through the use of
unproduced (‘original’ or ‘primary’) factors: land (possibly, different types of land) and labour
(which we will generally assume homogeneous, but we could also assume different types of
labour). To describe the general equilibrium of this economy it will suffice to consider production
possibility sets of individual firms and of the entire economy, that specify what flows of
consumption goods per time period can be produced from given flows of services of ‘original’
factors per time period. Note that the production of a consumption good, e.g. of cooked potatoes,
may require that, initially, unassisted labour and land services produce some intermediate good, e.g.
firewood and raw potatoes found in the wild, which then (with the aid of further labour and land
services, e.g. to produce fire and to roast the potatoes) produce the final good cooked potatoes; but
we will assume that whatever intermediate-stage good is produced by the initial unassisted labour
and land remains inside the firm’s productive process and is entirely consumed inside it in order to
produce the final consumption good: the raw potatoes and woodfire are not sold to other firms
(production is “vertically integrated”); so the production process of each consumption good can be
described as a finite succession in time of exclusively labour and land services, without needing to
specify in detail what is produced and then transformed as the production process proceeds[1].
Always for the purpose of determining the general equilibrium of such an economy, we will further
assume: that the interest rate is zero, so production costs of the potato-producing firm are the same
as if all labour and land services were paid at the same moment the output of cooked potatoes
comes out; that the total time length between initial labour and land use, and output of the
consumption good, is one period – one year –; and that production is in separate yearly cycles,
labour and land are employed during the year and consumption goods come out at the end of the
1
The non-specification of how the productive process develops in time is an inevitable necessity. All noninstantaneous production processes gradually transform something into something else and then into
something else still; for example potato cultivation produces small potato plants, then bigger potato plants,
then potato plants with small potato bulbs, then with bigger potatoes ... ; production of a sofa produces first
the various things that are then assembled into the final sofa; and so on. To specify all details of such
processes would be impossible, and economically not important. Intermediate goods (sometimes called
‘work-in-progress’) is the name given to the things that appear in these intermediate stages, to be then
transformed further, up to the final output of the firm’s production process. But the term is ambiguous:
sometimes the production process can be split into successive stages operated by separate firms, and then
what would have been a proper, ‘internal’ intermediate good in the process carried out entirely by one firm
becomes an output sold to be an input into another firm’s production, that is, becomes a marketed circulating
capital good; sometimes it is called an ‘intermediate’ good in this case too, but this usage risks generating
confusion. For example, car engines, that are proper, ‘internal’ intermediate goods if produced by the same
firm that produces the finished car, are sometimes called intermediate goods also when bought from a
separate firm. But in the latter case the theory must determine their price, which is unnecessary for ‘proper’
intermediate goods.
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year[2]. With these assumptions, inputs are always different from outputs of production processes,
and it is unnecessary to refer to time in the specification of production possibilities, except to make
clear to what time length, what ‘period’, the flows refer. On the contrary, in an economy that
produces with capital goods, for example that produces corn with labour and seed-corn as inputs, it
is necessary to specify that produced corn and seed-corn, although physically the same good, are
separated in time, seed-corn preceding corn output. The absence of such a need, plus the absence of
a rate of interest, in the economy studied in this chapter, has favoured the neoclassical habit of
calling it the ‘atemporal’ economy, and to interpret production in such an economy as ‘timeless’, a
very unclear notion that has given rise to confusions as explained below. In fact the labour-land
economy is not ‘atemporal’; production takes time. As a consequence, disequilibrium adjustments
take time too, and the general equilibrium we will study must be conceived as the normal (or longperiod) situation which the economy gravitates towards on the basis of factor endowments, tastes,
and technical knowledge, that either remain constant, or change sufficiently slowly (relative to the
presumable speed of gravitation toward the equilibrium) for the change to be negligible.
There are two types of agents: consumers, with given endowments of factors and given
preferences; and firms. In the background, there must be an institutional setup ensuring respect of
private property and of contracts, something like a state apparatus with laws, police, courts, and
other employees; these require resources; but for the moment we neglect this aspect.
Firms have production possibility sets. The production possibility set Y of a firm is the set
of all combinations of inputs and outputs that are possible for the given firm in the given time
period. These combinations are called production processes or production plans. A production
process is a vector, of physical quantities of inputs and of outputs.
As announced, in this chapter we want to consider more general production possibility sets
than needed for the ‘atemporal’ economy, hence also with capital goods among the inputs and
outputs. Remember that any produced input is a capital good, even if it lasts only a few minutes
from the moment it is produced to the moment it disappears in some production process. But the
capital goods that are produced and re-employed inside the same firm without being marketed do
not need that a price be specified for them, and so need not appear among the goods produced in
the economy; but their price could be determined if necessary, and sometimes (see below) it can be
convenient to make this price appear explicitly, by imagining the firm to sell to good to itself.
2
This is called a flow-input, point-output production process. The amount produced of consumption goods is
still a flow, because it is per year, although when it comes out at the end of a year it can be treated as a stock.
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Durable capital goods provide a flow of services per time unit, e.g. a leased printer produces a
number of printed pages per day, and what is relevant for cost calculations is the gross rental per
time period that must be paid for its use; circulating capital goods, e.g. ink, or parts to be assembled
to obtain a product (e.g. wood planks, screws, nails, handles etc. assembled to produce a piece of
furniture)[3], or seed-corn, disappear in a single usage, and what is relevant is their purchase cost. If
there is a positive interest rate, and if costs must be borne before the revenue from the sale of the
output is obtained, then interest must be added on top of these costs for the relevant time interval.
For example in the single-production economy studied in ch. 2, all inputs apart from labour are
circulating capital goods bought one time period before revenue is obtained, and a rate of interest
(rate of profit) is added on their purchase cost; production is implicitly supposed to be of the pointinput, point-output type (inputs coming in at a precise moment, and output at another precise
moment); note that labour services may well be spread out over the period, but if they are all paid at
precise moments, then they can be equivalently treated as if all used at those moments, so pointinput again. Inputs temporally precede outputs, even if only by a few seconds; but in some dynamic
models[4] in continuous time it may be convenient to neglect the lag and to imagine production to
be a flow contemporaneous to the utilization of the services that produce it; then it will necessarily
be production of a continuous finite flow of output per time unit, due to a contemporaneous flow of
services provided by a stock of durable factors[5], so the passage of time is anyway necessary to
obtain a nonzero amount of output. The finite output flow depends on the finiteness of the flows of
productive services. These can often be represented by the dimension of a stock, implicitly
assuming that the flow of productive services supplied by a stock of a factor is proportional to the
dimension of the stock. For example, production of eggs by a big hen factory can be seen as
approximately a continuous flow, produced by a mass of hens (a stock) with the help of input flows
of labour services, electricity, hen nutrients, young hens that replace the ones that die, etc. A typical
such formalization is the aggregate Solow growth model, where a single output, Y, is produced as a
continuous flow per time unit by the flow of services supplied by a stock of capital K and a stock of
workers L. (With labour, the assumption that the flow of labour services is proportional to the stock
of workers L requires a given average labour intensity and a given average number of labour-hours
3
If a car engine is bought by a car factory and assembled into a new car, it is a circulating capital good for
this firm, because it ‘disappears’ into the product, differently from the electric screwers that mount it.
4
That is, models that attempt to determine the rates of change of some variables.
5
E.g. workers in a quarry produce a continuous flow of raw minerals with a contemporaneous flow of labour
services. The stocks are the number of workers, and the quarry.
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per period per worker.)
It is opportune to assume that inputs precede outputs; by choosing the length of a unitary time
interval short enough, it is possible to assume that inputs applied at instant or date t (that is, at the
beginning of period t) or during period t (if inputs are a flow of services) produce the output at
instant t+1. A one-period production process is then a vector in two parts (a1, a2,...,an; b1, b2, ..., bn)
where ai is an input and bj is an output that comes out one period later. This representation admits
joint production; also, it admits that the same good can appear as an input and as an output, like
corn used as seed to produce corn; this will never be the case in our labour-land economy without
capital goods, where inputs are services of ‘primary’ factors and outputs are consumption goods,
but it is a frequent case in economies with capital goods. If the production process of a good takes
longer than one time period, one splits the production process into a sequence of successive
processes of unitary length, that produce intermediate goods (circulating capital goods) which are
then inputs to the next stage, up to the production of the final good (the reader can easily visualize
the method e.g. for wine production). One consequence of this assumption is that the outputs of
inputs applied at time t cannot be used within period t as inputs in another process. This avoids
some dangers into which other representations of production possibilities occasionally incur, and
which we pass to illustrate.
In modern general equilibrium theory, the most frequently adopted formalization of a
production process is as a vector of netputs (short for net outputs), a vector y with N elements if the
economy has N goods and services, where negative numbers indicate net inputs and positive
numbers indicate net outputs of goods. The production possibility set of the economy is the set of
all netput vectors available to each firm, or to several firms combined, or even to all firms
combined (that is, to the entire economy). Note that the notion of netput implies that outputs are
necessarily different goods from inputs, because each element of a netput indicates either an input
or an output. This creates no problem if production does not use capital goods, because then inputs
are services of ‘original’ factors and outputs are consumption goods (then the qualification ‘net’ is
superfluous). But netputs are also used to describe the available production methods in economies
that use and produce capital goods. Then cases like seed-corn used to produce corn can be
accommodated by distinguishing goods according to their date of availability, so that corn at date t
is a different good from corn at date t+1. This is the representation of goods adopted in
intertemporal equilibrium theory (see ch. 8). In this case to talk of net outputs is useful for situations
like the following: consider a production plan including an output of 100 units of a circulating
capital good at date t, and also the re-utilization of 80 of those units at date t to obtain other
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products at date t+1; one says then that the planned netput of that capital good at time t is 20 (the
other 80 units are ‘proper’ intermediate products); it’s what the firm can sell of that capital good to
others. Netputs are convenient because (admitting now the possibility of nonzero interest rates) if y
is a netput vector and p the vector of discounted prices of the goods in y, the inner product p·y
yields the discounted (neoclassical) profit from adopting that production plan, with negative netput
entries (amounts of inputs) contributing to (discounted) cost and positive netput entries (amounts of
outputs) contributing to (discounted) revenue[6].
But the netting out of the use of a good as input from the production of the same good in the
same period is also sometimes considered appropriate to represent the production possibilities of the
economy, and then one ends up with absurdities. This will be shown through an example, a slight
modification of an example from p. 64 of the renowned treatise on general equilibrium by Arrow
and Hahn (1971): assume the economy has two industries and three goods: 2 products that are
consumption goods and also circulating capital goods, and labour services; in one period, industry
A produces 2 units of commodity 2 (iron) from 1 unit of commodity 1 (corn) and 1 unit of labour
services, and industry B produces 2 units of corn from 1 unit of iron and one unit of labour services;
with labour as good 3, this is represented by netputs (-1, 2, -1) and (2, -1, -1). This means that the
entire economy, that is the two industries jointly considered, by using 1 unit of corn plus 1 unit of
iron plus 2 units of labour can produce 2 units of each good; this, it is argued, means that the
economy’s production possibilities include the netput vector (1, 1, -2). Since the production
possibilities set of the economy is defined to consist of netputs, we conclude that this economy is
capable of producing one unit of each good by using, as inputs, only 2 units of labour. The trouble
is, that the production processes adopted by all firms together are needed to determine the demands
for inputs and the supplies of products of the firm sector, to be then confronted with the supplies of
inputs to this sector, and the demands for products from it, so as to determine whether the economy
is in equilibrium; now, the netput vector (1, 1, -2) does not represent these demands and supplies, it
only tells us the net productions of the period (according to the usual definition of net products);
suppose for example that both production processes take 1 period, this means that the firm sector
demands, besides labour, 1 unit of each good as inputs at the beginning of the period, and supplies 2
units of each good at the end of the period; there will be equilibrium then if there is the
corresponding supply of inputs at the beginning of the period, and the corresponding demand for the
products at the end of the period. On the contrary, the netput representation is interpreted as
6
Discounted prices are discussed in ch. 8.
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meaning that equilibrium needs only the supply of 2 units of labour at the beginning of the period,
and the demand for 1 unit of each product at the end.
Note that the determination of the beginning-of-period demand for corn and iron as inputs
need not be the one indicated above, because the assumption need not be made – although it makes
life so much simpler – that within the unit time period only one production cycle takes place. But
suppose that the two productions of our example result from the repetition 100 times, within the
period, of a cycle consisting of a production process producing 0.02 units of corn with 0.01 units of
iron and of labour followed by a production process producing 0.02 units of iron with 0.01 units of
corn and of labour; still, the first production cycle needs a positive initial endowment of 0.01 units
of iron. An initial endowment of some produced input besides labour is ineliminable. On the
contrary, according to the literature that uses netputs, for example according to Arrow and Hahn,
this economy does not need initial endowments of corn or iron to carry out those productions; it
only needs the two units of labour. Arrow and Hahn try to justify this thesis by assuming “that
production and all other economic activity is timeless; inputs and outputs are contemporaneous” (p.
53). The meaning of the nebulous adjective ‘timeless’ is not made any clearer by being
accompanied by ‘contemporaneous’, a notion that implies the flow of time. Since on p. 64 the
authors write: “Alternatively, if production takes time and differently dated commodities are
distinguished...”, one concludes that in their example production is instantaneous, in zero time iron
production can be used by the corn industry to instantaneously produce corn, and again in zero time
this corn production can be used by the other industry to instantaneously produce iron, and the thing
can be repeated indefinitely; then, were it not for the constraint due to the limited supply of labour,
even an extremely small initial endowment of corn or iron would allow, by infinitely fast infinitely
repeated production cycles, the production of any amout of output. Unfortunately this hypothesis,
absurd as it is, anyway does not save the thesis that only labour is needed, because if the initial
endowments of corn and iron are truly zero, neither industry can start producing, so production is
zero. So labour cannot be the only initial endowment; the aggregate economy’s netputs do not
correctly indicate the firm sector’s demand for inputs. And anyway production takes time, and
transferring a product to another firm again takes time. In Appendix 5 to Ch. 8 it will be explained
that the absurd notion of a ‘timeless’ economy has caused much confusion about the meaning of the
so called nonsubstitution theorem. In this book, production takes time, and inputs are at an earlier
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date than the date of their outputs[7]. Netputs, when used, will refer to dated goods, with outputs of
a later date than their inputs.
5.2.2. In the same way as for consumer theory, inputs and outputs can also be distinguished
by the location in which they are available, and by the state of nature with which they are
associated. When goods are dated, it will be generally legitimate to assume that if Y includes a
productive process with netputs distinguished by their dates, (yt,..,yT), then it also includes the same
sequence with all netputs’ dates increased by the same number, indicating that all that matters is the
lag between inputs and outputs, and not the moment when the productive process is started;
furthermore it is natural to assume that outputs cannot precede their inputs in time.
When the vector of outputs to be produced by a firm is given, the input requirement set is
the set of input vectors that allow the production of (at least) that vector of outputs. In the vectors of
the input requirement set the inputs are measured as positive quantities. If one assumes that some of
the quantities of inputs can be left idle[8], the input requirement set for a certain output vector
includes all vectors x of inputs that allow producing at least that output vector, plus all vectors x’≥x.
Firms will generally only utilize efficient production processes. In terms of netputs we say
that yY is efficient if there is no other y'Y such that y'≠y and y'≥y; in other words it is not
possible to produce the same outputs with less of some input, or to produce more of some output
with the same inputs. If the output vector z≥0 is given, an input vector x (inputs being measured
now as positive quantities) is efficient if no vector x’≤x exists that allows the production of z.
When one considers production processes that produce only one output, it is often assumed
that the economically relevant production processes that produce that output can be described by a
production function q=f(x1,…,xn)=f(x) where output q is the maximum output obtainable from the
vector of inputs x=(x1,…,xn), the latter measured as positive quantities. In economics when one
speaks of output obtainable from certain inputs, one means the maximum output. The set of input
vectors that a production function q=f(x) associates with a given output q° is called the isoquant
associated with q°. Note that an isoquant does not necessarily include only efficient input vectors:
suppose that production of one unit of corn requires labour and land in the fixed proportions 2:1, in
The exception will be, in ch. 8, Solow’s growth model, where stocks of capital K and of labour L produce a
flow of services which produces instantaneously a flow of output. As noted in the text, production being a
finite flow, time is anyway necessary to obtain any finite amount of output.
8
This qualification is due to the possibility that an increase in the amount actually employed of an input may,
after a certain point, cause decreases of the amounts produced, as explained already in ch. 3.
7
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the sense that if land input is 1, increasing the labour input above 2 units leaves output unaltered at
1 unit, and the same holds for land input above 1 unit if labour input is 2 units; graphically, in a
Cartesian plane with labour input in abscissa and land input in ordinate (the reader is invited to
draw the figure), the 1-unit-of-corn isoquant includes, besides the (2,1) point, the non-efficient
points on the vertical halfline above this point and on the horizontal halfline to the right of this point
(this is a so-called L-shaped isoquant, or Leontief isoquant).
It is sometimes assumed that in the short period the quantity of some inputs cannot be varied;
these are called fixed inputs. The short-period production function has two alternative
representations: it can include among the inputs the given quantities of the fixed factors, but for
brevity it can also describe output as a function of the sole variable inputs.
The equivalent of the production function for production processes that produce several
outputs
jointly
is
a
transformation
function
implicitly
defined
by
an
equation
T(x1,…,xn;q1,...,qm)=0, where, again, inputs are measured as positive quantities[9]. The equation
renders each output an implicit function of the quantities of inputs and of the other outputs.
5.3. When the production possibilities set Y is a set of netput vectors, some axioms that may
be postulated on it are:
1
0Y, inactivity is one possibility
2
Y∩Rn+ = 0, no production of outputs without inputs
3
Y∩–Y = Ø, production is irreversible
4
Y is convex
5
Y is bounded above
6
for any good i, and any positive scalar q, the vector y=(0,...,0,–qi,0,...0) is Y (free
disposal).
Axioms 1, 2 and 3 are unproblematic and are assumed in what follows. Axiom 4 implies
perfect divisibility of all inputs and outputs; its connection with returns to scale will be discussed
presently. Axiom 5 is convenient in a first stage of some mathematical proofs, and it is justified by
referring to limited factor endowments that do not allow producing more than certain maximum
quantities, but this means mixing up endowments with technologies, so it will not be assumed in
what follows. Axiom 6 postulates that for each good there is available a process that uses that good
alone as an input and produces nothing, so one can always get rid of any amount of any good
9
Exercise: Obtain the transformation-function representation of a production function.
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without any cost; it is called the free disposal assumption. When the free disposal assumption is
made, then any firm can couple any production process with free disposal processes, and the result
is a property or assumption sometimes called monotonicity of the production possibilities set(10):
if yY, then y' such that y'≤y is also Y,
because y' is a process that employs at least as much of each input as y ( 11), and produces not more
of each output than y, and it is always possible to obtain y' from y by use of free disposal processes.
q=f(x)
q
O
x
Fig. 5.1. Production possibilities set resulting from a production function q=f(x) plus free disposal.
Free disposal may appear a questionable assumption in many situations (it is often costly to
get rid of goods, or to prevent some output from coming out), but it can be argued to be
fundamentally harmless. An ability costlessly to get rid of excess inputs is not needed as long as
these inputs have a positive cost, because then firms will not buy them to start with, so a free
disposal assumption of excess costly inputs is superfluous but then also harmless; and if inputs are
costless and with a negative marginal product, they can be left idle and all one needs is to
distinguish the technological from the economic production method (as was done in §3.3.4). As to
undesired outputs, if they must not be produced it can be assumed that, whatever disposal process is
necessary in order not to produce them (or in order to dispose of them), its inputs (and costs) are
10
Some authors call monotonicity the following slightly different assumption: let q stand for a vector of
outputs and let V(q) stand for the set of input vectors (measured as positive quantities) that allow the
production of at least q; if an input vector x is in V(q), and x'≥x, then x'V(q).
11
Remember that inputs are negative numbers, so a greater use of an input means a greater absolute value
of a negative number i.e., algebraically, a smaller number indicating input use.
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included among the inputs (and costs) of the desired output. When there is a choice about how
much to produce of an undesired side product, then a formalization can usually be found in which
the abatement of its production is counted as an output, and then the problem becomes again the
usual problem of cost minimization and profit maximization, this time with joint production. Thus
the free disposal assumption is not really necessary but it is also generally harmless.
5.4. Axiom 4 stipulates that if yY and y’Y, then for 0<a<1 all netputs y”=ay+(1–a)y’ are
also Y; strict convexity of Y means that if y and y’ are efficient, then y” is not efficient.
Convexity requires divisibility of inputs and of outputs.
Exercise: α) Suppose a single output q is produced by two inputs x1 and x2 according to the
production function q=ax1+bx2, a,b>0. Draw some isoquants and prove that the production
possibility set is convex.
β) Suppose two outputs z1 and z2 are jointly produced by a single input x with transformation
function z12+z22–x=0. Prove that the production possibility set is convex and that for fixed x the
locus of possible efficient combinations of z1 and z2 (the transformation curve) is concave.
γ) Suppose a single output q is produced by two inputs x1 and x2 according to the production
function q=x12+x22; draw some isoquants and prove that the production possibility set is not convex.
An assessment of Axiom 4 requires a discussion of increasing, constant and decreasing
returns to scale, a notion given different meanings in the history of economic thought.
Technological returns to scale are respectively increasing, constant or decreasing according
as, when all physical inputs are increased by the same percentage, physical output increases
respectively in a greater, the same, or a lesser percentage. It requires divisible inputs.
Firm returns, still in terms of output, to the scale of total cost, are increasing, constant or
decreasing according as an increase in total cost by 1% raises output by more than, exactly, or less
than 1%. This notion is especially helpful in the presence of indivisibilities; it takes input prices as
fixed; constant returns in this sense means that the firm requires a doubling of total cost in order to
double output; but the doubling of total cost may mean a radical change in input composition.
A still different meaning is industry returns in terms of physical output from total industry
costs, a very complex Marshallian notion, historically important in partial equilibrium analysis, that
takes into account (i) technological returns to scale at the firm level, (ii) what happens to the prices
of the industry's inputs when the industry expands its total output, and (iii) external effects, when
the industry's output changes(12); this notion of returns (in this meaning the appendage ‘to scale’ is
12
This is the notion of returns in the famous controversy originated by Piero Sraffa with his 1925 and
1926 articles. ‘External effects’ (externalities) internal to the industry are changes in the efficiency of
(cont. next page →)
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usually absent) will be discussed when we arrive at the industry's supply curve; let it suffice now to
say that in this third, Marshallian meaning, a constant returns industry is an industry that (within
limits) can expand long-period output without change in average cost, and therefore has a longperiod horizontal supply curve.
Production functions allow a quick definition of technological returns to scale. Let x be the
vector of inputs to a production function f(x). We say that f(x) exhibits constant returns to scale
(CRS) if f(tx)=tf(x) for t>0;
increasing returns to scale (IRS) if f(tx)>tf(x) for t>1;
decreasing returns to scale (DRS) if f(tx)<tf(x) for t>1.
Returns to scale can be variable: for example a production function might exhibit returns to
scale that are increasing at first, then constant, then decreasing. Local returns to scale for a given
vector of inputs x are ascertained by checking which one of the three inequalities holds for small
variations of t around 1 (cf. §5.6.1 below).
Technical returns to scale can also be defined in terms of netputs and of frontier of the
production possibility set F(Y). I give a taste. If netput yF(Y) implies tyF(Y), with t any positive
scalar in a neighbourhood of 1, we say that Y exhibits local CRS. There are increasing returns to
scale at least locally, if there is some yF(Y) and some scalar t>1, such that tyY and ty  F(Y),
that is,  y’Y and ≠ty such that y’≥ty.
Note that if there are increasing returns to scale at least locally, Y is not convex.
Proof: by contradiction. By Axiom 1, y=(0,...,0)Y, so convexity requires that for any yY also t'y
with 0<t'<1 is in Y; then consider netputs yF(Y), ty and y'≥ty of the above definition of locally increasing
returns in terms of netputs: if Y is convex, y"=t'y' with 0<t'<1 is in Y; choose t’=1/t, then y"≥y and y"≠y, so y
cannot be in F(Y), contradicting the assumptions; hence Y cannot be convex.
■
This shows that Axiom 4 excludes increasing returns to scale. Now, increasing returns to
scale are argued by many economists to be often present in reality; therefore Axiom 4 is a
production and more generally in the unit costs of the several firms in an industry, due to changes of the
dimension of the industry: an example of positive external effects might be the greater ease with finding
competent repair men or spare parts when the industry is larger; an example of negative external effects
might be air pollution depending on the industry’s output and obliging firms to use air filters. Externality or
external effect is the term denoting an influence of consumption or production decisions on the utility
function or production function of other consumers or firms. Listening to loud rock music, for example, may
reduce the utility of neighbours and it is then a negative consumption externality. External effects in
production need not be internal to the industry, but are then depending on the general scale of aggregate
production and only little affected by changes in the scale of a single industry.
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restrictive assumption; in spite of this, we will generally assume it because it is necessary for the
standard theory of general equilibrium, whose presentation is now our main aim.
As an Exercise, you are asked to prove that a convex production possibility set with a single
output implies a quasiconcave production function. (Hint: prove first that the input requirement set
is convex. The definition of quasiconcavity was given in Ch. 4, §4.5)
5.5. Some clarification on the notions of productive process, inputs, indivisibilities can be
useful. A production process usually starts with certain inputs and lasts some time, during which
time the inputs become transformed into a series of other things before reaching the final form of
the saleable product. There is therefore some arbitrariness in the description of a production
process; one might always break it into two successive processes, the first one producing
intermediate products which are the inputs to the second one. For example, the production of a sofa
from wood planks, springs and fabric might be broken down into a series of stages, each one
producing an incomplete sofa closer and closer to completion. The intermediate products might
themselves be priced, as made evident by the fact that sometimes a process originally performed in
its entirety by a firm is broken down into processes performed by different firms, for example
production of a car has historically been more and more broken down among separate firms which
produce the seats, the windshield, the tires, the brakes, etc., parts which are finally assembled into a
car by a different firm. Which inputs are initial inputs that appear in the production function, and
which are intermediate stages that are only implicitly considered (but might become explicit if the
process were broken down into successive subprocesses performed by different firms), depends on
the organization of production and, from the theorist's point of view, is largely arbitrary. This makes
the notion of intermediate input ambiguous. Flour is an intermediate input in the production of
bread from wheat, but it is an initial input in the production of bread from flour.
The term 'intermediate input' is sometimes used as synonym of circulating capital good,
which means a capital good that is fully destroyed in a single utilization (one might say, that
disappears into the product). These two notions are best kept distinct; it is best to reserve the term
'intermediate inputs' for the products produced and re-utilized inside a production process, and
therefore not appearing among the inputs of the production function (they must not be paid for).
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The inputs appearing in the production function are all the 'initial' inputs that must be paid for( 13);
when they are capital goods, they can be either circulating capital goods, or durable capital goods;
in the latter case they reappear among the outputs, obviously with the alterations caused by
utilization(14).
We will generally assume that all inputs and outputs are perfectly divisible, i.e. can be
represented by continuous variables. This is a good approximation for land and for labour time[15],
but it is obviously unrealistic for capital goods and for many products; however, if the analysis
deals with big quantities the assumption may still be acceptable if the indivisibilities are small
relative to total input use or total output.
Much more debatable is the assumption of differentiability of the production function, and
here we come to a very important issue. In most industries a different productive process requires,
not different proportions among the same capital goods, but different capital goods, and for each
ensemble of capital goods a rather rigid labour input per unit of output. The amount of labour
services needed to assemble a car, for example, is rather strictly determined by how mechanized the
production process is; and there is no substitutability between the parts to be assembled, a car needs
exactly one engine, five tyres, etc., each one of these parts needing in turn, to be produced, rigid
quantities of material inputs and rigid amounts of labour determined by the machinery used. With
the only exception of some agricultural production (where one can vary amounts of irrigation, of
fertilizers etc.), there generally is no, or nearly no, variability of proportions among the same inputs
when these include specialized capital goods: hence, generally differentiability of the production
function is a very unrealistic assumption. But then how come the differentiability assumption is so
widely accepted?
The reason would appear to be a historical one, namely the Marshallian habit, shared by the
great majority of economists up to the 1960s, and still widespread to-day, to describe production
functions as combinations of labour, land and capital treated as a single factor K, measured as an
amount of value; implicitly, the prices of the several capital goods are treated as given, and the
13
In the case of intertemporal production functions, 'inital' inputs are not necessarily entering the process
at the same date, the term 'initial' must be interpreted as meaning ‘not produced by the production process
itself’.
14
It is also possible to imagine cases in which a durable capital good is an intermediate good, because the
production process considered lasts a number of periods, and produces itself a durable capital good which is
then entirely utilized during the remainder of the process.
15
Labour time is, physically speaking, perfectly divisible, but regulations often limit this divisibility; for
example a firm may be obliged to hire full-time labour only. Even then, if the firm employs 1000 labourers a
perfect divisibility assumption may be an acceptable approximation.
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production function is determined as follows: the firm is assumed to determine, for each given
vector of non-capital inputs and each given K, the vector of capital goods of value K that maximizes
production; thus, given the amounts of labour and land, small increases of K can well imply a
totally different vector of capital goods; along the total productivity curve of capital so conceived,
capital changes not only in quantity (an amount of exchange value) but also in ‘form’ (physical
composition)[16]. With such a specification of the capital input, the assumption of smooth
variability of proportions between the inputs is no longer totally implausible: increasing only the
number of tyres certainly does not increase the output of cars; on the contrary if what can be
increased is the value of the capital goods used, which can be associated with a change in the capital
goods used, then it is likely that a way can be found to use the capital increase so as to increase the
number of cars produced by a given number of workers. Unfortunately in more recent times this
origin of the use of ‘smooth’ production functions and of nicely decreasing marginal product curves
appears to have been lost sight of; in modern microeconomic theory and general equilibrium theory
this treatment of capital has gone out of fashion[17], inputs are all measured in technical units, the
several capital goods are each treated as a separate factor; but continuity and differentiability of
production functions are still commonly assumed. I must follow now this usual practice in order to
introduce the readers to this literature; but it is important to realize that we are encountering here
one instance of survival, in a context no longer justifying them, of assumptions that were originally
justified by the treatment of capital as a single value factor of variable ‘form’. (Several other
instances of this type will be met in later chapters.)
So unless differently specified, we assume production functions to be continuous and
differentiable functions of perfectly divisible inputs.
5.6. Returns to scale.
5.6.1. In Ch. 4 we met homogeneous functions (cf. Ch. 4 fn. 35??). CRS implies that the
production function is homogeneous of degree 1 [18]. If the production function is homogeneous of
16
Analogously an isoquant in terms of, say, labour and K indicates, for each level of labour, the minimum
value of capital required to produce the given output; again, small movements along the isoquant can well
mean a passage to a production method requiring very different capital goods.
17
It is not difficult to understand why. The treatment of capital as a quantity of value in the firm’s
production function is only legitimate if the prices of capital goods are given. But these prices cannot be
taken as given when the purpose of the analysis is not a partial-equilibrium one but rather the determination
of income distribution – which affects relative prices.
18
Two properties of homogeneous functions (already briefly indicated in footnote 35 of ch. 4) are of great
relevance for CRS production functions. A continuous function f(x1,...,xn) is homogeneous of degree k if,
(cont. next page →)
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degree k>1, it has increasing returns to scale; if homogeneous of degree k<1 it has decreasing
returns to scale. Even when a function f(x) is not homogeneous, we can determine its local degree
of homogeneity by increasing all independent variables by a common small percentage, say, .1%,
and observing whether f(x) increases by more or less than .1%. This indicates the local
technological returns to scale (of course, under perfect divisibility of all inputs). Their most
widely used measure is the scale elasticity of output (or simply elasticity of scale) with respect to
inputs. Let x be the vector of inputs in an initial situation with q=f(x), and consider f(tx) with t>0;
the scalar t measures scale, and the scale elasticity of output in q=f(x) is defined as
e = [df(tx)/f(tx)]/(dt/t) = (∂q/∂t)∙(t/q) = ∂ ln q / ∂ ln t evaluated in t=1, with x fixed.
Note that this definition does not require differentiability of f(x), it only requires
differentiability of f(tx) with respect to t, i.e. with respect to proportional variations of all inputs,
therefore it is applicable to production functions where inputs, or some of them, are perfect
complements. But if f(x) is differentiable, then by the derivative rule of a function of function it is
1
n
e=
[xi∙∂f/∂xi].

f ( x ) i 1
By Euler's theorem on homogeneous functions, if the production function has constant returns to
scale then ∑i(xi∙∂f/∂xi)=f(x) so e=1. According as e is equal to, more than, or less than, 1, the
production function exhibits locally constant, locally increasing, or locally decreasing returns to
scale(19).
with t a positive scalar, f(tx1,...,txn) = tkf(x1,...,xn). The first property is that if a function homogeneous of
degree k is differentiable, then its partial derivatives are homogeneous of degree k–1; the proof is by
differentiating both sides of f(tx1,...,txn) = tkf(x1,...,xn) with respect to xi, and indicating with ∂f(tx)/∂(txi) the
partial derivative of f relative to the i-th independent variable, calculated in tx: one obtains
t
f (tx)
f ( x)
 tk
which implies that the partial derivative calculated in tx is the partial derivative
 (txi )
xi
calculated in x multiplied by tk–1; thus if f is a CRS production function, hence homogeneous of degree 1, its
marginal products are homogeneous of degree zero i.e. depend only on factor proportions (hence the
expansion path, the locus of tangencies between isoquants and isocosts, is a ray from the origin). For the
second property, differentiate both sides of f(tx1,...,txn)=tkf(x1,...,xn) with respect to t, obtaining
n
f (tx)
 ( (tx )  x )  kt
i 1
i
k 1
f ( x) , a result sometimes called Euler’s theorem for homogeneous functions, and
i
assume t=1, that is, perform the differentiation at x: one obtains ∑i(xi∙∂f/∂xi)=kf(x); for CRS production
functions it is k=1, hence if each factor is paid its physical marginal product the payment to factors exhausts
the product, a result called the product exhaustion theorem.
19
The extension of these definitions to transformation functions is left to the reader (rays of outputs will
replace the single output; the sole complication is that, since with transformation functions generally a given
vector of inputs does not uniquely determine the vector of outputs, it is now possible to imagine cases where
returns to scale differ according to which output ray one considers).
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If all relevant inputs are taken into account in the specification of the production function,
then an exact doubling of all inputs allows the exact replication of the same plant twice and
therefore it should permit at least a doubling of production: for integer t>1, f(tx)≥tf(x), i.e. there
must be at least constant returns to scale. Why do I say 'at least'? Because there may be some
indivisibility of the technological process, which was not fully exploited at the original scale, and
can be better exploited at a bigger scale, allowing for increasing returns to scale. Thus consider the
production function of a firm that extracts and transports oil and also produces all the pipes for the
pipeline. The pipes are intermediate products in the overall production process and appear neither
among the inputs nor among the outputs of the oil production-and-transport function. The inputs
include, for example, the steel needed to make the pipes. Now, up to a point the carrying capacity of
pipes increases more than proportionally with the increase in the steel utilized to make pipes,
because the steel utilized is – within certain limits – roughly proportional to the diameter of the pipe
but the carrying capacity is proportional to the square of the diameter. Doubling the amount of
produced and transported oil may then require pipes of double diameter that use less than double the
steel, with less than double the cost. A similar issue arises with tanks. This example shows that,
when the production function reflects vertically integrated production processes which include the
production and utilization of intermediate goods, a doubling of all inputs need not correspond to a
replication of the same production method twice, and a doubling of output need not require a
doubling of inputs(20). Still, the replication of the same production method twice (the building of a
second plant identical to the first one) is always possible and therefore returns to scale for integer
t>1 are at least constant. What allows the existence of increasing technical returns to scale is the
existence of indivisibilities (either of goods, or of processes) which are not fully taken advantage of
at small scales of production.
The result f(tx)≥tf(x) need not hold for fractional increases in scale, again owing to
indivisibilities. It may be impossible to increase all inputs by, say, 30% if some inputs are
indivisible; or the indivisibilities can be in the production process, e.g. the production process might
be vertically integrated and include the internal production and utilization of a large indivisible
capital good. Below I will assume that this problem is of minor importance, we will see that when
one considers an entire industry with free entry it is generally plausible to treat the industry output
The same can happen if capital goods are aggregated into the single factor ‘(value of) capital’; then
doubling the capital and the non-capital inputs need not correspond to purchasing twice as many of the same
capital goods, it may for example mean the use of a different fixed plant that costs twice as much but allows
more than double the production. Cf. below, §5.20, the notion of scale returns to cost.
20
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as coming from a CRS production function even when at the firm level there are relevant
indivisibilities
5.6.2. Plausibly, even if indivisibilities cause increasing returns at small scales of production,
at each stage of technical knowledge there is a finite production scale beyond which returns to scale
are no longer increasing: we can call it minimum optimal scale. Beyond a certain dimension larger
pipes and tanks are no longer convenient because they require special reinforcing structures. In a
perfectly competitive industry with free entry firms must produce at minimum average cost and
therefore will tend to adopt plants of at least minimum optimal scale, possibly several of them. As
long as this efficient scale of production is small relative to total industry output, it will be
approximately true that the aggregate of firms composing an industry can be seen as having a
production function exhibiting constant returns to scale – where the constancy is generated by
variations in the number of identical efficient plants. For this reason, below we generally assume
constant technical returns to scale for industries. This assumption may appear valid for only a very
restricted set of industries, given the observable tendency of firms in many industries to grow as
large as they can. But the advantages of size can be due to many other reasons besides increasing
technical returns to scale, reasons that do not concern us now(21). Anyway globalization has
increased competition in many industries where minimum efficient plant size is very large; for
example, one can buy cars produced all over the world, Korean cars compete with German and
USA cars, thus even for industries where minimum plant size is very large there often appears to be
sufficient competition for the assumption of price equal to average cost to be broadly acceptable for
many analyses.
5.6.3. What about decreasing technical returns to scale? They are difficult to defend if the
inputs appearing in the production function are really all relevant inputs, because then, as argued
above, identical duplication of plant and process should yield double output. Decreasing returns to
scale can be admitted only if some relevant inputs are fixed in quantity and do not appear in the
production function. This is the case in short-period analysis when only variable inputs are made to
appear in the production function; but for long-period analysis, the sole way which appears
21
We only mention at this stage the possible advantages in terms of funds for marketing expenditures, or
for research and development (R&D) expenditures, and the possibility to obtain discounts on some input
prices .
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acceptable to arrive at decreasing returns to scale is to postulate a greater difficulty of control and
co-ordination over the performance of subordinates[22]. It might be answered that if one duplicates
not only the plant but also the managers, there is no reason why there should be a greater difficulty
of control. The objection to this is that managers must be controlled too, and the owner of the firm
or top manager may have greater difficulty in avoiding subordinate managers shirking or
embezzling when she must control many managers; furthermore, the information that the top
manager must process increases, and its reliability decreases, as it must pass through a greater
number of bureaucratic layers, making correct decisions more difficult. Whether this objection is
sufficient to justify an optimal size of price-taking firms is still an object of disagreement among
theoreticians[23]. But for the purposes of value theory what is important is the behaviour of
industries, and then the moment one admits free entry the difficulties of control may explain why
individual firms are limited in size, but industry output can be varied by variation in the number of
firms; so at the industry level there will be CRS anyway in long-period analysis as long as one can
assume that minimum efficient size is small relative to total demand.
5.6.4. Finally a word on the difference between production process and production method
when there are CRS. Any netput vector in the production possibility set is a production process.
With CRS, if (–x,q) is the netput representation of a single-output production process then (–tx,tq)
with t>0 is a production process too, and if the first one is efficient so is the second. One means
then by production method a set of ratios between inputs and outputs: a vector (–x,q)Y and a
vector (–tx,tq), t≠1, are considered two production processes representing the same (CRS) method
22
Decreasing economic returns to scale (i.e. profits that increase less than proportionately with output)
can arise because an output increase raises the rentals of some inputs; but this is best kept separate from the
issue of technical returns to scale.
23
For example, Scherer and Ross p. 106 agree with the traditional position well represented by EAG
Robinson ??ref on the greater difficulty of control as a cause of ultimately decreasing returns to scale, and
their arguments on the difficulty of control increasing with size are prima facie convincing. Edith Penrose on
the contrary wrote: “We do not know how effective the decentralization of authority can be as a means of
keeping costs per unit of output from rising as a firm expands. Reliable empirical evidence does not exist and
all studies of the matter are inconclusive, but there is no evidence that a large decentralized concern requires
supermen to run it....Neither is there significant evidence that the ability to fill the higher administrative
positions is excessively rare or that the demands on the men occupying these positions exceed their ability to
cope with them effectively.” E. Penrose, “Limits to the growth and size of firms”, AER 1955, vol. 45 (2),
May, 531-43, p. 542. Cases supporting Penrose, for example MacDonald’s or CocaCola or United Fruits,
easily come to mind. Perhaps in many cases the advantages of increasing size are so great (especially when
one considers the scale economies in marketing, R&D, transport costs, employee training, etc.), as to more
than counterbalance the increasing difficulties of co-ordination; also, tying decentralized managers’ pay to
results may be often sufficient to motivate them.
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activated at two different activity levels.
5.7. Activity analysis
The distinction between production process and production method comes useful when, given
the general lack of smooth substitutability between different capital goods, one esteems that it is a
closer approximation to reality to represent the production possibilities of a firm as including only a
finite number of CRS fixed-proportions production methods, which can be activated at any activity
level (we still assume divisibility); the firm can also activate linear combinations of processes
achievable from those methods, obtaining other processes. In this case the production possibility set
is a polyhedral convex cone, with the efficient methods as edges and the vertex in the origin; the
isoquants are said to be of the activity-analysis type. These isoquants appeared in Appendix 2 to
chapter 3, cf. Fig. 3.15?? which is reproduced here for ease of reference as Fig. 5.1bis??.
Availability of a single method to produce a good (without joint production) implies L-shaped
isoquants, corresponding to a production function of type q=min{αx1,βx2} in the two-inputs case
(represented in red in Fig. 5.1bis); if this is the case for all goods, one has a Leontief technology.
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method 1
method 2
isocost
A
• B'
method 3
B
C
O
labour
Fig. 5.1bis (Fig. 3.15). Activity-analysis isoquant with three alternative methods. The red
broken line is the isoquant associated with method 2 alone.
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5.8. Marginal product, isoquant, transformation curve
5.8.1. We assume now that the firm produces a single output and its efficient production
possibilities are represented by a production function f(x) which is twice continuously differentiable
(that is, it has partial derivatives that are differentiable, and their partial derivatives are continuous).
Then we can define the marginal product of factor xi as MPi≡∂f/∂xi; and the locus of input
combinations which yield an assigned level of output, the isoquant, is a differentiable surface (in
Rn if there are n inputs) by the implicit function theorem.
If some factors are fixed, the locus of combinations of the remaining inputs which yield the
given output is called a restricted isoquant or short-period isoquant (note that it may be empty).
x2
q2
MRT
isoquant
transformation curve
TRS
x1
q1
Fig. 5.2 Standard isoquant and standard transformation curve in R2.
Suppose we consider an isoquant restricted to the two inputs i and j. The condition that
production be equal to an assigned quantity q^ and that all inputs be given except for x i and xj:
f(x1,...,xi,...,xj,...,xn) = q^ , with xh given except for h=i,j
implicitly makes xj a function of xi. The slope of this function xj=xj(xi), i.e. the slope of the
restricted isoquant when xi is treated as the independent and xj as the dependent variable, is the
equivalent in production theory of the consumer's marginal rate of substitution; it is called the
technical rate of substitution of factor j for factor i, to be indicated as TRSji; by the derivative rule
for differentiable implicit functions the TRSji is given by TRSji ≡ dxj/dxi = − (∂f/∂xi)/(∂f/∂xj) = −
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MPi/MPj, the well-known result of first-year textbooks. The TRS is a negative quantity if both
marginal products are positive; economists sometimes speak of the TRS as a positive quantity, then
they are implicitly referring to its absolute value.
In Chapter 3 we saw that marginal products need not be positive, and therefore technological
isoquants need not be downward-sloping. (The reader is invited to re-read in Chapter 3 the
distinction between technological and economic isoquants.) We supply now a numerical example.
Assume
q = (x1x2 − .8x12 − .2x22)1/2
The reader can check that this production function has CRS. Marginal products are given by
1
1
∂q/∂x1= 1 / 2 (x2 – 1.6x1), ∂q/∂x2 =
( x1 – .4x2); provided it is q>0, which will be the case as
2q
2q1 / 2
long as 1<x2/x1<4, the marginal product of factor 1 is positive as long as x 2>1.6x1, and the marginal
product of factor 2 is positive as long as x2<2.5x1. Hence both marginal products are positive and
isoquants are negatively sloped only as long as 1.6<x2/x1<2.5. Outside this fairly restricted range of
factor proportions, one of the two marginal products is zero or negative, implying an upwardsloping isoquant.
For a differentiable transformation function T(x1,…,xn;q1,...,qm)=0, since there is more than
one output, an input has several marginal products, one for each output (Exercise: write the
expression for them from the rule of derivation of implicit functions). The determination of the TRS
between two inputs requires that all other inputs and all outputs be fixed. If all inputs, and all
outputs but two, are fixed, the locus of efficient combinations of the two remaining outputs (where
efficiency means that one output is maximized when the other one is given) is called a
transformation curve(24) and its slope is called the marginal rate of transformation, MRT, and
is given by MRTji ≡ dqj/dqi = –(∂T/∂qi)/(∂T/∂qj).
With constant returns to scale, marginal products only depend on factor proportions and not
on the scale of production. This is because the partial derivatives of a function homogeneous of
degree one are homogeneous of degree zero, i.e. do not vary for equiproportional variations of all
independent variables. Thus along a ray from the origin all isoquants have the same TRS.
5.8.2. The reader will have noticed the resemblance between isoquants and indifference
24
It is also called a production possibility frontier for the firm (in order to distinguish it from the
production possibility frontier for the entire economy).
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curves, and between marginal products and marginal utilities. Thus, for example, convex isoquants
imply that the production function is quasiconcave. However, there are some differences.
A first difference reflects the cardinal character of production possibilities versus the ordinal
nature of preferences. Any strictly increasing transformation of an increasing utility function
represents the same preferences; on the contrary, any transformation of a production function alters
the production possibilities. Therefore, quasiconcavity is not as important a notion as in utility
theory: in production theory we also want to know returns to scale, which are arbitrary in ordinal
utility theory.
Also, as long as CRS and perfect divisibility are assumed, an isoquant is necessarily convex.
This is because under these assumptions if x and x' are two input vectors belonging to the isoquant
associated with output q, then the firm's production possibilities set Y includes all netput vectors (–
tx,tq) and (–tx',tq) for any t>0, so the firm can produce q by using any convex combination of the
two processes (–ax,aq)+(–(1–a)x',(1–a)q), where 0  a  1, and therefore the isoquant cannot go
above the segment connecting any two points of it, cf. Fig. 5.3b. On the contrary, in utility theory it
would be a most special case if all linear combinations of two consumption bundles on the same
indifference curve yielded the same utility (the consumption goods would be locally perfect
substitutes).
x2
A
B•
x2'●
C
●
x1'
Figure 5.3a Short-period restricted isoquant with
incompatibility between the two variable factors.
x1
Fig. 5.3b. With divisibility and CRS, isoquants
cannot be strictly concave, point B of input use
allows the same production as A or C through a
linear combination of the activities (processes)
corresponding to A and C.
For the same reason, under divisibility and CRS, production functions are concave because,
given any two efficient vectors (-x',f(x')) and (-x",f(x")), it is possible to produce at least any linear
combination of these vectors and therefore f(ax'+(1-a)x")≥af(x')+(1-a)f(x").
This is no longer necessarily the case with restricted, or short-period, isoquants. For example,
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with a given machine for chemical reactions, it might be possible to produce a given output by
using a certain chemical process, or another chemical process based on different components, but
any joint use (or even alternate use) of the two processes on the same machine might be impossible
because the residues of the components of one process would react with the components of the
other process and destroy the machine. In this case, assuming free disposal, the restricted isoquant
would have the shape shown in Figure 5.3a. The meaning of such an isoquant is that, given the
other inputs, the desired output can be produced either with the quantity x1' of input 1, or with the
quantity x2' of input 2, but not with both. However, cases such as this one can be considered highly
unusual and therefore we leave them aside.
5.9. Profit maximization and WAPM
The most common assumption about the behaviour of firms is that they aim to maximize
profit. The meaning of 'profit' here and in the entire chapter is the marginalist/neoclassical one, i.e.
in this chapter 'profit' stands for what is left to the ‘entrepreneur’ (the owner of the firm) after
paying all costs including interest on capital advances(25). Even when the apparent aim of the firm
is another one, e.g. sales maximization or maximization of growth rate, a case can usually be made
that this does not entail significantly different choices from the ones aimed at maximizing long-run
profit. More relevant is the possibility of inefficiency, but as long as management strives for profit
maximization the fact that the goal is only imperfectly realized does not alter the broad pattern of
industry behaviour. For example, the tendency to invest more in the industries that offer better
profitability prospects will still exist even if on average management is not very good at minimizing
costs. The first-year textbook short-period supply curve of the firm, coinciding with the marginal
cost curve, most probably remains upward-sloping and therefore suggests an increase in output if
the product price rises, even if marginal cost reflects inefficiencies. And the occasional episodes of
managers pursuing strategies of personal enrichment at the expense of the profitability of their firm
usually end up rather quickly in the disappearance of the firm, whose market shares are absorbed by
better run firms. We accept profit maximization as broadly valid as a survival condition, especially
in competitive environments[26]. A monopolist entrepreneur not threatened by takeovers might
25
And including an allowance for risk too; but we are not considering risk for the moment. (The reader
may be surprised by our mentioning interest here; but as we said, the treatment of firms aims to be general,
the assumption that there are no capital goods and no interest will only be made when we come to the
general equilibrium model to be studied at the end of this chapter.)
26
A popular textbook states: “From the voluminous and often inconsistent evidence, it appears that the profit
maximization assumption at least provides a good first approximation in describing business behavior.
(cont. next page →)
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indulge in other aims, e.g. to have political influence, or play golf, or be generous toward
employees; firms in competitive environments and under threat of takeovers end up being taken
over or going bankrupt if they do not struggle to survive, and this requires profit maximization.
In what follows the prices of outputs are indicated as pj, j=1,...,m; the rentals (prices of the
services[27]) of inputs are vi, i=1,...,n. The vectors p=(p1,...,pm) and v=(v1,...,vn) are to be conceived
as row vectors unless otherwise indicated, and the vectors of inputs x and of outputs q of a
production process as column vectors. At given prices (p,v), for each production process with
netput (–x,q) total cost is vx and profit is π:=pq–vx. With the netput notation y=(–x,q), one can put
all prices – of inputs as well as of outputs – into a single vector; if one uses the symbol P=(v,p) for
this encompassing row vector, then profit can be represented more compactly as π:=Py. If inputs
and outputs are paid at different dates, the several prices of the previous expression are to be
intended as discounted or capitalized to a common date: for the moment, the date at which output is
sold. If, as I will mostly assume in what follows, the firm produces a single output, then vector q
has only one nonzero element.
A result, somewhat analogous to the weak axiom of revealed preferences, that derives
immediately from profit maximization is the following: if at prices P°=(v°,p°) the firm chooses a
netput vector y°, the profit must be at least as great as with any other netput in Y, i.e. P°y°≥P°y, y;
This is sometimes called the Weak Axiom of Profit Maximization, WAPM. It has the following
implication: if netput y° is chosen at prices P° and netput y* is chosen at prices P*, it must be
P°y°≥P°y* and P*y*≥P*y°>; re-write these inequalities as P°(y°–y*)≥0, –P*(y°–y*)≥0 and add
them to get (P°–P*)(y°–y*)≥0, which is often written
ΔP∙Δy≥0.
In words: the inner or dot product of the vector of price changes and the vector of netput
changes is non-negative and generally positive; geometrically, the two vectors form an acute angle.
The interpretation requires to remember that inputs are negative numbers in the netput
representation. Thus if price i is the only one to change and it increases, the WAPM implies that
netput i must increase or at least not decrease: if netput i is an input, an algebraic increase of its
negative quantity means a smaller utilization of that input.
Deviations, both intended and inadvertent, undoubtedly exist in abundance, but they are kept within more or
less narrow bounds by competitive pressures, the self-interest of stock-owning managers, and the threat of
managerial displacement by important outside shareholders or takeovers.” (Scherer and Ross ?? p. 52).
27
As pointed out in Ch. 3, it is preferable to speak of input rentals to mean price of the services of inputs;
but ‘input price’ is so often used to mean input rental, that in this chapter we use the two terms
interchangeably.
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5.10. Optimal employment of a factor
Let us first consider the decision of how much to employ of a single input, when the
quantities of the other inputs are given. The firm is competitive, i.e. price taker, and produces a
single output. The production function is q=f(x), differentiable. Profit is =pq–vx=pf(x)–vx where q
is output (a scalar), p its price, x the vector of inputs (positive quantities). Interior maximization of
 with respect to xi under xi>0 requires the first-order condition p·∂f/∂xi – vi = 0, i.e. the equality
between marginal revenue product of the factor and 'price' (i.e. rental) of the factor, where the
marginal revenue product of a factor is the derivative of revenue pq with respect to the employment
of the factor, i.e. (under price-taking) p·∂f/∂xi. With the usual symbols:
p·MPi=vi.
This can also be expressed as equality between marginal product of the factor, and real rental
of the factor measured in terms of the product, MPi=vi/p.
The second-order condition is that the marginal revenue product must be decreasing in xi. The
increase in profit if the firm employs one more small unit of factor i is p∙MPi–vi, and if it is positive,
or if it becomes positive for further increases of the factor, the firm finds it convenient to expand the
use of the factor, so the optimal level of factor employment must be where one more unit of the
factor no longer increases profit and further units would only make things worse.
Careful: there may be no positive solution to this maximization problem; in other words, it
may happen that no positive value of xi, however small, avoids a marginal revenue product inferior
to the given rental. In this case, since it must be xi ≥ 0, the firm reaches a 'corner solution' with xi=0
and p·MPi ≤ vi. It is possible, although a fluke, that at xi=0 it is p·MPi=vi.
5.11. Cost minimization
5.11.1. If two variable factors i and j are both demanded in positive amounts, then p·MPi=vi
and p·MPj=vj imply the well-known condition MPi/MPj=vi/vj; however, this last equality can be
satisfied when the two other ones are not, and this will mean, as we now show, that a different
problem is being solved: cost minimization.
A necessary condition for profits to be maximized is that the total cost of producing the profitmaximizing output be minimized. Profit maximization can be achieved in two steps: first, for each
level of output, minimize cost, and find how this minimized cost varies with output, i.e. find the
cost function; second, maximize profit by finding the level of output that maximizes the difference
between revenue, and the minimized cost.
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The cost function is the minimum-value function
c(v,q) = min vx s.t. f(x)≥q .
This will be a short-run cost function if some factors cannot be varied; usually the other
ones, called variable factors, are the sole factors that are made to appear in x.
Assume there is a unique cost-minimizing solution x for each given (v,q). Let x=H(v,q) be the
vector function indicating how the solution changes with v and q; H(v,q) is a vector function called
the conditional input demand function. Then c(v,q)=vH(v,q). The word 'conditional' derives from
the fact that these are the input demands conditional on the level of output.
There is a strict similarity between the cost function in production theory and the expenditure
function in consumer theory[28], and between the firm's conditional input demand function and the
consumer's compensated (or Hicksian) demand function: mathematically they are just the same
thing (this is why we use the same symbol H to indicate the conditional factor demand function
too). Therefore we need not prove the properties we now list because the proofs are the same as for
the expenditure function, cf. chapter 4.
As long as f(x) is continuous and x is such that f(x) is strictly increasing in a neighbourhood
of x, the cost function has the following properties:
(1) c(v,q) is nondecreasing in vi
(2) c(v,q) is homogeneous of degree 1 in v
(3) c(v,q) is continuous in vi, for v>>0.
(4) c(v,q) is strictly increasing in q as long as q>>0
(5) c(v,q) is concave in vi.
If the production function is continuous we can replace the constraint f(x)≥q with the
constraint f(x)=q; if it is also differentiable, for interior solutions (x>>0) we can use the Lagrangian
approach with equality constraint (the Kuhn-Tucker conditions are the more general necessary firstorder conditions). Formulating the problem as one of maximization of –c(v,q), the Lagrangian
function is –vx–(q–f(x)) where q and v are given(29). The first-order conditions for an interior
Indeed a number of economists call ‘cost function’ the consumer’s expenditure function.
In the consumer maximization problem we write the constraint in the Lagrangian function as –(px–m),
here we write it as –(q–f(x)): the difference derives from the fact that, in order to obtain a non-negative
value for the Lagrange multipliers in Kuhn-Tucker theory, the constraint (the expression in parenthesis) must
be written in such a way that it is constrained to be non-positive (if there is a minus sign before the λ): in the
case of the consumer, the constraint is m≥px, in the case of the firm it is f(x)≥q. Another way of putting the
thing is, that in the consumer problem a relaxation of the constraint requires an increase of m, in the cost
minimization problem a relaxation of the constraint requires a decrease of q.
28
29
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solution yield
–vi = –∂f/∂xi,
i=1,...,n
from which one derives the well-known condition
vi/vj = MPi/MPj .
This is interpretable geometrically. Assume all input levels apart from those of inputs i and j
to be given and to cause a cost B=∑s≠i,j(vsxs). For each given total cost C, the expression
vixi+vjxj=C–B makes xj a linear function of xi. This function is called a restricted (two-dimensional)
isocost. Its geometrical representation is a downward-sloping straight line in (xi,xj)-space, with
slope equal to –vi/vj and intercepts (C–B)/vi on the abscissa and (C–B)/vj on the ordinate axis. It is
the locus of quantities employed of the two factors that cause the same total cost C. Increases in C
induce a parallel outward shift of the isocost line. Cost minimization requires that the isocost be as
close as possible to the origin under the condition that it has a point in common with the given
isoquant corresponding to the desired output. If the isoquant is ‘smooth’, the condition vi/vj =
MPi/MPj imposes that the isocost be tangent to the i,j-isoquant associated with q. In order for this
tangency actually to indicate a point of minimum cost, any isocost closer to the origin must have no
point in common with the isoquant. This is ensured if the isoquant is convex.
5.11.2. The partial analogy with the utility maximization problem is graphically clear in the
two-factors case: in both cases we have a map of straight lines and a map of curves; the difference
is that in order to maximize utility we look for the point, on a given straight line (the budget line),
that touches the curve (the indifference curve) farthest from the origin; while in order to minimize
cost we look for the point, on a given curve (the isoquant), that touches the straight line (the isocost)
closest to the origin.
x2
isocosts
isoquant
x1
Figure 5.4
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With n factors, the isoquant is a surface of dimension n–1 in Rn, the isocost is a hyperplane;
the first-order conditions for an interior solution imply tangency between isoquant and isocost, and
the second-order sufficient condition is that the isoquant surface be convex. The thing is graphically
evident with two factors. As in the UMP, convexity of the isoquants ensures that, when the firstorder conditions are satisfied, the solution is a global maximum of –C, that is to say, a global
minimum of C. Mathematically, this second-order condition can be expressed, as in the UMP
problem, as the condition that at the solution x* it is d2(–C)>0 for displacements from x* that
satisfy the constraint f(x)=q°, and it can be again shown that this corresponds to the condition that
the leading, or naturally ordered, principal minors (starting from the third one) of the bordered
Hessian[30] alternate in sign starting from positive (remember that the problem is formulated as the
maximization of –C[31]).
For the reasons explained earlier, convexity of the isoquant surface always obtains when CRS
and divisibility are assumed and all factors are variable.
The condition vi/vj = MPi/MPj can be realized without the conditions MPi=vi/p, MPj=vj/p
being realized; when so, factor employments are not optimal, the output level is not profitmaximizing. Cost minimization is only one part of what is necessary for profit maximization: one
must also choose the optimal output level. This latter choice can also be examined in terms of cost
function and revenue function, see below. But before, we explore the cost function a little more.
5.12. WACm; Kuhn-Tucker conditions and cost minimization; Shephard’s lemma.
5.12.1. Cost minimization has an implication analogous to the WAPM. If for a given output q
and given input rentals v the firm finds it optimal to utilize an input vector x°=H(v,q), it must mean
that any other input vector capable of producing q (or more) must cost at least vx°, in other words,
vx°≤vx for all x such that f(x)≥q°, a result sometimes called weak axiom of cost minimization,
WACm for short. If at input prices v° the firm chooses input vector x° and at input prices v* the
firm chooses input vector x* to produce the same output, proceeding in the same way as for the
WAPM one reaches the conclusion (v°–v*)(x°–x*)≤0, more often expressed as
Δv∙Δx≤0.
30
Exercise: derive the bordered Hessian in the two-factors case and show that its determinant is positive
if and only if the analogous bordered Hessian of the UMP in the two-goods case has a positive determinant.
31
If the problem is formulated as the minimization of C, then the leading principal minors of the bordered
Hessian must be all negative.
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This implies, for example, that if only one input price changes, the demand for that input must
change in the opposite direction. (Note that if inputs were measured as negative quantities the
inequality sign would be reversed, still this is not the same result as was derived from the WAPM,
because here output is kept fixed.)
In general, not all inputs will be used in positive amounts by a firm; the condition vi/vj =
MPi/MPj must hold for inputs both used in positive quantities; the more general necessary firstorder conditions for cost minimization are derivable from the Kuhn-Tucker theorem. In this case,
with q° the given output, the function to be maximized is –vx and the constraints are q°–f(x)≤0,
and –xi≤0, i=1,...,n. The first-order conditions are therefore (multiplying both sides by –1):
vi = 0∂f/∂xi + i,
i=1,...,n, with the complementary slackness condition that if i>0,
that is to say, if vi>0∂f/∂xi, then xi=0.
The Lagrange multiplier 0 in this problem can be interpreted through an application of the
Envelope Theorem. We define the marginal cost function MC(v,q) as the derivative of the cost
function with respect to output: MC(v,q)=∂c(v,q)/∂q . It tells us by how much total cost increases if
the firm increases output by one (small) unit. Now let M be the value function of the maximization
problem with constraint
m a x ( vx ) s . t . f  x   q ;
x
hence M = –c(v,q); the Lagrangian function of the maximization problem is
–vx–0(q–f(x)),
and the Envelope Theorem implies
∂M/∂q= –0, i.e. ∂c/∂q=0:
the Lagrange multiplier λ0 in the cost minimization problem is the marginal cost.
Another way to prove this result is the following, where fi≡∂f(x)/∂xi:
 f dx  ...  0 f n dx1
c( v ,q ) v1 dx1  ...  vn dx1

 ( from the first  order conditions ) 0 1 1
 0 .
q
f 1 dx1  ... f n dxn
f 1 dx1  ... f n dxn
5.12.2. Let us now consider again the function – c(v,q) as the value function of the
maximization problem with constraint, m a x (vx) s. t. f  x   q . Let us apply the Envelope
x
Theorem to the derivative of this value function with respect to factor rentals; we obtain (now we
indicate the Lagrange multiplier as simply ):
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cv , q 
f  x 
f ( x )
  xi  
  xi (because
 0 );
vi
vi
vi
the quantity xi that appears in this expression is in fact the conditional demand for input i, because it
is the quantity of this input demanded when output is kept fixed at the level q; so we obtain:
Shephard's Lemma for the firm:
c( v ,q )
 xi ( v , q ) .
vi
In words: The conditional factor demands are the partial derivatives of the cost function with
respect to factor rentals.
This is the original Shephard's Lemma, which was later extended to consumer theory.
5.13. The profit function and Hotelling's Lemma
5.13.1. The profit function (p,v) is defined as the value function of the (unconstrained)
maximization problem
maxx pf(x)–vx,
which asks to find the input vector that maximizes profit, as a function of output price and input
rentals. (The function π(q)=pq–C(q), profit as a function of output under cost minimization, is not
called ‘profit function’.) In short-period analysis the inputs are the variable ones.
The profit function is only defined when the condition p=MC yields a definite optimal output.
This is not the case if the production function has constant returns to scale (or if there is perfect
replicability of plants, cf. §5.??): then average and marginal cost coincide and are constant, and for
a price-taking firm, if p>AC, profits grow endlessly with increases in output, so there is no optimal
output, while if p=AC there is an infinity of solutions all yielding zero profit. Thus the profit
function requires sufficiently decreasing returns to the scale of variable inputs[32]; this can be
justified in short-period analyses by taking some inputs as fixed; on the contrary, a long-period
profit function requires assumptions on returns to scale (a U-shaped LAC curve) that not all
economists find plausible. But precisely in long-period analyses the profit function is irrelevant
even when it can be defined, because entry will anyway maintain profits equal to zero, so the
determination of equilibrium industry output does not require consideration of the profit function, as
we explain later. As to the short-period profit function, it is based on a rather misleading definition
of profit as revenue minus variable cost, neglecting the need to include among costs the quasirents
Exercise 5.2: Why the stress on ‘sufficiently’? Is decreasing returns to scale without qualifications a
sufficient condition for the existence of a profit function? Try exploring the case in which C(q)
asymptotically approaches from below a function a+bq, with a,b>0.
32
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of fixed factors as opportunity costs. These quasirents, if included in total cost and if the
entrepreneur is neither better nor worse than other entrepreneurs at maximizing profit, would
always bring profit to zero even in short-period analysis because they are the opportunity cost of
using the fixed plant instead of renting it out to other entrepreneurs (these would be ready to pay for
the use of the fixed plant a maximum amount equal precisely to what would bring down their profit
to zero). Therefore the ‘profit’ of the short-period profit function is the sum of true profit and of the
quasirent to be attributed to the fixed factors. As a consequence, it may be the case that this ‘profit’
is positive but the entrepreneur would do better to sell the firm because other entrepreneurs would
get more out of that set of fixed factors and so they value the fixed factors more than she does.
However, the profit function is widely used in microeconometric practice, so we list its main
properties:
Properties of the profit function. Suppose that the production function f: Rn+→R+ is
continuous, strictly increasing and strictly quasiconcave and that the profit function (p,v), i.e. the
value function of the problem maxx pf(x)–vx, is well defined for given (p',v') and continuous in (p,v)
in a neighbourhood of (p',v'); then in that neighbourhood (p,v) is
(1) increasing in p
(2) decreasing in v
(3) homogeneous of degree one in (p,v)
(4) convex in (p,v)
(5) differentiable in (p,v)>>0.
PROOF: We only prove the least intuitive of these properties, convexity, i.e. that for any scalar a such
that 0≤a≤1 it is a(p,v)+(1–a)(p',v')≥[ap+(1–a)p',av+(1–a)v']. Define pa≡ap+(1–a)p' and va≡av+(1–a)v'.
Let x, x', xa and q=f(x), q'=f(x'), qa=f(xa) be the solution inputs and outputs associated respectively with
(p,v), (p',v') and (pa,va). Then it is
(p,v) = pq–vx ≥ pqa–vxa
(p',v') = p'q'–v'x' ≥ p'qa–v'xa
which imply a(p,v)+(1–a)(p',v') ≥ a(pqa–vxa)+(1–a)(p'qa–v'xa) = paqa–vaxa = (pa,va).
■
5.13.2. Now let us apply the Envelope Theorem to the profit function. The latter is the value
function of a maximization problem without constraints, so we obtain:
∂/∂p = f(x)=q i.e. the partial derivative of the profit function with respect to the output price
yields the optimal output;
∂/∂vi = –xi i.e. the partial derivative of the profit function with respect to input rental vi
yields the (unconditional) demand for input i measured as a negative number (i.e. in accord with
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the netput notation).
These two results constitute Hotelling's Lemma (sometimes called the Derivative Property).
Thus the partial derivatives of the profit function, when the latter exists, give us the output
supply function and (the negative of) the input demand functions[33]. Since the profit function is
convex, we reach the result that, when the profit function is well defined, both ∂/∂p and ∂/∂vi are
non-decreasing and generally increasing in p, respectively in vi, that is to say:
– supply is a non-decreasing, and generally an increasing, function of output price (because
convexity of π implies ∂/∂p is non-decreasing and generally increasing in p, but ∂/∂p=q);
– the unconditional demand for an input, when measured as a positive quantity, is a nonincreasing, and generally a decreasing, function of the input own rental: there are no Giffen inputs.
5.14. Conditional and unconditional factor demands, inferior inputs, rival inputs,
substitution effect and output effect.
5.14.1. When the profit function is well defined, for each output price p and vector of factor
rentals v there is an optimal output and an associated vector x of optimal factor utilizations (the
latter vector need not be unique, but I will assume it is). In this case we can define the supply
function (of the individual firm) S(v,p) that indicates how optimal output changes with factor
prices v and the output price p, and the (vectorial) unconditional factor demand function x(v,p)
that indicates the associated optimal factor employments; it is
x(v,p)=H(v,S(v,p)).
If the profit function is defined for the short period, i.e. with some inputs fixed, then only the
variable inputs appear in x and in v. When the profit function does not exist, the output supply
function and the input demand functions do not exist either: no unique profit-maximizing output
exists. When these functions can be defined, what about the sign of their partial derivatives?
If profit is considered a function of q, its maximization requires solving the problem
maxq pq–C(v,q),
whose first-order necessary condition is the equality of product price p, and marginal cost
MC(v,q):=∂C/∂q; the second-order sufficient condition is that MC must be rising at the optimal q.
Hence
33
Exercise: Prove that if the firm is a multiproduct one and profit depends on inputs and outputs, then
Hotelling’s Lemma, again derived from the Envelope Theorem, generalizes to equality between the partial
derivative of the profit function relative to any one output price, and supply function of that output.
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∂S(v,p)/∂p>0
the (inverse) supply curve (optimal q as a function of p, with p in ordinate and q in abscissa) is
increasing (as long as to increase output is possible), a result actually already reached via
Hotelling’s Lemma plus the convexity of the profit function; but it can be useful to see the same
result from different perspectives.
We cannot reach a result on the sign of ∂xi(v,p)/∂vi directly from the condition vi=p∙MPi,
because xi is not the only input use that will change when vi changes; but Hotelling’s Lemma and
the convexity of the profit function imply ∂xi(v,p)/∂vi≤0: the own-rental effect is nonpositive (and
generally negative).
5.14.2. Does the above result on input use imply ∂S(v,p)/∂vi ≤ 0 ? in other words, is it always
the case that the optimal output, when it exists, decreases if the price of a factor (in positive use)
rises? Perhaps surprisingly, not always. When the price or rental of a factor rises, supply rises when
the factor is an inferior input, defined as an input whose conditional demand falls as output
increases, that is, such that ∂xi(v,q)/∂q<0 (at least at the given input rentals and in a neighbourhood
of the initial q), meaning that an increase of q is best achieved by increasing some input other than i
and decreasing input i.
pq
xj
total cost,
revenue
expansion path
c’
c
xi
Fig. 5.4bis(a)
q
Fig. 5.4bis(b)
The possibility of inferior inputs is shown in Fig. 5.4bis(a), where, assuming two variable
factors with given prices, several isoquants are drawn and for each one the tangency with an isocost
is shown; the locus of tangency points is called the expansion path of the firm for given input
prices; this expansion path can exhibit a decrease of the utilization of one factor as higher isoquants
are reached. As the drawing helps to grasp, input inferiority is necessarily not a global property, it
can only hold within some range of output levels: in order for optimal input to decrease as output
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increases, it must first have increased with output. An example of inferior input can be labour in
some agricultural productions on a given specialized land where, as long as the required output is
not large, it can be optimal to achieve it with abundant use of labour nearly unassisted by
machinery, but when output becomes large, it becomes convenient to mechanize production which
reduces the needed amount of labour. Another example can be the use of fuel in a chemical
production process (carried out in a fixed plant) that requires heat, such that when the process is run
on a small scale, heat must be provided by flames under the cauldron, which means consumption of
fuel, but the greater the scale on which the process is run, the more heat is produced by the chemical
reaction itself, and the need for fuel consumption decreases: fuel is an inferior input[34].
The relevance of inferior inputs for the sign of ∂S(v,p)/∂vi comes from the fact that locally the
increase in the rental of an inferior input shifts the marginal cost curve downwards, so the p=MC
condition is achieved for a greater q although at a smaller profit, as in Fig. 5.4bis(b) where the
increase in the price of an inferior factor shifts the cost curve from c to c’. This means that if factor i
is inferior, then ∂S(v,p)/∂vi >0, or equivalently
∂S(v,p)/∂vi ≤ 0 only if factor i is not inferior.
The proof that the marginal cost curve shifts downwards, that is, ∂MC/∂vi<0 if input i is
inferior, is an interesting application of the notions studied thus far. The cost function is assumed
twice continuously differentiable, so second-order cross partial derivatives coincide.
Proof. Let input i be inferior; differentiate both sides of xi(v,p)=Hi(v,S(v,p)) with respect to p, and
apply Shephard's Lemma and the coincidence of cross partials to obtain:
xi ( v, p) H i (v, q) S (v, p)  c(v, q) S (v, p)
 c(v, q) S (v, p) MC S (v, p)








.
p
q
p
q vi
p
vi q
p
vi
p
S( v, p)
Since
> 0 and we are assuming that xi is inferior, the conditional demand for factor i decreases
p
MC
x i ( v, p)
when output increases with given input prices, so it is
< 0, hence it must be
< 0. Or also,
v i
p
H i (v, q ) MC

the above shows that
, and the left-hand side is <0 by the definition of inferior input.■
q
vi
I will not go into the intricacies of inferior inputs except to prove that when the production
function is differentiable a necessary (but not sufficient) condition for an input xi to be inferior is
34
Exercise: suppose the chemical process in the text produces output q with chemical input x and gas y as
variable factors, according to the production function {q=f(x,y) with q=x 1/2 if y≥1/x, q=0 if y<1/x}. Assume
px=1/3, py=3, and fixed cost (due to the presence of fixed machinery) is 2. Confirm that y is an inferior input;
derive the cost function and prove that minimum average cost is reached for q=3.
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that it be rival of some other input xj, which means that an increase in xj decreases the marginal
product of xi[35]; this, both to introduce rival inputs, that are interesting and will deserve some
further consideration, and to give a further taste of the formal developments that the neoclassical
theory of production can originate.
Proof. The proof will consist of showing that if no inputs are rival then the rise of the rental of an input
cannot raise optimal output. Let w = 1/p · v be the vector of real factor rentals in terms of the firm's output;
f(x)=q is the firm's production function. Assume that marginal cost is locally strictly increasing so the profit
function π(w) is well defined at the optimal q. Maximization of π(w) determines the unconditional factor
demands x(w). These satisfy the well-known condition wi = ∂f(x)/∂xi, all i, that for brevity I indicate as
Dxf(x) = w,
where Dxf(·) means the vector of partial derivatives of f(·) with respect to the variables in x. Let us assume
that x(w) is invertible, i.e. to each input vector x there corresponds a unique vector w that renders it optimal.
Let w(x(w)) be this inverse function[36]. Since w = Dxf(x), it is Dx(w(x(w)) = Dx2f(x(w)). And by the
derivative rule for inverse vectorial functions, the Jacobian Dx(w(x(w)) is the inverse of the Jacobian
Dwx(w):
Dx(w(x(w)) = [Dwx(w)]–1 = Dx2f(x(w)).
But then, inverting everything:
Dwx(w) = [Dx(w(x(w))]–1 = [Dx2f(x(w))]–1.
Now we use a result in the theory of matrices (cf. e.g. Takayama, Mathematical Economics, 1973, p. 393,
Theorem 4.D.3) that states that if the off-diagonal elements of a square nonsingular matrix are all nonnegative then its inverse is non-positive. If no inputs are rival, the marginal product of no input decreases
when some other input is increased, so the Hessian of the production function has non-negative off-diagonal
elements, hence its inverse is non-positive, and therefore Dwx(w) is non-positive.
Now fix the product price p so we can return from w to v. It is ∂S(v,p)/∂vi=
(
j
f ( x) x j (v, p )
) <0
x j
vi
because we have shown that the second term inside the parenthesis is non-positive for all j. Therefore
absence of rival inputs implies that supply cannot increase when an input price increases. ■
35
Rivalry can arise, for example, if there is some third input x h, whose services co-operate with either xi
or xj, and such that when xj increases it is convenient to allocate xh’s services to co-operate mainly with xj;
or, one input can have direct negative side effects on the efficiency of another input, for example, owing to
chemical interactions, fertilizers might decrease the marginal productivity of pesticides in fruit production.
Of course when the production function is twice continuously differentiable the cross partial derivatives
coincide so rivalry is reciprocal.
36
As I have used x(w) to represent the function that makes x depend on w, its inverse should be actually
represented as w=x–1(x).
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5.14.3. The possibility of rivalry among inputs has some relevance for the marginalist or
neoclassical approach to income distribution. As explained in Chapter 3, according to this approach
each factor tends to earn, in the long period, a rental equal to (the value of) its full-employment
marginal product. In the long period, competitive industries behave like firms with CRS production
functions, and this prevents rivalry if factors are only two, as I prove below; but if factors are more
than two, an increase in the equilibrium use of a factor because of an increase in its supply can
cause a decrease of the full-employment marginal product of a rival factor in fixed supply: thus the
marginal approach does not exclude the possibility that an increase in the supply of a factor causes a
decrease of the equilibrium rental of another factor. However, cases of rivalry appear rare and
specific to certain industries, so marginalist/neoclassical economists unanimously consider the
likelihood of the decrease of equilibrium rental just illustrated to be zero for factors, like most types
of labour, the demand for which comes from very many industries.
Proof that with two factors and CRS there cannot be rivalry. If factors are only two, call them x and y,
since marginal products only depend on the proportion x/y owing to CRS[37], if an increase of x with y fixed
causes a decrease of MPy it must mean that an increase of y with x fixed causes an increase of MP y, and this
means a non-convexity of output as a function of y with x fixed; but such a non-convexity is excluded by
profit maximization plus CRS: in Fig. ??, where the curve represents output as a function of y with factor x
fixed, all points on the segment AB can be reached by a linear combination of the factor vectors (y A,x) and
(yB,x); this generalizes what was explained in Chapter 3 on the difference between technological and
economic total productivity functions or isoquants, and we conclude that with CRS and two factors,
increasing marginal products are impossible. This excludes rivalry. ■
output
B
A
yA
yB
Fig. ??
37
Cf. footnote 11.
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The possibility just mentioned of a decrease of the equilibrium rental of a factor when supply
of another factor increases is due to rivalry, not inferiority: an input is inferior if the demand for it
decreases when output increases at given factor rentals, but in this case with CRS all inputs increase
in the same proportion as output, and I have argued that constant returns to scale industries is the
only legitimate assumption for the long-period behaviour of competitive industries with possibility
of plant replication, while inferior inputs require non-constant returns to scale[38]. Now, the
determination of equilibrium factor rentals is a long-period issue, owing to the complex, timeconsuming adjustments (changes in outputs, shifts of labourers across firms, etc.) required for
equilibrium to be approached on factor markets; therefore input inferiority, differently from rivalry,
is irrelevant for the marginalist theory of income distribution[39].
Input rivalry can also cause ‘perverse’ effects of shifts in the composition of demand for
consumption goods on equilibrium factor rentals. As illustrated in Ch. 3, if one leaves aside
possible ‘perverse’ income effects then a shift in the composition of demand in favour of goods that
use a factor in a higher-than-average proportion tends to raise that factor’s equilibrium rental if
technical coefficients are fixed, and if there is technical substitutability the effect on the factor rental
is normally considered to be of the same sign, only weaker. But if a factor is specialized and used
only by one industry and is rival of other inputs in that industry, then when demand for the
industry’s product rises the industry increases the use of other inputs in order to satisfy the
increased demand and this can cause a decrease of the marginal product of the specialized factor
and hence a decrease of the demand for it if its rental remains the same: the excess supply of the
factor will then cause the rental of the factor to decrease; so it is not impossible that a rise in the
equilibrium output of the industry, although associated with a higher output price, be associated
with a lower equilibrium rental of the specialized factor. This possibility looks exceptional, but it
cannot be excluded.
5.14.4. Finally, we consider ∂xi(v,p)/∂vj. It should be intuitive that its sign is ambiguous: if
factor j is not inferior, optimal output decreases when vj increases, and this would reduce the
38
In the Exercise in footnote 27?? the industry would increase output by increasing the number of optimal
plants (owned by the same firms, or by newly formed firms), each one producing 3 units, thus at the industry
level the inferiority of factor y disappears. Indeed constant returns to scale imply that expansion paths are
rays from the origin.
39
Except possibly for specialized inputs to be associated with other specialized inputs, e.g. special fertilizers
to be used on very special lands for the production of specific products.
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demand for xi if factor proportions were fixed, on the contrary these proportions change (and in
directions that depend on the specific production function) both because optimal factor proportions
change with q even with fixed relative factor prices, cf. Fig. 5.4bis(a), and because relative factor
prices change and this alters optimal factor proportions for each level of q. The possibility that
factor j be inferior adds further uncertainty by rendering the sign of the change in optimal q
uncertain. All this can be analyzed more formally (but without great gains in clarification).
Differentiating x(v,p)=H(v,q(v,p)) with respect to vj, one gets
Hi (v, q) S(v, p)
= (cross substitution effect) + (output effect).

q
v j
This shows that the direction of change of xi is the composition of two directions of change:
∂xi(v,p)/∂vj = ∂Hi(v,q)/∂vj +
along the original isoquant as indicated by the first term on the right-hand side, that indicates the
change in xi if output were kept fixed; and along the expansion path owing to the change in q at
given factor rentals, as indicated by the second term. Both directions are of uncertain sign. The sign
of ∂S/∂vj depends on whether input j is inferior; the sign of
H i ( v, q)
depends on whether input i is
q
inferior. But even assuming that neither input is inferior (so the sign of
Hi (v, q) S(v, p)
is

q
v j
negative), still the sign of ∂xi(v,p)/∂vj for j≠i is uncertain, because the sign of the cross substitution
effect ∂Hi(v,q)/∂vj is uncertain as long as there are more than two factors. The reason is that when vj
rises and hence xj(v,p) decreases, the changes in other inputs are affected by two forces: first, it is
necessary to restore q to its given level, and this would tend to increase the demand for the inputs
other than xj; but second, the optimal proportions among the inputs other than xj will change in
favour of the factors whose marginal product relative to the other marginal products rises when x j
decreases: now, it is possible that the decrease of xj affects the marginal product of xi strongly and
negatively, so much so that the optimal proportion among the factors other than xj changes against
xi so much that the compensated demand for xi decreases.
5.15. Elasticity of substitution.
Cost minimization requires that firms which intend to produce a given output locate
themselves on the point of the corresponding (convex) isoquant where the absolute slope equals the
ratio between factor rentals. A change in the relative rental of two factors may induce no
substitution, little substitution, extensive substitution. For example, if factors are perfect
complements then isoquants (with two factors) are L-shaped and cost-minimizing firms will always
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locate themselves at the kink: changes in relative factor ‘prices’ induce no substitution. In order to
measure the effect of changes in relative factor rentals on the proportions in which firms find it
optimal to combine two factors, economists use the elasticity of substitution. We limit ourselves to
the two-factors case. We have already met this notion in chapter 4. When applied to production
functions, this elasticity – to be indicated now with the symbol σ – is the absolute value of the ratio
of the percentage change in x1/x2 along a given isoquant to the percentage change of the ratio of
their rentals, or of their marginal products[40]:
d ( x1 / x2 )
x1 / x2
 ( x1 / x2 ) v1 / v2
 

=–
.
d (MP1 /MP2 )
 (v1 / v2 ) x1 / x2
MP1 /MP2
The second of the above two expressions for the elasticity of substitution is based on the
assumption that the firm chooses the factor proportion that satisfies –TRS21=v1/v2; thus the
elasticity of substitution measures the sensitivity, of the proportion in which factors are demanded,
to relative factor rentals; its usefulness lies above all in that it gives an indication of what happens to
the relative shares of factors in total cost as relative factor rentals vary. The relative share of factor
1 in total cost is given by (v1x1)/(v2x2), which can be re-written as (x1/x2)·(v1/v2) or (x1/x2)·|TRS21|.
When v1/v2 increases, x1/x2 decreases; an elasticity of substitution equal to 1 means that relative
factor shares in total cost do not change because the two changes neutralize each other. An elasticity
of substitution less than 1 means that when factor 1 becomes relatively more expensive, x1/x2
decreases less than in proportion, so the relative share of factor 1 in cost increases.
This result is used in one-good, two-factor general equilibrium neoclassical models to derive
predictions on factor shares from the elasticity of substitution. In these models the economy
produces with a CRS production function, and income distribution is determined by fullemployment marginal products, hence the product exhaustion theorem holds (cf. footnote 11), and
total factor cost is also total revenue. Changes in factor supplies will then alter factor shares in a
direction that depends on the elasticity of substitution. Thus assume the factors are labour and land,
Definition of σ as positive is nearly universal, and justified by the absence of ambiguity as to the (certainly
negative) sign of the effect of a rise of v1/v2=MP1/MP2 on x1/x2 if there is any effect at all; there is less
consensus on referring directly to the absolute value for the elasticity of demand for a consumption good,
because it is of uncertain sign, owing to the possibility of Giffen goods. (However, Varian 1962 prefers to
define σ without the minus sign, that is, as negative.) Note that the minus sign in the expression for σ
disappears if we replace x1/x2 with x2/x1, while the percentage change remains unaffected: σ12=σ21.
40
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rigidly supplies and fully employed. Suppose labour immigration raises labour supply, causing the
real wage to decrease relative to land rent. Check your understanding of the issues with the
following questions. If the elasticity of substitution is less than one, the percentage decrease of the
ratio of wage to rent, needed to ensure the full employment of the increased supply of labour, will
have to be greater or smaller than the percentage increase in labour supply? and the share of wages
in national income will decrease or increase? The correct answers are in a footnote on the next page.
(The popularity of the Constant-Elasticity-of-Substitution (CES) production function among
macroeconometricians derived from the claim that the share of labour in national income did not
change much for many decades in the USA. The empirical evidence was however immediately
disputed, and certainly looks much less convincing now, because after the 1980s the share of wages
has decreased considerably; furthermore there are formidable aggregation problems behind any
attempt to study an economy as if it were producing a single output; also, it is totally unclear why
technical progress should not alter the elasticity of substitution over the decades; and last but not
least, the validity of the marginal/neoclassical approach to income distribution can be disputed with
strong arguments, as will be explained in later chapters.)
5.16. Integrability of conditional factor demands
We touch very briefly on the duality between some of the notions explained in this chapter.
We have seen that cost function and conditional factor demands stand to the production function in
exactly the same relationship as expenditure function and Hicksian (or compensated) consumer
demands stand to the utility function. Therefore the result reached in consumer theory, that the
expenditure function or the Hicksian consumer demands allow the reconstruction of the utility
function (more precisely, of its convexification), also holds for production theory: the cost function
contains the same economically relevant information as the production function, and from it one
can recover the (convexified) isoquants. Of course this is only possible if the chosen function really
is a cost function, i.e. if there exists a production function that generates it; the conditions
guaranteeing it are listed in the following proposition (we omit the proof, cf. Varian, 1992, p. 85):
Let c(v,q) be a differentiable function which is
(i) non-negative if (v,q) is non-negative,
(ii) non-decreasing in (v,q),
(iii) concave in v, and
(iv) satisfying homogeneity of degree 1 in v ;
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then c(v,q) is the cost function of a production function.
It can be convenient, in applied work, to start directly from a cost function rather than from a
production function[41].
As to conditional factor demands, they can be 'integrated' to yield the production function that
generated them, with the same procedure that derives the utility function from a system of
compensated demands for consumption goods.
5.17. Functional separability: Leontief separability (and weakly separable utility). Suppose
we can separate inputs into two sub-vectors, x=(x1,...,xm) and y=(y1,...,yn), respectively with prices v
and w, and that the production function satisfies the following condition: f(x,y)=f(z(x),y), where z(∙)
has the characteristics of a production function: it is as if inputs x produced a single intermediate
good which then produces the final output in combination with inputs y. This can reflect a true
production of an intermediate good, or be simply a property of the production function.
If z(x) is differentiable and f(z,y) is also differentiable, then ∂f/∂xi=(∂f/∂z)∙(∂z/∂xi); as a result,
the Leontief weak separability condition holds: the marginal rate of substitution between any two
x-goods is independent of the amounts of y-goods:
MRSxj,xi = – (∂f/∂xi)/(∂f/∂xj) = – (∂z/∂xi)/(∂z/∂xj).
Vice-versa if the Leontief weak separability condition holds, then a differentiable f(x,y) can be
written as f(z(x),y) where z(x) is a scalar function. (We omit the proof.)
Under weak separability, the firm can adopt a two-stage cost-minimization procedure: it can
first determine the cost-minimizing input combination of the x-inputs for each level of z, and the
resulting cost of z; and then it can determine the cost-minimizing input combination of (z,y) for
each level of output. If f(∙) has constant returns to scale, so does z(∙); then the cost function for the
‘good’ z can be written as ε(v)z, with ε(v) representing the unit price of z.
The production function must be additively separable if one wants that not only the marginal
rate of substitution between two inputs, but also the marginal product of an input, be independent of
the amounts of other inputs. It is not easy to think of realistic examples to which such an
assumption might apply, except when a firm uses physically separate processes that produce the
41
Answers to the questions posed on the previous page: greater, decrease. Note that the decrease in the share
of wages in national income does not imply a decrease of the total wage bill, because the increased supply of
labour raises total output, it is therefore possible that total wage payments increase, but by a lower
percentage than the increase of total output so that the share of wages decreases.
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same good using inputs of different quality, and one treats the inputs of each process as a single
input because in each process they are combined in fixed proportions: this might perhaps be the
case for some agricultural or mineral product produced on lands, or by mines, of different quality.
5.18. Supply curves: Short-period Marshallian analysis.
5.18.1. A traditional approach distinguishes short-period from long-period profit
maximization and short-period from long-period supply curves of firms. The difference consists in
the fact that in the short period a firm is unable to alter the quantities of some of the inputs.
This can derive from contracts that it would be too costly to modify, for example contracts
that oblige a firm to use the services of a certain accounting firm for a number of years; or from
time constraints, for example moving out of certain buildings to a new location may be a complex,
time-consuming affair. In the end time constraints are the essential element, because, given
sufficient time, everything can be modified.
However, the argument most commonly used to distinguish the short-period from the longperiod supply curve of the firm is the least convincing. This is that once a firm has a fixed plant, the
latter is given for the firm until it becomes so old as to justify scrapping and replacement with a new
fixed plant. This argument is not convincing because firms can always sell or rent out their fixed
plants to other firms, or buy or rent the fixed plants of other firms (a firm is a legal entity, not to be
confused with its plants). If one re-reads the inventor of the short-period/long-period distinction,
Alfred Marshall, one finds that what he had in mind was referred not to the individual firm but to
the industry: the short period was the analytical period within which there was not enough time
significantly to change the amounts available to the entire industry of specialized durable inputs
necessary to the industry and requiring considerable time for their production. Thus Marshall's
short-period analysis of the fishing industry took as given the number of fishing ships and of
experienced fishermen, not the number of firms nor how the fishing ships were divided among
different firms (Marshall ?? p.??). Marshall’s approach permits the determination of short-period
supply curves of industries, without requiring a fixity of factors for firms that is much more difficult
to justify.
However, for completeness we also explain the usual textbook approach. This can be seen as
a pedagogical first step toward a better appreciation of what long-period analysis means, a first step
that also allows a treatment of indivisibilities. We cover this ground quickly because we do not have
much to add to what is learned on these issues in introductory economics courses.
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5.18.2. So let us take the 'fixed plant' of the firm as given in the short period, where the fixed
plant includes all inputs whose (maximum) quantity is treated as given[42]; this means that one can
even omit the fixed plant from the inputs appearing in the production function; the other inputs, the
sole ones affecting production in the short period, will be called variable inputs; the corresponding
production function is called the short-period, or restricted, production function. Cost
minimization can only operate on the variable inputs, and the short-period cost function results from
the minimization of the cost of variable inputs.
Correspondingly, there will be fixed costs and variable costs. Variable cost is the total cost of
variable inputs. The costs that are fixed (in the sense of not depending on the quantity produced) but
only exist as long as the firm exists, i.e. disappear if the firm is closed down, are called quasi-fixed
costs. Fixed costs proper are those costs independent of the quantity produced, that must be borne
by the owners of the firm even if production is discontinued and the firm is closed down. Fixed
costs are due to irrevocable contracts that oblige the owners of the firm to pay them in all instances,
e.g. the repayment of debts. Fixed costs proper do not include, for example, those overhead labour
costs (manager's secretary and analogous accounting labour etc.) independent of the quantity
produced but which can be eliminated by closing down the firm and firing all workers. Fixed costs
do not necessarily coincide with the cost of fixed factors. A firm may have a debt to be repaid, that
causes a fixed interest cost but is due to past expenses and has no connection with the firm’s present
fixed plant.
For the question whether the firm should close down when profit is negative, fixed cost
proper should not make a difference since the owners of the firm must still bear it even if the firm
closes down; quasi-fixed cost on the contrary does make a difference and therefore it must be
included in the variable cost. But for simplicity in what follows there are no quasi-fixed costs.
The (short-period) variable cost function, to be indicated as VC(v,q), results from the
choice of variable inputs that minimizes variable cost for each assigned level of output. Since there
are fixed factors, the short-period production function will exhibit decreasing returns to scale at
least after a certain level of output; as a result, at least beyond a certain level of output VC(v,q) will
increase more than in proportion with output. This is formalized by assuming that the short-period
marginal cost, i.e. the derivative of variable cost with respect to output,
MC(v,q):=∂VC/∂q
42
It is possible not to use the entire amount of a fixed factor; this will be just one instance of the difference
between technological and economic production function.
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is an increasing function of q at least beyond a certain level of output. In what follows I take v as
given and for brevity I often drop the indication of the functional dependence on q. Let us now
define (short-period) average variable cost as
AVC(v,q) ≡ VC/q;
it is possible that initially AVC is a decreasing function of output, indicating that the fixed plant
was planned to be optimal for a certain level of production and up to that level variable cost
increases less than in proportion with output. If all these magnitudes are continuous functions of
otuput we can study the relationship among them. For q=0 it is AVC=MC because for the first
small unit of output average variable cost coincides with the increase in cost. Then if AVC is
initially a decreasing function of q, MC is a decreasing function of q too and is less than AVC (in
order for the average cost to decrease, the additional units of output must cause an additional cost
lower than the average). AVC remains a decreasing function of output as long as MC<AVC. But if
MC becomes an increasing function of q at least from a certain level of output onwards, then sooner
or later it becomes equal to AVC, and from that level of output onwards MC>AVC and AVC
becomes an increasing function of q (because the additional units of output cause an additional cost
greater than the average). It follows that AVC reaches its minimum where its curve crosses the MC
curve; if one knows the two functions, this minimum can be determined simply by solving
MC=AVC for q>0. However, if MC is increasing from the very start, then the minimum AVC is
reached for q=0. (The mathematical proof of these statements is easy and left to the reader as
Exercise; be sure to check the second-order conditions.)
Now define short-period average (total) cost as AC=(FC+VC)/q=AFC+AVC. AFC is
average fixed cost, defined as FC/q. If FC>0 then AC>AVC; the vertical distance between the AC
curve and the AVC curve decreases as q increases, because it measures AFC. For the same reason
as for AVC, AC is a decreasing function of q as long as it is greater than MC, and an increasing
function of q as long as it is smaller than MC, and as a result it too reaches a minimum where it
crosses the MC curve. All these relationships are shown in Fig. 5.5: a particularly important point is
X, where the AC and the MC curve cross each other; this point determines the minimum average
cost, MinAC, associated with the given fixed plant and the given factor rentals, and the
corresponding quantity of output q^. As long as the price at which the firm sells its output is greater
than MinAC, the firm makes a positive profit.
How does the firm maximize profit in the short period? By equalizing marginal cost and
output price. This can be shown as follows. Let R=pq stand for the firm's revenue, and SC(q) for its
short-period total cost function, whose derivative with respect to q is the marginal cost MC(q). Then
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=R–SC(q)=pq–SC(q)
and the first-order condition for a maximum is p–MC(q)=0. The second-order condition is
–dMC(q)/dq<0,
i.e. MC must be increasing where it equals the given output price.
The condition p=MC(q) implies a supply curve of the firm which coincides with part of the
MC curve (except that now the independent variable is the one on the vertical axis) if on the vertical
axis we also measure the output price. The part of the MC curve which coincides with the supply
curve is the part above the AVC curve. The reason is that the firm finds it convenient to go on
producing even if p<AC, as long as p>AVC, because in this way it can at least minimize its loss:
the excess of revenue over variable cost compensates at least partially for the fixed cost which must
be borne anyway. But if p<AVC then the firm minimizes its loss by not producing at all( 43).
43
In concrete situations, a firm may decide to go on producing even when the price is below minimum
average variable cost, if it esteems that this is a temporary situation and that the interruption would damage
profit more than continuing to produce at a loss for a time.
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p
MC=MVC
AC
AVC
p*=MinAC
X
AFC
O
q^
q
Fig. 5.5. Average (short-period) cost AC and average variable cost AVC when marginal cost MC is
initially decreasing; average fixed cost AFC is a rectangular hyperbola and equals AC–AVC. Note that if
AVC included some quasi-fixed costs then the AVC curve would not start at the same level as the MC curve,
but would have initially a shape similar to that of the AFC curve.
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5.18.3. The idea that a firm will continue to operate even when making a loss, as long as
revenue covers at least the variable costs, can derive from considering fixed cost proper a sunk cost,
to be paid whether one continues to produce or not, for example interest on the debt contracted to
start production; but this kind of explanation is endangered by the variable confines of the firm as a
legal entity, for example the uncertain existence of fixed costs proper in the presence of limited
liability. A clearer theory is obtained if one reformulates the question as regarding, not the survival
of the firm, but rather the survival of a fixed plant. The key is to consider the cost of using a fixed
plant a residually determined quasirent. The analogy with land is clarificatory. Imagine that
someone buys a land with borrowed money in order to rent it to firms, and then discovers that the
rent she can earn is insufficient to repay the interest on the debt; this is no reason not to rent the land
out to firms: as long as rent is positive, it is not convenient to leave the land idle; perhaps our owner
will go bankrupt, but then the land will be bought (for a price appropriate to its rent-earning
capacity), and utilized or rented out to firms, by someone else. Fixed plants are like lands in that,
once created, it is best to utilize them as long as they can earn a positive rental. The rental earned by
the fixed plant is not made explicit in the usual formalization of short-period firm cost and profit,
but it can be derived from it because it is the difference between revenue and cost of variable
factors. Indeed, a firm might lease its fixed plants to other entrepreneurs; what maximum rental will
an entrepreneur be ready to pay for the right to use a fixed plant she does not own? A rental equal to
the maximum residual obtainable after subtracting all other variable costs[44] from the revenue one
can earn by operating the fixed plant; such a rental would reduce profit to zero. If the rental is less
than that, the profit of the leasee is positive, and entrepreneurs must be expected to compete for the
right to use the fixed plant, therefore the rental will rise to the zero-profit residual just discussed. If
the entrepreneur is also the owner of the plant, she should include in the costs the opportunity cost
of the use of the fixed plant – the revenue the owner gives up when deciding not to lease the fixed
plant to other entrepreneurs –, and this opportunity cost is the maximum rental thus determined,
called quasirent by Alfred Marshall because of its analogy with the rent of land (the difference is
that fixed plants deteriorate). The entrepreneur who first purchases the fixed plant is in the same
position as the person who purchases a land; she may go bankrupt if the plant's quasirent falls below
the level expected at the time of purchase rendering it impossible to repay the debt incurred to
purchase the plant, but as long as quasirent is positive the plant will not be shut down, it will be
44
Inclusive of quasi-fixed costs if these are positive.
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bought by some other entrepreneur at its new value (the present value of its new expected quasirents
for its remaining economic life(45)) and will be kept in operation.
The considerations just developed have an interesting implication. If one neglects those fixed
costs which are, in the short period, ineliminable and therefore not an opportunity cost (e.g.
payments for debts contracted in the past), then a correct imputation of costs, inclusive of
opportunity costs and hence of quasirents of fixed plants, renders the short-period profit always
equal to zero, even when it appears positive with the usual formalization: the reason why it appears
positive is that short-period total cost as usually defined does not include the quasirent earned by the
fixed factors.
5.18.4. The above considerations also show that in order to determine the short-period supply
curve of an industry what is necessary is the supply curves of the several fixed plants in the
industry; how their property is subdivided among firms is not relevant (as long as efficiency is
independent of ownership). The short-period supply curve of the industry is given by the horizontal
sum of the parts, of the short-period marginal cost curves of the single plants, which lie above the
respective AVC curves.
price
p
a
b
c
q1
d
e
q2
f
q1+q2
Fig. 5.6. Horizontal sum of the short-period supply curves of two price-taking firms having different
minimum AVC. Aggregate supply at price p, the segment ef, is the horizontal sum of the supplies of the two
firms, the two segments ab and cd.
(In order to determine the short-period supply curve of the price-taking firm the consideration
of fixed costs, as well as of the AC curve, is not necessary, but these notions become necessary for
45
We are here introducing a rate of interest. This part of the chapter is concerned with the theory of the
firm, and we want to present this theory so that it is applicable also to economies where there is a rate of
interest.
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long-period analysis, and this is why they have been introduced.)
5.19. From short-period to long-period supply
5.19.1. The short-period AC and MC curves were derived for a given fixed plant and given
fixed and quasi-fixed costs. We can now extend the analysis to the long period i.e. to the analytical
situation where we treat all inputs as variable, by imagining that the firm can choose among a set of
fixed plants, and for each one of them it can derive the AC and MC curves and find the minimum
average cost and the associated output level. We need not assume perfect divisibility of the
elements which go to form a fixed plant: the firm can be confronted with a finite number of
alternative fixed plants, for each one of which it can determine the AC and MC curve. The smallest
of the minimum average costs associated with the several alternative fixed plants is the true
minimum average cost, to be indicated as MinLAC; let us indicate the associated output as q*.
The long-period average cost curve LAC is the inferior envelope of the short-period average
cost curves, cf. Fig. 5.7. Thus each point of the LAC curve is associated with a type of fixed plant,
but not generally with that fixed plant’s minimum-average-cost output. If the fixed plant consists of
a single factor whose amount can be varied continuously (no indivisibilities), then each short-period
supply curve has a point in common with the long-period supply curve, where the two curves have
the same slope.
LAC
q
Fig. 5.7
Proof. Assume the fixed factor is factor n; consider the unconditional long-period demand xn(v,q) and
assume v is given; for a given level q° of output, cost minimization determines a demand xn(v,q°) for factor
n; if its fixed amount is just xn*=xn(v,q°), then for q different from q° long-period cost cannot be greater than
short-period cost SC, because the additional short-period constraint (the fixed amount of xn) cannot possibly
permit a lower cost and will generally imply a higher cost, hence c(q)≤SC(q,xn*); and for q=q° long-period
and short-period cost coincide; hence the long-period average cost curve LAC coincides at q° with the shortperiod average cost curve based on xn=xn*, and is not above it (and generally below it) for q different from
q°. For each q°, there will be a xn*=xn(v,q°) for which one can repeat the above reasoning; if both kinds of
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average cost curves are ‘smooth’, at the q° for which the given xn* is optimal the two curves must be tangent
■
to each other. Repeating the reasoning for all q° completes the proof.
Note that some of the short-period AC curves in Fig. ?? have been drawn as having no point
in common with the LAC curve. The reason is that if the fixed plant consists of several factors
combined in fixed amounts, the combination may be a suboptimal one for all output levels; then the
AC curve corresponding to that fixed plant will be everywyere strictly above the LAC curve.
If there is perfect divisibility and constant returns to scale, then MinLAC can be reached for
any q; if there are indivisibilities and replicability of plant, then there is a minimum efficient scale
of output q* that allows the firm to achieve an average cost equal to MinLAC, and the firm can
reach the same minimum average cost by producing 2q* with two fixed plants identical to the one
which produced q*, or by producing 3q* with three fixed plants, etc. The AC and MC curves with
two fixed plants are the curves with one fixed plant, 'stretched' rightwards so as to reach the same
value on the ordinate for a double value on the abscissa. In Fig. 5.7b we see the AC curves and MC
curves with one, two and three fixed plants of the same type. A price-taking firm considers that it
can sell any amount of product at the given price, therefore as long as the output price p is greater
than MinLAC, the firm finds it convenient to grow without limits by replicating infinite times the
plant associated with MinLAC; if p=MinLAC, then the firm's maximum profit is zero and the firm
is indifferent between producing q*, 2q*, 3q* etcetera; if p<MinLAC, in the long period the firm
does not produce.
AC
MC1
AC1
MC2
AC2
MC3
AC3
MinLAC
q*
2q*
3q*
Fig. 5.7b. Average and marginal cost curves with one, two or three identical plants.
5.19.2. Let us see how these considerations connect with profit maximization.
Mathematically, the problem is
maxq (q)=R(q)–C(q)=pq–C(q)
where C(q) is the long-period cost function, and R=pq is revenue. For levels of production requiring
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the use of a high number of fixed plants, if replication of plants does not cause a decrease in
efficiency it is legitimate to treat C(q) as proportional to q, because average cost remains nearly
constant as q varies owing to the fact that the variability in the number of plants ensures that, even
when q is not an integer multiple of q*, each plant produces a quantity very close to q*; for example
in betweeen q’=100q* and q”=101q* the production of each plant differs from q* by at most 1%
and therefore – remembering the U-shape of the average cost curve of a plant – average cost is very
nearly equal to MinLAC; in Fig. 5.7 this is evident already with three plants. Therefore in this case
it is legitimate to treat the long-period average cost as constant, equal to MinLAC. It is then also
equal to the long-period marginal cost LMC, the derivative of the long-period cost function c(q).
Therefore if p>LMC=MinLAC, no q satisfies the condition p=LMC for a price-taking firm; the
profit (q)=(p–MinLAC)q increases without limit by increasing q; the problem max q =pq–c(q) has
no solution. If p=MinLAC, supply is indeterminate because π=0 for any output level; only
p<MinLAC yields a determinate solution, q=0. Therefore in this case there is no supply function of
the firm, the firm’s supply is either zero, or infinite, or indeterminate.
But this fact does not create problems to the theory. All one needs to assume is that if
p>MinLAC the firm will plan to expand productive capacity (i.e. cost-minimizing output) by
expanding or replicating plant, and if p<MinLAC the firm will plan to reduce productive capacity;
but variations of productive capacity take time, and since the firm will not be the only one to take
such decisions, and since it is unlikely that there be perfect synchronization of the decisions of the
several firms (possibly including new entrants), it is legitimate to assume that generally the
expansion or contraction of industry productive capacity will be gradual; thus the short-period
supply curve shifts gradually, and the short-period equilibrium price, determined by the intersection
of the demand curve with the short-period supply curve, tends toward MinLAC[46]. If the minimum
average cost is not the same for different firms, the less efficient firms will be eliminated by
competition, and only the firms with the least MinLAC will survive: in the long period, competition
enforces productive efficiency in the sense of minimization of average cost. The total number of
plants will be such as to bring output price as close as possible to MinLAC without falling below it.
The industry supply curve is derived as follows. Let LMCn(q) stand for the horizontal sum of
the long-period marginal cost curves of n identical efficient plants, with q their total output and q*
46
Synchronized decisions to alter productive capacity might cause phenomena akin to cobweb cycles (cf.
Ch. 6 §??) but for long-period decisions such phenomena are less likely because there is time to revise
decisions in the light of information on what other producers in the industry are doing.
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the single-plant minimum-average-cost output; the supply curve consists of a discontinuous series
of upward-sloping segments yielding a saw-like shape, the n-th segment being the portion of the
LMCn(q) curve corresponding to the semi-open interval [nq*, (n+1)q*). All segments start at a
height equal to MinLAC; as n increases the segments become flatter, the supply curve approaches a
continuous horizontal line at a level equal to MinLAC. If the downward-sloping demand function
for this good crosses this supply curve more than once, the equilibrium intersection is the one
corresponding to the greatest output at a price not less than MinLAC, qE in Fig. 5.8.
LMC1
demand curve
LMC2
MinLAC
q*
2q*
3q* etc.
qE
q
Fig. 5.8. Long-period industry supply curve with plants that reach minimum
average cost at output level q*, and a demand curve crossing the supply curve twice. In
this case in equilibrium there is room for 6 plants. The LMC curves are drawn as
straight-line segments only for simplicity.
Let us for example suppose that, at p=MinLAC, the demand for the industry's output falls
between 100q* and 101q*. There is therefore room in the industry for 100 plants, each one
producing slightly more than q*, and therefore having a long-period marginal cost just slightly
above MinLAC. The equilibrium price will be so close to p=MinLAC, that to assume that the
equilibrium price is equal to MinLAC is an excellent approximation. There is no room in the long
run for 101 plants because LMC101(q) crosses the downward-sloping demand curve at a price below
MinLAC, and firms would make losses. The conclusion is that, with perfect replicability of plants,
as long as the output of a single plant is only a small fraction of total output, the long-period
industry supply curve can be treated, to all relevant purposes, as a horizontal straight line (if input
prices are given) at a level equal to minimum average cost.
However, one may feel uneasy with the fact that the theory does not determine the size of
individual firms. We pass to discuss this issue.
5.20. The size and the number of firms
Economists still discuss on the issue of the long-period equilibrium size of competitive firms.
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When there is product differentiation, the size of a firm is limited by the market for its product(s),
the question becomes what determines the size of that market, and answers are not difficult to find:
the quality of the product relative to the tastes of consumers, if it is a consumption good; the
number of firms that use technologies needing that product, if it is a productive input; the extent of
competition from rival products, and the marketing strategies of the firm and of its rivals. Now, it is
important to understand that perfect product homogeneity is a very rare occurrence; usually
location, or the importance for customers of past tradings that have built confidence in their usual
suppliers, are sufficient to render it costly for a firm to extend its sales by subtracting customers
from other firms, although often the divergence from perfectly competitive behaviour remains small
enough that one may continue to apply the long-period theory of the perfectly competitive industry.
In other cases the size of the firm is determined by the nature of the product: rock bands are firms
too, and their product requires a certain size of the ‘workforce’; there is no possibility of replicating
the ‘plant’ identically within the same firm. Similar considerations apply to all firms based on a
strict, creative interaction among few persons. Apart from these cases, one finds the disagreement
among economists mentioned in §5.6.3, with some (e.g. Edith Penrose) arguing that in many
instances firms are able to grow to enormous sizes without any increase in average cost and
therefore the limits to size must be found either on the demand side or on the need for own capital
or collateral, and others arguing that the general case is U-shaped LAC curves because of coordination difficulties that increase with size.
In the discussion whether LAC curves are U-shaped or not, we find here the second meaning
of returns to scale mentioned in §5.4, returns to the scale of total cost or, briefly, (scale) returns to
cost, obviously a notion that assumes given input prices. These returns are defined by the elasticity
of output to total cost, and need for their definition neither that all inputs be increased in the same
proportion, nor differentiability of the production function, nor divisibility of all inputs; as total cost
increases, there may well be discontinuous changes in the quantities employed of some inputs, e.g.
some capital goods may be replaced by capital goods of a different type, the fixed plant may
change, or fixed plants may be indivisible and be discretely increased from one to two, three etc.;
there may then be some discontinuities in maximum output as total cost increases, and the point
elasticity of maximum output to total cost will not be defined at those points; but everywhere else,
and everywhere for discrete changes in total cost, the elasticity of output to cost will be well
defined, and therefore returns to cost is a more general notion than technical returns to scale. When
returns to cost are constant, output increases in the same proportion as total cost, and therefore
average cost is constant; when returns to cost are increasing, successive increases in total cost yield
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increasing returns i.e. bigger and bigger increases in output, so average cost is a decreasing function
of output; when returns to cost are decreasing, average cost is an increasing function of output.
‘Constant returns’ is also used for the case when MinLAC is reached only for the optimal outputs
corresponding to replication of indivisible plants. ‘Scale economies’ is another term used to indicate
increasing returns to cost.
Assuming now that maximum output is a continuous function of total cost and that the LAC
curve is U-shaped, MinLAC is reached where the returns to costs thus defined, in passing from
locally increasing to locally decreasing, are locally constant. If the production function is
differentiable with respect to all inputs[47], then the locally constant returns to costs at MinLAC,
that is, the equality of average and marginal cost, imply locally constant technical returns to scale. I
v
prove this for the two-factors case. At the point of minimum average cost it is MC = 1 =
MP1
v x v x
vx
v x
v
v x v x
= 2  AC  1 1 2 2 ; this can be re-written f(x1,x2)= 1 1 2 2  1 1  2 2 =
v1
v2
MP2
f ( x1 , x2 )
MC
MP1 MP2
MP1x1+MP2x2 , which implies that the production function is locally homogeneous of degree 1.
This implies that, if p=MinLAC, the payment to each factor of its marginal revenue product
exhausts revenue. Thus the fact that the long-period cost curve is U-shaped entails no contradiction
between assuming zero profits of competitive firms in equilibrium, and assuming that each factor is
paid its marginal revenue product.
If the firm's long-period AC curve is U-shaped, the firm's dimension is no longer
indeterminate: profit is maximized when LMC(q)=p. When this is the case, the long-period industry
supply curve is derived in the way already shown, with ‘firms’ replacing ‘plants’ in the reasoning:
the number of firms is endogenous, because competition also means free entry, and in the long
period there is time for entry. The conclusion is again that, as long as the minimum optimal
dimension of firms is small relative to total industry output, if factor prices (factor rentals) are given
47
Then the local elasticity of scale e(x) = [df(tx)/f(tx)]/(dt/t) evaluated at t=1, can be defined (§5.6.1), and if
the input vector x is cost-minimizing at the given factor rentals v then e(x)=AC/MC. Proof: rewrite the

n
i 1
elasticity of scale as e(x) =
f (x)
xi
xi
. Since x is cost-minimizing, it satisfies vi = pf(x)/xi, where the
f ( x)

product price satisfies p=MC. Therefore e(x) =
n
i 1 i i
vx
pf (x)

c( v, q) / q AC

. This confirms that returns to
p
MC
scale are locally increasing or decreasing depending on whether AC is greater or less than MC.
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then to all practical effects the long-period supply curve of the industry is horizontal at a price equal
to minimum average cost.
But even when the U-shaped cost curve is not accepted, the supply curve of the industry
remains horizontal at the MinLAC level; the size of the firms composing the industry is then simply
irrelevant. One can for example take it as determined by historical accidents, or by limits to the
growth of individual firms deriving from limits to possible indebtedness.
In conclusion both when there are, and when there aren’t, constant returns (to cost) by firms,
the assumption of competition with free entry implies an essentially horizontal long-period industry
supply curve once input prices are given, as long as the minimum quantity that allows a firm to
minimize average cost is small relative to the total demand forthcoming at a price equal to that
average cost. When this is the case, since in the long period competition eliminates inefficient firms,
one can assume a common technology within the industry. We can therefore treat the industry's
long-period aggregate production function as exhibiting constant technical returns to scale even
when we cannot do so for single firms.
This conclusion will allow us, when in Part III of the chapter we formulate the neoclassical
competitive general equilibrium with production, to assume product prices equal to minimum
average costs, and that at those prices supply adapts to the demand forthcoming at those product
prices (and at the factor prices that determine them).
So far in this long-period analysis we have taken all input prices as given. In this case, except
for flukes only one production method will be the cost-minimizing one for the production of each
product. However, when we come to determining factor rentals endogenously, it may well happen
that two (or more) methods will co-exist in the production of the same product: the different factor
employments need not imply differences in average cost if factor rentals, for some of the factors,
adapt so as to ensure the same average cost with both methods. One typical such case is that of
extensive differential land rent: when the same product is produced on lands of different fertility, a
differential rent will arise on more fertile lands, that will make the utilization of different lands
equally convenient. Here we need not add to what was said on this topic in Chapters 1 and 3.
5.21. Aggregation
5.21.1. When the number of firms in an industry is given, and for each firm a profit function,
and therefore a supply function, exists, then the industry’s competitive supply can be determined as
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if forthcoming from a single multiplant price-taking firm that operates all the individual production
functions[48]. Formally, the aggregability condition is that, given the individual production
possibility sets (whose elements are netput vectors) Y1, ... , Yj, ... ,YJ of the individual firms (where
J is the number of firms), the aggregate production possibility set be
Y = Y1 + ... + YJ = {yRn: y = ∑j yj for some yjYj, j=1,...,J}.
In words, the aggregate firm must have no additional production possibilities at its disposal
beyond a simultaneous activation of the production processes available to the individual firms, at
most one per firm. In this case, the aggregate firm can do no better in terms of profits than the sum
of the individual firms' profits, because it can do no better than copy what the individual firms
would choose autonomously.
When this is the case then, conversely, the firms in the industry maximize aggregate profits;
since profit maximization requires cost minimization, this also shows that the allocation of the
industry output among the several firms in the industry is cost-minimizing. Indeed, since for each
firm output price equals marginal cost, marginal cost is the same in all active firms, which is an
obvious efficiency condition: if marginal cost were different in two (active) firms, a small transfer
of production to the firm with the lower marginal cost would decrease aggregate cost. Analogously,
for each variable (and hence, transferable) factor the marginal product will be the same in all firms
where the factor is used, because equal to the factor rental divided by the output price( 49), again an
obvious efficiency condition: if the marginal product of a factor were not the same in all firms (and
lower in the firms not using it), transferring a small amount of the factor to the firm where it has the
greater marginal product would increase total production. These considerations show that the
aggregate firm obeys the conditions for profit maximization when each individual firm does.
5.21.2. In long-period analysis with free entry, again the industry’s behaviour can be derived
as coming from the decisions of a single firm, a constant-returns-to-scale firm which, for each
48
This is true as long as the possibility is excluded that a single management of all the factors of the
individual firms (including the fixed factors which need not explicitly appear in the short-period production
functions) would achieve cost reductions. For example, the fusion of five small farms into a single big farm
might permit the utilization of big agricultural machinery which was uneconomical for each individual farm;
or there might be unnecessary duplication of some indivisible factors (for example, each separate farm might
need to buy its own tractor if no sharing is allowed, while four shared tractors would suffice for the five
farms). Only when one excludes such phenomena can one conclude that the industry behaves in the same
way as if a single firm were to operate all the individual production functions.
49
Assuming that marginal products can be defined. The reader is reminded that factor rentals must equal
marginal revenue products.
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vector of factor rentals and each output level, adopts the factor employments corresponding to
minimum average cost[50]. Competition eliminates less efficient firms and thus causes the
production function to become the same for all firms. If the single firms have a CRS production
function, then the industry acts like a giant firm with that same production function. If the
individual firms’ production function yields U-shaped average cost curves, then (assuming a
sufficiently small minimum efficient size relative to aggregate output) because of the possibility of
replication of plants or firms the industry acts like a single giant CRS firm, with a production
function which, for each vector of relative factor rentals, yields the same optimal factor
employments per unit of output as the average-cost-minimizing choice of the individual firms.
We illustrate with two numerical examples. Assume the firm's production function is q=xαyα,
then returns to scale are locally increasing, constant or decreasing according as α is greater, equal or
less than 1/2; we want a U-shaped long-period average cost curve, hence α must be initially greater
than 1/2 and decreasing in q; suppose then that α = 4/10 + 10/(xy), which yields α>1/2 if xy<100
and α<1/2 if xy>100. Minimum average cost is achieved where returns to scale are locally constant
i.e. at α=1/2, xy=100; all firms in the long period produce 10 with such amounts of x and y that
xy=100, while of course the ratio x/y in which the factors are employed depends on their relative
rentals. The industry has production function q=(xy)1/2. Second example: assume the firm’s
production function is q = 2(1+x1–1x2–1)–1 . If both factors are multiplied by a scalar t, we obtain q(t)
2
=
; it is convenient to put x1x2=1/A; then q(t) = 2t2/(t2+A); and the scale elasticity of
1
1 2
t x1 x2
output (remember that it is evaluated at t=1) is e=2A/(1+A) which is ⋛1 according as A⋛1. Thus
for each given factor proportion this function produces a U-shaped cost curve, because it exhibits
locally increasing returns to scale when x1x2<1, locally decreasing returns to scale when x1x2>1,
and locally CRS (and minimum average cost) when x1x2=1 i.e. when q=1. Therefore in the long
period each firm produces one unit of output, and the isoquant map of the industry’s long-period
production function is the radial expansion of the isoquant x2=1/x1 associated with q=1; in order to
know how much an industry input vector (x1,x2) produces, we must view it as t times the vector
(x1*,x2*) that produces 1 unit of output with the same factor proportion as (x1,x2), where t is
determined by x1=tx1*, x2=tx2* such that x1*x2*=1; in other words, inputs (x1,x2) produce t units of
output where t2 = x1x2. Hence the industry’s production function is, again, q = (x1x2)1/2.
50
Here as elsewhere in this chapter it is assumed that the average-cost-minimizing factor proportions are
uniquely determined.
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We can use the last example to clarify a formal problem arising with U-shaped cost curves of
individual firms in long-period analysis with free entry. Suppose that factor rentals are v1=v2=1;
then in this example MinLAC=1. Suppose that the demand curve is decreasing, and at the product
price p=1, demand is 100.5 units. There is room for 100 firms; if there were 101 firms, price would
go below 1 and all firms would make negative profits. But with 100 firms producing 1 unit each,
demand is less than supply, so the equilibrium price must be slightly above 1, each firm will
produce slightly more than 1 unit when setting long-period marginal cost equal to the equilibrium
price, and profits will be positive; hence, if we assume that firms enter as long as p>MinLAC, there
will be entry; no equilibrium exists if we define it as simultaneously requiring rigorously
demand=supply and profits=0. However, if the aim is to determine a long-period equilibrium (the
average situation around which the economy oscillates), this is not a problem because even if firms
do enter and bring the total number of firms above 100, the number will subsequently decrease, and
we can still assume that the average around which the price oscillates is 1. Long-period equilibrium
only aims at determining the average around which actual market variables oscillate. (Furthermore
it is plausible that potential entrants try to make an estimate of the effect of their entry, and
generally they will realize that it is likely that the small profits associated with 100 firms will
disappear if they enter, and hence will not enter; but the excess of the equilibrium price over
MinLAC will be negligible.)
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PART II
5.22. PARTIAL EQUILIBRIUM
5.22.1. As a first step toward the study of the marginalist/neoclassical competitive general
equilibrium of production and exchange, we discuss in greater detail the construction that, more or
less explicitly, we have been using in this chapter (for example in Fig. 5.8?? and in the text
discussing it): the determination, via the intersection of a supply curve and a demand curve, of the
so-called particular equilibrium, as it was originally called, or (its current usual denomination)
partial equilibrium, of a single market studied in isolation. The prices and quantities on other
markets are taken as given and are considered essentially unaffected by changes in the market under
study: this is called the assumption of coeteris paribus (latin for 'other things remaining the same').
A famous 1926 article describes this approach as follows:
This point of view assumes that the conditions of production and the demand for a
commodity can be considered, in respect to small variations, as being practically
independent, both in regard to each other and in relation to the supply and demand of all
other commodities. It is well known that such an assumption would not be illegitimate
merely because the independence may not be absolutely perfect, as, in fact, it never can be;
and a slight degree of interdependence may be overlooked without disadvantage if it
applies to quantities of the second order of smalls, as would be the case if the effect (for
example, an increase of cost) of a variation in the industry which we propose to isolate
were to react partially on the price of the products of other industries, and this latter effect
were to influence the demand for the product of the first industry. (Sraffa 1926, p. 538)
The partial equilibrium approach can also be used for the study of imperfectly competitive
markets, for example a monopolistic market.
Sometimes it may be legitimate to isolate not one, but two interdependent markets: one
example is the study of the effects of changes of the supply conditions and hence of the price of one
product on the demand and hence on the equilibrium price of a complementary or of a substitute
product; another example is the determination of the supply curve of an industry that uses a
specialized factor, whose rental rises when, owing to a rise in the demand for the industry’s product,
the industry’s output rises.
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5.22.2. The partial-equilibrium supply curve of a competitive industry can be a long-period or
a short-period one. The long-period supply curve is horizontal if factor rentals are given[51]. The
short-period supply curve is upward-sloping even with given factor rentals, owing to the given
amounts of some factors. Their derivation has been illustrated already; we add some considerations
on their legitimacy and on the analogy between them.
Besides needing the price-taking assumption, the use of a partial-equilibrium long-period
supply curve is legitimate in two cases. The first one is when the industry uses only a small fraction
of the total supply of each one of the factors of production it utilizes; then a variation in the quantity
produced by that industry will not exert an appreciable influence on its factors’ rentals, because it
will cause a very small percentage variation of the demand for them. As a result, factor rentals can
be taken as given. If the influence on the rental of some factor were significant, this would alter the
cost conditions of other products too and then their prices would change, rendering the coeteris
paribus assumption illegitimate.
The second case is when there is a specialized factor demanded only by the industry one is
isolating. This might be for example a special type of land indispensable to (and only demanded
for) the production of one product, say a famous wine. In this case the specialized factor’s rental is
determined endogenously; it will rise as demand for the product rises, so as to maintain the
producers’ profit at zero; the marginal product of the remaining factors (whose rentals are given)
decreases as output increases; the supply curve is upward-sloping. The independence between
supply curve and demand curve, necessary for partial equilibrium analysis, additionally requires
that the changes in the incomes of the owners of the specialized factor do not appreciably influence
the demand for the product of the industry under analysis.
Marshall generalized this second case to include cases where the supply to the industry of
some specialized factors, differently from the supply of specialized land, is variable in the long
period because those factors are produced factors, but it varies sufficiently slowly relative to the
supply of the other factors for it to be treated as given in shorter-period equilibration processes.
Some types of fixed plants may indeed take a long time to be built; this authorizes treating their
supply as fixed for equilibration processes on time horizons of, say, a few months. (As pointed out
earlier, what is needed for the determination of the short-period supply curve and hence for short-
51
This assumes that if there are profits to be made, there will always be someone (existing or new firms)
ready to add further plants in the industry. This assumption appears strongly confirmed by experience when
adequate account is taken of barriers to entry, that we are assuming absent here and will be discussed in
Chapter 11.
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period partial equilibrium analysis is that the supply of fixed factors to the industry, not to each
firm, be given.)
The supply curve is valid only for comparatively small variations of the quantity produced,
because for considerable variations the coeteris paribus condition becomes doubtful[52]. Particular
difficulties arise when one attempts comparative statics of partial equilibria of a capital good: a
change in the supply conditions of a capital good (due for example to the discovery of a new,
cheaper production method, or to a tax) alters the costs and hence the prices of all goods for whose
production that capital good is used, and many of these may be in turn inputs to the production of
that capital good, so the supply curve of the capital good shifts for more reasons than the direct
effect of the first change[53]; and the derivation of the demand curve is not easy to conceive.
5.22.3. Alfred Marshall attempted to argue that the long-period partial-equilibrium supply
curve of a product can also be downward-sloping, owing to two effects of increases of the
dimension of an industry: first, the possibility better to exploit scale economies; second, an increase
in external effects or externalities. He was thus trying to make room in the theory of partial
equilibrium for a phenomenon no doubt often observed, an association between increase in
production and decrease in price of a product produced by an industry where it was difficult to deny
the existence of competition among producers. But in two articles, in 1925 and 1926, Piero Sraffa
showed that the decreasing supply curve is incompatible with competitive partial equilibrium. He
remembered that the existence of unexploited scale economies is incompatible with competition
with undifferentiated products, because competition requires firms to be rather small relative to total
industry demand, and the perfect substitutability for the buyer among the products of the different
firms in the industry implies that any small price reduction by a firm will attract to the firm enough
buyers to make it able to sell the increased output that allows the exploitation of scale
economies[54]. As to externalities, he noticed that the positive external effects due to increases of
52
Marshall Principles p. 384 fn. of the original 8th edition (1920; p. 318 fn. in the after-1949 reset
editions): “the ordinary demand and supply curves have no practical value except in the immediate
neighbourhood of the point of equilibrium.” The reason is partly different for demand curves, cf. below in
the text and also §??(consumer surplus).
53
Changes in input use can be sometimes very surprising when these interrelations are taken into account;
their exploration has started only recently and is still proceeding, cf. Opocher and Steedman ??
54
Marshall had argued that unexploited scale economies exist, but require time to be exploited because firms
are slowed down in their expansion by the need for collateral and therefore for accumulated profits, and this
prevents firms from becoming indefinitely large because the founders of successful firms pass the firm to
their children who are much less competent and cause the firm to decline and die; he compared the firms in
an industry to trees in a forest, some of which are growing, while others are dying. This picture is
(cont. next page →)
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economic activity, e.g. greater ease in finding repairmen or transportation firms or skilled workers,
are very seldom internal to an industry, they generally concern large groups of firms belonging to
different industries but connected by common location or by similar needed skills; these effects
cannot be admitted in the partial equilibrium analysis of one industry because they extend to other
products, altering their prices, and therefore violating the coeteris paribus condition. Sraffa
concluded, and subsequent economists have admired the cogency of his critique, that competitive
long-period partial equilibrium theory can admit only constant-cost industries, or increasing-cost
industries in the sole case of an industry being the sole demander of a specialized factor. (When the
expansion of an industry affects the rental of a factor also used by other industries, then the costs of
these other industries are affected as much as in the first industry, the prices of the products of those
other industries are relevantly affected, and again the coeteris paribus condition does not hold.) This
does not mean that unexploited scale economies do not exist, it only means that their causes and
effects require a different approach: Sraffa suggested to abandon the assumption of perfect
competition and admit that the general case is rather one of differentiated products, which, if
coupled with free entry, ensures nonetheless a broad tendency of prices toward average costs[55].
5.22.4. The demand curve is a function that specifies the quantity demanded of the product
under investigation as a function of its price. The legitimacy of assuming the existence of a partialequilibrium demand curve requires:
1) Given prices of other goods56; this implies a given income distribution. (This in turn
implies that partial equilibrium analyses cannot study changes in the price and quantity of a product
induced by changes in income distribution.)
2) Given incomes of consumers. This requires, in addition to a given income distribution, a
occasionally confirmed by facts, but nowadays more and more it is the case that firms are owned by many
shareholders and run by hired managers, and a decline due to incompetent owner-entrepreneur heirs is rare;
therefore a decline before scale economies can be exploited is generally implausible; the more so, because
there are more and more giant firms, conglomerates which have the financial potential to set up very large
firms from the start.
55
The reading of both Sraffa’s articles, the 1925 and the 1926 one??refs in English, is strongly recommended
as they are excellent examples of penetrating reasoning attentive to the economic justification of theoretical
constructs.
56
It is also possible to consider the demand curve as derived under an assumption that the prices of some
strongly interconnected goods change when the quantity demanded of the first good changes; for example if
one considered probable that a big increase of taxes on gasoline would considerably decrease the demand for
cars, any estimate of the demand curve for gasoline not restricted to the very short period should try to take
into account the effect on the car market, including the possible effect on the price of cars. Fortunately, for
most industrially produced goods the price is rather insensitive to demand, cf. Chapter 11.
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given level of utilization of resources (in particular, a given level of labour employment): this was
traditionally justified by an assumption of full utilization of resources, the tendential result of the
working of market economies according to the marginal approach, as we know from Chapter 3.
3) Given preferences, unaffected by actual non-equilibrium consumptions.
These three sets of conditions too are subsumed under the expression coeteris paribus. These
givens are, and must be, assumed not to change (or more precisely, to change only negligibly)
during the adjustment processes toward the partial equilibrium, otherwise the equilibrium would
lack the persistence necessary to give a good indication of the average behaviour of the market; it
would be impossible to assume, for example, that the demand curve has the persistence that allows
a monopolistic firm to form a reasonably correct idea of its position and slope (a necessary premise
to the derivation of the marginal revenue curve).
It is then clear that the partial equilibrium method requires that income distribution and
aggregate demand can be considered given (although not necessarily determined according to the
marginal/neoclassical approach), and that preferences can be considered sufficiently unaffected by
the disequilibrium adjustments. On this last issue, we have remembered at the beginning of chapter
4 Marshall’s own admission that experience irreversibly affects tastes. We may add here that when
a price changes significantly, obliging consumers to a relevant change in consumption habits, it
seems plausible that people will not know in advance how their own behaviour is going to change;
they will experiment and discover and develop new consumption habits that could not be predicted
from past evidence, and that often can only be described as due to a formation, or discovery, of
preferences until then undefined[57]. As a consequence, the assumption of a well-defined demand
curve for prices different from the prevailing one may be questioned, as again admitted by Marshall
himself (cf. above ch. 4 §4.22): hence the assumption (that will be met frequently in Chapter 11)
that firms know the demand curve facing them must be treated with suspicion[58].
But we must introduce the reader to the dominant analyses, so we do not further question the
57
The dependence of preferences on experience can be used for a further criticism of the marginal
approach. It is not only that if consumption of a certain good has never been experienced, the preference for
it cannot but be vague, and open to modification by experience. There is also the fact that repeated
experience can permanently alter preferences (e.g. people can develop a taste for listening to certain kinds of
music, for drinking good wine, for practicing certain sport activities). Now, whether and how many times a
good is experienced can depend on prices. This questions the assumption of preferences independent of
prices, on which the marginalist/neoclassical determination of equilibrium is based.
58
Firms must have had the possibility to explore how demand depends on price in an economic situation
undergoing very little change in the variables impounded in the ceteris paribus clause. This may be an
acceptable assumption in some cases, but not in many other ones; for the latter cases, one will need theories
explaining firm behaviour without an assumption that there is a well-defined and known demand curve.
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notion of partial-equilibrium demand curve; however, let us not forget that this notion can be
considered reasonably well defined only for consumption goods (and perhaps for some non-basic
capital goods), only for rather small departures from the until then prevailing price, and only as long
as the incomes of consumers (hence income distribution and the aggregate level of activity of the
economy) are given.
5.23. Stability of partial equilibria.
5.23.1. If the good is homogeneous (undifferentiated) then on average all units of the good
must sell at the same price. Of course this can only be approximately true but, as already pointed
out several times, the equilibrium can only aim at describing the average resulting from the trialand-error higgling of the market. We can speak therefore of ‘the’ price of the good.
Given a supply curve and a demand curve, that represent supply price and demand price as
functions of the quantity of the good[59], equilibrium obtains where aggregate demand for the good
equals aggregate supply, or equivalently where supply price and demand price coincide.
Let us examine the stability of equilibrium. Apart from the implausible case of Giffen goods
the demand curve for a consumption good can be assumed to be downward-sloping. (For capital
goods the issue is more complex, and as argued earlier the partial equilibrium method is seldom
acceptable, but an argument can still be put forward to the effect that an increase in the price of a
capital good, with other factor prices given, will tend to reduce the demand for that capital good,
both because of technical substitution, and because of the rise in cost and hence in relative price of
the consumption goods using that capital good as an input.) The supply curve is either horizontal, or
upward-sloping[60]. In the latter case the stability of equilibrium is clear, under the assumption that
The Marshallian preference for considering price the dependent variable – thus supply price is the price
necessary to induce a given supply to be forthcoming, and demand price is the price that induces a given
demand – has the advantage that one can still speak of a supply function even when the supply curve is
horizontal.
60
Actually, Marshall considered at length the possibility that the long-period supply curve of a product be
decreasing, owing to economies of scale achieved by an average increase of firm dimension with the growth
of industry size, or owing to cost reductions due to economies of scale external to the firm but internal to the
industry, a special case of externalities. The first cause was soon judged incompatible with the perfect
competition assumption, that must assume that in the long period firms are at the minLAC size; what
Marshall was implicitly admitting was demand limits to the expansion of individual firms, and this can only
be discussed by abandoning price-taking and turning to theories of imperfect competition. An example of the
second cause might be the increasing average skill of specialized skilled labour when the dimension of an
industry grows and with it grows the number of workers who have acquired high skills owing to work
experience, and the result is a community where expertise is greater, with a reduction in costs. But such
externalities can only act very slowly, on a time scale superior to that of the adjustments contemplated in
short-period or long-period equilibration: the time scale one considers when one discusses economic growth.
(cont. next page →)
59
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price tends to rise if demand exceeds supply, and tends to decrease if supply exceeds demand.
When the supply curve is horizontal it is a long-period supply curve, and the adjustment goes on in
a succession of short-period situations, in each one of which the number of plants is given and the
supply curve is a short-period, upward-sloping one. The short-period equilibrium price is stable, and
if higher than minLAC it induces in the long period an increase in the number of plants in the
industry, that is, a shift of the short-period supply curve to the right that causes the short-period
equilibrium price to decrease and thus to tend toward minLAC[61]; the reverse process will go on if
the short-period equilibrium price is lower than minLAC. Thus we obtain stability of the longperiod equilibrium too.
5.23.2. The analysis suggests that it seems legitimate to assume a tendency of the quantity
produced of the several products to adapt to the demand for them, at prices that tend to equal
minimum average cost. In Chapter 6 it will be seen that the consideration of adjustment lags can
raise doubts on this conclusion, but it will be argued there that the difficulties are not very serious.
So we have some justification for believing that the assumption that will be made in the formulation
of the general equilibrium equations in Part III of this chapter, of equilibrium product prices equal
to minimum average costs and of quantities produced equal to the demand for them at those prices,
reflects actual tendencies. However, the stability of product markets thus assumed rests on given
factor prices and given demand curves; therefore it does not prove the stability of the general
equilibrium of production and exchange, which requires in addition the stability of factor markets
(and, to such an end, cannot assume given demand curves for products because changes in factor
rentals change incomes and demands). This will be discussed in chapter 6.
5.24. Welfare analysis
5.24.1. Let us now prove that in a partial-equilibrium framework the competitive equilibrium
of a single market, determined by the intersection of demand curve and supply curve, is a Pareto-
Furthermore externalities external to firms but internal to an industry are very rare, generally positive
externalities do not reduce costs only in one industry and are therefore incompatible with partial equilibrium.
61
Note that for the process to push the price toward minLAC it is not necessary that all firms have optimal
plants; it suffices that there be entry of optimal plants until supply obliges the price to equal minLAC; this
process will generally be faster than the process of closure and replacement of older plants (the economic life
of plants and durable capital goods is generally much longer than the time required by their production), so
one must consider it normal that a price very close to minLAC co-exists with plants that are not optimal and
earn residual quasirents.
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efficient allocation. But we must clarify, allocation of what? Of the quantity produced, x, among
consumers, and of income among consumers and producers, under a trade-off (analogous to a
production function) between income (=cost) and x. We consider two groups of maximizers:
consumers maximize utility, producers maximize their income i.e. profit. The given prices of all
other goods allow their treatment as a Hicksian composite commodity, whose price can be made
equal to 1 and whose quantity therefore can be legitimately identified with expenditure on goods
other than x, or ‘residual income’ y; thus consumer h’s utility depends on xh and yh, and when a
consumer pays p for a unit of x-good she is giving up p units of y-good; the producer gives away
amounts of y-good as cost to produce the unit of x-good; thus it is as if x were produced by using
quantities of y as input. In equilibrium, consumers have the same MRS between x-good and y-good;
through an opportune choice of units for utility the equilibrium marginal utility of income can be
rendered equal to 1 for each consumer. This means p*, the equilibrium price of x, equals the
marginal utility of x for all consumers active on the market; producers are interested in maximizing
their income i.e. profit, so they equalize price and marginal cost; hence MC(x*)=MU(x*) where x*
is equilibrium output.
Proof of the Pareto efficiency of the perfectly competitive partial equilibrium. We first prove that
production of a quantity different from the equilibrium quantity x* cannot be Pareto efficient. Pareto
efficiency means that a Pareto improvement (a change that makes somebody better off without making
anybody worse off) is impossible. If x<x*, there will be some consumer ready to pay for one more unit of x
a ‘demand price’ pd greater than the equilibrium price p*, and there will be some producer who can produce
one extra unit at a marginal cost not greater than p* (we admit the possibility of CRS and constant MC) and
would be therefore ready to sell it at a ‘supply price’ ps≤p*, hence these two agents can strike a mutually
advantageous bargain (to produce and exchange one extra unit at any p intermediate between pd and ps)
without making anybody else worse off. If x>x*, there will be some producer with a marginal cost not
smaller than p* who is ready to pay a sum greater than or equal to p* for the right to reduce production by
one unit, and there will be some consumer ready to renounce one unit of x for a recompense less than p*,
hence again these two agents can strike a mutually advantageous bargain. Thus x=x* is a necessary condition
for Pareto efficiency.
Let us now prove that no different allocation of the production of x* among producers can be a Pareto
improvement, and no different allocation of x* among the consumers can be Pareto efficient. A different
allocation of the production of x* among firms either leaves marginal costs unchanged, or causes an increase
in marginal cost in at least one firm, whose profit decreases: in either case there isn’t a Pareto improvement;
in the first of these two cases Pareto efficiency does not uniquely determine the allocation of the production
of x* among firms. A different allocation of x* among consumers means that at least one has more x-good
and at least one has less x-good than in equilibrium; this means that their MRS’s differ so they can strike a
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■
mutually advantageous exchange.
Consumer and producer surplus and welfare changes.
§5.24.2. We have seen in ch. 4 the definition of consumer surplus in an industry. Producer
surplus is analogously defined as the area above the supply curve up to the horizontal price line, cf.
the triangle ADC in Fig. 5.10. It intends to measure the maximum amount of money that
‘producers’, that is, the ensemble of entrepreneurs and factor suppliers involved in producing the
good, would be ready to pay in the aggregate rather than forgo the possibility to produce and sell
the good at the given price. It too assumes a constant marginal utility of income.
In the short period, the producer surplus of a single firm at a given product price and
quantity supplied is defined as total revenue minus total variable cost or equivalently, as pure profit
plus total fixed cost. (Remember that fixed cost is by definition a cost that the producer must bear
whether she produces or not. Thus in the short period the entrepreneur finds it convenient to
produce as long as she is able to more than cover variable cost.) That this is equivalent to the area
above the short-period supply curve up to the horizontal price line is easily proved by remembering
that total variable cost is the integral of marginal cost and therefore it is the area under the supply
curve, cf. Fig. 5.??. Considering fixed cost to be a sunk cost which cannot be avoided, the
entrepreneur, rather than be excluded from the market, will be ready to pay up to the amount that
would leave her with what is just sufficient to cover variable cost, amount which is measured by
that area. The sum of these areas is the area above the industry’s supply curve.
price
B
Industry supply curve
D
C
competitive price p*
A
demand curve
quantity
Fig. 5.10. Marshallian total surplus is the area of triangle ABC, the sum of consumer surplus (the area of
triangle DBC) and producer surplus (the area of triangle ADC).
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MC
AVC
p, MC, AVC
p
A
D
B
O
C
E
output
Fig. 5.11. Graphical proof that for a firm the area above the supply curve up to the price line (trapeze
ABDp) equals revenue minus variable cost. When price is p and therefore the supply of the firm is OE,
revenue is rectangle OEDp, and the area under the supply curve OABD of the firm equals total variable cost,
because the area of rectangle OABC is variable cost up to output OC, and, by integration, the area under the
marginal cost curve from B to D is the addition to variable cost caused by increasing output from OC to OE.
In long-period analysis all cost is variable; if the industry’s supply curve is horizontal
because all factor rentals are given, producer surplus is zero. But if the industry uses a specialized
input whose rental rises with industry supply, the long-period industry supply curve is upwardsloping, so producer surplus is positive. However, profits are zero all the same, because at each
point of the supply curve the specialized input’s price is given and each firm treats input prices as
given and produces the quantity that minimizes average cost, i.e. that causes average and marginal
cost to coincide[62]. How is the positive producer surplus reconciled with the zero profits? The point
is that the rise in the rental of the specialized factor due to the expansion of production implies an
income gain for the suppliers of that factor, so they would be ready to pay rather than see
production of the good forbidden. The maximum amount they would be ready to pay, again
misleadingly called producer surplus (it is in fact a consumer surplus of the consumers who supply
the factor), is actually measured by the area above the factor’s supply curve (in the graph with
factor supply and factor rental on the axes) for reasons similar to those defining the consumer
surplus on the demand side[63]; for example if the supply of the factor is rigid (implying a zero
reservation price of the factor owners), the producer surplus is the area of the rectangle formed by
62
Thus a long-period industry supply curve has average and marginal cost coincide at all points even when it
is upward-sloping.
63
Again, under the assumption of constant marginal utility of money – less easily justifiable in this case! It is
very unusual that income from ownership of a specialized factor of production be a very small part of one's
overall income.
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the axes, the vertical supply line, and the factor rental horizontal line. This area coincides with the
area above the industry’s long-period supply curve because the area below the latter curve is the
payment to the other factors: this is made clear by noticing that, if total production came from a
single firm with the specialized input as fixed factor, the supply curve would be the marginal cost
curve of this firm, and the area under it would measure its variable cost.
Proof. We prove it for a simple case. Assume that the good is produced by unspecialized labour L,
whose wage w is constant and for simplicity equal to 1, over a specialized land T in rigid supply, whose
surface, again for simplicity, is measured in such units as to make it equal to 1; the CRS differentiable
production function is q=f(L,T); each marginal product is a decreasing function of the ratio of the factor to
the other factor, and becomes negative above a sufficiently high ratio, in which case as explained in ch. 3 the
factor will not be fully utilized, only the amount yielding a zero marginal product will be utilized. Land is
fully employed if possible, so we consider q a function of L only, q=f(L) which we assume invertible,
L(q)=f-1(q) whose derivative is 1/MPL(q). Optimal labour employment requires w=MPL∙p where p is the
product's price; since w=1 is constant, it must be p=1/MPL, and an output increase requires p to rise if MPL is
decreasing; this causes the land rental β to rise too, because land receives its marginal revenue product too,
and MPT rises as labour employment rises, except initially when land is not fully utilized and its marginal
product is zero. Thus β is a function of q, β=β(q), and initially it is zero: the opportunity cost of land T is
zero, the entire land revenue is 'rent' or producer surplus. At each q, supply price is defined by equality with
average cost, p(q)=(β(q)+L(q))/q, and the function p(q) thus defined is the supply curve. We want to show
that, given the quantity produced q*, total industry revenue p(q*)q* minus the area under the supply curve
equals β(q*). So what we need to show is that the area under the supply curve is L(q*). Now, p also satisfies
p(q)=1/MPL(q); therefore the area under the supply curve is the definite integral
L(0)=L(q*).

q*
0
1
MPL ( q )
dq =L(q*)–
■
Exercise: Assume wine V is produced by labour L over 1 unit of specialized land in fixed supply
according to the production function V=T1/2L1/2. Labour’s wage in terms of other goods is fixed and equal to
w. Prove that the industry’s long-period inverse supply curve is p=2wV and that the area below it for any
given V equals the wage payments to the labour employed to produce that V.
Actually, short-period producer surplus in a competitive industry is determined in the same
way, because as argued in §5.14, the entrepreneurs' short-period profits should be seen as earnings
accruing to the fixed factors.
If for some reason (e.g. rigid factor contracts stipulated in the past under different
conditions) there are profits, then this is a subtraction of part of the surplus from the factor
suppliers, but the area above the marginal cost curve still measures producer surplus, only now
consisting partly of entrepreneurial profits and partly of factor suppliers' surplus.
Marshallian aggregate producer surplus, or simply producer surplus for brevity, is the sum
of the individual producer surpluses in a market. And the sum of consumer and producer surplus is
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called the Marshallian aggregate total surplus. It is the area of the triangle formed by demand
curve, supply curve, and ordinate axis.
It is easy to see graphically that the competitive partial equilibrium maximizes total surplus.
Any price or quantity different from the equilibrium ones would ration either buyers or sellers,
because there would not be enough production or enough demand.
It is important to be aware of the limits of the above conclusion: the argument bringing to it
has neglected externalities, has taken incomes and factor property as given, and concerns the market
for a consumption good.
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PART III
ATEMPORAL GENERAL EQUILIBRIUM
5.24. Price-taking and perfect competition
5.24.1. We have studied the short-period and long-period behaviour of price-taking firms and
industries. We proceed to the (neoclassical) theory of the general equilibrium of the competitive
production-and-exchange economy (still without capital goods), where ‘competitive’ means here
price-taking consumers and firms. Price taking is usually identified with perfect competition, but it
need not be: a discussion of the notion of perfect competition will clarify the issue.
PERFECT COMPETITION: DIFFERENT NOTIONS
A perfectly competitive market is the term used in neoclassical economics to designate a
market for a good or service (usually against money), in which
(i) the equilibrium price is the same for all units of the good or service,
(ii) all agents active in the market, both buyers and sellers, voluntarily act as price takers
relative to the equilibrium price, i.e. take the equilibrium price as a parameter in their
optimizations[64].
This is also expressed by saying that no agent in that market has market power. This
definition of perfectly competitive market must be distinguished from its possible foundations or
justifications, of which we will discuss two: negligibile size of agents, and free entry. Whichever
the foundation, for product markets the implication of price taking is that a firm in a perfectly
competitive market considers the price at which it can sell its product as independent of the quantity
offered for sale[65]; this is often expressed by saying that the demand curve facing the individual
firm is horizontal.
The price-taking assumption is central both to the standard neoclassical theory of consumer
behaviour (cf. chapter 4) and to the standard theory of the firm in general competitive equilibrium.
In particular, it implies that product price and marginal revenue coincide.
A perfectly competitive economy is a multimarket economy where all markets are perfectly
64
The adverb 'voluntarily' excludes the cases where agents are obliged by law to treat the price as given.
"Perfect competition is that state of affairs where the individual firm can sell 'as much as it likes' at a
price which the market determines independently of this firm's output" (Shackle 1967 p. 13).
65
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competitive and there is perfect mobility of agents and resources across markets i.e. there is free
entry. Thus free entry is not part of the definition of a perfectly competitive market and is not
necessarily part of the justification of price taking, but it is part of the definition of a perfectly
competitive economy: in the latter, if competition is given time fully to work out its effects, all units
of a production input earn the same rental independently of where they are employed.
The usual textbook presentations of perfect competition characterize it differently, through a
list of conditions that include most or all of the following:
1- homogeneity of the good on sale (indifference of buyers among the goods offered by the
several sellers; the good is also said undifferentiated);
2- readiness of agents to take advantage of every opportunity to improve their condition by
switching to trading with different agents and/or proposing a different price (costlessness of such
switches is also often added);
3- anonymity (indifference as to whom one is exchanging with);
4- perfect information about the prices charged by other sellers or offered by other buyers;
thus buyers try to buy from the sellers asking for the lowest price, and sellers try to sell to the
buyers accepting the highest price; a firm charging a price higher than other firms is unable to sell
anything (at least, as long as the supply of the firms selling at a lower price is capable of satisfying
demand - see below);
5- atomistic market (numerosity and negligibility of agents, i.e. smallness of their demands
and supplies relative to the size of the market), hence a change in the quantity demanded or supplied
by a single agent has only a negligible effect on total demand or supply and hence a negligible
effect on price;
6- absence of collusion;
7- no transaction costs for resale of the good (this is to make arbitrage easy and hence price
discrimination or discounts for large purchases impossible), which, together with homogeneity, is
taken to imply linear price (bad jargon meaning that expenditure on a good, or revenue from
selling the good, is a linear function of the quantity bought or sold);
8- free entry and free access to state-of-the-arts technology, if the good is a product.
This list of conditions, some subset of which is generally used to define perfect competition,
is argued to imply price taking and a uniform unit price but, as pointed out by Kreps (1990, p. 265),
the implication is based on intuition; a rigorous proof that these characteristics are sufficient, or
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necessary, to such an end would require a careful specification of the market institutions, which is
generally not supplied[66]. Price taking and ‘linear price’ are the real assumptions, the ones that
distinguish the equations of a competitive equilibrium from those of the equilibria of other market
forms. The list also suffers from an insufficient distinction between short-period and long-period
analysis: most intermediate textbooks include free entry among the defining characteristics of a
perfectly competitive market, without realizing that then they have no right to talk of perfectly
competitive equilibria for markets where entry of new firms is excluded, which is the usual
assumption in short-period analyses. Furthermore, it is seldom made clear that, as will be explained,
price taking by all agents only makes sense with respect to equilibrium prices.
5.24.2. Some reference to historical developments can be useful. The notion of perfect
competition is a marginalist modification of the earlier classical notion of free competition. A freely
competitive market was defined by Adam Smith as one in which there is no impediment to free
contracting and to free entry and exit of productive resources; the implication was that in a
competitive industry the reward to be paid to each employed unit of a certain type of productive
resource (be it a kind of labour, or a kind of land, or capital) would tend to be the same as in other
competitive industries. Two elements of the classical notion of free competition are argued by an
increasing number of economists to be very reasonable and in need of full recuperation.
First, free competition was a long-period notion. This made it possible to reach definite
conclusions on the tendential results of its operation without the notion of definite demand curves.
In the analyses of classical authors there appears no notion of a demand curve specifying a precise
relationship between price and quantity demanded; there is only the notion of effectual demand (the
quantity demanded at the normal – or long-period, or natural – price) together with the thesis that if
the quantity supplied is less than the effectual demand, the market price will be above the natural
price, and vice-versa, a thesis representable as in Fig. 5.?? (from Garegnani AER??). The sign of the
difference between natural price and market price suffices to stimulate changes in industry capacity
that will push the quantity supplied toward equality with effectual demand, and thus the market
price toward equality with the natural price. The absence of a need for a precise indication of how
demand will change with price can be seen as an advantage, the moment one remembers the doubts
raised in §?? about the notion of demand curve.
66
Except when the Walrasian tâtonnement is assumed, see below in the text.
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price
natural
price
effectual demand
quantity
Fig. 11.1. According to classical authors when the quantity supplied differs from the effectual demand
the quantity-price point will be in the shaded areas but generally nothing more precise can be stated, nor need
be stated, because the sign of the difference between natural price and market price suffices to stimulate
capital movements that will push the quantity supplied toward equality with the effectual demand.
Second, it was not part of the notion of free competition that all units of the supply of the
productive resource labour would earn the same reward – this would have meant full employment,
which was not assumed by classical economists. Thus free competition did not mean indefinite
wage reduction as long as labour supply was greater than demand; its operation embodied the forces
and social constraints determining the division of the social product between wages and profits; free
competition among labourers only entailed a tendency for equal types of labour to earn the same
wage; it implied uniformity of wages, but nothing about their level. Only for product prices[67] did
67
Actually in Ricardo there is one distributive variable, differential rent, for which free competition
allows conclusions about levels: the competition among landowners offering 'marginal' land (in excess
(cont. next page →)
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free competition allow clear conclusions about their levels, because, given the natural rates of rent,
wages and profits (cf. ch. 1), free competition implied that product prices would tend toward their
natural levels i.e. toward the levels allowing paying labour, land and capital their natural rates: a
price different from its natural level would entail that either the rents, or the wages, or the profits
paid out of that price would be different from their natural levels but then forces would be set in
motion tending to restore them to their natural levels and thus tending to bring product prices to
their natural levels (cf. again ch. 1).
In such a conception, the stronger the competition in a product market, the closer the price of
the product got to its natural level; and free competition in product markets resulted in firms being
obliged to sell, on an average over long periods, at a price equal to the natural price. Note that the
notion implied no restriction of competition to price competition: active rivalry could include
marketing efforts, R&D, process and product innovation (especially stressed by Marx); still, the
tendency, due to free entry, toward a uniform rate of pay for each type of input – and thus, centrally,
toward a uniform rate of profit – would be operative.
In order to understand why this notion of free competition was later replaced by the notion of
perfect competition, one must remember the new analytical needs connected with the marginal
approach to value and distribution. In this approach, changes of prices in response to excess
demands have the role of eliminating disequilibria by altering excess demands; the decisions of
firms and of consumers must therefore be determined by prices, and this requires that prices be
parameters in the agents' optimizations[68]. Finding a foundation for price taking becomes an
important part of theoretical research. Furthermore, the claim that all prices, both product and factor
prices, are determined by the same basic mechanism (tendency toward an equilibrium between
supply by definition) brings the rate of rent on this land to zero, and competition enforces differential rents
on better lands. This shows that competition among suppliers of productive services was seen by classical
authors to operate differently for land, labour, and capital.
68
If prices are not parameters, excess demand functions no longer exist, and the determination of welldefined equilibria, not to speak of univocal comparative-statics results, becomes extremely difficult; thus
Shackle (1967 p. 54) could write that in the 1930s with the abandonment of perfect competition in favour of
monopolistic competition "the theory of value ... had turned from mechanics into taxonomy. So far as there
was anything that could still be called an industry, the demand for its output could take an infinite variety of
forms and could change in an infinite variety of ways. On the cost side a similar case prevailed. The output
and prices of the commodity were the upshot of an interaction between these two sets of conditions or
changes of conditions, and only a card-index would suffice to give a truly comprehensive picture of the main
sorts of things that could happen"; Hicks in Value and Capital (1946 p. 84) refused to abandon the perfect
competition assumption because “the threatened wreckage is that of the greater part of general equilibrium
theory”. More recently it has been pointed out (Roberts and Sonnenschein 1977) that the demonstration of
existence of an imperfectly competitive general equilibrium encounters considerable additional difficulties.
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supply and demand) requires a justification of price taking applicable to all kinds of markets,
including exchange markets (i.e. without entry of firms) and markets of non-produced factors, and
also applicable (given the widespread acceptance of Marshall's approach) to short-period
equilibration processes. Thus the need arises for a justification of price taking independent of free
entry of firms, and this is found in the assumption of 'atomistic' market, which means agents of
negligible size: each agent’s demand or supply is unable significantly to affect the equilibrium price
because it is always such a small fraction of total supply or demand that its modification leaves the
equilibrium price very nearly unaffected, so much so that it makes no practical difference for the
agent to consider it as completely unaffected by her choices.
This assumption of negligible impact of each agent on the equilibrium price has been
formalized by Aumann (1964) for exchange economies by postulating a continuum of infinitesimal
agents. A presentation of the resulting theory would require very advanced mathematics and its
payoff in terms of a better understanding of how actual markets work is doubtful, so it is not
attempted in this book[69].
5.24.3. One point about price taking deserves clarification. The notion of perfect competition
is frequently criticized and even dismissed as irrelevant because it is interpreted as paradoxically
implying the absence of any competitive behaviour: all agents passively accept the ruling price.
This criticism is correct in so far as it points to the exclusion of non-price competition (e.g. through
marketing expenses or innovation) from the theory; but for price competition it is based on a
misunderstanding. The price with respect to which all agents act as price takers in a perfectly
competitive market is, and can only be, the equilibrium price, and it is the active price competition
of buyers and sellers that tends to bring such a price about. It is best to approach the issue
historically, this will clarify why this fact (that will be proved below) is often lost sight of. The
foundations of marginalist/neoclassical theory underwent a very important shift after the 1930s (I
anticipate here issues whose relevance will be discussed in greater depth in chapter 7): the
traditional centrality of long-period analysis was increasingly abandoned in favour of very-shortperiod general equilibria that included given endowments of each capital good among their data;
these endowments were susceptible of very rapid change and required therefore – in order to
conceive the equilibrium corresponding to those data as indicating market realizations with
sufficient approximation – that the equilibrium should be reached extremely quickly, before
69
Cf. Ellickson ?? for an introduction including the necessary mathematics.
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disequilibrium productions could relevantly alter the endowments of the several capital goods. The
reaching of equilibrium was then increasingly described as brought about by the so-called
Walrasian tâtonnement (‘groping’ in French). The picture is as follows. Economic activity is
suspended, all agents meet in a big stadium or connect through Internet, and there is a central
institution, called the auctioneer, which proposes prices for all goods; at these prices the agents
(who are price takers) write down ‘bons’, promises of excess demands which become binding only
if they generate an equilibrium, and pass these written pledges to the auctioneer who calculates the
excess demand for each market and then, if a general equilibrium has not been achieved, declares
the pledges not valid and proceeds to announce new prices, raising them where demand has
exceeded supply, and lowering them where supply has exceeded demand. This process only stops
when an equilibrium is reached. Only then do pledges become binding and exchanges are allowed
and only then, in an economy with production, production processes are started[70].
The widespread recourse to the tâtonnement tale appears to have favoured an interpretation of
perfect competition as meaning price taking always, i.e. also at non-equilibrium prices (the
accusation of passivity would then be justified!). But, as noticed by Arrow (1959), if agents were
always price takers the theory would be unable to explain how price adjusts toward equilibrium in a
market, because there would be no one left to change the price (the fairy tale of the auctioneer is
just that, a fairy tale with no correspondence with real markets). Markets (with the – rare –
exception of auctions[71]) are continuous, and during the flow of transactions an agent can always
propose a different price, so if the agent acts as price taker it must be in her interest to do so; now, at
a non-equilibrium price some sellers or some buyers cannot possibly carry through their intended
transactions and then they will not find it optimal to accept the current market price; for example
when demand exceeds supply a buyer unable to purchase the good can improve her situation by
offering a slightly higher price to some seller to get him to sell to her first (thus, in disequilibrium,
price will not be uniform). Therefore, first, price taking must be justified; second, price taking can
be legitimate for all agents only in equilibrium. So it must be shown that in equilibrium no buyer
finds it convenient or possible to try to lower the price (she has no incentive to raise the price, of
course, since in equilibrium she can satisfy her demand), and no seller finds it convenient or
70
If one assumes the auctioneer-guided tâtonnement then there is no reason why firms cannot be born during
it; the stress earlier in this chapter on the long-period nature of entry wanted to underlie that it takes time to
build new plants, that is, that realistic adjustments take considerable time; but the fairy-tale ultrafast
tâtonnement establishes in advance what is to be done by the agents, so it may well include the formation of
new firms with new plants, as well as changes of property of existing plants.
71
Where anyway agents are not price takers.
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possible to try to raise the price. This is achieved by noting (i) that in equilibrium a buyer would not
get anywhere by proposing a lower price because the seller she is buying from would then be able
to turn to other buyers and by distributing her supply among them she would be able to sell with
only an infinitesimal price reduction; and (ii) that the price change that the buyer can achieve by
reducing her demand is too small to make the utility or profit improvement worthwhile; the agent
realizes this, and does not try[ 72]. A symmetrical reasoning applies to sellers. As pointed out, this
requires a negligible impact on the equilibrium price of any plausible alteration of one agent's
demand or supply, i.e. an assumption that all agents's demands and supplies are a very small
fraction of the total – rigorously, that they are infinitesimal.
Outside equilibrium, inevitably agents must be assumed not to be price takers, and to actively
look for the best exchange opportunities, proposing different and changing prices. The
marginalist/neoclassical approach argues − but this is part of the approach, not of the general notion
of competition − that in each competitive market, be it a product market or a factor market, the
readiness of buyers to offer a slightly higher price if they are unable to purchase what they desire
for want of supply, the readiness of sellers to be content with a slightly lower price if they are
unable to sell all their supply for want of demand, and the readiness of buyers and sellers to switch
to different exchange partners if in this way they can fetch a better price, cause the average price to
increase as long as demand is greater than supply, and to decrease as long as supply is greater than
demand; hence the sole possible state of rest (equilibrium) of the price in each market is when
supply and demand are equal (or supply is greater than demand but the price has become zero and
cannot further decrease). The other elements of the above list of 8 characteristics of perfectly
competitive markets (except the last one) are intended to guarantee that this disequilibrium higgling
and bargaining does not stop until equilibrium, with a uniform price for all transactions, is achieved.
Under the mentioned influence of the shift to very-short-period notions of equilibrium, these
characteristics are often interpreted as meaning that the more perfect the competition, the faster the
agents discover more favourable exchange opportunities and exploit them (e.g. the faster a seller
loses his custom if he charges a higher price than other sellers), with the implication that the more
perfect the competition, the faster the average market price adjusts in response to inequalities
between supply and demand; thus one finds occasional statements that perfect competition implies
72
There is here, it would seem, an implicit assumption that altering one's demand or supply and waiting
for the higgling and bargaining of the market to find a new equilibrium price is not without costs for the
agent – which is perhaps an acceptable assumption.
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that equilibrium between supply and demand is reached instantaneously, an extreme view of little
fruitfulness.
5.24.4. What we have been discussing can be called intramarket perfect competition in that it
characterizes the interaction of the agents active within a market. Free entry is logically superfluous
for such a conception of perfect competition based on negligible agents. But historically the notion
of competition was associated with the notion of a tendency of input rewards to equalize across
markets, and, for such a notion of inter-markets competition, free entry is essential.
Actually, as rediscovered in recent decades, free entry of firms[73] can be a foundation of
absence of market power in product markets alternative to negligibility of agents (Novshek and
Sonnenschein 1987[74]). Owing to indivisibilities (of means of production or of processes) there
will always be a minimum scale of production, below which average cost is higher; but if this
minimum efficient size of firms is small relative to total demand, then free entry will cause the
product price to tend toward minimum average cost, with at most a negligible upward deviation
from it when demand at a price equal to minimum average cost does not correspond to an integer
multiple of minimum efficient production. Then as long as input prices can be taken as given, the
supply curve of the good is practically horizontal in spite of a non-negligible size of firms, firms are
obliged by entry to charge a price equal to minimum average cost[ 75]; and then each buyer acts as
price taker too, because a change in her demand does not alter the supply price. Of course if it is
accepted that entry and exit of firms takes time, then the free-entry foundation of price taking
concerns long-period choices, i.e. choices connected with changes in the quantities in existence of
the several capital goods, as Marshall assumed. Actually, even in the instantaneous fairy-tale
tâtonnement the possibility of formation of new firms should be admitted, with the new firms
demanding existing capital goods – including fixed plants – in competition with other firms, and
prices tending to equal average costs that will include quasi-rents of existing fixed plants analogous
to rents of lands; but since in real economies adjustments do take time, below I follow the
Marshallian approach.
73
We will concentrate on entry of firms, but free entry can be applied to any agent. Free entry of
labourers, for example, will prevent the wage in an industry from going above the average wage elsewhere.
74
They used a Cournot approach; we will come back on this after explaining Cournot. However, they did
not make it clear that the analysis had to be a long-period one owing to the time required by entry and exit.
75
“...signifcant entry barriers are the sine qua non of monopoly and oligopoly, for as we shall see in later
chapters, sellers have little or no enduring power over price when entry barriers are nonexistent.” Scherer
and Ross p. 18
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Assume a U-shaped average cost curve of efficient firms, the same for all firms because
competition eliminates inefficient producers. Let q be the efficient scale of prodution at the given
input prices. As industry supply increases from nq to something less than (n+1)q, the long-period
supply curve is the horizontal sum of the long-period marginal cost curves of n efficient plants; its
slope is 1/n times the slope of the long-period marginal cost curve of an efficient plant; thus for
example with 20 single-plant firms, the supply curve rises to what the long-period marginal cost
becomes when plant production increases 5% beyond minimum efficient scale, then it falls again to
the minimum average cost when ε becomes zero, cf. §5.15??.
Now, it is highly plausible that the long-period marginal cost curve of a firm is rather flat at
the efficient output, cf. Fig. 5.??, because in long-period analysis the only element capable of
causing decreasing returns to scale is the increasing difficulty of supervision and management, and
there is no reason to think that this difficulty is increasing sharply at the efficient scale; also, many
economists esteem that its increase is counterbalanced for a considerable interval by increasing
economies of scale associated with advantages of bulk production, bulk transport, etcetera; for
which reason it is usually admitted that a U-shaped long-period average cost curve will be nearly
flat for some considerable interval around its minimum point. But then the maximum deviation of
the long-period supply curve from the minimum average cost is likely to become insignificant
rather quickly as the maximum number of efficient plants allowed for by the size of demand
increases. Thus it seems plausible that, in most industries, room for 10 minimum-efficient-size
firms is already sufficient for the approximation to a horizontal long-period supply curve to be
nearly perfect. (Cf. also Fig. 5.7 in §5.19.) (The frequent case of nearly inexhaustible increasing
returns to scale requires a different analysis, to be sketched later, in chapter 11).
AC
LMC
MinAC
q*
Fig. 5.??. Plausible long-period average and marginal cost curves of a firm
LAC
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Whenever the industry supply curve can be assumed to be practically horizontal at the price
equal to minimum average cost, the assumption of negligible size of all agents is no longer
necessary to obtain absence of market power in long-period analysis. Even a single buyer, a
monopsonist, cannot affect the equilibrium price because, at a lower price, supply would disappear
(all firms would make losses). And even a (multi-plant) firm supplying nearly the entire industry
output would have no market power because if it tried to raise the price above minimum average
cost it would cause entry or expansion of other firms.
This alternative approach to absence of market power is compatible with large constantreturns-to-scale firms or, more generally, with minimum efficient firm sizes that, although not very
large relative to market size, cannot be considered negligible. Also, it makes it easier to justify,
rather than simply to assume, absence of collusion: any collusion by existing firms can be
undermined by entry of new firms.
Furthermore, and particularly important, a long-period free-entry characterization of price
taking allows a different interpretation of the horizontal demand curve that, according to the usual
characterization of perfect competition, each firm faces. This interpretation reconciles it with the
feeling of businessmen that "contrary to economic theory, sales are by no means unlimited at the
current market price" (Arrow 1959 p. 49). The approach based on entry does not have the
implication that a firm would have no difficulty with selling more at an unchanged price; it only
implies that whatever different quantity from to-day's a firm may sell will be sold at the same price,
once the market has gone back to equilibrium: but in order for one firm to sell more, some other
firm must accept to sell less or exit the market, which is not going to be a painless process[ 76]. Thus
price taking with respect to the long-period price is perfectly compatible with a short-period
inability to sell more at that price, and therefore also with price wars, and more generally with
active competitive behaviour in its traditional classical sense.[77]
76
A similar consideration applies to the short-period horizontal demand curve faced by each perfectly
competitive firm according to the usual textbook presentation. The usual justification for this horizontal
demand curve is that, owing to product homogeneity, it suffices that the firm reduces infinitesimally its price
below the current price, and the demand for the firm's product will coincide with the entire market demand.
The defect of this reasoning is that one is implicitly assuming that the other firms do not match the price
reduction – but why shouldn’t they? So the horizontal demand curve for the single firm only means that, if
some other firm decreases its supply, our firm can increase its supply by the corresponding amount and
obtain the same price, because its product is a perfect substitute for the other producer's product. The
alternative justification based on the negligible agent approach is much less defensible for firms than for
consumers.
77
But this long-period definition of perfect competition, as inability in the long period to sell at a price
higher than minimum average cost because of free entry, fits uneasily with the current very-short-period
(cont. next page →)
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Another advantage of this approach is that one is reminded that one should not confuse
tendencies with actual states. That there is a tendency toward the elimination of less efficient firms
does not mean that one can expect to find that at any moment the firms in an industry will be of
equal efficiency. That there is a tendency toward the Law of One Price (Jevons’s “Law of
Indifference”) does not mean that in fact in any given period the price will be perfectly uniform.
And so on.
Also, the long-period nature of the analysis makes it easier to extend the analysis to markets
where there isn't complete product homogeneity and where competition is also exercised through
foundations of neoclassical value theory; thus in the current literature the basis of price taking predominantly
remains the negligibility of agents.
It may be useful to comment here on some criticisms of the notion of perfect competition. It has been
pointed out in the text that it is not true that perfect competition means passive agents. But other criticisms
are better founded, in particular that price competition is only one form of competition, and that perfect
competition excludes the active attempts to increase one's welfare or profits by product design, advertising,
innovation, activities that – the critics argue – characterize most industries and markets. These issues will
indeed need discussion in Chapter 11. Another important frequent criticism is aimed at the short-period
perfectly competitive model: it is often false – many economists have insisted – that, in the short run,
differences between supply and demand cause changes in price; especially in manufacturing, the more
common behaviour is alteration of production without nearly any alteration of price (cf. e.g. Lee 1998). This
issue too will be discussed in Chapter 11. However, the criticism of the assumption of perfect competition in
product markets need not question the basic neoclassical view of the working of market economies, or a
rejection of the latter view may go together with the rejection of perfect competition but may not be
motivated by that rejection. The Neo-Austrian school rejects perfect competition and in particular the
auctioneer, and yet it shares the neoclassical view of the long-period working of market economies as
fundamentally efficient (indeed, more so than simple neoclassical theory would have us believe, because
competition is argued to stimulate technical progress), reflecting consumer choices and assigning to each
agent the contribution of her/his sacrifices to social welfare (Kirzner 1981). Non-neoclassical schools, e.g.
Marxian, Sraffian, Post-Keynesian, reject the neoclassical approach to value and distribution, but not because
of their rejection of perfect competition as a reasonable approximation to the working of most markets; the
reason for rejection of the neoclassical 'vision' is basically a different view of the determinants of income
distribution and of aggregate demand (Petri 2004). In particular, the rejection of perfect competition does not
generally entail the rejection of free competition as characterizing most product markets; indeed it has been
argued (Clifton 1977) that competition is stronger nowadays than in 19th century capitalism, owing to the
increasing capacity of gigantic firms to enter any industry, and that therefore the classical idea of a tendency
toward a uniform rate of return on investment in all industries owing to free entry is even more valid to-day;
in such a perspective, the reason why Toyota, Exxon or Nestle do not enter the computers or pharmaceutical
industries is not special barriers to entry or collusion, but rather that the rate of return in the latter industries
is already sufficiently in line with the average rate of return elsewhere as not to justify entry. On this, few
economists would disagree, it would seem. Thus when the issue is normal, or long-period, product prices,
differences on the validity of the perfect competition assumption do not appear to imply important
differences on the existence or not of a tendency of rates of return toward uniformity as long as entry is
possible, and what is found fundamentally lacking in the perfect competition model is the absence of
marketing expenses and innovation as causes of costs that do enter normal average cost.
The issue is different with respect to factor markets. Here the acceptance or denial of perfect
competition in labour markets does make a big difference to the view of the working of market economies.
But this is for Chapter 12.
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advertising, product design, servicing, location, innovation. The tendency toward a uniform rate of
return on investment is not eliminated, for example, by the need for advertising or for continuous
innovation, when these are necessary for survival; in those markets the costs of advertising and
innovation will enter the calculations on the convenience of entry. But these topics are for ch. 11.
5.25. The number of firms in modern GE
5.25.1. We have mentioned in previous chapters and will further clarify in chapter 7 that
general equilibrium theory was originally intended to determine long-period or ‘normal’ positions;
this is the approach we follow here. Indeed the adjustments on factor markets (e.g. the tendency
toward a uniform rent rate for lands of the same type, or toward a uniform wage rate for labourers
of the same skills) necessarily requires considerable time, both because factor rentals are usually
fixed by contract for some time length (and therefore are alterable only at time intervals often of
considerable length, even more than a year), and because any change in the average rental of a
factor alters relative product costs and hence prices, hence product demands, hence factor demands;
so the theory must admit that any equilibrium temporarily reached on a factor market will be
disturbed again and again before a general equilibrium can be sufficiently approached. During this
time there is time for firms to alter their fixed plants, and for the number of fixed plants or firms in
an industry to be altered. It seems then inescapable that the analysis of equilibrium on factor
markets must be a long-period analysis, the demand for factors being derived from the long-period
decisions of industries as to how much and how to produce.
This has an important consequence: the long-period supply of a product cannot be treated as a
function of prices; once factor prices are given, because of free entry the supply curve is horizontal
(i.e. supply is indeterminate) at a product price equal to minimum average cost; if profit is positive,
supply tends to grow indefinitely; only if profits are negative supply is well-defined, because equal
to zero. Thus, at given factor and product prices, when the supply of an industry is not zero then it is
either indeterminate, or infinite. The same is true for the demands for factors coming from that
industry. The implication is that, differently from demand functions for consumer goods, one cannot
define supply functions of products, nor – as a consequence – can one define excess demand
functions for produced goods.
The traditional solution, the one to be found, for example, in Walras and in Wicksell, was to
admit free entry and to argue that the adjustment of the number and/or dimension of firms would
cause product supply to adjust so as to become equal to the demand forthcoming at a product price
equal to minimum average cost. No need was therefore felt for supply functions of firms or
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industries; industries were treated as having constant returns to scale, and it was assumed that, in
equilibrium, supplies of produced products would equal demands at zero-profit product prices.
In modern works on competitive GE, following the lead of Arrow and Debreu, it is on the
contrary more common to assume a given number of (potential) firms, each one with its production
function or production possibility set[78]. Decreasing returns to scale are not excluded, indeed they
are often assumed. Then the payment to each factor of its marginal revenue product can leave a
profit[79]. These profits go to firm owners. It is assumed that firms are owned by one or more
consumers according to given share ownerships, and the profit of each firm is distributed among its
share owners and enters their income.
The original inspiration for this treatment of the number of firms must probably be found in
Hicks’s Value and Capital (1939), where some (not very explicit) sentences suggest that individuals
are not only consumers but also potential entrepreneurs, each one with different ‘entrepreneurial
capacities’ that influence the production functions of the firms they own or might set up. Depending
on prices, the potential firm a consumer might set up may be able to make nonnegative profits, in
which case it will be active (if not already in existence, it will be born); or it may not be able to
make profits, in which case the individual will be only a consumer. Hicks also argues that the
increase in co-ordination difficulties as firm size increases causes firms to have U-shaped average
cost curves. Depending on prices, individuals may choose different industries in which to employ
their entrepreneurial capacities and set up firms. An implication is that even with given input prices
and all factors variable (a long-period partial-equilibrium analysis), as the price of a product rises its
supply never becomes infinite, the supply function is the sum of the [gradually growing number of]
marginal cost functions of the firms that gradually decide to become active in the industry. Another
implication is the possibility of positive firm profits in equilibrium.
This approach appears unacceptable. It treats ‘entrepreneurship’ as a factor of production, but
this is an illegitimate assimilation of ‘entrepreneurship’ to engineering knowledge required for
organizing the production process, or to managerial capacities: skills that can be acquired on the
market, and if supplied by the owner should be included among the costs as if the owner had hired
them. What ‘entrepreneurship’ can mean is itself vague, but as Kaldor noted in 1934 it would seem
78
However, this assumption is far from universally adopted. McKenzie and Morishima, for example, have
remained close to Walras by assuming constant-returns-to-scale industries, thus implicitly admitting free
entry.
79
Exercise: Prove it for the case of decreasing returns to scale, using Euler’s theorem on homogeneous
functions. ‘Profit’ here stands for the neoclassical notion of profit.
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to refer essentially to the taking of decisions concerning the need to adapt the firm to novelties,
therefore to disequilibria; in competitive long-period equilibrium, there is nothing left to do for
‘entrepreneurs’ other than what salaried personnel can do[80]. The idea of the production function
differing depending on who is the owner of the firm is therefore unacceptable in equilibrium
analysis. Technical know-how can be bought; if the owner has valuable know-how the firm’s costs
should include the market payment for its use. A particularly good manager is like a particularly
fertile land, she/he will be able to earn a differential rent, and the competition for good managers
will raise the differential rent of the good manager until the firm hiring her will make zero profit,
precisely like in the case of agriculture and particularly fertile lands. Ownership as such cannot
contribute to efficiency, therefore there is also no limit to the number of firms that an entrepreneur
qua owner can set up. There is therefore no limit to the number of equally efficient firms that can
enter an industry if they can get the same quality of inputs; and if they cannot, differential rents will
anyway cause all firms to have the same minimum average cost; there is therefore no reason why
there should be positive profits in equilibrium. Actually, since there will generally be a considerable
supply of good engineers and of good managers, if we leave specialized lands and increasing
returns to scale aside[81] the partial-equilibrium assumption that it is possible in any industry to
have entry of further firms as efficient as existing firms appears quite plausible.
Essentially the same criticism applies to the Arrow-Debreu approach, that treats firms as
joint-stock companies, given in number, each one with a given production possibility set, that may
derive from resources owned by the firm (i.e. by its owners) which do not appear among the inputs
“For the function which lends uniqueness and determinateness to the firm – the ability to adjust, to coordinate – is an essentially dynamic function; it is only required so long as adjustments are required; and the
extent to which it is required (which, as its supply is " fixed," governs the amount of other factors which can
be most advantageously combined with it) depends on the frequency and the magnitude of the adjustments to
be undertaken. It is essentially a feature not of "equilibrium" but of "disequilibrium"; it is needed only so
long as, and in so far as, the actual situation in which the firm finds itself deviates from the equilibrium
situation. ... With every successive adjustment to a given constellation of data, the number of " co-ordinating
" tasks still remaining becomes less and the " volume of business " which a given unit of co-ordinating
ability can most successfully manage becomes greater; until finally, in a full long-period equilibrium (in
Marshall's stationary state), the task of management is reduced to pure "supervision," "co-ordinating ability"
becomes a free good and the technically optimum size of the individual firm becomes infinite (or
indeterminate).” (Kaldor, 1934, pp. 70-71) As Kaldor also notes, perfect foresight or complete futures
markets have the same effect. Then a single ‘entrepreneur’ can also set up several firms without loss of
efficiency in equilibrium. A reading of Kaldor’s article is strongly recommended.
81
And we assume therefore that input prices are fixed and do not rise with increases in the industry’s output.
Remember that in partial equilibrium analysis the rise in the price of a non-specialised input as the industry
expands must be assumed negligible, because otherwise it would affect the costs of many other industries
and hence violate the assumption that other prices are given.
80
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of the firm’s possible netputs: these only list the inputs the firm must buy on the market. It is left
unclear what these resources can be, and why further firms cannot be born. If these resources are
tradeable (and the above arguments àpropos Hicks suggest that they are[82]), they are in fact factors
owned by the owners of the firm, and they might be sold or lent to other firms, so they should
appear in the netputs and their opportunity costs should appear in the firm’s cost; also, there is no
reason to consider them as only utilizable by that firm; with the implication that other firms may
buy or rent them, so even if they were essential to the birth or efficiency of firms, there is no reason
why further firms might not be competing with the ones originally listed[83].
5.25.2. Because, starting from Arrow-Debreu, formulations of general equilibrium based on a
given number of firms with possibly positive profits are very common, a description will be now
supplied of these versions, but it will be brief – just so that the reader can recognize them when she
meets them – because, as argued, the given number of firms is not acceptable.
Consumers own factors, and shares in the property of firms; they demand factor services for
direct consumption, and consumption goods. Firms (given in number) may own resources, which
do not appear in the firm-specific production function (so they are implicity assumed not to be
tradeable); firms hire the factor services not provided by their own resources; the more usual
formalization, especially in textbooks, assumes that there are sufficiently decreasing returns (to the
scale of hired factor services) for firms to have definite supply functions; but in other formalizations
the possibility of firm-level constant returns to scale is admitted. The ‘profit’ of a firm includes the
implicit payment to (i.e. the opportunity cost of) the firm’s own resources. Firms are active only if
they are able to make non-negative ‘profit’. Netputs are used to describe firms’ choices, so firms’
supplies of outputs appear as positive numbers in the determination of excess demands, while firms’
demands for inputs are negative numbers. The excess demand on any market is defined as the
aggregate excess demand of consumers for that good minus the aggregate excess supply of firms;
this works for factor markets too, the consumers’ aggregate demand for a factor is a negative
number if it means a positive supply, the firms’ aggregate ‘supply’ of a factor is negative too but it
has a minus sign in front in the determination of excess demand and thus it becomes a positive
82
Also, the description of firms as owned by shareholders suggests that the property of firms is alienable,
and there is nothing in the model specifying that if the owners change the production possibilities set of a
firm changes; this suggests that the firm’s productive capacities are independent of its owners, and then the
entire firm can be sold, and whatever special productive capacity may be connected with the firm will then
earn a differential rent.
83
Cf. also McKenzie 2008 p. 589-590.
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number, representing the demand for the factor. It is assumed that there are S consumers with index
s = 1,..,S with endowments ωs, and with vector excess demand functions zs(p), and H firms with
index h=1,…,H, each one with a well defined and continuous[84] excess supply function yh(p)
(where if yjh < 0 it is the demand for input j); then the aggregate excess demand for good i is:
zi(p) = Σszis(p) – Σhyih(p).
Each consumer s has a right to a percentage Tsh of the profit Πh of firm h, proportional to her
share in the ownership of the firm – her share in the property of the resources owned by the firm.
(Obviously for most consumers at most a few Tsh will not be zero.) The budget constraint of a
consumer states that her income equals the value of her endowments plus the value of the profits
she receives from her property of firms.
(Note the need for given property shares Tsh of firms: otherwise the budget constraints of
consumers are indeterminate. And yet in real economies there is continuous exchange of shares on
the stock market; therefore the group of data of this type of equilibrium consisting of the
consumers’ share ownerships in firms might change considerably during the trial-and-error
disequilibrium adjustment processes that should cause the economy to gravitate toward equilibrium;
some assumption must be made of very little influence of these changes upon prices, quantities and
income distribution, in order for the equilibrium to be considered a centre of gravitation
fundamentally unaffected by the details of the gravitation. This assumption would be the form,
taken in this formalization of the equilibrium, of the traditional marginalist assumption that changes
in the endowments of consumers due to disequilibrium exchanges of properties of factors can be
normally neglected, because their effects are zero if the consumers’ wealth is not affected, and are
still nearly insignificant in the aggregate even when causing some redistribution of wealth, because
the latter, being due only to disequilibrium transactions, will be of very limited entity.)
It is possible to prove, with techniques that have recourse to Kakutani’s fixed-point theorem,
that an equilibrium exists for such a formalization of the production economy.
5.25.3. The reader might perhaps object that the assumptions of a given number of firms and
of given property shares reflect the intention to formalize a short-period equilibrium, and therefore
it is off the mark to criticize these assumptions because incompatible with a long-period
equilibrium. But those assumptions are incompatible with a short-period equilibrium as well!
84
Fig. 5.11 in §5.24.2 shows a discontinuity in the supply function, that is neglected in these models with
little justification.
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Property shares in firms, that is stocks, can change hands in a few minutes on the stock exchange.
As to the number of firms, even if adjustments were the unrealistic ultrafast ones assumed in the
fairy tale of the auctioneer-guided tâtonnement[85], there is no reason why the formation of new
firms should not be one of the possible economic decisions taken during the tâtonnement.
The assumption of a given number of firms is probably justified, in the minds of some
economists, by the fact that in short-period analysis fixed plants are given and are owned by firms;
from this premise the conclusion is inferred that firms are given too. Wrong! As stressed earlier,
firms (incumbent or newly formed) can demand or supply the services of fixed plants too (and if
anything, the assumption of the auctioneer that implies zero transaction costs makes the tradeability
of the services of fixed plants even easier, and therefore impossible to exclude from the analysis).
Even if the number of fixed plants is given, the number of potential demanders for their services is
indefinite and entry will cause the rentals of fixed plants to absorb all profits.
We conclude that even under the assumption that the equilibrium is a very-short-period one
reached by an auctioneer-guided tâtonnement, the number and/or dimension of firms must be
considered variable. All the more so, then, for long-period equilibria. Thus the traditional
formalization (that we will now present) is the sole sensible one[86].
5.26. The equations of equilibrium
We come now to a formalization of the general competitive equilibrium of production and
exchange that avoids the mistakes of the Hicksian-type formalizations. It is based on prices equal to
85
The Walrasian tâtonnement is discussed extensively in the next chapter. We have anticipated in §5.24.3
part of the description to be given there.
86
. The difference of the modern formal analysis from the one of the traditional approach might be
thought to disappear if it is assumed that firms have CRS technologies and are price takers (which entails
that if profit is positive they want to become infinitely large). Then, at given factor rentals, in each industry
only the firms survive which are able to produce at the MinAC, product supplies are horizontal at a price
equal to that MinAC, and profit must be zero if firms are price-takers else profit-making firms become
infinitely large. Since profits are zero, the property shares Tsh become irrelevant. But now a little-noticed new
problem arises: the price-taking assumption becomes generally illegitimate. The modern analysis we are
discussing treats technological capabilities (production possibility sets) as specific to each firm. The general
case will therefore be that, for any given vector of input prices, in each industry the lowest MinAC is
associated with only one firm; this firm will notice that, at a price equal to its MinAC, it is the sole producer,
and it will then stop behaving like a price-taker. This firm is therefore a monopolist for the product price
range between its MinAC and the higher one of the second most efficient potential firm in the industry, let us
indicate it as MinAC2. If the monopoly price falls in that range, it will be established; if it is above MinAC2,
the most efficient firm’s profit is the higher the closer the product price is to MinAC2, and an oligopolistic
situation arises for a product price equal to, or less than, MinAC2 (as long as MinAC3 is still lower), with a
danger of no maximizing solution for the most efficient firm; and if there are more than one, but still few,
firms sharing the same lowest MinAC, oligopoly is again the outcome. The perfect competition assumption
collapses.
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minimum average costs and therefore on free entry. We will show (leaving joint production aside
for simplicity) that it is ultimately reducible to an equilibrium of pure exchange, more precisely, of
indirect exchange of factor endowments. Our interpretation of the equilibrium as the long-period
equilibrium of an acapitalistic economy, with (possibly heterogeneous) labour and land as factors
that produce a variety of consumption goods, justifies an assumption that factors and (produced)
consumption goods do not overlap; there are m products (consumption goods) and n factors
(inputs); a reservation demand for factor services by consumers is admitted, but to help intuition I
assume that this reservation demand need not be made explicit, all one needs is that for each factor
there be an aggregate supply function of its services by the aggregate of consumers to firms;
correspondingly, the demand for factor services that will appear in the model is the one coming
from firms. (This makes it easier to consider the case of rigid factor supplies to firms, which is the
simplest one to visualize and therefore can be an aid to intuition.) These assumptions are only for
simplicity and can be removed with no consequence for the results to be presented.
The factor endowments of consumers are to be interpreted as per-period endowments of
services of the factors they own, for example if I own a hectare of type-B land and if the period
length is a year, I have an endowment of 365 ‘days of use of one hectare of type-B land’ per period.
The symbols are:
pj, j=1,...,m the price of product j (a consumption good)
qj the aggregate supply of good j
vi, i=1,...,n the rental (i.e. price of the services) of factor i
xi the aggregate demand for factor i (by firms)
aij the technical coefficient of factor i in the production of good j, i.e. the quantity of factor i
employed per unit of product j
Qj(p,v)=Qj(p1,...,pm,v1,...,vn) the aggregate demand function for good j, derived from the
choices of consumers, non-negative
Xi(p,v) the aggregate supply function of factor i to firms, derived from the choices of
consumers, and non-negative too.
I come to the equilibrium equations. Once a vector of factor rentals v=(v 1,...,vn) is assigned,
cost minimization plus the assumptions of a common technology for all firms in the same industry
univocally determine the minimum long-period average cost MinLACj for each consumption good
j, which – because of the assumption that industry production functions have CRS (constant returns
to scale) – is independent of the aggregate quantity produced by the industry and only depends on
factor rentals, so we can represent the minimum average cost of good j as MinLAC j(v), a function
homogeneous of degree one. Since we assume strictly convex isoquants, the technical coefficients
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aij are also uniquely determined by the tangency between unit isoquant and isocost, and we can
represent them as aij(v), functions homogeneous of degree zero; obviously MinLACj(v)=Σiviaij(v).
Profits in equilibrium must be zero, hence product prices must equal minimum average costs. Thus
(A) pj = MinLACj(v) = Σiviaij(v),
j=1,...,m.
Thus, once v is given, p is given too, and we have all that is needed to determine consumer
choices; hence the aggregate product demands Qj(p,v) and factor supplies Xi(p,v) are determined.
The adjustment of the quantities supplied to the quantities demanded of consumption goods
determines the quantities to be produced::
(B)
qj = Qj(p,v), j=1,...,m.
Since the technical coefficients are also determined, the quantities produced determine the
aggregate demands for factors by firms:
(C) xi = Σjaij(v)qj,
i=1,...,n.
All that remains is to specify that there must be equilibrium on factor markets i.e. that the
aggregate demands for factors must equal (or be not greater than) the aggregate supplies:
(D)
Xi(p,v) ≥ xi, i=,...,m,
and if for a factor the inequality is strict, then the corresponding
rental is zero.
These four groups of equations[87] are the conditions that the equilibrium of production and
exchange must satisfy. They are 2n + 2m equations, one of which is not independent of the other
ones owing to Walras’ Law[88]; hence 2n+2m–1 independent equations in the 2n+2m variables
(p,q,x,v), but the equations are homogeneous of degree zero in (p,v) so only relative prices count,
hence we can add one more equation fixing the price of the numéraire commodity or basket of
commodities as equal to 1, and then the number of independent equations is equal to the number of
Actually reducible to three by replacing the xi’s in equations (D) with the sums that determine them in
equations (C), and then abolishing equations (C).
88
Exercise: Prove Walras’ Law for the present production economy (Hint: use the fact that firm profits are
zero).
87
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variables.
Obviously this is only an initial check of consistency, and a rigorous demonstration that an
equilibrium exists requires more than this. But the rigorous demonstration – cf. Ch. 6 – will require
little more than for the pure exchange case, and it is useful to understand why.
5.27. The 'reduction' to an exchange economy
This system of equations can easily be reduced by substitution to a system of equations
describing an equilibrium of pure exchange of factors; hints in this direction were given in the way
the equilibrium equations (A) to (D) were introduced. Suppose v is given. Then p is determined by
equations (A) and so we can consider p a function of v and write p(v). Then consumer demands for
products and supplies of factors become functions of v only: in vector notation, Q(p(v),v) and
X(p(v),v) can be re-written as Q*(v), X*(v). But then by equations (B) the vector of quantities to
be produced q can be considered a function of v, that we can write q*(v); and therefore using (C)
the demands for factors can be made into functions of v only: x i*(v)≡ Σjaij(v)qj*(v). Hence both
supplies and demands on factor markets are determined once v is assigned, and equations (D)
suffice to determine the equilibrium if re-written as
(D*)
X*(v) ≥ x*(v).
If we define the aggregate excess demand vector function on factor markets as
ζ(v) ≡ x*(v) – X*(v)
then equations (D*) become the standard equilibrium conditions of a pure exchange equilibrium:
(D**) ζ(v) ≤ 0.
However, excess demands in the pure exchange economy were defined as demands minus
endowments, while in (D*) there appear no endowments. But define the reservation demand for
factor services by consumer s as the vector of differences between her endowments of factor
services and her supply of these services to firms, ωs–Xs*(v) ; then the aggregate reservation
demand, RX(v), is defined by
S


R X v    ω s  X s * ( v) = Ω – X*(v)
where Ω is the aggregate endowment vector.
s 1
We can then define the aggregate direct and indirect demand for factor services by consumers
as the sum of their reservation demands, and of their indirect demands for factors deriving from
their demands for consumption goods and expressed therefore by x*(v). Let DX(v) be this aggregate
direct and indirect consumer demand for factors:
DX(v) = x*(v) + RX(v).
Now by using the fact that X*(v)= Ω – RX(v), aggregate excess demand becomes aggregate
direct and indirect factor demand minus endowments:
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ζ(v) ≡ [x*(v) + RX(v)] – Ω ≡ x*(v) – X*(v) .
Thus if by demand for factors one means direct and indirect demand for factors by consumers,
the equilibrium condition (D*) comes out to be perfectly equivalent to the equilibrium condition of
a pure exchange economy, demand ≤ endowments. Because of its derivation from consumer choice
and firm choices, ζ(v) can be called the derived excess demand for factors.
The reducibility of the equations of the production-and-exchange equilibrium to those of a
pure
exchange
equilibrium
can
be
given
a
concrete
interpretation.
Suppose
that
consumers/households are also entrepreneurs and, exploiting perfect divisibility and constant
returns to scale, each household sets up minifirms producing all and only the consumption goods it
demands. Thus households do not purchase consumption goods from other agents; they only
demand the factor services they are unable to supply, and offer some factor services to other
households in exchange. Factor services would be the only things exchanged in such a hypothetical
economy; equation (D**) would be the standard representation of the equilibrium conditions. We
see in this way that, by demanding consumption goods, consumers indirectly demand factor
services, and thus that the production-and-exchange economy is ultimately an economy of indirect
exchange of factor services.
5.28. The role of demand in determining product prices: why general-equilibrium product
supply curves are upward-sloping.
We can now clarify the role of the composition of demand in the determination of prices.
Actually the issue has been already discussed in chapter 3, §3.6.5. We summarize the results of that
discussion in terms of their relevance for the shape of supply curves of products.
In this theory, demand is able to influence product prices only to the extent to which it is able
to influence the rentals of factors (i.e. income distribution). If the vector of factor prices is given,
product prices are given independently of demand: the supply curve of a product is horizontal. Even
in the short period and in partial equilibrium analysis, we have seen earlier (??§§5.14, 5.23) that the
rising short-period supply curve is upward-sloping only because of a rise in the implicit rentals of
fixed factors as the quantity produced rises (which implies that even in short-period analysis profits
are always zero when correctly computed). The influence of the quantities demanded on product
prices operates through the effect that the demand for consumption goods has on factor rentals
owing to its influence on the demand for factors.
This means that, if one abandons the given-factor-rentals assumption of partial-equilibrium
analysis (which results in a constant product price) and asks about the general-equilibrium effect of
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a shift in preferences on the price of a product, one can no longer assume a horizontal supply curve,
but one must consider the repercussions of the shift in the composition of demand on factor rentals.
This is rather simple if one assumes rigid factor supplies[89]. We are interested in equilibrium on all
factor markets before and after the change. Then, as long at least as the indirect substitution
mechanism works ‘well’, the general-equilibrium supply curve of a product will be upward-sloping
(if the numéraire good is produced with an average factor proportion while the product in question
is not). The reason is that, as the quantity demanded and hence produced of a product shifts
upwards, relative factor rentals cannot remain unchanged: if they did, technical coefficients would
remain unchanged and the change in the composition of demand for products would alter the
composition of the demand for factors, causing disequilibrium on factor markets. The only way to
maintain equilibrium on factor markets is to cause such a change in technical coefficients and/or in
the composition of the demand for other products as will bring the demands for factors back to their
unchanged equilibrium levels, and this requires a rise in the rental of the factor(s) used in a greaterthan-average proportion in the production of the good whose ‘demand function’[90] has shifted
upwards. Thus let us assume two factors, labour and land, that produce a variety of consumption
goods, and let us assume that the good whose demand increases, say corn, is more labour-intensive
than the average. If, starting from an equilibrium situation, corn demand rises owing to a change in
tastes, the attempt to increase the production of corn by transferring some land to the corn industry
raises the demand for labour. Hence the wage rate rises relative to the land rent rate. This will
induce two processes of factor substitution that, by 'freeing' some labour, will allow the corn
industry to expand: technical (or direct) substitution in favour of land in all industries, and/or
indirect substitution owing to a change in the dimension of the industries producing the other
consumption goods.
The indirect substitution process can be grasped with the aid of an example: assume that
besides corn there are two more consumption goods, cloth and meat, and assume that cloth is more
labour-intensive than meat; the rise in the price of labour will cause the price of cloth to rise relative
to the price of meat, causing (if the indirect substitution mechanism works ‘well’, aiding rather than
endangering stability) a shift in favour of meat in the composition of that part of demand which
89
Nothing fundamental would change if factor supplies are increasing in own rentals; on the contrary,
much can change with ‘backward-bending’ factor supplies, but here we leave these complications aside, and
furthermore we assume uniqueness of equilibrium.
90
By ‘demand function’ for a consumption good I mean the function indicating the total quantity demanded
of the good for each vector of factor and product prices.
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goes to cloth and meat; this will reduce the average labour-land proportion on the land employed in
productions other than corn.
The effect of both factor substitution processes is to counter the initial increase in the demand
for labour, and the rise in the price of labour will go on until equilibrium is re-established in factor
markets in the face of the increased corn production. But since the initial assumption was that corn
is relative-labour-intensive, the effect is to raise the relative price of corn. Thus the generalequilibrium supply curve of corn, that is, derived by assuming equilibrium on factor markets[91], is
increasing even in the long period[92]. However, if the good only absorbs a very small portion of
total factor use, moderate changes in the demand for it will be accommodated by only very small
changes in relative factor prices, so the supply curve will be almost horizontal and the partialequilibrium horizontal supply curve can be considered a reasonable approximation.
5.29. International trade
The basic insights the neoclassical approach reaches in the pure theory of international trade,
i.e. in the explanation of what determines relative prices of commodities produced in different
nations, derive from the application of reasonings similar to those presented above to cases in which
factor supplies are not freely transferable across national boundaries.
Thus, let us assume that good A is produced in one nation and good B in another nation; each
nation uses labour and land and is incapable of producing the other nation's good (perhaps because
of climate differences), and there is no international factor mobility. (The analysis is very similar to
the analysis of an economy with four different factors, two of them only useful for the production of
good A, the other two only useful for the production of good B.) We assume substitutability,
otherwise the simultaneous full employment of labour and land in either nation would be generally
impossible and one would reach the implausible result of all income in each nation going to only
91
The general-equilibrium supply curve of a product is an intuitive notion not given analytical precision
by marginalist authors. The idea is that it must trace the supply price of the good (in terms of the chosen
numéraire) as a function of the quantity produced of it, assuming equilibrium on all markets and
parametrically varying the demand for the good. Of course there is some arbitrariness in how one specifies
how the demand for the other goods, and possibly the supply of factors, is affected by the resulting changes
in prices. But this kind of analysis can seldom hope to reach more than conclusions on the signs of the
changes, and the conclusion that the general-equilibrium supply price of a product is an increasing function
of the quantity produced appears hardly controvertible, as long as one assumes a unique and stable
equilibrium (or at least, a locally stable equilibrium and only a small change in preferences).
92
Exercise 5.4: discuss why for this result it is necessary that the numéraire good is produced with an
average labour-land ratio. Exercise 5.5: discuss what would happen in the example if the consumption goods
are only two, corn and meat, and there is no technical choice: fixed coefficients in both industries.
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one factor. For simplicity we assume rigid factor supplies. The basic argument then is, that in each
nation technical substitution induced by changes in the wage-rent ratio will ensure the full
employment of the nation's factor supplies; the supplies of good A and of good B are therefore
determined, and the relative price pA/pB in international exchange must be such as to ensure that
they are absorbed by total demand. If we can assume that, as p A/pB varies, in each nation the
composition of demand varies continuously in the opposite direction, then there will be a unique
value of pA/pB ensuring equilibrium, and the equilibrium will also be stable(93). The real factor
rentals in each nation will be determined in terms of the nation's product by the full-employment
marginal products of the two factors; the purchasing power of these factor rentals in terms of the
good not produced by the nation is determined by the equilibrium p A/pB ratio. The possibility to
consume both goods raises the welfare in each nation relative to autarchy, if the consumers in a
nation allocate part of their income to purchase the good produced by the other nation it must be
because they prefer it to consuming only the good produced by their nation.
Let us now assume that both nations can produce both A and B, again with variable
coefficients, but because of differences in the quality of their labour and/or land the production
functions are not the same in the two nations. There are therefore four factors, labour and land of
nation I, and labour and land of nation II, not transferable across nations but transferable across
industries within each nation. For each nation one can draw a production possibility curve between
goods A and B, associated with the full employment of factors, and the slope at the point on it
where the nation locates itself determines the relative price pA/pB in the nation. We assume that
transport costs are negligible so pA/pB must be the same in both nations if trade is allowed. In each
nation, as pA/pB increases, the composition of supply shifts in favour of good A. Thus if we can
assume that the composition of the joint demand of the two nations shifts against good A as p A/pB
increases, again there will be a unique equilibrium relative product price, and in each nation relative
factor prices will be the ones associated with the point on the production possibility curve
determined by the equilibrium pA/pB.
This case confirms the favourable view of free international trade, characteristic of
marginalist/neoclassical economists. Let us compare the autarchic and the free-trade equilibria.
Both are full-employment equilibria, but, as long as in the autarchic equilibria the p A/pB ratio differs
in the two nations, the introduction of free trade is equivalent to an outward shift of the production
93
Here we neglect the possibility that however low the relative price of one of the two goods becomes,
the composition of demand never becomes equal to the composition of supply.
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possibility frontier: the trade with the other nation means that the economy reaches a point outside
its production possibility frontier.
B
■
■
Y
K
Z■
X■■H
A
Thus consider for simplicity the case of two nations having identical production possibility
frontiers but different tastes. In Fig. ?? the autarchic equilibrium of nation I is at point X, the one of
nation II is at point Y. Thus pA/pB is higher in nation I; the initial impact of free trade, by allowing
trade at some intermediate pA/pB, is to render it convenient for nation I to exchange B against A of
nation II. The change in pA/pB will cause a change in the composition of production in both nations,
and the free trade equilibrium will determine a pA/pB intermediate between the one at X and the one
at Y. Thus nation I will locate itself on some point such as Z and will exchange part of its
production of good B against good A produced by nation II, reaching for example point H.
Analogously, nation II will reach a point like K. If one can neglect redistribution effects, for
example by supposing that one can treat the consumers of each nation as a single representative
consumer, there is an indubitable increase in welfare in both nations, shown by the higher
indifference curve that both representative consumers reach. (In Fig. ?? indifference curves of the
representative consumer of nation I have been drawn as dotted curves.) Actually, redistribution
effects may be relevant: the change in the composition of production in each nation alters relative
factor prices and it may well be that some factor rental decreases, making the owners of that factor
poorer. But it remains possible to argue that if the nation can correct this effect with redistributive
policies, e.g. through changes in taxation of personal incomes, then free trade makes it possible to
increase everybody's welfare.
These arguments are heavily dependent on the marginal approach to distribution, and in
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particular on the thesis that the economy is always close to the full employment of resources. If, as
argued in other approaches, market economies do not spontaneously tend to the full employment of
labour, then the possibility arises that opening the nation to free trade may cause an increase in
unemployment, due for example to firms moving their production plants to other nations and
importing what previously they produced domestically[ 94]: then it would be unclear that free trade is
preferable to autarky for the nation suffering the increase of unemployment.
94
If a car-making firm closes down a plant in nation A and opens up an identical plant in nation B that
uses unemployed labour, capital and land of nation B, there has been no international factor movement, so
the assumption of absence of international factor mobility is not violated.
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3.A.3. APPENDIX 3. Marginal indirect utility of factors with fixed technical coefficients.
In the discussion of Pareto efficiency we introduced the notion of marginal indirect utility of a
factor, for the case in which physical marginal products of factors are well defined. Can the
marginal indirect utility of a factor be defined for cases where production requires fixed factor
proportions? Will it equal the factor rental?
Let us refer again for simplicity to an economy where labour and land, both in fixed supply,
produce consumption goods (no joint production); let us now admit any number of consumption
goods. Let q1,...,qn be the quantities of n different consumption goods produced in this economy,
with good i produced with fixed technical coefficients Li, Ti. Assume the economy is in
equilibrium, with labour and land both fully utilized, w and β the equilibrium wage rate and land
rate in terms of the chosen numéraire, and all the n consumption goods produced in the positive
quantities in which they are demanded by consumers; the price of good i is wLi+βTi. Suppose
labour supply increases by an infinitesimal amount dL while land supply is unchanged. The
composition of production must change so as to utilize that much extra labour with the unchanged
supply of land; a little land must be transferred from land-intensive to labour-intensive industries.
The equilibrium change in quantities produced will be achieved by changes in the composition of
consumer demand induced by changes of the relative prices of consumption goods, brought about
by changes in income distribution; but if the change in labour supply is very small (infinitesimal),
the change in the composition of consumption will be infinitesimal too and will therefore require
only an infinitesimal change in income distribution. So we can treat w and β as unchanged (this can
be proved rigorously with the Envelope Theorem, but for now this intuitive motivation will suffice),
and then since product prices equal cost, the total value of production increases by wdL. Consider
the effect on any one consumer who were to bear all the marginal changes in production, and who
has a differentiable utility function. Choose as unit for this consumer’s utility her equilibrium
marginal utility of money; then for her the price of each good equals its marginal utility; we can
conclude that for this consumer the increase in utility is wdL; hence w measures the marginal
indirect utility of labour for this consumer, who would indeed be ready to pay an exchange value
wdL in order to obtain those changes in consumption (or any other marginal changes in quantities
produced capable of raising labour employment by dL[ 95] and having therefore total value wdL).
Alternatively, one can consider the change in quantities produced to be spread among
consumers, and one can ask what exchange value each consumer would be ready to pay in order to
obtain her marginal changes in consumption, and sum these values; again the sum will be wdL.
This does not mean that each individual gets what she/he contributes to the welfare of society
95
This allows us to admit that the consumer may have zero marginal utility for some of the goods.
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in the sense of the welfare increase brought about by her total factor supply, because the above
equality holds at the margin; if an individual supplies many units of labour, or of land, only the last
small unit supplied receives its contribution to the welfare of society. Unless the supply of factors of
each individual is infinitesimal, it is impossible to give each individual what would be lost if her
total factor supply were withdrawn, because the sum of these losses exceeds total output: as an
extreme example, if a two-persons economy produces a single output with two factors and a CobbDouglas production function, and each factor is entirely supplied by one individual, the ‘total’
contribution of each individual, in the sense of the loss in production if her factor is entirely
withdrawn, is the entire production. But if the individual’s factor contribution is very small relative
to total factor supply, the indirect marginal utility of the ‘intramarginal’ factor units supplied by the
individual is very close to the one of the last small unit, and then what the individual receives is
close to what society would lose if the individual’s entire factor supply were withdrawn, and to say
that each individual receives her contribution to society’s welfare is roughly correct.
3.A.4. APPENDIX 4. Again on the effects of consumer preferences on prices.
Let us first extend the study of §3.6.5 of the effect of changes of consumer demand on prices
and income distribution in the absence of direct factor substitutability to three consumption goods 1,
2 and 3, produced in fixed-coefficients industries in an economy with rigid supplies of labour L*
and of land T*. Suppose that, starting from an initial equilibrium where the quantities demanded are
Q1, Q2, Q3, tastes shift in favour of consumption good 1 while leaving relative preference for the
other two goods unchanged, that is, at the initial factor and product prices Q1 increases, and Q2 and
Q3 decrease while Q2/Q3 does not change. Assume that good 1 is more labour-intensive than the
average, i.e. the labour-land ratio in its production is higher than the ratio of aggregate labour
supply to aggregate land supply, which is the average labour-land ratio in production in the initial
equilibrium owing to full factor employment. Therefore initially the production of goods 2 and 3
jointly considered has a lower labour-land ratio than the average. The increase in demand for good
1 raises the demand for labour, so w/β tends to increase. Suppose that good 2 is more labourintensive than good 3; then p2/p3 rises, and then the standard argument is that the Q2/Q3 ratio in
demand will generally tend to decrease. Assuming the outputs of industries 2 and 3 to adjust to
demand, the average labour-land ratio in industries 2 and 3 jointly considered decreases, creating
room for an expansion of industry 1. At the same time the price of good 1 rises (relative to that of a
hypothetical good produced with the economy-wide average labour-land ratio, that even if nonexisting can still be chosen as theoretical numéraire: the full-employment output, whose cost of
production is necessarily wL*+βT* whatever its composition). The change in income distribution in
favour of wages will go on until the expansion of industry 1 coupled with the rise in the price of
good 1 re-establish equilibrium between supply and demand for it. If the change in quantities and in
relative prices is not enough, decrease is not enough to counterbalance the tendency to an increase
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of the average economy-wide labour-land ratio caused by the expansion of production of good 1,
there will still be excess demand for labour, and the wage/rent ratio will keep increasing, until the
demand for good 1 is sufficiently discouraged and the q2/q3 ratio has sufficiently decreased for the
simultaneous full employment of labour and land to be again possible. Now the shift of tastes in
favour of good 1 is able to cause an expansion of its equilibrium production because by raising the
relative price of labour it induces a lesser utilization of labour per unit of land in the rest of the
economy. Exercise: discuss the effects on distribution and on the composition of demand in this
three-goods economy if there is a shift of tastes in favour of good 1 but this good is produced with
exactly the average labour-land proportion.
Now an addition to the analysis with two goods and technical factor substitutability. Suppose
tastes change in favour of one good. In the main text it was not shown that, when the factor used
more intensively in the expanding industry rises in relative rental, and this causes a change in factor
proportions in all industries, the effect on relative product prices is still a rise in the relative price of
the good whose demand and hence production have increased.
intl 1.5
T
T
T
unit isoquant of A
unit isoquant of B
unit isoquant of B
unit isoquant of A
K
▪X
H
L
L
(a)
(b)
1/
L
(c)
Fig. 3.8bis
Here is a graphical proof. Assume that consumption goods A and B are produced by
inelastically supplied labour and land, with CRS production functions that yield smooth, strictly
convex isoquants. Assume initially that good A is more labour-intensive than good B at all wagerent ratios (then isoquants can only cross once, see Fig. 3.8bis(a)). We want to study the effect on
pA/pB of a change in general equilibrium due to a shift of preferences in favour of good A. Choose
units for the two goods, such that in the initial equilibrium their costs of production are the same,
pA/pB=1. This is represented in Fig. 3.8bis(a)?? as the same isocost being tangent to both the unit
isoquant of good A and the unit isoquant of good B; the tangency points are H and K. The isoquants
are assumed here to be such that for all slopes of isocosts the optimal labour-land ratio is always
higher in the production of good A. Assume now that the wage-rent ratio increases: the new
tangency between unit isoquants and (dotted) isocosts causes the isocost to be farther from the
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origin for good A, indicating a higher cost of production and hence pA/pB>1.
Now le us admit the possibility of factor intensity reversal, that is, that the optimal labourland ratio can be higher for good A or for good B depending on the wage-rent ratio. This will be the
case if isoquants of the two production functions can cross twice, as shown in Fig. 3.8bis(b). Then
they can also be chosen tangent to each other. Let us choose such units for the two consumption
goods that the unit isoquants are indeed tangent to each other, as shown in Fig. 3.8bis(c), and let us
call γ the w/β ratio equal to the absolute slope of the two isoquants at their tangency, and α the
labour-land ratio associated with it. When w/β=γ, the two goods cost the same. Let us suppose that
good A’s isoquant is the less convex one. Then Fig. 3.8bis(c) makes it clear that as w/β moves away
from γ in either direction, good A becomes cheaper than good B, and more importantly, for any
w/β<γ the labour-land ratio is greater than α for both goods but more so for A which is the labourintensive good, and for any w/β>γ the labour-land ratio is less than α for both goods, and B is the
labour-intensive good. Does this dependence of relative labour intensity on w/β imply that shifts of
demand in favour of a good need not cause the equilibrium relative price of that good to rise? If in
the initial equilibrium w/β is precisely γ (which must mean that the ratio of labour supply to land
supply L*/T* is α), then changes in the composition of demand cause no change in distribution nor,
hence, in pA/pB. This cannot be the case if the ratio between factor supplies is not α. Suppose
L*/T*>α (e.g. the endowment point is X), then w/β≥γ would cause excess supply of labour; at least
in one industry it must be L/T>α but this requires w/β<γ so L/T>α in both industries; hence
equilibrium requires w/β<γ; then pA<pB, and more importantly, A is the labour-intensive good.
Then a shift of preferences in favour of good A causes a rise in the demand for labour, hence a rise
of w/β, steeper isocosts, and a leftward movement along the unit isoquants that causes the
difference in unit costs to decrease, indicating that pA/pB increases. If L*/T*<α, in the initial
equilibrium it must be w/β>γ, so it is again p A<pB but now A is the less labour-intensive good; a
shift of preferences in favour of A causes w/β to decrease, so again p A/pB rises. Conclusion: with
rigid factor supplies, a shift in preferences from an initial equilibrium causes the good, for which
demand rises, to rise in price.
3.A.1. APPENDIX 1. Concavity of the Production Possibility Frontier.
This Appendix studies in somewhat greater detail why the PPF is concave, and generally
strictly concave.
Assume an economy with two factors, labour L and land T, rigidly supplied, and used in two
industries that produce different consumption goods with constant-returns-to-scale production
functions x1=F(L1,T1), x2=G(L2,T2), with smooth and strictly convex isoquants. Draw an Edgeworth
production box with L in abscissa and T in ordinate and with side lengths equal to the factor
supplies, and draw in it the isoquants of industry 1 (with the lower-left corner A as origin) and of
industry 2 (upside, with the upper-right corner B as origin). Because of CRS, for each industry the
slope of isoquants is constant along a ray from the origin.
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Thus if a point of the Pareto set different from the origins is on the diagonal connecting the
origins, then the Pareto set coincides with that diagonal, and this means that the two goods are
produced by production functions that yield the same TRS, and therefore by the same production
function for an opportune choice of units for the two goods; then the PPF is a straight line. If a point
P of the Pareto set is below the diagonal, then the production functions are different and the Pareto
set must be entirely below the segments AP and PB (except for the extreme points of these
segments). Prove it yourself, noting that, as you move along the ray AP from P toward A, the slope
of good 1’s isoquants does not change but the slope of good 2’s isoquants decreases in absolute
value because L/T decreases. An analogous reasoning holding if P is above the diagonal allows us
to conclude that when the production functions are different, the Pareto set is either strictly convex
(and entirely below the diagonal except at the origins), or strictly concave (entirely above), and,
moving along it, the common TRS changes.
intl1.5
B
D
land T
P
A
labour L
C
Fig. 3.18
We can say more: if for example the Pareto set is strictly convex, the equality of TRS requires
that the L/T ratio be lower in industry 1 than in industry 2, so industry 1 is the more labourintensive one along the Pareto set, and as one increases the amount produced of good 1 the ratio
L/T decreases in industry 1, which implies that MP L/MPT increases in both industries: since in
equilibrium it must be w/β= MPL/MPT, this means that if consumer choices change in favour of
good 1, equilibrium w/β rises. If the Pareto set is strictly concave, then industry 1 is the less labourintensive one, and increases of the demand for good 1 have the opposite effect: w/β decreases. The
result is that in both cases as the equilibrium production of good 1 rises, the change in w/β causes
its relative price to rise, because w/β rises when good 1 is the more labour-intensive one, and
decreases when good 1 is the less labour-intensive one. This proves that p1/p2 rises along the PPF as
one increases the amount of good 1; since, as shown in §3.8.2, the slope of the PPF equals –p1/p2,
this proves that the PPF is strictly concave (unless the two goods are produced by the same
production function in which case the PPF is a straight line).
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A different proof is as follows. Assume rigid factor supplies, and two consumption goods.
The set of nonnegative points on or inside the PPF is called the production possibility set; it
represents the set of feasible vectors of productions of the two goods, given the factor endowments
and the available technologies. Consider two points Y=(x1,x2) and Z=(x1’,x2’) on the PPF. Assume
that the two goods are produced by different production functions. Then factor proportions in the
production of x1 may be different from the ones in the production of x1’, and the same may be true
for x2 and x2’. The production of either vector employs the full supplies of labour and of land;
because of constant returns to scale, the production, with the same factor proportions as in Y, of
αY=α(x1,x2), 0<α<1, will employ the fraction α of factor supplies, and the production, with the
same factor proportions as in Z, of (1−α)Z=(1−α)(x1’,x2’) will employ the remainder of factor
supplies, hence producing α(x1,x2)+(1−α)(x1’,x2’) is feasible. Therefore the segment connecting
points Y and Z (cf. Fig. 3.19) belongs to the production possibility set, which is thus shown to be a
convex set. This means that the PPF is concave.
intl1.5
x2
Y
Z
O
x1
Fig. 3.19
What about strict concavity of the PPF? In the Edgeworth production box of Fig. 3.18,
interpret points like D and C as indicating the allocations of labour and of land between industries 1
and 2 that produce the two production vectors Y and Z just mentioned; since in Y and Z there is
efficiency, points C and D are on the Pareto set of the Edgeworth box. As we vary α from 1 to 0, the
corresponding factor allocations move along the segment from C to D (cf. the red dotted segment in
Fig. 3.18); the factor allocations on this segment are not Pareto optimal, so it is possible to produce
more, hence in Fig. 3.19 the segment joining Y and Z is strictly interior to the production possibility
set (except for point Y and Z), so the PPF is strictly concave. It is not strictly concave (it is a
straight line) only if the production functions of the two goods are such that both industries always
use labour and land in the same proportion.
■
The above discussion relies on CRS production functions. If there are increasing returns to an
industry, the PPF may be at least in part strictly convex.
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EXERCISES
5.11 The production function is Cobb-Douglas with CRS, q=Axαy1-α. Let factor prices be vx
and vy. Find the cost function C(vx,vy,q) and confirm Shephard’s Lemma by showing that indeed
∂C/∂vx=x.
5.12 Explain why constant returns to scale and divisibility imply that isoquants are convex.
5.13. Consider the production function (in terms of variable factors only) q = (x+y)α – βy2,
where 0<α, β>1. Prove that for some level of output and some prices, input y is inferior. (If you
have difficulties, try G. S. Epstein and U. Spiegel, “A Production Function with an Inferior Input”,
The Manchester School, vol. 68 no. 5, September 2000, 503-515, and “Comment” by C. A. Weber,
The Manchester School, vol. 69 no. 6, December 2001, 616-622.)