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for the attainment of a social optimum (i.e. an efficient allocation of
the production function is ˜linear-homogeneous™ (e.g. doubling all inputs
society™s resources that allows for all parties to be maximizing), it is
will exactly double output) “ this is usually called ˜constant returns to
curious that many model-builders so glibly assume the existence of
scale™. As stated, this assumption is not a necessary assumption for the
constant returns to scale. If competition is to matter, the production
attainment of a long-run equilibrium since the existence of such an
function cannot be everywhere linear-homogeneous. It is the external
equilibrium only requires the existence of a point on the production
pressure of competition that eventually produces the condition of zero
function which is locally linear-homogeneous [see again Baumol 1977, p.
profit (if profits are positive there is an incentive for someone to enter the
578]. However, it is not uncommon for a long-run model-builder to assume
competition from outside the industry).
that the production function is everywhere linear-homogeneous.
At this stage of the discussion,5 an important general limitation
Parenthetically, let us note that a production function will necessarily be
regarding assumptions [5.1], [5.2a], [5.2b] and [5.3] should also be noted.
linear-homogeneous if all inputs are unrestrictedly variable. 4 But, if any
Specifically, whenever any three of the statements are true, the fourth must
input is fixed (such as space, time available, technological knowledge,
also be true. For example, this means that even when it is impossible to
management talents, etc.) or cannot be duplicated, then the relationship
vary the amount of capital used and yet the production function is
between the other inputs and the output will not usually be everywhere
everywhere linear-homogeneous, if there is enough time for a short-run
linear-homogeneous.
equilibrium and for competition to force profits down to zero, the firm will
For now, let us examine the properties of everywhere-linear-
unintentionally be maximizing profit with respect to its fixed capital. 6
homogeneous production functions. First let us note that the homogeneity
Similarly, even if there is no reason for the production function to be
of such a function implies Euler™s theorem holds, that is, for any function
everywhere linear-homogeneous, maximization and competition will force
X = f (L, K ) it will be true that:
the firm to operate at a point where the production function is at least
X = MPPL…L + MPPK …K at all L, K and X = f (L, K ). [5.1] locally linear-homogeneous.
Now I shall show that when one adds to this assumption that the firm is
in an intermediate-run equilibrium one automatically obtains the necessary
conditions for a long-run equilibrium. The intermediate-run equilibrium
© LAWRENCE A. BOLAND
70 Principles of economics Axiomatic analysis of disequilibrium states 71
LINEAR HOMOGENEITY WITHOUT PERFECT COMPETITION tion function, whenever we apply the conditions of profit maximization in
the intermediate run to this, namely [5.2a′] and [5.2b′], we get:
Note that what is accomplished with the assumption that the firm is a
(W/Px )…L (Pk /Px )…K
perfect competitor is to allow Px to be used as it is in [5.2a]. That is, if P x is
X = ________ + ________
given, Px is both average revenue (AR) and marginal revenue (MR). Thus, 1 + (1/µ) 1 + (1/µ)
[5.2a] can be rearranged according to the definition of marginal cost (MC) 7 or rearranging,
Px…X…[1 + (1/µ)] = W…L + Pk…K
to obtain:
Px = MC. [5.2c] or further,
Px…X = (W…L + Pk…K ) “ (Px…X/µ).
Equation [5.2c] is merely a special case of the more general necessary
condition of profit maximization: Since “ ∞ < µ < 0 (because the demand curve is negatively sloped) we can
conclude that whenever MR is positive (i.e. µ < “1) it must be true that:
MR = MC. [5.2c′]
Px > (W…L + Pk…K )/X ≡ AC
Now whenever the firm is not a perfect competitor and instead faces a
demand curve for its product rather than just a single demand price, [5.2c′]
or in other words there will be an excess profit of
is the operative rule for profit maximization. Facing a (positive-valued)
TP = “ (Px…X/µ) > 0.
downward sloping demand curve means that the price will not equal
Thus we can say that if the firm is not a perfect competitor but is a profit
marginal revenue “ the price will only indicate average revenue. And
maximizer with respect to all inputs (as well as facing a linear-
further, the downward slope means that average revenue is falling with
homogeneous production function), then total profit will be positive “ that
rising quantity and thus at all prices
is, a long-run equilibrium is impossible. 11
MR < AR ≡ Px .
Given the value of the elasticity of demand relative to price changes, µ, and
POSSIBLE ALTERNATIVE MODELS OF THE FIRM
given a specific point on the curve with that elasticity, we can calculate the
marginal revenue as
Now let us look at all this from a more general viewpoint by recognizing
MR ≡ AR…[1 + (1/µ)]
the four separate propositions that have been considered.
which follows from the definition of the terms. 8 If we take into account that
[A] The production function is everywhere linear-homogeneous (i.e.
price always equals AR and that for profit maximization MC = MR and we
[5.1]).
recognize that a firm™s not being a perfect competitor in its product market
does not preclude that market from setting the output price, 9 then we can [B] Total profit is maximized with respect to all inputs (i.e. [5.2a′] and
[5.2b′]).
determine the relationship between price and marginal cost:
[C] Total profit is zero (TP = 0).
Px = MC / [1 + (1/µ)]. [5.2c″]
[D] The firm™s demand curve is negatively sloped (“ ∞ < µ < 0).
And if the firm is still a perfect competitor with respect to input prices 10
then the idea expressed by [5.2a] still holds and thus the necessary
We just saw at the end of the last section that a conjunction of all four of
conditions for profit maximization with respect to both inputs are now:
these is a contradiction “ that is, if [A], [D] and [B] are true then
MPPL = (W/Px ) / [1 + (1/µ)] [5.2a′] necessarily [C] is false. We also saw before that if [A] and [B] hold, [C]
also holds if [D] does not hold (i.e. when the price is given).
MPPK = (Pk /Px ) / [1 + (1/µ)]. [5.2b′]
In fact, more can be said. When any three of these propositions are true
Next I want to show how these last two equations affect our assump-
the fourth must be false. To see this let us first note that the traditional dis-
tions regarding the production function. Recall that if the production func-
cussion of imperfect competition with a few large firms usually considers a
tion of the firm is linear-homogeneous, then [5.1] holds, that is,
long-run equilibrium where total profit is forced to zero (by competition
X = MPPL…L + MPPK …K.
from new firms or competing industries producing close substitutes). With
If we assume the imperfect competitor has a linear-homogeneous produc these traditional models, then, [C] will eventually hold. But it is usually
© LAWRENCE A. BOLAND
72 Principles of economics Axiomatic analysis of disequilibrium states 73
»n“1 “ 1 ≡ (1/β)…»n“1.
also assumed that the firms are all profit maximizers ([B] holds) even when
For later reference, note that β can be considered a ˜measure™ of the
facing a downward sloping demand curve (i.e. even when [D] holds). All
this implies that [A] does not hold, that is, the production function cannot closeness to constant returns (i.e. to linearity). The greater the degree of
increasing returns, the smaller will be β.
be everywhere linear-homogeneous. Specifically, the firm must be at a
The reason why I have chosen this peculiar way of expressing » n“1 will
point where there are increasing returns to scale.
So far I have only discussed the properties of everywhere-linear- be more apparent a little later, but for further reference let me re-express
[5.1′] using β rather than » :
homogeneous production functions. To see what it means to imply
increasing returns to scale, let us now examine a production function which
X / [1 “ (1/β)] = MPPL…L + MPPK …K. [5.1″]
is homogeneous but not linear. If a production function is homogeneous, it
Let us put these considerations aside for now except to remember that a
is of a form that whenever the inputs are multiplied by some arbitrarily
positive factor » (i.e. we move outward along a ray through the origin of an production function which gives increasing returns to scale will be
expressed with 0 < β < ∞ or equivalently with (1/β) > 0. A few paragraphs
iso-quant map), the output level will increase by some multiple of the same
» or, more generally, for X = f (L, K ): ago it was said that [A] is denied whenever we add [C] to [D] and [B]. Let
us consider the more general case where all that we know is that [D] and
»n…X = f (»…L, »…K ).
[H]
[B] hold “ that is, the profit-maximizing firm is facing a downward sloping
Note that a linear-homogeneous function is then just a special case, demand curve in an intermediate-run equilibrium situation. First let us
namely where n = 1. When n > 1 the function gives increasing returns to calculate its total cost (TC):
outward movements along the scale line since the multiple » n is greater TC ≡ W…L + Pk…K.
than ». Note also that this is just one example of increasing returns “
Assuming [D] and [B] hold allows us to use [5.2a′] and [5.2b′] to get
increasing returns do not require homogeneity. Nevertheless, it is often
TC = Px…[1 + (1/µ)]…(MPPL…L + MPPK …K ).
convenient to assume that the production function is homogeneous because
Now we can add [C]. Since total revenue is merely Px…X, zero profit means
the question of whether returns are increasing or decreasing can be reduced
to the value of the single parameter n. Moreover, in this case, we can use that
X = [1 + (1/µ)]…(MPPL…L + MPPK …K )
the particular property of any continuous function that allows us to
calculate the changes in output as linear combinations of the changes in or more conveniently,
inputs weighted by their respective marginal productivities. By recognizing
X / [1 + (1/µ)] = MPPL…L + MPPK …K . [5.4]
that at any point on any continuous function it is also true that:
Now we can make the comparison which reveals an interesting
dX = MPPL…dL + MPPK …dK.
[E]
relationship between imperfect competition and increasing returns. First
If we also assume [H] holds, then if using [E] we set dL = »…L and note that equations [5.1″] and [5.4] have the same right hand side thus their
dK = »…K, it follows that left hand sides must be the same as well. Thus whenever [B], [C] and [D]
dX = »n…X , hold, we can say that
or in a rearranged equation form: 1 “ (1/β) = 1 + (1/µ)
»n“1…X = MPPL…L + MPPK …K. [5.1′] or more directly,
β = “µ ! [5.5]
We see here again that equation [5.1] is the special case of [5.1′] where
n = 1. While we have obtained [5.5] by assuming that the imperfect competitor
I now wish to put [5.1′] into a form which will be easier to compare with is in a long-run equilibrium (and an intermediate-run equilibrium), this is
some later results and to do so I want to express » n“1 differently. Since we really the consequence of the mathematical relationship between the
really are only interested in the extent to which » n“1 exceeds 1, let us marginal and the average given the definition of elasticity. 12 Equation [5.5]
calculate this directly. There are many ways to do this but let us calculate shows that there is no formal difference between the returns to scale of the
the fraction, 1/β, which represents the portion of the multiple » n“1 that production function (its closeness to constant returns) and the elasticity of
exceeds 1, that is, let the firm™s demand curve in long-run equilibrium.
© LAWRENCE A. BOLAND
74 Principles of economics Axiomatic analysis of disequilibrium states 75
Again we can see how special the linear-homogeneous production func- a long-run equilibrium, it does not matter whether the firm is a profit
tion is. Proposition [A] is consistent with [B] and [C] “ that is, with a long- maximizer (i.e. [5.2b] holds) or thinks it is an ANP K maximizer (i.e. [5.6]
run equilibrium “ but this is true only when µ = “ ∞ (that is, when the price holds) with respect to capital. Now earlier we said that if [A] but not [D]
is given, MR = AR ≡ price). Equation [5.5] shows this by noting that in this holds the intermediate run implies a long-run equilibrium. Thus, if we only
case β = ∞ or (1/β) = 0 which implies that the production function is (at know that TP > 0, we can say that whenever [A] holds, [B] cannot hold
least locally) linear-homogeneous. except when [D] also holds. Alternatively, when TP > 0 whenever [D] does
Finally, note that the existence of ˜increasing returns™ is often called the not hold, [A] cannot hold if [B] does.
case of ˜excess capacity™ “ that is, where the firm is not exploiting the full
capacity of its (fixed) plant which if it did it could lower its average cost (in
ON BUILDING MORE ˜REALISTIC™ MODELS OF THE FIRM
other words, it is to the left of the lowest point on its AC curve). All this
leads to the conclusion that when [D] holds with profit maximization, that Now all this leads us to an argument that we should avoid assuming linear-
is, with [B], either we have ˜excess profits™ (viz. when there are constant homogeneous production (i.e. assumption [A]) and thereby allow us to deal
returns to scale) or we have ˜excess capacity™ (viz. when TP = 0). with the intermediate-run equilibrium with or without profit maximization.
In particular, I think a realistic model of the firm will focus on the
properties of an intermediate-run equilibrium which is not a long-run
PROFIT MAXIMIZATION [B]
equilibrium, or on the excess capacity version of imperfect competition,
Note that so far we have always assumed profit maximization. Let us now both of which require that the firm™s production function not be
consider circumstances under which [B] does not hold. First let us assume everywhere linear-homogeneous. Neither assumption denies the possibility
that the firm is a perfect competitor, that is, that [D] does not hold. But this that the production function can be locally linear-homogeneous at one or
time we will assume the firm in the intermediate run is maximizing the more points. This latter consideration means that the intermediate-run view
˜rate of return™ (r) on its capital 13 or what amounts to the same thing, is of the firm offers the opportunity to explain internally the size of the firm
maximizing the average-net-product of capital (ANP K) which is defined as, in the long-run equilibrium. Size is impossible to explain if [A] holds
ANPK ≡ [X “ (W/Px )…L] / K. (unless we introduce new ideas such as the financial endowments of each
firm). Furthermore, it is again easy to see that competition is unimportant
And since average productivity of capital (APP K) is simply X/K,
when [A] is assumed to hold and [D] does not. That is, the traditional
ANPK ≡ APPK “ (W/Px )…(L/K ). argument that ˜competition™ is a good thing would be vacuous when [A]
Moreover, when ANPK is maximized in the intermediate run, the following and [B] hold but [D] does not hold. This is because [A] and [B] alone (i.e.
holds:14 without the additional assumption that competition exists) imply [C] which
MPPK = [X “ (W/Px )…L] / K ≡ ANPK [5.6] was one of the ˜good things™ explicitly promised by long-time advocates of
free-enterprise capitalism or more recently implicitly by advocates of the
MPPL = W/Px . [5.2a]
privatization of government-owned companies. So, again, if economists are
First let us see what this means if we assume [A] holds but not [C], such
to argue that competition matters, they must avoid [A].
as when TP > 0. From the definition of TP, TR and TC, when TP > 0 we
get:
Px…X > W…L + Pk…K USING MODELS OF DISEQUILIBRIUM
or, rearranging,
Now with the above elementary axiomatization of the Marshallian theory
[X “ (W/Px )…L] / K > Pk /Px .
of the firm in mind, let us return to the consideration of how such a theory
Since by [5.6] the left side of this last inequality is equal to MPP K if the can be used to explain states of disequilibrium. To do this we need only
firm is maximizing ANPK, the firm cannot also be maximizing profit with consider each of the four models we will get when we decide which of the
respect to capital (because TP ≠ 0). However, had we assumed that TP = 0, assumptions [A] to [D] we will relax (since, as I explained, the four
we would get the same situation as if [5.2a] and [5.2b] were the governing assumptions cannot all be true simultaneously).
rules rather than [5.2a] and [5.6]. That is to say, if we assume the firm is in
© LAWRENCE A. BOLAND
76 Principles of economics Axiomatic analysis of disequilibrium states 77
Model 1. Dropping assumption [D] sulting from the limited amount of time available for competition to pro-
duce either zero profit or the optimum use of all inputs. The phenomena are
Dropping the notion that the firm can affect its price (by altering the
suboptimal only in comparison with long-run equilibrium. Once one recog-
quantity it supplies to the market) merely yields the old Marshallian theory
nizes that there has not been enough time, as long as the firm is maximiz-
of the price-taking firm (see Figure 5.1). Nevertheless, it does give us the
ing with respect to every variable input, nothing more can be expected. In
opportunity to explain various states of disequilibrium. Let us consider
other words, disequilibrium phenomena may be long-run disequilibria and
various attributes of disequilibrium. If the firm is not at the point where the
short-run equilibria.
production function is locally linear-homogeneous, there can be several
interpretations of the situation depending on whether or not we assume [B]
or [C] holds. If [C] does not hold but [B] does, there could be either Model 2. Dropping assumption [B]
positive or negative profits. If we wish to explain the absence of zero
Dropping assumption [B] leads us astray from ordinary neoclassical
profits, we can always claim that this is due to our not allowing sufficient
models since [B] says that the firm is a maximizer. What we need to be
time for competition to work. If [B] does not hold but [C] does, then there
able to explain is the situation depicted in Figures 5.2(a) and 5.2(b), again
must be something inhibiting the firm from moving to the optimum point
depending on whether or not we are assuming a long-run situation. In
where price equals marginal costs. In comparative-statics terms, we can
either case it is clear that the firm is setting price equal to marginal cost 15
explain either type of disequilibrium state by noting that since the last state
which means that MPPL equals W/Px and thus cannot be satisfying
of equilibrium was reached certain exogenous givens have changed. For
equation [5.2b′] which is the necessary condition for profit maximization
example, tastes may have changed in favour of one good against another,
when [D] holds. An exception is possible if we assume the owner of the
thus one firm will be making profits and another losses or the firm has not
firm is not very smart and attempts to maximize the rate of return on
had enough time to move along its marginal cost curve. Similarly, it could
capital rather than profit. For a maximum ANP K, all that would be required
be that technology has changed. Any such explanation thus would have to
is that ANPK equals MPPK. There is nothing inconsistent since it is still
be specific about the time it takes to change variables such as capital as
possible for [D] and [A] to hold so long as ANPL equals MPPL and this is
well as specify the changes in the appropriate exogenous variables.
the case. But again, maximizing rates of return to either labour or capital is
Hopefully, such an explanation would be testable.
not what we would normally assume in a neoclassical explanation.
$ $ $
MC

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