Optimum and Regulation in Monopolistic Competition
Henrik Vetter
Statsbiblioteket,
Universitetsparken DK 8000 Aarhus C
Abstract: Chamberlain speaks of monopolistic competition as an ideal that combines
elements of a monopoly and features of competition between firms experiencing economies
of scale. In this paper, we ask if monopolistically competitive markets regulate themselves in
a reasonably efficient way, which would lead us to conclude that it is better not to regulate, or
if they can be improved using instruments like unit taxes and subsidies. Our conclusions
suggest that monopolistically competitive markets function well under a unit tax, and in some
cases they function well even without regulation.
Jel.no.: D43, D60
Keywords: monopolistic competition, constrained efficiency, regulation.
1
Introduction
The issue of the market’s performance in the presence of scale economies is an old one.
Chamberlain speaks of monopolistic competition where multiple firms produce an almost
identical product with diverse quality and price, and individual firms are still able to cover
their variable as well fixed production costs (Chamberlain 1933, 110–113). In the large group
case, in symmetric equilibria, oligopolistic interaction is ignorable; that is, the individual
firm’s demand curve is downward sloped and stays fixed unless all of the firms change
output simultaneously. Profit maximization thus calls for equality of marginal revenue and
marginal cost while full long-run equilibrium involves elimination of (super-normal) profit.
The basic issues in such conditions are (1) whether and to what degree the market is efficient
and (2) is there is a need for regulatory intervention.
In this paper, we look at the efficiency of monopolistically competitive markets to see if they
can be improved using instruments like unit taxes and subsidies or if the market functions in a
reasonably efficient way, which would lead us to conclude that it is better not to regulate.
Following Dixit and Stiglitz (1977) and more generally Guesnerie (1975) we can ask either
about the optimum that is constrained by technology only (denoted the unconstrained
equilibrium) and compare this to the situation under monopolistic competition, or,
alternatively, we can compare the monopolistically competitive market’s allocation to the set
of equilibria that allows each firm a non-negative profit in the absence of lump-sum transfers.
The difference between the comparisons has to do with the kind of policies that are practical.
Plainly, when each firm experiences economies of scale at the unconstrained optimum the
price must exceed marginal cost for firms to break even and this violates efficiency, or if the
price is to be kept equal to the marginal cost firms must be given a sump-sum payment. The
2
alternative is to look at the regulated market’s best performance when lump-sum instruments
are dismissed. That is, the allocation under monopolistic competition is compared to the
allocations that a planner can reach by using franchise, unit taxes and subsidizes (which are
instruments that allow for decentralization).
Our concern is whether to regulate monopolistically competitive markets or not, and in the
affirmative case ask about the appropriate kind of regulation – it has been established
elsewhere that the market can supply too few products (Spence 1976 and Dixit and Stiglitz
(1977) or too many products (Hart 1985a, 1985b). We use a constrained-regulated optimum
as a benchmark, the constraint being that firms survive without the use lump-sum instruments
as they are not practical; for example, they are based on the presumption that the regulator is
essentially not constrained by informational limitations. Even if concerns about information
were somehow easily dismissed, Baumol and Bradford (1970) and later Baumol (1979) alone
argue that lump-sum transfers are ruled out by the economy-wide budget when production
lies at a point of increasing returns to scale. For such reasons, it seems more practically
interesting and relevant to judge the efficiency of the monopolistically competitive market
based on a comparison with a range of realizable equilibria in a decentralized, nonconvex
private-ownership economy. As noted above, restricting attention to equilibria with nonnegative firm profits allows decentralization through a unit tax or subsidy.
Discussing whether there for practical purposes is a need for regulation of monopolistically
competitive markets, and what kind of instruments can be used, we initially assume that firms
are symmetric. The assumption that firms are symmetric together with the assumption that
firms experience economies of scale through fixed costs is problematic when we consider
issues of efficiency since production cost minimization is a necessary condition for
3
efficiency. Bergstrom and Varian (1985a, 1985b) show that a fixed aggregate output is
produced most cheaply (in some circumstances) when firms have different marginal costs.
Salant and Shaffer (1999) subsequently demonstrate that in a two-stage Cournot duopoly,
where the first-period technology choice affects the second-period marginal costs, that
rearranging symmetric investments to make them asymmetric can reduce aggregate
production costs. When firms face a fixed cost, like for example advertising expenses, that do
not affect production, asymmetric behavior will not save costs. However, in the long run, all
production factors can be changed, and the choice of capital or capacity is a decision on the
position of the long-run average cost curve characterized by scale economies. In this case,
which we briefly discuss, symmetric investments in the unregulated market can be a source
of inefficient market performance.
This paper proceeds as follows. In section 1, we set forth a familiar model of monopolistic
competition. In section 2, we discuss efficiency and show that regulation in the absence of
lump-sum transfers is either not called for, or that a simple unit-tax scheme implements the
best possible allocation. Section 3 extends the arguments to include possible cost savings
when the fixed cost is unevenly distributed between firms. Section 4 concludes.
1. Monopolistically Competitive Equilibrium
Following Dixit and Stiglitz (1977) and Spence (1976), we assume that there is one
aggregative good and a huge range of potential varieties of goods in a monopolistically
competitive sector. All of the varieties enter symmetrically into the utility function of a
representative consumer, and firms are supposed to have access to identical technologies. A
firm in the monopolistically competitive sector produces one variety out of the large number
of possible varieties, or the firm does not produce at all. In this case product variety is simply
4
the number of firms that survives in the market. This number, called n, is determined by a
zero-profit condition.
Production of each variety requires a fixed cost. When the fixed cost has to do with firm size
or the firm’s choice of capital quality, it is natural to assume that the fixed cost changes shortrun costs. That is, the short-run cost function is generally written as ψ ( x, k )x , where k is the
representative firm’s expenses on size or equipment quality and x is the output of the firm.
Of course, the firm is uncommitted in the long run, but, once it has finished planning and
decided on the size or quality of capital, the cost function is fixed. Alternatively, if the fixed
cost is advertizing costs or administrative costs for example, the short-run cost is ψx, like in
Dixit and Stiglitz (1977) and Spence (Spence 1976), or the short-run cost function is given by
ψ ( x )x.
When the total production cost for a typical variety is ψ ( x, k )x + k , average cost is
ψ (x, k ) + k x . It is assumed that marginal output costs are positive and increasing, that is,
ψ x > 0,ψ xx > 0, and that capital lowers running costs but at a decreasing rate, that is, ψ k < 0 ,
and ψ kk > 0 . Moreover, an increase in the firm’s fixed cost lowers the firm’s marginal output
cost when ψ xk ( x, k ) x +ψ k ( x, k ) < 0 . For a fixed capacity, the relationship between firm
output and average cost is U-shaped according to ψ x ( x, k ) − k x 2 , and long-run average cost
is the U-shaped envelope of the short-run average cost functions. The long-run average cost
curve becomes ψ (k ) + k x or ψ + k x if, for some reason or another, the firm experiences a
fixed marginal cost that is either dependent or independent, respectively, of its investment. In
such cases, long-run average costs are decreasing in firm output and meet the fixed marginal
cost asymptotically as output grows.
5
With respect to demand and consumers’ tastes for variety, we follow Spence (1976) and Dixit
and Stiglitz (1977) and assume that the utility function of the representative consumer is
u ( x ) = G (m ) + x 0 , m =
i
φi ( xi ) , where x0 is a numeraire good, G and φi are concave
functions, xi is the consumption of the i’th variety, and x is the bundle of varieties actually
consumed. The term m =
i
φi (xi ) is sometimes referred to as the market-congestion index,
and this term is used here. Of course, we have u ( x ) = G (m ) + x 0 , m = nφ ( x ) under
conditions of symmetry, where symmetry as noted also implies that different firms have
access to the same set of technological possibilities. With additive utility, we can focus on
partial equilibrium (see Spence 1976).1 The φ - function describes a preference for variety as
follows: Pick out two goods and notice that only the sum of the amounts of the two goods
matters for utility when the φ - function is linear. Conversely, when the φ - function is
concave, replacing one good with the other gives a loss since diversity is reduced while the
quantity is unchanged.
When consumers take prices as given, utility maximization defines a set of demand functions
given by p = G '
( m ) φ '( x ) . When a firm can safely assume that the market-congestion index
stays put regardless of its own actions, the firm is a monopolist in the production of its own
variety. On the other hand, when all of the firms move together or if enough firms change
their actions simultaneously, the congestion index is affected, and the demand of any specific
variety is changed. Whether equilibrium in one market directly affects the demand for goods
in other markets depends on the relation between the equilibrium in any specific market and
the congestion index. The congestion index’s elasticity with respect to output of the typical
6
variety is ∂ log m ∂ log x = xφ '
(x ) nφ (x ) . Moreover, actions of the individual firm leave the
congestion index unchanged when the elasticity is vanishing. For example, if
φ ( x ) = xb ,0 < b < 1 , the elasticity is bx b (nx b ) = b / n . In turn, demand for one specific
variety is not affected by the equilibrium in markets for other varieties when b n → 0 , that
is, when the initial endowment in the economy sufficiently supports many varieties (see Hart
1985a).2 Long-run equilibrium in the market calls for equality between marginal revenue and
marginal cost, that each firm’s capacity choice minimizes production cost, and finally that the
profit of the representative firm is zero since otherwise there would be entry or exit of firms.
(nφ (x ))φ '(x )x − (ψ (x, k )x + k ) , and the conditions
The profit of the representative firm is G '
for long-run equilibrium are:
(1)
G'
(nφ (x ))[φ '(x ) + xφ ''(x )] − ψ (x, k ) −ψ x (x, k )x = 0
(2)
ψ k ( x, k )x + 1 = 0
(3)
G'
(nφ (x ))φ '(x )x = ψ (x, k )x + k
The first optimum condition simply says that the marginal revenue of the representative firm
should equal marginal costs, the second is a cost minimizing condition, and the third
optimum condition says that the profit of the marginal firm is zero. The solution to equations
{
}
(1), (2), and (3) is called xˆ, kˆ, nˆ .
1
Dixit and Stiglitz (1977) analyze a model that allows for substitution between sectors.
Symmetry rules out neighboring goods. If one would like to study asymmetric equilibria, an
additional assumption that rules out neighboring goods must be added. One way to accomplish this is assume
that a consumer is interested in only a subset of all potential product varieties, see Hart 1985a, 1985b.
2
7
2. Welfare, Markets, and Regulation
As noted by Spence (1976), consumers’ wants in general are not measured adequately by
revenue, but, in perfectly competitive markets prices guide the allocation towards an efficient
point. First, any fixed aggregate output is produced as cheaply as possible; that is, it is not
possible to provide consumers with some fixed amount of any good at a lower cost. Second,
the market selects the best way to distribute resources between different activities; that is, it is
impossible to imagine some policy that redistributes resources between sectors and improves
consumers’ utility. In monopolistically competitive markets, the utility consumers derive
from the consumption of goods in the monopolistically competitive sector is increasing in the
congestion index, and the monopolistically competitive market is constrained efficient when
it guides the economy to an allocation that minimizes the cost of producing some fixed value
of the congestion index. Clearly, constrained efficiency is a necessary condition for Pareto
optimality, but sufficiency also requires that the marginal benefit of expanding the
monopolistically competitive sector is equal to the marginal social cost of doing so. The
monopolistically competitive market is unconstrained efficient—or Pareto efficient—when
the kinds and quantities of varieties combine to provide a value of the congestion index for
which the marginal cost of an increase in the index is equal to the marginal benefit (see
Spence 1976 for further discussion).
2.1.
Markets and Welfare
Clearly, for an allocation to maximize welfare it is necessary that production costs are
brought to their minimal value. With respect to the question of how to minimize aggregate
production cost and the issue of how scale economies should be utilized when consumers
value product variety, we can fix the congestion index, say at m = nφ ( x ), and ask how m is
produced most cheaply. It is straightforward that the aggregate cost of providing some fixed
8
value of the congestion index is (m φ ( x ))(ψ ( x, k )x + k ) , and minimizing with respect to
output and capacity per firm we have the first-order conditions:
(4)
(ψ (x, k ) + ψ x (x, k )x )φ (x ) − (ψ (x, k )x + k )φ '(x ) = 0
(5)
ψ k ( x, k )x + 1 = 0
From the cost minimization condition in equation (4), proposition 1 limits how far scale
economies can be utilized at a social optimum.
Proposition 1.
At the unconstrained optimum, the average cost must exceed the marginal cost.
Proof.
The relationship between average and marginal costs at constrained optimum can be found
from equation (4) to satisfy
(6)
φ (x ) x
(ψ (x, k )x + k ) x .
=
φ '( x ) ψ ( x, k ) + ψ x ( x, k )x
Since the φ (x ) − function is concave, the left-hand side is always greater than one, and the
relationship in equation (6) thus shows that average costs exceed marginal costs at
unconstrained optimum.
End of proof.
The proposition is concerned with how to provide as cheaply as possible some fixed value of
the congestion index and the result reaffirms that it is not logical that the welfare-maximizing
size of each firm is coincident with full exploitation of all of the economies of scale (see, for
example, Chamberlain 1951, especially pages 345 and 349) when product variety matters. In
9
a model of optimal location, Starrett (1974) shows a somewhat similar result; that is, at the
optimum, firms’ production lies in the range where there are increasing returns. Of course,
when the average cost exceeds the marginal cost, it is possible to reduce the aggregate cost by
reducing the number of firms and let each remaining firm produce more. However, the
significance of tastes for variety is that reducing the number of firms drives down welfare by
reducing diversity. That is, economies of scale can only be fully utilized at the cost of
reducing variety to a level that is too low.
It is a consequence of proposition 1 that Pareto optimum is at a point of increasing returns
and the existence of decentralized equilibrium with equality between price and marginal cost
is foreclosed. This follows since competitive equilibrium is incompatible with increasing
returns. On the other hand, if a competitive equilibrium exists, then it is also a Pareto
optimum. Thus the existence of competitive equilibrium must fail when Pareto optimum
requires that each firm produces with economies of scale. However, it is not a consequence of
proposition 1 that a monopolistically competitive market fails to minimize the cost of
producing some fixed level of the congestion index. Thus, the monopolistically competitive
market might be constrained efficient. To address this issue we can compare the cost
minimization conditions, equations (4) and (5), with the cost that is implied by the long run
equilibrium conditions, equations (1) through (3). Now, combine (1) and (3) in order to
(nφ (x )) to give
eliminate G '
(7)
φ '( x )x
ψ ( x, k )x + k
=
.
φ '( x ) + xφ ''( x ) ψ ( x, k ) + ψ x ( x, k )x
To see whether firm behavior actually minimizes the cost of attaining some fixed level of the
congestion index notice that the equations giving the cost minimizing capacity choice are
10
identical, cf. equations (2) and (5). In turn, if it so happens that the function x(k ) solves
equations (4) and (7) simultaneously, it is clear that the price system guides the allocation to
its cost minimizing position. Proposition 2 follows immediately from equations (4) and (7).
Proposition 2.
For the class of utility function that satisfies
φ '( x )x
φ (x )
=
,
φ '( x ) + xφ ''( x ) φ '( x )
(8)
the market is constrained optimal in the sense that some fixed level of the congestion index is
produced at the lowest possible cost.
When a utility function satisfies proposition 2, rearranging the mix of kinds and quantities
and keeping the value of the congestion index fixed will not lower aggregate production
costs. The only market failure in this case is that the size of the monopolistically competitive
sector is too large or too small. An illustration of proposition (2) is that, in the case of
φ (x ) = x α , 0 < α < 1 , equations (4) and (7) give the same solution for output per firm as a
function of capacity, and the market solution minimizes costs for some fixed value of the
congestion index. Dixit and Stiglitz (1977, Section I.c) use the utility function
( {
u (x ) = G x0 ,
xα
i i
}
1α
),
0 < α < 1, to analyze how monopolistically competitive markets
perform, and they speak about “a rather surprising case where monopolistic competition
equilibrium is identical with the optimum constrained by the lack of lump sum subsidies.”
(Dixit and Stiglitz 1977, 301) Since the aggregate costs of reaching some value of the
congestion index with this kind of utility function are minimized by the (unregulated) market
solution, proposition 2 makes precise why this is perhaps not so surprising.
11
2.2.
Regulation and Welfare
It is a straightforward consequence of proposition 1 that maximum welfare in relation to
technically limited opportunities are not realized without regulation. As noted, when the
firm’s average cost exceeds its marginal cost, scale economies should not be fully exploited
at the technology constrained optimum. In turn and this of course is at odds with
decentralized equilibrium without lump-sum transfers, marginal cost pricing results in a loss.
Clearly, were lump-sum transfers feasible, they could be used to attain the first best: A unit
tax or subsidy can be used to fix the right amounts of output and investment per firm,
subsequently the number of varieties is fixed using a lump-sum transfer. Following Dixit and
Stiglitz (1976) lump-sum instruments are generally not practical instruments. Moreover, an
argument due to Baumol (1979) makes it unclear if lump-sum transfers are feasible in general
even if we suppose that some inescapable poll tax is practical. Essentially, when all firms
operate on a point of increasing scale, they need a subsidy to break even. Now suppose that
consumers are the owners of production factors. When production lies at a point of increasing
returns, prices that are equal to marginal cost provide the firm with insufficient revenue to
cover costs, inclusive of the costs of labor and capital, and even is all income is taxed away
there is insufficient tax revenue to cover costs. Simply put, there is a difference between
consumer and producer surplus and the amount of cash that can be taxed when we consider
the economy in the aggregate.
For such reasons, we focus on policies that are constrained by the non-negativity of each
firm’s profit. Such kinds of intervention can be based on “regulation, or by excise or
franchise taxes and subsidies.” (Dixit and Stiglitz 1977, 300). Excluding lump-sum taxation
or subsidization that regulates the number of firms (and keeps unchanged the mix of kinds
and quantities) the upshot of proposition 2 is that there are conditions where monopolistically
12
competitive markets function in a reasonably efficient way, which lead us to conclude that it
is better not to regulate. On the other hand, when utility functions do not satisfy proposition
2 we can look at the efficiency of monopolistically competitive markets by asking if they can
be improved by practical regulation. Proposition 3 makes precise when the unregulated
market is as efficient as a regulated market.
Proposition 3.
When the unregulated market is constrained efficient, a constrained-regulated monopolistic
market (constrained by the absence of lump-sum subsidies) is identical to the unregulated
market. That is, if we denote by x*, k *, and n * output, capacity, and the number of firms
{
}
that the regulator fixes, we have xˆ, kˆ, nˆ = {x*, k *, n *}.
Proof.
Social benefits are measured by W = G (nφ ( x )) − n(ψ ( x, k )x + k ) , but we find it convenient to
write the Lagrange function in terms of m, x and k . The Lagrange function is:
(9)
ℑ = G (m ) −
m
(ψ (x, k )x + k ) + λ {G '(m)φ '(x )x −ψ (x, k )x − k }
φ (x )
The first-order conditions for x*, k *, m * and λ are:
(10)
(11)
m (ψ (x, k ) + ψ x ( x, k )x )φ ( x ) − φ '( x )(ψ ( x, k )x + k )
φ (x )
φ (x )
+ λ {G '(m )(φ ''( x )x + φ '( x )) − ψ ( x, k ) − ψ x ( x, k )x} = 0
−
−
m
(ψ k (x, k )x + 1) − λ (ψ k (x, k )x + 1) = 0
φ (x )
13
1
(ψ (x, k )x + k ) + λG''(m )φ '(x )x = 0
φ (x )
(12)
G'
(m ) −
(13)
G'
(m )φ '(x )x − ψ (x, k )x − k = 0
Clearly, equations (11) and (13) are the same as equations (2) and (3), respectively. Suppose
now that the solution for output in equations (10) and (12) is the same as the solution for
output in equation (1). This implies that the second term on the left-hand side of equation
(10) equals zero. This is possible when the assumption that the market solution minimizes
costs is satisfied because this assumption implies that the first term on the left-hand side of
equation (10) is also zero, cf. equation (4). Equation (12) then must determine the value of
the multiplier.
End of proof.
Chamberlain (1933) speaks about “a sort of ideal” equilibrium that involves elements of
competition and elements of monopoly that are present because of consumers’ taste for
varieties of goods. Proposition 3 shows that, in the absence of lump-sum transfers, the
monopolistically competitive market can be improved upon only if unregulated firms fail to
minimize the cost of achieving some degree of consumer satisfaction. Proposition 2 then
makes clear when Camberlain’s (1933) ideal allocation is actually reached by the markets on
its own. When monopolistically competitive firms fail to cost minimize, it is an immediate
implication of propositions 2 and 3 that a simple nonlinear tax supports Chamberlain’s “sort
of ideal” market. To see this, inspect the case in which firm output in laissez-faire is greater
than the constrained social optimum output, xˆ > x * . In this case, an output tax defined by
t = 0 for any output less than or equal to x * and by t = T , where T is prohibitive, makes
the firm’s output choice coincident with the socially optimal output per firm. Since the
14
condition defining optimum capacity is left unchanged by the introduction of the output tax
(or subsidy), it follows that the firm chooses k * when it produces x * . As the tax does not
extract revenue, it follows straightforward that it implements the constrained optimum. When
xˆ < x * , the output tax is defined by t = 0 for any output greater than or equal to x * , and
when x < x * the total tax is T ( x * − x ), where once again T is prohibitive. This tax makes
the firm’s output choice coincide with the socially optimal output per firm, and, as in the case
of xˆ > x * , it follows immediately that capacity and the number of firms are at their socially
optimal levels. We summarize this as proposition 4.
Proposition 4.
The constrained social optimum is either reached by the market on its own, or it can be
attained by a simple nonlinear per-unit tax.
3. Welfare and Asymmetric Firms
When firms are symmetric the upshot of propositions 2 and 3 is that the market, in spite of
monopolistically competitive characteristics, directs firm output to a point that minimizes the
aggregate cost of reaching some level of consumer satisfaction for a class of utility functions.
Proposition 4 demonstrates how a simple revenue-neutral tax is the basis for the required kind
of production efficiency should the market fail to minimize costs. Of course, the idea of
similar (a large number of) but nonidentical and nonidentifiable firms provides a neat
justification for the downward sloping demand curve for an individual firm that
simultaneously
allows
one
to
distinguish
between
monopoly
and
oligopolistic
interdependence between firms. However, when the set-up cost changes firms’ marginal cost
functions, the restriction to a symmetric fixed cost need not be consistent with cost
minimization. This is demonstrated in the context of oligopoly by Salant and Shaffer (1999)
15
and initially by Bergstrom and Varian (1985a, 1985b) showing that, for a fixed sum of
constant marginal costs, production costs go down with increases in the variance of marginal
costs. With constant marginal costs, a perturbation that keeps the marginal cost sum
unchanged will not change aggregate output. However, a firm increases its market share
when marginal costs go down, and it loses market shares when marginal costs go up. In this
way, a perturbation in marginal cost shifts production from a “high cost” to “low cost”
producer and reduces aggregate variable production while aggregate output and thus
consumer satisfaction is constant.
To spell out whether or not the result that it might be beneficial that ex ante identical firms
behave differently has implications in relation to propositions 2, 3, and 4 we should be
precise about the effects of the changes in capacity and the form of the cost function. Of
course, when the fixed cost is advertising costs, as suggested by Spence (1976), marginal
costs cannot be changed by a different distribution of the firms’ fixed costs and there are not
benefits to asymmetric investments. However, when the fixed cost is capacity or quality of
capital and the marginal cost is a function of the fixed cost and possibly also a function of the
firm’s output - cost functions of the form ψ ( x, k )x + k 3 - our conclusion that the market is
constrained efficient, or easily regulated to this position, is also valid when firms can invest
in different levels of capital. To show this take as a starting point the symmetric allocation in
which active firms produce x̂ with a capacity of k̂ . Let us denote aggregate output by X̂ ,
and, likewise, the sum of investments is denoted K̂ . Pick out two firms and suppose that it is
possible to redistribute capacity between them and manipulate their production to leave intact
aggregate production. That is, aggregate capacity and output are by construction unaffected.
3
Salant and Schaffer (1999) and Bergstrom and Varian (1985a, 1985b) analyze cases of either a fixed
marginal cost or a linear marginal cost where the intercept is a function of investment.
16
{
}
Now, for any set of fixed costs for firms 1 and 2, k1 , Kˆ − k1 , output and capacity must be
manipulated so that the firms’ marginal costs are the same since it would otherwise be
possible to reallocate production between them and save costs. That is, we have the
restriction that
(13)
(
)
(
)(
ψ ( x1 , k1 ) + ψ x ( x1 , k1 )x1 = ψ Xˆ − x1 , Kˆ − k1 + ψ x Xˆ − x1 , Kˆ − k1 Xˆ − x1
)
Clearly, we have more unknowns than equations, thus we can always find a range of
simultaneous values of output and capacity that satisfies equation (13). This kind of
perturbation affects firms 1 and 2 only, outputs and costs of the remaining firms are left
undisturbed. In order to see whether the redistribution of output and capacity drives
production costs up or down, notice that the aggregate cost for the two firms is
(
)(
)
Ω = ψ ( x1 , k1 )x1 + k1 + ψ Xˆ − x1 , Kˆ − k1 Xˆ − x1 + Kˆ − k1 . For any fixed value of k1 , the first-
order condition for cost minimization, dΩ dx1 = 0, is
(14)
(
)
(
)(
)
ψ ( x1 , k1 ) + ψ x ( x1 , k1 )x1 − ψ Xˆ − x1 , Kˆ − k1 + ψ x Xˆ − x1 , Kˆ − k1 Xˆ − x1 = 0 .
( )
Of course, this is nothing but equation (13) repeated, and xˆ, kˆ is a critical point, however,
there is evidently more than a single critical point. Whether the production costs of firms 1
( )
and 2 are minimized (locally) by the symmetric allocation xˆ, kˆ depend on the second
derivative of Ω . We have
(15)
(
)
(
)(
)
Ω''= 2ψ ( x1 , k1 ) + ψ xx ( x1 , k1 )x1 + 2ψ Xˆ − x1 , Kˆ − k1 + ψ xx Xˆ − x1 , Kˆ − k1 Xˆ − x1 .
( )
( )
( ) ( )
( )
Evaluating this at xˆ, kˆ , we have Ω'
'xˆ , kˆ = 4ψ x xˆ, kˆ ψ xx xˆ, kˆ Xˆ . Now, the sign of Ω'
'xˆ , kˆ
determines whether the total of the two firms’ costs is at a local maximum or at a local
17
( )
minimum at the critical point xˆ, kˆ . The optimum properties of symmetric and asymmetric
investments, respectively, are spelled out in propositions 5 and 6.4
Proposition 5.
( )
( )
The critical point xˆ, kˆ minimizes aggregate cost locally if Ω'
'xˆ , kˆ > 0.
Proposition 5 along with propositions 2 and 3 spell out when a symmetric monopolistically
competitive market is constrained efficient—meaning that the mix of the kinds and quantities
of varieties cannot be produced more cheaply. Moreover, when propositions 4 and 5 are
fulfilled a simple nonlinear unit-tax scheme guides the market to an allocation that minimizes
the cost of reaching some value of the congestion index.
Proposition 6.
( )
( )
The critical point xˆ, kˆ maximizes aggregate cost locally when Ω xˆ, kˆ ''
< 0.
It is an upshot of proposition 6 that, even if propositions 2 and 3 are satisfied or if a unit-tax
scheme is used in accordance with proposition 4, the monopolistically competitive market
fails to minimize the costs of achieving some fixed level of the congestion index. In theory, a
slight disturbance of investment behavior that forces ex ante identical firms to choose
asymmetric capacity would reduce the cost. The practical implication of this observation is
however another matter. Picking out any two firms, we know that the symmetric allocation
4
Salant and Schaffer (1999) give a different condition on the optimum properties of symmetric
investments, which has to do with the fact that they (primarily) study firms with a fixed marginal cost. In this
case, production costs net of investment costs are minimized when the firm with the lower marginal cost
produces all of the output (regardless of the type of investment—symmetric or asymmetric), however, keeping
18
(xˆ, kˆ ) is a critical point of the cost function; that is, k
1
= k 2 = kˆ and x1 = x 2 = xˆ results in
equality between marginal costs of the two firms, cf. equation (14). In long-run equilibrium,
the firms’ profits must be zero and, since the firms face the same price, this in turn implies
that their average costs should be equal. When firm 1 uses capacity k1 and produces x1 and
firm 2 uses capacity Kˆ − k1 and produces Xˆ − x1 , average costs are equal when
(15)
ψ ( x1 , k1 ) +
k1
Kˆ − k1
.
= ψ Xˆ − x1 , Kˆ − k1 +
x1
Xˆ − x1
(
)
Minimization of aggregate production costs calls for equality between marginal costs, cf.
equation (14), and the zero-profit condition calls for equality between average costs, cf.
equation (15). Equations (14) and (15) are equations with two unknowns, and the solution is
k1 = k 2 = kˆ and x1 = x 2 = xˆ. In turn, if one selects other values than x̂ and k̂ for production
and capacity that satisfy the condition for minimizing costs, it is necessary with a lump-sum
transfer scheme to ensure that the zero-profit condition is satisfied simultaneously.
Considering lump-sum transfers to be impractical, the implication is that one should not
advocate regulation even though the market’s performance would improve by asymmetric
capacity investments.
4. Conclusion
In this paper, we have argued that monopolistically competitive markets function in a
reasonably efficient manner in spite of the presence of scale economies: For a class of utility
functions, the market allocation cannot be improved by a planner unless the planner can use
lump-sum transfers. For other utility functions, the market’s allocation with a simple
output unchanged drives costs up or down depending upon the convexity of the fixed cost. We have modeled
19
nonlinear tax is coincident with the allocation that a planner would chose when lump-sum
transfers are out of the question. It is obvious that a lump-sum transfer can support the firstbest allocation when scale economies are due to a fixed cost. One way to look at the
exclusion of lump-sum instruments is that they are not practical because of the general
problem of obtaining the information needed to use them. However, there is also the problem
that the concept of lump-sum transfers is problematic when there are scale economies.
As argued by Baumol and Bradford (Baumol 1970) and later by Baumol (1979), if there are
economies of scale, the value of output is less than the cost of inputs, and marginal cost
pricing would violate Walras’ law. Thus, there cannot be general equilibrium with marginal
cost pricing; that is, the economy-wide budget restriction cannot be satisfied. Turning to
regulation that can be decentralized, for example, a unit tax, and the question of what the
market optimizes in the presence of scale economies, the answer is that the market sometimes
maximizes what a planner would maximize (when the conditions in proposition 2 are
satisfied by the utility functions), or it maximizes what a planner who uses a unit tax would
maximize (proposition 4).
When a fixed cost can affect variable costs, Salant and Shaffer (1999) have shown that
production cost can go down when ex ante identical firms choose different investments. Our
discussion of the efficiency of monopolistically competitive markets is focused on the issue
of cost minimization. Therefore, we must ask how the proposition on cost reduction through
asymmetric investment can affect the results on how efficient monopolistically competitive
markets are. Of course, (as demonstrated in proposition 6), under proper circumstances, it
would also be possible to reduce aggregate costs if it is somehow possible to induce firms to
fixed cost more simply and variable costs more generally when compared to Salant and Schaffer (1999).
20
choose different capacities. Clearly, the market is symmetric and will not realize the cost
savings. When lump-sum transfers are out of the question, the zero-profit condition along
with a requirement for equal marginal costs is consistent only with symmetric behavior. That
is, without lump-sum transfers, neither the regulator nor the market is able to take advantage
of the potential for reducing the aggregate production cost.
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