Generalized Comparative Statics under Monopolistic Competition:
Anti-competitive Paradox, Immiserizing Growth, Catastrophes
S. Kokovin, E. Zhelobodko∗
Novosibirsk State University
Abstract
Several common wisdoms of economic geography and trade theories rely on specific technical assumptions,
notably, CES utilities. Krugman’s (1979) general approach to monopolistic competition avoid this limitation, but
has too narrow use. We expand it now to a family of multisector models and to additional effects of comparative
statics. It turnes out that under market growth the price for varieties can go up or down, depending upon ArrowPratt measure of concavity of the utility function, does it decrease or increase. Welfare and number of firms
also can increase or decrease. There can be asymmetric equilibria, multiple equilibria and related catastrophic
shocks.
Introduction
Various models of monopolistic competition started from Chamberlin’s basic idea to model a firm – owner of a
brand – as a price-maker, under free entry to the industry and both fixed and variable costs. This idea became
really productive in industrial organization and other areas after the famous paper by Dixit and Stiglitz [12],
who invented an approach to practically find equilibria in such models, using, typically, the constant-elastisityof-substitution (CES) class of utility functions. This novelty enabled these models to become a cornerstone, or a
main construction element, for such important areas as New theory of economic growth (see Aghion, Howitt [1]),
New theory of international trade (see Helpman, Krugman [16]), New economic geography (see Fujita, Krugman,
Venables [13], Fujita, Thisse [14], Combes, Mayer, Thisse [10]), as well as for other applications of industrialorganization methods (see the reviews of these models usage in [4] and in [7]).
General features or assumptions that characterize an industry in monopolistic-competition approach are as
follows: (1) increasing returns to scale in a firm, usually as a result of combining the fixed costs to start business
and the constant marginal costs, same for all firms in the industry; (2) each firm produces only one “variety”
or brand of the industry’s “commodity” or service (say, produces one brand of automobiles), and behaves as a
monopolistic price-maker, but takes into account a specific demand for its brand, infuenced by the competition of
other brands; (3) the demand curve is a result of maximization of a specific utility function reflecting “preference
for many varieties” excibited by the representative consumer; (4) the number of firms is big enough for a firm to
behave non-strategically, ignoring its own infuence on the industry/economy as a whole, i.e., ignoring the influence
on other prices within and outside the industry; (5) free entry into the industry drives all profits to zero (in spite
of increasing returns within each firm).
∗
We gratefully acknowledge the generous support of EERC and very valuable advices of its experts, first of all Shlomo Weber,
Richard Ericson, Olexandr Shepotilo, Olexandr Skiba, David Tarr.
1
Quite standard is to model the representative consumer’s preferences by a superposition of a high-level utility
function U (.) operating on two or several “industries” (aggregated goods) and a low-level function u(.) operating on
´
varieties within a diversified industry, i.e., on a differentiated good, the superposition taking the form U ( u(.), ...).
For instance, the high-level function can reflect preferences for such aggregates as “food”, “housing”, “transportation”, and determines the budget share allocated for each aggregate (each industry). In contrast, the low-level
utility function reflects sort of substitution/complimentarity in consumption amongst varieties within such industry. Typically, in hundreds of papers, the high-level function is a Cobb-Douglas one, while the low-level one is
a CES-function, that means a power function over the sum of power functions. The reason is technical: CESfunction allows for closed-form demand and for direct derivation of all the needed comparative-statics statements,
i.e, finding the direction of shifts in quantities and utilities when the market increases (because population grows,
or international trade opens, that pose quite meaningful questions), while the price remains constant.
However, despite its convenience, CES-function is a serious restriction resulting in some effects that look quite
artificial, notably the “neutral-competitive effect.” It means that in the short-term equilibrium (before the free
entry annihilates profits) the increasing number of firms in the industry does not influence prices, surprizingly for
economists. In general equilibrium this neutrality manifestates itself by the price indifferent to the market size
(population size), that we also call as neutral-competitive effect further. It also looks strange under increasing
returns. Economic intuition would rather prefer a model that predicts the effect called “pro-competitive”, i.e.,
decreasing average price in an industry in response to growing market size or/and number of firms. Unsatisfaction
with limitations of CES models and related neutrality is expressed by the leading economic-geography theorists
in [13] and [10]. They point out the realism of the pro-competitive effect, supported also by some empirical
papers. In particular, Campbell and Hopenhayn [8] show that a bigger market has smaller mark-up and biger total
consumption. On the other hand, anti-competitive effect also can be realistic in some specific circumstances as we
argue below. At least there are papers in oligopoly theory that show possibility of such effect in some settings:
[2] and especially [21] (discussed in more detail in Section 2 when explaining this effect), or even claim the effect
empirically discovered ([9, 15, 20]). We are unaware of any mentioning of such effect possibility in monopolisticcompetition situation, though we show in Section 4 that it does not contradict intuition, unlike neutrality. Thus,
we focus on the open question of pro- and anti-competitive effects in general monopolistic-competition setting,
previously weakly cleared because of dominating CES approach, and this question worth studying because theory
should meet economic intuition.
We should say that there are some rare papers fighting this gap between theory and intuition through stepping
aside from CES-functions. Even in the pioneering Dixit-Stiglitz paper there is one property derived in general form,
without specific functions: the equilibrium is Pareto-inefficient. Another seminal paper of the kind is Krugman
[17], discussed in detail after mentioning few others.
More recently, the approach taken in [19], uses quasi-linear function U and the quadratic valuation of commodity,
i.e, linear demand. This really results in pro-competitive effect, but for the price of absent income-effect (like all
quasi-linear functions), that also looks artificial in general equilibrium. Behrens and Murata in their recent paper [6]
invented one more functional class (in addition to CES and quasi-linear functions) that allows for deriving the closedform solutions of equilibria: exponential utilities, somewhat similar to CES, also resulting in the pro-competitive
effect.1 Another study without CES is Anderson [3] with a quasi-linar function U , exhibiting preference for varieties
itself, without lower-level function u. Somewhat similar is Benassy [5] who introduced some definition of the
functional property of the upper function U named “preference for varieties” and derived the welfare consequences
1
The propeties of the short-term equilibria studied in the latter two papers with specific functions are similar: the price and the
mark-up are decreasing with the number of firms, and consumption of each variety decreases; gross consumption of all varieties increses
as they become more numerous. The latter paper also makes a step towards deriving the equilibria effects without assuming the specific
functional form, but ends with some specific fuction.
2
from increasing the number of varieties (firms) from this assumption. Besides, Das [11] study linear utility u and
homothetic U .
Up to our knowledge, there are no other attempts to derive the equilibria effects without assuming the specific
functional forms. The need for such study stems from the question: how much the important predictions of all
monopolistic-competition models and related theories are robust against varying the specific assumptions? Aren’t
they artefacts like the neutral-competitive effect? Which effect is born by which assumption?
To fill this lacuna, the present investigation continues the seminal Krugman’s study [17] of a one-sector
monopolistic-competition Dixit-Stiglitz model with arbitrary low-level function u(.) and continuum of varieties
(upper level is inessential here). Krugman assumes decreasing elasticity of demand for individual variety, i.e., the
00
Dp characterising the utility function u.
increasing Arrow-Pratt’s “relative risk aversion” ru (x) := − xu
u0 = −1/ε
This assumption is shown sufficient for pro-competitive effect. Krugman’s graphical proof is very intuitive, but it
needs adding mathematical rigor, because some conditions on u, ensuring profit concavity and correct derivation
of the equilibrium equations are missed, the derivation itself being only schematic, ignoring possible multiplicity,
discontinuities and inexistence of equilibria (see discussion after Theorem 2). Besides, from Krugman’s graphical
reasoning it was not clear how to extend his method of building comparative statics to many sectors or factors
and to other extensions of the model, demanded by theorists. Maybe, these limitations explain why this seminal
fruitfull approach was not extended and generalized (as we do). Surprizingly, the named result on pro-competitive
effect is rarely mentioned, being unmentioned even in the Helpman-Krugman’s book [16]! In a survay dedicated to
30-th anniversary of the DS model Neary (2007) expresses oppinion that “... In any case, the [general Krugman’s]
specification ... has not proved tractable, and from Dixit and Norman (1980) and Krugman (1980) onwards, most
writers have used the CES specification in (2), with its unsatisfactory implications that firm size is fixed by tastes
and technology, and all adjustments in industry size (required for example by changes in trade policy) come about
through changes in the number of firms.”
In contrast, this paper makes the Krugman’s approach more tractable and extends it to quite general multisector Dixit-Stiglitz models, restricting only the number of production factors (one). The technical trick for such
generalization is inroducing the expenditure-for-varieties function E that relplaces the upper-level utility U as a
basic primitive, also reflecting preferences among the aggregated goods. We rigorously derive conditions on both
functions E(.), u(.), necessary and sufficient for pro-competitive, neutral- and anti-competitive effects when the
market increases, and show paradoxes to be explained now.
At the first stage, in Section 3, Propositions 1–4 state conditions for validity of two crucial equilibrium equations,
and explain a method of finding equilibria from these roots, the method being valuable by itself. Remarkably, the
equilibrium price and quantity of each variety can be found solely from one exogeneous parameter, “relative size
of the market,” and from relative-risk-aversion function (concavity measure) ru (.) of low-level utility u. Further,
price and quantity allow to find equilibrium number of firms/varieties from the second equation, reflecting the
upper-level utility function U and other sector(s) of economy. It turns out, that such convenient and economically
important independence between diversified/non-diversified sectors (known for CES models), stems only from the
two-level construction of utility, not being an artefact of CES assumption like the neutral-competitive feature (as
explained in Section 4).
In Section 4, Theorem 1 states comprehensive local comparative-statics: changes of price, quantity and varieties’
e The latter is number of population divided by fixed cost and
number w.r.t. the relative size of the market L.
multiplied by the variable cost of production. Respectively, impact of any combination of these parameters onto
the equilibria can be described through the same theorem. Specifically, markets are classified into pro-, neutral-,
and anti-competitive ones according to increasing, neutral, or decreasing Arrow-Pratt’s measure ru (.) of concavity
of the utility function u(.) , respectively (see Table 1). In the first class of markets price decreases but a firm’s
3
production increases in market size, while in the anti-competitive class the impact is the opposite, neutrality being
revealed only for CES functions. An individual variety’s consumption by a household decreases in all cases, as well
as the varieties’ number, the latter requiring additional reasonable condition on upper-level utility U expressing
some sort of complimentarity between the diversified and non-diversified goods.
Most interesting in our classification of markets is the anti-competitive effect named “price paradox.” It is
explained in special motivating Section 2 through related examples and finaly discussed after the theorems. “Price
paradox” means that equilibrium price of each variety grows with the relative size of the market (say, with population
size) though the number of competitors increases! Example 2 shows that such strange-behaving equilibria really
√
exist, under neo-classical utility function u(x) = x + x, which looks quite natural, and for a broad class of
functions with a linear component. How this strange effect can be economically understood? As shown on Fig.3,
the increasing number of competitors pushes down the demand for i-th variety, but at the same time it changes
the slope of the demand curve, that gives room for increasing monopolistic price.
Section 5 is devoted to changes in welfare w.r.t. the market size. To this end, Krugman [17] makes “evident”
conjecture that in pro-competitive market (ru0 (x) > 0) always ensures improving welfare of a consumer w.r.t. the
size of the market. However, we prove it to be true only under additional normalization condition u(0) = 0 not
mentioned by Krugman. When this condition is violated, we have have found a counterexample due to “envy
paradox,” explained in Subsection 5.2, that may seem somewhat artificial (new varieties can bring dissatisfaction
of consumers not buying these varieties when u(0) < 0). In contrast, another, “price-and-utility paradox” of this
kind (see Subsection 5.1) occurs in many anti-competitive markets and looks more natural. It means that each
consumer may suffer from the population (and size of the market) growth:
∂U ∗
∂L
< 0. Again, the related utility
function looks natural and the subsection explains why the effect itself can be economically understandible.
Finally, Section 6 is devoted to possibly multiple and asymmetric equilibria. Proposition 5 states conditions
for equilibria uniqueness/multiplicuty. It turns out that multiple equilibria can occur if and only if the marginalrevenue function xu00 (x) + u0 (x) is non-monotone and the kink in it occurs substantially above the marginal cost.
In this case there is an interval of asymmetric equilibria, and it is always accompanied by two symmetric equilibria,
which are the ends of the interval. Related Example 5 demonstrates existence of such effect and the structure of
equilibria set in this case. On this basis, Theorem 2 states the global comparative statics of price, quantity and
number of firms mappings for the case of multiple symmetric/asymmetric equilibria. Multiplicity can occur only
e under assumptions of Proposition 5, otherwise, and at other points, equilibria are
at one value of the market size L
symetric and unique and generally behave like in Theorem 1. At the unique point of discontinuity of each mapping
there occur catastrophic changes in prices, quantities and number of firms in response to smooth changes of the
relative size of the market, demonstrated by the extended Example 5. Such abrupt changes, not contemplated
previously in monopolistic competition, may be interpreted as possiblity of revolutionary events in the market
under slight shifts in population or technology.
Conclusion summarises and Appendix presents all proofs and details.
1
Reduced model of a Dixit-Stiglitz economy
1.1
Basic model and notations
To compare the following general analysis with the traditional one, we start now with the standard Dixit-Stiglitz
model of one-sector one-production-factor economy, as in ([16]), but with non-specific utility function, like in Krugman (1979).
There is a continuum, i.e., an interval [0, N ] of firms (N > 0). They produce as much as [0, N ] varieties or
4
brands of a diversified commodity, say, many kinds of clothes. Each firm, identified by index s ∈ [0, N ], produces
only one brand (see Intro for basic assumptions). There are L identical consumers, and a consumer’s index is
dropped. Each of them chooses an infinite-dimensional consumption vector x = xs∈[0,N ] , which is an element of the
space Ξ[0, N ] of all non-negative Lebesgue-integrable functions on interval [0, N ]. It means solving the following
utility-maximization program:
ˆ
ˆ
N
u(xs )ds →
0
max
x(.)∈Ξ[0,N ]
N
ps xs ds ≤ .
w.r.t.
0
Here u : R+ → R+ denotes a neoclassical utility function. The infinite-dimensional price vector is p ∈ Ξ[0, N ], p :
[0, N ] → R+ , its component ps ≡ p(s) is the price for s-th variety, and xs ≡ x(s) is the individual demand for s
-th variety. We further use symbol s as an index
s
or argument (s) interchangable.2 Symbol > 0 denotes the
expenditure for all varieties. In the basic model equals the consumer’s income, but generaly it can vary. All
consumers are identical and have 1 unit of labor (or any other resource) inelastically supplied to the market in
exchange for wages w > 0. The consumers equally own the firms, and the expenditure for all varieties satisfies
´N
≤ w + L1 0 πs ds = w, because anyway zero profit π = 0 results at long-run equilibrium.
We denote by x̌s (p, , N ) = x̌s : Ξ[0, N ] × R2+ → R+ the s-th component of a solution to the above problem,
and by x̌ = x̌s∈[0,N ] whole consumer’s solution, i.e., demand of any consumer for s-th variety. This x̌s depends
upon infinite-dimensional price-vector p, budget and the scope N of the varieties, called also “number of varieties”.
Hoping for no confusion, we sometimes do not display further some arguments when they are fixed, but have in
mind the same demand function: x̌s (p, , N ) ≡ x̌s (p, N ) ≡ x̌s (p).
On the supply side, each (s-th) of the identical producers knows her gross demand function x̌Σ
s (.) =
´L
0
x̌s ds =
Lx̌s , and solves her monopolistic profit-maximization program:
ps Lx̌s (ps , p-s , , N ) − cwLx̌s (ps , p-s , , N ) − wF → max .
ps ∈R+
Here, as usual, p-s denotes all other prices besides ps . The variable cost c > 0 and the fixed cost of starting business
F > 0 are measured in labor under normalized wages w = 1, so totally the producer spends cLx̌s + F units of labor
(some other joint normalizations of w, c, F are also suitable, but we stick to w = 1 ).
Now, to generalize the Krugman’s approach and findings, we are going to supplement this basic model with
additional sectors of economy, described implicitely through some expenditure function E(.) that replaces constant
in the consumer’s problem. The derivation of this function from other primitives U, c, F and the extensive form
of the general model are in Appendix. Instead of this derivation, in the next subsection we directly introduce the
reduced form of the multisector model. It seemingly describes only one diversified sector and looks like partialequilibrium model, but really it models general equilibrium.
1.2
Reduced general model: consumer’s expenditure function
In this subsection the above classical basic model of a diversified sector is combined with other sectors. It is done
through making the expenditure on varieties dependent upon current variables in the form = E(p, N ). This
function E : Ξ[0, N ] × R+ → R+ becomes a new primitive of our model, supplementing initial parameters u, c, F, L
and implicitely describing the economy outside our sector of interest.
2
Representing the argument s of demand/price functions as an index: xs ≡ x(s), ps ≡ p(s) economizes many useless parentheses.
Another motivation for´ such notationPis possible direct application of the below analysis to discrete-varieties models: just understand
N
the integral as a sum: 0 u(xs )ds = N
0 u(xs ), and apply all our reasoning and results.
5
To comprehend the nature of the expenditure function E, see the extensive form of the general model in
Appendix. Each state of the economy is described by a bundle (p1 , N1 , p2 , N2 , ...) of prices pi and scopes Ni
for all n sectors of economy. Observing this information, a consumer optimizes her consumption, and as a byproduct finds the bundle of auxiliary variables (Ẽ1 , Ẽ2 , ..., Ẽn ) denoting expenditures for each sector, dependent
upon (p1 , N1 , p2 , N2 , ...). Our approach to comparative statics of equilibria is based on preliminary reducing the
system of related equilibrium equations to only two equations with two variables p1 , N1 . In doing so, we get rid off
p2 , N2 , ..., pn , Nn expressing them through p1 , N1 . Thereby, all arguments of the expenditure function Ẽ1 of the first
sector become expressed through vector (p1 , N1 ), which takes into account equilibrium response of other sectors
to this (first) sector. Thus we get the expenditure function E1 (p1 , N1 ) = Ẽ1 (p1 , N1 , p2 (p1 , N1 ), ...pn (p1 , N1 )) used
further throughout, dropping the index 1 of the sector that we study.
After dropping this index, the consumer’s sub-program related to this sector becomes:3
ˆ
ˆ
N
u(xs )ds → max
0
N
ps xs ds ≤ = E(p, N ).
w.r.t.
x(.),
0
This formulation differs from the basic model only in the upper bound E, which is not a constant from the
comparative statics viewpoint; it depends now upon prices and varieties’ number. In contrast, for deriving FOC
this = E(p, N ) can be percepted as a fixed magnitude. Therefore, like in the basic model, the resulting individual
demand function x̌s for s-th variety includes constant = E(p, N ) as an exogeneous argument:
x̌s = x̌s (p, , N ) = x̌s (ps , p-s , , N ).
From now on we look on the expenditure function E(.) in sector #1 as on a primitive. For our goal (expansion of
the Krugman’s method to multi-sector models) it seems more practical to study comparative statics of the reduced
model, rather than extensive one. It turnes out that it is sufficient to impose reasonable restrictions on this function
E for describing sector #1 within comparative statics of the whole economy, without complicated direct derivation
of demand. In this sense, the reduced model is a lucky finding.
It overcomes the limitations of the closed-form-demand approach, typical so far. To get a closed form, the
´
P 1
upper level utility function must be a Cobb-Douglas one: U (X1 , ..., Xn ) = m
uj (zj ) and the
j αj Log(Xj ), Xj =
β
lower-level function must be CES: uj (zj ) = zj j .4 In this simple case the budget share allocated to each sector is
just a constant Ej (pj , Nj ) ≡ j = αj , that looks restrictive.
In contrast, we allow for more general upper- and lower-level utility functions, becoming now suitable for
analysing many questions of international trade and economic geography without any explicit demand function.
Generally, studying the equilibrium equations does not require this explicit function.
3
One can also think of this consumer’s model in terms of two-stage optimization. First, a consumer chooses her general budget
strategy: which budget share allocate to which sector (like food, clothes, housing, etc.) in each possible state of economy. Second,
having fixed the expenditure function Ej (.) for each (j-th) sector, the consumer solves the lower-level utility-maximization sub-program
for this sector, similarly to the basic model.
4
Only combination of these two restrictive assumptions is sufficient for constant , otherwise the budget share can vary. This fact is
not noticed in Helpman and Krugman ([16], p. 190) where the constant-share assumption is just added to seemingly-general functions
analyzed.
6
1.3
Reduced general model: producer and equilibrium
The producer’s profit-maximization program remains as in the basic model. Knowing the individual demand
function x̌s (.) for her variety and the competitors’ price vector p-s , the s-th producer chooses her price ps ∈ R+ as
ps Lx̌s (ps , p-s , , N ) − cwLx̌s (ps , p-s , , N ) − wF → max .
ps ∈R+
(1)
Here L > 0 denotes the number of identical consumers in the economy, wage is normalized as w = 1. Fixed cost
F > 0 and marginal cost c > 0 are measured in labor, and total cost is cLx̌s + F . The producer percepts the budget
for this sector, other prices and scopes as constants, because her own influence on the sector is infinitesimally
small. This idea motivates the following notion for (short-term) sub-equilibrium.
Definition 1. A symmetric sub-equilibrium of a diversified sector under given N, , L is a Nash equilibrium
(xis , ps )i≤L,s≤N among L consumers and the producers’ population [0, N ], with symmetric quantities and prices:
x̌is = x∗ (N, , L) ∈ R+ ∀i, s, p̌s = p∗ (N, , L) ∈ R+ ∀s.5
Thus we have defined a symmetric sub-equilibrium functions p∗ (N, , L) ∈ R+ , x∗ (N, , L) ∈ R+ . Now, to
define the long-term symmetric equilibrium, we take into account that our expenditures function E(p, N ) should
determine the equilibrium expenditure magnitude ¯, and free entry should determine the equilibrium number of
competitors N̄ through the zero-profit condition. Therefore, the equilibrium vector (p̄, x̄, N̄ , ¯) must satisfy four
equations:
p∗ (N̄ , ¯)Lx∗ (N̄ , ¯) = cLx∗ (N̄ , ¯) + F ,
(2)
¯ = E(p̄, N̄ ), p̄ = p∗ (N̄ , ¯), x̄ = x∗ (N̄ , ¯).
(3)
The zero-profit condition (2) can be replaced at equilibrium with the labor balance in the economy:6
N (cLx∗ (N ) + F ) = LE(p∗ (N ), N ).
Besides, both these conditions and the budget constraint entail the Walras identity:
w (cLx∗ (N ) + F ) = p∗ (N )Lx∗ (N ) = L
w
.
N
These considerations motivate
Definition 3. The quadruple (p̄, x̄, N̄ , ¯) ∈ R4+ solving under given c, L, F, u(.), E(.) the consumers’/producers’
problems and satisfying the balance conditions (2)-(3) is called as (symmetric) long-term equilibrium of the reduced
model of economy.7
To guarantee nice tractable demand properties and existence of the producer’s solution, we maintain throughout
classical conditions on utility and a reasonable additional restrictions on expenditure and on price elasticity εP x ,
formulated as
Assumption 1. Lower-level utility function u : R+ → R+ is neoclassical one (strictly concave, strictly
increasing), three times differentiable, normalized as u(0) = 0, and −1/εP x (x) = −xu00 (x)/u0 (x) < 1 at x = 0.
The expenditure for diversities E = E(p, N ) is two times differentiable.8
5
Following the tradition, we focus here on symmetric equilibria only, though asymmetry is possible and we touch it in another paper.
Here E plays the role of labor share for our Dixit-Stiglitz sector within the economy, which, as shown later on, coincides at the
equilibrium with a consumer’s budget share for this commodity.
7
When this kind of a model is applied to some economy consisting of two or more trading countries, our long-term equilibrium
becomes “integrated equilibrium”. This term is traditional for international trade literature to mark free flow of commodities/factors
among the countries. The same model describes autarky also.
8
Condition −xu00 (x)/u0 (x) < 1 on the demand, in application to a standard monopoly, would mean just positivity of the marginal
6
7
The above reduced model of the whole economy is our main object of study below, though for illustrations
we use also the one-sector model, that means E ≡ 1. We present some of such examples now, before the general
theorems, for better comprehension of the model and the results.
2
Examples of pro- and anti-competitive effects
To motivate subsequent derivation of necessary and sufficient conditions for positive, zero and negative price effects,
we show in this section that all these do exist in the model.
Definition. Anti-competitive entrance effect or “price paradox” is the following property of a symmetric equilibrium: when the size L of the market increases, either both number of firms (varieties) and the price increase, or
both decrease, i.e.,
∂ N̄ ∂ p̄
∂L ∂L
in the opposite direction:
> 0. In contrast, pro-competitive entrance effect means that price and varieties change
∂ N̄ ∂ p̄
∂L ∂L
< 0, while neutral-competitive effect means no price changes:
∂ p̄
∂L
= 0.9
So far, the literature focused mostly on CES utility functions, and related neutral-competitive effect. Neutrality
means that price of each variety is not influenced by the size of the market and the number of competitors, which
seems rather artificial. The pro-competitive effect in general-form one-sector model is explored only in Krugman
(1979). As to anti-competitive effect, it was never contemplated for a monopolistic-competition model, to the best
of our knowledge.
Following Behrens-Murata [6], Krugman and others unsatisfied with neutrality, we step aside from CES assumption, and demonstrate in this section: (1) one more, in addition to Behrens-Murata’s one, quite natural utility
function yielding pro-competitive effect; (2) a natural utility function yielding anti-competitive effect. Besides,
Example 2 illustrates our method of indirect derivation of equilibria.
2.1
Pro- and anti-competitive effects, “price paradox”
Now we show the utility functions yielding all three effects defined, i.e., existence of neutral, positive and negative
price reactions to population growth.
´
As to neutral competitive-effect, power functions like u(z) = z α , 1 > α > 0 (called CES after summation
u(zs )ds), are well known to bring it. Besides, we prove in our Theorem below that it is the only function class
yielding such effect everywhere.
Example 1. Pro-competitive effect. As to pro-competitive effect, recetly Behrens and Murata has shown an
example of such utility function: u(z) = 1-e−αz , α > 0, and, in addition, this function allows for closed-form
derivations. We show now another example with similar pro-competitive effect, also allowing for direct solution.
Our new convenient function with pro-competive effect is u(z) = ln(z + 1). One can analyse also more general
argument z + a, and use the below formulae in various models with a diversified sector.
revenue at x = 0, which is a weak restriction.
9
Thereby, anti-competitiveness in terms
of subequilibrium means that the price increases, surprizingly, w.r.t. the number N of the
∗
varieties/competitors in the industry: ∂p
> 0. This effect or “price paradox” is not mentioned so far in the Dixit-Stiglitz model, but it
∂N
is reported in several other oligopolistic settings, briefly reviewed in Rosenthal [21]. In these papers mainly increasing competition leads
to increasing price either because of increasing returns, or through reputation-and-search cost, or through other specific cost structures.
In contrast, Rosenthal’s setting, like ours, derives “price paradox” only from the demand structure. Namely, each of his oligopolists
face two kinds of markets: “domestic” one, where she has a monopoly power, and a “third-country” market, where all compete in prices
in Betranian way, so a unique price emerges in the third country. Besides, somehow each firm is forced to sell for the same price at
home and outside (that can be realistic when “domestic market” means loyal customers of a brand, while “third-country” means new
or unloyal customers). When a new competitor enters this market system, she takes away a 1/(n + 1)-th share of the “ third-country.”
Thereby the residual demand curve of each incumbent changes in the direction closer to monopoly, because more steep slope arises,
tempting Nash-type incumbents to increase prices. Somewhat similar is our “price paradox,” connected with more steep demand slope,
as explained in this Section.
8
To derive an equilibrium, see that the consumers’ maximization entails
xs + 1
pk
pk
=
⇒ xs = (xk + 1) − 1.
xk + 1
ps
ps
Substitute this into the budget constraint to get the demand function:10
ˆ
ˆ
N
N
ps xs ds = 1 ⇔
ps
0
0
´N
ps ds
pk
1
(xk + 1) − 1 ds = 1 ⇒ x̌k (p, N ) =
+ 0
− 1.
ps
pk N
pk N
Then the producer’s optimization problem becomes
L
+L
pk Lx̌k (p, N ) − cLx̌k (p, N ) − F =
N
´N
0
´N
ps ds
ps ds
Lc
− Lpk −
− cL 0
+ cL − F → max .
pk
N
pk N
pk N
The related FOC give us the producer’s reaction to other prices and to the number of competitors:11
´N
s
ps ds
c
−1 + 2 + c 0 2
= 0 ⇒ pk =
pk N
pk N
c
+c
N
´N
0
ps ds
.
N
The equilibrum’s symmetry is obvious, and sub-equilibrium formula is derived as
r
P =
c+
c
+ cP ⇔ N P 2 − N cP − c = 0 ⇒ p∗ (N ) =
N
p
c2 + 4c/N
,
2
because only the positive root applies.
The pro-competitive effect is obvious in such subequilibrium: the price has negative reaction to the number of
firms. Coming to integrated equilibrium through the labor balance, we get the number of firms:
N̄ (F + cLx∗ (N̄ )) = L ⇔ N̄ =
F+
L
√
LF c
.
The resulting comparative statics w.r.t. market size L is shown in Fig.1. Interestingly, function N̄ = N (L) is
concave here, unlike anti-competitive case shown later on.
Now we come to most interesting anti-competitive effect.
Example 2. Anti-competitive effect (price paradox). It is sufficient to take a utility u(x) =
√
x + x, or any
other CES function with a linear addition. Such neo-classic functions look quite natural. We were surprized to see
√
that this innocent modification of so popular CES utility function x brings anti-competitive effect everywhere!
Indeed, the consumer’s FOC yields a relation
√1
2 xk
√1
2 xs
+1
+1
=
pk
1
⇒ xs = 2 .
ps
ps √1
+
2
−
2
pk
xk
10
Remark that the demand function here turns out rather similar to one derived in Ottaviano, Tabuchi and Thisse [19] for quasi-linear
utility.
´
11
As to the producer’s second-order condition, − p2cL
3 N − 2c
k
concave.
N
0
ps ds
p3
N
k
9
≤ 0, it is definitely true for the function taken, so our profit is
Figure 1: Comparative statics with pro-competitive effect: p̄(L), x̄(L), N̄ (L),U (L) for utility u(z) = ln(z + 1).
Substituting this xs into the budget we get
ˆ
ˆ
N
N
ps xs ds = 1 ⇔
0
0
ps
ps
pk
√1
xk
2 ds = 1.
+2 −2
This expression cannot be resolved directly for xk , but we can link xk with p under the assumption of symmetric
other-firms prices equal all to some P > 0, as follows
ˆ
N
0
√
1 + 2 xk
2 ds = 1 ⇔ pk (xk , P, N ) = P √ √
xk ( P N + 2)
+2 −2
ps
ps
pk
√1
xk
Now we can solve the producer’s problem by maximizing her (obviously concave) profit function w.r.t. xk :
√
xk + 2xk
P2
L P √
− cxk − F → max ⇒ xk = √
2 .
xk
( P N + 2)
4 c( P N + 2) − 2P
Then k-th optimal response to other prices is
√
2 c( P N + 2) − P
√
pk =
.
PN + 2
Any equilibrium is obviously symmetric here, so pk = P and we can derive the sub-equilibrium condition as
√
√
2 c( P N + 2) − P
√
√
⇔ P P N = 2 c( P N + 2) − 2P .
P =
PN + 2
10
Figure 2: Comparative statics of anti-competitive paradox: p̄(L), x̄(L), N̄ (L), U (L) for utility u(z) =
√
z+z
Now from the labor-market balance we express P N and derive the equilibrium condition
r
N (F + cLx) = L ⇔ N F + cLN x = L ⇔ P
In principle, this cubic equation w.r.t
√
r
P L − cL
P L − cL
= 2c(
+ 2) − 4P.
F
F
P can be resolved directly. But we prefer to present now the graphs of
comparative statics of equilibrium P (L) and other magnitudes calculated numerically under assumption F = c = 1,
as shown in Fig.2.
Interestingly, function N (L) is convex here, unlike pro-competitive case. Utility increases in both examples.
But the main observation now is “price paradox”: the price for each variety grows w.r.t. the size of the market L
together with the number of competitors N .
Our numerical study of functions like u(z) = z a + bz shows that there is a large region of parameters (a, b)
bringing such “price paradox,” which appears therefore generic. Thus, unlike degenerate neutral-competitive effect,
our anti-competitive effect should not be logically excluded from analysis. But, is it realistic economically? Its
natural explaination in terms of demand, supply and elasticities is postponed to Section 4.
3
Equilibrium equations in terms of demand elasticity
We have just shown through examples that the effects studied do exist. In Section 4 below we are going to classify
all utility functions bringing these effects. Now, to prepare tools for such general comparative statics, we should
characterize demand/supply functions and formulate general equilibrium equations. In our reasoning, we exploit
00
(z)
and develop the seminal Krugman’s technique based on Arrow-Pratt measure ru (z) = − zuu0 (z)
= −1/εP x (z) of
concavity (inverse to demand elasticity), becoming below the main element of all expressions and conditions.
11
3.1
Demand and supply characterization
Now we formulate main demand and supply properties (all proofs and other properties are in Appendix). From
the consumer’s FOC one can easily get
Proposition 1 The k-th demand elasticities w.r.t. each price are characterized as:
Xp
Ekk
=
pk ∂xk
1
ps ∂xk
z u00 (z)
= 0, where ru (z) = − 0
=−
,
.
xk ∂pk
ru (xk ) xk ∂ps
u (z)
(4)
Xp
One can observe here, like in quasi-linear IO settings, that the demand elasticity Ekk
w.r.t. its own price is
inverse to the measure of concavity (but for the sign) and another price is irrelevant. The difference from IO is
that any measurable interval of other prices does influence k-th demand (see Appendix). We use these elasticities
to analyse the k-th monopolist profit-maximization program:
πk = (pk − cw)Lx̌k (pk , p−k , Rm (pk , p−k , N ), N ) − wF → max .
pk
Now we show conditions for this function concavity, the producer’s solution uniqueness and symmetry.
Proposition 2 (i) Condition 2 > ru0 (xk ) is necessary and sufficient for strict concavity of the marginal utility at
point xk , and for strict concavity of the profit function at points satisfying FOC. (ii) If this upper bound 2 > ru0 (xk )
holds everywhere on R+ , then profit is generally strictly concave and has a unique maximum. In this case any
sub-equilibrium is symmetric.12
It is useful to formulate the producer’s FOC in terms of markup Mk and concavity measure ru .
Proposition 3 Necessary first-order condition (FOC) for profit maximization is:
Mk =
(pk − cw)
1
= − ∂ x̌ p = ru (x̌k ) < 1.
k
k
pk
∂p x̌
k
k
We observe that markup (Lerner’s index) is equal to inverse-demand elasticity and should be less than 1. So,
any equilibria can occur only at points with inelastic demand, i.e., with moderate concavity, like in traditional IO
with quasi-linear utilities.
3.2
Methods of finding equilibria and “independence” of sectors
Recall that definition of the symmetric long-term equilibrium supplements the above profit-maximization expressions by the labor balance (or the zero-profit condition). Then the symmetric-equilibrium price p, markup M and
number N of firms can be found from the equations:
M :=
(F + cL
p−c
E(p, N )
= ru (
)
p
pN
E(p, N )
)N = E(p, N )L.
pN
12
The condition 2 > ru0 (xk ) here, unlike quasi-concave economies, is not necessary for profit concavity and equilibrium symmetry.
For instance, the Behrens-Murata’s function mentioned before Example 1 satisfies this condition not everywhere, but shows concavity
and symmetry everywhere. It is because the demand function of a Dixit-Stiglitz industry differs in income-effect from demand of a
similar-utility quasi-concave sector. Only their elasticities are similar.
12
It is convenient to reformulate these equations in terms of crucial constant L̃ called in this paper as relative size of
the market:
L̃ :=
Lc
.
F
(5)
This L̃ includes the number of consumers fixed and variable cost. Therefore our comparative statics in terms of L̃
captures altogether all possible shocks of population and technology, influencing M and N .
The necessary conditions of equilibria in terms of L̃ and M (L̃), N (L̃) follow.
Proposition 4 Each symmetric equilibrium of the general model satisfies the following equilibrium equations:13
M = ru (
1−M
1−M
F
·
) = ru (
)
L·c
M
L̃ · M
N · c = E(
(6)
c
, N ) · M · L̃,
1−M
(7)
First observation here is that the equations are quite simple, unexpectedly for a complicated model. Somewhat
similar equations for one-sector model are in Krugman (1979), but our version turns out, by this proposition,
applicable to multi-sector models also without any complications.
Second observation is independence of the diversivied sector under consideration from other sectors (like in
quasi-linear IO). Indeed, note that we can find markup M solely from the first equation. The markup yields
equilibrium price p =
c
1−M ,
consumption x =
1−M
L̃M
and production xL =
F
c
·
1−M
M
of each variety, expressed in
only one exogeneous constant L̃. The markup influences the number of firms N through the second equation but
is not inflenced by it. Thereby, price and quantity of our varieties do not depend upon the expenditure function
E(·) describing other sectors of the economy, and preferences for other commodities!
Looking very strange, such independence can be explained as follows. Each producer in the model is sufficiently
small and ignores second-order impacts of her price, she takes into consideration only the demand elesticity. The
demand elasticity, in turn, depends only upon the degree of substitution among varieties of our diversified sector
expressed in function ru . Other properties of preferences are completely separated from this elasticity by the
assumption of two-level utility U (u(·), ...).
Initially we supposed such indepedence in various models to be the artefact of CES utilities, but now it turns out
to be a general feature of two-level representation of consumer’s “utility-tree” (similar effect in usual quasi-linear IO
models has somewhat different nature: income-neutrality). However, under heterogeneous consumers’ population
with varying endowments, independence disappears, as we have checked. So, independence can be an artefact of
modelling or a feature of reality, depending upon realism of two-level utility and homogeneity of population.
Luckily for computation and analysis, this independence allows to study price/quantity reaction of varieties to
the market size regardless of other sectors! In particular, our Examples 1,2 of increasing/decreasing price under
growing market can be directly generalized from one-sector economy to any multi-sector economies (but not to
multi-factor models, as we have checked).
Thirdly, the equilibrium equations suggest three methods of studying the equilibria. (1) It can be through
the closed-form solutions, and we have found that for many utility functions, not only for CES, our equations
allow for such direct way without explicitely deriving the demand (which can be harder than solving directly these
two equations). (2) For comparative statics, we prefer instead to study all utility functions indirectly through the
13
Similar two equations with function r(.), though posed in terms of quantities instead of prices/markup, are formulated in Krugman,
without complete rigorous derivation.
13
implicit function theorem or through global method as in Theorem 2 below. (3) For numerical simulations, the
equations can be solved through stationary-points and other methods, one after another, finding first M , then N .
Again, deriving the demand and the supply is not necessary.
However, we must remember the conditions on demand and supply used for deriving our equations. In particular,
there can be several roots to the equations, only some of them being the equilibria (see Proposition ??). There
can be multiple equilibria, but some roots can be irrelevant. Of course, under condition ru0 (x) < 2 ∀x which means
global profit concavity, any non-boundary root is a true equilibrium (bounds are 0 < M < 1, 0 < N < ∞). But
when global concavity is not guaranteed, only a root that satisfies condition ru0 (x∗ (M )) < 2 can be an equilibrium,
otherwise it is not a local maximum of profit. Besides, to ensure global maxima of profit, for each root p = c/(1−M ),
we must check, whether this price level ps = p is really a global maximum of each producer’s profit πs (ps , p̄), other
prices being fixed at p̄ = p. If not, it is not a symmetric equilibrium, but it can be an asymmetric one (more
detailed discussion of equilibria existence, symmetry, and multiplicity or uniqueness is postponed to Section 6 and
Proposition 5).
The next section derives the comprehensive comparative statics of unique or multiple equilibria, without any
closed-form solutions.
4
Comparative statics: classifying pro- and anti-competitive markets through
utilities
To generalize the Krugman’s findings and the above examples, we seek for necessary and sufficient conditions on
the utility function for pro- and anti-competitive effects. Studying the impact of the relative size of the market onto
prices and quantities, we have in mind that this growth can result from opening international trade, or population
growth, or income increase, or changes in technologies (costs). Luckily, all these important economic questions can
be answered altogether through our Theorem 1.
To get the theorem, we just apply the implicit function theorem to equilibrium equations (6). As we have said,
this approach revitalizes the seminal Krugman’s graphical study of somewhat similar two equations.14
4.1
Local and global impact of relative market size on prices, quantities and varieties
Local comparative statics of our monopolistic competition model, illustrated by the above examples, is summarized in the below comprehensive Theorem 1 and in Table 1. The theorem uses the following assumption on
expenditure function E and (implicitely) on the upper-level utility U .
Assumption 2. Expenditure function’s elasticities satisfy bounds 1 >
∂E(p,N )
∂N
·
N
E
and 1 >
∂E(p,N )
∂p
·
p
E
> 0.
The assumption is formulated in too general form here, but it is expressed in basic terms of the two-sector
model in Appendix (see Lemmas 5,6). Mainly, it expresses sufficient complimentarity in consumption between the
two types of goods.
We recall that ru (x) := −xu00 (x)/u0 (x) and formulate the main theorem and its corrolaries for those points
where equilibria exist (see Appendix).
Theorem 1
15 Under
our Assumption 1, at any symmetric equilibrium the increase in the relative size of the market
L̃ can have three impacts: (i) price decrease ( ∂∂pL̃ < 0) is equivalent to increasing concavity: ru0 (x) > 0; (ii) price
14
More specifically, Krugman’s equations were expressed in quantities instead of prices. Our version is more convenient for applying
the implicit function theorem and for generalizations. Maybe, inconvenience was the reason for ignoring Krugman’s fruitful approach
by theorists through decades.
0
15
Local conditions ru0 (x) < 2, ru (x) < 1, EN (.) < 1 used for applicability of the implicit-function theorem, are guaranteed at
equilibria points by Assumptions 1,2. The existence of derivatives ∂∂pL̃ , ∂N
follows from this theorem.
∂ L̃
14
increase ( ∂∂pL̃ > 0) is equivalent to decreasing concavity: ru0 (x) < 0; (iii) neutrality ( ∂∂pL̃ = 0) is equivalent to
ru0 (x) = 0 (that means CES or power function u).
Corollary T1.1. Under additional Assumption 2, the varieties’ number N increases w.r.t. relative size of
the market:
( ∂∂pL̃ ·
∂N
∂ L̃
> 0. Then pro-competitive effect ( ∂∂pL̃ ·
∂N
∂ L̃
< 0) holds under ru0 (x) > 0, anti-competitive effect
> 0) is equivalent to ru0 (x) < 0; (iii) neutral-competitive effect ( ∂∂pL̃ ·
∂N
∂ L̃
∂N
∂ L̃
= 0) is equivalent to ru0 (x) = 0.
Corollary T1.2. Each equilibrium’s local comparative statics and bounds on elasticities are as in Table 1, where
↑ means increasing, ↓ means decreasing, o means no change, and S means that any sign is possible (ambiguity).
When relative market size
L̃ =
Lc
F
increases, its
impact on variables is:
Equilibrium price p
Under increasing
Under
Under decreasing
concavity16
neutrality
ru0 (x)
ru0 (x) = 0
o : ∂∂pL̃ · L̃p = 0
o : ∂∂yL̃ · L̃y = 0
↓: ∂∂xL̃ · L̃x = −1
↑: ∂N
· L̃ > 0
∂ L̃ N
>0
∂p L̃
· <
∂ L̃ p
0 < ∂∂yL̃ · L̃y < 1
−1 < ∂∂xL̃ · L̃x < 0
∂N
· L̃ > 0
∂ L̃ N
↓: −M <
A firm’s production Lx
↑:
A variety’s consumption x
↓:
0
(CES)17
concavity
ru0 (x) < 0
∂p L̃
· S1
∂ L̃ p
−1 S ∂∂yL̃ · L̃y <
∂x L̃
· < −1
∂ L̃ x
∂N
· L̃ > 0
∂ L̃ N
↑: 0 <
↓:
↓:
0
Varieties’ number*18 N
↑:
↑:
Remark. Similar global comparative statics obviously follows from these local results when the equilibrium is
unique: under ru0 (x) > 0 or monotone marginal-revenue function u00 (x)x + u0 (x) (see Proposition 5).
The proof of this theorem, its corollaries and Remark goes through direct algebraic transformations, using the
implicit function theorem and the above propositions (see Appendix).
Observing these results one may object, that they only locally characterize each root of the equilibrium equations, but generally the roots can be multiple! So, it remains unclear, what happens with all equilibria within
the comparative statics, especially when some equilibria appear or disappear. Section 6 below clears these delicate
questions and global comparative statics under possible multiplicity of equilibria, not relying on local characteristics
and any derivatives.
4.2
Discussion of comparative statics results
1) Comparing our Theorems 1,2 with somewhat similar Krugman’s result, we should say that Krugman [17] derives
two equilibrium equations in terms of x̄ and N̄ and geometrically establishes two facts: (i) the pro-competitive
sufficient condition is ru0 (x) > 0; (ii) under ru0 (x) ≥ 0 utility increases in the size of the market L.
Statement (i) is similar to sufficiency of r0 > 0 for pro-competitive effect in our Theorem 1 and Corollary 1,
but for specific one-sector model. We have more than this and for general model. Besides, Krugman’s graphical
argumentation is not quite rigorous in some respects. Namely, he skips discussing existence of equilibria, possible multiplicity of equilibria, and two important conditions for rigorous comparative statics are missed in his
formulation: (a) condition 2 > ru0 (x) serving for concavity of profit function, for unique sub-equilibrium and for
symmetry (actually, Krugman’s equilibrium equations become invalid for some functions but it was not noticed);
(b) non-mentioned normalization condition u(0) = 0 is needed for utility-increase fact (ii), otherwise utility may
decrease as we show below.
Our more comprehensive Theorem 1 provides rigorous necessary and sufficient conditions for positive, negative
and neutral impacts of the relative market size on prices and quantities, for pro- and anti-competitive effects.
Besides, Table 1 shows elasticities, namely, in pro-competitive markets the impact on prices and quantities is
weaker, in per cent terms, than stimulating increase of the market, but for anti-competitive markets it can be
stronger.
Our technique works through equilibrium price p∗ (L) instead of quantity x∗ (L), more conveniently for using
the implicit-function theorem (instead of graphical proofs) and for generalized model needed in all “new” economic
15
xk
xk
xk
p0k
pk
b) 0 < ru0 − ru < 1
a) ru ≥ ru0
p0k
pk
pk
c) 1 < ru0 − ru
Figure 3: Various possible shifts of s-th demand curve under increasing number of varieties N .
theories (maybe, it is the limitation of one-sector model that explain why Krugman’s approach to comparative
statics did not become popular so far). In our view, the most important Krugman’s finding is that the only
essential property of the utility is concavity r(.). This notion enables to get simple equilibrium formulae and
comparative statics formulae from frightful huge derivatives and equations. Such simplification probably seemed
impossible for many researches, who seeked instead the (unnecessary now) closed-form demand formulae.
2) Discussing the assumptions of the theorems, we should explain now three conditions on elasticity of expenditure E(p, N ): 1 >
∂E
∂N
·
N
E,
1>
∂E
∂p
·
p
E
> 0. In Appendix we show in Lemmas 3-6 that this condition is rather
reasonable, at least for two-sector general model. There the condition is expressed in terms of cross-derivatives of
the upper-level utility function U and function u. In particular, when the upper-level utility is either quasi-lenear,
or additively separable, or homogeneous of degree 1, then these three conditions on E mean some kind of complimentarity (in utility terms) between the diversified and the “traditional” sectors. Maybe, in real economy it is not
always the case. When the three conditions are violated, the number of firms may decrease under growing market,
as we explain below.
3) Intuitive explanation of pro- and anti-competitive effects. Examples 1,2 show and Table 1 classifies markets
with pro- and anti-competitive effects, based on increasing or decreasing concavity of utility u. Why concavity
is connected with these effects? Why anti-competitiveness may happen? It can be explained in terms of usual
demand-supply Marshalian diagram like Fig.3 below (similar to quasi-linear IO settings). Namely, when the varieties
(competitors) multitude N increases, the demand curve for a single variety s, naturally, decreases. But at the same
time, the demand curve changes its slope. These two effects may either enforce or hamper each other in affecting
the price. When the slope decreases strongly enough to outweigh the downward demand shift, we get the anticompetitive effect as shown in Fig.3(c). Here, as usual, the rectangle area depicts the amount of profit. To maximize
it, the monopolist choses a longer rectangle under smaller slope.
Generally, as Fig.3 shows, the slope may (a) increase, or (b) slightly decrease, or (c) strongly decrease. One can
easily derive the three formulae connecting r and increasing/decreasing slope:
∂ 2 xk
∂pk ∂N
> 0 ⇔ ru (xk ) > ru0 (xk ) (see
(14) in Appendix). Besides, we can connect the strongly decreasing slope of x̌(p) with anti-competitive condition
ru0 < 0. Using ru0 x = (1 − ru0 + ru ) ru we get ru0 < 0 ⇔ 1 < ru0 − ru . Thus the slope change is formally connected
with anti-competitiveness.
In other words, decreasing elasticity ru (xs ) = εp (xs ) under decreasing value x̄s (N ) brings more rigid demand
curve x∗s (.), therefore monopolist increases price under decreasing demand. Such situation does not seem unrealistic
to our economic intuition.
Further, natural or not are pro- and anti-competitive effects from the mathematical viewpoint? We recall that
the examples of natural utility functions with both effects are numerous and generic (unlike neutrality effects). For
16
Figure 4: Pro-, neutral- and anti-competitive utilities
instance, pro-competitive condition ru0 (x) > 0 holds for u(x) = 1 − e−ax , u(x) = b ln(x + a), and many other utility
functions. Price paradox (ru0 (x) < 0) also holds for very broad and natural class of utilities as we show now.
000
We have found that adding a linear component to a neoclassic utility function with − uu00z > 1 typically turns it
into the anti-competitive class. Indeed, we have seen that the condition for anti-competitive effect is ru0 (z)−ru (z) =
000
− uu00z +
u00 z
u0
> 1. The first summand here, ru0 (z, a) > 1 does not change when a linear component az is added to
initial function u(z), because u00 , u000 does not change. In contrast, the negative summand −ru (z, a) =
goes up, closer to 0, under increasing parameter a > 0. Then, for fixed
u00 , u0 z
u00 z
u0 +a
<0
the second term approaches
zero, and a sufficiently high a can guarantee the anti-competitive condition at a point z for any function with
000
ru0 (z) = − uu00z > 1, and there are many such functions. Of course, for anti-competitive effect, the equilibrium point
z(a) should also fit the region where ru0 (z, a) > 1, but our examples show that it is possible.
Now, which utilities look more natural: pro-, neutral- or anti-competitive? Looking on Fig.4 we cannot find
any reason to prefer one or another!
Left, middle and right panels show three functions: u(x) = log(x+1) (pro-competitive), u(x) = log(x) (neutral),
u(x) = log(x) + x (anti-competitive), respectively. In the upper panel there are indifference curves for log(x1 +
1) + log(x2 + 1), log(x1 ) + log(x2 ), log(x1 ) + x1 + log(x2 ) + x2 , an below are the initial curves u (solid), their
0
derivatives u (dashed) and their Arrow-Pratt measures ru of concavity (thick dash-lines). The right panel, anticompetitive case, shows more rapidly increasing substitution than other two cases, when x1 , x2 increase. It means
that when the equilibrium quantities x1 , x2 decrease under increasing market, the substitution decreases rapidly but
complimentarity increases. Thus, each demand curve xs (ps , .) becomes inelastic to price, its elasticity |εDp
ss | = 1/ru
decreases, and the monopolist increases prices in response. Summarizing, we cannot find any reason, why anticompetitive effect should be considered less natural than the pro-competitive one. Is it more realistic or not, is
only an empirical question.
4) Verifyability. We argue now that most of the effects described in our theorems and corollaries can be verified
through economic observations. In particular, increasing/decreasing ru is equivalent to increasing/decreasing elasticity of the inverse demand for the diversified good. In principle, the elasticity can be measured econometrically.
Elasticity of expenditure E(p, N ) for the diversified good w.r.t. price also can be statistically observable, through
17
value of sells or otherwise. Generally, in our oppinion, our claims connect some observable magnitudes with other
observable magnitudes, so their predictions, and the model itself, can be supported or falsified by observations.
For instance, our prediction that under pro-competitive utilities the increase in the market size for 1% makes
the price falling for less than 1%, should be supported or rejected empirically.
5
Harmful growth and harmful trade
The two subsections below demonstrate two different reasons for possible negative impact of growth on welfare.
5.1
Harmful growth and trade because of price paradox
Now we show that very strong anti-competitive effect can result in a surprizing decrease in welfare, not contemplated
so far for monopolistic-competition models, up to our knowledge.
Definition. Harmful population growth (entailing also harmful international trade) means the decrease in each
consumer’s equilibrium utility under increasing population L, i.e,
∂U (x̄(L))
∂L
< 0.
Example 3. Harmful population growth. We take the utility function from Behrens and Murata (2007), modify
it with a linear component to get anti-competitive effect for some region of parameters:

 1 − e−z + z, if
u(z) =
 1 − e−z + 2, if
z≤2
.
z>2
On the upper interval z > 2 the function is constructed as initial one, to get concave profit function everywhere,
but this region is inessential for our equilibria lying below 2.
To derive the demand, from FOC with z < 2 we get
e−xs + 1
ps
=
⇒ xs = − ln
−x
k
e
+1
pk
ps −xk
e
+1 −1 .
pk
Substituting this into the budget constraint, we get
ˆ
ˆ
N
N
ps xs ds = 1 ⇔
0
ps ln
0
ps −xk
+ 1 − 1 ds = −1.
e
pk
If all the competitors of k-th producer maintain same prices equal to P we get her inverse-demand function
(e−xk +1)P
dependent upon P as pk (xk ) = 1
. The related profit-maximizing program is:
e− P N +1
(e−xk
P
1
e− P N
− P1N
−xk + 1) x − cx
P
(e
e
+
1
k
k
+ 1)
xk − (F + cxk ) =
− F → max .
1
xk
+1
e− P N + 1
It yields FOC:19
1
P e−xk + 1 − P e−xk xk − c e− P N + 1 = 0.
Combining this FOC with the labor balance we have numerically found series of equilibria (P̄ (L), x̄(L), N̄ (L)) for
constants F = c = 1 and various L. At each of these points we calculated the utility level and thereby got the
comparative-statics diagrams presented in Fig. 5.
19
It is easy to check that under x ≤ 2 the second-order conditions show concavity of the profit function, so FOC is valid.
18
Figure 5: Comparative statics with harmful growth: equilibria functions p̄(L), x̄(L), N̄ (L), U (L) under utility
u(z) = 1 − exp(−z) + z
Observe that prices here behave non-monotonically, showing first anti-, then pro-competitive effects. Within the
anti-competitive region (below L = 1.7), there exist a sub-region approximately from L = 1.34 to L = 1.35 with
decreasing utility.20 It means harmful growth. Interpreting this example within international trade framework,
here a small country of size L1 = 0.01 can joins a bigger country of size L2 = 1.34, making the bigger country
worse off, under these primitives u, c, F . The region of parameters bringing the paradoxical effect is small but
non-degenerate, and the utility function in this example looks quite reasonable.
Discussion. This strange effect occurs in the model studied in spite of the increasing returns to scale and
labor being the only important resource! This happens only due to very strong “price paradox”, under additional
non-trivial restrictions on utility. Price paradox itself does not bring harmful growth necessarily, as we have seen
in Example 2 and here outside interval [1.34, 1.35] for L.
To explain the nature of this negative impact of growth on utility, we may tell that it works through anticompetitive effect as follows. When population grows, a single consumer has the same 1 unit of labor to sell,
irrespectively, does additional population arise or not. But new population pushes up the number of firms/varieties
and prices. On one hand, the increase in varieties itself is beneficial for a consumer due to a broader choice. On
the other, growing prices may outweigh these benefits by too significant decrease in consumption of each variety.
Then utility paradox emerges.
From the welfare viewpoint, this loss in utility is explained by the increasing deadweight loss. The inefficiency
of monopolistic-competitive trade relations found in Dixit and Stiglitz [12], is seriously aggravated in situations like
Example 3 by the population growth, that makes the essence of this effect.
Arguing similarly, but only under pro-competitive condition ru0 > 0, Krugman ([17]) notes that the price goes
down under increasing market and varieties’ number grows (see our Table 1 also). This shift is twice favorable
for consumer’s welfare, because she can buy the same bundle as before and even more (below we question this
20
By the way, the utility derease in this example is accompanied by a decrease in total production of varieties.
19
conclusion when utility is not normalized at u(0) = 0 but now we focus on prices). In contrast, under anticompetitive condition ru0 < 0, the varieties’ scope N and prices work oppositely, the possibility not considered by
Krugman.
5.2
Harmful growth because of envy effect
Now we consider the case when utility function is not normalized at zero as u(0) = 0 but negative: u(0) < 0, that
can also bring negative welfare effect under growing market.
Why zero consumption of some varieties can be unpleasant and welfare can decrease? It can be the case under
envy or jealousy growing together with the broader consumer choice. When new varieties appear in the market
(say, caviar), which a person is not buying, she can be dissatisfied with new situation (eating only her usual bread).
From the formal side, if the consumer somehow stick to her previous (still admissible) consumption bundle, new
´N
welfare becomes less than previously, because the integral 0 u(x) is taken over more broad interval Ñ > N , that
is why new unbought varieties bring negative utility. More generally, even when the consumer optimally spend her
money and buy all varieties, still the negative impact of broader choice can outweigh other effects as one can see
through the following simple example. The reason is that, for such non-normalized function there is a very small
constant ε > 0 that also (like 0) brings negative utility (discomfort) when consumed.
√
Example 4. The non-normalized CES function u(x) = x − 1 suffices for such effect. Indeed, within related
comparative statics of one-sector model, the same budget w = 1 is spent for more equilibrium varieties N̄ (L) =
0.5L/F under constant prices for all market sizes L. The consumtion x̄s (L) = F/(cL) of each vriety tends to zero
´ N̄
under growing L, so, at some stage the integrand and total utility 0 u(x) becomes negative.
Moral from this example is that, unlike quasi-linear IO, ordinal approach to utilities in welfare analysis of
monopolistic competition is not innocent. Therefore, condition u(0) ≥ 0 is unjustly dropped in Krugman’s reasoning
about welfare increase under pro-competitive price effect.
Are examples of price-driven or envy-driven welfare decrease economically significant or not? It is an empirical
question to exclude or confirm such counter-intuitive welfare phenomena in reality, we only have shown them
logically possible.
6
Multiple and asymmetric equilibria, catastrophes
6.1
Multiple roots of equilibria equations
Recall that our comparative statics in Section 4 is incomplete because of possible multiplicity of equilibria postponed
to this section. For this issue, it is useful to reformulate our equilibrium equations in terms of two functions, ξ and
ζ equalized to zero, as follows.
1−M
c
) − M = 0; ζ(N, M, L̃) = E(
, N ) · M · L̃/c − N = 0.
(8)
1−M
L̃ · M
Each of these functions can intersect zero one or more times, for instance, ξ generally decreases either monoξ(M, L̃) := ru (
tonically or not, as in two cases of Fig. 6.
By Proposition ?? from Appendix, the monotone-decreasing case like in the left panel occurs under global
profit-concavity condition ru0 (x) < 2 ∀x > 0 or/and global condition ru0 (x) > 0 ∀x > 0. Then (adding reasonable
boundary conditions on expenditure E and boundary conditions ru (+∞) > 0, ru (0) < 1 for 0 < ξ(0, L̃) ∀L̃ > 0 and
ξ(1, L̃) < 0 ∀L̃ > 0) a unique symmetric equilibrium exists, because the continuous function ξ going from positive
to negative values should have a root and concavity ensures this root to be an equilibrium.
20
ξ(M)
ξ(M)
1
0
1
0
Figure 6: Various cases of equilibrium equation for markup M .
However, when profit concavity is not global, like in the right panel, there can be multiple roots of these
equations and even multiple equilibria, but not every root is an equilibrium. Under the above boundary conditions
on ru (+∞), ru (0), function ξ goes from positive to negative regions, so, with probability 1 it has odd number of
roots. At those roots where ξ crosses 0 from above, such point can be an equilibrium, in the opposite case it cannot
because the local maximum of profit requires condition ru0 (x) < 2 which is equivalent to local decreasing of this
curve (see Appendix). So, only downward, not upward, intersections are candidates for equilibria, and there can
be either one or even number of such roots. But each point should be checked additionally, is it a global maximum
of profit or not (see Example 5 below).
6.2
Characterization of multiple and asymmetric equilibria
We are going to characterize now multiple equilibria together with asymmetric ones.
Asymmetric equilibrium for one-sector economy is an interval [0, N̄ ] of firms, a number n̄ of its sub-intervals
P
[0, N̄1 ], [N̄1 , N̄1 +N̄2 ], [N̄1 +N̄2 , N̄1 +N̄2 +N̄3 ], ..., [ n−1
k=1 N̄k , N̄ ] and prices and quantities ((p̄1 , x̄1 ), (p̄2 , x̄2 ), ...(p̄n , x̄n )) ∈
R2n̄ different among these n groups of producers, such that: (1) this step-wise consumption function x maximizes
the consumer’s utility under this step-wise price function p̄, (2) each p̄k maximizes the k-th producer’s profit, equal
to zero, and (3) labor balance is satisfied in the form L̄1 + L̄2 + ... + L̄n = L, where L̄k denotes the equilibrium
labor applied in k-th group.
Now we introduce the necessary and sufficient conditions on utility u for multiple and asymmetric equilibria,
in terms of characteristics M Ru (.) and πu (., c̃) of the utility.
Assumption MR. The utility function u(.) is neoclassic, it generates the non-monotone marginal-revenue
00
function M Ru (x) := u (x)x + u0 (x), and the proft function πu (x, c̃) = x(u0 (x) − c̃) with at least two global maxima
x̌, x̃ ∈ arg maxx πu (x, c̃), x̌ 6= x̃ for some cost level c̃ > 0.21
Proposition 5 Given a utility function u(.), related one-sector Dixit-Stiglitz economy with asymmetric equilibrium
exists (i.e., some parameters F, L, c yielding such equilibrium can be chosen) if and only if utility u(.) satisfies
Assumption MR.
Corollary 1. Multiple symmetric equilbria (x̌, p̌, Ň ) 6= (x̂, p̂, N̂ ) in such economy always coexists with an
asymmetric equilibrium and vice verse: any asymmetric equilibrium is always acompanied by at least two symmetric
equilibria.22
Next corollary describes the structure of the equilibria set when they are multiple: the set is an interval of a
line.
21
22
In terms of r(.), non-monotone M R(.) means that there are points x, z such that, ru0 (x) > 2, ru0 (z) < 2.
Probably, an asymmetric equilibrium can generate also 3 or more symmetric equilibria, but we leave aside this topic.
21
Corollary 2. Let Assumption MR be satisfied, with exactly two argmaxima x̌ > x̂. Then any asymmetric
equilibrium in such economy has the specific structure: some Nx̌ ≥ 0 firms produce quantity x̌ and some Nx̂ ≥ 0
firms produce quantity x̂, these Nx̌ , Nx̂ satisfying the equation 1 = Nx̌ 1−rx̌cu (x̌) + Nx̂ 1−rx̂cu (x̂) . Any non-negative
Nx̌ , Nx̂ satisfying it comprise an asymmetric equilibrium (x̌, x̂, Nx̌ , Nx̂ ) under same quantities (x̌, x̂). This means
existence of an interval (Nx̌ , Nx̂ ) ∈ [(0, Ňmax ), (N̂max , 0)] of asymmetric equilibria, while the couple of symmetric
equilibria ((x̌, Ňmax ), (x̂, N̂max )) presents the ends of the interval. These ends can differ in consumer’s utility.23
Remark 3. Similar statements can be reformulated for multi-sector Dixit-Stiglitz economy under appropriate
assumptions on expenditure function E(.) guaranteeing equilibria existence.
The proofs are in Appendix. The following example and Figures 7,8 illustrate the proposition with its corollaries
and the structure of the equilibria.
Example 5. Asymmetric and multiple equilibria of monopolistic competition.24
An example of asymmetric and multiple equilibria was constructed by the method described in Proposition 5
and explained in its proof. First we have found a utility function satisfying Assumption MR, namely
u(x) = x +
√
x + 1.4 arctan(2x + 0.05),
which is, obviuosly, concave and increasing. We took F = 0.0025, L = 10, and through computer algebra, by
the method described, found the cost level c ≈ 0.000264937336989 needed for mutiple equilibria. Upper panels
in Fig.7 characterize the properties of the functions ru , ru0 , ξ important for equilibria. The solid curve ru in the
left panel is non-monotone, that allows both for pro- and anti-competitive effects, while another curve intersects
level 2 that violates the sufficient condition for profit concavity. The upper right panel shows the solid curve ξ(M )
from equation (8) for finding the equilibrium markup M . It intersects zero (horizontal axis) three times like in the
right panel of Fig.6, but only the leftest and the rightest intersections, from above downward, are candidates for
equilibria, by Proposition ??. The dashed line supports this claim showing the condition for local profit concavity
only at the boundary intersections.
To finally ensure validity of these two roots, we calculated (x̌, x̂, p̌, p̂, Nx̌ , Nx̂ ) from the equilibria equations
and checked global maxima of profit and equal-profits conditions holding. The lower panels show that profits are
really equal at both equilibria (marked by dashed vertical lines) and at alternative points. The left one shows the
profit function of a producer when all her competitors produce about x = 0.652644, and we observe her being
indifferent between joining this pattern or switching to high quantity x = 12.8961. The same indifference holds
under equilibrium x = 12.8961 in the right panel. The left and right curves are identical, illustrating Corollary 2
to Proposition 5 about the structure of equilibria.
When rounding the numbers to 12 digits (from 60-digit precision used in calculation), the high-quantity and
low-quantity symmetric equilibria are as follows.
M
p
Small-price
x
N
U
0.068181933646
0.000284323031025
12.8961077968
272.727734586
5081.514933136
0.591142507799
0.0006479943306508
0.652644395317
2364.570031198
6549.082752038
equilibrium
Big-price
equilibrium
By Corollary 2, whole interval of asymmetric equilibria lie between these two symmetric ones, which are the
ends of the interval. Along this interval, the high and the low quantities remain the same, while the line connecting
N1 and N3 in this example is N3 = 2364.570031198 − 8.670075431758N1 .
23
Actually, they are non-equivalent with probabilty 1, up to our knowledge, but proving it here would be excessive.
We are indebted for Alexei Gorn for developing computations of this example in his graduating diploma, for our request, and for
commenting our proofs also.
24
22
Figure 7: Example of multiple equilibria.
The utility levels were found as
ˆ
Ǔ =
ˆ
Nx̌
u(x̌)dt = 5081.514933136
Nx̂
u(x̂)dt = 6549.082752038.
Ǔ =
0
0
Interestingly, the utility levels are different at different equilibria, as proposed by Corollary 2.
6.3
Global comparative statics under multiple equilibria
Now consider global comparative statics under possible multiplicity of equilibria illustrated by Example 5. We have
noted in Section 4 that local comparative statics developed there can be inapplicable under multiplicity, at least at
the multiplicity points.
Now, the below Theorem 2 get global conclusions similar to Theorem 1 and Table 1 (though without elasticities)
directly, without relying on local behavior of our functions and any derivatives, because derivatives for set-valued
mapping p̄(L̃), N̄ (L̃) are not helpful. Theorem 2 applies an approach to comparative statics from Milgrom and
Roberts [18] based only on boundary conditions for functions ξ, ζ from the equilibrium equations (8).
The below auxiliary Lemma 1 is a simplified, and rewritten in our terms, version of Milgrom and Roberts’s
Theorem 1 (p.446) about monotone comparative statics of left and right roots of any equation ξ(., .) = 0.
Lemma 1 Let a function ξ(., .) = ξ(M, L̃) : [0, 1] × R → R be continuous and satisfy boundary conditions ξ(0, L̃) ≥
0, ξ(1, L̃) ≤ 0 ∀L̃. Then there exists a solution M to equation ξ(M, L̃) = 0. The lowest root is defined as Ml (L̃) =
inf{M |ξ(M, L̃) ≤ 0} and the highest root is Mh (L̃) = sup{M |ξ(M, L̃) ≥ 0}. If ξ(M, L̃) ∀M is monotone increasing
(non-decresing) in L̃, then both roots Ml (L̃), Mh (L̃) are monotone increasing (non-decresing) for all L̃.
23
Using Proposition 5 we can apply now this lemma to the equilibrium equation in the form ξ(M, L̃) = 0, and
derive a global theorem about price effects, pro- and anti-competitive effects in terms of set-valued mapping of
equilibria.
Theorem 2 Suggest Assumption 1, Assumption MR and additionally, that marginal revenue has only one bounded
interval of increase, but decreases outside this interval (near 0 and at ∞). Then: (a) No more than one relative
market size L̃∗ > 0 yields two symmetric equilibria, at other L̃ the equilibrium is unique.25 (b) The mapping of
quantity x̄(L̃) at the discontinuity point L̃∗ (if any exists) jumpes down and globally decreases.26 (c) The price
mapping p̄(L̃) jumps up at L̃∗ , so, under globally decreasing concavity ru the price globally increases but can be
non-monotone in other cases.
Corollary T2.1. Under monotone decreasing marginal revenue M R, equilibrium is unique and the equilibrium
magnitudes p̄(L̃), x̄(L̃) behave like in Table 1.
Corollary T2.2. At any interval Iˆ 63 L̃∗ excluding the discontinuity point, the equilibrium magnitudes
p̄(L̃), x̄(L̃) behave like in Table 1.
Now we turn to the number of varieties/firms. Lemma 1 applied to function −ζ(N,˜ L) from the equilibrium
equations (8) yieds
Corollary T2.3. Under Assumption 2, equilibrium firms’ number N̄ (L̃) monotonically increases.
Now we illustrate the theorem and its corollaries, proving also that the price can behave non-monotonically.
Example 5 (continued). Comparative statics of multiple equilibria.
The mappings of equilibrium consumption, price, number of varieties and utility in Example 5 behave like in
Fig.8 below (calculated numerically).
Here, at the discontinuity point, when the relative market size is L̃ = 10 , the interval [272.7, 2364.6] of
equilibrium number of firms describes all asymmetric equilibria (see Proposition 5), but not two symmetric equilibria
N̄ = 272.7, N̄ = 2364.6. This point is really a point of jumps in competition, consumption of each variety, price
and utility, that means a catastrophic change. Small gradual changes in population, technology (or gevernmental
regulation, absent in this model) can, theoretically, result in great abrupt changes in economy!
From mathematical point of view, Proposition 5 says that these effects occur under rather broad class of
functions u satisfying Assumption MR: just a non-monotone marginal revenue with a kink substantially above the
marginal cost. Such inverse-demand functions u0 are not a degenerate case because all small modifications of such
function mainain this property.
Are such revolutionary changes generated by monopolistoc competition realistic or not, is an empirical question.
It amounts to realism or not of demand curves allowing for two maxima of monopolistic profit, only here the whole
sector of economy coherently switches from one maximum to another. The more heterogeneous are the firms in
real life, the less abrupt can be the revolution, but the nature of this effect can remain.
7
Conclusion
This paper studies comparative statics of quite general multi-sector Dixit-Stiglitz model of monopolistic competition
with one production factor and arbitrary upper- and lower-level utilities.
A method (generalizing the Krugman’s approach) is obtained to study all similar models through deriving
quite simple system of equilibrium equations in terms of Arrow-Pratt’s measure ru (.) of utility’s concavity (price
25
The structure of these two equilibria is illustrated in Fig.8 and analysed in Proposition 5, which states also an asymmetric equilibria
at the same point L̃∗ and excludes more than two symmetric equilibria.
26
A mapping f is called decreasing when x > z ⇒ f 0 < f ” for all f 0 ∈ f (x), f ” ∈ f (z).
24
Figure 8: Discontinuous comparative statics under multiple equilibria
elasticity), and function E(.) describing expenditure for varieties and, implicitely, the rest of economy. We add a
complete analysis of the roots to this system and their relation to unique or multiple, symmetric or asymmetric
equilibria.
Comprehensive comparative-statics in Theorems 1,2 describes local and global changes of price, quantity and varieties’ number w.r.t. the “relative size of the market.” The latter describes the size of the population related to fixed
cost and variable cost. The theorems classify markets into pro-, neutral-, and anti-competitive ones according to
increasing, neutral, or decreasing measure ru (.) of concavity, respectively. These and other effects are shown to exist
and explained through examples. Most interesting and important for international-trade and economic-geography
studies are the price paradox, harmful growth, multiple equilibria and discontimuity (catastrophic changes) unknown
previously for monopolistic-competition. These effects worth further studying and allow for empirical testing.
Extensions. Our method of equilibria derivation for arbitrary function u(.) without the closed-form demand,
allows for generalization to several models. First of all, we are intersted in directions of non-linear costs and/or
multiple production factors, to seek for the same effects.
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26
Appendix
7.1
Derivating the reduced model from the explicit multi-sector model
In Section 1 we have introduced the “basic” or standard Dixit-Stiglitz model of one-sector one-production-factor
economy, and then generalized it to “reduced” multisector model where all but one sectors are described through
the expenditure function E. The next subsection shows how to derived this expenditure function E from some
upper-level utility function U , reflecting prefernces over other aggregated goods.
7.1.1
General model of a Dixit-Stiglitz one-factor economy
Now we introduce our main object of study in extensive form. The basic model of a diversified sector is supplemented
by additional diversified or/and homogeneous sectors (aggregated commodities). First we explain the general model
and its equilibrium, and then derive the reduced version.
There are m1 diversified sectors (aggregated commodities) indexed as i = 1, ..., m1 and m2 homogeneous sectors
k = m1 + 1, m1 + 2, ..., M (M = m1 + m2 ). Each of L identical consumers/employees solves the following utilitymaximization problem:
U (X1 , ..., Xm1 , y1 , ..., ym2 ) → max
w.r.t.,
x(.),y
m1
X
i +
i=1
ˆ
M
X
ˆ
Ni
pis xis ds,
i =
pk yk ≤ 1,
where
k=m1 +1
Ni
u(xis )ds.
Xi :=
0
0
Here intermediate variable i ≥ 0 denotes expenditure for i-th sector, Xi ≥ 0 is the aggregate utility from this
sector, and yk ≥ 0 is consumption of k-th homogeneous good. Labor endowment is normalized to 1, as well as
wage, and income from profits is normalized to zero because of zero-profit concept of all sectors. Upper-level utility
is U . Note that now the demand depends upon a longer price vector p then previously, including prices p1 , ..., pM
of all sectors, which are scalars for the homogeneous goods but functions for the diversified goods. This utility
maximization generates some demand function x̌(p, N ).
Each ((is)-th) monopolist out of Ni producers in i-th diversified sector, is already described by profit-maximization
program (1). It remains the same but for more indicies and enlarged argument p = (p1 , ..., pM ) of the demand
function. Index (is) should replace more simple index s when describing the price pis , quantity xis , variable cost
ci and fixed cost Fi of the producer. As to homogeneous sector, it is supposed to be a competitive one, with
linear technology, so it is sufficient to introduce marginal costs ck , measured in labor, for each sector (number
of firms does not matter here). This profit maximization generates some response function p̌is (p-is , N ) depending
upon all outside prices p-is and the total vector of scopes of sectors as N = (N1 , ..., Nm1 ) ∈ RM . Now we denote the symmetrized vector of prices as p̃ = (p̃1 , ..., p̃M ) ∈ RM , because assuming symmetry within each sector.
This means same price p̃i = p̌is (p̃-i , N ) for each (is-th) producer in the sector and symmetric production quantity
x̃i (p̃-i , N ) = x̌is (p̌is (p̃-i , N ), p∗-i , N ) that exploits the demand function depending upon stepwise (symmetric) other
prices and i-th response to these other prices (we maintain the same notations x̌, p̌ for demand and price response
though these function formally depend upon simpler arguments than before). Solving the equations system for
Nash price-responses of all producers to each other, we define the subequilibrium price vector p∗ (N ) as a function
of all scopes, that yeilds also sub-equilibrium quantities x∗ (N ). From the consumer’s program we get also the
27
sub-equilibrium demands yk∗ (N ) for homogeneous goods as a function of N under the pricess found and trivial
prices pk = ck of homogeneaous goods.
To link the sectors to each other, we add the zero-profit (free-entry) assumption for each diversified sector and
the labor balance in the economy:
p∗i (Ni )Lx∗i (Ni )
=
(ci Lx∗i (Ni )
+ Fi ) ,
Ni (ci Lx∗i (Ni )
+ Fi ) = l i ,
m1
X
li +
i=1
M
X
ck yk∗ (N ) = L.
k=1+m1
The consumer’s budget equality follows from these equations at equilibrium.
Definition 2. An integrated symmetric equilibrium ((x̄, ȳ), p̄, N̄ ) ∈ R3M of such economy is a vector that
satisfies optimization programs of consumers and producers (i.e., symmetric subequilibrium requirements), and the
above balances of profit and labor.
Looking on this model through our goals, it seems impractical to study comparative statics of such complicated
equilibria directly. Instead, we use a reduced formulation of such an economy, and study mainly this reduced
formulation, which is focused on one diversified sector within the economy. The reduced model looks like partial
equilibrium, but really it is not.
7.1.2
Reduced version of the general model
As soon as our goal is expansion of the Krugman’s method to multi-sector models, we would like to analyze each
(i-th) Dixit-Stiglitz diversified sector as a separate entity, connected with the rest of the economy only through
variables of expenditures ei and labor li applied in i-th sector. It is easy to understand that these two vectors
are interconnected, being proportional to each other at equilibrium in the sense l = (l1 , ..., lM ) = (e1 , ..., eM )L,
because zero-profit condition equalizes budget shares and labor shares. So, we need only to reduced the “rest of the
world” to some response function Ei (pi , Ni ) of outside sectors to this sector. Ideologically, this function denotes
the economy’s long-run equilibrium reaction to shifts in this diversified sector. But technically, we can look on it as
on a method to derive the integrated equilibrium defined above. We should show that step by step we can derive
such function Ei from all primitives U, u, c, F, L.
Without serious restriction of generality, we describe now such recursive derivation for the case of two diversified
sectors i = 1, 2 and only one homogeneous (competitive) sector k = 3. Suppose, we are mainly interested, what
happens with equilibrium price, quantity and number of firms in sector #1 when the size of the market L grows
(similarly we can study sector #2).
Step 1. For the homogeneous sector, its equilibrium price, trivially, equals costs: p̄3 = c3 . We find also its
per-consumer quantity as a symmetric demand function x̃3 (p1 , p2 , N1 , N2 ) ∈ R depending upon upon (unknown so
far) infinite-dimensional price vectors p1 , p2 of the first and second sectors, and upon numbers of varieties. Other
components of the demand are out of use so far. Finding the demand practically for some utilities is a hard task,
but we try only to explain now why our expenditure function E is a reasonable concept.
Step 2. For the second sector, using the equilibrium equations, we derive its response symmetric functions
p̃2 (p1 , p̄3 , N1 ), x̃2 (p1 , p̄3 , N1 ), Ñ2 (p1 , p̄3 , N1 ) depending upon p̄3 (already found) and upon (unknown so far) infinitedimensional price vector p1 and number of firms N1 in the first sector.
Step 3. For the first sector, we substitute the symmetric price obtained and the functions p̃2 (.), x̃2 (.) to get
the general-form expenditure function
E1 (p1 , N1 ) = 1 − p̃2 (p1 , p̄3 , N1 )x̃2 (p1 , p̄3 , N1 ) − p̄3 x̃3 (p1 , p̃2 (p1 , p̄3 , N1 ), N1 , Ñ2 (p1 , p̄3 , N1 ))
28
of the first sector. It depends upon the unknown infinite-dimensional price p1 and the scalar number of varieties
N1 , expressing the equilibrium expenditure for varieties of this sector, used further in all derivations.
It turnes out that it is sufficient to impose reasonable restrictions on this function E for finding out what
happens to sector #1 within comparative statics of the whole economy, without complicated direct derivation
of demand. It is one of our lucky findings. It overcomes the limitation in typical papers (obsessed with the
closed-form demand) where the upper level utility function for technical reasons is a Cobb-Douglas one: U (X, y) =
PM
Pm1
β 27 Then the budget share allocated
k=m1 +1 ak Log(yk ), with CES lower-level function u(z) = z .
i ai Log(Xi )+
to each sector is just constant Ei (pi , Ni ) ≡ i = ai . We allow for more general functions, becoming now available.
7.2
How to derive the demand properties?
For general multi-sector economy we derive now the consumer’s demand for any variety, and its properties.
According to the two-level concept of consumer’s behavior, in choosing varieties a consumer percepts her expenditures E = E(p, N ) for the whole diversified sector as already chosen. Respectively, her optimization problem
is similar to that of the basic model:
ˆ
ˆ
N
u(xs )ds → max
x
0
N
ps xs ds ≤ E.
w.r.t.
0
The FOC (first-order conditions) of the Lagrangian w.r.t. each variety xs are
u0 (xs ) = λps ⇒
u0 (xs )
ps
= .
0
u (xk )
pk
To express the demand we denote the inverse of the marginal utility as ϕ = (u0 )−1 (following Behrens and Murata’s
notation). Then
xs = ϕ(ϕ−1 (xk )
ps
).
pk
(9)
Now our departure from the classics starts. At this stage, the Dixit-Stiglitz tradition (including Behrens and
Murata), instead of hoping for general solutions, turnes to analysing specific functions ϕ, i.e., specific utilities.
Instead, inserting this expression for demand xs into the budget equality we get
ˆ
N
ps ϕ(ϕ−1 (xk )
0
ps
)ds = R.
pk
(10)
We can look on this equality as on an indirect function, yielding the dependence x̌k (p, E, N ) of consumer’s optimal
choice upon prices, income and number of varieties N (which is the length of the varieties interval in continual
model). Further we drop the “check” and these arguments of this function, denoting it as xk (.) but have them in
mind, and have in mind its optimality. Assuming “very large” N for the discrete version of the model, or using the
same idea in continual model, we get negligibility of spillovers between single varieties:
∂xk
∂ x̌k (p, E, N )
≡
=0
∂ps
∂ps
for all s 6= k. This absence of any complimentarity or gross substitutability (GS) between commodoties is due to
negligible influence of each variety on the total demand in the continual model. In discrete monopolistic-competition
27
Only combination of these two restrictive assumptions is sufficient for constant , otherwise the budget share can vary. This fact is
not noticed in Helpman and Krugman ([16], p. 190) where the constant-share assumption is just added to seemingly-general functions
analyzed.
29
model this equality becomes an additional assumption. Similarly, we have ignored in differentiating the dependence
of expenditures E(p, N ) upon a single price, because of its zero measure among all prices.
On the other hand, if we look on reaction of demand for k-th commodity to changes in a whole group of
other prices with positive measure, it is not negligible. To express this reaction we introduce two most important
notations:
ru (xk ) := −
xk u00 (xk )
xk u000 (xk )
0 (xk ) := −
;
r
,
u
u0 (xk )
u00 (xk )
This characteristic ru (.) of u means Arrow-Pratt’s relative risk aversion measure, or elasticity of the inverse
demand. It turnes out to be the crucial characteristics of preferences, most tightly connected with all important
features of demand.
In particular, it shows the borderline between complimentarity and substitutability among measurable groups
of varieties. In microeconomics, for neo-classical separable utilities the sufficient condition for strict gross substitutability (GS) is known as ru (xk ) < 1, while ru (xk ) = 1 relates to absent sensitivity (Cobb-Douglas utility) and
ru (xk ) > 1 means some degree of complimantarity. Now, for fixed E, we give a direct proof for the same gross
substitution (GS) effect between a given (zero-measure) variety xk and the “price level” P that means all other
prices when they are uniform (same) and derive the demand elasticity:
ˆ
N
P ϕ(ϕ−1 (xk )
0
P
P
P
E
)ds = N P ϕ(ϕ−1 (xk ) ) = E ⇔ ϕ(ϕ−1 (xk ) ) =
⇒
pk
pk
pk
NP
∂xk 0 −1 0 P
ϕ
1
∂xk P
1 − ru (X)
ϕ ϕ
= − − ϕ0 ϕ−1
⇔
=
.
∂P
pk
P
pk
∂P xk
ru (xk )
Here the denomenator is positive (by concavity of u), so the numerator expresses the condition 1 − ru (X) > 0
sufficient for strict GS, which plays some role further for equilibrium existence.
Demand dependence upon its own price. As to the demand derivative
∂xk
∂pk
w.r.t. its own price, it need not be
negligible. Differentiating equation (9) w.r.t. ps we get
∂
ps
∂xs
ps
ps ∂xk
1
0
[xs − ϕ(ϕ−1 (xk ) )] =
− ϕ0 (ϕ−1 (xk ) )(ϕ−1 (xk )
+ ϕ−1 (xk ) ) = 0 ⇒
∂ps
pk
∂ps
pk
pk ∂ps
pk
∂xk
ϕ−1 (xk )
ϕ−1 (xk )
xk
= ϕ0 (ϕ−1 (xk ))
= −1 0
=−
,
∂pk
pk
r
(x
(ϕ ) (xk )pk
u k )pk
Here we have used equation (9) to substitute xk , pk for xs , ps . Besides, like for
∂xk
∂ps ,
(11)
we have neglected in differentiating
the dependence E(p, N ) of the expenditures upon a single price, for the same reason.
Demand dependence upon the number of varieties. This dependence will not be used in further proofs, serving
rather for interpretations of the direct effect of growing competition (ignoring equilibrium spillovers), so we stick
to fixed-expenditure case (E = const) in this paragraph. The effect studied is the increase or decrease in the slope
of the demand curve, already discussed in Examples 1,2 but now shown in general form, to give more intuitions for
anti-competitive paradox.
We can study the demand derivative
∂xk
∂N
taken w.r.t. broadness of the varieties interval (similar estimate in the
discrete model is applicable only at the equilibrium, but here it goes easy). We differentiate equation (10) w.r.t.
parameter N , incoming both into the integral’s limits and into demand’s xk (p, E, N ) arguments, and like before
take the constants out of the integral to get
30
pN ϕ(ϕ
−1
pN
∂xk −1 0
(xk ) ) +
ϕ
(xk )
pk
∂N
ˆ
N
0
p2s 0 −1
ps
ps
ϕ (ϕ (xk ) )ds = 0 ⇒ (using ϕ−1 (xs ) = ϕ−1 (xk ) )
pk
pk
pk
∂xk
pN xN
< 0.
=−
´ N p2s
0
∂N
ϕ0 (ϕ−1 (xs ))ds
(ϕ−1 ) (xk )
(12)
pk
0
The sign is due to ϕ = (u0 )−1 > 0, u00 < 0, and can be interpreted as “market-crowding”. When more varieties
become available, a consumer, naturally, starts buying less quantity of each. In the particular case of symmetric
equilibrium (xs = xk , ps = pk ) this exression amounts to
∂xk
∂N
= −xk /N .
As we have explained in Examples 1,2, growing number N of competitors not only shifts down the demand
curve, but also changes its slope. Indeed, we can differentiate expression (11) w.r.t. N :
ϕ−1 0 (xk ) ∂xk
ru0 (xk )
∂ 2 xk
1
∂xk
00 −1
−1
0 −1
= ϕ (ϕ (xk ))ϕ (xk ) + ϕ (ϕ (xk )) ·
·
= 00 −
+1
.
∂pk ∂N
pk
∂N
u
ru (xk )
∂N
⇒
∂ 2 xk
> 0 ⇔ ru (xk ) > ru0 (xk ).
∂pk ∂N
(13)
(14)
Thus, depending upon relation between concavity of u and concavity of u’, the slope of demand can increase or
decrease.
7.3
Proofs for producer’s FOC and SOC
For considering the (standard) Dixit-Stiglitz producer, we should turn from individual demand to total demand,
and explain it. In CES approach it is standard to just multiply the individual demand x̌k (p) for each variety by the
quantity of labor L to get x̌Σk (p) = Lx̌k (p). This simple aggregation rests on homogeneity and related existence of
a representative consumer. But in our case these assumptions are not imposed, so for similar simple aggregation
we should (and did) assume that each consumer has the same income, equal to her one unit of labor multiplied by
the wage, w = 1.
The k-th monopolistic producer maximizes her profit π w.r.t. per-consumer output xk , and price pk taking into
account the total demand function for this variety:
π = (pk − cw)Lx̌k (pk , p−k , E(pk , p−k , N ), N ) − wF → max .
pk
Linearity of variable cost allows to simplify FOC (first-order conditions) as
πp = (pk − cw)
∂ x̌k
+ x̌k = 0
∂pk
(15)
and SOC (second-order conditions) or profit-concavity condition amounts to
πpp = 2
∂ x̌k
∂ 2 x̌k
+ (pk − cw) 2 ≤ 0.
∂pk
∂pk
So, like in traditional IO with quasi-linear utilities, any equilibria can occur only at points with elastic inverse
demand. Indeed, FOC and profitablity pk > cw entail the following simple bound on markup and on r (similar to
Lerner’s index):
31
(pk − cw)
1
= − ∂ x̌ p = ru (x̌k ) < 1.
k k
pk
∂p x̌
k
k
We see that equilibria can occur only at points with moderate concavity (related to gross-substitution property
as we shall see). This neat expression of the optimal markup in terms of the utility’s risk-aversion ru , common
in quasi-linear contexts of IO, helps very much in the following simple proposition stating the producer’s solution
uniqueness (but not existence, which is unclear).
Proposition 6 (i) If concavity of the marginal utility is restricted as 2 > ru0 (xk ) at point xk , then each producer’s
profit function is strictly concave at this point; at points satisfying FOC this restriction is necessary and sufficient
for strict concavity; (ii) If this upper bound on ru0 (xk ) holds everywhere on R+ , then profit is generally strictly
concave and has a unique maximum; and any sub-equilibrium is symmetric.28
Proof. Let us express the condition on the profit’s second derivative in terms of the consumer’s parameter ϕ:
∂ 2 xN
∂
=
2
∂pN
∂pN
0
ϕ (ϕ
−1
ϕ−1 (xN )
(xN ))
pN
00
= ϕ (ϕ
−1
(xk ))
ϕ−1 (xk )
pk
.2
Here we have used the identity ϕ0 (ϕ−1 (xN )) = 1/((ϕ−1 )0 (xN )). More generally, the profit concavity condition
2
∂ xk
k
2 ∂x
∂pk + (pk − cw) ∂p2 ≤ 0 at point of producer’s optimum is equivalent to
k
(pk − cw) ∂ 2 xk
(pk − cw)
2+
=2+
ϕ00 (ϕ−1 (xk ))
−1
2
∂xk
0 (ϕ−1 (x )) ϕ (xk )
∂p
ϕ
k
k
∂pk
pk
ϕ−1 (xk )
=2−
00
−1
xk ϕ (ϕ (xk )) pk
∂xk
ϕ0 (ϕ−1 (xk ))
=2−
∂pk
=2+
ϕ00 (ϕ−1 (xk ))
xk
ϕ0 (ϕ−1 (xk )) ϕ0 (ϕ−1 (xk ))
ϕ−1
00
(xk )xk
0
(ϕ−1 ) (xk )
=2+
ϕ−1 (xk )
pk
2
=
2 (ϕ−1 )00 (xk )
= 2 + xk (ϕ−1 )0 (xk )
3 =
((ϕ−1 )0 (xk ))
u000 (xk )xk
= 2 − ru0 (xk ) ≥ 0.
u00 (xk )
(16)
Thus, the optimal price found from FOC is unique.
Let us show equilibrium symmetry. From the producers’ and consumers’ FOC we have
u0 (xk )
pk
1 − ru (xs )
=
=
⇔
0
u (xs )
ps
1 − ru (xk )
u0 (xk ) (1 − ru (xk )) = u0 (xs ) (1 − ru (xs )) .
The obtained function u0 (x) + u00 (x)x increases because of 2 > ru0 (x), therefore solution symmetry is guaranteed:
xk = xs . Here we give the full version of the proposition about uniqueness and existence of equilibria.29
28
The condition 2 > ru0 (xk ) found here, unlike quasi-concave economies, is not necessary for profit concavity and equilibrium
symmetry, say, the Behrens-Murata’s function mentioned before Example 1 satisfies this condition not everywhere, but shows concavity
and symmetry everywhere. The point is that the demand function for a Dixit-Stiglitz sector differs from that of a quasi-concave sector
having similar u (because showing some income-effect), only their elasticities are similar.
29
As to equilibria inexistence, there can be neoclassic utility functions without any equilibria, for instance, u(x) = ln x, or, more
interestingly, function from Example 2 but unrestricted from above: u(x) = 1 − e−x + x. The latter utility generates a profit function
π(ps , p̄) which has local maxima and solutions to equation (6) for some parameters L, F, c, N , but profit is unrestricted in direction
ps → 0, not having a global maxima.
32
Proposition 7 Let the risk-aversion ru (.) be continuous and satisfy boundary conditions: ru (+∞) > 0 (i.e.,
0 < ξ(0, L̃) ∀L̃ > 0) and ru (0) < 1 (i.e., ξ(1, L̃) < 0 ∀L̃ > 0). Then: (i) For any L̃ > 0 equation (6) has a root
0 (M̄ , L̃) < 0; (ii) In any such root
M̄ satisfying bounds 0 < M̄ < 1 where ξ intersects 0 downwards in the sense ξM
M̄
M̄
two conditions are satisfied: local gross substitution ru ( 1−
) < 1 and local concavity of profit ru0 ( 1−
) < 2, while
L̃·M̄
L̃·M̄
any upwards root lacks local concavity and cannot be an equilibrium; (iii) Under global sufficient profit-concavity
condition ru0 (x) < 2 ∀x > 0 or/and global condition ru0 (x) > 0 ∀x > 0 the root M̄ is unique and it is definitely an
equilibrium value of markup;30
c
c
(iv) If expenditure function E( 1−M
, N ) is continuous and bounded for all M ∈(0, 1) having limN →0 E( 1−M
, N) >
0 ∀M ∈ (0, 1), then for any M equation (7) has a solution N̄ = N̄ (M ) > 0. Then, any such N̄ (M ) obtained from
any equilibrium markup M̄ : 0 < M̄ < 1 is a symmetric monopolistic-competition equilibrium (M ,N ), which
thereby, exists under global profit concavity and boundary conditions on expenditure E and concavity ru .
Proof. Consider function ξ(M, L̃) := ru ( 1−M
) − M . By assumptions, ξ(0, L̃) = ru (+∞) > 0 and ξ(1, L̃) =
L̃·M
ru (0) − 1 < 0. Then existence of a root in (i) is a standard result for continuous ξ, changing signs on an interval,
continuity following from Assumption 1 (see Fig.6.). For (ii), note that the derivative whose negative sign show
downward intersection is
0
ξM
(M, L̃) = −ru0
1
1
−1=−
ru0 x̄ + M (1 − M ) ∀L̃ > 0.
2
M (1 − M )
L̃ · M
It is evidently negative everywhere under condition ru0 (x) > 0 ∀x > 0, then ξ decreses. For a weaker conclusion
under weaker condition 2 > ru0 this magnitude at points M = ru ( 1−M
) of roots to ξ(M, L̃) = 0 can be expressed
L̃·M
through r as
0
ξM
(M̄ , L̃) = −
1
1
2 − r u0
ru0 x̄ + ru (1 − ru ) = −
(ru (1 − ru0 + ru ) + ru (1 − ru )) = −
.
ru (1 − ru )
ru (1 − ru )
1 − ru
This magnitude at intersection points is negative if and only if 2 − ru0 because the denominator is always positive
at intersections on the interval 0 < M < 1 (local GS condition). This equivalency and local GS evidently imply
(ii), (iii). As to (iv) it is proved similarly due to continuity of E from Assumption. 7.4
Proofs for comparative statics
Proof of Theorem 1. The proof goes through direct algebraic transformations based on the implicit function theorem. We use the implicit dependence of markup M upon the relative market size as M = ru ( 1−M
). Differentiating
L̃M
it we get
∂M
1 1−M
1
1 ∂M
0
= ru −
·
− · 2
⇒
M
∂ L̃
L̃2
L̃ M ∂ L̃
∂M L̃
ru0 x̄
ru0 x̄ (1 − ru )
·
=−
=
−
.
1
1
ru (1 − ru ) + ru0 x̄
ru (1 + ru0 L̃ · M 2 )
∂ L̃ M
00
0
00
2
u000 (x)x2
u00 (x)x
u (x)x
Note that ru0 x = − uu0(x)x
x
=
−
−
+
−
= −ru0 ru + ru + ru2 = ru (−ru0 + 1 + ru ) .
0
0
0
(x)
u (x)
u (x)
u (x)
Therefore
∂M L̃
ru (−ru0 + 1 + ru ) (1 − ru )
(ru0 − 1 − ru ) (1 − ru )
·
=−
=
.
ru (1 − ru ) + ru (−ru0 + 1 + ru )
(2 − ru0 )
∂ L̃ M
30
We are not stating here thatξ(., L̃) is monotone decreasing, but it cannot intersect 0 from below due to (ii). Besides, we are not
stating here the general equilibrium existence yet, because the second equation is not studied.
33
One can see that
∂M
∂ L̃
·
L̃
M
> − (1 − M ). Indeed,
1
∂M L̃
(ru0 − 1 − ru )
(ru0 − 2)
·
·
=
>
= −1.
1 − M ∂ L̃ M
(2 − ru0 )
(2 − ru0 )
Now look on all other variables:
p=
x=
1
M
∂x L̃
1
∂M L̃
·
⇒
· = −1 −
·
·
1 − M ∂ L̃ M
L̃ 1 − M
∂ L̃ x
y=
Now let ru0 > 0, then
∂M
∂ L̃
∂M
∂ L̃
·
L̃
M
∂p L̃
M
∂M L̃
c
⇒
·
· =
·
1−M
p
1
−
M
∂ L̃
∂ L̃ M
F 1−M
∂y L̃
1
∂M L̃
·
⇒
· =−
·
·
c
M
y
1
−
M
∂ L̃
∂ L̃ M
< 0, therefore
∂p
∂ L̃
L̃
·M
> − (1 − M ) obtained, but combining it with
∂p
∂ L̃
guarantees that
·
L̃
p
=
M
1−M
·
∂M
∂ L̃
·
L̃
M
> −M and
∂y
L̃
L̃
∂x L̃
p < 0 and ∂ L̃ · y > 0. Note that ∂ L̃ · x < 0 because of
∂M L̃
· < 0 we have −1 < ∂∂xL̃ · L̃x < 0. Besides, this property
∂ L̃ M
∂y
1
L̃
· L̃ = − 1−M
· ∂M
·M
< 1. The other cells of the table
∂ L̃ y
∂ L̃
·
related to output, consumption and price are filled evidently by same reasoning.
To get equilibrium number of firms we exploit the following implicit dependence of markup M from the market
size L:
N F = LM (L)E(p(L), N ).
Differentiating it as implicit function we get
∂N L̃
∂M L̃
∂E p ∂p L̃ ∂E N ∂N L̃
·
=1+
·
+
· ·
· +
·
·
·
⇒
∂p E ∂ L̃ p
∂N E ∂ L̃ N
∂ L̃ N
∂ L̃ M
∂N L̃
∂E N
∂E p
M
∂M L̃
· (1 −
· )=1+ 1+
· ·
·
·
.
N
∂N
E
∂p
E
1
−
M
∂ L̃
∂ L̃ M
Now, if 1 >
∂E
∂N
·
N
E
and 1 >
∂E
∂p
·
p
E
> 0, then
∂N L̃
∂E N
∂E p
M
∂M L̃
∂E p
M
∂R p
· (1−
· ) = 1+ 1 +
· ·
·
·
≥ 1+ 1 +
· ·
·(M −1) = M (1−
· )
∂N E
∂p E 1 − M
∂p E 1 − M
∂p R
∂ L̃ N
∂ L̃ M
that is
> 0. Thus, under assumptions taken, the necessary and sufficient condition for pro (anti)-competitive
effect
∂N L̃
·
∂ L̃ N
0
is ru >
0 (ru0 < 0). 7.5
Impact of price and diversity on expenditures for varieties in two-sector case
Assume, like in many papers, that our economy have two sectors: the diversified industry producing many varieties
of “machinery”, and usual competitive agriculture producing homogeneous “food”. Labor is mobile among sectors
and wage normalized as previously to w = 1. Productivity in agriculture is also normalized to 1, so eqilibrium price
of “food” is pa = 1.
The consumer’s problem with constant income E takes the form
ˆ
U(
N
u(x(s))ds, a) → max
0
34
ˆ
N
p(s)x(s)ds + pa a ≤ E.
0
Upper-level utility function U here determines complimentarity/substitution between the two sectors. One can
persept the consumer’s choice as two-level one: first she allocate her budget between food and machinery, and
then specify spendings on each variety in the diversified sector. The lower-level problem with given expenditure for
machinery magnitude Em takes the form, including the Lagrange multiplier λ from the uper-level problem:
ˆ
N
u(x(s))ds → max
0
ˆ
N
p(s)x(s)ds ≤ Em λ
0
Denote the optimal value of the objective function u as ν(p, Em , N ). Then the upper level can be expressed as
U (ν(p, Em , N ), Ea ) → max
Em + Ea ≤ E ⇔
U (ν(p, Em , N ), E − Em ) → max
The upper level maximization is explained already above. Introducing the second sector brings a new economic
force or tendency: substitution of manufacturing goods in spendings by “food” under changing varieties’ number or
prices. To clear this question, look on the expenditures E(p, N ) on manufacture under changing (p, N ). The FOC
are
0
U10 (ν(p, Em , N ), E − Em )νR
(p, Em , N ) = U20 (ν(p, Em , N ), E − Em ).
The second-order conditions are
00
0
U11
νE
2
00 0
00 0
00
00
− U12
νE − U21
νE + U22
+ U10 νEE
≤ 0.
Here U (ν(p, p∗ , Em ), E − Em ) is concave w.r.t. Em when both upper- and lower-level functions are concave.
Lemma 2 Let upper- and lower-level functions U, u are concave, then
00
0
U11
νE
2
00 0
00 0
00
00
− U12
νE − U21
νE + U22
+ U10 νEE
≤ 0.
Proof. The expression
00
0
U11
νE
2
0
00
00
00
− U12
+ U21
νE + U22
0 . The discriminant is
is a polynom w.r.t. νE
00
00
U12
+ U21
2
00 00
00
− 4U11
U22 = 4 U12
2
00 00
00 00
00 00
− 4U11
U22 = 4 U12
U21 − U11
U22
00 < 0, we have U 00 (ν 0 )2 − (U 00 + U 00 ) ν 0 + U 00 < 0.
and concavity makes this experssion negative. Based on U22
11 E
12
21 E
22
´N
0
00 , we find ν 00 . From FOC of lower-level optimization, u0 (x ) = λp . Using the budget
To see the sign of U10 νEE
s
s
EE
ps xs ds = E we get
´N
λ=
0
νR
=
0
u0 (xs )xs ds
> 0.
E
35
Now find
∂λ
00
= νEE
=
∂E
´N
0
s
(u00 (xs )xs + u0 (xs )) ∂x
∂E ds
−
E
´N
∂λ
= ´N
∂E
0
0
u0 (xs )xs ds
=
E2
λ
u0 (xs )
u00 (xs ) ps ds
∂λ
∂R
´N
0
(u00 (xs )xs +u0 (xs ))
ps ds
u00 (xs )
E
−
λ
⇔
E
<0
00 < 0, so negative is also the whole expression U 00 (ν 0 )2 − U 00 ν 0 − U 00 ν 0 + U 00 + U 0 ν 00
Therefore νEE
11 E
12 E
21 E
22
1 EE that means
concavity of U (ν(p, Em , N ), E − Em ) w.r.t. Em . This proposition justifies validity of using FOC further. Now find elasticity of expenditures w.r.t. prices and
varieties’ number.
Lemma 3 Equilibrium elasticity of expenditures Rm w.r.t. varieties’ number N is
00 ν 0 ν + U 00 (ν + ν 0 E ) − E U 00
−U11
∂Em N
m 22
21
E
E m
.
−1= ·
2
∂N Em
00 ν 0
00 ν 0 − U 00 ν 0 + U 00 + U 0 ν 00
U11
−
U
1 EE Em
22
21 E
12 E
E
Proof. Hereafter the second derivative is denoted SOC =
d2 U
2 .
dEm
(17)
Looking on the FOC as on implicit function R(N )
we differentiate it and get
00 − U 00 ν 0
00 )ν 0 + U 0 ν 00
U 00 ν 0 ν 0 + U10 νEN
(U 00 ν 0 − U21
∂Em
21 N
1 EN
N
= − 11 E N
= − 11 E
.
∂N
SOC
SOC
Using it we have
00 ν 0 − U 00 )ν 0 + U 0 ν 00
(U11
∂Em N
N
1 EN
21 N
E
·
−1=−
·
−1=
2
00 ν 0
00 ν 0 − U 00 ν 0 + U 00 + U 0 ν 00
∂N Em
Em
U11
−
U
12 E
21 E
22
1 EE
E
00 ν 0 N ν 0 + E (ν 0 )2 + U 00 (N ν 0 + 2ν 0 E ) − U 0 (N ν 00 + E ν 00 ) − E U 00
−U11
m EE
m 22
m E
21
1
N
E m
EN
E
N
.
00 ν 0 2 − U 00 ν 0 − U 00 ν 0 + U 00 + U 0 ν 00
E
U11
m
22
1 EE
21 E
12 E
E
As previously, the denominator is negetive due to second order condition. Before studying the numerator sign note
that under normalizing u(0) = 0, the function
ν(p, Em , N ) = N u(
Em
)
pN
0 =
is homogeneous of degree 0 w.r.t. (p, Em ) of degree 1 w.r.t. (N, Em ), while its derivative νE
neous of degree 0 w.r.t. (Em , N ) of degree -1 w.r.t. (p, Em ).
Therefore
00
0
0
0 2
00 0
0
0
00 0
−U11
νE
N νN
+ Em νE
= −U11
νE N νN
+ Em νE
= −U11
νE ν
00
00
−U10 N νEN
+ Em νEE
=0
00
0
0
00
0
Em
U21
N νN
+ 2νE
Em = U21
ν + νE
that allows to express elasticities as
00 ν 0 ν + U 00 (ν + ν 0 E ) − E U 00
−U11
∂Em N
m 22
21
E
E m
·
−1= ,
2
∂N Em
00
0
00
0
00
0
00
00
U11 νE − U12 νE − U21 νE + U22 + U10 νEE
Em
36
m)
u0 ( E
pN
p
is homoge-
as we needed. Lemma 4 Equilibrium elasticity of expenditure on varieties w.r.t. price level p is
0 + U 00 E ν 0 − E U 00
U10 νE
∂Em p
m 22
21 m E
.
−1=
·
2
∂p Em
00
00 ν 0 + U 00
0
0
0
00
00
Em U11 νE − U12 νE + U1 νEE − U21
22
E
(18)
Proof. Let us study impact of price level on the expenditure. Differentiating FOC as in previous proposition, we
get
00 0 0
00
0
U11
νp νE + U11
νE
2 ∂Em
00 0 ∂Em
00
00 ∂Em
− U12
νE
+ U10 νEp
+ U10 νEE
=
∂p
∂p
∂p
00 0
00 0
U21
νp + U21
νE
∂Em
00 ∂Em
− U22
∂p
∂p
00 ν 0 ν 0 − U 0 ν 00 + U 00 ν 0
−U11
∂Em
p E
1 Ep
21 p
=
2
00
0
00
0
0
00
00 ν 0 + U 00
∂p
U11 νE − U12 νE + U1 νEE − U21
22
E
As previously, the elasticity of expenditure w.r.t. price is expressed as
00 ν 0 ν 0 − U 0 ν 00 + U 00 ν 0
−U11
p
∂Em p
p E
1 Ep
21 p
·
·
−1=
−1=
2
00
0
00
00
0
0
00
0
00
∂p Em
E
m
U11 νE − U12 νE + U1 νEE − U21 νE + U22
00
0
00
0
00 pν 0 ν 0 + E (ν 0 )2 − U 0 pν 00 + E ν 00
−U11
m EE + U21 pνp + 2Em νE − Em U22
m E
p E
1
Ep
=
.
00 ν 0 2 − U 00 ν 0 + U 0 ν 00 − U 00 ν 0 + U 00
Em U11
12 E
1 EE
21 E
22
E
Again, SOC
0-homogeneity of νestimates
denominator. As to the numerator,
the first summand
give the negative
2
00
0
0
0
0
00 + E ν 00
00
0
0
0
= −U11 νE pνp + Em νE = 0. The second summand is −U1 pνEp
=
as −U11 pνp νE + Em (νE )
m EE
0
0
0
00
0
0
00
0
U1 νE > 0, because of homogeneity of νE . The third summand becomes U21 pνp + 2Em νE = U21 Em νE . Therefore
0 + U 00 E ν 0 − E U 00
U10 νE
∂Em p
m 22
21 m E
.
·
−1=
2
∂p Em
00
0
00
0
0
00
00 ν 0 + U 00
Em U11 νE − U12 νE + U1 νEE − U21
22
E
00 ≥ 0 and
Lemma 5 Let U21
∂Em
∂p
·
p
Em
> 0, then
0<
00 ≥ 0 and
Proof. From U21
∂Em
∂p
·
p
Em
∂Em p
·
<1
∂p Em
> 0 we have
0<
Further, from
∂Em
∂p
·
p
Em
∂Em N
·
< 1.
∂N Em
∂Em p
·
< 1.
∂p Em
> 0, and negativity of the denominator (ensured in Lemma 3), we have
00 0 0
00
00 0
− U21
νp > 0.
U11
νp νE + U10 νEp
00 use
To find the sign of νEp
0
νE
=
m
u0 ( E
Np )
p
37
> 0,
to see
00
νEp
=−
m
Em u00 ( E
Np )
N p3
Analogeously, νp0 = −
m)
Em u0 ( E
Np
2
p
−
m
u0 ( E
Np )
p2
m
Em u00 ( E
Np )
=−
N p3
−
m
u0 ( E
Np )
p2
=−
m
u0 ( E
Np )
p2
Em
1 − ru (
) < 0.
Np
00 ν 0 − U 00 ) ν 0 > −U 0 ν 00 > 0, hence
< 0. Therefore we have (U11
21 p
1 Ep
E
00 0
00
U11
νE − U21
< 0.
This, in turn, entails
∂Em N
< 1.
·
∂N Em
Lemma 6 Elasticity of expenditures on the diversified goods is positive (i.e.
Proof. Note that νp0 = −
00 (X, Y )X
U11
U 00 (X, Y )Y
− 210
0
U1 (X, Y )
U2 (X, Y )
v 0 ( NEp )E
,
p2
0 =
νE
v 0 ( NEp )
p
∂Em
∂p
·
p
Em
> 0) if
v 0 (x)x
v 00 (x)x
+1+ 0
<0
v(x)
v (x)
00 = −
and νEp
v 0 ( NEp )
p2
−
v 00 ( NEp )E
.
N p3
Then we can modify expression
00 ν 0 ν 0 − U 0 ν 00 + U 00 ν 0
−U11
∂Em
p E
1 Ep
21 p
=
2
00
0
00
0
0
00
00 ν 0 + U 00
∂p
U11 νE − U12 νE + U1 νEE − U21
22
E
to get
00 ν 0 ν 0 + U 0 ν 00 − U 00 ν 0
U11
∂Em
p E
21 p
1 Ep
=
=
2
∂p
00
0
00
0
0
00
00
0
00
− U11 νE − U12 νE + U1 νEE − U21 νE + U22
=
The sign of
∂Em
∂p
00
− U11
00
00 ν 0 − U 0 νEp + U 00
−νp0 −U11
0
1 νp
21
E
.
2
0
00
0
0
00
00
0
00
νE − U12 νE + U1 νEE − U21 νE + U22
00
00 ν 0 − U 0 νEp + U 00 . The latter expression amounts to
coincides with the sign of −U11
0
1 ν
21
E
p
ν 00
0 Ep
00 0
−U11
νE − U1
U0
=− 1
E
νp0
00
00
+ U21
= −U11
00 N v( E )
U11
Np
U10
−
v 0 ( NEp )
p
00 N v( E )
U21
Np
−
0
− U1
!
v 0 ( NEp )
v 00 ( NEp )E
−
2
p
N p3
E
0
v(
)E
− Np2p
v 0 ( NEp ) NEp
U20
v( NEp )
+1+
00
+ U21
=
v 00 ( NEp )E
!
v 0 ( NEp )N p
.
U0
Using − E1 < 0 it follows that
∂Em
> 0 ⇐⇒
∂p
00 N v( E )
U11
Np
U10
−
00 N v( E )
U21
Np
U20
The lemma is proved.
38
!
v 0 ( NEp ) NEp
v( NEp )
+1+
v 00 ( NEp )E
v 0 ( NEp )N p
< 0.
7.6
Proofs for multiple equilibria
Proof of Proposition 5 and its Corollaries. For all these three statements we exploit the analogy between a
standard quasi-linear monopolist facing the inverse-demand function P (x) = u0 (x), and a Dixit-Stiglitz monopolist
facing the per-consumer inverse-demand Pλ (x) = u0 (x)/λ, as illustrated by Fig. 6. Here λ denotes the Lagrange
multiplier of the consumer’s budget constraint at the equilibrium. In other words, the standard monopolist maximizes profit function π M (x, c̃) = x(u0 (x) − c̃) with FOC u00 (x)x + u0 (x) = c̃, while the Dixit-Stiglitz monopolist
maximizes very similar per-consumer net-of-fixed-cost profit function πs (xs , c, λ) = xs (u0 (xs )/λ − c) = π M (x, cλ)/λ
with similar FOC u00 (xs )xs + u0 (xs ) = cλ. The latter yields the same solution as if a standard monopolist were
facing inverse demand P but having cost c̃ = cλ. Though the multiplier λ = λ(N, p) is dependent upon the number
of varieties and on their prices, but this λ takes the same value at all (symmetric or asymmetric) equilibria, as we
show below. This similarity is the essence of our proof.
(i) Sufficiency of Assumption MR for asymmetric equilibrium existence, for Corollary 2.
Let us use the two global maxima of profit x̌ 6= x̂ and cost c̃ from the standard monopolist’s optimum suggested by Assumption MR, to construct the needed parameters c, F, L, N1 , N2 for Dixit-Stiglitz economy with some
asymmetric equilibrium and same utility u(.). We are constructing the specific kind of asymmetric equilibrium
described in Corollary 2. Namely, we take only two quantities x̌ > x̂ (sign is taken without loss of generality) and
two multitudes Nx̌ , Nx̂ of firms producing these quantities. So, the total number of firms will be Nx̌ + Nx̂ . We just
check equilibrium equations holding for such point (x̌, x̂, Nx̌ , Nx̂ ).
Denoting as λ̌ = λ̂ the equilibrium’s Lagrange multiplier for the budget constraint, the asymmetric equilibrium
equations are:31
u00 (x̌)x̌ + u0 (x̌) = c̃ = λ̌c,
u00 (x̂)x̂ + u0 (x̂) = c̃ = λ̂c
(19)
(FOC of both producers’ types),
Nx̌ x̌p̌ + Nx̂ x̂p̂ = 1, u0 (x̌) = λ̌p̌, u0 (x̂) = λ̂p̂
(consumer’s budget and FOC),
(p̌ − c)x̌ = F/L = (p̂ − c)x̂
(producers’ zero-profit condition),
(cLx̌ + F )Nx̌ + (cLx̌ + F )Nx̂ = L
(labor balance, that can be derived from the latter two equations).
The symmetric-equilibria equations are similar, only equality λ̌ = λ̂ is not guaranteed ad hoc but derived, and
either Nx̌ = 0 or Nx̂ = 0.
Now we reformulate this symmetric-equilibrium system using our notation r(.) and excluding λ̌:
p̌ = c/(1 − ru (x̌)),
p̂ = c/(1 − ru (x̂)),
x̌(p̌ − c)
x̌ru (x̌)
F
x̂ru (x̂)
=
=
=
,
c
1 − ru (x̌)
cL
1 − ru (x̂)
31
Additional condition for profit concavity follows from the global maxima in Assumption MR.
39
(20)
(21)
1 = Nx̌
x̌c
x̂c
+ Nx̂
.
1 − ru (x̌)
1 − ru (x̂)
(22)
Now, for our given quantities x̌, x̂ and any fixed F/L > 0 we can find cost c satisfying two equations (21).
Both can be satisfied simultaneously, because by Assumption MR profits are equal at two standard monopolist’s
global optima: π M (x̌, c̃) = x̌(u0 (x̌) − c̃) = π M (x̂, c̃) = x(u0 (x̂) − c̃) > 0. So, at both problems of the Dixit-Stiglitz
monopolists, related per-consumer net-of-fixed-cost profits π̄ := πs (xs , c, λ̌) = π M (x̌, cλ̌)/λ̌ = π M (x̂, cλ̌)/λ̌ > 0 are
also equal when cλ̌ = c̃. Then, under special c =
F
L
·
1−ru (x̌)
x̌ru (x̌)
> 0, λ̌ = c̃/c this value π̄ > 0 is exactly sufficient to
cover the per-consumer fixed cost of both producers in the sense π̄ = F/L, that means both equations satisfied.
Further, using values c, λ̌ obtained, we calculate prices
p̌ = λ̌p̌M = c/(1 − ru (x̌)),
p̂ = λ̌p̂M = c/(1 − ru (x̂)),
from equations (20) (denoting here as p̂M ,p̌M the standard monopolist’s prices for low quantity x̂ and high quantity
x̌ respectively). Positivity of p̂M ,p̌M ensures positivity of p̂,p̌. What remains is to take any couple of multitudes
(Nx̌ , Nx̂ ) of big-quantity firms and small-quantity firms that satisfy the last equation (22). This pair can be chosen
non-uniquely, since any non-negative couple satisfying this equation is appropriate for the equilibrium. In particular
cases when Nx̌ = 0 or Nx̂ = 0 we get two symmetric equilibria satisfying all requirements. Thus we have built an
interval of asymmetric equilibria and two symmetric ones from Assumption MR.
(ii) Necessity of MR for asymmetric equilibrium existence just follows from using our logic in reverse direction.
Indeed, we take the equilibrium magnitudes (x̌, x̂, Nx̌ , Nx̂ ) satisfying equilibrium equations (20)-(22),(19) and construct cost c̃ = λ̌c > 0 for the standard monopolist’s problem. Obviously, this c̃ and same x̌, x̂ satisfy FOC and
SOC for two equivalent global maxima of standard monopolist’s problem.
(iii) Now consider necessity of MR for two symmetric equilibria existence. Each equilibrium (for xs = x̌ and for
xs = x̂) is characterized by the same system of equations as for asymmetric equilibrium, but for assuming one of
two types of producers absent; some Nxs = 0.
u00 (xs )xs + u0 (xs ) = λs c,
Nxs xs ps = 1, u0 (xs ) = λs ps ,
(ps − c)xs =
cxs ru (xs )
= F/L.
1 − ru (xs )
(23)
We should take two solutions (x̌, λ̌, Nx̌ ), ( x̂, λ̂, Nx̂ ) to these equations under parameters (c, F, L) and build c̃
suitable for two (equivalent to each other) standard-monopoly global maxima. The proof that such c̃ exists, is
similar to (ii). Indeed, equilibria quntities x̌, x̂ are the two roots of equation (23). Each gives the related price
ps , from the same equation. Then, taking some λ and c̃ = λc, p̂M = λp̂, p̌M = λp̌ we see that standard-monopoly
equations at both points x̌, x̂ are satisfied and both these solutions give the same profit π M = xs (pM
s − c̃) =
x̌(λp̌ − λc) = x̂(λp̂ − λc) = F/L under these costs.
What remains is ensuring that Lagrange multipliers λ̌,λ̂ are the same at equilibrium x̌ and at equilibrium x̂,
i.e., that equations
u00 (x̌)x̌ + u0 (x̌) = λ̌c,
u00 (x̂)x̂ + u0 (x̂) = λ̂c
have the same λ̌ = λ̂. Indeed, each (s-th) Dixit-Stiglitz monopolist maximizes w.r.t. xs the per-consumer profit
function πs (xs , c, λ) = xs (u0 (xs )/λ − c). Its optimal value can be denoted π ∗ (λ) = x∗ (λ)(u0 (x∗ (λ))/λ − c) , where
x∗ (λ) denotes the optimal x under λ. It decreases w.r.t. λ by the Envelope theorem, which says that total derivative
40
amounts to direct derivative
∂ ∗
∂λ π (λ)
= −x∗ u0 (x∗ )/λ2 < 0 while indirect effects are negligible. This monotonicity
is true even for non-differentiable optimal function and multiple local optima (which is our case) where expression
∂ ∗
∂λ π (λ)
is invalid, because expression x∗ u0 (x∗ )/λ > 0 with positive numerator and the function optimized decreases
w.r.t. λ. So, its optimal value also decreases. Therefore there is unique value λ = λ̂ = λ̌ such that global maximum
of this function equals F/L. Thus we have proved as a by-product that all equations studied in item (i) are satisfied
and our two symmetric equilibra are accompanied by the whole interval od asymmetric eqilibria.
(iv) The structure of asymmetric and symmetric equilibria is already derived.
(v) To see that the different coexisting equilibria can differ in the consumer’s utility, see Example 5.
This completes the proof of Proposition 5 and its Corralries.
Proof of Theorem 2.
(i) Unique equilibrium under monotone marginal revenue is guaranteed because it is equivalent to global concavity of the profit function. For item (ii) we need
Lemma MR. When Assumption MR holds in specific version: marginal revenue has ony one interval of increase,
there is unique level of cost c̄ yielding two global argmaxima x̂ < x̌ of profit, for other C 6= c̄ the argmaximum is
unique. Under small cost C < c̄ the biggest quantity x̌ brings the global maximum, and under big cost C > c̄ the
lowest quantity x̂ is optimal, and both local argmaxima x̂(C) < x̌(C) decrease w.r.t. C.
Proof of Lemma MR. Denoting xi = x̌(C) or xi = x̂(C), by the Envelope theorem, the objective-function
πi (C) = xi (u0 (xi ) − C) value at any argmaxima decreases w.r.t. C. Indeed, when a function decreases at any point,
its maximal value also decreases. Moreover, the big-quantity value π̂(C) = x̂(C)(u0 (x̂(C)) − C) decreases faster
than the small-quantity value π̌(C) = x̂(C)(u0 (x̂(C)) − C), for the same reason. Therefore these two function has
only one intersection c̄ and to the left from it (C < c̄) the big quantity is optimal, and reverse. The direction of
changes follows from Lemma 1, whose boundary condition on marginal revenue ξ(x, C) = xu00 (x)−u0 (x) is satisfied.
Q.E.D.
Let us consider the comparative statics in L̃ (dropping its accent) for any two roots: (x̂(L), λ̂(L)) and (x̌(L), λ̌(L)),
of system (21)-(20) of equilibrium equations. By Lemma MR, there is unique value of λ(L) bringing the same level
F/L of per-consumer net-of-fixed-cost profit at both roots, λ(L) is monotone decreasing, and small root (x̂(L), λ̂(L))
is valid global optimum for big λ (so, big L), and vice verse: big root (x̌(L), λ̌(L)) is valid for small L. This proves
uniqueness of L where both roots of basic equilibrium equation ξ(M, L) = 0 are valid as equilibria. Besides, it
proves the direction of “jump” in quantities: x̌ switches downward to x̂ when L passes L̄.
To ensure Corollary 1, we can just apply Lemma 1 to function ξ(., .) = ξ(M, L̃) : [0, 1] × R → R which is
continuous (because r is continuous) and satisfy border conditions ξ(0, L̃) ≥ 0, ξ(1, L̃) ≤ 0 ∀L̃ by Assumption 1
on r. It guarantees monotonicity for border roots, which coincide under uniqueness. The difference between the
pro-competitive (r0 (x) > 0) and anti-competitive case (r0 (x) < 0) lies only in the direction of influence of L̃ on
ξ(M, L̃), the results are opposite. 41