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, buthas too narrow use. We expand it now to a family of multisector models and to additional effects of comparativestatics. It turnes out that under market growth the price for varieties can go up or down, depending upon Arrow-Pratt measure of concavity of the utility function, does it decrease or increase. Welfare and number of firmsalso can increase or decrease. There can be asymmetric equilibria, multiple equilibria and related catastrophicshocks.

Introduction

Various models of monopolistic competition started from Chamberlin’s basic idea to model a firm – owner of abrand – as a price-maker, under free entry to the industry and both fixed and variable costs. This idea becamereally 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-elastisity-of-substitution (CES) class of utility functions. This novelty enabled these models to become a cornerstone, or amain 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 industrial-organization 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 asfollows: (1) increasing returns to scale in a firm, usually as a result of combining the fixed costs to start businessand 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 amonopolistic price-maker, but takes into account a specific demand for its brand, infuenced by the competition ofother brands; (3) the demand curve is a result of maximization of a specific utility function reflecting “preferencefor many varieties” excibited by the representative consumer; (4) the number of firms is big enough for a firm tobehave non-strategically, ignoring its own infuence on the industry/economy as a whole, i.e., ignoring the influenceon other prices within and outside the industry; (5) free entry into the industry drives all profits to zero (in spiteof 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.

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Quite standard is to model the representative consumer’s preferences by a superposition of a high-level utilityfunction U(.) operating on two or several “industries” (aggregated goods) and a low-level function u(.) operating onvarieties 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”, “transporta-tion”, and determines the budget share allocated for each aggregate (each industry). In contrast, the low-levelutility function reflects sort of substitution/complimentarity in consumption amongst varieties within such indus-try. Typically, in hundreds of papers, the high-level function is a Cobb-Douglas one, while the low-level one isa CES-function, that means a power function over the sum of power functions. The reason is technical: CES-function 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 quiteartificial, notably the “neutral-competitive effect.” It means that in the short-term equilibrium (before the freeentry annihilates profits) the increasing number of firms in the industry does not influence prices, surprizingly foreconomists. 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 increasingreturns. 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. Unsatisfactionwith limitations of CES models and related neutrality is expressed by the leading economic-geography theoristsin [13] and [10]. They point out the realism of the pro-competitive effect, supported also by some empiricalpapers. In particular, Campbell and Hopenhayn [8] show that a bigger market has smaller mark-up and biger totalconsumption. On the other hand, anti-competitive effect also can be realistic in some specific circumstances as weargue 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 effectempirically discovered ([9, 15, 20]). We are unaware of any mentioning of such effect possibility in monopolistic-competition 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 theoryshould meet economic intuition.

We should say that there are some rare papers fighting this gap between theory and intuition through steppingaside 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 allquasi-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 closed-form solutions of equilibria: exponential utilities, somewhat similar to CES, also resulting in the pro-competitiveeffect.1 Another study without CES is Anderson [3] with a quasi-linar function U , exhibiting preference for varietiesitself, without lower-level function u. Somewhat similar is Benassy [5] who introduced some definition of thefunctional property of the upper function U named “preference for varieties” and derived the welfare consequences

1The propeties of the short-term equilibria studied in the latter two papers with specific functions are similar: the price and themark-up are decreasing with the number of firms, and consumption of each variety decreases; gross consumption of all varieties incresesas they become more numerous. The latter paper also makes a step towards deriving the equilibria effects without assuming the specificfunctional form, but ends with some specific fuction.

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from increasing the number of varieties (firms) from this assumption. Besides, Das [11] study linear utility u andhomothetic U .

Up to our knowledge, there are no other attempts to derive the equilibria effects without assuming the specificfunctional forms. The need for such study stems from the question: how much the important predictions of allmonopolistic-competition models and related theories are robust against varying the specific assumptions? Aren’tthey 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-sectormonopolistic-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., theincreasing Arrow-Pratt’s “relative risk aversion” ru(x) := −xu′′

u′ = −1/εDp characterising the utility function u.This assumption is shown sufficient for pro-competitive effect. Krugman’s graphical proof is very intuitive, but itneeds adding mathematical rigor, because some conditions on u, ensuring profit concavity and correct derivationof 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 graphicalreasoning it was not clear how to extend his method of building comparative statics to many sectors or factorsand to other extensions of the model, demanded by theorists. Maybe, these limitations explain why this seminalfruitfull approach was not extended and generalized (as we do). Surprizingly, the named result on pro-competitiveeffect is rarely mentioned, being unmentioned even in the Helpman-Krugman’s book [16]! In a survay dedicated to30-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, mostwriters have used the CES specification in (2), with its unsatisfactory implications that firm size is fixed by tastesand technology, and all adjustments in industry size (required for example by changes in trade policy) come aboutthrough changes in the number of firms.”

In contrast, this paper makes the Krugman’s approach more tractable and extends it to quite general multi-sector Dixit-Stiglitz models, restricting only the number of production factors (one). The technical trick for suchgeneralization is inroducing the expenditure-for-varieties function E that relplaces the upper-level utility U as abasic primitive, also reflecting preferences among the aggregated goods. We rigorously derive conditions on bothfunctions E(.), u(.), necessary and sufficient for pro-competitive, neutral- and anti-competitive effects when themarket 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, theequilibrium price and quantity of each variety can be found solely from one exogeneous parameter, “relative sizeof 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 theupper-level utility function U and other sector(s) of economy. It turns out, that such convenient and economicallyimportant independence between diversified/non-diversified sectors (known for CES models), stems only from thetwo-level construction of utility, not being an artefact of CES assumption like the neutral-competitive feature (asexplained in Section 4).

In Section 4, Theorem 1 states comprehensive local comparative-statics: changes of price, quantity and varieties’number w.r.t. the relative size of the market L. The latter is number of population divided by fixed cost andmultiplied by the variable cost of production. Respectively, impact of any combination of these parameters ontothe 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 concavityof the utility function u(.) , respectively (see Table 1). In the first class of markets price decreases but a firm’s

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production increases in market size, while in the anti-competitive class the impact is the opposite, neutrality beingrevealed only for CES functions. An individual variety’s consumption by a household decreases in all cases, as wellas the varieties’ number, the latter requiring additional reasonable condition on upper-level utility U expressingsome 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 isexplained in special motivating Section 2 through related examples and finaly discussed after the theorems. “Priceparadox” means that equilibrium price of each variety grows with the relative size of the market (say, with populationsize) though the number of competitors increases! Example 2 shows that such strange-behaving equilibria reallyexist, 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 changesthe 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 (r′u(x) > 0) always ensures improving welfare of a consumer w.r.t. thesize of the market. However, we prove it to be true only under additional normalization condition u(0) = 0 notmentioned by Krugman. When this condition is violated, we have have found a counterexample due to “envyparadox,” explained in Subsection 5.2, that may seem somewhat artificial (new varieties can bring dissatisfactionof consumers not buying these varieties when u(0) < 0). In contrast, another, “price-and-utility paradox” of thiskind (see Subsection 5.1) occurs in many anti-competitive markets and looks more natural. It means that eachconsumer may suffer from the population (and size of the market) growth: ∂U∗

∂L < 0. Again, the related utilityfunction 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 conditionsfor equilibria uniqueness/multiplicuty. It turns out that multiple equilibria can occur if and only if the marginal-revenue function xu′′(x) + u′(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 ofequilibria set in this case. On this basis, Theorem 2 states the global comparative statics of price, quantity andnumber of firms mappings for the case of multiple symmetric/asymmetric equilibria. Multiplicity can occur onlyat one value of the market size L under assumptions of Proposition 5, otherwise, and at other points, equilibria aresymetric and unique and generally behave like in Theorem 1. At the unique point of discontinuity of each mappingthere occur catastrophic changes in prices, quantities and number of firms in response to smooth changes of therelative size of the market, demonstrated by the extended Example 5. Such abrupt changes, not contemplatedpreviously in monopolistic competition, may be interpreted as possiblity of revolutionary events in the marketunder 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-Stiglitzmodel of one-sector one-production-factor economy, as in ([16]), but with non-specific utility function, like in Krug-man (1979).

There is a continuum, i.e., an interval [0, N ] of firms (N > 0). They produce as much as [0, N ] varieties or

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brands of a diversified commodity, say, many kinds of clothes. Each firm, identified by index s ∈ [0, N ], producesonly one brand (see Intro for basic assumptions). There are L identical consumers, and a consumer’s index isdropped. Each of them chooses an infinite-dimensional consumption vector x = xs∈[0,N ], which is an element of thespace Ξ[0, N ] of all non-negative Lebesgue-integrable functions on interval [0, N ]. It means solving the followingutility-maximization program:

ˆ N

0u(xs)ds → max

x(.)∈Ξ[0,N ]w.r.t.

ˆ N

0psxsds ≤ ε.

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 ε > 0 denotes theexpenditure for all varieties. In the basic model ε equals the consumer’s income, but generaly it can vary. Allconsumers are identical and have 1 unit of labor (or any other resource) inelastically supplied to the market inexchange for wages w > 0. The consumers equally own the firms, and the expenditure for all varieties satisfiesε ≤ w + 1

L

´ N0 πsds = w, because anyway zero profit π = 0 results at long-run equilibrium.

We denote by xs(p, ε, N) = xs : Ξ[0, N ] × R2+ → R+ the s-th component of a solution to the above problem,

and by x = xs∈[0,N ] whole consumer’s solution, i.e., demand of any consumer for s-th variety. This xs dependsupon infinite-dimensional price-vector p, budget ε 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 inmind the same demand function: xs(p, ε, N) ≡ xs(p, N) ≡ xs(p).

On the supply side, each (s-th) of the identical producers knows her gross demand function xΣs (.) =

´ L0 xsds =

Lxs, and solves her monopolistic profit-maximization program:

psLxs(ps, p-s, ε, N)− cwLxs(ps, p-s, ε, N)− wF → maxps∈R+

.

Here, as usual, p-s denotes all other prices besides ps. The variable cost c > 0 and the fixed cost of starting businessF > 0 are measured in labor under normalized wages w = 1, so totally the producer spends cLxs +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 withadditional sectors of economy, described implicitely through some expenditure function E(.) that replaces constantε in the consumer’s problem. The derivation of this function from other primitives U, c, F and the extensive formof the general model are in Appendix. Instead of this derivation, in the next subsection we directly introduce thereduced form of the multisector model. It seemingly describes only one diversified sector and looks like partial-equilibrium 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 donethrough making the expenditure on varieties ε dependent upon current variables in the form ε = E(p, N). Thisfunction 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.2Representing the argument s of demand/price functions as an index: xs ≡ x(s), ps ≡ p(s) economizes many useless parentheses.

Another motivation for such notation is possible direct application of the below analysis to discrete-varieties models: just understandthe integral as a sum:

´ N

0u(xs)ds =

∑N0 u(xs), and apply all our reasoning and results.

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To comprehend the nature of the expenditure function E, see the extensive form of the general model inAppendix. 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 by-product finds the bundle of auxiliary variables (E1, E2, ..., En) denoting expenditures for each sector, dependentupon (p1, N1, p2, N2, ...). Our approach to comparative statics of equilibria is based on preliminary reducing thesystem of related equilibrium equations to only two equations with two variables p1, N1. In doing so, we get rid offp2, N2, ..., pn, Nn expressing them through p1, N1. Thereby, all arguments of the expenditure function E1 of the firstsector become expressed through vector (p1, N1), which takes into account equilibrium response of other sectorsto this (first) sector. Thus we get the expenditure function E1(p1, N1) = E1(p1, N1, p2(p1, N1), ...pn(p1, N1)) usedfurther 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 N

0u(xs)ds → max

x(.),εw.r.t.

ˆ N

0psxsds ≤ ε = E(p, N).

This formulation differs from the basic model only in the upper bound E, which is not a constant from thecomparative statics viewpoint; it depends now upon prices and varieties’ number. In contrast, for deriving FOCthis ε = E(p, N) can be percepted as a fixed magnitude. Therefore, like in the basic model, the resulting individualdemand function xs for s-th variety includes constant ε = E(p, N) as an exogeneous argument:

xs = xs(p, ε, N) = xs(ps, p-s, ε, N).

From now on we look on the expenditure function E(.) in sector #1 as on a primitive. For our goal (expansion ofthe Krugman’s method to multi-sector models) it seems more practical to study comparative statics of the reducedmodel, rather than extensive one. It turnes out that it is sufficient to impose reasonable restrictions on this functionE for describing sector #1 within comparative statics of the whole economy, without complicated direct derivationof 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, theupper level utility function must be a Cobb-Douglas one: U(X1, ..., Xn) =

∑m1j αjLog(Xj), Xj =

´uj(zj) and the

lower-level function must be CES: uj(zj) = zβj

j .4 In this simple case the budget share allocated to each sector isjust a constant Ej(pj , Nj) ≡ εj = αj , that looks restrictive.

In contrast, we allow for more general upper- and lower-level utility functions, becoming now suitable foranalysing many questions of international trade and economic geography without any explicit demand function.Generally, studying the equilibrium equations does not require this explicit function.

3One can also think of this consumer’s model in terms of two-stage optimization. First, a consumer chooses her general budgetstrategy: 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-programfor this sector, similarly to the basic model.

4Only combination of these two restrictive assumptions is sufficient for constant ε, otherwise the budget share can vary. This fact isnot noticed in Helpman and Krugman ([16], p. 190) where the constant-share assumption is just added to seemingly-general functionsanalyzed.

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1.3 Reduced general model: producer and equilibrium

The producer’s profit-maximization program remains as in the basic model. Knowing the individual demandfunction xs(.) for her variety and the competitors’ price vector p-s, the s-th producer chooses her price ps ∈ R+ as

psLxs(ps, p-s, ε, N)− cwLxs(ps, p-s, ε, N)− wF → maxps∈R+

. (1)

Here L > 0 denotes the number of identical consumers in the economy, wage is normalized as w = 1. Fixed costF > 0 and marginal cost c > 0 are measured in labor, and total cost is cLxs +F . The producer percepts the budgetε for this sector, other prices and scopes as constants, because her own influence on the sector is infinitesimallysmall. This idea motivates the following notion for (short-term) sub-equilibrium.

Definition 1. A symmetric sub-equilibrium of a diversified sector under given N, ε, 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:xis = x∗(N, ε, L) ∈ R+ ∀i, s, ps = p∗(N, ε, L) ∈ R+ ∀s.5

Thus we have defined a symmetric sub-equilibrium functions p∗(N, ε, L) ∈ R+, x∗(N, ε, L) ∈ R+. Now, todefine the long-term symmetric equilibrium, we take into account that our expenditures function E(p, N) shoulddetermine the equilibrium expenditure magnitude ε, and free entry should determine the equilibrium number ofcompetitors N through the zero-profit condition. Therefore, the equilibrium vector (p, x, N , ε) must satisfy fourequations:

p∗(N , ε)Lx∗(N , ε) =(cLx∗(N , ε) + F

), (2)

ε = E(p, N), p = p∗(N , ε), x = x∗(N , ε). (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) = Lw

N.

These considerations motivateDefinition 3. The quadruple (p, x, N , ε) ∈ 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 reducedmodel of economy.7

To guarantee nice tractable demand properties and existence of the producer’s solution, we maintain throughoutclassical conditions on utility and a reasonable additional restrictions on expenditure and on price elasticity εPx,formulated as

Assumption 1. Lower-level utility function u : R+ → R+ is neoclassical one (strictly concave, strictlyincreasing), three times differentiable, normalized as u(0) = 0, and −1/εPx(x) = −xu′′(x)/u′(x) < 1 at x = 0.The expenditure for diversities E = E(p, N) is two times differentiable.8

5Following the tradition, we focus here on symmetric equilibria only, though asymmetry is possible and we touch it in another paper.6Here 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.7When 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/factorsamong the countries. The same model describes autarky also.

8Condition −xu′′(x)/u′(x) < 1 on the demand, in application to a standard monopoly, would mean just positivity of the marginal

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The above reduced model of the whole economy is our main object of study below, though for illustrationswe use also the one-sector model, that means E ≡ 1. We present some of such examples now, before the generaltheorems, 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 equi-librium: when the size L of the market increases, either both number of firms (varieties) and the price increase, orboth decrease, i.e., ∂N

∂L∂p∂L > 0. In contrast, pro-competitive entrance effect means that price and varieties change

in the opposite direction: ∂N∂L

∂p∂L < 0, while neutral-competitive effect means no price changes: ∂p

∂L = 0.9

So far, the literature focused mostly on CES utility functions, and related neutral-competitive effect. Neutralitymeans that price of each variety is not influenced by the size of the market and the number of competitors, whichseems 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 bestof our knowledge.

Following Behrens-Murata [6], Krugman and others unsatisfied with neutrality, we step aside from CES as-sumption, and demonstrate in this section: (1) one more, in addition to Behrens-Murata’s one, quite natural utilityfunction 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 negativeprice reactions to population growth.

As to neutral competitive-effect, power functions like u(z) = zα, 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-formderivations. 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 generalargument z + a, and use the below formulae in various models with a diversified sector.

revenue at x = 0, which is a weak restriction.9Thereby, 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∗

∂N> 0. This effect or “price paradox” is not mentioned so far in the Dixit-Stiglitz model, but it

is reported in several other oligopolistic settings, briefly reviewed in Rosenthal [21]. In these papers mainly increasing competition leadsto 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 oligopolistsface two kinds of markets: “domestic” one, where she has a monopoly power, and a “third-country” market, where all compete in pricesin Betranian way, so a unique price emerges in the third country. Besides, somehow each firm is forced to sell for the same price athome and outside (that can be realistic when “domestic market” means loyal customers of a brand, while “third-country” means newor 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.

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To derive an equilibrium, see that the consumers’ maximization entails

xs + 1xk + 1

=pk

ps⇒ xs =

pk

ps(xk + 1)− 1.

Substitute this into the budget constraint to get the demand function:10

ˆ N

0psxsds = 1 ⇔

ˆ N

0ps

(pk

ps(xk + 1)− 1

)ds = 1 ⇒ xk(p, N) =

1pkN

+

´ N0 psds

pkN− 1.

Then the producer’s optimization problem becomes

pkLxk(p, N)− cLxk(p, N)− F =L

N+ L

´ N0 psds

N− Lpk −

Lc

pkN− cL

´ N0 psds

pkN+ cL− F → max

pk

.

The related FOC give us the producer’s reaction to other prices and to the number of competitors:11

−1 +c

p2kN

+ c

´ N0 psds

p2kN

= 0 ⇒ pk =

√c

N+ c

´ N0 psds

N.

The equilibrum’s symmetry is obvious, and sub-equilibrium formula is derived as

P =√

c

N+ cP ⇔ NP 2 −NcP − c = 0 ⇒ p∗(N) =

c +√

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:

N(F + cLx∗(N)) = L ⇔ N =L

F +√

LFc.

The resulting comparative statics w.r.t. market size L is shown in Fig.1. Interestingly, function N = N(L) isconcave 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 seethat this innocent modification of so popular CES utility function

√x brings anti-competitive effect everywhere!

Indeed, the consumer’s FOC yields a relation

12√

xk+ 1

12√

xs+ 1

=pk

ps⇒ xs =

1(ps

pk

(1√xk

+ 2)− 2)2 .

10Remark that the demand function here turns out rather similar to one derived in Ottaviano, Tabuchi and Thisse [19] for quasi-linearutility.

11As to the producer’s second-order condition, − 2cLp3

kN− 2c

´N0 psds

p3k

N≤ 0, it is definitely true for the function taken, so our profit is

concave.

9

Figure 1: Comparative statics with pro-competitive effect: p(L), x(L), N(L),U(L) for utility u(z) = ln(z + 1).

Substituting this xs into the budget we get

ˆ N

0psxsds = 1 ⇔

ˆ N

0

ps(ps

pk

(1√xk

+ 2)− 2)2 ds = 1.

This expression cannot be resolved directly for xk, but we can link xk with p under the assumption of symmetricother-firms prices equal all to some P > 0, as follows

ˆ N

0

ps(ps

pk

(1√xk

+ 2)− 2)2 ds = 1 ⇔ pk(xk, P, N) = P

1 + 2√

xk√

xk(√

PN + 2)

Now we can solve the producer’s problem by maximizing her (obviously concave) profit function w.r.t. xk:

L

(P

√xk + 2xk

(√

PN + 2)− cxk

)− F → max

xk

⇒ xk =P 2

4(c(√

PN + 2)− 2P)2 .

Then k-th optimal response to other prices is

pk =2(c(√

PN + 2)− P)

√PN + 2

.

Any equilibrium is obviously symmetric here, so pk = P and we can derive the sub-equilibrium condition as

P =2(c(√

PN + 2)− P)

√PN + 2

⇔ P√

PN = 2(c(√

PN + 2)− 2P)

.

10

Figure 2: Comparative statics of anti-competitive paradox: p(L), x(L), N(L), U(L) for utility u(z) =√

z + z

Now from the labor-market balance we express PN and derive the equilibrium condition

N(F + cLx) = L ⇔ NF + cLNx = L ⇔ P

√PL− cL

F= 2c(

√PL− cL

F+ 2)− 4P.

In principle, this cubic equation w.r.t√

P can be resolved directly. But we prefer to present now the graphs ofcomparative 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) = za + 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? Itsnatural 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 classifyall utility functions bringing these effects. Now, to prepare tools for such general comparative statics, we shouldcharacterize demand/supply functions and formulate general equilibrium equations. In our reasoning, we exploitand develop the seminal Krugman’s technique based on Arrow-Pratt measure ru(z) = − z u′′(z)

u′(z) = −1/εPx(z) ofconcavity (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). Fromthe consumer’s FOC one can easily get

Proposition 1 The k-th demand elasticities w.r.t. each price are characterized as:

EXpkk =

pk∂xk

xk∂pk= − 1

ru(xk),

ps∂xk

xk∂ps= 0, where ru(z) = −z u′′(z)

u′(z). (4)

One can observe here, like in quasi-linear IO settings, that the demand elasticity EXpkk 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 isthat any measurable interval of other prices does influence k-th demand (see Appendix). We use these elasticitiesto analyse the k-th monopolist profit-maximization program:

πk = (pk − cw)Lxk(pk, p−k, Rm(pk, p−k, N), N)− wF → maxpk

.

Now we show conditions for this function concavity, the producer’s solution uniqueness and symmetry.

Proposition 2 (i) Condition 2 > ru′(xk) is necessary and sufficient for strict concavity of the marginal utility atpoint xk, and for strict concavity of the profit function at points satisfying FOC. (ii) If this upper bound 2 > ru′(xk)holds everywhere on R+, then profit is generally strictly concave and has a unique maximum. In this case anysub-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)

pk= − 1

∂xk∂pk

pkxk

= ru(xk) < 1.

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 IOwith 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 expres-sions by the labor balance (or the zero-profit condition). Then the symmetric-equilibrium price p, markup M andnumber N of firms can be found from the equations:

M :=p− c

p= ru(

E(p, N)pN

)

(F + cLE(p, N)

pN)N = E(p, N)L.

12The condition 2 > ru′(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 concavityand symmetry everywhere. It is because the demand function of a Dixit-Stiglitz industry differs in income-effect from demand of asimilar-utility quasi-concave sector. Only their elasticities are similar.

12

It is convenient to reformulate these equations in terms of crucial constant L called in this paper as relative size ofthe market :

L :=Lc

F. (5)

This L includes the number of consumers fixed and variable cost. Therefore our comparative statics in terms of L

captures altogether all possible shocks of population and technology, influencing M and N .The necessary conditions of equilibria in terms of L and M(L), N(L) follow.

Proposition 4 Each symmetric equilibrium of the general model satisfies the following equilibrium equations:13

M = ru(F

L · c· 1−M

M) = ru(

1−M

L ·M) (6)

N · c = E(c

1−M, N) ·M · L, (7)

First observation here is that the equations are quite simple, unexpectedly for a complicated model. Somewhatsimilar 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 inquasi-linear IO). Indeed, note that we can find markup M solely from the first equation. The markup yieldsequilibrium price p = c

1−M , consumption x = 1−MLM

and production xL = Fc ·

1−MM of each variety, expressed in

only one exogeneous constant L. The markup influences the number of firms N through the second equation butis not inflenced by it. Thereby, price and quantity of our varieties do not depend upon the expenditure functionE(·) 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 sufficientlysmall and ignores second-order impacts of her price, she takes into consideration only the demand elesticity. Thedemand elasticity, in turn, depends only upon the degree of substitution among varieties of our diversified sectorexpressed in function ru. Other properties of preferences are completely separated from this elasticity by theassumption 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 outto be a general feature of two-level representation of consumer’s “utility-tree” (similar effect in usual quasi-linear IOmodels has somewhat different nature: income-neutrality). However, under heterogeneous consumers’ populationwith varying endowments, independence disappears, as we have checked. So, independence can be an artefact ofmodelling 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 tothe market size regardless of other sectors! In particular, our Examples 1,2 of increasing/decreasing price undergrowing market can be directly generalized from one-sector economy to any multi-sector economies (but not tomulti-factor models, as we have checked).

Thirdly, the equilibrium equations suggest three methods of studying the equilibria. (1) It can be throughthe closed-form solutions, and we have found that for many utility functions, not only for CES, our equationsallow for such direct way without explicitely deriving the demand (which can be harder than solving directly thesetwo equations). (2) For comparative statics, we prefer instead to study all utility functions indirectly through the

13Similar 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, theequations 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 ??). Therecan be multiple equilibria, but some roots can be irrelevant. Of course, under condition ru′(x) < 2 ∀x which meansglobal profit concavity, any non-boundary root is a true equilibrium (bounds are 0 < M < 1, 0 < N < ∞). Butwhen global concavity is not guaranteed, only a root that satisfies condition ru′(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, p), otherprices being fixed at p = p. If not, it is not a symmetric equilibrium, but it can be an asymmetric one (moredetailed discussion of equilibria existence, symmetry, and multiplicity or uniqueness is postponed to Section 6 andProposition 5).

The next section derives the comprehensive comparative statics of unique or multiple equilibria, without anyclosed-form solutions.

4 Comparative statics: classifying pro- and anti-competitive markets throughutilities

To generalize the Krugman’s findings and the above examples, we seek for necessary and sufficient conditions onthe utility function for pro- and anti-competitive effects. Studying the impact of the relative size of the market ontoprices and quantities, we have in mind that this growth can result from opening international trade, or populationgrowth, or income increase, or changes in technologies (costs). Luckily, all these important economic questions canbe 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 sum-marized in the below comprehensive Theorem 1 and in Table 1. The theorem uses the following assumption onexpenditure 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 thetwo types of goods.

We recall that ru(x) := −xu′′(x)/u′(x) and formulate the main theorem and its corrolaries for those pointswhere equilibria exist (see Appendix).

Theorem 1 15Under our Assumption 1, at any symmetric equilibrium the increase in the relative size of the marketL can have three impacts: (i) price decrease ( ∂p

∂L< 0) is equivalent to increasing concavity: r′u(x) > 0; (ii) price

14More specifically, Krugman’s equations were expressed in quantities instead of prices. Our version is more convenient for applyingthe implicit function theorem and for generalizations. Maybe, inconvenience was the reason for ignoring Krugman’s fruitful approachby theorists through decades.

15Local conditions ru′(x) < 2, ru(x) < 1, E′N (.) < 1 used for applicability of the implicit-function theorem, are guaranteed at

equilibria points by Assumptions 1,2. The existence of derivatives ∂p

∂L, ∂N

∂Lfollows from this theorem.

14

increase ( ∂p

∂L> 0) is equivalent to decreasing concavity: r′u(x) < 0; (iii) neutrality ( ∂p

∂L= 0) is equivalent to

r′u(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 ofthe market: ∂N

∂L> 0. Then pro-competitive effect ( ∂p

∂L· ∂N

∂L< 0) holds under r′u(x) > 0, anti-competitive effect

( ∂p

∂L· ∂N

∂L> 0) is equivalent to r′u(x) < 0; (iii) neutral-competitive effect ( ∂p

∂L· ∂N

∂L= 0) is equivalent to r′u(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 Under increasing Under Under decreasingL = Lc

F increases, its concavity16 neutrality (CES)17 concavityimpact on variables is: r′u(x) > 0 r′u(x) = 0 r′u(x) < 0

Equilibrium price p ↓: −M < ∂p

∂L· L

p < 0 o : ∂p

∂L· L

p = 0 ↑: 0 < ∂p

∂L· L

p S 1

A firm’s production Lx ↑: 0 < ∂y

∂L· L

y < 1 o : ∂y

∂L· L

y = 0 ↓: −1 S ∂y

∂L· L

y < 0

A variety’s consumption x ↓: −1 < ∂x∂L· L

x < 0 ↓: ∂x∂L· L

x = −1 ↓: ∂x∂L· L

x < −1

Varieties’ number*18 N ↑: ∂N∂L· L

N > 0 ↑: ∂N∂L· L

N > 0 ↑: ∂N∂L· L

N > 0Remark. Similar global comparative statics obviously follows from these local results when the equilibrium is

unique: under r′u(x) > 0 or monotone marginal-revenue function u′′(x)x + u′(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 equa-

tions, but generally the roots can be multiple! So, it remains unclear, what happens with all equilibria withinthe comparative statics, especially when some equilibria appear or disappear. Section 6 below clears these delicatequestions and global comparative statics under possible multiplicity of equilibria, not relying on local characteristicsand 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] derivestwo equilibrium equations in terms of x and N and geometrically establishes two facts: (i) the pro-competitivesufficient condition is r′u(x) > 0; (ii) under r′u(x) ≥ 0 utility increases in the size of the market L.

Statement (i) is similar to sufficiency of r′ > 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 graphicalargumentation is not quite rigorous in some respects. Namely, he skips discussing existence of equilibria, pos-sible multiplicity of equilibria, and two important conditions for rigorous comparative statics are missed in hisformulation: (a) condition 2 > ru′(x) serving for concavity of profit function, for unique sub-equilibrium and forsymmetry (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 maydecrease as we show below.

Our more comprehensive Theorem 1 provides rigorous necessary and sufficient conditions for positive, negativeand 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 isweaker, in per cent terms, than stimulating increase of the market, but for anti-competitive markets it can bestronger.

Our technique works through equilibrium price p∗(L) instead of quantity x∗(L), more conveniently for usingthe implicit-function theorem (instead of graphical proofs) and for generalized model needed in all “new” economic

15

pk

xk xk xk

pk pk

b) 0 < ru′ − ru < 1a) ru ≥ ru′

p′k p′

k

c) 1 < ru′ − 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 comparativestatics did not become popular so far). In our view, the most important Krugman’s finding is that the onlyessential property of the utility is concavity r(.). This notion enables to get simple equilibrium formulae andcomparative statics formulae from frightful huge derivatives and equations. Such simplification probably seemedimpossible 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 expen-diture E(p, N): 1 > ∂E

∂N · NE , 1 > ∂E

∂p ·pE > 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 ofthe 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 compli-mentarity (in utility terms) between the diversified and the “traditional” sectors. Maybe, in real economy it is notalways 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 marketswith pro- and anti-competitive effects, based on increasing or decreasing concavity of utility u. Why concavityis connected with these effects? Why anti-competitiveness may happen? It can be explained in terms of usualdemand-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 sametime, the demand curve changes its slope. These two effects may either enforce or hamper each other in affectingthe price. When the slope decreases strongly enough to outweigh the downward demand shift, we get the anti-competitive effect as shown in Fig.3(c). Here, as usual, the rectangle area depicts the amount of profit. To maximizeit, 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 caneasily derive the three formulae connecting r and increasing/decreasing slope: ∂2xk

∂pk∂N > 0 ⇔ ru(xk) > ru′(xk) (see(14) in Appendix). Besides, we can connect the strongly decreasing slope of x(p) with anti-competitive conditionr′u < 0. Using r′ux = (1− ru′ + ru) ru we get r′u < 0 ⇔ 1 < ru′ − ru. Thus the slope change is formally connectedwith anti-competitiveness.

In other words, decreasing elasticity ru(xs) = εp(xs) under decreasing value xs(N) brings more rigid demandcurve x∗s(.), therefore monopolist increases price under decreasing demand. Such situation does not seem unrealisticto our economic intuition.

Further, natural or not are pro- and anti-competitive effects from the mathematical viewpoint? We recall thatthe 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 r′u(x) > 0 holds for u(x) = 1− e−ax, u(x) = b ln(x + a), and many other utilityfunctions. Price paradox (r′u(x) < 0) also holds for very broad and natural class of utilities as we show now.

We have found that adding a linear component to a neoclassic utility function with −u′′′zu′′ > 1 typically turns it

into the anti-competitive class. Indeed, we have seen that the condition for anti-competitive effect is ru′(z)−ru(z) =−u′′′z

u′′ + u′′zu′ > 1. The first summand here, ru′(z, a) > 1 does not change when a linear component az is added to

initial function u(z), because u′′, u′′′ does not change. In contrast, the negative summand −ru(z, a) = u′′zu′+a < 0

goes up, closer to 0, under increasing parameter a > 0. Then, for fixed u′′, u′z the second term approacheszero, and a sufficiently high a can guarantee the anti-competitive condition at a point z for any function withru′(z) = −u′′′z

u′′ > 1, and there are many such functions. Of course, for anti-competitive effect, the equilibrium pointz(a) should also fit the region where ru′(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 findany 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), theirderivatives u

′ (dashed) and their Arrow-Pratt measures ru of concavity (thick dash-lines). The right panel, anti-competitive case, shows more rapidly increasing substitution than other two cases, when x1, x2 increase. It meansthat when the equilibrium quantities x1, x2 decrease under increasing market, the substitution decreases rapidly butcomplimentarity 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 anti-competitive effect should be considered less natural than the pro-competitive one. Is it more realistic or not, isonly an empirical question.

4) Verifyability. We argue now that most of the effects described in our theorems and corollaries can be verifiedthrough economic observations. In particular, increasing/decreasing ru is equivalent to increasing/decreasing elas-ticity 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 otherobservable 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% makesthe 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 contemplatedso far for monopolistic-competition models, up to our knowledge.

Definition. Harmful population growth (entailing also harmful international trade) means the decrease in eachconsumer’s equilibrium utility under increasing population L, i.e, ∂U(x(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:

u(z) =

1− e−z + z, if z ≤ 2

1− e−z + 2, if 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 + 1e−xk + 1

=ps

pk⇒ xs = − ln

(ps

pk

(e−xk + 1

)− 1)

.

Substituting this into the budget constraint, we get

ˆ N

0psxsds = 1 ⇔

ˆ N

0ps ln

(ps

pk

(e−xk + 1

)− 1)

ds = −1.

If all the competitors of k-th producer maintain same prices equal to P we get her inverse-demand functiondependent upon P as pk(xk) = (e−xk+1)P(

e−1

PN +1

) . The related profit-maximizing program is:

P (e−xk + 1)(e−

1PN + 1

)xk − (F + cxk) =P (e−xk + 1) xk − cxk

(e−

1PN + 1

)(e−

1PN + 1

) − F → maxxk

.

It yields FOC:19

P(e−xk + 1

)− Pe−xkxk − c

(e−

1PN + 1

)= 0.

Combining this FOC with the labor balance we have numerically found series of equilibria (P (L), x(L), N(L)) forconstants F = c = 1 and various L. At each of these points we calculated the utility level and thereby got thecomparative-statics diagrams presented in Fig. 5.

19It 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 p(L), x(L), N(L), U(L) under utilityu(z) = 1− exp(−z) + z

Observe that prices here behave non-monotonically, showing first anti-, then pro-competitive effects. Within theanti-competitive region (below L = 1.7), there exist a sub-region approximately from L = 1.34 to L = 1.35 withdecreasing 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 countryworse off, under these primitives u, c, F . The region of parameters bringing the paradoxical effect is small butnon-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 andlabor being the only important resource! This happens only due to very strong “price paradox”, under additionalnon-trivial restrictions on utility. Price paradox itself does not bring harmful growth necessarily, as we have seenin 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 anti-competitive 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/varietiesand prices. On one hand, the increase in varieties itself is beneficial for a consumer due to a broader choice. Onthe 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 inefficiencyof monopolistic-competitive trade relations found in Dixit and Stiglitz [12], is seriously aggravated in situations likeExample 3 by the population growth, that makes the essence of this effect.

Arguing similarly, but only under pro-competitive condition r′u > 0, Krugman ([17]) notes that the price goesdown under increasing market and varieties’ number grows (see our Table 1 also). This shift is twice favorablefor consumer’s welfare, because she can buy the same bundle as before and even more (below we question this

20By 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 anti-competitive condition r′u < 0, the varieties’ scope N and prices work oppositely, the possibility not considered byKrugman.

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, thatcan 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 underenvy 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, newwelfare becomes less than previously, because the integral

´ N0 u(x) is taken over more broad interval N > N , that

is why new unbought varieties bring negative utility. More generally, even when the consumer optimally spend hermoney and buy all varieties, still the negative impact of broader choice can outweigh other effects as one can seethrough the following simple example. The reason is that, for such non-normalized function there is a very smallconstant 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 relatedcomparative statics of one-sector model, the same budget w = 1 is spent for more equilibrium varieties N(L) =0.5L/F under constant prices for all market sizes L. The consumtion xs(L) = F/(cL) of each vriety tends to zerounder growing L, so, at some stage the integrand and total utility

´ N0 u(x) becomes negative.

Moral from this example is that, unlike quasi-linear IO, ordinal approach to utilities in welfare analysis ofmonopolistic competition is not innocent. Therefore, condition u(0) ≥ 0 is unjustly dropped in Krugman’s reasoningabout welfare increase under pro-competitive price effect.

Are examples of price-driven or envy-driven welfare decrease economically significant or not? It is an empiricalquestion to exclude or confirm such counter-intuitive welfare phenomena in reality, we only have shown themlogically 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 postponedto this section. For this issue, it is useful to reformulate our equilibrium equations in terms of two functions, ξ andζ equalized to zero, as follows.

ξ(M, L) := ru(1−M

L ·M)−M = 0; ζ(N,M, L) = E(

c

1−M, N) ·M · L/c−N = 0. (8)

Each of these functions can intersect zero one or more times, for instance, ξ generally decreases either mono-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 globalprofit-concavity condition ru′(x) < 2 ∀x > 0 or/and global condition r′u(x) > 0 ∀x > 0. Then (adding reasonableboundary conditions on expenditure E and boundary conditions ru(+∞) > 0, ru(0) < 1 for 0 < ξ(0, L)∀L > 0 andξ(1, L) < 0∀L > 0) a unique symmetric equilibrium exists, because the continuous function ξ going from positiveto negative values should have a root and concavity ensures this root to be an equilibrium.

20

0 1 0 1

ξ(M) ξ(M)

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 theseequations and even multiple equilibria, but not every root is an equilibrium. Under the above boundary conditionson ru(+∞), ru(0), function ξ goes from positive to negative regions, so, with probability 1 it has odd number ofroots. At those roots where ξ crosses 0 from above, such point can be an equilibrium, in the opposite case it cannotbecause the local maximum of profit requires condition ru′(x) < 2 which is equivalent to local decreasing of thiscurve (see Appendix). So, only downward, not upward, intersections are candidates for equilibria, and there canbe either one or even number of such roots. But each point should be checked additionally, is it a global maximumof 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, N ] of firms, a number n of its sub-intervals

[0, N1], [N1, N1+N2], [N1+N2, N1+N2+N3], ..., [∑n−1

k=1 Nk, N ] and prices and quantities ((p1, x1), (p2, x2), ...(pn, xn)) ∈R2n different among these n groups of producers, such that: (1) this step-wise consumption function x maximizesthe consumer’s utility under this step-wise price function p, (2) each pk maximizes the k-th producer’s profit, equalto zero, and (3) labor balance is satisfied in the form L1 + L2 + ... + Ln = L, where Lk denotes the equilibriumlabor 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 MRu(.) and πu(., c) of the utility.

Assumption MR. The utility function u(.) is neoclassic, it generates the non-monotone marginal-revenuefunction MRu(x) := u

′′(x)x+u′(x), and the proft function πu(x, c) = x(u′(x)− c) with at least two global maxima

x, x ∈ arg maxx πu(x, c), x 6= x for some cost level c > 0.21

Proposition 5 Given a utility function u(.), related one-sector Dixit-Stiglitz economy with asymmetric equilibriumexists (i.e., some parameters F,L, c yielding such equilibrium can be chosen) if and only if utility u(.) satisfiesAssumption MR.

Corollary 1. Multiple symmetric equilbria (x, p, N) 6= (x, p, N) in such economy always coexists with anasymmetric equilibrium and vice verse: any asymmetric equilibrium is always acompanied by at least two symmetricequilibria.22

Next corollary describes the structure of the equilibria set when they are multiple: the set is an interval of aline.

21In terms of r(.), non-monotone MR(.) means that there are points x, z such that, ru′(x) > 2, ru′(z) < 2.22Probably, 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 x > x. Then any asymmetricequilibrium in such economy has the specific structure: some Nx ≥ 0 firms produce quantity x and some Nx ≥ 0firms produce quantity x, these Nx, Nx satisfying the equation 1 = Nx

xc1−ru(x) + Nx

xc1−ru(x) . Any non-negative

Nx, Nx satisfying it comprise an asymmetric equilibrium (x, x, Nx, Nx) under same quantities (x, x). This meansexistence of an interval (Nx, Nx) ∈ [(0, Nmax), (Nmax, 0)] of asymmetric equilibria, while the couple of symmetricequilibria ((x, Nmax), (x, Nmax)) 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 appropriateassumptions on expenditure function E(.) guaranteeing equilibria existence.

The proofs are in Appendix. The following example and Figures 7,8 illustrate the proposition with its corollariesand 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 5and 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, bythe method described, found the cost level c ≈ 0.000264937336989 needed for mutiple equilibria. Upper panelsin Fig.7 characterize the properties of the functions ru, ru′ , ξ important for equilibria. The solid curve ru in theleft panel is non-monotone, that allows both for pro- and anti-competitive effects, while another curve intersectslevel 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 theright panel of Fig.6, but only the leftest and the rightest intersections, from above downward, are candidates forequilibria, by Proposition ??. The dashed line supports this claim showing the condition for local profit concavityonly at the boundary intersections.

To finally ensure validity of these two roots, we calculated (x, x, p, p, Nx, Nx) from the equilibria equationsand checked global maxima of profit and equal-profits conditions holding. The lower panels show that profits arereally equal at both equilibria (marked by dashed vertical lines) and at alternative points. The left one shows theprofit function of a producer when all her competitors produce about x = 0.652644, and we observe her beingindifferent between joining this pattern or switching to high quantity x = 12.8961. The same indifference holdsunder equilibrium x = 12.8961 in the right panel. The left and right curves are identical, illustrating Corollary 2to Proposition 5 about the structure of equilibria.

When rounding the numbers to 12 digits (from 60-digit precision used in calculation), the high-quantity andlow-quantity symmetric equilibria are as follows.

M p x N U

Small-price 0.068181933646 0.000284323031025 12.8961077968 272.727734586 5081.514933136equilibriumBig-price 0.591142507799 0.0006479943306508 0.652644395317 2364.570031198 6549.082752038

equilibriumBy 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 connectingN1 and N3 in this example is N3 = 2364.570031198− 8.670075431758N1.

23Actually, they are non-equivalent with probabilty 1, up to our knowledge, but proving it here would be excessive.24We are indebted for Alexei Gorn for developing computations of this example in his graduating diploma, for our request, and for

commenting our proofs also.

22

Figure 7: Example of multiple equilibria.

The utility levels were found as

U =ˆ Nx

0u(x)dt = 5081.514933136 U =

ˆ Nx

0u(x)dt = 6549.082752038.

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 havenoted in Section 4 that local comparative statics developed there can be inapplicable under multiplicity, at least atthe 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-valuedmapping p(L), N(L) are not helpful. Theorem 2 applies an approach to comparative statics from Milgrom andRoberts [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’sTheorem 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, L) : [0, 1]×R → R be continuous and satisfy boundary conditions ξ(0, L) ≥0, ξ(1, L) ≤ 0∀L. Then there exists a solution M to equation ξ(M, L) = 0. The lowest root is defined as Ml(L) =inf{M |ξ(M, L) ≤ 0} and the highest root is Mh(L) = sup{M |ξ(M, L) ≥ 0}. If ξ(M, L) ∀M is monotone increasing(non-decresing) in L, then both roots Ml(L), Mh(L) are monotone increasing (non-decresing) for all L.

23

Using Proposition 5 we can apply now this lemma to the equilibrium equation in the form ξ(M, L) = 0, andderive a global theorem about price effects, pro- and anti-competitive effects in terms of set-valued mapping ofequilibria.

Theorem 2 Suggest Assumption 1, Assumption MR and additionally, that marginal revenue has only one boundedinterval of increase, but decreases outside this interval (near 0 and at ∞). Then: (a) No more than one relativemarket size L∗ > 0 yields two symmetric equilibria, at other L the equilibrium is unique.25 (b) The mapping ofquantity x(L) at the discontinuity point L∗ (if any exists) jumpes down and globally decreases.26 (c) The pricemapping p(L) jumps up at L∗, so, under globally decreasing concavity ru the price globally increases but can benon-monotone in other cases.

Corollary T2.1. Under monotone decreasing marginal revenue MR, equilibrium is unique and the equilibriummagnitudes p(L), x(L) behave like in Table 1.

Corollary T2.2. At any interval I 63 L∗ excluding the discontinuity point, the equilibrium magnitudesp(L), x(L) behave like in Table 1.

Now we turn to the number of varieties/firms. Lemma 1 applied to function −ζ( ˜N,L) from the equilibriumequations (8) yieds

Corollary T2.3. Under Assumption 2, equilibrium firms’ number N(L) 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 L = 10 , the interval [272.7, 2364.6] of

equilibrium number of firms describes all asymmetric equilibria (see Proposition 5), but not two symmetric equilibriaN = 272.7, N = 2364.6. This point is really a point of jumps in competition, consumption of each variety, priceand utility, that means a catastrophic change. Small gradual changes in population, technology (or gevernmentalregulation, 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 offunctions u satisfying Assumption MR: just a non-monotone marginal revenue with a kink substantially above themarginal cost. Such inverse-demand functions u′ are not a degenerate case because all small modifications of suchfunction 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 wholesector of economy coherently switches from one maximum to another. The more heterogeneous are the firms inreal 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 competitionwith 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 derivingquite simple system of equilibrium equations in terms of Arrow-Pratt’s measure ru(.) of utility’s concavity (price

25The structure of these two equilibria is illustrated in Fig.8 and analysed in Proposition 5, which states also an asymmetric equilibriaat the same point L∗ and excludes more than two symmetric equilibria.

26A mapping f is called decreasing when x > z ⇒ f ′ < f” for all f ′ ∈ 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 acomplete analysis of the roots to this system and their relation to unique or multiple, symmetric or asymmetricequilibria.

Comprehensive comparative-statics in Theorems 1,2 describes local and global changes of price, quantity and va-rieties’ number w.r.t. the “relative size of the market.” The latter describes the size of the population related to fixedcost and variable cost. The theorems classify markets into pro-, neutral-, and anti-competitive ones according toincreasing, neutral, or decreasing measure ru(.) of concavity, respectively. These and other effects are shown to existand explained through examples. Most interesting and important for international-trade and economic-geographystudies are the price paradox, harmful growth, multiple equilibria and discontimuity (catastrophic changes) unknownpreviously 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/ormultiple production factors, to seek for the same effects.

References

[1] Aghion, P. and P. Howitt, 1998, Endogenous Growth Theory. Cambridge: MIT Press

[2] Amir, R. and V.E. Lambson, 2000, On the effects of entry in Cournaut markets, The Review of EconomicStudies 67(2), 235-54

[3] Anderson, F.J., 1991, Trade, firm size, and product variety under monopolistic competition, Canadian Journalof economics 24, 12-20

25

[4] Benassy, J.-P., 1991, Monopolistic competition. In Handbook of Mathematical Economics. Volume 4, ed W.Hildenbrand and H. Sonnenschein. Amsterdam: North-Holland, 1997-2045

[5] Benassy, J.-P., 1996, Taste for variety and optimum production patterns in monopolistic competition, Eco-nomics Letters 52: 41-47

[6] Behrens, K. and Y. Murata, 2007, General equilibrium models of monopolistic competition: a new approach,Journal of Economic Theory 136(1): 776-87

[7] Brakman, S. and B.J. Heijdra, 2004, The Monopolistic Competition Revolution in Retrospect. Cambridge:Cambridge University Press

[8] Campbell, J.R. and H.A. Hopenhayn, 2005, Market size matter, Journal of Industrial Economics LIII:1-25

[9] Chen, Y. and S.J. Savage, 2007, The effects of competition on the price for cable modem internet access, NetInstitute,Working paper 07-13

[10] Combes, P.-P., Mayer, T. and J.-F. Thisse, fortcoming, Introduction to Economic Geography

[11] Das S.P., 1982, Economies of scale, imperfect competition, and the pattern of trade, The Economic Journal92, 684-93

[12] Dixit, A.K. and J.E. Stiglitz, 1977, Monopolistic competition and optimum product diversity, American Eco-nomic Review 67: 297-308

[13] Fujita, M., Krugman, P. and A.J. Venables, 1999, The Spatial Economy: Cities, Regions, and InternationalTrade. Cambridge: MIT Press

[14] Fujita, M., and J.-F. Thisse, 2002, Economics of Agglomeration, Cities, Industrial Location and RegionalGrowth. Cambridge: Cambridge University Press

[15] Goolsbee, Syverson, 2004, How do incumbents respond to the threat of entry? University Chicago workingpaper

[16] Helpman, E. and P.R. Krugman, 1985, Market Structure and Foreign Trade: Increasing Returns, ImperfectCompetition, and the International Economy. Cambridge: MIT Press

[17] Krugman, P., 1979, Increasing returns, monopolistic competition, and international trade, Journal of interna-tional economics 9, 151-75

[18] Milgrom, P. and J. Roberts, 1994, Comparing Equilibria, American Economic Review 84: 441-59

[19] Ottaviano, G.I.P., Tabuchi, T. and J.-F. Thisse, 2002, Agglomeration and trade revisited, International Eco-nomic Review 43: 409-36

[20] Perloff, J.M., V. Y. Suslow and P.J. Seguin, 1996, Higher prices from entry: pricing of brand-name drugs,Discussion paper 778

[21] Rosenthal, R.W., 1980, A model in which an increse in the number of sellers leads to a higher price, Econo-metrica 48(6): 1575-79

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-factoreconomy, and then generalized it to “reduced” multisector model where all but one sectors are described throughthe expenditure function E. The next subsection shows how to derived this expenditure function E from someupper-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 supplementedby additional diversified or/and homogeneous sectors (aggregated commodities). First we explain the general modeland its equilibrium, and then derive the reduced version.

There are m1 diversified sectors (aggregated commodities) indexed as i = 1, ...,m1 and m2 homogeneous sectorsk = m1 + 1,m1 + 2, ...,M (M = m1 + m2). Each of L identical consumers/employees solves the following utility-maximization problem:

U(X1, ..., Xm1 , y1, ..., ym2) → maxx(.),y

w.r.t.,

m1∑i=1

εi +M∑

k=m1+1

pkyk ≤ 1, where

εi =ˆ Ni

0pisxisds, Xi :=

ˆ Ni

0u(xis)ds.

Here intermediate variable εi ≥ 0 denotes expenditure for i-th sector, Xi ≥ 0 is the aggregate utility from thissector, and yk ≥ 0 is consumption of k-th homogeneous good. Labor endowment is normalized to 1, as well aswage, and income from profits is normalized to zero because of zero-profit concept of all sectors. Upper-level utilityis 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 utilitymaximization generates some demand function x(p, N).

Each ((is)-th) monopolist out of Ni producers in i-th diversified sector, is already described by profit-maximizationprogram (1). It remains the same but for more indicies and enlarged argument p = (p1, ..., pM ) of the demandfunction. Index (is) should replace more simple index s when describing the price pis, quantity xis, variable costci and fixed cost Fi of the producer. As to homogeneous sector, it is supposed to be a competitive one, withlinear technology, so it is sufficient to introduce marginal costs ck, measured in labor, for each sector (numberof firms does not matter here). This profit maximization generates some response function pis(p-is, N) dependingupon all outside prices p-is and the total vector of scopes of sectors as N = (N1, ..., Nm1) ∈ RM . Now we de-note the symmetrized vector of prices as p = (p1, ..., pM ) ∈ RM , because assuming symmetry within each sector.This means same price pi = pis(p-i, N) for each (is-th) producer in the sector and symmetric production quantityxi(p-i, N) = xis(pis(p-i, N), p∗-i, N) that exploits the demand function depending upon stepwise (symmetric) otherprices and i-th response to these other prices (we maintain the same notations x, p for demand and price responsethough these function formally depend upon simpler arguments than before). Solving the equations system forNash price-responses of all producers to each other, we define the subequilibrium price vector p∗(N) as a functionof all scopes, that yeilds also sub-equilibrium quantities x∗(N). From the consumer’s program we get also the

27

sub-equilibrium demands y∗k(N) for homogeneous goods as a function of N under the pricess found and trivialprices pk = ck of homogeneaous goods.

To link the sectors to each other, we add the zero-profit (free-entry) assumption for each diversified sector andthe labor balance in the economy:

p∗i (Ni)Lx∗i (Ni) = (ciLx∗i (Ni) + Fi) , Ni(ciLx∗i (Ni) + Fi) = li,

m1∑i=1

li +M∑

k=1+m1

cky∗k(N) = L.

The consumer’s budget equality follows from these equations at equilibrium.Definition 2. An integrated symmetric equilibrium ((x, y), p, N) ∈ R3M of such economy is a vector that

satisfies optimization programs of consumers and producers (i.e., symmetric subequilibrium requirements), and theabove balances of profit and labor.

Looking on this model through our goals, it seems impractical to study comparative statics of such complicatedequilibria directly. Instead, we use a reduced formulation of such an economy, and study mainly this reducedformulation, which is focused on one diversified sector within the economy. The reduced model looks like partialequilibrium, 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 throughvariables of expenditures ei and labor li applied in i-th sector. It is easy to understand that these two vectorsare 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 theworld” to some response function Ei(pi, Ni) of outside sectors to this sector. Ideologically, this function denotesthe economy’s long-run equilibrium reaction to shifts in this diversified sector. But technically, we can look on it ason a method to derive the integrated equilibrium defined above. We should show that step by step we can derivesuch 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 diversifiedsectors i = 1, 2 and only one homogeneous (competitive) sector k = 3. Suppose, we are mainly interested, whathappens 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: p3 = c3. We find also itsper-consumer quantity as a symmetric demand function x3(p1, p2, N1, N2) ∈ R depending upon upon (unknown sofar) infinite-dimensional price vectors p1, p2 of the first and second sectors, and upon numbers of varieties. Othercomponents 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 functionsp2(p1, p3, N1), x2(p1, p3, N1), N2(p1, p3, N1) depending upon p3 (already found) and upon (unknown so far) infinite-dimensional 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 p2(.), x2(.) to getthe general-form expenditure function

E1(p1, N1) = 1− p2(p1, p3, N1)x2(p1, p3, N1)− p3x3(p1, p2(p1, p3, N1), N1, N2(p1, p3, N1))

28

of the first sector. It depends upon the unknown infinite-dimensional price p1 and the scalar number of varietiesN1, 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 whathappens to sector #1 within comparative statics of the whole economy, without complicated direct derivationof demand. It is one of our lucky findings. It overcomes the limitation in typical papers (obsessed with theclosed-form demand) where the upper level utility function for technical reasons is a Cobb-Douglas one: U(X, y) =∑m1

i aiLog(Xi)+∑M

k=m1+1 akLog(yk), with CES lower-level function u(z) = zβ .27 Then the budget share allocatedto each sector is just constant Ei(pi, Ni) ≡ εi = 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 ex-

penditures E = E(p, N) for the whole diversified sector as already chosen. Respectively, her optimization problemis similar to that of the basic model:

ˆ N

0u(xs)ds → max

xw.r.t.

ˆ N

0psxsds ≤ E.

The FOC (first-order conditions) of the Lagrangian w.r.t. each variety xs are

u′(xs) = λps ⇒ u′(xs)u′(xk)

=ps

pk.

To express the demand we denote the inverse of the marginal utility as ϕ = (u′)−1 (following Behrens and Murata’snotation). 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 andMurata), 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

0psϕ(ϕ−1(xk)

ps

pk)ds = R. (10)

We can look on this equality as on an indirect function, yielding the dependence xk(p, E, N) of consumer’s optimalchoice upon prices, income and number of varieties N (which is the length of the varieties interval in continualmodel). Further we drop the “check” and these arguments of this function, denoting it as xk(.) but have them inmind, and have in mind its optimality. Assuming “very large” N for the discrete version of the model, or using thesame idea in continual model, we get negligibility of spillovers between single varieties:

∂xk

∂ps≡ ∂xk(p, E, N)

∂ps= 0

for all s 6= k. This absence of any complimentarity or gross substitutability (GS) between commodoties is due tonegligible influence of each variety on the total demand in the continual model. In discrete monopolistic-competition

27Only combination of these two restrictive assumptions is sufficient for constant ε, otherwise the budget share can vary. This fact isnot noticed in Helpman and Krugman ([16], p. 190) where the constant-share assumption is just added to seemingly-general functionsanalyzed.

29

model this equality becomes an additional assumption. Similarly, we have ignored in differentiating the dependenceof 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 ofother prices with positive measure, it is not negligible. To express this reaction we introduce two most importantnotations:

ru(xk) := −xku′′(xk)

u′(xk); ru′(xk) := −xku

′′′(xk)u′′(xk)

,

This characteristic ru(.) of u means Arrow-Pratt’s relative risk aversion measure, or elasticity of the inversedemand. It turnes out to be the crucial characteristics of preferences, most tightly connected with all importantfeatures of demand.

In particular, it shows the borderline between complimentarity and substitutability among measurable groupsof varieties. In microeconomics, for neo-classical separable utilities the sufficient condition for strict gross substi-tutability (GS) is known as ru(xk) < 1, while ru(xk) = 1 relates to absent sensitivity (Cobb-Douglas utility) andru(xk) > 1 means some degree of complimantarity. Now, for fixed E, we give a direct proof for the same grosssubstitution (GS) effect between a given (zero-measure) variety xk and the “price level” P that means all otherprices when they are uniform (same) and derive the demand elasticity:

ˆ N

0Pϕ(ϕ−1(xk)

P

pk)ds = NPϕ(ϕ−1(xk)

P

pk) = E ⇔ ϕ(ϕ−1(xk)

P

pk) =

E

NP⇒

∂xk

∂Pϕ′(ϕ−1

)′ P

pk= −ϕ

P− ϕ′ϕ−1 1

pk⇔ ∂xk

∂P

P

xk=

1− ru(X)ru(xk)

.

Here the denomenator is positive (by concavity of u), so the numerator expresses the condition 1 − ru(X) > 0sufficient 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 benegligible. Differentiating equation (9) w.r.t. ps we get

∂

∂ps[xs − ϕ(ϕ−1(xk)

ps

pk)] =

∂xs

∂ps− ϕ′(ϕ−1(xk)

ps

pk)(ϕ−1′(xk)

ps

pk

∂xk

∂ps+ ϕ−1(xk)

1pk

) = 0 ⇒

∂xk

∂pk= ϕ′(ϕ−1(xk))

ϕ−1(xk)pk

=ϕ−1(xk)

(ϕ−1)′ (xk)pk

= − xk

ru(xk)pk, (11)

Here we have used equation (9) to substitute xk, pkfor xs, ps. Besides, like for ∂xk∂ps

, we have neglected in differentiatingthe 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, servingrather for interpretations of the direct effect of growing competition (ignoring equilibrium spillovers), so we stickto fixed-expenditure case (E = const) in this paragraph. The effect studied is the increase or decrease in the slopeof the demand curve, already discussed in Examples 1,2 but now shown in general form, to give more intuitions foranti-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 beforetake the constants out of the integral to get

30

pNϕ(ϕ−1(xk)pN

pk) +

∂xk

∂N

(ϕ−1

)′ (xk)ˆ N

0

p2s

pkϕ′(ϕ−1(xk)

ps

pk)ds = 0 ⇒ (using ϕ−1(xs) = ϕ−1(xk)

ps

pk)

∂xk

∂N= − pNxN

(ϕ−1)′ (xk)´ N0

p2s

pkϕ′(ϕ−1(xs))ds

< 0. (12)

The sign is due to ϕ = (u′)−1 > 0, u′′ < 0, and can be interpreted as “market-crowding”. When more varietiesbecome available, a consumer, naturally, starts buying less quantity of each. In the particular case of symmetricequilibrium (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 :

∂2xk

∂pk∂N=(ϕ′′(ϕ−1(xk))ϕ−1(xk) + ϕ′(ϕ−1(xk))

)·(ϕ−1

)′ (xk)pk

· ∂xk

∂N=

1u′′

(−ru′(xk)

ru(xk)+ 1)

∂xk

∂N. (13)

⇒ ∂2xk

∂pk∂N> 0 ⇔ ru(xk) > ru′(xk). (14)

Thus, depending upon relation between concavity of u and concavity of u’, the slope of demand can increase ordecrease.

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 xk(p) for each variety by thequantity of labor L to get xΣk(p) = Lxk(p). This simple aggregation rests on homogeneity and related existence ofa representative consumer. But in our case these assumptions are not imposed, so for similar simple aggregationwe should (and did) assume that each consumer has the same income, equal to her one unit of labor multiplied bythe wage, w = 1.

The k-th monopolistic producer maximizes her profit π w.r.t. per-consumer output xk, and price pk taking intoaccount the total demand function for this variety:

π = (pk − cw)Lxk(pk, p−k, E(pk, p−k, N), N)− wF → maxpk

.

Linearity of variable cost allows to simplify FOC (first-order conditions) as

πp = (pk − cw)∂xk

∂pk+ xk = 0 (15)

and SOC (second-order conditions) or profit-concavity condition amounts to

πpp = 2∂xk

∂pk+ (pk − cw)

∂2xk

∂p2k

≤ 0.

So, like in traditional IO with quasi-linear utilities, any equilibria can occur only at points with elastic inversedemand. Indeed, FOC and profitablity pk > cw entail the following simple bound on markup and on r (similar toLerner’s index):

31

(pk − cw)pk

= − 1∂xk∂pk

pkxk

= ru(xk) < 1.

We see that equilibria can occur only at points with moderate concavity (related to gross-substitution propertyas we shall see). This neat expression of the optimal markup in terms of the utility’s risk-aversion ru, commonin quasi-linear contexts of IO, helps very much in the following simple proposition stating the producer’s solutionuniqueness (but not existence, which is unclear).

Proposition 6 (i) If concavity of the marginal utility is restricted as 2 > ru′(xk) at point xk, then each producer’sprofit function is strictly concave at this point; at points satisfying FOC this restriction is necessary and sufficientfor strict concavity; (ii) If this upper bound on ru′(xk) holds everywhere on R+, then profit is generally strictlyconcave 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:

∂2xN

∂p2N

=∂

∂pN

(ϕ′(ϕ−1(xN ))

ϕ−1(xN )pN

)= ϕ′′(ϕ−1(xk))

(ϕ−1(xk)

pk

).2

Here we have used the identity ϕ′(ϕ−1(xN )) = 1/((ϕ−1)′(xN )). More generally, the profit concavity condition2∂xk

∂pk+ (pk − cw)∂2xk

∂p2k≤ 0 at point of producer’s optimum is equivalent to

2 +(pk − cw)

∂xk∂pk

∂2xk

∂p2k

= 2 +(pk − cw)

ϕ′(ϕ−1(xk))ϕ−1(xk)

pk

ϕ′′(ϕ−1(xk))(

ϕ−1(xk)pk

)2

=

= 2− xk∂xk∂pk

ϕ′′(ϕ−1(xk))ϕ−1(xk)

pkϕ′(ϕ−1(xk))

= 2− xkϕ′(ϕ−1(xk))

ϕ′′(ϕ−1(xk))ϕ′(ϕ−1(xk))

= 2 + xk

((ϕ−1)′(xk)

)2 (ϕ−1)′′(xk)

((ϕ−1)′(xk))3 =

= 2 +

(ϕ−1

)′′ (xk)xk

(ϕ−1)′ (xk)= 2 +

u′′′(xk)xk

u′′(xk)= 2− ru′(xk) ≥ 0. (16)

Thus, the optimal price found from FOC is unique.Let us show equilibrium symmetry. From the producers’ and consumers’ FOC we have

u′(xk)u′(xs)

=pk

ps=

1− ru(xs)1− ru(xk)

⇔

u′(xk) (1− ru(xk)) = u′(xs) (1− ru(xs)) .

The obtained function u′(x) + u′′(x)x increases because of 2 > ru′(x), therefore solution symmetry is guaranteed:xk = xs. �

Here we give the full version of the proposition about uniqueness and existence of equilibria.29

28The condition 2 > ru′(xk) found here, unlike quasi-concave economies, is not necessary for profit concavity and equilibriumsymmetry, say, the Behrens-Murata’s function mentioned before Example 1 satisfies this condition not everywhere, but shows concavityand symmetry everywhere. The point is that the demand function for a Dixit-Stiglitz sector differs from that of a quasi-concave sectorhaving similar u (because showing some income-effect), only their elasticities are similar.

29As to equilibria inexistence, there can be neoclassic utility functions without any equilibria, for instance, u(x) = ln x, or, moreinterestingly, function from Example 2 but unrestricted from above: u(x) = 1− e−x + x. The latter utility generates a profit functionπ(ps, p) which has local maxima and solutions to equation (6) for some parameters L, F, c, N , but profit is unrestricted in directionps → 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, L)∀L > 0) and ru(0) < 1 (i.e., ξ(1, L) < 0∀L > 0). Then: (i) For any L > 0 equation (6) has a rootM satisfying bounds 0 < M < 1 where ξ intersects 0 downwards in the sense ξ′M (M, L) < 0; (ii) In any such roottwo conditions are satisfied: local gross substitution ru(1−M

L·M ) < 1 and local concavity of profit ru′(1−ML·M ) < 2, while

any upwards root lacks local concavity and cannot be an equilibrium; (iii) Under global sufficient profit-concavitycondition ru′(x) < 2 ∀x > 0 or/and global condition r′u(x) > 0 ∀x > 0 the root M is unique and it is definitely anequilibrium value of markup;30

(iv) If expenditure function E( c1−M , N) is continuous and bounded for all M ∈(0, 1) having limN→0 E( c

1−M , N) >

0 ∀M ∈ (0, 1), then for any M equation (7) has a solution N = N(M) > 0. Then, any such N(M) obtained fromany equilibrium markup M : 0 < M < 1 is a symmetric monopolistic-competition equilibrium (M ,N), whichthereby, exists under global profit concavity and boundary conditions on expenditure E and concavity ru.

Proof. Consider function ξ(M, L) := ru(1−ML·M ) − M . By assumptions, ξ(0, L) = ru(+∞) > 0 and ξ(1, L) =

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 showdownward intersection is

ξ′M (M, L) = −r′u1

L ·M2− 1 = − 1

M(1−M)(r′ux + M(1−M)

)∀L > 0.

It is evidently negative everywhere under condition r′u(x) > 0 ∀x > 0, then ξ decreses. For a weaker conclusionunder weaker condition 2 > ru′ this magnitude at points M = ru(1−M

L·M ) of roots to ξ(M, L) = 0 can be expressedthrough r as

ξ′M (M, L) = − 1ru(1− ru)

(r′ux + ru(1− ru)

)= − 1

ru(1− ru)(ru(1− ru′ + ru) + ru(1− ru)) = −2− ru′

1− ru.

This magnitude at intersection points is negative if and only if 2− ru′ because the denominator is always positiveat 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 the-orem. We use the implicit dependence of markup M upon the relative market size as M = ru(1−M

LM). Differentiating

it we get∂M

∂L= r′u

(− 1

L2· 1−M

M− 1

L· 1M2

∂M

∂L

)⇒

∂M

∂L· L

M= − r′ux

ru(1 + r′u1L· 1

M2 )= − r′ux (1− ru)

ru(1− ru) + r′ux.

Note that r′ux =(−u′′(x)x

u′(x)

)′x = −u′′′(x)x2

u′(x) − u′′(x)xu′(x) +

(−u′′(x)x

u′(x)

)2= −ru′ru + ru + r2

u = ru (−ru′ + 1 + ru) .

Therefore∂M

∂L· L

M= − ru (−ru′ + 1 + ru) (1− ru)

ru(1− ru) + ru (−ru′ + 1 + ru)=

(ru′ − 1− ru) (1− ru)(2− ru′)

.

30We are not stating here thatξ(., L) is monotone decreasing, but it cannot intersect 0 from below due to (ii). Besides, we are notstating here the general equilibrium existence yet, because the second equation is not studied.

33

One can see that ∂M∂L

· LM > − (1−M). Indeed,

11−M

· ∂M

∂L· L

M=

(ru′ − 1− ru)(2− ru′)

>(ru′ − 2)(2− ru′)

= −1.

Now look on all other variables:

p =c

1−M⇒ ∂p

∂L· L

p=

M

1−M· ∂M

∂L· L

M

x =1L· M

1−M⇒ ∂x

∂L· L

x= −1− 1

1−M· ∂M

∂L· L

M

y =F

c· 1−M

M⇒ ∂y

∂L· L

y= − 1

1−M· ∂M

∂L· L

M

Now let r′u > 0, then ∂M∂L

· LM < 0, therefore ∂p

∂L· L

p < 0 and ∂y

∂L· L

y > 0. Note that ∂x∂L· L

x < 0 because of∂M∂L

· LM > − (1−M) obtained, but combining it with ∂M

∂L· L

M < 0 we have −1 < ∂x∂L· L

x < 0. Besides, this property

guarantees that ∂p

∂L· L

p = M1−M · ∂M

∂L· L

M > −M and ∂y

∂L· L

y = − 11−M · ∂M

∂L· L

M < 1. The other cells of the tablerelated 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 marketsize L:

NF = LM(L)E(p(L), N).

Differentiating it as implicit function we get

∂N

∂L· L

N= 1 +

∂M

∂L· L

M+

∂E

∂p· p

E· ∂p

∂L· L

p+

∂E

∂N· N

E· ∂N

∂L· L

N⇒

∂N

∂L· L

N(1− ∂E

∂N· N

E) = 1 +

(1 +

∂E

∂p· p

E· M

1−M

)· ∂M

∂L· L

M.

Now, if 1 > ∂E∂N · N

E and 1 > ∂E∂p ·

pE > 0, then

∂N

∂L· LN

(1− ∂E

∂N·NE

) = 1+(

1 +∂E

∂p· p

E· M

1−M

)·∂M

∂L· L

M≥ 1+

(1 +

∂E

∂p· p

E· M

1−M

)·(M−1) = M(1−∂R

∂p· pR

)

that is ∂N∂L· L

N > 0. Thus, under assumptions taken, the necessary and sufficient condition for pro (anti)-competitiveeffect is r′u > 0 (r′u < 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 varietiesof “machinery”, and usual competitive agriculture producing homogeneous “food”. Labor is mobile among sectorsand wage normalized as previously to w = 1. Productivity in agriculture is also normalized to 1, so eqilibrium priceof “food” is pa = 1.

The consumer’s problem with constant income E takes the form

U(ˆ N

0u(x(s))ds, a) → max

34

ˆ N

0p(s)x(s)ds + paa ≤ E.

Upper-level utility function U here determines complimentarity/substitution between the two sectors. One canpersept the consumer’s choice as two-level one: first she allocate her budget between food and machinery, andthen specify spendings on each variety in the diversified sector. The lower-level problem with given expenditure formachinery magnitude Em takes the form, including the Lagrange multiplier λ from the uper-level problem:

ˆ N

0u(x(s))ds → max

ˆ N

0p(s)x(s)ds ≤ Em λ

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 economicforce or tendency: substitution of manufacturing goods in spendings by “food” under changing varieties’ number orprices. To clear this question, look on the expenditures E(p, N) on manufacture under changing (p, N). The FOCare

U ′1(ν(p, Em, N), E − Em)ν ′R(p, Em, N) = U ′

2(ν(p, Em, N), E − Em).

The second-order conditions are

U ′′11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE ≤ 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

U ′′11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE ≤ 0.

Proof. The expressionU ′′

11

(ν ′E)2 − (U ′′

12 + U ′′21

)ν ′E + U ′′

22

is a polynom w.r.t. ν ′E . The discriminant is

(U ′′

12 + U ′′21

)2 − 4U ′′11U

′′22 = 4

(U ′′

12

)2 − 4U ′′11U

′′22 = 4

(U ′′

12U′′21 − U ′′

11U′′22

)and concavity makes this experssion negative. Based on U ′′

22 < 0, we have U ′′11 (ν ′E)2 − (U ′′

12 + U ′′21) ν ′E + U ′′

22 < 0.

To see the sign of U ′1ν

′′EE , we find ν ′′EE . From FOC of lower-level optimization, u′(xs) = λps. Using the budget´ N

0 psxsds = E we get

λ = ν ′R =

´ N0 u′(xs)xsds

E> 0.

35

Now find

∂λ

∂E= ν ′′EE =

´ N0 (u′′(xs)xs + u′(xs)) ∂xs

∂E ds

E−´ N0 u′(xs)xsds

E2=

∂λ∂R

´ N0

(u′′(xs)xs+u′(xs))u′′(xs)

psds

E− λ

E⇔

∂λ

∂E=

λ´ N0

u′(xs)u′′(xs)

psds< 0

Therefore ν ′′EE < 0, so negative is also the whole expression U ′′11 (ν ′E)2 −U ′′

12ν′E −U ′′

21ν′E + U ′′

22 + U ′1ν

′′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 andvarieties’ number.

Lemma 3 Equilibrium elasticity of expenditures Rm w.r.t. varieties’ number N is

∂Em

∂N· N

Em− 1 =

−U ′′11ν

′Eν + U ′′

21 (ν + ν ′EEm)− EmU ′′22(

U ′′11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE

)Em

. (17)

Proof. Hereafter the second derivative is denoted SOC = d2UdE2

m. Looking on the FOC as on implicit function R(N)

we differentiate it and get

∂Em

∂N= −

U ′′11ν

′Eν ′N + U ′

1ν′′EN − U ′′

21ν′N

SOC= −

(U ′′11ν

′E − U ′′

21)ν′N + U ′

1ν′′EN

SOC.

Using it we have

∂Em

∂N· N

Em− 1 = −

(U ′′11ν

′E − U ′′

21)ν′N + U ′

1ν′′EN

U ′′11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE

· N

Em− 1 =

−U ′′11

(ν ′ENν ′N + Em (ν ′E)2

)+ U ′′

21 (Nν ′N + 2ν ′EEm)− U ′1 (Nν ′′EN + Emν ′′EE)− EmU ′′

22(U ′′

11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE

)Em

.

As previously, the denominator is negetive due to second order condition. Before studying the numerator sign notethat under normalizing u(0) = 0, the function

ν(p, Em, N) = Nu(Em

pN)

is homogeneous of degree 0 w.r.t. (p, Em) of degree 1 w.r.t. (N, Em), while its derivative ν ′E =u′(Em

pN)

p is homoge-neous of degree 0 w.r.t. (Em, N) of degree -1 w.r.t. (p, Em).

Therefore−U ′′

11

(ν ′ENν ′N + Em

(ν ′E)2) = −U ′′

11ν′E

(Nν ′N + Emν ′E

)= −U ′′

11ν′Eν

−U ′1

(Nν ′′EN + Emν ′′EE

)= 0

U ′′21

(Nν ′N + 2ν ′EEm

)= U ′′

21

(ν + ν ′EEm

)that allows to express elasticities as

∂Em

∂N· N

Em− 1 =

−U ′′11ν

′Eν + U ′′

21 (ν + ν ′EEm)− EmU ′′22(

U ′′11

(ν ′E)2 − U ′′

12ν′E − U ′′

21ν′E + U ′′

22 + U ′1ν

′′EE

)Em

,

36

as we needed. �

Lemma 4 Equilibrium elasticity of expenditure on varieties w.r.t. price level p is

∂Em

∂p· p

Em− 1 =

U ′1ν

′E + U ′′

21Emν ′E − EmU ′′22

Em

(U ′′

11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

) . (18)

Proof. Let us study impact of price level on the expenditure. Differentiating FOC as in previous proposition, weget

U ′′11ν

′pν

′E + U ′′

11

(ν ′E)2 ∂Em

∂p− U ′′

12ν′E

∂Em

∂p+ U ′

1ν′′Ep + U ′

1ν′′EE

∂Em

∂p=

U ′′21ν

′p + U ′′

21ν′E

∂Em

∂p− U ′′

22

∂Em

∂p

∂Em

∂p=

−U ′′11ν

′pν

′E − U ′

1ν′′Ep + U ′′

21ν′p

U ′′11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

As previously, the elasticity of expenditure w.r.t. price is expressed as

∂Em

∂p· p

Em− 1 =

−U ′′11ν

′pν

′E − U ′

1ν′′Ep + U ′′

21ν′p

U ′′11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

· p

Em− 1 =

=−U ′′

11

(pν ′pν

′E + Em (ν ′E)2

)− U ′

1

(pν ′′Ep + Emν ′′EE

)+ U ′′

21

(pν ′p + 2Emν ′E

)− EmU ′′

22

Em

(U ′′

11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

) .

Again, SOC give the negative denominator. As to the numerator, 0-homogeneity of νestimates the first summandas −U ′′

11

(pν ′pν

′E + Em (ν ′E)2

)= −U ′′

11ν′E

(pν ′p + Emν ′E

)= 0. The second summand is −U ′

1

(pν ′′Ep + Emν ′′EE

)=

U ′1ν

′E > 0, because of homogeneity of ν ′E . The third summand becomes U ′′

21

(pν ′p + 2Emν ′E

)= U ′′

21Emν ′E . Therefore

∂Em

∂p· p

Em− 1 =

U ′1ν

′E + U ′′

21Emν ′E − EmU ′′22

Em

(U ′′

11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

) .

�

Lemma 5 Let U ′′21 ≥ 0 and ∂Em

∂p · pEm

> 0, then

0 <∂Em

∂p· p

Em< 1

∂Em

∂N· N

Em< 1.

Proof. From U ′′21 ≥ 0 and ∂Em

∂p · pEm

> 0 we have

0 <∂Em

∂p· p

Em< 1.

Further, from ∂Em∂p · p

Em> 0, and negativity of the denominator (ensured in Lemma 3), we have

U ′′11ν

′pν

′E + U ′

1ν′′Ep − U ′′

21ν′p > 0.

To find the sign of ν ′′Ep use

ν ′E =u′(Em

Np )

p> 0,

37

to see

ν ′′Ep = −Emu′′(Em

Np )

Np3−

u′(EmNp )

p2= −

Emu′′(EmNp )

Np3−

u′(EmNp )

p2= −

u′(EmNp )

p2

(1− ru(

Em

Np))

< 0.

Analogeously, ν ′p = −Emu′(Em

Np)

p2 < 0. Therefore we have (U ′′11ν

′E − U ′′

21) ν ′p > −U ′1ν

′′Ep > 0, hence

U ′′11ν

′E − U ′′

21 < 0.

This, in turn, entails∂Em

∂N· N

Em< 1.

�

Lemma 6 Elasticity of expenditures on the diversified goods is positive (i.e. ∂Em∂p · p

Em> 0) if(

U ′′11(X, Y )XU ′

1(X, Y )− U ′′

21(X, Y )YU ′

2(X, Y )

)�v′(x)xv(x)

+ 1 +v′′(x)xv′(x)

< 0

Proof. Note that ν ′p = −v′( E

Np)E

p2 , ν ′E =v′( E

Np)

p and ν ′′Ep = −v′( E

Np)

p2 −v′′( E

Np)E

Np3 . Then we can modify expression

∂Em

∂p=

−U ′′11ν

′pν

′E − U ′

1ν′′Ep + U ′′

21ν′p

U ′′11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

to get∂Em

∂p=

U ′′11ν

′pν

′E + U ′

1ν′′Ep − U ′′

21ν′p

−(U ′′

11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

) =

=−ν ′p

(−U ′′

11ν′E − U ′

1

ν′′Ep

ν′p+ U ′′

21

)−(U ′′

11

(ν ′E)2 − U ′′

12ν′E + U ′

1ν′′EE − U ′′

21ν′E + U ′′

22

) .

The sign of ∂Em∂p coincides with the sign of

(−U ′′

11ν′E − U ′

1

ν′′Ep

ν′p+ U ′′

21

). The latter expression amounts to

−U ′′11ν

′E − U ′

1

ν ′′Ep

ν ′p+ U ′′

21 = −U ′′11

v′( ENp)

p− U ′

1

−v′( E

Np)

p2 −v′′( E

Np)E

Np3

−v′( E

Np)E

p2

+ U ′′21 =

= −U ′1

E

((U ′′

11Nv( ENp)

U ′1

−U ′′

21Nv( ENp)

U ′2

)v′( E

Np) ENp

v( ENp)

+ 1 +v′′( E

Np)E

v′( ENp)Np

).

Using −U ′1

E < 0 it follows that

∂Em

∂p> 0 ⇐⇒

(U ′′

11Nv( ENp)

U ′1

−U ′′

21Nv( ENp)

U ′2

)v′( E

Np) ENp

v( ENp)

+ 1 +v′′( E

Np)E

v′( ENp)Np

< 0.

The lemma is proved.�

38

7.6 Proofs for multiple equilibria

Proof of Proposition 5 and its Corollaries. For all these three statements we exploit the analogy between astandard quasi-linear monopolist facing the inverse-demand function P (x) = u′(x), and a Dixit-Stiglitz monopolistfacing the per-consumer inverse-demand Pλ(x) = u′(x)/λ, as illustrated by Fig. 6. Here λ denotes the Lagrangemultiplier of the consumer’s budget constraint at the equilibrium. In other words, the standard monopolist max-imizes profit function πM (x, c) = x(u′(x) − c) with FOC u′′(x)x + u′(x) = c, while the Dixit-Stiglitz monopolistmaximizes very similar per-consumer net-of-fixed-cost profit function πs(xs, c, λ) = xs(u′(xs)/λ− c) = πM (x, cλ)/λ

with similar FOC u′′(xs)xs + u′(xs) = cλ. The latter yields the same solution as if a standard monopolist werefacing inverse demand P but having cost c = cλ. Though the multiplier λ = λ(N, p) is dependent upon the numberof varieties and on their prices, but this λ takes the same value at all (symmetric or asymmetric) equilibria, as weshow 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 x 6= x and cost c from the standard monopolist’s optimum sug-

gested by Assumption MR, to construct the needed parameters c, F, L,N1, N2 for Dixit-Stiglitz economy with someasymmetric equilibrium and same utility u(.). We are constructing the specific kind of asymmetric equilibriumdescribed in Corollary 2. Namely, we take only two quantities x > x (sign is taken without loss of generality) andtwo multitudes Nx, Nx of firms producing these quantities. So, the total number of firms will be Nx + Nx. We justcheck equilibrium equations holding for such point (x, x, Nx, Nx).

Denoting as λ = λ the equilibrium’s Lagrange multiplier for the budget constraint, the asymmetric equilibriumequations are:31

u′′(x)x + u′(x) = c = λc, u′′(x)x + u′(x) = c = λc (19)

(FOC of both producers’ types),

Nxxp + Nxxp = 1, u′(x) = λp, u′(x) = λp

(consumer’s budget and FOC),

(p− c)x = F/L = (p− c)x

(producers’ zero-profit condition),

(cLx + F )Nx + (cLx + F )Nx = L

(labor balance, that can be derived from the latter two equations).The symmetric-equilibria equations are similar, only equality λ = λ is not guaranteed ad hoc but derived, and

either Nx = 0 or Nx = 0.Now we reformulate this symmetric-equilibrium system using our notation r(.) and excluding λ:

p = c/(1− ru(x)), p = c/(1− ru(x)), (20)

x(p− c)c

=xru(x)

1− ru(x)=

F

cL=

xru(x)1− ru(x)

, (21)

31Additional condition for profit concavity follows from the global maxima in Assumption MR.

39

1 = Nxxc

1− ru(x)+ Nx

xc

1− ru(x). (22)

Now, for our given quantities x, x 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’sglobal optima: πM (x, c) = x(u′(x) − c) = πM (x, c) = x(u′(x) − c) > 0. So, at both problems of the Dixit-Stiglitzmonopolists, related per-consumer net-of-fixed-cost profits π := πs(xs, c, λ) = πM (x, cλ)/λ = πM (x, cλ)/λ > 0 arealso equal when cλ = c. Then, under special c = F

L ·1−ru(x)xru(x) > 0, λ = c/c this value π > 0 is exactly sufficient to

cover the per-consumer fixed cost of both producers in the sense π = F/L, that means both equations satisfied.Further, using values c, λ obtained, we calculate prices

p = λpM = c/(1− ru(x)), p = λpM = c/(1− ru(x)),

from equations (20) (denoting here as pM ,pM the standard monopolist’s prices for low quantity x and high quantityx respectively). Positivity of pM ,pM ensures positivity of p,p. What remains is to take any couple of multitudes(Nx, Nx) of big-quantity firms and small-quantity firms that satisfy the last equation (22). This pair can be chosennon-uniquely, since any non-negative couple satisfying this equation is appropriate for the equilibrium. In particularcases when Nx = 0 or Nx = 0 we get two symmetric equilibria satisfying all requirements. Thus we have built aninterval 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 (x, x, Nx, Nx) satisfying equilibrium equations (20)-(22),(19) and con-struct cost c = λc > 0 for the standard monopolist’s problem. Obviously, this c and same x, x satisfy FOC andSOC 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 = x and forxs = x) is characterized by the same system of equations as for asymmetric equilibrium, but for assuming one oftwo types of producers absent; some Nxs = 0.

u′′(xs)xs + u′(xs) = λsc,

Nxsxsps = 1, u′(xs) = λsps,

(ps − c)xs =cxsru(xs)1− ru(xs)

= F/L. (23)

We should take two solutions (x, λ, Nx), ( x, λ, Nx) to these equations under parameters (c, F, L) and build c

suitable for two (equivalent to each other) standard-monopoly global maxima. The proof that such c exists, issimilar to (ii). Indeed, equilibria quntities x, x are the two roots of equation (23). Each gives the related priceps, from the same equation. Then, taking some λ and c = λc, pM = λp, pM = λp we see that standard-monopolyequations at both points x, x are satisfied and both these solutions give the same profit πM = xs(pM

s − c) =x(λp− λc) = x(λp− λc) = F/L under these costs.

What remains is ensuring that Lagrange multipliers λ,λ are the same at equilibrium x and at equilibrium x,i.e., that equations

u′′(x)x + u′(x) = λc, u′′(x)x + u′(x) = λc

have the same λ = λ. Indeed, each (s-th) Dixit-Stiglitz monopolist maximizes w.r.t. xs the per-consumer profitfunction πs(xs, c, λ) = xs(u′(xs)/λ− c). Its optimal value can be denoted π∗(λ) = x∗(λ)(u′(x∗(λ))/λ− c) , wherex∗(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 ∂∂λπ∗(λ) = −x∗u′(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∂∂λπ∗(λ) is invalid, because expression x∗u′(x∗)/λ > 0 with positive numerator and the function optimized decreasesw.r.t. λ. So, its optimal value also decreases. Therefore there is unique value λ = λ = λ such that global maximumof this function equals F/L. Thus we have proved as a by-product that all equations studied in item (i) are satisfiedand 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 con-

cavity of the profit function. For item (ii) we needLemma MR. When Assumption MR holds in specific version: marginal revenue has ony one interval of increase,

there is unique level of cost c yielding two global argmaxima x < x of profit, for other C 6= c the argmaximum isunique. Under small cost C < c the biggest quantity x brings the global maximum, and under big cost C > c thelowest quantity x is optimal, and both local argmaxima x(C) < x(C) decrease w.r.t. C.

Proof of Lemma MR. Denoting xi = x(C) or xi = x(C), by the Envelope theorem, the objective-functionπi(C) = xi(u′(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 π(C) = x(C)(u′(x(C)) − C) decreases fasterthan the small-quantity value π(C) = x(C)(u′(x(C))− C), for the same reason. Therefore these two function hasonly one intersection c and to the left from it (C < c) the big quantity is optimal, and reverse. The direction ofchanges follows from Lemma 1, whose boundary condition on marginal revenue ξ(x,C) = xu′′(x)−u′(x) is satisfied.Q.E.D.

Let us consider the comparative statics in L (dropping its accent) for any two roots: (x(L), λ(L)) and (x(L), λ(L)),of system (21)-(20) of equilibrium equations. By Lemma MR, there is unique value of λ(L) bringing the same levelF/L of per-consumer net-of-fixed-cost profit at both roots, λ(L) is monotone decreasing, and small root (x(L), λ(L))is valid global optimum for big λ (so, big L), and vice verse: big root (x(L), λ(L)) is valid for small L. This provesuniqueness of L where both roots of basic equilibrium equation ξ(M,L) = 0 are valid as equilibria. Besides, itproves the direction of “jump” in quantities: x switches downward to x when L passes L.

To ensure Corollary 1, we can just apply Lemma 1 to function ξ(., .) = ξ(M, L) : [0, 1] × R → R which iscontinuous (because r is continuous) and satisfy border conditions ξ(0, L) ≥ 0, ξ(1, L) ≤ 0∀L by Assumption 1on r. It guarantees monotonicity for border roots, which coincide under uniqueness. The difference between thepro-competitive (r′(x) > 0) and anti-competitive case (r′(x) < 0) lies only in the direction of influence of L onξ(M, L), the results are opposite. �

41