Non-concave Prot, Multiple Equilibria and Catastrophes in Monopolistic Competition ∗

Sergey Kokovin,

†

Alexey Gorn,

Evgeny Zhelobodko

Discussion paper, 14.02.2014

‡

§

Abstract

We study the classical Dixit-Stiglitz model with unspecied preferences but relax requirements of monotone marginal revenue (prot quasi-concavity), typically violated under distinct consumer groups. Then, equilibria still exist but can become asymmetric, set-valued. Under non-monotone marginal revenue, asymmetric equilibria must emerge during population growth; here rms ambiguously split into small producers and big producers. Related catastrophic jumps in consumption, price, and mass of rmsstill maintain unambiguous direction of changes. In the case of heterogeneous rms, the rms may split endogenously into high-productivity group and low-productivity group, with a gap in between; the cuto behaves as under concave prot.

JEL codes: L11, L13, F12, F15. Keywords: monopolistic competition, non-concave prot, multiple equilibria, catastrophes, discontinuous comparative statics.

1

Introduction

Our

motivation for studying non-concave producers' prot function includes:

(a) extending the market theory to

unexplored situations that may be realistic; (b) nding new eects, namely, jumps and gaps in market reactions to parameters. So far, global prot concavity remains a dominating technical assumption in any modeling, though it lacks clear empirical support or intuitive motivation. For instance, when the demand curve results from combining the curves of two distinct consumer groups having a chock-price, prot typically appears non-concave. Even one consumer group can generate non-concave prot when its demand is rather at, then makes a turn and again becomes at, which is not unnatural. We would like to model such situations, usually excluded for the sake of technical convenience. Can we just drop usual concavity assumption and thus expand the applicability of monopolistic competition concept?

positively

Would typical conclusions about equilibria existence, uniqueness and comparative statics remain true? This paper answers more or less

but changes the equilibrium concept.

∗ Sobolev Institute of Mathematics SBRAS, Novosibirsk State University, NRU Higher School of Economics (Russia).

Email:

[email protected]

† Bocconi University. Email: [email protected] ‡ 25.09.197327.03.2013. Novosibirsk State University, NRU Higher School of Economics. § We gratefully acknowledge grants RFBR 12-06-00174a and 11.G34.31.0059 from the Russian Government.

This study was also

supported by Economics Education and Research Consortium (EERC, funded by Eurasia Foundation, USAID, World Bank, GDN, and the Government of Sweden) grant No 08-036-2009. We are indebted to Jacques-Francois Thisse for advices and attention.

1

To outline the theoretical context, we recall that monopolistic competition model has become the work-horse of international trade, economic geography, growth theory, and other elds. Its key assumptions are price-making rms, free entry, increasing returns to scale and consumers' love for variety (incomplete substitution). Pioneered by Dixit and Stiglitz (1977), it mainly exploited constant elasticity (CES), then expanded to other tractable specications: quadratic (Ottaviano et al., 2002) and exponential (Behrens and Murata, 2007). At some stage, Krugman's (1979) general approach was revitalized independently by Bertoletti et al. (2008), Zhelobodko et al. (2010), Dhingra and Morrow (2011), Mrazova and Neary (2012), Zhelobodko et al.

1

ndings in comparative statics.

(2012, called further ZKPT)achieving various

Keeping this strand of theory, the present paper further expands monopolistic

competition modeling onto a broader range of markets. Namely, we extend ZKPT in the direction of possibly nonconcave prot. This case was never studied under monopolistic competition, up to our knowledge. How multiple or/and asymmetric equilibria arise? How can they change when market size increases because of trade or increasing population? These questions looking technical, still worth clarication: rst, to make all theoretical predictions robust against

always

any changes in utilities; second, to uncover new eects possible. To argue that non-concave prot function is not something too peculiar and unrealistic, we repeat that it

has a kink

arise when the demand curve is piece-wise smooth,

including realistic combinations of several demand curves having a chock-price. Essentially, such aggregate demand curve

, that yields non-monotone marginal revenue, i.e, non-concave prot. One should not think that

such non-monotonicity can be cured by small demand variations smoothing such kinks.

generic

No, non-concave prot

remains for all smooth approximations of kinked demands. The family of non-monotone marginal revenues is as much

and realistic, as the opposite family.

Summarizing, we cannot see any reasons to exclude non-concave prot from theory.

One more reason for

considering it in further study may arise from the supply side, because endogenous R&D (Vives, 2008) can make prot non-convex. Indeed, small R&D often brings too small eect, higher scale makes prot jumping up (yet, in this publication we conne ourselves to simple linear cost, no R&D). Probably, non-concave prot situations were neglected by theory only for technical hardshipshardships to be overcome now. Our

setting repeats ZKPT, but without prot concavity and without non-linear costs.

Namely, we study a closed

economy with one diversied sector, one homogenous production factorlabor, homogeneous monopolisticallycompetitive rms. The representative consumer's elementary utility is unspecied, satisfying weak natural restrictions, and gross utility is the sum of elementary utilities.

function

results characterizing such equilibria, we nd the weakest conditions when monopolistically-competitive

do exist

Among equilibria

.

Existence is guaranteed mainly by a natural boundary condition on elementary utility

xu0 (x) called elementary revenue

suitable for monopolistic competition

u(x):

must become zero at the origin (unlike logarithm utility), and suciently

decrease at innity. Thus we dene utilities

modeling. This is a very broad

class, avoiding some doubtful restrictions imposed typically. However, the equilibria studied may become set-valued and asymmetric. Asymmetry means coexistence of two or more kinds of equally-protable behavior of rms: those with big outputs and ones with small outputs. The masses of both types remain ambiguous, up to their weighed sum, satisfying the labor market clearing. Such set-valued asymmetric equilibrium is the main conceptual novelty of this paper.

To analyze such equilibria, we use ordinal technique of comparative statics from Milgrom and

Shannon (1994), applied to monopolistic competition in the manner like in Mrazova and Neary (2011). Further, to achieve more denite equilibrium structure, we impose a regularity restriction on our elementary utility

u(x):

two

there must be a nite number of kinks in the elementary inverse demand

non-degenerate, i.e., there cannot be more than and small outputs.

u0 (x)

and they must be

kinds of equally-protable rm behavior types: big outputs

Using the regularity assumption, Proposition 3 states the structure of a set-valued equilibrium. We show that

1 The

rst draft of this paper is Alexey Gorn's diploma (2009) at Novosibirsk State University, completed under S.Kokovin's super-

vision.

2

couple

the equilibrium value of the intensity of competition (marginal utility of income) is always can be multiple equilibrium outputs/prices, namely, a

unique

. However, there

together

of possible outputs and a couple of prices, whereas all

possible masses of rms constitute an interval. Equilibria multiplicity and asymmetry always arise

.

Further, we turn to comparative statics w.r.t. the market size. We consider complete path of evolution, when

if and only if utility u(x) generates non-monotone elementary marginal revenue the multiple-asymmetric equilibria do arise

the economy population continuously grows from zero to innity (or, equivalently, the evolution can result from a monotone decrease in costs). Proposition 4 shows that ,

at some moment during the market growth. Somewhat

non-trivial here is the idea, that monotonic decrease in marginal revenue everywhere is equivalent to prot

given

concavity in all possible market situations and simultaneously equivalent to prot quasi-concavity (by contrast, in a

must

situation, quasi-concavity is a weaker assumption). Rather surprising is also the necessity side of Proposition

5: that equilibria multiplicity

arise under non-monotone marginal revenue during market evolution, though

being a degenerate situationspoints on the path of growth. The explanation lies in continuity of such comparative statics. Intuitively, such market evolution works through the entry of new competitors and resulting gradually growing intensity of competition. When marginal revenue is non-monotone, at some moment rising marginal cost must hit

must

this interval of non-monotonicity, and thereby sooner or later equalizes two local maxima of prot. Here the rms split into groups with asymmetric behavior. Indeed, we give such numerical examples: multiplicity/asymmetry

situations do arise under some reasonable utilities.

jump catastrophic

Importantly, comparing situations before and after this split of rms into groups, we see that outputs and prices make a nitely-big

in response to innitely-small increase in the market size. In our numerical example, the

mass of rms (varieties) suddenly jumped up as much as 10 times and the price doubled! changes

We call such abrupt

, though these jumps are good for consumers under expanding market, bad under shrinking

market. Moreover, all asymmetric equilibria are non-equivalent for consumers' welfare. Studying in more detail the growing market, Proposition 6 establishes the direction of changes in prices/outputs.

up

Relying on the ordinal technique, we extend the comparative statics conclusions from ZKPT onto set-valued equilibria.

down

Namely, under growing market the mass of rms anyway goes

individual consumption of each variety always goes

up

, with or without jumps, and the

. As to prices, they jump

during a catastrophe, that

looks paradoxical under increasing competition of rms. Explaining similar paradox in smooth comparative statics, ZKPT exploits elasticity of the inverse demand. Decreasingly-elastic demands, called also super-convex (Mrazova and Neary, 2011), necessarily make the prices go up together with the population and mass of rms. similarly, the

increase

demand curve at the zone of any kink appears too much convex. That's why the upward jump of the equilibrium mass of rms brings a paradoxical

in prices. Firms' outputs, in spite of jumps, follow the same patterns

as in ZKPT. Namely, most realistic sub-convex demands show increasing outputs, whereas super-convex demands display the opposite pattern, CES utility being the neutral borderline. Extending our concept onto rms' heterogeneity as in Melitz (2003), we nd that if the interval of operating rms' types intersects with the area of any kink, rms population splits into two or more distinct clusters, with a gap between them. Namely, highly productive rms choose to produce high quantity while low productive rms choose low quantity and there is a signicant gap in production quantity of these two groups. Further, in spite of non-concave prot novelty, we nd that equilibrium cuto reacts to market size exactly like in ZKPT: minimal productivity increases under increasingly-elastic demand and decreases in the opposite case.

practical

As theorists, we are satised that most market regularities, found under global prot concavity and symmetric equilibrium, become now extended to more general situations.

Can a

economists also learn something

from our ndings? It depends upon realism of our new eect found: market jumps. If non-monotone marginal

small causes can bring great market consequences

revenue (very convex interval of the demand curve) is not excluded in reality, then our paper means a warning is that

, abrupt changes.

The next section introduces the model, developing all the notions of set-valued equilibria and motivates non-

3

concave prot. Section 3 displays existence of multiple and asymmetric equilibria, Section 4 presents comparative statics of set-valued equilibria. Section 5 presents implications of our concept under rms' heterogeneity.

2 Model and examples We study one-sector closed economy with monopolistic competition a'la Dixit-Stiglitz but with general (unspecied) utility function. In doing so, we follow ZKPT but impose less restrictions. Our economy involves one production factorlabor. of rms/varieties with same index

i ∈ [0, n]

One dierentiated good is split into continuum

[0, n]

because each variety is produced by one single-product rm (n is

endogenous).

Demand.

Labor is chosen as the numéraire. There are

inelastically, so that consumption vector constraint:

2

1 is both x = xi≤n

L identical consumers who supply each one unit of labor

a worker's income and expenditure. Every consumer chooses an innite-dimensional (a measurable consumption function) to maximize her utility subject to the budget

ˆ

ˆ

n

max U ≡

u(xi )di

x(.)

pi xi di = 1.

0

(1)

0

p : [0, n] → R+ ,

Here we use innite-dimensional price vector

n

s.t. where

pi ≡ p(i)

is the price of

i-th

variety;

xi ≡ x(i)

denotes i-th consumption. One of our goals is to formulate the weakest restriction on utilities suitable for possibility

non-Inada conditions

of monopolistic competition in the market, as follows.

Assumption

1 (

dierentiable and strictly concave on

).

Elementary utility function

(0, ∞),

denotes (nite or innite) argmaximum of function utility

u(·) : R+ 7→ R+ is thrice continuously [0, xmax ), where xmax ≤ ∞

increasing on some non-empty interval

Ru (x) ≡ xu0 (x),

called elementary revenue.

At the origin,

u(·) is normalized as u(0) = 0, and generates increasing strictly concave bounded revenue Ru (0), in the sense lim Ru (x) = 0, M R ≡ lim Ru0 (x) > 0,

At the elementary revenue's argmaximum point either

MR ≡

lim Ru00 (x) < 0.

x→+0

x→+0

lim

x→xmax

xmax ,

we require concavity:

Ru0 (x) = 0

or

lim

c→+u0min

3

Strict concavity of

u implies that gross utility U

limx→xmax Ru00 (x) < 0

and condition:

max[Ru (x) − cx] = ∞.

We call all such utilities suitable for monopolistic competition modeling throughout.

(2)

x→+0

(3)

x

(MC-suitable)

and maintain this assumption

displays some love for variety, because, as in risk-taking theory,

strictly concave elementary utility entails strictly convex preferences. Thereby under uniform prices across varieties, the consumer strictly prefers buying a mixture of varieties rather than any single variety. Like in risk-taking, to express this preference for mixture, we can exploit Arrow-Pratt measure

ru (xi ) ≡ −

ru

of concavity for any function

xi u00 (xi ) > 0. u0 (xi )

2 Prots are not included into income, because they vanish under free entry. 3 Assumption 1 rules out some neoclassical utilities like log(x) ± bx or −1/x. The

(4)

assumption is not technical: the functions excluded

really do not suit any monopolistic competition model, because related prot may remain positive and increasing at at innite

x.

x → 0,

or increases

But, unlike restrictive Inada conditions, our formulation allows for all utilities useful in monopolistic competition models:

quadratic utilities

u = ax − bx2 that have a satiation point x ˇ0 = 0.5a/b and chock-price u0 (0) < ∞, CES utilties u(x) = xρ : ρ < 1 that 0, AHARA utilties u(x) = (a+x)ρ −aρ ±bx that may have positive limiting derivative u0min ≡ limx→∞ u0 (x) =

have innite derivative at

b > 0,

u:

and many others.

4

relative love for variety inverse of the elasticity of substitution across varieties σ(x), i.e., r (z) = 1/σ(z) Applied to utility,

ru

in ZKPT is called

of the elasticity of substitution

σ,

. Importantly for price-making, at

u

xi the demand elasticity ru = 1 − ρ ≡ σ1 ; 0 < ρ < 1).

any

σ(xi ).

is also equal to

(In particular, CES utility

u(xi ) = xρi

Further, using FOC of consumers' optimization and denoting the Lagrange multiplier as demand function

the RLV is the

(hereafter, RLV). Using the standard denition

it is easy to show that at a symmetric consumption pattern

implies constant RLV

λ, we obtain the inverse

p∗ : p∗ (xi , λ) = u0 (xi )/λ.

Thus, marginal utility of income

λ

(5)

makes each demand shrinking, and therefore measures the intensity of competi-

tion.

Supply.

We assume identical rms: to produce output

c is the marginal cost and f is the xed cost of p∗ (xi , λ) for i-th variety and given λ, i-th rm maximize where

π(xi , λ, c) ≡ (

competition

We see that current marginal utility of money

λ

qi = Lxi

business.

each rm

i

must spend

cqi + f

units of labor,

Taking as given the inverse demand function

its per-consumer operational prot

4

π

w.r.t. quantity:

u0 (xi ) − c)xi → max . xi ∈R+ λ

(6)

becomes the single market statistic expressing

, like price index under CES modeling.

Now we can introduce notations for the maximal value

πu∗

intensity of

of the per-consumer prot function, for the set

Xu∗

of

prot-maximizers, and formulate the rm's survival or free-entry condition:

Xu∗ (λ, c) ≡ arg max π(xi , λ, c). xi ∈R+

πu∗ (λ, c) ≡ max π(xi , λ, c), xi ∈R+

f ≤ Lπu∗ (λ, c). Note that both mappings

among rms' decisions x

πu∗

and

Xu∗

do not have index

i

Xu∗

mR : mR (xi , λ) ≡

prot is strictly concave

u.

Still, the program above

(8)

across

is not a singleton. Such possible asymmetry is the essence of our paper. To

explain it, we recall usual FOC for maximizing prot marginal revenue

same allows for asymmetry

because the optimization program is the

rms, so both mappings are just characteristics of function

i when

(7)

π(x, λ, c)

formulated as equality between marginal cost and

xi u00 (xi ) + u0 (xi ) = c. λ

(9)

is strictly decreasing

As to SOC, the second derivative of prot (6) being the rst derivative of marginal revenue (9), it standardly follows that at

x.

In particular, when

u000

at

x if and only if the elementary marginal revenue mR (x, 1) u0 looks

exists, such restriction on utility in terms of concavity of

ru0 (x) ≡ −

like (see ZKPT):

xu000 (x) < 2. u00 (x)

4 Standardly, prot maximization w.r.t. price gives an equivalent result. Also, equivalent is maximization of gross prot

Lπ(qi /L, λ, c) − f

w.r.t. output

qi .

5

(10)

Figure 1: Non-monotone marginal revenue and two peaks of prot under utility

u(x) = x +

√

x + 1.4 arctan(2x +

0.05) − 1.4 arctan(0.05). In ZKPT this requirement is imposed globally (∀x

> 0),

being a condition for unique symmetric equilibrium, but

our goal here is the opposite and we do not require it (it naturally holds locally at local optima). Now we are ready to construct a complicated notion of set-valued equilibria. But to motivate it, we rst present an example illustrating everything throughout. In some sense, it immediately presents all main ideas of our paper.

Example 1, introducing non-concave prot. u(x) = x + Algebraically, this function

u

√

To illustrate our reasoning, we use throughout the utility

x + 1.4 arctan(2x + 0.05) − 1.4 arctan(0.05).

(11)

looks rather exotic, but nevertheless it satises Assumption 1, i.e., it is increasing at

0, strictly concave, smooth, etc. The only specic feature diering from textbook examples like CES or quadratic functions, is that related marginal revenue becomes non-monotone and thereby related prot is non-concave. Specifically, in Fig.1, we take parameters

,

L/f ≈ 10.04 λc ≈ 0.00038 suitable for two global argmaxima of prot (see Section

xi (p∗ (xi , λ) − c) is not locally strictly M R(x) ≡ [xi p∗ (xi , 1)]0 = u0 (xi ) + xi u00 (xi ) is

4 for numerical details). One can see that the producer's operational prot concave only at those points where the normalized marginal revenue increasing: approximately from 0.9 to 4.8. Here, as usual, the marginal revenue

MR

intersects the normalized marginal cost

the prot function could reach local maxima or minima. the local minimum.

M C = λc

at points where

The middle intersection, approximately

x ≈ 2.5,

odd

is

For us important are only the leftmost and the rightmost intersections  local argmaxima

x ˆ(λc) ≈ 0.65 < x ˇ(λc) ≈ 12.9.

More generally, under

K

intersections, local maxima must be among the

FOC equation (9), which can be reformulated as:

roots of

(u00 (x) + xu0 (x) − λc) = 0. Under some value

λc,

indierent

like in this picture, two or more local maxima become global, bringing the same prot.

Then each producer becomes

which optimal quantity to produce:

In the case of such ambiguity, we denote by output), and by

n ˇ

n ˆ

the mass of rms choosing the bigger output

of both rm types as

(cˆ xL + f ), (cˇ xL + f ),

x ˆ

or

x ˇ.

the unknown mass of rms who chose the left optimum

x ˇ.

x ˆ

(small

Using these notations, we express the total costs

and formulate the labor balance (labor market clearing condition):

(cˆ xL + f )ˆ n + (cˇ xL + f )ˇ n = L. 6

(12)

To dene a certain kind of market equilibrium, this balance equation should be combined with free entry, consumers' and producers' FOC. Luckily, the equilibrium equations are not simultaneous. One can rst nd the equilibrium value of marginal utility of income

λ

from inequality (8) turning into the free-entry equation:

πu∗ (λ, c) = f /L.

(13)

λ, we nd equilibrium consumptions (ˆ x, x ˇ) from (9), then both prices are determined from (5). Finally, n ˇ and n ˆ from the labor balance (12). Here some diculty arise: nding two variables from one equation is impossible. Thereby some indeterminacy always remains in the rms' masses (ˆ n, n ˇ ). In other words, there exists the whole interval of possible couples (ˆ n, n ˇ ) that satisfy all equilibrium conditions, Then, having

we seek for the masses of groups of rms

set of equilibria n ˆ, n ˇ asymmetry and multiplicity of equilibria always come together

ambiguity cannot be excluded. Thereby, instead of a single equilibrium arising under strictly concave prot, in any

x ˆ
situation with two argmaxima

we get a continuous

dierently

: any couple

satisfying the labor

balance (and therefore the consumers' budget constraint) satises the idea of asymmetric equilibrium. Thus, we see that identical rms may behave

. Besides,

.

Now we can generalize this example from two argmaxima to nitely-many prot argmaxima

1,

and formulate related notion of a set-valued equilibrium.

Asymmetric equilibria and set-valued equilibria.

5

X = (x1 , ..., xK ) K ≥

Consider K ≥ 1 types of rms' behavior, and a bundle λ > 0  the level of competition, X = (x1 , ..., xK ) ∈ RK +  the vectorK of consumptions K bringing maximal prot, P = (p1 , ..., pK ) ∈ R+  the vector of prices, and N = (n1 , ..., nK ) ∈ R+  the masses of rms' types. This z is called a (free-entry) equilibrium when λ satises the free-entry equation (13), X satisfy 0 optimization necessary conditions (FOC:  (9)) and sucient conditions (SOC: mR (x, 1) < 0), prices t the demand ∗ ˆ rule pk = p (xk , λ) and masses of rms t the labor balance:

z = (λ, X, P, N )

consisting of

K X

(cxk L + f )nk = L.

(14)

k When

K ≥ 2, 0 < xi < xj ∃i, j ,

this equilibrium is called

the named conditions under given exogenous parameters case when

K=1

this

Z = {z}

asymmetric ;

set-valued equilibrium. unique and symmetric

then simplex

(u, c, L, f ) is called a

Z

of all bundles satisfying all In the special

becomes a singleton and such equilibrium is called

.

After stating the existence theorem, in our comparative statics we would like to reduce generality of possible equilibrium structures to

K ≤ 2.

Assumption 2 (κ-regular

Then the following regularity restriction on utility

u

will be used to rule out any

degenerate outcomes like three or more global maxima, lying on the same line.

u).

(i) Under

λ = 1, ∀c,

the number of prot argmaxima do not exceed 2:

#| arg max π(x, 1, c)| ≤ 2 ∀c x∈R+ π(x, 1, 0) ≡ xu0 (x) more than at two points simultaneously, dominating the elementary revenue in the sense f (x) ≥ π(x, 1, 0) ∀x). (ii) There can be only a nite number κ ≥ 0 of magnitudes ck≤κ : #| arg maxx∈R+ π(x, 1, ck )| = 2 bringing multiplicity of argmaxima (i.e., there can be only a nite number κ ≥ 0 of the dominating lines with double tangency to π(x, 1, 0)). Such utility is called κ .

(i.e., any line

f (x) = ax + b

cannot be tangent to the elementary revenue

-regular

5 An

extension of this denition is possible: instead of nite

can use the same denition for innite-dimensional

X

K,

and nite-dimensional argmaxima vector

X = (x1 , x2 , x3 , ...),

one

arising when prot function includes linear intervals. Then the summation in

condition (14) should be understood as an integral. Probably, we can include this innite-dimensional case into our further theorems without changing anything in formulations and proofs but the extension is not too important.

7

u

2.0

x u’ 1.5 MR=xu’’+u’

profitHx1L

u’

1.0 c

0.5

xHu’-cL x1

x2 x

0.5

1.0

1.5

Figure 2: Three-peaks non-concave prot under utility

Geometrically,

κ = 0,

2.0

u(x) = 2x − x2 + 0.5x3 − 0.1x4 +

1 3600

sin[4px].

κ ≥ 0 is the number of sags (non-concave intervals) in the curve of elementary revenue;

the revenue appears strictly concave.

the elementary revenue

xu0 (x),

whenever

More precisely, we can take the convex hull of the undergraph of

and dene function

Rconv

as the upper envelope of this convex hull

Runder ≡

0

conv{(x, r)|r ≤ xu (x)}. Then, each sag is the at (linear) interval of

Rconv ,

and

κ

is the number of such intervals. These notions and

the regularity assumption help us in the next section to economize notations by ensuring specic dimensionality of all asymmetric equilibria

7 ˆ X, P, N ) = (λ, ˆ (ˆ (λ, x, x ˇ), (ˆ p, pˇ), (ˇ n, n ˆ )) ∈ R+

Example 2, explaining Assumption 2.

revenue is not necessarily the number

κ

for all values (Lc/f ).

Figure 2 shows that the number of increasing intervals in the marginal

of sags, i.e., possible asymmetry situations.

1 u(x) = 2x − x2 + 0.5x3 − 0.1x4 + 3600 sin[4px]. So, the (normalized) inverse demand u decreases. Here the cost value c = 0.883965 crosses marginal revenue (MR) as much as times, but only of these crossings relate to global argmaxima {ˆ x, x ˇ} = {0.54966, 1.40512} of 0 operational prot xu (x) − cx. This function is the red curve, whereas cost c(x) = 0.883965 is the solid blue line, 0 and the sloped dashed line shows the tangent line ax + b dominating the elementary revenue π(x, 1, 0) = xu (x). In Fig.2 we use exotic but regular and concave utility function

ve

two

This revenue has

global

of prot

ˆ = 3). (k

two

0

However, there is only

prot maxima

one essential

intervals where it is not concave (i.e., where MR increases), that generates

x ˆ, x ˇ (K = 2).

because the dashed line tangent to

sag in the revenue graph (κ

Indeed, the middle peak in the revenue curve

xu0 (x)

three local two inessential

maxima

= 1) and it generates xu0 (x) is too small,

only

cannot become global prot maxima are inessential.

degenerate.

always

To understand regularity of asymmetric equilibria phenomenon, note that this twin-peak example is

some

c

Indeed, any small changes in utility

yielding twin maxima

#|X ∗ (c)| = 2.

necessity

,

does not touch the middle peak. In other words, those prot peaks that

u

do not change the essence of the picture: there

non-

exists

Similarly, our guiding example is non-degenerate. More generally, all

functions generating non-monotone MR is a broad class, where each function generates some asymmetric equilibria under some

c

degenerate.

Example 3,

with

.

In contrast, any example with 3 equivalent peaks

motivating kinked demand.

#|X ∗ (c) | = 3

for some

c

is

Having discussed non-monotone marginal revenue, we would like

8

one

to support its realism by two or more distinct consumer groups. heterogeneity, we think of

To avoid a complicated model of consumers'

economic agent representing a household containing several members. Let it contain

two members: Adam and Eve, who jointly spend their income, maximizing the sum of their utilities

uA

and

uE

from various apples in the following form:

ˆ max

xA (.),xE (.)

ˆ

n


E E uA (xA i ) + u (xi ) di

U≡

E

a >a




di

= 1.

0

0

Consider a particular case with quadratic utility functions:

A

n E p i xA i + xi

s.t.

  uj xji = aj xji −

bj 2

 2 xji , j ∈ {A, E},

. In this case we can easily nd the FOCs for consumption of each member of the household:

j

a

such that

−bj xji

= pi λ.

A symmetric equilibrium can display a corner solution where only Adam consume any apples:

aE ≤ aA − bA

E pn

Rearranging the last inequality we express the choke price for Eve:

pE =

bA E (aA − ac ) n

Now we can nd the inverse demand function for a single variety:

( ∗

p (x, λ) =

aA −bA x λ aA bE +aE bA −bA bE x (bA +bE )λ

A

E

if x ≤ a b−a A aA −aE if x > bA

One can check that resulting marginal revenue is not monotone and even dis-continuous, consisting of two linear pieces. Here the prot function has two local maxima. Thus, we argue that distinct consumer groups like husbands and wives, or parents and children can be the source of demand kinks that we study. In this case, a producer faces two distinct choices: either to serve both groups or one of them, and switching between these two strategies may generate market jumps that we are interested in. Our further plan is to explore existence of equilibria, reveal their structure, and study their responses to the market expansion, e.g., population growth, or countries' integration, or technological shocks.

3 Equilibrium existence and structure We start with the properties of the prot argmaxima

Xu∗ (λ, c)

and optimal per-consumer prot

(7). Rather obviously, for determining these argmaxima under given

λ, c,

πu∗ (λ, c),

dened in

the following formulations are equivalent:

Xu∗ (λ, c) ≡ arg max π(x, λ, c) = arg max λπ(x, λ, c) : x∈R+ x∈R+

(15)

λπ(x, λ, c) ≡ π(x, 1, λc) ≡ [u0 (x) − λc]x.

(16)

∗ ∗ This reformulation allows us to use further the single-argument mapping Xu (λc) ≡ Xu (λ, c) interchangeably with ∗ two-argument one (with a little abuse in notation). Such trick simplies our reasoning about reaction of Xu to changing arguments competition

λ

λ

or

c,

k -times the intensity of k -times the marginal cost

or both. Economically, this equivalency means that increasing

aects the rm's optimal output

Lx

exactly in the same way as increasing

c. 9

arg maxx∈R+ [u0 (x) − λc]x means the following. Slope λc given, the rm should 0 vary a for nding the highest line a + λcx tangent to the elementary revenue curve Ru (x) ≡ xu (x) (painted purple ∗ in Fig.2, whereas a+λcx is painted dashing-blue). Then the tangency points are the argmaxima Xu (λc). Obviously, Geometrically (see Fig.2), nding

they

exist if and only if λc < M R,

At smaller

λc > M R ≡

the objective function

Ru0 (x).

lim

x→xmax

[u0 (x) − λc]x

(17)

is unbounded when

x → ∞;

even under

λc = M R > 0

this function goes to innity by Assumption 1. Now consider the argmaxima comparative statics. When we increase the slope our (set-valued) argmaxima, irrespectively, is the undergraph of

monotone nonincreasing

[u0 (x) − λc]x

λc,

we always induce

convex or not.

decrease

in

X: R→ ¯>λ ˜ implies x ∧ x ˜ and x ∨ x ¯ for λ ˜ ∈ X(λ) ˜ ∈ X(λ) ∨ is maximum, thereby the extreme members do

To express this idea rigorously, let us dene three kinds of decreasing mapping (set-valued function)

2R .

X(λ) ¯ and x ˜ x ∈ X(λ) ˜ ∈ X(λ)

We call

every

not increase).

6

if a bigger argument

∧ denotes minimum X(λ) (strictly)

decreasing

(where

We call a mapping

¯>λ ˜ ⇒ min < min λ ¯ x∈X(λ)

We call

X

strongly decreasing

and

, when its extreme members decrease in the sense

and

˜ x ˜∈X(λ)

max < max .

¯ x∈X(λ)

(18)

˜ x ˜∈X(λ)

, when all its selections decrease in the sense

¯>λ ˜⇒x ¯ ∀˜ ˜ λ ¯
X

(19) single-valued everywhere, excluding

isolated points (downward jumps). To reveal, step by step, all these types of monotonicity (in the cost were by

¯ X(λ)

c=1

(λc))

of our argmaximum

(we just economize notation, the same logic works for any

c

Xu∗ ,

we shall argue as if

or any changes in

λc).

We denote

all roots of the FOC equation:

¯ X(λ) ≡ {x ≥ 0| u0 (x) + xu00 (x) − λ = 0}, this set including global argmaxima

¯ , Xu∗ (λ) ⊂ X(λ)

(20)

and maybe some other extrema.

Now we can apply the

7

following lemma, which is a version of a theorem from Milgrom and Roberts (1994, Theorem 1).

x ˆ≤x ˇ

It predicts

Assume a partially ordered set Λ, some bounds x > x of the domain and a parametrized function g(., .) = g(x, λ) : [x, x] × Λ → R which is continuous and weakly changes the sign, in the sense [g(x, λ) ≥ 0 & g(x, λ) ≤ 0 ∀λ ∈ Λ]. Then for all λ ∈ Λ: (i) there exist some non-negative roots of equation g(x, λ) = 0, including the lowest solution xˆ ≡ sup{x|g(x, λ) ≥ 0 and the highest solution x ˇ ≡ inf{x|g(x, λ) ≤ 0}, these can coincide; 8 (ii) if our function g(x, λ) is non-increasing w.r.t. λ everywhere, then both extreme roots xˆ(λ), xˇ(λ) are non¯ increasing w.r.t. λ, i.e., mapping X(λ) is a nonincreasing one; (iii) if, moreover, g(x, λ) is decreasing in λ and strictly¯ changes the sign [g(x, λ) > 0 & g(x, λ) < 0 ∀λ ∈ Λ], then both extreme roots xˆ, xˇ are decreasing, i.e., mapping X(λ) is a decreasing one. monotone comparative statics of both extreme roots

Lemma 1.

6 This than

of any equation

g(x, λ) = 0

λ.

(Monotone roots, Milgrom and Roberts):

terminology follows Milgrom and Shannon (1994), who use it for lattices: strong set order

X ),

with a parameter

if for every

x∈X

and

y ∈ Y, x∧y ∈ X

and

x∨y ∈ Y.

≤s

X ≤s Y (Y is higher (λ, X) ⊂ R2 to be a lane

says that

Such order requires our simple mapping

without any increases in its extreme members. Similar is the notion of nondecreasing mapping, actually used in Milgrom and Shannon.

7 Their

original Theorem 1 diers in using function

[x, x] = [0, 1].

8 Naturally,

g(x, t)

non-decreasing in

t,

continuous but for upward jumps, and domain

This makes a minor dierence. when the roots are nite,

x ˆ(λ) = min{x|g(x, λ) = 0}, x ˇ(λ) = max{x|g(x, λ) = 0}).

10

The intuition behind this lemma is simple: when we shift down any continuous curve whose left wing is above zero and the right one is belowthe roots should decline.

More subtle fact is that when some isolated root

x ˇ

disappears or emerges, the jump goes in the same direction as all continuous changes, i.e., downward. We apply this lemma to the (continuous) auxiliary function

g

gained from FOC of

π(x, 1, λ):

g(x, λ) ≡ [u0 (x) + xu00 (x) − λ], ¯ is nonincreasing. We would like to enforce this property; Λ = [0, ∞). We conclude that mapping X ¯ to nd decreasing X at those λ and domains [x, x], where we can apply claim (iii). Locally, this task is easy: at a ´ > 0 of any positive local argmaximum x ´|g(´ x, λ) = 0whenever given λ, we can apply (iii) to any vicinity (x, x) 3 x 0 00 strict SOC holds. The latter means that u (x) + xu (x) decreases at x ´, i.e., it is an isolated argmaximum. Thereby, x ´=x ´(λ) λ. ¯ , on a positive ray we would like to identify a subinterval (λmin , λmax ) ⊂ Searching for globally decreasing X [0, ∞) where claim (iii) is applicable. This amounts to nding an area where all roots of equation (20) are positive using domain

any positive local argmaximum

satisfying strict SOClocally decreases w.r.t.

and nite, under Assumption 1.

Lowest

yielding

Ru (x) = xu0 (x) has a nite global argmaximum xmax (that implies satiable demand). Then, obviously, all positive λ enable solutions 0 00 to (20), i.e., we can take the lower bound of the needed interval as λmin = M R = u (xmax ) + xmax u (xmax ) = 0 0 (using notations from (3)). Similarly, under insatiable demand (xmax = ∞) but zero limiting value limx→∞ (u (x) + 00 xu (x)) = 0, the inmum of all λ bringing nite roots is λmin = M R = 0. The√third possible case is when at 0 innity Ru remains positive: our parameter M R > 0 (an example is utility u = x + x, in such cases this M R becomes the parameter delimiting the situations λ where mapping X(λ) is nite). However, the outcome is the same and we conclude that anyway we must take (zero or positive) lower bound λmin = M R when we search for an interval (λmin , λmax ) bringing positive nite roots of function g . Highest λ yielding x ∈ (0, ∞). Recall notation M R from (2) and consider the case of nite derivative at the origin (M R < ∞), that implies chock-price. Then all suciently high parameters λ ≥ M R should bring zero solutions x ˆ(λ) = x ˇ(λ) = 0 to (20), for lower parameters the solutions are positive. In the case of innite derivative M R = ∞ all λ must bring positive x. We conclude that anyway we must take nite or innite λmax = M R as an λ

x ∈ (0, ∞).

nite

Consider the case when our elementary revenue

upper boundary, that determines the open interval

ˆ ≡ (λmin , λmax ) ≡ (M R, M R), Λ which brings positive nite roots of concavity of

xu0 (x)

at 0 and at

xmax

g.

It also strictly decreases in

x

at both boundaries (x, x), because of strict

(Assumption 1). This ensures that outside interval

ˆ Λ

the roots of

g

cannot be

positive and nite, that we use in Theorem 1.

ˆ , because our function g(x, λ) ≡ [u0 (x) + xu00 (x) − λ] takes positive Λ ˆ ). value M R − λ > 0 at the lower boundary x = 0 and negative value M R − λ < 0 at x = xmax (for all λ ∈ Λ Moreover, g remains strictly decreasing in λ. Thus, our function g(x, λ) satises the boundary conditions and monotonicity conditions needed for Lemma 1-(iii). This implies x ˆ(λ) ≤ x ˇ(λ) on ˆ. Λ It must be added that both extreme roots x ˆ(λ) ≤ x ˇ(λ) of (20) are the local maxima (not minima) of function π(x, 1, λ) ≡ xu0 (x) − λx, because of SOC. Indeed, by denition of x ˆ, x ˇ, function g(x, λ) > 0 must be (strictly) decreasing in some left vicinity of the left point x ˆ, and in some right vicinity of x ˇ. Using continuous dierentiability 0 ˆ, x ˇ. This of xu (x) (Assumption 1) we expand this decrease to complete (left and right) vicinities of each point x 0 decrease of g(x, λ) ≡ π (x, 1, λ) means SOC. We can summarize our arguments as follows. Proposition 1 (Monotone local argmaxima). π(x, 1, λ) λ≥0 Now we can apply claim (iii) to this interval

strict decrease of the extreme roots

nonincreasing w.r.t. parameter

Each local argmaximum of the normalized prot is . Moreover, the local argmaximum decreases when being positive and nite, 11

which is guaranteed only on interval Λˆ ≡ (M R, M R). In the case of (suciently small) positive parameters λ ∈ (0, M R] all argmaxima are innite, under (suciently big) nite λ ∈ [M R, ∞) all argmaxima are zero global X

. ∗ u we use single crossing notion and Theorems 4, 4' from Milgrom and Shannon (1994) simplied here for our case of real parameter t and unidimensional Now, to establish similar monotonic behavior of

real domain

S(t)

of maximizers.

g

satises the

property means

g : R2 → R.

g(x0 , t”) ≥ g(x”, t”)

strict single crossing

Consider a function then

argmaxima set

If

implies inequality

property w.r.t.

(x; t)

couple

g(x0 , t0 ) > g(x”, t0 ) ∀(x0 > x”, t0 > t”),

of arguments.

[g(x0 , t”) ≥ g(x”, t”) ⇒ g(x0 , t0 ) ≥ g(x”, t0 ) ∀(x0 > x”, t0 > t”)

single crossing

Similarly,

and

g(x0 , t”) > g(x”, t”) ⇒ g(x0 , t0 ) > g(x”, t0 ) ∀(x0 > x”, t0 > t”)] (essentially, in these two versions of single-crossing notion, parameter

t

strictly or weakly amplies monotonicity of

Lemma 2 Consider a domain S(t) : R → 2 which is nonshrinking w.r.t. t (nondecreasing by inclusion) and a function g : R → R. If g satises the single crossing property in (x; t), then arg max g(x, t) is monotone nondecreasing in t If g satises the strict single crossing property in (x; t), then every selection x (t) from arg max g(x, t) is monotone nondecreasing in t g

w.r.t.

x,

alike supermodularity).

R

(Monotone argmaxima, Milgrom and Shannon).

2

.

x∈S(t)

∗

.

x∈S(t)

Unfortunately, this lemma does not predict strict increase that we need. But, it is important for us that the

open

latter claim about all selection implies that all points of multi-valued no

g(., t)

are isolated, in the sense that there is

interval of multi-valuedness (single-valuedness holds everywhere except isolated points). The third result

that we needis similar to envelope Theorems 1, 2 from Milgrom and Segal (2002) but formulated here for more

Lemma 3 (Monotone maxima, Milgrom, Segal) Consider a compact choice set X , a continuous function g(x, t) :

simple conditions as (trivial) Lemma 3.

and its maximal value π (t) = sup g(x, t) If π(x, t) continuously decreases in t for all x ∈ X, t ∈ , then its maximal value π (t) continuously decreases in t ∈ (0, 1) ∗

X × [0, 1] → R (0, 1)

Now, using our notations

.

x∈X

∗

.

M R, M R, xmax

and new notions

λcmin ≡ M R/c, λcmax ≡ M R/c, Rmax ≡ xmax u0 (xmax ),

Proposition Consider some given c > 0 and parameter λ increasing in the open interval Λ(c) = (λ , λ ). Then on Λ(c) the argmaxima set X (λc) ≡ arg max [xu (x) − λcx] is non-empty and strongly decreases from x to 0 (i.e., all its selections decrease); simultaneously the objective function π (1, λc) ≡ x[u (x) − λc] continuously decreases from R to 0. Outside Λ(c), under smaller parameter λ≤λ all argmaxima and maxima remain innite, under bigger λ ≥ λ all argmaxima and maxima remain zero (that can become innite) we formulate and prove the result we were long driving to. 2. (Monotone argmaxima and maxima):

cmin

∗ u

cmax

x

0

max

∗ u

0

max

cmin

cmax

.

Proof. Using S ≡ R+ , we can apply Lemma 2 to our auxiliary function g(x, t) ≡ π(x, 1, −t) with argument λc = −t > 0 because evident is strict single crossing property: it means increase of πx0 (x, 1, −t) = u0 (x) + xu00 (x) + t ∗ w.r.t. t. Thereby, whenever Xu (λc) exists, x∗ (λc) from Xu∗ (λc) is monotone when ∗ x > 0. This yields almost-everywhere single-valued Xu (λc), i.e., absence of any open intervals for λ maintaining ∗ ∗ multi-valued Xu (λc). In other words, Xu (λc) is single-valued, except for some isolated downward jumps. In 0 0 essence, this fact follows from smoothness of xu (x) (Assumption 1). Smoothness makes function πx (x, 1, −t) single-

every selection

nonincreasing

valued and strict single crossing property applicable (geometrically, the reason for strictly decreasing argmaximum

xu0 (x)cannot

strongly decreasing X (λc)

λc at a given point x). ∗ ∗ on interval Λ(c) (at nite positive Xu ), u we apply Proposition 1 used for any local maximum. Since global maxima should be among the local ones, in the is that a smooth setundergraph of

To transform the monotonicity found into

have multiple tangent slopes

12

Xu∗ = x ˆ=x ˇ they must strictly decrease.

downward

intervals of single-valued

strongly

The remaining isolated points of multi-valued

∗ are the points of jumps, as we have found. Thus, we conclude that mapping Xu (λ) on interval Λ(c), remaining innite for smaller λ and remaining zero for higher λ. Now we turn to the value function and apply Lemma 3 to ensure monotonicity of maximal

Xu∗

decreases

πu∗ (1, λc).9

Indeed,

πu (x, 1, λc) continuously decreases w.r.t. λ everywhere under positive x. Thereby its optimal ∗ continuously decreases when positive, i.e., on our interval (λcmin , λcmax ). The optimal value πu → 0

the objective function value when

πu∗ also λ → λcmax

because of monotonicity and zero lower bound of prot found in Proposition 1. So, continuity at

Λ(c) is maintained. Similar logic proves continuity at the lower boundary when M R > 0, the continuity of πu∗ becomes more delicate, questionable at point be violated for a utility like u(x) = ln(x + a) − ln(a) + bx (a, b > 0), because here from above for all λ ≥ M R/c = b/c but abruptly jumps to innity under

the upper boundary of our interval

M R = 0. Only λcmin = M R/c.

in special case when Continuity could

the maximal prot value remains any lower parameter

λ < M R/c.

bounded

X ≡ [0, x ¯]

exotic utilities. Using Lemma 3 with any compactied domain response of

πu∗ (1, λc)

any x¯

continuous

However, revenue unboundedness requirement (3) in Assumption 1 excludes such

to decreasing

λ → 0.

under

Therefore, if there were a jump of

πu∗

, we guarantee a

X ≡ [0, ∞) it ∗ value of πu (1, λc)

under open domain

λc ∈ [0, ∞) Q.E.D continuously M R The value of maximal per-consumer prot π (λ, c) ≡ x [u (x )/λ − c] continuously decreases, changing from value M R/λ to 0 on interval Λ(c); it remains zero on [λ , ∞) (whenever this interval exists); it remains innite on [0, λ ) (whenever this interval exists); and anyway would occur also on some compactied domain, that contradicts our ndings. Thus, the maximal

decreases

to 0 under increasing

from

Corollary 1.

. This completes the proof.

∗ (λ)

∗ u

cmin

0

∗ (λ)

cmax

cmin

lim πu∗ (λ, c) = ∞, lim πu∗ (λ, c) = 0.

λ→0

(21)

λ→∞

This corollary is obvious, we just transform function

πu∗ (1, λc)

into

πu∗ (λ, c).

. Under Assumption 1 and any cost/population parameters (c > 0, L > 0, f > 0) there exists a unique equilibrium value λˆ that generates some nonempty set (λ,ˆ X, P, N ) of (possibly-asymmetric) equilibria. Each equilibrium in this set is positive in the sense λˆ > 0, π (λ,ˆ c) > 0, X 63 0. Based on these facts, equilibrium existence can be stated, together with uniqueness of the equilibrium

Proposition Proof.

λ.

3

∗ u We can apply Lemma 1 to decreasing maximal per-consumer prot function

g(λ) ≡ πu∗ (λ, c) − f /L

on interval (λcmin , λcmax ), based on g continuity and its bounds of change (Corollary 1). So, some positive root ˆ ∈ (λcmin , λcmax ) of free-entry equation must exist. In essence, it equalizes a positive investment f with optimal λ ∗ ˆ ˆ is positive. However, to apply Lemma 1, we must c) = f > 0, that's why λ operational prot in the sense Lπu (λ, ∗ ensure that all equilibrium notions are valid. Indeed, on (λcmin , λcmax ) the inverse demand p (x, λ, c) is well-dened, as well as prot argmaxima and maxima.

positive

positive

ˆ c)  0, one can calculate Xu∗ (λ, equilibrium ˆ prices pk = p (xk , λ, c) ∀xk ∈ X , using the inverse demand function. The remaining element of the equilibrium denition is such a vector N = (n1 , ..., nK ) that satisfy the labor balance (14). All coecients of this linear equation being positive, we conrm existence of of admissible vectors (n1 , ..., nK ), which is truncated to an admissible polygon by the positivity requirement (n1 , ..., nK ) ≥ 0. Thus, all equilibrium components exist. ∗ ˆ follows from Uniqueness of equilibrium λ monotonicity of πu at points with positive nite prot: 0 < ∗ πu (λ, c) < ∞. Q.E.D. To reveal positivity of all equilibrium variables, from

∗

a hyperplane strict

two

interval (ˆn, nˇ)

Based on these facts and regular utility assumption, it is easy to establish now more denite equilibrium structure: unique

9 For

ˆ, λ

groups of rms and an

revealing monotonicity of

points where

X∗

∗ (λ, c) πu

of rms' masses.

we cannot use more standard envelope theorems since

make jumps (see Section 4).

13

∗ πu

appears non-dierentiable at the

Proposition Under Assumption 1 and Assumption 2 ( κ-regular u), the set-valued equilibrium contains one or two types of rms behavior. It consists of a unique marginal utility of income λˆ, a unique couple ((ˆx, pˆ), (ˇx, pˇ)) of quantity-price bundles: xˆ ≤ xˇ, pˆ ≥ pˇ, and an interval of rms' masses (ˆn, nˇ) : nˆ ≥ nˇ, namely, all the masses that satisfy the labor balance (12) with the10couple ((ˆx, pˆ), (ˇx, pˇ)). This interval of masses degenerates into a point under coincidence of points (ˆx, pˆ) = (ˇx, pˇ). 4.

Proof.

The uniqueness of

ˆ is already stated. We also had discussed already (in connection with the equilibrium λ (n1 , ..., nK ) of rms' masses. What remains is to ensure the dimensionality

denition) a polygon of admissible vectors

K ≤ 2:

not more than two global maxima of prot, i.e., not more than two types of rms' behavior. This fact was

already explained when discussing Example 2: it amounts to Assumption 2 on regular utility. Thus, the polygon

Q.E.D.

of masses is an interval.

4 Comparative statics under growing market: monotonicity and catastrophes This section shows an example of multiplicity and asymmetry, nds how numerous can be the asymmetry moments under growing population (or decreasing xed cost) and claries the direction of changes in consumption, price and variety.

Number of jumps .

κ-regular

Based on

utility function, the following theorem establishes the number of

jumps (catastrophes) in consumption and price during the complete path of comparative statics, i.e., when the relative market size

L/f

grows from zero to innity. Surprisingly, under non-globally concave prot such jumps are

guaranteed, being accompanied by equilibria asymmetry. After proving this, we explain the underlying behavior of

Proposition 5. Let Assumption 1 and Assumption 2 hold, with κ ≥ 0. Then under any constant marginal cost c, there are exactly κ critical values of relative market size L/f ∈ (0, ∞) that bring equilibria multiplicity and asymmetry, being also points of discontinuity of equilibria w.r.t L/f . Corollary (i) For equilibrium uniqueness and symmetry under all c, f, L > 0, sucient is strict concavity of the elementary revenue function xu (x) (i.e., κ = 0). (ii) This strict concavity condition is also ˆnecessary, in the sense that under non-concave xu (x), for any cost c > 0 there exists some relative market size L/f > 0 that generates equilibria asymmetry, multiplicity and discontinuity. ˆ ˆ the local argmaxima, and turn to equilibria monotonicity during such market evolution.

2.

0

0

Proof.

We have shown already that the equilibrium value

ˆ λ(L/f ) : R+ 7→ R+ changes from λcmin ≥ 0

and exists, i.e., the equilibrium mapping

ˆ λ(L/f )

λ = λ(L/f )

of marginal utility of income is unique

is well-dened and single-valued. Now we would like

L/f ∈ [0, ∞). Obviously, ˆ π ¯u,c (L/f ) ≡ πu∗ (λ(L/f ), c) ≡ f /L, ∗ inversely dependent on the market size. Our Proposition 2 claims that function πu (λ, c) continuously decreases in λ. Then, we can apply Lemma 1 to equation g(λ, L/f ) ≡ πu∗ (λ, c) − f /L = 0 (with L/f as a parameter and λ ˆ as argument). Therefore, the unique solution λ(L/f ) to this zero-prot equation must w.r.t. L/f from ˆ value λcmin ≥ 0 to λcmax ≤ ∞, without reaching these borders (using Theorem 1). Additionally, λ(L/f ) increases ∗ continuously, because the maximal value function πu (λ, c) continuously decreases. Therefore, λ takes values from (λcmin , λcmax ). Further, to get the number of jumps, we can use equivalency to ensure that

continuously

to

λcmax

over the domain

well-dened and continuous is the (equilibrium per-consumer) prot mapping

increase

all

(15) between the real prot maximization and the maximization of the auxiliary function (16). number of possible multiplicity instances directly from Assumption 2 on below). Corollary is evident.

10 Thus, n

Q.E.D.

κ-regularity

becomes a point only when equilibrium becomes symmetric but under asymmetry

mass of rms

n=n ˆ+n ˇ

remain ambiguous.

14

We obtain the

(see also geometry reasoning

x ˆ
the masses

(ˆ n, n ˇ)

and the total

Figure 3: Two locally-optimal

Let us explain now non-dierentiability of of

λ

λ)

prot (15), i.e., auxiliary functions

λ-adjusted

prot functions

π ˆ ∗ (λc), π ˇ ∗ (λc)

.

∗ (L/f ) and the geometry of our comparative statics. π ¯u,c

For any value

that brings two local maxima, we can dene two functions, which are locally-maximal values of (multiplied by

11

π ˆ ∗ (λc) ≡ λπ(ˆ x(λc), λ, c), The rst function uses the left

local

argmaximum

exploits the right local argmaximum

x ˇ.

x ˆ

π ˇ ∗ (λc) ≡ λπ(ˇ x(λc), λ, c).

(the leftmost solution to FOC described in Lemma 1) while

Somewhat loosely, we use the same notations

x ˆ, x ˇ

π ˇ∗

without specifying are

they argmaxima or not. To describe the behavior of

π ˆ ∗ (.), π ˇ ∗ (.),

we apply the usual envelope theorem, and conclude (standardly) that

the absolute value of the derivative of prot w.r.t. costis equal to the demand value whenever there are two local argmaxima

x ˆ
x

at the point studied. I.e.,

it must be that

dˆ π ∗ (λc) dˇ π ∗ (λc) = −ˆ x(λc) > = −ˇ x(λc). dλ dλ

Example 1 continued

upper envelope

slower

π ˇ ∗ (λc). ∗ . Such decrease is illustrated by Fig. 3, where the global maximum of λπu (λ, c) is the of these two locally-optimal loci π ˆ ∗ (λc), π ˇ ∗ (λc).

In other words, the left locally-optimal value

π ˆ ∗ (λc)

(22)

everywhere decreases

than the right value

once

ˆ (this proves the number κ of at some point λ ˆ jumps). Besides, for all small λ < λ the bigger root x = x ˇ of FOC remains the true global argmaximum, whereas ˆ the smaller root x for all big λ > λ ˆ becomes the global argmaximum. Then the jump goes downward. Thus, under ˆ generating multiple prot argmaxima during the evolution regular utility with κ = 1 there λ of λ from 0 to ∞. It is also evident why our prot function is not dierentiable at this point. Because of unequal slopes (22), these two loci can cross only

is exactly one point

Equilibria monotonicity . must

decrease

11 As

w.r.t.

Generally, both valid roots of equation (9), i.e., both local argmaxima

λ (Proposition 1).

x ˆ(λ), x ˇ(λ)

Really, one can observe such negative monotonicity of both local argmaxima

we have seen, there is one-to-one correspondence between the argmaxima of three functions

λπ(x, λ, 1) ⇔ x ∈ arg maxx∈R+ π(x, λ, 1).

15

x ∈ arg maxx∈R+ π(x, 1, λ) =

Figure 4: Dependence of consumption and price upon population

L.

Figure 5: Changes in welfare and mass of rms.

x ˆ(λ), x ˇ(λ) in our example in Fig. 4. Here, we illustrate multiplicity of equilibria 0.0002555. To derive comparative statics, we vary population L from 0 to ∞.12 Resulting comparative statics of consumption

x

of each variety w.r.t.

L

with costs

f = 0.0025, c =

is presented in Figures 4, 5 together with

related evolution of price and mass of rms:

Specically, we have found the unique switching point interval of them:

ˆ x (λ, ˇ, x ˆ, pˇ, pˆ, n ˆ , 0)

and

ˆ x (λ, ˇ, x ˆ, pˇ, pˆ, 0, n ˇ ),

ˆ L/f ≈ 10.04

and related two boundary equilibria among the

diering only in vector

(ˆ n, n ˇ)

and gross utility

U:

(1)

x ˆ = 0.652644395317, x ˇ = 12.8961077968, pˆ = 0.000284323031025, n ˆ 1 = 272.727734586, U1 = 5081.514933136;

(2)

x ˆ = 0.652644395317 ,x ˇ = 12.8961077968, pˇ = 0.0006479943306508, n ˇ 2 = 2364.570031198, U2 = 6549.082752038. There is an interval of asymmetric equilibria between these two extreme equilibria under this

this interval low and high consumptions

x ˆ, x ˇ

ˆ ) (L/f

and related

ˆ. λ

On

remain the same, whereas the set for masses of rms has the form:

N = {(ˇ n, n ˆ ) ≥ 0| n ˇ = 2364.570031198 − 8.670075431758 n ˆ }. 12 Calculation technique is

mR (xj ). Inverting it on its intervals of monox ˆ(λc), x ˇ(λc). Substituting these x ˆ(λc), x ˇ(λc) into the operational prot function (13) we obtain two branches of the adjusted prot function, π ˆ ∗ (λc), π ˇ ∗ (λc) and their upper envelope π ∗ (λ, c) = max{ˆ π ∗ (λc)/λ, π ˇ ∗ (λc)/λ}. ˆ It appears monotone decreasing and continuous, so from equation (13) we calculate the unique root, equilibrium value λ(L/f ) which is an increasing continuous function of L/f , and derive related consumptions and prices. On each interval of L we select the valid (small or big) consumption value relying on smaller or bigger π ˆ ∗ (λc) ≶ π ˇ ∗ (λc). as follows: we rst compute related marginal revenue function

tonicity, we nd two locally-optimal quantities

16

Here some rms choose producing high quantity

monotonically

x ˇ

whereas other rms choose low quantity

x ˆ.

In this example one can observe that growing market and related entry of new rms push per-variety consumption down

market size

with only one jump (discontinuity) at some point

L/f < 10.04,

relatively low. But when

ˆ ≈ 10.04 L/f

of this evolution. Under small

the global prot maximum is attained at higher consumption

L/f

x ˇ

and the mass of rms is

exceeds 10.04, all rms switch to producing smaller quantity

x ˆ

a general rule

because their mass

in all situations with non-concave prot

jumps up. By Theorem 3, such discontinuous behavior of equilibria is not a degenerate example, but .

Turning from this example of market size impact to general case, we formulate now a theorem about set-valued

X(L) strongly Under Assumption 1, an increase in the relative market size L/f induces several changes in the ˆ set-valued ¯equilibria: (i) the marginal utility of money λ(L/f ) increases with elasticity r (x); (ii) the consumption mapping X (L/f ) for each variety is a closed one and strongly decreases; (iii) the set-valued total mass of rms (n + n + ...) : ∀n ∈ N strongly increases; (iv) the price p(L/f ) strongly decreases on any interval of single-valued ¯ (L/f ) where r (x) > 0, and conversely, price strongly increases on any interval where r (x) < 0; (v) at any point X (L/f ) with multi-valued X (L/f )the price always jumps up (vi) whenever a rm's output Lx is single-valued, it changes oppositely to price, whereas at the jump it jumps down.13 monotone comparative statics, similar to single-valued comparative statics in ZKPT. We use again our monotonicity notions for mappings:

decreases when all its selections decrease.

Proposition 6.

u

u

1

2

0 u

u

Proof.

0 u

∗ u

;

Claim (i) about increasing

E

elasticity L λ(L) of equilibrium

λ.

λ

has been veried when proving Theorem 3. Moreover, we can derive the

We totally dierentiate w.r.t.

π=q We can ignore

0 qL

L

the free entry condition

u0 (q/L) − cq − f = 0. λ(L)

due to the envelope theorem. We get

0 πL = −q 2

λ0 (L) u00 (q/L) u00 (q/L) qu0 (q/L) 0 − λ (L) = 0 ⇒ L = −q 0 , 2 2 λ(L)L λ (L) λ(L) u (q/L)L

and obtain the needed elasticity

E L λ(L)

=L

λ0 (L) q u00 (q/L) =− · 0 = ru (q/L). λ(L) L u (q/L)

ˆ ¯ u (L/f ) = X ∗ (λ(L/f X )), claim (ii) actually becomes equivalent to strongly decreasing mapping of u ∗ ∗ argmaxima Xu (λ) which was stated in Proposition 2. Additionally, mapping Xu (λ) under our assumptions

Now, using prot

is closed (upper-semi-continuous on every compact set), being the argmaximum of a continuous function on a compact set, or, more correctly, a set

ˆ ¯ u (L/f ) = Xu∗ (λ(L/f X ))

[0, ∞)

compactiable under any given

λ.

Since

ˆ λ(L/f )

is continuous, so,

is closed too.

Claim (iii) about the rms' masses follows from isolated points of multi-valued argmaximum property being transferred to

¯ u (L/f ). X

Xu∗ (λ),

the same

Respectively, at the open intervals of single-valuedness, unique

n=N (n1 , n2 , ...)

decreases in such a way, that (using labor balance (12) related mass

x ¯(L/f )

increases, by Proposition 2 from ZKPT.

N of admissible masses of dierent rms behave at any point of ¯ f ≡ L/ ¯ f¯. It is sucient to note that to the left and to the right from such point L ¯ f , unique L consumption x ¯(L/f ) satises the labor balance in the form c¯ x + f /L = 1/n, and x ¯ makes a jump. At the ¯ f remains essentially the same. Thereby, same time, parameter f /L in the left and right vicinities of this point L the jump in unique equilibrium n in these vicinities is in the sense n ≡ limL/f →L ¯ ≡ limL/f →L¯ f (+) . ¯ f (−) < n What remains is to nd how vector

jump, denoted here

upward

13 Additionally,

the equilibrium markup

M = (p − c)/p = ru (x)

always behaves like price.

17

downward

¯ = (n1 , n2 , ...) of admissible N ¯ ¯ masses of rms at the limiting point Lf itself. Therefore, amongst all N = (n1 , n2 , ...) satisfying the labor balance ¯ f ) also must belong to the right and the ¯ f ) and the inmum n = 1/(cˇ x + 1/L (12), the supremum n ¯ = 1/(cˆ x + 1/L left limiting values of N , respectively. ¯ f ) at the (iv) Similar reasoning with limits can be applied to prices, therefore the set of possible prices P (L

Using closedness of the equilibrium mapping, both these limits belong to the vector

jumping pointcontains the limits of prices taken from the left and from the right. The conclusion about prices is

Xu∗

simple on any intervals where

is a singleton, because the direction of price changes just follows from Proposition

2 from ZKPT: prices go down when

ru0 > 0

14

and up in the opposite case.

(v) By contrast, at the multi-valued situation, any price jump always occurs consumptions are compared as

x ˆ
upward

, because two (equi-protable)

whereas the inverse demand function decreases, so,

Thus, under growing market, this jump goes from the point

pˇ to

the higher point

pˆ = p∗ (ˆ x) > pˇ = p∗ (ˇ x).

pˆ.

(vi) As to the behavior of output, on intervals of single-valuedness it is revealed in ZKPT:

ru0 > 0

jump at a multi-valued point, we recall that

q

q(L)

increases under

(increasingly-elastic demand) and decreases under opposite condition. Further, to establish the direction of

q = Lx and L remains constant at a point,

whereas

x jumps down.

So,

jumps down. This completes the proof.

Q.E.D.

anti-competitive eect

Let us explain again the strange direction of the price changes (following ZKPT and Intro). Under observe some counter-intuitive

ru0 < 0

we

: growing market attracts more rms but still all prices go

same

up. The explanation lies in the demand convexity: (only) whenever convexity is too strong, the rms compensate their decreasing output with growing prices. Now we extended the

abrupt

mechanism of anti-competitive eect to

set-valued equilibria also. Indeed, convexity is even stronger at the points of jumps. In addition, at such points the price increase becomes

.

Such discontinuity of equilibria evolution also seems a paradoxical outcome. consumptions and prices in response to smooth shifts in exogenous parameters. typically only at one point among the continuum of parameters

guaranteed

L

It means

catastrophic jumps

in

Though the jump itself occurs

determining the market evolution, but we have

the probability of catastrophes is not negligible

found that under non-monotone marginal revenuehitting such point sooner or later during complete evolution is ! Even on any nite interval

[L, L] of changing parameters

.

Conrming or excluding catastrophes appears now as an empirical question. Economically, the possibility of

non-monotone marginal revenue looks quite plausible. Mathematically, any demand curve that reminds a piece-wise

must generate a non-monotone

linear function (at, then having a kink and again at) must generate a non-monotone marginal revenue. Then, any

marginal revenue

gross demand summed up from linear demands of two distinct consumer groups

. These considerations increase our faith in the possibility of catastrophic eects reactions in a

monopolistically-competitive market. They explain, that a jump may happen, in particular, when a large group of consumers coherently comes out of the market in response to changes (not in our homogenous model), for example after imposing high trade barriers.

5 Heterogeneous rms Now we consider important case of rms' heterogeneity a'la Melitz (2003) but with variable elasticity of substitution. Will dierent rms respond dierently to kinks in revenue? We nd that rms may split endogenously into two

14 It

seemingly contradicts the picture in Fig.4: here price increases before the jump but decreases after it. However, actually function

ˆ to the left from the jump, and to increase on the ru (x) can be checked to decrease at all arguments x(L/f ) : L < L 0 < 0, and on the ˆ . Thereby on the left interval including the jump point L/f ˆ L>L ≈ 10.04 the claim holds true with ru 0 (x) = 0) it holds true either, with r 0 > 0. (excluding the jump point where ru u

18

right interval right interval

groups with high and low production with signicant gap between these two groups. As to market size eect found in ZKPT, it remains: cuto productivity grows (decreases) conditional on IED (DED) property.

5.1 Model Timing and goods.

L identical consumers, one diversied good, and continuum of potential c being distributed according to some continuous ´t [c0 , ∞), c0 > 0, higher c denoting higher cost (Γ(t) = c0 γ(c)dc is cumulative probability).

The economy includes

businessmen with heterogeneous abilities, their type-parameter density

γ(c)

dened on

For conciseness, we use the one-period timing proposed by Melitz and Ottaviano (2008), which is a proxy for many similar independent periods. In the beginning of the period whole population of businessmen is newly born without knowing their abilities, but knowing the market outcome of typical period: prices, prots, etc. Among indenitely many potential businessmen (rms), only

N

cuto type cˆ

copies of each rm-type decide to try entering the market.

revealing their productivity, only rms with costs lower than certain

After

survive as protable. These

N, cˆ

are endogenously determined by expected and real protability. Each rm produces one rm-specic variety and total number of varieties thereby amounts to

Consumers.

may

N

´ cˆ

c0

γ(c)dc.

From consumer's point of view, all varieties bring similar satisfaction without being perfect

substitutes, so all rms with equal costs

charge equal prices and get same purchase sizes

xc .

Under kinks of

marginal revenue, such symmetry is not guaranteed but asymmetry is possible, like in the previous section. But it has no impact on consumer because of kink-type rms' zero measure, and therefore ignored. So, rms will be identied only by type, skipping an index of rm's personality.

Then, the problem of representative consumer

becomes

ˆ

ˆ

cˆ

u(xc )dΓ(c) → max

U =N

s.t.

xi

c0

cˆ

pc xc dΓ(c) = w ≡ 1,

N c0

[c0 , cˆ] is the set of available types of varieties, while xc is the individual consumption of each variety of type c ∈ [c0 , cˆ]. Then, as before, the inverse demand function depends on the Lagrange multiplier λ and purchase size xc as pc (xc ) = u0 (xc )/λ. (23) where

Operating rms. total cost is

We have assumed that rms are ordered by their marginal costs, having same xed cost, and

C(c, q) = f + cq .

The per-variety operating prot of a type-c rm is given by

 u0 (xc ) − c xc ≥ f /L. π(c, xc , λ) = λ 

The

c − th

producer prot maximization w.r.t.

of (possibly multiple) argmaxima

x∗c ∈ Xc∗ ,

xc

is equivalent to maximization w.r.t. output and, installing any

still gives unique optimal per-variety operating prot function:

π ∗ (c, λ) ≡ max xc

This optimal prot function

L.

∗

π (c, λ) = π(c, xc (λ), λ)



 u0 (xc ) − c xc ≥ f /L. λ

is continuous (see Milgorm-Segal, 2002) as a function of

Further, we formulate the condition for cuto producer, i.e. such producer type

(so that all rms, operating in the market, have a type weakly smaller than

π ∗ (ˆ c, λ) = f /L.

19

cˆ and

cˆ,

λ

and

that her prot equals zero

typically earn positive prots): (24)

When

π∗

single-valued continuous decreasing

if optimal prot is not dierentiable, its strict decreasing in determines some

previous section. The solution

cˆ(λ)

Γ(c)

λ cˆ(λ), because

and

of lemma on monotone roots from

We assume that costs are assigned randomly, according to some distribution

[c0 , ∞). Prior to entering the some fe > 0  xed and known

on

ad hoc spends

c

function

can be non-dierentiable, because of revenue kinks.

Experimenting entrepreneurs.

function

∗ ∂π ∗ (c,λ) (c,λ) < 0, ∂π ∂λ < 0. Even ∂c is also obvious. Thereby, the equation above

is dierentiable, it is easy to nd the signs of prot partial derivatives

experimenting cost

market, each entrepreneur does not know her actual cost

c.

to study one's productivity (for example,

She

fe

is

the cost of a business plan). All rms make decisions simultaneously in Nash fashion, correctly anticipating the consumer's demand function and the expected competition intensity competition intensity

λ

λ.

Then rms enter until the equilibrium

drives expected prot to zero:

ˆ Π(λ, L) ≡

c

ˆ ˆ(λ)

h i ˆ − f dΓ(c) = fe . Lπ ∗ (c, λ)

(25)

c0

ˆ cˆ) of competition and cuto is dened by the system of two above equations with two Equilibrium couple (λ, ˆ cˆ = ˆc(λ)

variables,

and

ˆ. λ

Other variables  outputs

qc ,

prices

pc

and mass

N

of experimenting entrepreneurs

(copies of each type) are the consequences derived as in previous section, but using new labor balance:

ˆ N

c

ˆ

(f + cqc )dΓ(c) + N fe = L.

c0 The latter equation ensures that economy is closed, whole labor is spent for production and experimenting, and only labor makes consumer's income. Hence, the mass of operating rms is given by For w.r.t.

equilibrium existence and uniqueness,

λ,

c

prot

reach value

fe

uniqueness

π ∗ (c, λ). This gives ∗ goes from limλ→0 π (c, λ) = ∞

decreasing integrand for each

Π(λ, L), the decreasing in λ for

the same goes for expected prot

Thereby, the left-hand side in (25) is

N Γ(ˆ c) ≤ N . Lπ ∗ (c, λ)

we recall that optimal operational prot

to

cuto type ˆ c

= ˆc(λ)

u

λ.

two reasons: decreasing upper limit of integration and

ˆ , if it exists. λ limλ→∞ π (c, λ) − f /L < 0. of equilibrium

∗

To ensure existence, note that

Π(λ, L) must π ∗ (c, λ) per se.

So, the integral

somewhere. The only obstacle for existence may be nonexistence of optimal prots

However, Assumption 1 imposed on

decreases

being a decreasing function of

for homogenous economy ensure existence for heterogeneous rms, as well as

for homogenous ones.

5.2 Comparisons between good and bad rms ∗ (c,λ) ∂π ∗ (c,λ) < 0, ∂q ∂c < 0) enables to immediately ∂c ∗ ∗ formulate the monotonicity properties of the equilibrium curves of outputs, prices and markups. Whenever π , q

Monotonicity of prot and optimal output

qi

w.r.t.

type (

are non-dierentiable, the same result follow from our lemmas on output reaction to prot-maximizing output responds strictly negatively to cost so we have

Proposition 7.

prices:

c,

c.

Indeed: we have found that

whereas price is a decreasing function of quantity,

At a given equilibrium, ecient rms have higher outputs, bigger consumptions and smaller c < c˜ ⇒ qc > qc˜, xc > xc˜, pc =

c c˜ < pc˜ = . 1 − ru (xc ) 1 − ru (xc˜)

This proposition extends similar one from ZKPT onto cases without dierentiability. Furthermore, DED case, which always occurs at kinks, show theoretical possibility of a paradoxical high markups for inecient rms. Then, small rms compensate low output with too high markup, whereas better rms use output-expanding strategy.

20

Now we are going to specify how a kink in the revenue function changes the structure of market outcome. In Proposition 2 we have established behavior of any rm's prot-maximizing set-valued output

LX ∗ (λc):

it strongly

Assume the revenue function xu (x) has some kink interval [x, x¯] and an equilibrium shows consumptions curve x(c) such that its lowest value x < x lies below interval [x, x¯], whereas the highest value x > x¯ lies above. Then the equilibrium shows distinct groups of rms' behavior: some interval [c , c¯) of high-productivity rms produces more than L¯ x, another interval (¯ c, cˆ] produces less than Lx but nobody has intermediate output q ∈ (Lx, L¯x) decreases and has downward jumps at kinks. This fact and notions from Assumption 2 allow to formulate

Proposition 8

0

(clusters of behavior).

cˆ

c0

0

.

As to borderline rms' type high-type producers

[c0 , c¯)

c¯,

they produce either Lx or L¯ x. Moreover, the c¯ , the low-type producers (¯ c , c ˆ ] charge prices above 1−ru (¯ x)

their behavior is ambiguous:

charge prices below

pc¯ ≡

c¯ 1−ru (x) > pc¯ but nobody in between. To check that the proposition is true and clusterization really take place, it is sucient to note that when

p¯c¯ ≡

revenue function has some kink interval

¯] [x, x

nobody would produce such quantity. The rest is just reformulation

of this idea in view of previous proposition. Such clusterization eect may manifest itself in reality as very distinct groups of producers' behavior.

For

example, sometimes one can see a gap in the product line supplied by the market: say, either high-price restaurants or poor canteens in some city, with nothing in between. Moreover, our model can be extended to generate similar gap between exporters and non-exporters. Melitz (2003), for modeling such distinction we need not assume additional xed cost for exporters.

Unlike

Marginal

transport/trade cost itself, combined with choke-price, is sucient to generate kinked demand stemming from two groups of consumers  home and foreign, with lower demand from foreign consumers. Then, self-selection to exporters would choose larger and more productive rms, with a gap in production from non-exporters, exactly as in our simpler setting, and similar to Melitz and Ottaviano (2008). Thus, modeling non-concave prot is useful also for important trade questions.

5.3 Growing market size comparative statics w.r.t.

For

population

L

(e.g., opening trade) we can study equation (25). Under dieren-

tiability, like in ZKPT one can use partial derivatives mentioned ( that equilibrium intensity of competition

ˆ λ

increases with

L,

∂Π(c,λ,L) ∂λ

< 0,

∂Π(c,λ,L) ∂L

> 0),

to see immediately

i.e.

ˆ dλ(L) > 0. dL More generally, without dierentiability, the same conclusion of increasing

ˆ λ

follows from monotonicity of roots

lemma. This increase allows us to nd important changes in equilibrium value of the cuto rm size

L.

cˆ(L)

w.r.t. market

Thus, expanding Proposition 3 of ZKPT to possibly non-concave prots, we formulate now main comparative

L

ˆ (λ(L), cˆ(L), ).

Proposition 9 Equilibrium intensity of competition increases with market size. The cuto type cˆ decreases when the cuto rm output belongs to IED interval ( ), increases in DED case (r (¯x ) < 0), remaining constant if r (¯x ) = 0

statics result under heterogeneity: reaction to a small change in .

0 u

Proof.

cˆ

at some equilibrium

ˆ λ 0 ru (¯ xcˆ) > 0

0 u

cˆ

.

It is sucient to apply monotonicity of

ˆ λ(L)

obtained to cuto function

ˆ cˆ(λ)

that was found already to

be decreasing. The question how cuto behaves when arising exactly at the kink interval is open, it needs further study.

21

productivity increases Thus, in

IED

IED

case, we have found that some group of least-ecient rms leaves the market, so, average , and the opposite happens in

DED

case. Again, CES case looks very peculiar. Most realistic

case shows increasing average productivity, that can be an important reason for gains from international trade

(markets integration), probably underestimated so far.

6 Conclusion First of all, this paper has satised our curiosity about robustness of the monopolistic competition theory to a non-traditional assumption: possibly non-concave prot, or, equivalently, non-monotone marginal revenue.

It

turns out that equilibria exist, but may become asymmetric and set-valued. Importantly, the comparative statics

similar

of equilibrium-set in response to growing market (for instance, population growth or trade integration) remains

catastrophes, must

to what we know about single-valed equilibria.

Second, new eects found are

i.e., jumps of outputs and prices in response to small shifts in

population or costs. Surprisingly, such jumps

happen whenever marginal revenue is non-monotone, i.e., the

demand has kinks. This case looks natural under distinct groups of demands. Thus, abrupt market reactions to small parameters shifts look now not quite unrealistic. Under heterogeneous rms, the same eect of catastrophes manifest itself in clusterisation of producers, showing a gap in behavior. Extending the model would show such gap between exporters and non-exporters, or just holes in a product line supplied by the market.

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