economic distributions in monopolistic competition · 2018-01-23 · economic distributions in...

Economic Distributions in Monopolistic competition

André de Palma, Ecole Nomale Supérieure de CachanCoauthored with Simon Anderson, University of Virginia, CEPR

2016 EAJ ConferenceClaude Lefèvre Day – Research conference

September 5, 2017

Former title: Economic distributions and primitive distributions in Industrial Organization and International Trade

Keywords: CES, Logit, IIA, monopolistic competition, primitive and economic distributions, Pareto, (log‐)normal, exponential,

mark‐ups, demand form, …

My collaboration with Claude Lefèvre

• Mathematical sociology: probabilistic decision models, search, cross‐nested, logit with interactions Social network & Search Logit models

• Transportation (with Ben‐Akiva, and P. Kanaroglou & N. Litinas): ; 1st day‐to‐day dynamic congestion modelNew paradigm in transportation

05/09/2016 Claude Lefèvre's Day 2016 3

My collaboration with Claude Lefèvre

• Marketing (with Droesbeke, Rosinski): diffusion of innovation with negative feed‐back

Information congestion• Urban Economics : compartmental analysis with (non)extensive interactions

MFD


This was for the dean, ….

truth is a bit different


(Mathematical) sociology


Transportation in practice



Shared modes

Synthetic measure of Claude’s friendliness


Autonomous vehicles – 1981


.. In 2009


Marketing


New urban economics


Jaipur


La machine magiqueAu XVIIIe siècle en Inde sous le règne du puissant maharajah Jai Singh, à l'ombre du fameux observatoire de Jaipur, un jeune brahmane raconte sa double éducation sensuelle et scientifique. Tandis que son maître l'initie à la cosmogonie traditionnelle, il découvre les mystères de l'amour avec une jolie gitane. Des savants d'une mission européenne lui ouvriront des horizons nouveaux. Alerte, bondissant comme son héros, ce roman nous conduit de harems en bibliothèques, d'une culture à l'autre avec une verve qui semble emprunter autant à Voltaire qu'à une fable paradoxalement moderne.La machine Magique, Luc Leruth, Gallimard


Khajuraho


Gange @ Varanasi


1er avril 2010, école obligatoire en Inde pour les enfants de 6 à 14 ans


1981: Les enfants privilégiés allaient à l’école


2016: Kalkeri School of Music (Karnataka)


Mallikriyan

“Statistical mechanics” approach in IO and Trade

Shift attention from standard IO & Trade models based on “point values” on an approach based on “distribution” How consumers and Firms (in an industry) can be described with distributions?Distribution of cost, quality Distribution of consumers’ preferencesDistribution of price, mark‐up, output, profits


From Physics to Statistical mechanics


Bose‐Einstein statistics


Fermi‐Dirac statistics

• Energy ; potential ; Boltzmann constant: k, temperature: T


exp 1

ii

i

gnA

kT

Two objectives (here)

1. Establish logical relations between distribution of quality ‐ costs of demand/preference

2. How moments (means, variance, …) of one distribution can explain moments of other distributions?


And a third one….

Economists should explain how to elicit consumer behavior, technological features, competition rules from large data sets ?

3. Back‐up demand functions

We propose an operational approach to do that!


Backdrop• Build or reverse engineer a model with distributions

• Primitive distributions (h&c; h&v‐c) beget economic ones (y, p‐c, )

• Distribution belong to “classes”, which are linkedE.g. price (p) & cost (c) distrns in same family, linked thru mark‐up; bridged to output (y) and profit (distrinsvia quality‐cost function (c)

• Illustrations : CES, Logit, etc.05/09/2016 Claude Lefèvre's Day 2016 28

Earlier• “Is the quality of consumer goods too low?”[JIE ~ 2001, Anderson & de Palma]• Asymmetric Bertrand logit oligopoly

• Underlying “ranking” Higher quality‐cost have higher mark‐up and higher equilibrium demand (hence profits)


Two simplifying hypothesis

• Monopolistic competition – OK since continuum of firms

• Unidimensional ranking of firms– E.g. 20% most efficient firms are also the 20% most profitable one


Our approach : summary

• Cost, prices, qualities, output are described as distributions.

• We will uncover – the logical links between these distributions– the basic distribution driving the others

• Scope of this research: provide theoretical foundations of empirical distributions?


Outline• Mon. competition with cost differences• Recovering demand from economic distribution

• Demand observed +? : deriving all distributions

• CES, Pareto Circle, breaking the circle • Mon. competition with q‐c differences• Logit : log‐linear model of mon. comp. • Comparative statics on Distributions• Extensions: long run, generalization…05/09/2016 32Claude Lefèvre's Day 2016

MONOPOLISTIC COMPETITION WITH COST DIFFERENCES


The demand function

= : price; : quality ; : - marAssumption 1: h . ; . 0, ' . 0, '' exists, 1 -conc

k-up

ave.h

y h p

p c h

h

p mh m cp c m m p c

h


Property of the monopolistic competition model with fixed mc


Theorem 1: Under Assumption 1, there is a unique mark‐up (c)>0, with ’(c)>‐1, log‐convex.’(c)>0, if h(.) is log‐convex’(c)<0, if h(.) is log‐concave Equilibrium demand h*(c)=h((c)+c) is decreasing in c as is profit, with *’(c)=‐h*(c).

Recover demand functionfrom mark‐ups


Theorem 2: consider a positive mark‐up, (c), with ’(c)>‐1. Then there exists an equilibrium demand, h*(c), h*’(c)<0, and a demand function, h(.) satisfying Assumption 1.h(.) is :

log‐convex if ’(c)>0log‐concave if ’(c)<0.

RECOVERING DEMAND FROM ECONOMIC DISTRIBUTION


Combining CFD with monotonicity (only for economists!)


2 1

2 1

X1 1 X2 2

1 2

2 2

2 2

Consider two distributions F and F strictly increasing.

Let X , then

F if g' . 0, and

F

Lemm

1 if

a

g

1

' . 0.

X X

X X

x x

g X

x F g x

x F g x

Recover demand (and the rest) from cost and price distributions

Theorem 3: There is a continuum of firms with demand satisfying Ass. 1, but not observable. Assume FC and Fp are known. Then demand, mark‐up and output and profits distributions are recovered (up to a positive multiplicative factor).


Recover demand (and the rest) from output and profit distributions

Theorem 4: There is a continuum of firms with demand satisfying Ass. 1, but not observable. Assume Fy and F are known. Then demand, mark‐up, and cost distributions are recovered (up to a shift factor for the scale of c).


Key relation

Using monotonicity, we have:


1

10

1We find:

c .

C Y

z

Y

F c F y F z

F rz c dr

F r

Recover demand (and the rest) from price and profit distributions

Theorem 5: There is a continuum of firms with demand satisfying Ass. 1, but not observable. Assume FP and F are known. Then demand, mark‐up, and the other distributions are recovered.


Recover demand (and the rest) from c&y,c& or p&y

Theorem 6: There is a continuum of firms with demand satisfying Ass. 1, but not observable. Assume {FC, FY} or {FC, F} or {FP,FY} is known.Then, demand, mark‐up, and the other distributions are recovered.


DEMAND OBSERVED + ? : DERIVING ALL DISTRIBUTIONS


Demand known: 1 distribution suffices!

Theorem 7: There is a continuum of firms with observed demand satisfying Ass. 1. Assume FC or FP or FY or F is known.Then, all the other distributions are recovered.


CES PARETO CIRCLE


MONOPOLISTIC COMPETITION WITH COST AND QUALITY DIFFERENTIATION


Notation quality‐cost and markup

,

.

: (1) . Equilibrim mark-up is .

This key formula relates demand (.) to mark-up

'

.

i i

i i i

i i i i i i i i

i

x m

i ii

i

h x mm

y h v p

h v p h v c p c h x m

FOC x

h

h x m

x


The demand function

A

[most of the time the ^ is dropped]

= ( ) : price;

ssumption 2:

0; ' . 0, '' exists, 1 -concave

Assumption 2':

0; ' . 0, ''

v: qual

exis

ity; : cost

ts, log -co

.

ncave

y

h v p h h

h v p h h

h v p

p c h v p mh x m x v cp c


From scale value h(x) toequilibrium mark‐up (x)

Assume Ass 2 holds. Under monopolistic competition there exists a unique >0, ' 1.

Then, equilibrium demand *

is increasing, as is * ,

Theorem 8:

* ' * .

x x

h x h x x

x x h x


Monotonicity matters!

Corollary 2If Ass 2 holds: higher quality‐cost (v) are associated with higher output (y) and higher profit (); If Ass 2’ holds: higher v are also have higher mark‐ups ().


From equilibrium mark‐up (x) to (equilibrium) demand h(x)


1

Consider an equilibrium mark-up , ,

' 1. Then the equilibrium demand is given by :

1 'h* =h* exp , , .

Let: . The demand function is:

h h*

Theorem

9

,

:

x

x

x x x x

x

x x

vx x dv x x x

v

u x

x

x x

and obeys Assumption 2.x

Hence, we recover first the maximized value function, then the primitive, on the support for which we have observations. Outside there, we only know it should not be too convex as to violate the observed profit maximum. Note we get Logit for µ’(x) = 0

Demand known: Two legs and one bridge


Assume that demand is known and satisfies A2. If 1 element is known from two of of the following 3 legs, they are all known: ( ) , ,( )( ) strict monotone relation between c a

Theorem 10:

nd x

X Y

C

i F F Fii Fiii

.If ' 0, then can be added to leg (ii),

where x(=v-c)= .Px F

c

LOGIT: LOG‐LINEAR MODEL MONOPOLITIC COMPETITION


The exponential h(.) function

0

In this case: exp .

exp, .

exp exp

The total number of consumers is normalized to one (N=1).

i i

i

uh u

v p

y iv p vd

05/09/2016 57Claude Lefèvre's Day 2016

Discussion

This is the most standard IIA function (along with the CES function).Start‐point in structural empirical IO (BLP)Here used as a Monopolistic Competition model.

Assume here that the mark‐up is known.


Monopolistic competition in non‐symmetric case

i

i

*

Profit: ,

, Monopolistic competition hypothesis

1 0, (Nash)

Equilibrium price: Absolute mark-up (versus Percentage mark-up):, .

Note for oli

i i i

i ii

i

i

i i

i

i

dy yd

p c y i

i

p cd y id

p

p

p c i

*ipololy: as y 1.

1i i ii

p c cy


Distribution of quality‐cost


*

Density: Quality-cost

0

* *

Equilibrium output

exp, mass of firms, where:

exp V

V exp 1 .

Equilibrium profits:

x

X Ox

O

v c

y MxM f x dx

v

y

Comparative statics

PROPOSITION 3 In the Logit model of monopolistic competition, all firms set the same absolute mark‐up. Higher quality firms have higher equilibrium outputs and higher profits

Cf. Melitz: all same % mark‐up in CES


LOGIT WITH QUALITY‐COST, OUTPUT AND PROFIT DISTRIBUTION


X

Y

number and X=Random variable = quality-cost =(V ): ; pdf: f .

Similar definitions for F and F .

exp expOutput : , with ,

D the Logit denomin

X

x CCDF F x x

y

x x

y y yD D

Notations and key relations

000

ator:

, where V =exp 1 .exp VXx x

x f u d vD M u


Quality‐costOutput and ProfitsPROPOSITION 4 : For the Logit Monopolistic Competition model, the distribution of quality‐costs, FX(x), generates the distribution of equilibrium…


O u t p u t : l n

P r o f i t : l n ,

w h e r e : i s t h e L o g i t d e n o m i n a t o r .

Y X

X

F y F y D

DF F

D

Output or Profits Quality‐cost

: The quality-cost distribution for the Logit model of Monopolistic Competition, , with V >0 can be generated

from the equilibrium output distribution via the relation:

X O

Y

F x

F y

F

PROPOSITION 4 (cn't)

1

1

exp , D V / 1 .

It can also be generated from the equilibrium profit distribution as:

exp , D V / 1 .

X Y y y O av

X O av

xx F D My

x MF x F D


COMPARATIVE STATICS ON DISTRIBUTIONS

Equilibrium Properties:


Increase in quality‐cost

Recall x =v‐c (quality minus cost)PROPOSITION 7 A f.o.s.d increase in x or a mean preserving spread in x increases mean output and mean profits.

(Strict increase if market not fully covered, i.e. V0 >0)

‐‐ Take more from outside option‐‐ Spread ‐ because exp(x) convex (plays more at top); better firms do much better than worse lose


Increase taste for variety, µ


* *i i

: The indirect utility function is ln .More heterogenity, , generates more consumer benefit:

ln

Lemma

P ln P 0 (Shannon measure)

D

d Dd

Ancillary result; note entropy associated to logit (“rational inattention” link)

Not clear a priori that more “noise” makes better off because Type 1 EV isn’t symmetric, and worse (negative) draws make worse off.

Here, more noise is better: expected value of Max goes up

Outside good and heterogeneityProposition 8:Higher V0 fosd decreases output and profits.

Higher fosd increases output and profit, for low xand………fosd deceases output and profit for high x.


SPECIFIC DISTRIBUTIONS


Truncated Pareto distribution


Let the quality-cost X [ , ) be distributedwith parameter >0. Then the

truncated Pareto truncated General profit, has a

distribution:

1 1 logF

ized Log-Pareto

1

x

x

xx

COROLLARY 2

, , .

where is the mimimum (resp. maximum) profit corresponding to the miniumum (resp. maximum) quality-cost (resp. ).Truncation guarentees that exists.

x xD

GENERATING THE PARETO DISTRIBUTION FOR PROFIT AND OUTPUT


Generating empirical Pareto distribution for profit and output


5 Let F =1- exp(- ( - )), 0, 0, [ , ), with µ>1. Th

x be exponetially distributed:

equilibrium p

en

(1) the 1rofit is Pareto

equilibrium output is Par

;

(2) the diste o rt

X x x xx x x

F

Proposition

ibuted:

1 , y y, , = .

A Pareto distribution for equilibrium output can only be generated by an exponential quality-cost distribution i > . 1f

yyF y y

Interpretation

• The condition needed to bound D is μ>1/: taste heterogeneity should be larger than the average quality‐cost.


Next….

• Free entry equilibrium: how?• Multi‐dimensional cases• Empirical validation,..


Thank you all for your attention

thank you Claude for such prolific academic career!


economic distributions in monopolistic competition · 2018-01-23 · economic distributions in...

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