A Note on The Analogies between

A Note on The Analogies between Empirical Models of Auctions

and of Di?erentiated Product Markets.?

Liran Einav?

August26,2003

PRELIMINARY AND INCOMPLETE

Comments are welcome

Abstract

Under standard equilibrium assumptions,a bidder in an independent private value auction

chooses her bid by trading o?between markup and probability of winning.Under similar

assumptions,a producer in a di?erentiated product market chooses its price by trading o?

between markup and residual demand.This similarity in the nature of the problem lands itself

to a striking similarity in the?rst order conditions,which form the basis for estimation.Despite

these similarities,the two literatures evolved in very di?erent directions;much of the auction

literature emphasizes nonparametric identi?cation and nonparametric estimation techniques,

while the demand literature concentrates on(parametric)ways to deal with endogeneity of

prices.This note tries to provide a uni?ed framework which nests the two literatures,thereby

clarifying the di?erences in assumptions that made the literatures go in such distinct directions.

Keywords:First-price auctions,Di?erentiated products,Estimation

JEL classi?cation:C1,C7,D8,L0

?I thank Tim Bresnahan,Estelle Cantillon,Ali Hortacsu,and participants in Stanford’s“Structural Lunch”for useful comments.The ususal disclaimer applies.

?Department of Economics,Stanford University,Stanford,CA94305-6072;Tel:(650)723-3704; Leinav@https://www.sodocs.net/doc/034435789.html,.

1Introduction

This note tries to create a conceptual link between two important strands of the recent empirical literature in Industrial Organization.The?rst focuses on estimation of demand and costs in di?erentiated product markets(Bresnahan,1987;Berry,Levinsohn,and Pakes,1995;Nevo,2001; and many others).The second attempts to estimate the cost/valuation distribution in auction markets(Paarsch,1992;La?ont,Ossard,and Vuong,1995;Guerre,Perrigne,and Vuong,2000; and many others).

A standard assumption in the di?erentiated-product literature is that prices are set as an outcome of a Nash Equilibrium behavior,typically with constant marginal costs.Under these assumptions,single-product?rms choose prices to solve

(p i?c i)D i(p i,p?i)(1)

max

p i

Consequently,in a Nash Equilibrium this choice satis?es the following?rst order condition:

c i=p i+μ?D i(p i,p?i)?p i??1D i(p i,p?i)(2)

Similarly,in independent-private-value(IPV)(procurement)auction,the literature typically assumes that bids are set as an outcome of a Bayesian Nash Equilibrium behavior.Under this assumption,bidders place bids to solve

(b i?c i)Pr(b i

max

b i

Consequently,in equilibrium this choice satis?es the following?rst order condition:

c i=b i+μ?Pr(b i

?b i??1Pr(b i

One common belief should be dismissed at the outset.Thinking about auctions as a homoge-neous product Bertrand competition has led many economists to believe that auctions are simpler; unlike multi-dimensional demand systems for di?erentiated products,so the argument goes,the allocation rule in auctions is known and simple:the lowest bid wins.It turns out that thinking about the mapping helps to clarify why this argument does not hold.Taking the original Bertrand case as a benchmark,the move towards an incomplete information version of it complicates the problem just as much,if not more,as the move towards product di?erentiation.I argue in this

note that the so-called simplicity of the auction literature,which allows nonparametric treatment, is achieved through a set of simplifying assumptions rather than through a fundamentally simpler problem.The key simplifying assumption is about the wedge between the information available to the econometrician and that available to agents.The demand literature assumes such a wedge exists,and takes the form of unobserved product quality.The auction literature assumes this wedge away,therefore faces no endogeneity problem,allowing it to be more?exible on other di-mensions.Which set of assumptions is more reasonable depends on the particular question and particular industry analyzed.It should not depend,however,on whether it is an auction or a demand application.

This note has several goals.First,the uni?ed framework should help in facilitating com-munication between the two literatures,which so far evolved quite independently of each other. Second,the uni?ed framework may help in thinking and evaluating the implicit trade-o?s made by making certain assumptions.Third,one may view certain markets as a combination of dif-ferentiated product markets and auctions.In such cases,using ideas from both literatures may be important.For example,one can think about di?erentiated product markets in which?rms cannot change prices too often,and therefore prices are set without complete information about opponents’prices.Alternatively,one can think about auctions in which the allocation rule is somewhat vague,and depends not only on prices.In such cases,the allocation rule cannot be assumed,but has to be estimated,just as any other demand system.

The rest of the note continues as follows.Section2provides a uni?ed framework.Section3 maps each literature to the uni?ed framework,and maps the assumptions made by each literature to an equivalent set of assumptions in the other.Section4discusses several related analogies, as well as di?erences in the nature of the problem,which may have made the literatures go in distinct directions.Section5concludes.

2An Innocent Model

The Model

Consider a simultaneous-move game of the following structure.There are N players.Each player has to choose an action x i∈[0,∞].The utility of player i from choosing x i and a(potentially random)action of e x?i for the other players is given by(x i?θi)E(e S i(x i,e x?i)).We assume that e S i(x i,x?i)is non-negative,is weakly decreasing in x i,and weakly increasing in x j for j=i.Let S i(x i)≡E(e S i(x i,e x?i)).We assume that S i(x i)is continuous and twice continuously di?erentiable in all its arguments.It is easy to think about(x i?θi)as markup,and about S i(x i)as a residual demand function.

Under these assumptions,player i solves

(x i?θi)S i(x i)(5)

max

x i

and,in equilibrium,the following?rst order condition is satis?ed

+S i(x i)=0(6)

(x i?θi)?S i(x i)

?x i

Second order conditions are assumed to be satis?https://www.sodocs.net/doc/034435789.html,ly,we assume that

(x i?θi)?2S i(x i)2

i +2

?S i(x i)

i

<0(7)

or,in words,that S i(x i)is not too convex.This also guarantees that the solution x i(θi)is increasing inθi(keeping?xed the opponents’strategies,e x?i).

We can rearrange the?rst order condition to obtain

θi=x i+μ?S i(x i)?x i??1S i(x i)(8) Estimation

Suppose now the econometrician observes many outcomes of such a game,and that for each game the econometrician observes the action,x i,for each player.If S i(x i)was known or estimable,one could use equation(8)to back outθi.If S i(x i)is unknown,however,this is not possible.A higher observed x i can be driven either by a higherθi or by a“more favorable”S i(x i).

To simplify the intuition,suppose all the information about a player can be characterized by a one-dimensional parameter.In particular,let S i(x i)=S(x i;λi,λ?i),with S(x i;λi,λ?i)increasing inλi and decreasing inλj for j=i.A player with a higherλis“stronger”:Ceteris paribus,she faces higher demand,and makes her opponents face lower demand.

Suppose also that S(x i;λi,λ?i)is known to the econometrician up to theλ’s parameters, which are unobserved(to the econometrician only;the players know everything).A player is now de?ned by a pair(θi,λi).The econometric task is to infer costs,θi’s,from observing x i’s.It is quite clear that there are“too many”degrees of freedom:without further assumptions,it is impossible to infer a two-dimensional error structure by observing only a one-dimensional set of variables.Thus,the system is not identi?ed.The only way to identify the system would be to either make parametric restrictions on(θi,λi),or to use other sources of data which would provide independent information about(θi,λi).

The key conceptual distinction between the two error terms is the way they show up in the objective function.θi is a non-strategic error term,as it a?ects player i utility,but has no e?ect on the utility of the other players,other than through the indirect e?ect on player i’s choice of x i.In contrast,λi is a strategic error:it directly a?ects the objective function of both player i and her opponents.

3Mapping to The Literature

Di?erentiated Product Demand

Consider?rst the case of di?erentiated product markets.It is easy to see that equation(1)maps itself directly to equation(5)of Section2:x is the price,θis the marginal costs,and S(·)is the residual demand function.To?x ideas,consider,as a benchmark example,a simple logit discrete-choice demand model(Berry,1994).The utility for consumer h from product i is given by u hi=δi?αp i+εhi whereδi is the average quality of product i,p i is its price,andεhi is an

idiosyncratic taste preference,distributed according to a type I extreme value,and is i.i.d across consumers and products.The mean utility from the outside good(good0)is normalized to zero. If the number of consumers in the market is M,this speci?cation gives rise to the well-known

logit demand function:

Q i(p i,p?i)=

exp(δi?αp i)

1+P j∈J exp(δj?αp j)(9)

This demand function satis?es the restriction thatδi is a su?cient statistic for player i,so all heterogeneity can be summarized by a one-dimensional parameter.In the language of the end of Section2,we can write S(x i;λi,λ?i)=exp(δi?αx i)

1+P j∈J exp(δj?αx j),whereλis nowδ.As already indicated, if we only observed prices,we will not be able to determine whether the price of a certain product is higher because of higher marginal costs or because of higher quality.Clearly,the distinction is of crucial importance for any counterfactual analysis.Loosely speaking,products of higher quality are good to have,and products of higher marginal costs are ine?ciently manufactured.In that sense,prices are endogenous:they are correlated with the(potentially unobserved)quality of the product.

Luckily,in demand models we can solve the indeterminacy problem by exploiting an additional source of data.We typically observe quantities,as well as prices.Quantities can identify theλi’s, and therefore prices can nonparametrically identify the marginal costs,θi.For example,in the logit case described above,with single-product?rms,equation(8)becomes

c i=p i?1

α(1?q i/M)

(10) and marginal costs can be backed out from information about prices and quantities,and the parametersαand M.Without quantity data,however,the system is not identi?ed,unless we know(or make assumptions about)the product qualities,δi’s.One should note,however,the in order to identify the system,one has to make the parametric assumptions which makeδi a one-dimensional su?cient statistic for player i.As we discuss below,such parametric assumptions may have strong implication in counterfactuals,so other alternatives may be worth exploring. IPV Auctions

Consider now an IPV procurement auction.A quick comparison of equation(3)and equation(5) reveals that one can think of the probability of winning as the demand function,S.In addition, just as before,x is the bid,andθis the costs(which are private information).In thinking about this mapping,it is important to emphasize what may make the probability of winning function di?erent from bidder to bidder.In demand models,this variation is due to quality di?erences among products.In auction models,this is due to di?erences among bidders,which are common https://www.sodocs.net/doc/034435789.html,mon knowledge di?erences are strategic:they make one bidder’s expectation about her probability of winning,given a bid,be di?erent than those expectation of other bidders.Cost variation which is private information is non-strategic:by construction,it does not enter the opponent’s optimization problem.

In an auction setting,the corresponding quantity data(i.e.the probability of winning)is not observable.Thus,the literature has solved the indeterminacy problem by imposing restric-

tions on the structure of the strategic error term,or,in other words,on the shape of possible

asymmetries across bidders within and across auctions.Guerre,Perrigne,and Vuong(2000),for

example,assumes that all variation among bidders is due to private information.This implies that

the demand function faced by each bidder varies only with the number of bidders,but not with

their identities.Thus,in the language of Section2,one can think of this symmetry assumption

as assuming thatλi=λj?i,j.This allows us to solve the indeterminacy problem.It“shuts down”one source of variation among bidders,and therefore allows us to map the one-dimensional

data(bids,in this case)to the remaining one-dimensional error term(theθi’s,which are private

information to each bidder).It is now easy to see the equivalence with the di?erentiated prod-

uct demand system.The corresponding assumption on demand would be that all products are

symmetric.A logit system(as above)with all product qualities being the same is one example.1

A Dixit-Stiglitz-Spence preferences is a di?erent example(Spence,1976;and Dixit and Stiglitz,

1977).These are parametric examples.With these assumptions and enough data,however,just

as in auctions,one can relax all parametric assumptions on the demand structure.Under these

assumptions prices vary only due to idiosyncratic shocks to marginal costs,and therefore are not

endogenous in an econometric sense.Thus,with enough independent markets,one can observe

prices and quantities for each(symmetric)product,and nonparametrically back out the demand

function.

Several auction papers relax the symmetry assumption by allowing bidders to di?er from

each other.Campo,Perrigne,and Vuong(2002)and Kransutskaya(2002)rely on the fact that

bidders can be categorized to a?nite set of types(i.e.,in the language of Section2,theλi’s can

take a?nite set of values),and that the econometrician knows which bidders are of the which

type.With enough data,one can then nonparametrically estimate S(x i;λi,λ?i)for any?nite combination of(λi,λ?i)and continue as before.Two comments are in place.First,in the context of demand models,these assumptions are equivalent to an assumption that the mean quality of a product does not vary across markets.In such a case,within a logit framework,using product ?xed e?ects would eliminate the endogeneity problem of prices.Moreover,the auction literature implies that and with enough markets and corresponding quantity data the demand function,as before,can be nonparametrically backed out.Second,one should note that the data requirements for nonparametric estimation increase exponentially with the number of values theλi’s can take, forcing the econometrician,in practice,to either rely on a very small number of types(Hortacsu, 2002)or to use stylized environments,in which the same set of bidders play against each other repeatedly(Bajari and Hortacsu,2003).

Finally,a di?erent set of papers(Bajari,1999;Bajari and Ye,2003;and Pesendorfer and

Jofre-Bonet,2002)assume that bidders’types are drawn from a continuous set,but are known

(exactly,or up to a small set of parameters)by the econometrician.Either way,this forces them

to rely on parametric assumptions,making it somewhat more similar to the demand literature.

1A nested-logit demand model with all inside goods being identical,and the outside good in a di?erent nest will

also work.See an application of such a model in Berry and Waldfogel(1999).

Discussion

As the analogy makes clear,the probability of winning is the ex-ante demand function faced by a bidder in an auction.Unlike di?erentiated product markets,in which the realization of demand (quantity)is observed,the realization of demand in auction markets,namely the probability of winning,is not an observable variable.Thus,we do not have a quantity analog in auction models, which will help us identify the two-dimensional error structure.At the same time,the absence of quantity data makes it harder to falsify the symmetry assumptions.This is,I believe,one of the key reasons that the auction literature made these di?erent set of assumptions.One should note,however,that one can identify the two-dimensional error structure by imposing parametric assumptions,which is the goal of an ongoing research project.This may lead to very di?erent counterfactual results,and is somewhat analogous to the logit demand model described in the beginning of the section.

To summarize,both demand and IPV auction models have very similar structure,which can be uni?ed by the uni?ed model of Section2.In order to identify this model with typically available data,one has to make certain assumptions.There are,in general,two sets of assumptions one may consider.The?rst,typically made in the demand literature,relies on parametric assumptions about the demand function,still allowing one-dimensional asymmetry across players.The second, which is popular in the auction literature,relies on stronger symmetry assumptions,but allows non-parametric estimation of the remaining heterogeneity.

More generally,however,the choice between these two sets of assumptions should not be linked with the demand/auction model,but rather with the application in hand.For example,as is well known,logit demand(or related techniques)restrict the relationship between demand and marginal revenue in a certain way.Therefore,when the focus of the analysis is on this relationship, one may want to consider relaxing the functional-form assumptions and put stronger restriction on the asymmetry among products(Hortacsu and Syverson(2003)is a recent example of such an approach).Analogously,in certain auction applications asymmetries and unobserved bidder-speci?c heterogeneity may be important,in which case one may consider imposing parametric distributional assumptions,while allowing a more?exible pattern of bidder asymmetries.As I already emphasized,it is the particular application and the particular research question that should guide us in making these modeling decisions.

4Other Analogies,and Di?erences

Few other analogies between the two literatures should be pointed out.First,multi-unit auctions are very much like multi-product?rms if winning more than one auction gives rise to some synergies(in costs or pro?ts).Such synergies,in the form of substitutability between units,arise by the structure of the allocation rule if combination bids are allowed(Cantillon and Pesendorfer, 2003).In such a case,an increase in the probability of winning one unit alone reduces the probability of winning the bundle.This type of auctions would be a direct analog to demand models if multi-product?rms could also price the bundle in a discount,which is something we rarely observe in applications.

Second,much of the demand literature has departed from the simple logit model presented in the previous section,in favor of extensions which allow for a less restrictive(but still parametric) substitution pattern(e.g.Berry,Levinsohn,and Pakes,1995).This is somewhat analogous to private-value auctions with a?liated values.Typical symmetric models of auctions with a?liated values allow for a symmetric structure of a?liation.With heterogeneous bidders,however,the a?liation structure is likely to be asymmetric.Two bidders may be more similar to each other than they are to a third one.Thus,their costs may be more a?liated to each other,than they are with the third bidder’s costs.Just as in demand models,allowing full?exibility for the a?liation pattern is not practical in typical data sets.Therefore,one could follow similar ideas to those in the demand literature,and parametrize the a?liation structure.Such parametrization would allow stronger a?liation between bidders who are more similar on observables.

Third,both literatures focus on testing the assumptions about the rules of the game.This amounts to testing between competition and collusion in the demand models,and to testing between competition and collusion and between private value and common value models in the auction literature.In demand models,these tests rely on exogenous rotation of the demand curve (Bresnahan,1989),e.g.by(exogenously)having di?erent sets of products o?ered in di?erent markets.In auction models the tests rely on a similar idea of exogenous rotation,driven by exogenous change in the number of bidders(Athey and Haile,2002;Haile,Hong,and Shum, 2003).

Despite these similarities,there are important di?erences between the two literatures.One has already been mentioned:data sets will be di?erent.The analog to quantity data is not observable in an auction setting,thereby forcing us to rely more on structural assumptions.A second di?erence is computational:while in demand models we solve for a Nash Equilibrium, auction models require us to solve(or to avoid solving)for a Bayesian Nash Equilibrium.Thus, we need to solve for the full equilibrium strategy rather than for the equilibrium price.This makes things somewhat more computationally intensive(e.g.we need to solve a system of ODE’s rather than a system of equations).Most important,the literatures di?er in the object of interest.In demand models,the demand system is taken as given and drives many of the counterfactuals, thus allowing for a?exible demand system is crucial.In auctions,we may in?uence the structure of the game by changing the rules of the auction(i.e.changing the“demand function”),thus we may be interested in a di?erent set of counterfactuals,and may need to be more?exible on other dimensions.

5Concluding Remarks

This note lays out a conceptual mapping between empirical models of di?erentiated product demand systems and empirical models of IPV auctions.It highlights the fact that the striking di?erences in the emphasis of the two literatures is mainly driven by the nature of the assumptions employed.The auction literature can emphasize nonparametric estimation because it makes stronger assumption about the data generating process.Most importantly,it assumes that the agents do not possess any informational advantage about the environment compared to that known

to the econometrician.The reasons,I believe,are twofold:the nature of the data(quantity data can both identify the demand shifters and,at the same time,falsify the restriction)and complexity of computations(solving for the asymmetric Bayesian Nash Equilibrium is harder than for the full-information Nash Equilibrium).

Is this di?erence in the set of assumptions driven by fundamental di?erences in the economics of these two types of markets?My belief is that in an abstract sense the answer is no,although certain cases may be“cleaner”and hence may satisfy this stronger set of restrictions.The goal of this note,however,is not to evaluate the di?erent sets of assumption,but rather to put them within the same framework.My hope is that it will bring closer together the two literatures,and will shed some light on questions such as:what can we get in di?erentiated demand estimation if we assume away the endogeneity problem but relax the functional-form assumptions,or what would be the main issues in the auction literature if we relax the symmetry assumptions.

In addition,the note makes us think about application in which problems from both literatures are combined.This can be thought of in the context of incomplete information Bertrand com-petition,when products are di?erentiated.This may be the case when prices cannot be changed costlessly as a response to opponents’prices.Equivalently,one can think of applications in which the allocation rule of the auction is unknown to the econometrician,as it depends not only on price,but on other variables.This will force the econometrician to estimate the allocation rule at the same time as it estimates the cost distribution.These two problems are interesting avenues for future research.

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