Lecture 1: Measuring Market Power and Collusion
Yiyi Zhou
Department of Economics
Stony Brook University
ECO 637: Empirical IO
Overview
1 Measuring Market Power
Genesove and Mullin (1998)
Bounds on Market Power
2 Collusion and Price Wars
Porter (1983)
Ellison (1994)
Bresnahan (1987)
Estimating cost functions without using cost data
During the 1960s and 1970s, IO economists were building methods
to estimate cost functions. Why?
I Return to scale,
I Learning by doing,
I Efficient scale, etc.
At the same time, many papers tried to test the SCP paradigm
using accounting data.
For instance, cross-industry comparisons were conducted to
estimate the causal effect of concentration on profitability or
prices:
Profits
jt
= + Concentration
jt
+ X
jt
+ u
jt
Critics started pointing out that,
1 Market structure is not exogenous
2 Accounting costs 6= economic costs
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Rosse (1970, Econometrica)
Combine economic theory assumptions with prices and output
data to estimate economic marginal cost functions.
One output y
j
example
p
j
= 0 + 1xj 2yj + uj
mc
j
= 0 + 1zj + 2yj + vj
E(u) = E(v) = 0
E(u v) = 0
E(x u) = E(x v) = 0
E(z u) = E(z v) = 0
The last two corresponds to short-run assumptions: Quality and
other sunk product characteristics are fixed in the short run.
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Conduct Assumption
Problems: mc
j
is unobserved & How can we estimate the return to
scale parameter 2?
Conduct Assumption: Each local newspaper is a local monopolist
and chooses y
j
to maximize profits.
Equilibrium condition:
MR
j
MC
j
= e
j
where e
j
is a mean-zero optimization/specification error
With linear demand and marginal-cost functions:
0 + 1xj 22yj + uj = 0 + 1zj + 2yj + vj + ej
$ p
j
= 0 + 1zj + (2 + 2)yj + vj + ej uj| {z }
=w
j
where E (w
j
(x
j
, z
j
)) = 0
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Identification and Estimation: GMM Set-Up
Theoretical and empirical moment conditions:
E(u
j
(x
j
z
j
)) = 0 )
1
n
X
j
u
j
(x
j
z
j
) = 0
E(w
j
(x
j
z
j
)) = 0 )
1
n
X
j
w
j
(x
j
z
j
) = 0
Identification?
Rank conditions: MC function is identified as long as z
j
contains
exogenous variables not included in x
j
to identify the demand
curve (and vice-versa).
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More generally, demand and supply relations can take non-linear
forms:
y
j
= f (x
j
, p
j
, u
j
| )
p
j
= g(y
j
, x
j
,w
j
| )
Takeaway: If firms are optimizing (i.e. conduct), observed actions
reveal the implicit opportunity cost of production. This leads to a
(now) standard revealed-preference estimation strategy.
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What about Oligopoly Markets?
Under a particular conduct assumption, the same insight can be
extended to markets with more than one firm.
Symmetric Cournot:
P(q
i,t , qi,t) + P
0(q
i,t , qi,t)qi,t = MC(qi,t)
$ P(Q
t
) = MC(Q
t
)
1
n
t
P
0(Q
t
)Q
t
[Summing across i ]
Asymmetric Cournot:
P(q
i,t , qi,t) + P
0(q
i,t , qi,t)qi,t = MCi (qi,t)
$ P(Q
t
) =
1
n
t
X
i
MC
i
(q
i,t)
1
n
t
P
0(Q
t
)Q
t
[Summing across i ]
Bertrand:
P(Q
t
) = MC(Q
t
)
Collusion:
P(Q
t
) + P 0(Q
t
)Q
t
= MC(Q
t
)
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Supply Relations Estimation
MC is identified under specific conduct assumptions. What
identifies firms conduct?
Most oligopoly models can be nested into a general supply relation
equation:
P(Q
t
) = MC(Q
t
) P 0(Q
t
)Q
t
where 2 (0, 1) is a measure of market power (i.e. conduct
parameter)
Example:
I = 0: Bertrand
I = 1/n: Symmetric Cournot
I = 1: Monopoly
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Two Justifications
1 Can be used to test particular models (e.g. H0 : = 1/n)
2 Theoretical foundation for 2 (0, 1) : Conjectural variation model
I CV Equilibrium
max
q
i
P
q
i
+
P
j 6=i Qj(qi ),X
q
i
C
i
(q
i
,Z)
F .O.C . P(Q,X ) + q
i
P 0(Q,X )
0
@q
i
+
X
j 6=i
@Q
j
(q
i
)
@q
i
1
A
| {z }
1+r
C
0
i
(q
i
,Z) = 0
I Averaging across firms: P
m
+ QP
0
m
(Q,X
m
) MC(Q
m
,Z
m
) = 0 where
= (1 + r)/n
I The conduct parameter then corresponds to the average
conjecture in the industry:
F
Bresnahan (1989): IF the conjectures are constant over time and
collusion breakdowns are infrequent, measures the average
collusiveness of conduct
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Identification of Market Conduct
Example: Linear demand/cost (symmetric)
P(Q
t
) =
x
x
t
q
Q
t
+ u
t
MC(Q
t
) =
z
z
t
+
q
Q
t
+ v
t
Two estimating equations:
Demand : P
t
=
x
x
t
q
Q
t
+ u
t
Supply relation : P
t
=
z
z
t
+ (
q
+
q
)Q
t
+ v
t
Negative result: The industry conduct parameter is not identified,
even with all the necessary exclusion restrictions. Why?
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The linearity of demand implies that the supply relation between
q
t
and p
t
is confounded with the possibility of a non-linear cost
function.
Note: This also implies that estimates of the pass-through of cost
shocks onto prices are not sufficient to test for market-power
(unless we assume a constant MC function).
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Lack of Identification in a Figure
When P(q
t
) is linear, variation in x
t
induces parallel shifts:
P(q
t
) = x
t
x
q
Q
t
+ u
t
This implied change from E1 to E2 can be explained by collusion or
competition
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Relevant Source of Variation
This is not the case when we can observe demand rotation.
I Example: P(q
t
) = x
t
x
q
y
t
Q
t
+ u
t
The implied change from E1 to E3 (caused by yt) can only be
explained by collusion
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Genesove and Mullin (1998): Sugar Cartel
Historical notes:
I Industry organized as Trust in 1887.
I Between 1887 and 1911, the industry alternated between periods of
collusion, and price war episodes triggered by the entry and
expansion of outside firms.
I The trust was dismantled in 1911 by the US government.
Objective: Validate the conduct estimation approach by comparing
predicted and observed estimates of marginal costs, under known
conduct.
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Direct Measure of Market-Power
With complete price and cost information, optimality condition
implies pricing relation:
P(c , ) =
c(P)
(P)
Therefore,
= (P)
P c
P
L
so, conduct parameter = elasticity-adjusted Lerner index L
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Step 1: Technology
Linear production technology:
MC
t
= c
t
= c0 + kPraw,t
where k = 1.075 (i.e., inverse rate of transformation)
Intercept: c0 2 (0.16, 0.26) from industry documents
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Step 2: Demand Estimation
Functional form:
Q
t
(P) =
t
(
t
P)t
where
t
, and
t
or
t
are allowed to vary by season (third quarter)
Instrument: Cuban imports (i.e., closest substitute)
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Step 2: Demand Estimation
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Step 3: Direct Conduct Estimates
Demand elasticity:
t
(P) =
t
t
(
t
P)t1
P
t
(
t
P)t
=
t
P
t
P
Prices:
P(c , ) =
c(P)
(P)
=
+ c
+
Conduct parameter = elasticity-adjusted Lerner index L
L = = (P)
P c
P
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Step 3: Direct Conduct Estimates
Mean L
n
is close to 0.10
I corresponds to static, ten-firm symmetric Cournot oligopoy
I reject both perfect competitive and monopoly pricing
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NEIO Conduct Parameter Estimation
Supply relation (linear model):
P
t
=
t
+ c
t
1 +
=
t
+ c0
1 +
+
k
1 +
P
raw ,t + ut
Using the Cuban imports as IV for the price of raw can sugar,
yields the following moment condition:
E [(1 + )P
t
t
c0 kPraw ,t | Zt ] = 0
where
t
=
low
1(t = Low season) +
high
1(t = High season)
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Identification of
Assumption: Unobserved changes in firms conduct (i.e.
u
t
= 4
t
+ e
t
) are independent of IVs.
This is a difficult assumption to satisfy
I E.g: Price wars during booms.
I Corts (1999): Correlation between conduct changes and demand
shocks invalidates standard instruments (downward bias)
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Key Result: is Under-Estimated
(1): unknown c0 and k ; (2) k = 1.075 is known
NEIO underetsimates , but this deviation was minimal
Still reject Monopoly ( = 1) and Cournot with nine or fewer firms
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Alternative Approach: Bounds on Market Power
Market conduct tests a la Bresnahan suffer from (at least) two
critics:
I Requires knowledge of demand curve: Functional form assumptions
can invalidate the results
I Necessitates variation in the slope of the demand curve (somewhat
arbitrary)
Sullivan (1985) and Ashenfelter and Sullivan (1987) construct an
upper bound on the degree of market power.
I Null hypothesis: Monopoly.
I Minimal assumptions on demand and cost functions
I Exploits observed (exogenous) shocks to marginal cost
I Key requirement: Shock must be separable (e.g. tax shock)
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Model Set-up
Homogenous goods: P(Q) = P
P
j
q
j
and P 0(Q) < 0Heterogenous cost functions: Cj(q) with C0j(q) 0Excise tax: Cj(q) + tq where Cj(q) is time-invariantProfit maximization condition give tax level tP(q(t)) + qi(t) P 0(q(t))| {z }=P0(t)/q0(t) = C0i(qi(t)) + tP(t) t mci(t)+ qi(t)P 0(t)Q 0(t)= 0where Q(t) =Piqi(t), Q 0(t) = dQ/dt and P 0(t) = dP(t)/dt is a market conjecture: @Q/@qiI Cournot conjecture: = 1I Collusion conjecture: = nYiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 25 / 55Necessary ConditionsNecessary equilibrium condition for conduct :P(t) t c+ qi(t)P 0(t)Q 0(t) 0, 8c < mci(t)Aggregating at the market level, this gives a lower bound on thenumber of equivalent Cournot competitors:n(t) =Xi1i n(t, c) =P 0(t)Q(t)(P(t) t c)Q 0(t)8c < mci(t)Why is it useful?I RHS depends only on observed variables P(t) and Q(t), andreduced form pass-through rates P 0(t) and Q 0(t)I If P 0(t) > 0 and Q 0(t) < 0 , n(t, 0) provides a useful lower boundon the industry conductI Allow to reject the monopoly model, but not the perfectcompetition assumption.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 26 / 55Application: Pass-through of cigarettes taxParametric reduced-form equations:qis(t) = exp1is+ g1t + h1(t t)2pis(t) = 2is+ g2t + h2(t t)2where jiscontrols for state FEs and time-trends.OLS results:I q0(t) = 2.93: consistent with the theory (i.e. tax increasesmarginal cost).I p0(t) = 1.089: significantly greater than 1, reject Bertrand withconstant mc (i.e. complete pass-through).Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 27 / 55Estimated n(t, c)n(t, c = 0) = 2.88: significantly different from 1, easily reject themonopoly model.The observed pass-through rates are consistent with a fairlyimportant level of competition.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 28 / 55Overview1 Measuring Market PowerGenesove and Mullin (1998)Bounds on Market Power2 Collusion and Price WarsPorter (1983)Ellison (1994)Bresnahan (1987)Collusion and Price WarsTextbook models of tacit collusion predict stable prices, andoff-equilibrium cheating.In most known cases of implicit or explicit collusions, we observealternating periods of high/low markups, price wars, cheating, etc.Price series: two regimes of pricing. Can periods of low pricingbe explained as price wars?Repeated game theory: view observed price series as realization ofequilibrium price processStandard repeated games (e.g., repeated Cournot game) withunchanging economic environment: equilibrium price path isconstant!I Need to specify punishment strategies which support collusiveequilibrium, but punishment is never observed on the equilibriumpath.I Need for model with nonconstant equilibrium price processYiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 29 / 55Two Models of Tacit Collusion1Demand fluctuations: Collusive prices must adjust to reflecthigher temptation to cheat when demand is highI Rotember and Saloner (1986). Price wars during booms2Imperfect information: Price wars discipline cartel membersprice cuts are not fully observedI Green & Porter (1984): Price wars during recessionsThese two models generate periods of low and high pricing on theequilibrium path. But low prices are not caused by cheating;rather they are manifestation of collusive behavior!Price wars, or more generally, equilibrium cheating behavior arestill phenomena that we dont understand very well (good researchtopic!).Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 30 / 55Rotemberg and Saloner (1986): Price Wars DuringBoomsEmpirical regularity: In many oligopolistic industries, prices ormarkups are counter-cyclicalI Cement (Rotemberg and Saloner 1986), refined sugar (Genesoveand Mullin 1998), gasoline (Borenstein and Shepard 1996), Groceryitems (Chevalier et. al 2003)Interpretation:With i.i.d. demand fluctuations, fixed discount rate, constantmarginal production costs, collusive prices will be lower in periodsof above-average demand (price wars during booms)Intuitively, cheat when (current) gains exceed (future) losses.Current gains highest during boom periods; reduce incentives tocheat by lowering collusive price.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 31 / 55Price Wars During Booms?Caveat 1: The model does not really predict price-wars, sincedeviations are not observed in equilibrium.I RS = Theory of countercyclical markups.Caveat 2: To test the prediction in the data, we need to be careful.Case 2 does not imply that p2 < pm1 . The predictions is aboutlowering the prices relative to the monopoly price (i.e. p2 < pm2 .).I Need to condition on demand/cost state variables.Caveat 3: When demand shocks are not IID, the predictions canbe reversed.I Important: Demand is expected to be low in the (near) future if itis very high today.I See Harrington and Haltiwanger (1991)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 32 / 55Collusion With Secret Price CutsThe idea that imperfect monitoring cause price wars dates back toStigler.First formalization: Green and Porter (1984)Same framework as RS(1986), but introduce imperfectinformation – firms cannot observe the output choices of theircompetitors, only observed realized market price. Prices can belower during periods of low demand (price wars duringrecessions).Note: market price can be low due to either (i) cheating; or (ii)adverse demand shocks. Firms cannot distinguish.Intuitively: equilibrium trigger strategies involve low priceregime when prices are low. That is, if the market price dropsbelow the trigger level, firms revert to one-shot Nash.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 33 / 55Porter (1983)Test the Green-Porter model: two regimes of behavior(cooperative vs. noncooperative/price war)I Subtle: noncooperative regime arises due to low demand, not dueto cheatingEmpirical problem: dont (or only imperfectly) observe when aprice war is occurring. How can you estimate this model then?Prices should be lower in price war periods, holding the demandfunction constant. Price war triggered by change in firm behavior.Estimate simultaneous-equation switching regression modelwith unobserved regimes.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 34 / 55Brief History of the JECLegal and public cartel formed in 1879.Control railroad eastbound shipments from Chicago to the Eastcoast.Historical evidence that the cartel used temporary price cuts topunish rumors of cheating by a member.Firms set rates individually and privately.Volume transported was surveyed weekly by the JEC.Prices are monitored only imperfectly, and firms only observedaggregate market shares.Recurrent price war episodes were documented by economichistorians.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 35 / 55Demand SideObserved data are market-level output (Qt) and price (pt) forweekly grain shipments between 1880 and 1886N firms (railroads), each producing a homogeneous product (grainshipments). Firm i chooses qitin period t.Market demand:logQt= 0 + 1logpt + 2Lt + U1t ,where Qt=Piqit, Ltis demand shifter: = 1 of Great lakes opento navigation (availability of substitute to rail transport)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 36 / 55Supply SideFirm i s cost fxn:Ci(qit) = aiqit+ FiFirm i s pricing equation:pt(1 +it1) = MCi(qit)where it= 0: Bertrand pricing; it= 1: Monopoly pricing;it= sit: Cournot outcomeYiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 37 / 55Aggregate Supply EquationAfter some manipulation, aggregate supply relation:logpt= logD ( 1)logQt log(1 + t/1)with empirical versionlogpt= 0 + 1logQt + 2St + 3It + U2t ,where It= 1: cooperative regime (joint profit maximization); It= 0:noncooperative regime (one shot Bertrand or Cournot); St: supplyshifters (dummies DM1, DM2, DM3, DM4 for entry by additional railcompanies)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 38 / 55Simultaneous EquationsDemand and Supply Equations:logQt= 0 + 1logpt + 2Lt + U1tlogpt= 0 + 1logQt + 2St + 3It + U2tAssume(U1t ,U2t)0 N(0,)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 39 / 55Matrix NotationByt= Xt+It+ Utyt= logQtlogpt!,Xt=0B@1LtSt1CA ,Ut= U1tU2t!B = 1 11 1!, = 03!, = 0 2 00 0 2!Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 40 / 55LikelihoodSuppose in each period you know whether firms are in the collusiveor non-collusive regime.Then, the likelihood conditional on It:L(Yt| It) =| |1/2| B | exp12(Byt XtIt)01(Byt XtIt)Problem: we dont observe the regime (It).How to proceed?Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 41 / 55LikelihoodTreat it as a nuisance parameter and integrate out over itsdistribution: L(Yt) =RL(Yt| It)g(It)dItAssume that Itfollows a discrete, two point distribution (fromstructure of GP equilibrium):It=8<:1, with prob 0, with prob 1 Treatment is exogenous. Endogeneity in this model arises fromsimultaneity of Qtand pt.So likelihood function:L(Yt) = L(Yt| It= 1) + (1 )L(Yt| It= 0)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 42 / 55Key ResultsEstimate of 3 is 0.545: prices 50% higher when firms are incooperative regimeIf we assume = 0 in non-cooperative periods, then3 = log(1 + /1) implies = 0.336 (close to Cournot) incooperative periods.Prices 66% higher and quantity 33% lower in cooperative regime.Price wars were not preceded by large negative demand residuals.Cartel earns $11,000 (11%) more in weeks when they arecooperatingYiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 43 / 55DiscussionWhat if Itis endogenous (arising from, eg., price wars more likelywhen demand is low, etc.)?If regime observed, 3 is the treatment effect of It . Possible touse other methods to estimate treatment effect?I Data structure is not DID: there are no control units for whichIt= 0 throughout the sample period. So cannot distinguish effect ofItapart from week dummiesI IV approach, with Ltas (discrete) instrument? But need anotherdemand shifter, because only one IV (Lt) for two endogenousvariables (Itadn Qt)Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 44 / 55Ellison (1994 RAND)Allow for variables to enter : theory gives guidance that pricewars precipitated by triggers: low demand (in GP model), largeshifts in market share. Allow Itto be an endogenous in this fashion.Allows unobserved Its to be serially correlatedProb {It+1 | It ,Zt} =eWt1 + eWtwhere Zt: set of predetermined variables at t, Wtcontains Itsoadds a Markov structure to the price wars (a constant independentof Itin Porter (1983))Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 45 / 55Ellison (1994 RAND)Also treats RS (1986) model:I alternative explanation for price wars (where serial correlation indemand, not unobserved firm actions, drives price fluctuations).I In equilibrium, prices lower in periods of relatively high demand,when demand is expected to fall in the future.Test for explicit cheating on the part of firms, which doesnthappen in the equilibrium of any of these models. Introducesadditional regimes to the model.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 46 / 55Ellison (1994 RAND): ResultsFind a greater degree of collusion.Regimes are not independent of each other.Unanticipated demand shock enters negatively in the price wartrigger probability (i.e. 6=RS).The RS story is not supported in this data-set.Find evidence of two-type of hidden regimes: (i) regular pricewars, and (ii) large unobserved demand shock (e.g secret price cut).Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 47 / 55Bresnahan (1987): The 1955 price war in the US automarketResearch question: Is the 1955 increase in production explained bya deviation from collusion?Focuses on static pricing models, among differentiated products.Evidence that manufacturers compete in 1955, but not insurrounding years.Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 48 / 55Quality Ladder ModelJ vertical differentiated products: xJ> x
J1 > > x1 > x0
Indirect utility given marginal utility for quality v
i
:
U(x
j
, v
i
,Y ) =
8
<:vixj+ Y Pj, if j 6= 0vix0 + Y E , if j = 0Distribution assumption: vi U([0,Vmax]) with density Market sharessj= Dj(pj, pj) =8>>><>>>:
1
V
max
PJPJ1
x
J
x
J1
if j = J
1
P
j+1Pj
x
j+1xj
PjPj1
x
j
x
j1
if 1 j < J1P1Ex1x0if j = 0Constant MC: mc(x) and mc 0(x) > 0
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 49 / 55
Conduct Assumptions
Individual product Bertrand-Nash:
s
i
+ (P
i
mc(x
i
))
@D
i
@P
i
= 0
Multi-product Bertrand-Nash:
s
i
+
j=i+1X
j=i1
ij
(P
j
mc(x
j
))
@D
j
@P
i
= 0
where
ij
= 1 if i and j are produced by the same firm
Collusion:
s
i
+
j=i+1X
j=i1
(P
j
mc(x
j
))
@D
j
@P
i
= 0
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 50 / 55
Identification in a Figure
If the marginal cost function is monotonically increasing in quality,
prices should be increasing in quality independently of the presence
of closed substitutes (under collusion).
If firms are competing, prices should also be function of the
presence of close substitutes.
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 51 / 55
Empirical Model
Deterministic quality and marginal cost functions:
x
i
=
p
0 + 0zi
mc(x) = exp(x)
where z
i
is a vector of car chars
Measurement errors:
p
i
= p
i
(x | H, ) + p
i
q
i
= q
i
(x | H, ) + q
i
where (p
i
,
q
i
) N(0,)
Likelihood function:
L(P ,Q | Z , ,H) =
X
i
ln (f (p
i
p
i
, q
i
q
i
| ))
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 52 / 55
Testing for Collusion
Non-nested hypothesis test: Likelihood ratio of H0 and H1,
evaluated under H0 (Cox statistic).
For instance, when evaluated using the parameters of the collusive
model (null), is the difference between collusion and competition
(alternative) likelihood large enough to reject the competition
model?
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 53 / 55
Conclusion: Bertrand-Nash cannot be rejected in 1955
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 54 / 55
Other Work on Collusion
Borenstein and Shepard (1996 RAND) find support for RS (1986)
theory in retail gasoline markets
Pakes and Ferschtmann (2000 RAND) examine the feasibility of
collusion in rich model of oligopolistic industry in which firms can
choose quality investment as well as price, and entry and exit can
occur.
Chevalier, Kashyap, Rossi (2003, AER ) tests RS models versus
other explanations of price wars during booms
Yiyi Zhou (Stony Brook) Market Power and Collusion ECO 637 55 / 55
Measuring Market Power
Genesove and Mullin (1998)
Bounds on Market Power
Collusion and Price Wars
Porter (1983)
Ellison (1994)
Bresnahan (1987)
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