(a) Assume thatNis °xed. For any given(p1; :::; pN);calculate demand for °rm 1. (Hint: to calculatethis, you need to identify two indi/erent consumers)(b) Find a symmetric Bertrand Nash equilibrium price as a function ofN; c;and±:(c) Assume thatNis endogenously determined by the free entry condition. LetN°denote the num-ber of °rms in the market, which is given by the free entry condition.CalculateN°and thecorresponding price level. (Hint: you can forget about the integer constraint)(d) We investigate if the free market produces a larger or a smaller variety than the optimal varietylevel. Consider the social planner who solves the following problem:minNW(F; ±; N) =NF+T(N; ±);whereNFis the total °xed cost incurred andT(N; ±)is the aggregate travel cost that the societyincurs.i. CalculateT(N; ±):(Hint: in a symmetric equilibrium, prices are equal so every consumer travelsto the closest store)ii. FindNSPthat minimizesW(F; ±; N):(Hint: you can forget about the integer constraint)iii. CompareN°andNSP:3. (30 points in total) Consider a spot market of electricity, wherenidentical °rms are producing a homoge-neous product (electricity) in a Cournot fashion. The anti-trust authority is concerned about conducts ofthosen°rms. In particular, they doubt thatn°rms are somehow colluding. Assume that the marginalcost of each °rm is given byMCi(Zt; qi) =³Zt+´qi+!t;whereZtis the strength of wind that shifts the marginal cost.!tsummarizes all the unobservablevariables that we assume are uncorrelated withZt. We assume that each °rm will produce the sameamount, so averaging the marginal cost over °rms gives the industry-level marginal cost:1NNXi=1MCi(Zt; q)=³Zt+°´Qs+!t=MC(Zt; Qs)(1)where°´=°N:I put a superscriptstoQto emphasize thatMCis a function of quantity supplied.Demand is given byQ(pt; Xt) =µXt+¶tpt+"t;(2)whereXtis the average temperature int;and"tsummarizes all the unobservable that are assumed tobe uncorrelated with temperature. The price coe¢ cient can take two di/erent values:¶t=8<:¶weekdayiftis a weekday¶weekendotherwise:(a) (3 points) Letting°=@Q@q;write down a °rm±s °rst order condition. Hint: write down the pro°t, andtake the °rst order condition with respect toqi:Your °rst order condition will includeMCi(Zt; qi);@p@Q;and°:
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- Spring '08
- Staff
- Economics, Game Theory, Bertrand Nash