University of California, Davis
Department of Agricultural and Resource Economics
Problem Set 3
1. Intel is a monopolist manufacturing computer chips with a competitive fringe of firms that act as price takers of the price Intel sets. The market willingness to pay for the P80 chip is given by= 400 − Q, with p in dollars p
per chip and Q in thousands of chips per month. The fringe marginal cost curve is MC= 40 + .5Q F (with Q F
and MC F also in thousands of chips and dollars per chip, respectively), and Intel's marginal cost of producing chips is constant at $50 per chip. Intel is planning its strategy to set the P80 chip price. (a) Determine the residual demand Intel faces after accounting for the quantity supplied by the competitive fringe for any level of price.
The residual demand that Intel can work with is the difference between total quantity demanded and the quantity supplied by the fringe at any given price. Total demand is= 400 − p, and fringe supply is Q f 2( p − 40), =
for prices above 40. Thus the residual demand is
Q r =Q − Q f =(400 − p ) − 2( p − 40) =480 − 3 p.
(b) How many P80 chips will Intel supply per month?
The inverse demand facing Intel is p r = − (1/ 3) ⋅ Q r , so Intel's marginal revenue is 160
MR r = − (2 / 3) ⋅ Q r . Setting this equal to Intel's MC, 160
MR r = 160 − (2 / 3) ⋅ Q r = 50 = MC r ,
Q r =− 50) /(2 / 3) =
Intel will supply 165 thousand chips per month.
(c) What is the resulting world P80 chip price?
The resulting world price is set from the Intel residual inverse demand: p = 160 − (1/ 3) ⋅165 = $105 per chip.
(d) How many P80 chips are supplied by the competitive fringe? The competitive fringe takes the price of oil as given, so from its supply function Q f =⋅ (105 − 40) = thousand chips per month.
2. You and another firm are a duopoly supplying the market for bread in Davis. The inverse aggregate demand you both face is p 10 − .001Q . You are firm 1, and your marginal cost of making bread is $1 per loaf, so =
MC1 = 1 ; while your competitor has marginal cost of MC2 = 2, or $2 per loaf. You are both engaged in Cournot competition.
(a) What is your reaction curve?
To get your own reaction curve, note that your profit expression is e
π 1= [10 − .001(q1 + q2 )] ⋅ q1 − 1 ⋅ q1 ,
which is maximized when
∂π 1 / ∂q1 = [9 − .002q1 − .001q2 )] = 0.
Solving this for q gives your reaction curve,
RC1 : = 4500 − .5q2 .
(b) What is your competitor's reaction curve?
The same logic is used for firm 2, whose profit is
π 2 = [10 − .001(q2 + q1e )] ⋅ q2 − 2 ⋅ q2 ,
which is maximized when
∂π 2 / ∂q2 =8 − .002q2 − .001q1e =0.
Solving for q gives 2's reaction curve:
RC2 : = 4000 − .5q1e .
(c) What are the equilibrium quantities supplied to the market by each firm in the Cournot equilibrium, and at what price?
The Cournot equilibrium quantities occur when what each rival expects the other to produce is in fact what the other produces, which is where the two reaction curves intersect. That is, RC and RC are two equations in the 1
two unknowns q and q , which represent the equilibrium Cournot quantities. Substituting RC into RC , we 1
q1n = − .5(4000 − .5q1n .),
or q n =
[4500 − .5(4000)](4 / 3) =
and substituting q n into RC , we get
q2 = − .5(3333.3) =
With these Cournot equilibrium quantities, the total supply to the market is Q = 5666.6, and the market price is
10 − .001(5666.6) = loaf.
(d) Suppose you collude with your competitor to maximize profit from bread in Davis; i.e., you act as a two-plant monopoly. What are the resulting total quantity supplied to the market, equilibrium price, and profits earned by each firm? (Assume that you and the competitor distribute the profits between yourselves so that each of you makes as much as under Cournot...
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