consider a Cournot duoply model Consider a Cournot duopoly model where the marke
ID: 1168891 • Letter: C
Question
consider a Cournot duoply model Consider a Cournot duopoly model where the market price is given by the inverse demand function: And suppose each firm produces at a cost of 25 per unit of production (i.e, total costs are given by 25 for i = 1,2). Find the payoff functions u1 (q1,q2) and u2 (q1,q2). Suppose player 1 believes that player 2 is equally likely to produce q2 = 10, q2 15, q2 18 and q2 22 units, and these are the only production values player 1 believes player 2 can choose. Let theta 2 denote these beliefs for player 1. Compute a, (q1, theta 2) for these beliefs theta 2. (c) Suppose player 2 believes that player 1 will produce q1, = 13 with probability 1/3, q1 = 10 with probability 1/4, and 1/4, q = 16 with probability 5/12, and these are the only production values player 2 believes player 1 can choose. Let theta 1, denote these beliefs for player 2. Compute u2(h=theta1,q2) for these beliefs theta 1.Explanation / Answer
a) pay off functions
u1(x1,x2)= revenue- cost = p.q1 - average cost .q1 =( 30-3q1 -q2 )q1 - 25 q1
u2(x1,x2)= revenue- cost = p.q2 - average cost .q1 =( 30-3q1 -q2 )q2 - 25 q2
b) player one thinks player 2 has 1/4 probabilty each of choosing q1= 10, 15, 18 ,22
as this is the case duopoly so decision of player 2 is going to effect player 1 payoff also ...Q2 denots this belief of 1 for 2 , so possible expected or average q2 based on these beliefs will b given by
q2 = 1/4*10+1/4*15+1/4*18+1/4*22 = 16.25
so u1(q1,Q2)= (30-3q1-16.25)q1-25q1
in in presence of Q2, this is pay off of player 1 and this needs to be maximised.
c)similarly now player two has some beliefs about player 1 decision, based on these beliefs Q1, now player 2 will get his payoff.
expected or average q1=1/3*13+1/4*10+5/12*16 = 13.5
so u2(Q1,q2)= (30-3(13.5)-q2)q2-25q2
in in presence of Q1, this is pay off of player 2 and this needs to be maximised.
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