I really help this problem. If you need more space, please leave a comment. 2. T
ID: 1125146 • Letter: I
Question
I really help this problem. If you need more space, please leave a comment.
2. There are two lemonade stands on the boardwalk. The lemonade from each stand tastes exactly the same. It is a hot day at the beach and demand for lemonade at the beach is Qd-500-p. The two stands have the same cost function: G (q) = 30+ 291 and C2 (q2) 30+242 (a) Solve for the equilibrium market price, the equilibrium quantity lemonade stand 1 pro- duces, the equilibrium quantity lemondade stand 2 produces, and the profit for each stand under Cournot competition. (9 points)Explanation / Answer
a) Find the profit functions of the two firms and maximize them. Demand function is taken in inverse form:
P =500 – Qd.
Profit (1) = R1 – C1
(1) = 500q1 – q1^2 – q1q2 – 30 – 2q1
d(1)/dq1 = 0
498 – 2q1 – q2 = 0
Profit (2) = R2 – C2
(2) = 500q2 – q2^2 – q1q2 – 30 – 2q2
d(2)/dq2 = 0
498 – 2q1 – q2 = 0
Solving these two equations will give a unique solution to the problem
q1 = q2 = q = 166
Price = 500 – 2*166 = $168
Profit earned by both firms is (1) = (2)
= 498q1 – q1^2 – q1q2 – 30 = 498*166 - 166^2 – 166*166 – 30
= $27526
b) Collusion has both firms jointly maximizing profit
= (500 – q1 – q2)(q1 + q2) – 30 – 2q1– 30 – 2q2
= 500(q1 + q2) – q1^2 – q2^2 – 2q1q2 – 60 – 2q1 – 2q2
d/dq1 = 0
498 – 2q1 – 2q2 = 0
d/dq2 = 0
498 – 2q2 – 2q1 = 0
Solving these two equations will give a unique solution to the problem
q1 = q2 = q = 124.5
Price = 500 – 2*124.50 = $251
Joint profit = 500*249 – (124.5)^2 – (124.5)^2 – 2*(124.5)^2 – 60 – 2*(249) = $61,941.
Profit earned by each firms is 61941/2 = $30970.50
c) Under a monopoly single output is produced. So the cost function is C = 30 + 2Q. MR = 500 - 2Q and MC = 2. This gives
MC = MR
500 - 2Q = 2
Q = 249
Price = 500 - 249 = 251
Profit = TR - TC = 251*249 - 30 - 2*249 = $61971.
d) Cournot firms produce more, charged a lower price and earn a lower profit. Colluded firm earns a higher profit by charging a higher price and producing a lower quantity. Monopoly has same quantity and price but a higher profit.