Number 5, show work for all parts please. 5. The market demand for a commodity i
ID: 1120114 • Letter: N
Question
Number 5, show work for all parts please. 5. The market demand for a commodity is given by: Q-200 - P. Suppose that this market is served by a duopoly. Assume that the two firms are able to produce according to the production function 0.5T 0.5 a. Set up the firm's cost minimization problem, and derive the total cost function associated with b. Find the Cournot solution for equilibrium price, firm output and firm profits. Q-KL. Further assume that w s1 and r-$2 and represent the prices of L and K, respectively. this production technology c. Find the Stackelberg solution for equilibrium price, firm output and firm profits, assuming that firm 1 is the Stackelburg leader. d Compare industry output,prices and profits in each ype of market.
Explanation / Answer
a) The cost minimization problem is:
Min C = L + 2K
subject to Q = K0.5L0.5
Find the tangency condition:
Marginal product of labor / Marginal product of capital = w/r
0.5 K0.5L-0.5/0.5 K-0.5L0.5 = 1/2
L/K = 1/2
K = 2L
Substitute the tangency condition in the production function:
Q = (2L)0.5L0.5
Q = 20.5 L
L = Q/20.5
Substitute the above expression in the tangency condition to get:
K = 20.5 Q
Substitute the above two expressions of K and L into the cost function to get:
C = (Q/20.5) + 2(20.5 Q)
C = 5Q/20.5
b)
From the cost function, we know that the marginal cost function is
MC = 5/20.5
The inverse demand function is
P = 200 – Q
P = 200 – Q1 – Q2
Calculate the best response function of firm 1:
MR1 = MC
dTR1/dQ1 = MC
d[(200 – Q1 – Q2) Q1] /dQ1 = MC
200 – 2Q1 – Q2 = 5/20.5
Q1 = 100 – (5/21.5) – (Q2/2)
Similarly, the best response function of firm 2 is:
Q2 = 100 – (5/21.5) – (Q1/2)
Solve the two response functions to get:
Q1 = Q2 = 65.49
The firm output is 65.49. The industry output is 130.98. The equilibrium price is $69.02 (= 200 – 130.98).
Each firm’s profits = (69.02)(65.49) – [3(65.49)/20.5] = $4,381.19.
(c) Substitute the best response function of firm 2 in the demand function to get the demand function faced by firm 1 in Stackelberg model.
P = 200 – Q1 – [100 – (5/21.5) – (Q1/2)]
P = 100 + (5/21.5) – (Q1/2)
Calculate the profit maximizing quantity of firm 1:
MR1 = MC
dTR1/dQ1 = MC
d{[100 + (5/21.5) – (Q1/2)]Q1} /dQ1 = MC
100 + (5/21.5) – Q1 = 5/20.5
Q1 = 98.23
Insert 98.23 at the place of Q1 in the best response function of firm 2 to get:
Q2 = 49.11
Industry output = 98.23 + 49.11 = 147.34
Equilibrium price = 200 – 147.34 = 52.66
Firm 1’s profits = (52.66)(98.23) – [5(98.23)/20.5] = $4,825.50.
Firm 2’s profits = (52.66)(49.11) – [5(49.11)/20.5] = $2,812.50.
(d) The industry output is higher in the case of Stackelberg model. It was only 130.98 in the case of Cournot model. The equilibrium price is lower in the case of Stackelberg model. In comparison to Counot model, Firm 1’s profit is higher (4,825.50 > 4,381.19) in the case of Stackelberg model because firm 1 is the leader. Firm 2’s profit is lower in the case of Stackelberg model.
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