[3] The market for lemonade in a town consists of two lemonade stands (i.e., fir

Question

[3]       The market for lemonade in a town consists of two lemonade stands (i.e., firms), 1 and 2. An agricultural economist estimates the following market demand for lemonade in this town:

Q = 250 - P,    where Q is the market quantity and P is the market price.

Total costs of the two firms are indicated below:    

TC1 = 50Q1     (such that marginal cost is constant at 50)

TC2 = 50Q2     (such that marginal cost is constant at 50)

Acting as Cournot competitors, find and graph the reaction functions of each firm (indicating appropriate horizontal- and vertical-axis intercepts). Determine the Cournot equilibrium output of each producer, as well as each firm’s profit.

Explanation / Answer

Q=q1+q2

P=250-Q

P = 250-q1-q2

For firm 1

R1 = 250q1-q1^2-q1q2

MR1 = 250-2q1-q2

MC=50

MR1=MC

2q1+q2=200 so q1 = 100-0.5q2 reaction function for firm 1

For firm 2

R2 = 250q2-q2^2-q1q2

MR2 = 250-q1-2q2

MC=50

MR2=MC

q1+2q2=200 so q2 = 100-0.5q1 reaction function for firm 2

Solving these functions we get q1=q2

so q1=q2=200/3

P=250-200/3-200/3 = 350/3

Profit for each firm = 350/3*200/3 - 50*200/3 = 40000/9 = 4444.44

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