Answer to Question #180640 in Microeconomics for MURIITHI MUCHEMI

Question

 There are only two farmers in Machakos producing milk. The local demand for milk is given by an inverse demand curve Q=1000-0.5P (P denotes price, Q denotes the total quantity). Both farmers have the same cost function given by C = 560Q + 80,000. Calculate the Cournot-Nash equilibrium. That is

(a) Output for each firm

(b) Total output (Q)

(c) The price (P)

(d) The profit for each firm

Answer 1

Solution:

Derive Total Revenue (TR):

First solve for P:

Q = 1000 – 0.5P

0.5P = 1000 – Q

P = 2000 – 2Q

TR = P*Q

TR = (2000 – 2Q) Q

TR = 2000Q – 2Q²

Derive marginal revenue:

MR = derivative of TR with respect to Q

$\frac{\partial TR}{\partial Q }$ = 2000 – 4Q

MR = 2000 – 4Q

Compute the profit maximizing output by setting MR = MC:

MC = derivative of TC with respect to Q

TC = 560Q + 80000

MC =$\frac{\partial TC}{\partial Q }$ = 560

MR = MC

2000 – 4Q = 560

2000 – 560 = 4Q

1440 = 4Q

Q = $\frac{1440}{4}$ = 360

Profit maximizing output = 360

a). Output for each firm = $\frac{360}{2 }$ = 180 each

Output for firm 1 = 180

Output for firm 2 = 180

b). Total Output (Q) = 360

 

c). Profit maximizing price = Substituting Q in the demand function

P = 2000 – 2Q

P = 2000 – 2(360)

P = 2000 – 720

P = 1280

Profit maximizing price for the firms = 1,280

 

d). Profit for each firm:

Profit = TR – TC

= (2000(180) – 2(180²)) – (560(360) +80000)

= (360000 – 64800) – (100800+80000)

= 295200 – 180800

= 114,400

Each firm will make a profit of 114,400