Question

A risk-neutral, price-taking firm must set output before it knows the market price. There is a 50 percent chance the market demand curve will be Qd = 10 – 2P and a 50 percent chance it will be Qd = 20 – 2P. The market supply curve is estimated to be QS = 2 + 2P.

a. Calculate the expected (mean) market price.
b. Calculate the variance of the market price.
c. If the firm's marginal cost is given by MC = 0.01 + 5Q, what level of output maximizes expected profits?

Answer 1

Given

Qd1=10-2P with 50% probability

Qd2=20-2P with 50% probability

Qs=2+2P

So for demand curve 1

Qd1=Qs

10-2P=2+2P

P1=$2

So for demand curve 2

Qd2=Qs

20-2P=2+2P

P2=$4.5

A)

Expected market price E(P)=50%*P1+50%*P2=50%*2+50%*4.5=$3.25

B)

Variance = 50%*(P1-E(P))^2+ 50%*(P2-E(P))^2=50%*(2-3.25)^2+50%*(4.5-3.25)^2=1.5625

C)

Given MC=0.01+5Q

For demand curve 1

Qd1=10-2P

P=5-0.5Qd1=5-0.5Q

TR1=P*Q=5Q-0.5Q^2

For demand curve 1

Qd1=20-2P

P=10-0.5Qd1=10-0.5Q

TR1=P*Q=10Q-0.5Q^2

E(TR)=50%*TR1+50%*TR2=50%*(5Q-0.5Q^2)+50%*(10Q-0.5Q^2)=7.5Q-0.5Q^2

for expected profit maximization

E(MR)=MC

E(MR)=dE(TR)/dQ=7.5-Q

E(MR)=MC

7.5-Q=0.01+5Q

Q=1.248 units