Question

3. The East Company and the West Company are the only two firms that produce and sell a specialized type of machinery. The market demand curve for the machinery is as follows: P 25-0.0005 a In this equation, P is the price of the machinery in dollars and Q is the total amount demanded. The total cost function for the East Company is as follows: TCEast 4Q+0.001 Q?East In this equation, TCEast is total cost in dollars and QEast is the output. The total cost function for the West Company is as follows: TCWest 10 Q+0.001Q2West In this equation, TCWest is total cost in dollars and Qwest is the output. a. If these two firms are successful in forming a cartel, determine (i) the profit-maximizing outputs for the East and West Companies, (ii) the profit-maximizing prices for the East and West companies, and (ii) the total profits for the East and West Companies.

Answer 1

Answer:

Demand Function P=25-0.0005Q

Q=50000-2000P

Where Q=Qe+Qw

TC for east company

TCe= 4Q+0.001Qe^2

TC for west company

TCw= 10Q+0.001Qw^2

Part A)

For cartel Total profit

Profit B=TR-TCe-TCw=Q*P-(4Q+0.001Qe^2)-(10Q+0.001Qw^2)

B=Q*(25-0.0005Q)-(4Q+0.001Qe^2)-(10Q+0.001Qw^2)

B=(Qe+Qw)*(25-0.0005(Qe+Qw))-(4*(Qe+Qw)+0.001Qe^2)-(10*(Qe+Qw)+0.001Qw^2)

for profit maximization partial derivative of profit with respect to Qe should be zero.

dB/dQe=(25-0.0005(Qe+Qw))-0.0005*(Qe+Qw)-(4+0.002Qe)-(10)=0

0.003Qe+0.001Qw=11 Eq 1

for profit maximization partial derivative of profit with respect to Qw should be zero.

dB/dQw=(25-0.0005(Qe+Qw))-0.0005*(Qe+Qw)-(10+0.002Qw)-(4)=0

0.003Qw+0.001Qe=11 Eq 2

So from equation 1 and 2 we get

Qe=2750 units

Qw=2750 units

Q=Qe+Qw=2750+2750=5500 units

Part B)

Profit Maximizing price

P=25-0.0005Q=25-0.0005*5500=$22.25

Part C)

Profit for east Be=TRe-TCe=Qe*P-(4Q+0.001Qe^2)

Be=2750*22.25-(4*5500+0.001*2750^2)=$31,625

Profit for west Bw=TRw-TCw=Qw*P-(10Q+0.001Qw^2)

Be=2750*22.25-(10*5500+0.001*2750^2)=-$1,375

Total profit B=TR-TCe-TCw=Q*P-(4Q+0.001Qe^2)-(10Q+0.001Qw^2)

B=5500*22.25-(4*5500+0.001*2750^2)-(10*5500+0.001*2750^2)=$30,250