Question

In: Economics

Suppose a profit-maximizing monopoly firm has short-run total cost function TC(Q)=5Q+300. It faces market demand Q=400-4P....

Suppose a profit-maximizing monopoly firm has short-run total cost function TC(Q)=5Q+300. It faces market demand Q=400-4P.

What price will it charge and what quantity will it sell if it acts as a one-price monopolist?

If the firm wishes to sell output in two different blocks at different prices, what are the prices it will charge and how much will it sell in each block?

Expert Solution

if it acts as one-price monopolists then. it would charge the price and supply the output where MR=MC.

therefore, we have Q = 400 -4P or P = 100 - 0.25Q and TC = 5Q +300

MR = 100 - 0.5Q and MC = 5

100 -0.5Q = 5 or Q = 190

P = 100 -0.25*190 = 100 -47.5 = $52.5

so, monopolist will charge $52.5 and sell 190 quantity.

when the firm want to sell in two different block . .

P1 = 100 - 0.25Q1 ( the price charged in the first block.)

TR1 = 100 - 0.25Q1( Q1)

TR1 = 100Q1 - 0.25Q1²

P2 = 100 -0.25Q2 ( the price charged in the second block.)

TR2 = 100 -0.25Q2 (Q2-Q1)

TR2 = 100Q2 - 100Q1 - 0.25Q2² + 0.25Q1Q2

profit = TR1 + TR2 - TC

profit = 100Q1 -0.25Q1² + 100Q2 - 100Q1 - 0.25Q2² + 0.25Q1Q2 - 5Q2-300

partial derivative of profit with respect to Q1 and put it to zero

= 100 - 0.5Q1 -100 +0.25Q2 = 0

Q2 = 2Q1

partial derivative of profit with respect to Q2 and put it to zero

= 100 -0.5Q2 +0.25Q1-5 = 0

0.25Q1 = -95 +0.5Q2

Q1 = -380 + 2Q2

Q2 = 2 ( -380 + 2Q2 )

Q2 = -760 + 4Q2

Q2 = 253

Q1 = -380 + 506 = 126

P1 = 100 - 0.25*126

P1 = $68.5

P2 = 100 - 0.25* 253

P2 = 36.75

Rahul Sunny answered 2 years ago

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