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.
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?
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