5) Suppose you are a monopolist operating two plants at different locations. Bot

Question

5) Suppose you are a monopolist operating two plants at different locations. Both plants produce the same product; Q is the quantity produced at plant 1, and Q2 is the quantity produced at plant 2. You face the following inverse demand function: P 500-20, where Q- 22. The cost functions for the two plants are f. What are your marginal revenue and marginal cost functions? g. To maximize profits, how much should you produce at plant 1? At plant 2? h. What is the price that maximizes profits? i. What are the maximum profits?

Explanation / Answer

(f)

Inverse demand function is as follows -

P = 500 - 2Q

Calculate Total Revenue -

TR = P * Q = (500-2Q)*Q = 500Q - 2Q2

Calculate marginal revenue -

MR = dTR/dQ = d(500Q - 2Q2)/dQ = 500 - 4Q

The marginal revenue function is 500 - 4Q.

Total cost fucntion of Plant 1 -

C1 = 25 + 2Q12

Calculate MC of Plant 1 -

MC1 = dC1/dQ1 = d(25 + 2Q12)/dQ1 = 4Q1

The marginal cost fucntion of Plant 1 is 4Q1.

Total cost fucntion of Plant 2 -

C2 = 20 + Q22

Calculate MC of Plant 2 -

MC2 = dC2/dQ2 = d(20 + Q22)/dQ2 = 2Q2

The marginal cost fucntion of Plant 2 is 2Q2.

(g)

In order to maximize profit, monopolist operating multi plants will produce that level of output at which marginal cost at both plants will become equal to each other.

Firstly, we have to ascertain the multi-plant marginal cost curve. This curve is equal to the horizontal summation of marginal cost curve of the individual plants such that,

MCp = MC1 = MC2

As we know that,

Q = Q1 + Q2

MC1 = 4Q1

Q1 = MC1/4

Q1 = MCp/4       [Since, MC1 = MCp]

MC2 = 2Q2

Q2 = MC2/2

Q2 = MCp/2       [Since, MC2 = MCp]

Q = Q1 + Q2 = MCp/4 + MCp/2

Q = 3MCp/4

MCp = 4Q/3

Equating MCp and MR to ascertain profit-maximizing quantity -

MCp = MR

4Q/3 = 500 - 4Q

Q = 93.75

MCp = 4Q/3 = (4*93.75)/3 = 125

Q1 = MCp/4 = 125/4 = 31.25

Q2 = MCp/2 = 125/2 = 62.50

To maximize profit, monopolist should produce 31.25 units at plant 1 and 62.50 units at plant 2.

(h)

Inverse demand function is as follows -

P = 500 - 2Q

Profit maximizing output, Q = 93.75

P = 500 - 2(93.75)

P = $312.5

The price that maximize profit is $312.5 per unit.

(i)

Calculate maximum profit -

Profit = TR - C1 - C2 = (P*Q) - (25+ 2Q12) - (20+Q22)

Profit = ($312.5*93.75) - (25 + 2*(31.25)2) - (20 + (62.50)2)

Profit = $23,392.5

The maximum profit are $23,392.5

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