A monopolist faces an inverse demand function of both quantity, Q, and advertisi

Question

A monopolist faces an inverse demand function of both quantity, Q, and advertising effort, A, P(Q, A) = 70 - Q + A1/2. The marginal cost of production is constant and 10, and the marginal cost of advertising is constant and 1.

(a) What is the monopolist's optimal choice of advertising and quantity?

(b) What is the monopolist's profit given this optimal choice?

(c) Find the optimal quantity given that advertising is equal to zero, A = 0.

(d) Find profit.

(e) Compare the profits from not advertising with the optimal advertising strategy.

Explanation / Answer

P = 70 - Q + A1/2

MC of production, MCP = 10

MC of advertising, MCA = 1

Total revenue, TR = P x Q = 70Q - Q2 + QA1/2

Total cost, TC = 10Q + A

Profit = TR - TC

Z = 70Q - Q2 + QA1/2 - 10Q - A

First-order conditions for profit maximizations:

dZ / dQ = 0, or

70 - 2Q + A1/2 - 10 = 0

2Q - A1/2 = 60   (1)

Or, A1/2 = 2Q - 60   (1 - A)

And

dZ / dA = 0

Q x 0.5 A- 1/2 = 1

Substituting from (1 - A),

Q x 0.5 / (2Q - 60) = 1

0.5Q = 2Q - 60

1.5Q = 60

Q = 40

And we have following relationship:

Q x 0.5 A- 1/2 = 1

Or, 20 x A- 1/2 = 1

A1/2 = 20

A = 400

(b) profit = 70Q - Q2 + QA1/2 - 10Q - A

= (70 x 40) - 1600 + 40 x 20 - (10 x 40) - 400

= 1200

(c) If A = 0:

TR = 70Q - Q2

MR = dTR / dQ = 70 - 2Q

MC (MCP) = 10

So, monopolist will equate MR with MC:

70 - 2Q = 10

2Q = 60

Q = 30

Profit = TR - TC

= (70Q - Q2) - 10Q

= 900

(e) Profit with advertising = 1200

Profit without advertising = 900

So, with advertising, profit is higher by 300.

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