Question

ASAP

1. Suppose the demand function facing a monopolist is given as p=50-q and the cost function is given as C=10q. Determine the total revenue function for the monopolist, and graph the relationship between total revenue, total cost and output. In your answer determine the profit maximizing level of output that would be chosen by the monopolist, and discuss mathematically how you know it is a profit maximum.

Answer 1

Given the demand curve of the monopolist p=50-q

Given the cost function of the monopolist C=10q

Total revenue of the monopolist= p x q= (50-q)q= $50q-q^{2}$

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C TR - 7abond /21/21 TE q=20 Output TR

TR is the total revenue curve, C is the total cost curve and TP is the total profit curve. The profit is maximum when the difference between TR and C is maximum.

The goal of the monopolist is to maximize profits. Profit is the difference between the total revenue and total cost.

$\pi = TR- C$

The first condition for maximum profit is that Marginal revenue should equal Marginal cost ( MR=MC).

Marginal revenue= d(TR)/dq= $\frac{d(50q-q^{2})}{dq}$ = 50-2q

Marginal cost= dC/dq= $\frac{d(10q)}{dq}$ = 10

We equate MR and MC

MR=MC

50-2q=10

40=2q

q= 20

The price charged by the monopolist is found by substituting q=20 in the demand function.

p=50-20

p=30

Total revenue= = 600

Cost= 10(20)= 200

The profit of the monopolist is $\pi = 600-200$ = 400

The second condition for maximum profit is slope of MC should be greater than slope of MR at the point of intersection.(MC should cut MR from below)

The second order conditions are:

-2<0

The second order condition is satisfied.

The maximum possible profit is 400 with profit maximising output and price equal to 20 and 30 respectively.