Minimize the average cost for total cost function

Question

Assignment:

1. [Revenue and Profit Maximization - I]

Maximize the following total revenue function (TR) and the profit π by finding out (a) the critical values (the output values); (b) checking the second order conditions and (c) the maximum values.

(a) Total Revenue

T R = 32Q-Q².

(b) Total Profit Function (π) π = -Q ² +11Q-24.

(c) Total Profit Function (π) π = - Q³ 3 -5Q ² +2000Q-326.

(d) Total Profit Function (π) π = -Q³ -6Q ² +1440Q-545.

2. [Minimum Average Cost]

Minimize the average cost for each of the following total cost function (TC) by finding out (a) average cost function, (b) the critical values (the output values) at which AC is minimized; and (c) the minimum average cost.

(a) Total Cost Function TC = Q³ -5Q² +60Q.

(b) Total Cost Function TC = Q³ -21Q² +500Q.

3. [Profit Maximization - II]

Given the following total revenue function (TR) and the total cost function (TC), maximize profit π by following steps (a) set up the profit function π = T R-TC,

(b) the critical values (the output values) where the profit is at a relative extremum; (c) checking the second order conditions and (d) the maximum profit value.

(a) T R = 1400Q-6Q² ; TC = 1500+80Q.
(b) T R = 4350Q-13Q² ; TC = Q³ -5.5Q² +150Q+675.
(c) T R = 5900Q-10Q² ; TC = 2Q³ -4Q² +140Q+845.

4. [Profit Maximization - III]

A seller has the choice of selling products at different prices in the domestic (market 1) and foreign markets (market 2).

Q₁ = 21-0.1P₁,

Q₂ = 50-0.4P₂.

The total cost for the firm is TC = 2000+10Q, where Q = Q₁ +Q₂.

What prices the seller should charge in order to maximize profit when it charges (a) the same prices in the two markets; (b) different prices in the two markets. Compare the total profit earned by the firm in the two cases above.

5. [Perfectly Competitive Firm]

A purely competitive firm has a single variable input L, with wage rate w0 per unit. Its fixed input costs the firm a total of $ F. The production function is given as Q = f(L) = L 0.5.

(a) Write the revenue function, cost function, and the profit function of the firm.

(b) What is the first order condition for profit maximization? Give an economic interpretation of this condition.