Question

1. In the market of cars, there are two firms operating. The Industry Demand Curve is a function of the outputs being produced by both firms, and is given as: P = 240−(X1+X2), where X1 and X2 are the outputs of Firm 1 and Firm 2 respectively. The Total Cost faced by Firm 1 is TC1 = 20X1 and by Firm 2 is TC2 = 20X2. Each firm maximizes its own profit by choosing its own output, while taking the output of the other firm as given. Write down the profit function of each firm, and find the level of output for each firm that will maximize its profit. What are the corresponding levels of prices and profits for each firm?

2. Suppose a personhas a utility functionU(x1,x2)=xa1+xa2 ,which she maximizes subject to her budget constraint, px1 + qx2 = m, where p, q, m are all positive. Use the Lagrangian method to solve the maximization problem, and find the demand functions for the consumer. Show that the demand functions are homogeneous of degree zero in prices (p, q) and income (m).

3. A consumer has a Cobb-Douglas Utility Function: U = x y . The price of good x is $2, the price of good y is $4, while the person has an income of $40. Find the solution to the consumer’s maximization problem, and verify that you have found the maximum point.

4. A firm produces a single output, Q, using the production function: Q = K 3 L 3 , where K is capital and L is labour. The cost of K per unit is $1 and L per unit is $2. Find the optimal choices of K and L that will minimize the cost of production for the firm when it intends to produce Q units of output.

Answer 1

As per HOMEWORKLIB RULES in case of multiple questions only the first question is to be answered

Kindly ask rest of the questions in a separate post

1.

Demand function: P = 240 - (X1+X2)

TC1 = 20X1

TC2 = 20X2

Firm 1:

Profit = PX1 - TC1

Profit = 240X1 - X1​​​​​2 - X1X2 - 20X1

Similarly for Firm 2:

Profit = 240X2 - X1X2 - X22 - 20X2

Profits are maximized at the point dProfit/dX = 0

Firm 1:

240 - 2X1 - X2 - 20 = 0

X1* = 110 - 0.5X2

Similarly for Firm 2:

X2* = 110 - 0.5X1

Substitute the two equations or best response functions of the two firms in each other to find values of X1 and X2:

This gives:

X1* = 73.3 units

X2* = 73.3 units

Corresponding P = 240-(73.3+73.3) = $93.4

Profit of each firm = PX - TC

Profit of each firm = (93.4*73.3) - (20*73.3) = $5380