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Chapter 10: Problem 8

1: C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\\ \text { Factory } \\# 2: C_{2}… # A firm has two factories, for which costs are given by: \\[ \begin{array}{l} \text { Factory } \\# 1: C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\ \text { Factory } \\# 2: C_{2}\left(Q_{2}\right)=20 Q_{2}^{2} \end{array} \\] The firm faces the following demand curve: \\[ P=700-5 Q \\] where \(Q\) is total output- \(i . e ., Q=Q_{1}+Q_{2}\) a. \(\mathrm{On}\) a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal cost of producing \(Q=Q_{1}+Q_{2}\) ). Indicate the profit-maximizing output for each factory, total output, and price. b. Calculate the values of \(Q_{1}, Q_{2}, Q,\) and \(P\) that maximize profit. c. Suppose that labor costs increase in Factory 1 but not in Factory \(2 .\) How should the firm adjust (i.e. raise, lower, or leave unchanged) the following: Output in Factory \(1 ?\) Output in Factory \(2 ?\) Total output? Price?

Short Answer

Expert verified
The profit-maximizing output for each factory are: Q1=0.25, Q2=0.125, total output Q=0.375, and the price P=681.25. With increased labor costs in Factory 1, output should decrease in Factory 1, stay unchanged in Factory 2, total output will decrease, and price will increase.

Step by step solution

01

Determine the marginal cost (MC)

The marginal cost function for the two factories can be derived from the cost functions using calculus. For Factory 1: MC1 = dC1/dQ1 = 20Q1. For Factory 2: MC2 = dC2/dQ2 = 40Q2.
02

Determine the average (AR) and marginal revenue (MR)

Since total output Q is equal to Q1 + Q2, we can substitute Q into the price function to get AR = P = 700 - 5Q. By taking the derivative of AR with respect to Q, the MR is obtained as MR = d(AR)/dQ = -5.
03

Plot the curves and determine the profit-maximizing output

On a graph plot the MC, AR and MR curves. Also plot the total MC which equals to MC1 + MC2. The point where MR cuts MC is the profit-maximizing output for each factory, total output and price.
04

Calculation of Q1, Q2, Q, and P

To maximize profit, set MR = MC. When MR = MC1, we get Q1 = (MR/20) = 5/20 = 0.25. When MR = MC2, we get Q2 = (MR/40) = 5/40 = 0.125. So, total output Q = Q1 + Q2 = 0.375. Substituting Q in the price function, P = 700 - 5Q = 681.25.
05

Understand the adjustments when the labor cost increases

If the labor cost increases in Factory 1, MC1 will increase. To maintain profit-maximizing conditions, the firm should reduce output in Factory 1. There is no change in Factory 2, so it should continue producing at the same level. The total output will decrease and due to the law of supply, the price will increase.

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Most popular questions from this chapter

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