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Chapter 10: Problem 8

1: } C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\\ \text { Factory #2: } C_{2}\lef… # 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 \(Q_{1^{\prime}} Q_{2^{\prime}} Q,\) 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 marginal cost curves for Factories 1 and 2 are \( MC_{1} = 20Q_{1} \) and \( MC_{2} = 40Q_{2} \) respectively. The profit-maximizing outputs corresponding to the quantities that make MC = MR are \( Q_{1^{\prime}} = 35 \) and \( Q_{2^{\prime}} = 17.5 \). The total output is \( Q = 52.5 \) and the corresponding price is \( P = 437.5 \). In case of an increase in labor costs in Factory 1, the output in Factory 1 should be reduced while that in Factory 2 should be increased. Total output should be reduced and the price will therefore go up.

Step by step solution

01

Understanding Factory Cost Equations

The cost functions for Factory 1 and Factory 2 are given by \( C_{1}(Q_{1}) = 10Q_{1}^{2} \) and \( C_{2}(Q_{2}) = 20Q_{2}^{2} \). The marginal cost for each factory is found by taking the derivative with respect to quantity. For Factory 1, the marginal cost is \( MC_{1} = 20Q_{1} \) and for Factory 2, it's \( MC_{2} = 40Q_{2} \).
02

Drawing Marginal Cost Curves

On graph paper, draw each of the marginal cost, average and marginal revenue, and the total marginal cost curve. Their axes are price on the y-axis and quantity on the x-axis. The curves are \( MC_{1} = 20Q_{1} \), \( MC_{2} = 40Q_{2} \), and \( MC = 20Q \). The marginal revenue curve is given by \( MR = 700 - 10Q \). Determine the price and quantities for Q1 and Q2 that makes MC = MR. Those points are the profit-maximizing outputs.
03

Calculating Values for Maximum Profit

By setting the marginal cost equal to the marginal revenue, we can solve for the quantities Q1 and Q2. For factory 1, \(20Q_{1} = 700 - 10Q \), which gives us \( Q_{1^{\prime}} = 35 \). Similarly for factory 2, \(40Q_{2} = 700 - 10Q \), which gives us \( Q_{2^{\prime}} = 17.5 \). The total output, \( Q \), is going to be the sum \( Q = Q_{1^{\prime}} + Q_{2^{\prime}} = 52.5 \). The price, \( P \), is given by substituting \( Q \) into the demand equation, \( P = 700 - 5*52.5 = 437.5 \). After those computations, substitute the quantity values into the cost functions to verify that Factory 1 and Factory 2 earn the maximum profit.
04

Dealing with Increased Labor Costs

Subsequently, when labor costs increase, the marginal cost of Factory 1 increases, but that of Factory 2 remains unchanged. Therefore, to reduce overall costs, the company should reduce the output of Factory 1 and increase the output of Factory 2, while the total output should be reduced to keep costs under control. As a result, with lower overall supply, the price increases according to the demand curve.

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Most popular questions from this chapter

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