Problem 1: Short-Run Profit Maximization [10 points]
Consider a firm that uses both capital (K) and labour (L) to produce a final product (Q) that it sells at the market price $5. The firm buys Labour at a cost of $4 per unit and capital at a cost of $10 per unit.
The firm is a price-taker for all prices with the following production function:
𝑄 = 4𝐾 ! “# 𝐿
! “#
This production function implies the following:
𝑀𝑃𝐾 = 2 𝐿
“#
𝐾 ! “#
𝑀𝑃𝐿 = 2 𝐾 ! “#
𝐿
! “#
Suppose also that the firm currently has 16 units of capital (K = 16) in the Short-Run.
For Q1 – Q6, assume that we are operating in the Short-Run:
Given the above production function, write down an expression for Labour employed as
a function of the quantity produced (i.e. L = f(Q)). [1 point]
Use the MPL and the wage rate to write down an expression for the Firm’s Marginal
Cost per unit of output (Q) produced as a function of Q. [3 points]
Show that the firm’s profit-maximizing quantity (Q*) is equal to 160 in the short-run.
[1 point]
What are the firm’s variable costs (VC) and fixed costs (FC) at Q*? [2 points]
What are this firm’s short-run profits in this example? [1 point]
What would be the firm’s Short-Run profits if it chooses to shut down? Should the firm keep producing in the short-run? How do you know? [2 points]
Problem 2: Long-Run Cost Minimization [8 points]
Use the same production function, MPK, MPL, and prices from Problem 1 (Q 1 – 6) for the
following Q7 – Q9.
Suppose also that we are now in the Long-Run, and that the firm has decided to still produce the same Q as in Problem 1. That is, the firm decides to set Q = 160.
If the firm wants to cost minimize, what must be the ratio of K to L that they employ given the prices in this market? [3 points]
Suppose that the firm wants to produce Q = 160. What is its cost-minimizing choice of K & L? [3 points]
What are the firm’s profits with this choice of K & L? Is this greater than or less than your answer to Q5? Why is this the case? [2 points]
Problem 3: Long-Run Cost Minimization [12 points]
Suppose that there is a firm that produces chairs and the firm receives an order for 80 chairs. The firm has two resources available to it. The first is a (human) worker, who must be paid $18 for each hour they spend producing chairs.
The second is a robot, that costs $15 of inputs (including electricity and maintenance) for each hour it works. Chairs produced by either method are identical and of equivalent quality.
Assume that the use of these two inputs is completely independent. This means that the number of hours of robot-work does not affect the productivity of the worker, and vice versa.
The production of chairs based upon the numbers of hours of each of the inputs used is given below. For example, 2 hours of robot time will produce 10 chairs. 7 hours of worker time will produce 54 chairs.
Robot
Hours
Robot
Production
Robot
Hours
Worker
Production
0 0 0 0
1 5 1 15
2 10 2 25
3 15 3 33
4 20 4 40
5 25 5 45
6 30 6 50
7 35 7 54
8 40 8 57
9 45 9 58
10 50 10 59
11 55 11 60
12 60 12 61
13 65 13 62
14 70 14 63
Do any of the inputs in this example exhibit diminishing returns to scale? If so, which and how do you know? If not, how do you know? [2 points]
Assume that the sale price of chairs is always sufficiently high that it is profitable to fulfill this 80-chair order. The firm needs to make 80 chairs to fulfill its order. Assume also that the firm is profit maximizing (& therefore cost minimizing).
What combinations of robot and worker hours must they use to minimize costs? Show your work. [Use the equation that must be true for cost minimization for full credit] [4 points]
Now suppose that the local economy increases the minimum wage, and the price of an hour of a worker’s time increases from $18 to $27.
What does the principle of substitution say should happen to the firm’s use of (i) worker hours and (ii) robot hours? Explain your answer. [2 points]
Continue to assume that it will be profitable to produce the 80 chairs and that the firm is profit maximizing.
With this new price for worker hours, what is the new combination of robot and worker hours that will minimize the cost of producing (at least) 80 chairs? Show your work. [Use the equation that must be true for cost minimization for full credit]