Reference no: EM13917168

1) Your job in this question is to explain a couple of puzzling facts about the US manufacturing sector over the last 10 years. Employment in US manufacturing is roughly 5.5 million jobs lower than it was in 1999. However, the real value of US manufacturing output rose 14% from 1999 to 2010. Naturally, making more output with fewer workers has shown up in the official statistics as a sharp increase in labor productivity.

Use the tools we've discussed so far in Module 3 to explain these trends in US manufacturing. Specifically, you need to address how it's possible to produce the same output with fewer workers - what's making US manufacturing workers more productive? A big hint: the rental price of capital follows long-term interest rates in the US pretty closely. If you argue that something causes firms to change their behavior, use a specific argument like "buck for the bang" to explain why. If you're not already familiar with what happened to long-term interest rates in the US over this period, google ^TNX and click the "max" link under the chart to see a graph of 10-Year Treasury Bond rates. Start with lower interest rates, think through what that means for input mix, and then connect it back to labor productivity.

2) You are the owner of a winery. Your production function is q^1/3 K^1/3 = 3√KL, where q measures the number of bottles of wine you can produce in a day. K measures how many "capital-hours" you rent today, and L is the number of hours of labor you hire today. It costs $10 to rent capital for an hour, and $5 to hire a worker for an hour. Each bottle of your wine sells for $45, regardless of how many you sell (i.e. you are a price taker in both your input and output markets).

a) First, let's look at the short run. Assume capital is fixed at K=27 units of capital, but you are free to adjust labor as necessary to maximize profits. Calculate the marginal revenue product of labor in this situation, and use it to determine the profit-maximizing amount of labor to hire. How many bottles of wine will you produce today? How much profit will you earn at this level of production? First, use the marginal revenue product rule to find L*. Then plug L* and K=27 back into the production function. Lastly, remember that C = wL +rK. The answers should all come out as integers.

b) Now it's the long run, so you can alter the amount of capital you are using. Assume you want to produce the same output as in part a. Calculate the cost minimizing level of capital and labor you should use.

Calculate the total cost of production using the cost minimizing combination of inputs, and the profit you would earn. Get a relationship between K* and L* from our cost minimizing rules, then substitute back into the production function with the q from 2a to get values for K* and L*. The answers here are not integers - sorry.

c) Draw an isoquant/isocost graph that illustrates the short-run input choice from part a, and the cost-minimizing long-run input choice in part b. You don't need to be too detailed in your intercepts, etc. Just draw a graph that shows the relative slopes of the isoquant and isocost line for 2a and 2b.

d) Consider the suboptimal input mix we used in the short run (2a). Use either the "buck for the bang" rule or the "equal slopes" rule to suggest a small shift in input mix that would allow the firm to produce the same amount at lower cost.

3) Suppose that your firm has a short run cost function of CSR = 250 + q³.

a) Calculate the marginal cost function (MC), the average variable cost function (AVC), and the average total cost function (AC).

b) Find the quantity at which MC=AC. Why is this an interesting point when thinking about the firm's profits?

c) Suppose that the price of this firm's output is P=$48. Calculate the profit maximizing level of production. Calculate the firm's profit at this level of production, first by calculating total revenue and total cost, then by using price and the average cost function.

d) Should this firm shut down in the short run? Answer this question in two ways: first, compare the profit of producing q = 0 to the profit from 3c. Next, use the AVC function. In both cases, use these results to explain why the firm should shut down or stay in business. If you decide that this firm should stay open in the short run, how low would the price have to drop for the firm to shut down immediately?

e) Lastly, draw the cost curve diagram for this problem: P, AC, MC, and AVC. Since these are polynomials I won't get picky - here's the level of detail I expect.

i) Draw your MC and AVC curves roughly to scale based on the coefficients.

ii) AC, MC, and AVC curves need to obey the rules we discussed in lecture.

iii) Label the following points: 1) q where MC=AC, and the associated MC.

2) Price of the good, and the profit maximizing quantity.

3) AC and AVC of the profit maximizing quantity.

4) The profit rectangle.