. Consider the one-period labor demand model we studied in class. The XYZ Corporation pro- duces semiconductor equipment with a production technology that is accurately
described by the production function
Y = 100K0.4 L0.6,
(a) What is the marginal product of labor (as a function of K and L)? Denote the price of
output Y by P and denote the wage rate by W, write down the profit maximization problem of the XYZ
Corporation. What is the first-order condition?
(b) We assume that the amount of capital stock is $100 million, (i.e., K = 100, 000, 000).
The price of the product is $11, and the wage for workers is $25,000 per year. What is the optimal
labor demand L? Round to the nearest integer.
(c) During the last year, South Korean manufactures have entered the U.S. semiconductor mar-
ket with extraordinary vigor. Because they are lower-cost producers, their entry has driven
the gross price of semiconductor equipment in the United States down from $11 to $9. What is
the optimal labor demand now? Round to the nearest integer.
(d) Continue from part (c). Suppose that the government wants to subsidize employment and
restore the labor demand to the previous schedule in (b). The subsidy takes the form that for every
dollar of wage bill, the government pays b dollars (b < 1). What should b be to restore the labor demand schedule to be the same as in part (b)? 2. Consider the simple model of unemployment dynamics that we considered in class. Specifically, let the job finding rate be denoted by f and let the job separation rate be denoted by s. If we interpret a period to be a month then reasonable values for these parameters based on US historical data are s = 0.02 and f = 0.33. (a) What are the average duration of unemployment and employment spells corresponding to these values of s and f. And solve for the steady state unemployment rate corresponding to these values of s and f. (b) Suppose that the economy were to start with an unemployment rate of 10%. How many months would it take for the unemployment rate to reach its steady state value, in the sense of being equal to the steady state value up to the first decimal point. (c) Consider a different economy that is less dynamic than the US economy, but which has the same steady state level of unemployment. In particular, assume that this economy has f = 0.05. Solve for the value of s in this economy that would give the same steady state unemployment rate as in the US. Now, repeat the exercise in part (b) for this economy, i.e., start the economy with an unemployment rate of 10% and solve for how long it takes to get to the steady state. Further more, how do your answers in part (b) and (c) compare in terms of time to reach steady state. Can you offer any intuition for the result? 3. Consider the same parameterization as in the previous question, i.e., s = .02 and f = .33. This question has you examine the responsiveness of the steady state unemployment rate to changes in the values of f and s. (a) In 2011 in the US economy, the value of f had decreased by more than one half, whereas the value of s had increased by about 5%. With this in mind, compute that implied steady state unemployment rate that results if f decreases to .15 and s increases to .021. (b) Now solve for the implied steady state unemployment rate that would result from only one of the two variables changing, i.e., first consider holding f fixed at .33 and increasing s to .021, and then hold s fixed at .02 and decrease f to .15. Compare the magnitude of the effects. What do you conclude about the relative importance of changes in f and s in accounting for cyclical changes in the US unemployment rate.