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325_PracticePS6_Soln - Department of Economics University...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Practice Problem Set 6 Suggested Solutions Professor Sanjay Chugh Spring 2011 1. Lags in Labor Hiring. Rather than supposing that the representative firm at the beginning of period t can decide how much labor it would like to hire for use in period t, suppose that labor used in period t must be chosen in period t-1. (That is, suppose n is a stock (aka state) variable.) As usual, capital for use in production in period t must be purchased in period t-1 because of the “time to build” surrounding capital goods. With this lag in labor hiring, construct the lifetime (in the two-period model) profit function of the firm, and show that the real interest rate now is a relevant price for labor as well as capital goods. Provide brief economic intuition. ( Hint: Make as close an analogy with our model of firm ownership of capital as you can – in particular, think of workers in this model as being “owned” (contractually obligated to) firms.) Solution: With employees being contractually bound to (“owned by”) firms, the period-t nominal profits of a firm are given by 1 1 ( , ) t t t t t t t t t t t t t t PR P f k n Pk Pw n Pk Pw n ± ± ± ± ² ² , in which labor used in production in period t, t n , is chosen in period t-1 (and thus labor used in production in period t+1, 1 t n ± , is chosen in period t. In analogy with our model with only capital pre-determined, the employees of a firm are a valuable “asset,” with total market value t t t Pw n -- notice that this term enters positively in period t profits, rather than negatively with non-pre-determined labor. What enters negatively in period t profits here is the “purchase” of period t+1 labor, namely the term 1 t t t Pw n ± ² . In the two period model, discounted nominal profits of the firm are therefore 2 2 2 2 2 2 2 2 2 3 2 2 3 1 1 1 1 1 1 1 1 1 2 1 1 2 ( , ) ( , ) 1 1 1 1 1 i i i i i P f k n P k P w n P k P w n PR P f k n Pk Pw n Pk Pw n i i i i i ± ± ² ² ± ± ± ² ² ± ± ± ± ± The usual zero-terminal-assets condition in this case means that 3 0 k and 3 0 n (the latter, again, because labor should be thought of as an “asset” here). Focusing attention on the choice of 2 n (since 1 n was chosen in period t-1), the first-order condition of the lifetime profit function with respect to 2 n is
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2 2 2 2 2 2 1 1 1 1 ( , ) 0 1 1 n P f k n P w Pw i i ² ± ± ± ± . This expression can be rearranged to yield (using the exact Fisher equation) 1 1 2 2 2 (1 ) ( , ) n r w f k n w ± ± . If the real wage were equal to one in each period, this condition would reduce to 1 2 2 ( , ) n r f k n , which would be almost identical to the condition we derived in class regarding capital demand (except of course in that case k f is the relevant marginal product rather than n f ). The expression 1 2 2 ( , ) n r f k n shows that if firms must choose labor for period 2 in period 1, the real interest rate between period 1 and period 2 is a relevant price to consider – which makes sense because there is now an interest opportunity cost associated with hiring labor (ie, “investment” in hiring).
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