7 the labor demand curve is now mpn s and the labor

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Unformatted text preview: labor where maximizes profits by choosing the quantity of labor where the slope of the revenue function MP w cost equals the slope Nof=the− s . function: MP = w. N In the bottom panel of Figure 4.7, the labor demand curve is now MPN + s , and the labor The firm’s demand for up. curve is the marginal product of labor schedule in the bottom demand curve has shifted laborThe subsidy acts to reduce the marginal cost of labor, and panel will hire 3. the firm of Figure more labor at any given real wage. Revenues, Costs wN d (w − s) N d zF ( K , N d ) Nd w MPN + s MPN Nd Figure 4.7 Figure 3: With an employment subsidy, the firm’s profits are given by: π = zF (K, N d ) − (w − s)N d where the term zF (K, N d ) is the unchanged revenue function, and (w − s)N d is the cost function. The subsidy acts to reduce the cost of each unit of labor by the amount of the subsidy, s. In the top panel of Figure 3, the subsidy acts to shift down the cost function for the firm by reducing its slope. As before, the firm will maximize profits by choosing the quantity of labor input where the slope of the revenue function is equal to the slope of the cost function, (w − s), so the firm chooses the quantity of labor where M PN = w − s. In the bottom panel of Figure 3, the labor demand curve is now M PN + s, and the labor demand curve has shifted up. The subsidy acts to reduce the marginal cost of labor, and the firm will hire more labor at any given real wage. 4 4. Consider a representative firm with the following technology: Y = zK α N 1−α , (2) where K = capital and N = labor. Assume the firm behaves competitively, that is it takes prices as given. (a) Show that the marginal product of labor is equal to M P N = (1 − α) Y . N (3) The marginal product of labor (MPN) is equal to: MPN = ∂Y N Y = (1 − α)zK α N −α = (1 − α)zK α N −α = (1 − α) . ∂N N N (b) Find the optimal demand for labor N if the stock of capital available to the firm is exogenously given and the firm takes as given the price of output p and the price of labor w. max{pzK α N 1−α − wN }. N The first order conditions for a maximum imply: (1 − α)pzK α N −α − w = 0. Solving for N we obtain: N= (1 − α)pzK α w 1/α = (1 − α)pz w 1/α K (c) What happens to N if z , K , or w increase? Given that α > 0, the above formula implies that N is an increasing function of z and a decreasing function of w. Moreover, N is an increasing function of K . Are these results intuitive? 5...
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