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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 Spring '14
 Economics

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