Macro assignment
Macroeconomic Theory
Week 6
1 Chapter 6: Firms
Households represented the demand side of the goods market and the supply side labor and �-
nancial markets. This behavior is represented by the consumption demand function, the labor
supply function and the private savings function respectively. Firms represent the opposite side
of the market than households: the supply side of the goods market, and the demand side of the
labor and �nancial market. Here we look at the �rms' decision-making process that determines
they optimal behavior.
1.1 Some Preliminaries on Firm Technology
⇒ Firms utilize labor hired from households, nt, along with their current stock of capital goods, kt, to produce output, qt, via a production function, f(kt,nt).
→ Unless otherwise speci�ed, f(kt,nt) will have the following properties:
1. f(·) is twice continuously di�erentiable in each argument
2. ∂f
∂kt , ∂f
∂nt > 0→ more inputs generate more output ⇒ positive marginal product (MP)
3. ∂2f
∂k2t , ∂2f
∂n2t < 0 → MP increases a decreasing rate ⇒ diminishing MP
Graphically, a production function that satis�es the properties of continuous, positive, and
diminishing marginal production looks like the following in two dimensions, in (q,{k,n}) space:
⇒ Capital is accumulated across time through a �ow of real net investment1:
invnett = kt+1 − (1−δ)kt (1)
where δ is the rate of economic depreciation on physical capital.
→ Capital has a `time-to-build' property, where resources spend on investment today do not become productive until the future period.
1The textbook incorrectly calls net investment in equation (1) 'gross investment'.
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1.2 Two-Period Firm Model
Figure 1: Timeline of Events for Model
© Sanjay K. Chugh 77 Spring 2008
Period 2Period 1
Begins with initial capital k1
Chooses quantity of labor to hire n1 and level of capital for
next period k2, and then produces output f(n1, k1)
Chooses quantity of labor to hire n2 and level of capital for
next period k3, and then produces output f(n2, k2)
End of the economy
Figure 25. Timing of events for the firm in the two-period model.
Before we study the firm’s investment decision, let’s briefly consider its decision about how many workers to hire. In fact, this is something with which you are already familiar from basic microeconomics. The firm’s demand for labor is a derived demand. To briefly review: in period 1, the capital 1k is fixed, so that hiring additional workers increases total output at an ever-decreasing rate according to the law of diminishing marginal product. The price 1P is determined by the market (think perfect competition here), so that marginal revenue product (defined as price of the output good times marginal product) declines the more labor the firm hires. The derived demand curve for labor is simply the marginal revenue product curve, as shown in Figure 26.
N om
in al
af te
r- ta
x w
ag e
hours of labor
Labor demand
1.2.1 Objective and Constraints
⇒ The Firm's objective is to choose inputs, kt and nt, to product output, f(kt,nt), at pro�t maximizing levels.
→ They take as given the nominal wage rate, Wt, the nominal interest rate, it, and the goods price Pt.
• Lifetime (or Intertemporal) Pro�t Function: expresses the present discounted value of pro�ts. In nominal terms:
Profit = P1f(k1,n1)−P1invnet1 −W1n1 + P2f(k2,n2)
1 + i − P2inv
net 2
1 + i − W2n2 1 + i
(2)
→ Unlike the household's optimization problem, we are setting up the �rm's problem such that the objective function and budget constraint are in one equation. One way to think about this is that
the �rm wants to produce the pro�t maximizing amount of output over their two-period planning
horizon subject to a constraint that relates present discounted value of output (available resources)
to present discounted value of expenses on investment and labor (possible expenditures), where
investment is de�ned in Equation 1.
→ The pro�t function can be expressed in real terms by application of the Fisher Equation.
1.2.2 Optimal Choice
The �rm chooses the optimal (k2∗,n1∗,n2∗) bundle that maximizes their lifetime pro�t function. → k2,n1, and n2 are the endogenous variables → P1, P2, W1, W2, and i are the exogenous variables
→ Initial and Terminal conditions: In order to get the full solution to the model, we have to make assumptions about k1 and k3 since Profit
∗ and inv∗t will depend on their values. However,
we can still gain economic intuition by characterizing the solution with optimality conditions and
assuming k1 and k3 are exogenous.
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→ For the �rm's problem, we will not be using the method of Lagrange multiplier (although one could). Instead we will be using the substitution method whereby we substitute the investment
�ow constraint from Equation (1) into the pro�t function in Equation (2).
max n1,n2,k2
P1f(k1,n1)−P1invnet1 −W1n1 + P2f(k2,n2)
1 + i − P2inv
net 2
1 + i − W2n2 1 + i
subject to: invnett = kt+1 − (1−δ)kt for t = 1,2
or alternatively:
max n1,n2,k2
P1f(k1,n1)−P1 (k2 − (1−δ)k1)−W1n1+ P2f(k2,n2)
1 + i − P2 (k3 − (1−δ)k2)
1 + i − W2n2 1 + i
(3)
Taking FOCs for n1 and n2:
n1 : P1 ∂f
∂n1 −W1 = 0 ⇒
∂f
∂n1 = W1 P1
(4)
n2 : P2 ∂f
∂n2 −W2 = 0 ⇒
∂f
∂n2 = W2 P2
(5)
⇒ Equations (4) and (5) are the Firm's Labor Optimality Condition. → This optimality condition has an important economic interpretation: The �rm hires labor
up until the point where the marginal product of labor is equal to the real wage (i.e. the marginal
cost of the last unit of labor hired). Notice that there is no intertemporal aspect of this condition,
and that it holds for each period t.
Taking the FOC for k2:
k2 : −P1 + P2 1 + i
∂f
∂k2 + P2(1−δ) 1 + i
= 0
∂f
∂k2 + (1−δ) =
P1 P2
(1 + i)
∂f
∂k2 + (1−δ) =
1 + i
1 + π2
∂f
∂k2 + (1−δ) = 1 + r2
⇒ ∂f
∂k2 = r2 + δ (6)
⇒ Equations (7) is the Firm's Capital Optimality Condition. → This optimality condition has an important economic interpretation: The �rm invests into
future capital up until the point where the marginal product of capital is equal to the real interest
rate plus the depreciation rate (i.e. the opportunity cost of the last unit of capital purchased).
Notice that since the �rm is choosing future capital, it is dated t + 1.
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1.2.3 Labor and Capital Demand
⇒ The Labor Demand Function gives the optimal quantity of labor, nt∗ chosen by the �rm at every possible real wage rate, wt.
→ It is characterized by the �rm's labor optimality condition for any period t:
wt = ∂f
∂nt
→ Since we're looking for the qualitative relationship between wt and nt at the �rm's optimal choice, we could di�erentiate the optimality condition:
∂wt ∂nt
= ∂2f
∂n2t < 0
which says that the �rm's demand curve is downward sloping in (nt,wt) space:
⇒ The Capital Demand Function gives the optimal quantity of future capital, kt+1∗ chosen by the �rm at every possible real interest rate, rt+1
→ It is characterized by the �rm's capital optimality condition for any period t + 1:
rt+1 = ∂f
∂kt+1 + δ
→ Since we're looking for the qualitative relationship between rt+1 and kt+1 at the �rm's optimal choice, we could di�erentiate the optimality condition:
∂rt+1 ∂kt+1
= ∂2f
∂k2t+1 < 0
which says that the �rm's demand curve is downward sloping in (kt+1,rt+1) space:
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⇒ The �rm's demand functions for capital and labor are considered derived demand func- tions, since the �rm's demand for these inputs arises from the demand for their output.
→ Given a functional form for the production function q = f(kt,nt), one could derive the derived demand functions for n∗t and k
∗ t in terms of q and other exogenous parameters. This is
done by substituting the speci�c functional form for the production function into each respective
optimality condition, and re-arranging for the relevant endogenous variable.
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- Chapter 6: Firms
- Some Preliminaries on Firm Technology
- Two-Period Firm Model