production, investment, and the current account roberto chang rutgers university april 2013
TRANSCRIPT
Motivation
• Recall that the current account is equal to savings minus investment.
• Empirically, investment is much more volatile than savings.
• Reference: chapter 6, section 3 of FT
Recall: The Savings Function
• Recall that we had derived a national savings function from a basic model of consumer choice
The Setup
• Again, we assume two dates t = 1,2
• Small open economy populated by households and firms.
• One final good in each period.
• The final good can be consumed or used to increase the stock of capital.
• Households own all capital.
Firms and Production
• Firms produce output with capital that they borrow from households.
• The amount of output produced at t is given by a production function:
Q(t) = F(K(t))
Production Function
• The production function Q(t) = F(K(t)) is increasing and strictly concave, with F(0) = 0. We also assume that F is differentiable.
• Key example: F(K) = A Kα, with 0 < α < 1.
• The marginal product of capital (MPK) is given by the derivative of the production function F.
• Since F is strictly concave, the MPK is a decreasing function of K (i.e. F’(K) falls with K)
• In our example, if F(K) = A Kα, the MPK is
MPK = F’(K) = αA Kα-1
Profit Maximization
• In each period t = 1, 2, the firm must rent (borrow) capital from households to produce.
• Let r(t) denote the rental cost in period t.
• In addition, we assume a fraction δ of capital is lost in the production process.
• Hence the total cost of capital (per unit) is r(t) + δ.
• In period t, a firm that operates with capital K(t) makes profits equal to:
Π(t) = F(K(t)) – [r(t)+ δ] K(t)
Profit maximization requires:
F’(K(t)) = r(t) + δ
F’(K(t)) = r(t) + δ
• This says that the firm will employ more capital until the marginal product of capital equals the marginal cost.
• Note that, because marginal cost is decreasing in capital, K(t) will fall with the rental cost r(t).
Investment
• The amount of capital in the economy at the beginning of period 2 is given by:
K(2) = (1-δ)K(1) + I(1)
• Hence investment in period one is
I(1) = K(2) - (1-δ)K(1)
Now recall
•K(1) is given as an initial condition
•K(2) is a decreasing function of r(2)
•Hence the equation
I(1) = K(2) - (1-δ)K(1)
implies that I(1) is a decreasing function of r(2)
The Investment Function
• But in an open economy, r(t) must be equal to the world interest rate r*
Investment in period 1 is a decreasing function of the world interest rate r*