Production, Investment, and the Current Account

Roberto Chang

Rutgers University

April 2013

Announcements

• Problem Set 3 available now in my web page

• Due: Next week (April 11th)

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

Savings

r*

S

S

The Savings FunctionInterestRate

S*

Savings

Interest Rate

S

S

An increase in savings.This may be due tohigher Y(1).

S’

S’

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.

Capital K

Output F(K)

F(K)

• 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

Capital K

MPK = F’(K)

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).

Capital K

MPK = F’(K)

Capital K(t)

MPK = F’(K)

r(t) + δ

Capital

MPK = F’(K)

r(t) + δ

K(t)

• Note that K(t) will fall if r(t) increases.

Capital

MPK = F’(K)

r(t) + δ

K(t)

Capital

MPK = F’(K)

r(t) + δ

K(t)

r’(t) + δ

K’(t)

A Fall in r:

r’(t) < r(t)

• K(t) will increase if MPK(t) increases

Capital

MPK = F’(K)

r(t) + δ

K(t)

Capital

MPK = F’(K)

r(t) + δ

K(t) K’(t)

An increase in MPK

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*

Investment

r*

The Investment FunctionInterestRate

I*

I

I

Investment

r*

An increase in investment, May be due to an increase in the future MPK

InterestRate

I*

I

I

I’

I’

I**