Production, Investment, and the Current Account

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# Production, Investment, and the Current Account - PowerPoint PPT Presentation

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 11 th ). Motivation. Recall that the current account is equal to savings minus investment.

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### 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

The Savings Function

Interest

Rate

S

r*

S

S*

Savings

Interest

Rate

An increase in savings.

This may be due to

higher Y(1).

S

S’

S

S’

Savings

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.

Output F(K)

F(K)

Capital 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

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).

MPK = F’(K)

r(t) + δ

Capital K(t)

MPK = F’(K)

r(t) + δ

K(t)

Capital

MPK = F’(K)

r(t) + δ

K(t)

Capital

MPK = F’(K)

A Fall in r:

r’(t) < r(t)

r(t) + δ

r’(t) + δ

K(t)

K’(t)

Capital

MPK = F’(K)

r(t) + δ

K(t)

Capital

MPK = F’(K)

An increase in MPK

r(t) + δ

K(t)

K’(t)

Capital

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*

The Investment Function

Interest

Rate

I

r*

I

I*

Investment

An increase in investment,

May be due to an increase in the future MPK

Interest

Rate

I’

I

r*

I’

I

I*

I**

Investment