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
MPK = F’(K)
Capital 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).
MPK = F’(K)
Capital K
MPK = F’(K)
r(t) + δ
Capital K(t)
MPK = F’(K)
r(t) + δ
K(t)
Capital
• Note that K(t) will fall if r(t) increases.
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
• K(t) will increase if MPK(t) increases
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