Announcements

1
Announcements

• Presidents Day No Class (Feb. 19th)
• Next Monday Prof. Occhino will lecture
• Homework Due Next Thursday (Feb. 15)

2
Production, Investment, and the Current Account

• Roberto Chang
• Rutgers University
• February 2007

3
Motivation

• Recall that the current account is equal to
  savings minus investment.
• Empirically, investment is much more volatile
  than savings.
• Reference here chapter 3 of Schmitt Grohe - Uribe

4
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.

5
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))

6
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 Ka, with 0 lt a lt 1.

7
Output F(K)
F(K)
Capital K
8

• 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 Ka, the MPK is
• MPK F(K) aA Ka-1

9
MPK F(K)
Capital K
10
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 d of capital is
  lost in the production process.
• Hence the total cost of capital (per unit) is
  r(t) d.

11

• In period t, a firm that operates with capital
  K(t) makes profits equal to
• ?(t) F(K(t)) r(t) d K(t)
• Profit maximization requires
• F(K(t)) r(t) d

12
F(K(t)) r(t) d

• 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).

13
MPK F(K)
Capital K
14
MPK F(K)
r(t) d
Capital K(t)
15
MPK F(K)
r(t) d
K(t)
Capital
16

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

17
MPK F(K)
r(t) d
K(t)
Capital
18
MPK F(K)
A Fall in r r(t) lt r(t)
r(t) d
r(t) d
K(t)
K(t)
Capital
19
Households

• The typical household owns K(1) units of capital
  at the beginning of period 1.
• The amount of capital it owns at the beginning of
  period 2 is given by
• K(2) (1-d)K(1) I(1)

20

• At the end of period 2, the household will choose
  not to hold any capital (since t 2 is the last
  period), and hence
• I(2) -(1-d) K(2)

21

• In addition, households own firms, and hence
  receive the firms profits.

22
Closed Economy case

• Suppose that the economy is closed. Then the
  households budget constraints are
• C(1) I(1) ?(1) K(1)(r(1) d)
• C(2) I(2) ?(2) K(2)(r(2) d)
• And, recall,
• K(2) (1-d)K(1) I(1)
• I(2) -(1-d) K(2)

23

• But all of these constraints are equivalent to
  the single constraint
• C(1) C(2)/(1r(2))
• ?(1) K(1)1 r(1) ?(2)/(1r(2))

24
Proof

• From
• C(1) I(1) ?(1) K(1)(r(1) d)
• and
• K(2) - (1-d)K(1) I(1)
• We obtain
• C(1) K(2) ?(1) K(1)1 r(1)

25

• Likewise,
• C(2) I(2) ?(2) K(2)(r(2) d)
• and
• I(2) -(1-d) K(2)
• yield
• C(2) ?(2) K(2)(1 r(2))

26

• Now,
• C(1) K(2) ?(1) K(1)1 r(1)
• C(2) ?(2) K(2)(1 r(2))
• can be combined to get the intertemporal budget
  constraint
• C(1) C(2)/(1r(2))
• ?(1) K(1)1 r(1) ?(2)/(1r(2))

27

• The households budget constraint
• C(1) C(2)/(1r(2))
• ?(1) K(1)1 r(1) ?(2)/(1r(2)) Z
• is similar to the ones we have seen before, with
  Z the present value of income.
• The household will choose consumption so that the
  marginal rate of substitution between C(1) and
  C(2) equals (1r(2)).

28
C(2)
Households Optimum
Z (1r(2))
C
C(2)
C(1)
O
Z
C(1)
29
C(2)
Households Optimum Here, Z ?(1) K(1)1
r(1) ?(2)/(1r(2)) is the present value of
income.
Z (1r(2))
C
C(2)
C(1)
O
Z
C(1)
30
C(2)
Households Optimum
Z (1r(2))
In the closed economy, the slope is (1r(2))
C
C(2)
C(1)
O
Z
C(1)
31
Productive Possibilities

• The resource constraints in the closed economy
  are
• C(1) I(1) F(K(1))
• C(2) I(2) F(K(2))
• But
• K(2) (1-d)K(1) I(1)
• I(2) -(1-d) K(2)

32

• The first and third equations give
• Y(1) F(K(1))(1-d)K(1) C(1) K(2)
• while the second and fourth give
• F(K(2)) (1-d)K(2) C(2)

33
Production Possibilities

• Since K(2) Y(1) C(1),
• C(2) F(K(2)) (1-d)K(2)
• F(Y(1) C(1)) (1-d)(Y(1) C(1))
• ? This gives the combinations (C(1),C(2)) that
  the economy can produce (the production
  possibility frontier)

34

• A special case is when d 1 (complete
  depreciation of capital), so the PPF is simply
• C(2) F(K(2)) F(Y(1) C(1))
• And its slope is
• ?C(2)/ ?C(1) -F(Y(1)-C(1))

35
C(2)
C(2) F(Y(1) C(1))
F(Y(1))
O
Y(1)
C(1)
36
Production Equilibrium

• Recall that the slope of the PPF is F(Y(1)-C(1))
  F(K(2)). But also, profit maximization
  requires
• (1r(2)) F(K(2))
• ? In equilibrium, production must be given by the
  PPF point at which the slope of the PPF equals
  1r(2)

I
I
I
37
C(2)
F(Y(1))
O
Y(1)
C(1)
38
C(2)
If r(2) is the rental rate, production
equilibrium is at P The slope of the PPF at P
is -(1r(2))
P
C(2)
C(1)
O
C(1)
39
Finally General Equilibrium in the Closed Economy

• In equilibrium in the closed economy, production
  must be equal to consumption.
• But we saw that both production and consumption
  depend on 1r(2).
• Hence r(2) must adjust to ensure equality of
  supply and demand.

40
C(2)
Households Optimum
Z(1r(2))
Slope - (1r(2))
C
C(2)
C(1)
O
Z
C(1)
41
C(2)
Production Equilibrium
Slope -(1r(2))
P
C(2)
C(1)
O
C(1)
42
C(2)
Equilibrium in the Closed Economy r(2) adjusts
to ensure the equality of production and
consumption in equilibrium.
Slope -(1r(2))
P C
C(2)
C(1)
O
C(1)
43

• Note that the rental rate r(2) must adjust to
  ensure equilibrium.

44
C(2)
P C
C(2)
C(1)
O
C(1)
45
C(2)
If r(2) were higher, production would be at P
and consumption at C, So markets would not clear.
C
P C
C(2)
P
C(1)
O
C(1)
46
Adjustment to an Income Shockin the Closed
Economy

• Suppose that Y(1) falls by ? (because, for
  example, there is less capital in period 1)

47
C(2)
P C
O
Y(1)
C(1)
48
C(2)
P
?
?
O
Y(1)
Y(1) - ?
C(1)
49
C(2)
P and P must have the same slope and their
horizontal distance is ?.
P
P
O
Y(1)
Y(1) - ?
C(1)
50

• Why is the horizontal distance between P and P
  equal to ??
• P and P correspond to the same value of C(2),
  and hence the same value of K(2). But K(2) Y(1)
  C(1), so if Y(1) is lower at P than at P by ?,
  C(1) must be lower by ? too.

51

• To see that P and P have the same slope, recall
  that the PPF must satisfy
• C(2) F(Y(1) C(1))
• So, since K(2) is the same at both P and P,
  Y(1) C(1) must also be the same.
• And, since, the slope of the PPF is
• ?C(2)/ ?C(1) -F(Y(1)-C(1))
• it is also the same at P and P.

52
C(2)
P and P must have the same slope
P
P
O
Y(1)
Y(1) - ?
C(1)
53
C(2)
Because C(1) and C(2) are normal, The new
consumption point would be a point such as C,
if r(2) stayed the same. But then markets would
not clear.
P
P
C
O
Y(1)
Y(1) - ?
C(1)
54
C(2)
Equilibrium is given by C P, where an
indifference curve is tangent to the PPF. The
slope of the PPF gives the new value of r(2),
which must be higher than before. C(1) falls by
less than ?.
P
P
CP
O
Y(1)
Y(1) - ?
C(1)
55

• Hence if Y(1) falls,
• The rental rate r(2) (the return on savings)
  increases.
• Consumption falls in both periods.
• Savings and Investment fall.

56
Open Economy

• Suppose that households can borrow and lend
  internationally at the interest rate r.
• Let W(t) denote the wealth of the typical
  household at the end of period t. Then, if B(t)
  denotes foreign assets at the end of t,
• W(t) K(t1) B(t)

57

• In addition, since the household can save either
  by holding capital or holding foreign bonds, the
  return on both kinds of assets must be the same,
  that is,
• r(t) r
• ? The world interest rate pins down the rental
  rate of capital.

58

• Hence, since the marginal product of capital is a
  function only of capital, K(2) is determined
  solely by the world interest rate.
• And, since K(2) (1-d)K(1) I(1), and K(1) is
  exogenously given, investment in period 1 (I(1))
  is also determined by the world interest rate.

59

• In particular, from
• F(K(2)) r(2) d
• It follows that
• F(K(2)) r d
• That is,
• K(2) K, where F(K) r d
• And I(1) K - (1-d)K(1).

60
MPK F(K)
r d
K(2) K
Capital
61

• Note that K(2) and I(1) then depend inversely on
  r . The previous graph can then be seen as an
  investment function.

62
rd
rd
I(1) K
Investment
63
The National Budget Line

• In the open economy case, the budget constraint
  is given by
• C(1) I(1) B(1) Y(1) (1r)B(0)
• C(2) I(2) Y(2) (1r)B(1)

64

• Assume again d 1, for simplicity. Then K(2)
  I(1). But we saw that K(2) K.
• Also, I(2) 0. Assuming that B(0) 0, the two
  constraints above reduce to
• C(1) K B(1) Y(1)
• C(2) F(K) (1r)B(1)
• Which imply
• C(1) K C(2)/(1r) Y(1) F(K)/(1r)

65

• In other words, the economys consumption
  possibilities in the open economy are given by a
  conventional budget line
• C(1) C(2)/(1r) Y(1) K F(K)/(1r)
• Z

66
C(2)
O
C(1)
67
C(2)
Z Y(1) K F(K)/(1r) (Recall that K is
uniquely defined by r)
O
Z
C(1)
68
C(2)
This is the national budget line Slope -(1r)
O
Z
C(1)
69
C(2)
By construction, B must be on the budget Line.
B
F(K)
Y(1) K
O
Z
C(1)
70
C(2)
Importantly, the PPF must go through B (since B
is feasible in the closed economy) and have
slope -(1r)
B
F(K)
Y(1) K
O
Z
C(1)
71
What determines consumption?

• Because (1r) is the return on savings, optimal
  consumption will require that the marginal rate
  of substitution between C(1) and C(2) equal
  (1r).

72
C(2)
Equilibrium consumption is at Point A.
B
F(K)
A
C(2)
Y(1) K
O
Z
C(1)
C(1)
73

• Note that the ability to borrow and lend
  internationally causes changes in consumption and
  production.

74
C(2)
In a closed economy, consumption and production
are at P and the return on savings is the
slope of the green line.
P
O
I
C(1)
75
C(2)
If the economy can borrow and lend at rate r
(cheaper than in the closed economy), there is
more investment and production moves to B.
F(K)
B
P
Y(1) K
O
C(1)
76
C(2)
International capital markets also allow an
optimal allocation of income between current and
future consumption, as in A.
F(K)
B
P
A
C(2)
Y(1) K
O
C(1)
C(1)
77
The Current Account Balance

• Budget constraints in each period are
• C(t) I(t) B(t) (1r) B(t-1) Y(t)
• Recalling that the current account is
• CA(t) B(t) B(t-1)
• rB(t-1) Y(t) C(t) I(t)
• savings - investment

78

• The trade balance is given by net exports
• TB(t) Y(t) C(t) I(t)
• Note that
• CA(t) TB(t) rB(t-1)

79

• In our example, in period 1 (recall B(0) 0 and
  I(1) K(2) K),
• CA(1) TB(1) Y(1) K - C(1)

80
C(2)

B
F(K)
A
C(2)
Y(1) K
O
C(1)
C(1)
81
C(2)

B
F(K)
A
C(2)
Y(1) K
O
C(1)
C(1)
Current Account Deficit
82
Adjustment to an Income Shockin the Open Economy

• Same Experiment as Before Suppose that Y(1)
  falls by ? (because, for example, there is less
  capital in period 1)

83
C(2)
Now we assume that the world interest rate is
such that, before the shock, trade is balanced.
P C
Slope -(1r)
O
Y(1)
C(1)
84
C(2)
Exactly as in the closed economy case, the PPF
shifts to the left.
P
?
?
O
Y(1)
Y(1) - ?
C(1)
85
C(2)
After the shock, the world interest rate is still
given by r. This means that the new production
point is P.
P
P
O
Y(1)
Y(1) - ?
C(1)
86
C(2)
The national budget line is given by the blue
line.
P
P
O
Y(1)
Y(1) - ?
C(1)
87
C(2)
Because C(1) and C(2) are normal, consumption
moves to a point such as C.
P
P
C
O
Y(1)
Y(1) - ?
C(1)
88
C(2)
Because C(1) and C(2) are normal, consumption
moves to a point such as C. Note that C(1)
falls by less than ?.
P
P
C
O
Y(1)
Y(1) - ?
C(1)
89

• Summarizing, the fall in Y(1)
• Leaves I(1) and K(2) unchanged (at K)
• C(2) must fall.
• C(1) falls, but by less than Y(1)
• If B(0) 0, this means that the trade balance
  and current account go into deficit in period 1

90

• Note, in particular, that a fall in Y(1)
• Does not affect I(1)
• Reduces savings in period 1 (S(1) Y(1) C(1))
• Causes a trade deficit and a current account
  deficit (CA(1) TB(1) S(1) Y(1))

91
Changes in World Interest Rate

• Now consider a change in the world interest rate
  an increase in r.

92
C(2)
Again, assume that the world interest rate is
such that, before the shock, trade is balanced.
P C
Slope -(1r)
O
Y(1)
C(1)
93
C(2)
Suppose that the world interest rate increases.
Then the national budget line would be the red
line, if production equilibrium remained at P.
P
O
Y(1)
C(1)
94
C(2)
Production, however, will change to P, where
the national budget line is tangent to the PPF.
I(1), in particular, must fall.
P
P
O
Y(1)
C(1)
95
C(2)
The new consumption point is C. Here, this means
that savings in period 1 increase. Since
investment falls, the trade balance goes into
surplus.
C
P
O
Y(1)
C(1)
96
C(2)
The adjustment can be regarded as the sum of a
substitution effect (C to C) and an income
effect (C to C)
C
C
CP
P
O
Y(1)
C(1)
97

• An increase the interest rate produces
• A substitution effect future consumption becomes
  relatively cheaper ? induces more savings
• An income effect production reallocation which
  increases the value of GNP ? induces less
  savings, if both goods are normal

98

• Finally, there is a wealth effect, ignored so
  far. If the country is initially a debtor, the
  cost of the debt increases, which reduces the net
  present value of income, and goes against the
  income effect.
• If the country is initially a creditor, the
  effect is the opposite, and the wealth effect
  reinforces the income effect.

99

• So, the impact of an increase in r on national
  savings is ambiguous.
• Our normal assumption will be that savings
  increase with the interest rate.
• The savings function (or schedule) relates
  savings to the interest rate, other things equal.

100
The Savings Function
Interest Rate
S
r
S
S
Savings
101
Interest Rate
An increase in savings. This may be due to higher
Y(1).
S
S
S
S
Savings