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Announcements
• Presidents’ Day: No Class (Feb. 19th)
• Next Monday: Prof. Occhino will lecture
• Homework: Due Next Thursday (Feb. 15)
Production, Investment, and the Current Account
Roberto Chang
Rutgers University
February 2007
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
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)
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-δ)K(1) + I(1)
• 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-δ) K(2)
• In addition, households own firms, and hence receive the firms’ profits.
Closed Economy case
• Suppose that the economy is closed. Then the household’s budget constraints are:
C(1) + I(1) = Π(1) + K(1)(r(1) + δ)C(2) + I(2) = Π(2) + K(2)(r(2) + δ)
And, recall,K(2) = (1-δ)K(1) + I(1)
I(2) = -(1-δ) K(2)
• But all of these constraints are equivalent to the single constraint:
C(1) + C(2)/(1+r(2)) =
Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2))
Proof
• From:
C(1) + I(1) = Π(1) + K(1)(r(1) + δ)
and
K(2) - (1-δ)K(1) = I(1)
We obtain
C(1) + K(2) = Π(1) + K(1)[1+ r(1) ]
• Likewise,
C(2) + I(2) = Π(2) + K(2)(r(2) + δ)
and
I(2) = -(1-δ) K(2)
yield
C(2) = Π(2) + K(2)(1+ r(2))
Now,C(1) + K(2) = Π(1) + K(1)[1+ r(1) ]
C(2) = Π(2) + K(2)(1+ r(2))can be combined to get the intertemporal
budget constraint:
C(1) + C(2)/(1+r(2)) = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2))
• The household’s budget constraint
C(1) + C(2)/(1+r(2)) =
Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(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 (1+r(2)).
C(1)
C(2)
O Z
Z (1+r(2))
C*(1)
C*(2) C*
Household’s Optimum
C(1)
C(2)
O Z
Z (1+r(2))
C*(1)
C*(2) C*
Household’s Optimum
Here, Z = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2)) is the present value of income.
C(1)
C(2)
O Z
Z (1+r(2))
C*(1)
C*(2) C*
Household’s Optimum
In the closed economy,the slope is –(1+r(2))
Productive Possibilities
• The resource constraints in the closed economy are:
C(1) + I(1) = F(K(1))C(2) + I(2) = F(K(2))
ButK(2) = (1-δ)K(1) + I(1)
I(2) = -(1-δ) K(2)
• The first and third equations give
Y(1) = F(K(1))+(1-δ)K(1) = C(1) + K(2)
while the second and fourth give
F(K(2)) + (1-δ)K(2) = C(2)
Production Possibilities
• Since K(2) = Y(1) – C(1),
C(2) = F(K(2)) + (1-δ)K(2)
= F(Y(1) – C(1)) + (1-δ)(Y(1) – C(1))
This gives the combinations (C(1),C(2)) that the economy can produce (the production possibility frontier)
• A special case is when δ = 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))
C(1)
C(2)
O
C(2) = F(Y(1) – C(1))
Y(1)
F(Y(1))
Production Equilibrium
• Recall that the slope of the PPF is F’(Y(1)-C(1)) = F’(K(2)). But also, profit maximization requires:
(1+r(2)) = F’(K(2))
In equilibrium, production must be given by the PPF point at which the slope of the PPF equals 1+r(2)
II
I
C(1)
C(2)
O Y(1)
F(Y(1))
C(1)
C(2)
O C*(1)
C*(2) P
If r(2) is the rental rate, production equilibrium is at P:
The slope of the PPF at P is-(1+r(2))
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 1+r(2).
• Hence r(2) must adjust to ensure equality of supply and demand.
C(1)
C(2)
O Z
Z(1+r(2))
C*(1)
C*(2) C*
Household’s Optimum
Slope = - (1+r(2))
C(1)
C(2)
O C*(1)
C*(2) P
Production Equilibrium
Slope =-(1+r(2))
C(1)
C(2)
O C*(1)
C*(2) P = C
Equilibrium in the Closed Economy:r(2) adjusts to ensure the equality of production and consumption in equilibrium.
Slope =-(1+r(2))
• Note that the rental rate r(2) must adjust to ensure equilibrium.
C(1)
C(2)
O C*(1)
C*(2) P = C
C(1)
C(2)
O C*(1)
C*(2) P = C
P’
C’
If r(2) were higher, production would be at P’ and consumption at C’,So markets would not clear.
Adjustment to an Income Shockin the Closed Economy
• Suppose that Y(1) falls by Δ (because, for example, there is less capital in period 1)
C(1)
C(2)
O
P = C
Y(1)
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
Δ
Δ
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
P and P’ must have the same slopeand their horizontal distance is Δ.
• Why is the horizontal distance between P and P’ equal to Δ?
• 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 Δ, C(1) must be lower by Δ too.
• 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’.
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
P and P’ must have the same slope
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
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.
C’
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’C’’=P’’
Equilibrium is given by C’’ = P’’, where an indifference curve istangent to the PPF.The slope of the PPF givesthe new value of r(2), which must be higher than before.C(1) falls by less than Δ.
Hence: if Y(1) falls,
• The rental rate r(2) (the return on savings) increases.
• Consumption falls in both periods.
• Savings and Investment fall.
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(t+1) + B*(t)
• 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.
• 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-δ)K(1) + I(1), and K(1) is exogenously given, investment in period 1 (I(1)) is also determined by the world interest rate.
• In particular, from
F’(K(2)) = r(2) + δ
It follows that
F’(K(2)) = r* + δ
That is,
K(2) = K*, where F’(K*) = r* + δ
And I(1) = K* - (1-δ)K(1).
Capital
MPK = F’(K)
r* + δ
K(2) = K*
• Note that K(2) and I(1) then depend inversely on r* . The previous graph can then be seen as an investment function.
Investment
r*+δ
r*+δ
I(1) = K*
The National Budget Line
• In the open economy case, the budget constraint is given by:
C(1) + I(1) + B(1) = Y(1) + (1+r*)B(0)
C(2) + I(2) = Y(2) + (1+r*)B(1)
• Assume again δ = 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*) + (1+r*)B(1)
Which imply:
C(1) + K* + C(2)/(1+r*) = Y(1) + F(K*)/(1+r*)
• In other words, the economy’s consumption possibilities in the open economy are given by a conventional budget line:
C(1) + C(2)/(1+r*) = Y(1) – K* + F(K*)/(1+r*)
= Z
C(1)
C(2)
O
C(1)
C(2)
O Z
Z = Y(1) – K* + F(K*)/(1+r*)
(Recall that K* is uniquelydefined by r*)
C(1)
C(2)
O Z
This is the nationalbudget line
Slope=-(1+r*)
C(1)
C(2)
O Z
By construction,B must be on the budgetLine.
F(K*)
Y(1) – K*
B
C(1)
C(2)
O Z
F(K*)
Y(1) – K*
B
Importantly, the PPF must go through B (since B is feasible inthe closed economy) and have slope-(1+r*)
What determines consumption?
• Because (1+r*) is the return on savings, optimal consumption will require that the marginal rate of substitution between C(1) and C(2) equal (1+r*).
C(1)
C(2)
O Z
F(K*)
Y(1) – K*
B
A
C*(1)
C*(2)
Equilibrium consumption is atPoint A.
• Note that the ability to borrow and lend internationally causes changes in consumption and production.
C(1)
C(2)
O I
P
In a closed economy, consumption and production are at Pand the return on savings is the slopeof the green line.
C(1)
C(2)
O
F(K*)
Y(1) – K*
B
P
If the economy can borrow and lend atrate r* (cheaper than in the closed economy),there is more investment and productionmoves to B.
C(1)
C(2)
O
F(K*)
Y(1) – K*
B
A
C*(1)
C*(2)P
International capital marketsalso allow an optimal allocation of income betweencurrent and future consumption,as in A.
The Current Account Balance
Budget constraints in each period are:
C(t) + I(t) + B(t) = (1+r*) B(t-1) + Y(t)Recalling that the current account is :
CA(t) = B(t) – B(t-1) = r*B(t-1) + Y(t) – C(t) – I(t)
= savings - investment
• The trade balance is given by net exports:
TB(t) = Y(t) – C(t) – I(t)
• Note that
CA(t) = TB(t) + r*B(t-1)
• 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)
C(1)
C(2)
O
F(K*)
Y(1) – K*
B
A
C*(1)
C*(2)
C(1)
C(2)
O
F(K*)
Y(1) – K*
B
A
C*(1)
C*(2)
Current Account Deficit
Adjustment to an Income Shockin the Open Economy
• Same Experiment as Before: Suppose that Y(1) falls by Δ (because, for example, there is less capital in period 1)
C(1)
C(2)
O
P = C
Y(1)
Now we assume that the world interestrate is such that, before the shock, trade is balanced.
Slope=-(1+r*)
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
Δ
Δ
Exactly as in the closed economy case,the PPF shifts to the left.
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
After the shock, the world interest rate isstill given by r*. This means that thenew production point is P’.
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
The national budget line is given by theblue line.
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
Because C(1) and C(2) are normal,consumption moves to a point such as C’.
C’
C(1)
C(2)
O
P
Y(1)Y(1) - Δ
P’
Because C(1) and C(2) are normal,consumption moves to a point such as C’. Note that C(1) falls by less than Δ.
C’
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
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))
Changes in World Interest Rate
• Now consider a change in the world interest rate: an increase in r*.
C(1)
C(2)
O
P = C
Y(1)
Again, assume that the world interestrate is such that, before the shock, trade is balanced.
Slope=-(1+r*)
C(1)
C(2)
O
P
Y(1)
Suppose that the world interest rateincreases. Then the national budget linewould be the red line, if production equilibrium remained at P.
C(1)
C(2)
O
P
Y(1)
Production, however, will change to P’, where the national budget line is tangent to the PPF. I(1), in particular, must fall.
P’
C(1)
C(2)
O Y(1)
The new consumption point is C’.Here, this means that savings in period 1increase. Since investment falls, the trade balance goes into surplus.
P’
C’
C(1)
C(2)
O Y(1)
The adjustment can be regarded as the sum of a substitution effect (C to C’’) andan income effect (C’’ to C’)
P’
C’
C=P
C’’
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
• 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.
• 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.
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’