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• 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’