The Intertemporal Approach to the Current Account

Professor Roberto Chang

Rutgers University

January 2007

• The so called Intertemporal Approach to the Current Account amount to the application of the basic principles behind decision theory to the question of how much an economy decides to borrow or lend internationally.

• Chapter 2 of Schmitt Grohe and Uribe.

A Small Economy

• Consider the problem of a resident of a small economy that can borrow or lend in international markets.

• Assume two periods (today vs tomorrow), one nonstorable good in each period.

• The typical agent in this economy has endowment Q1 in period 1 and Q2 in period 2

• Suppose that the typical agent can borrow or lend from the capital market at interest rate r.

• Let Bt = asset position at the end of period t. Then:

C1 + B1 = (1+r)B0 + Q1

C2 + B2 = (1+r)B1 + Q2

• However, no agent would hold a positive B2, and negative B2 will not be feasible. Hence B2 = 0.

• Assume that B0 = 0 here, for simplicity (SU allows nonzero B0). Then the two budget constraints above collapse to

C1 + C2/(1+r) = Q1 + Q2/(1+r) = I

• Suppose that the preferences of the typical agent are given by a utility function U = U(C1, C2)

• Then the problem is of the same form as before, with (1+r) = price of C1 relative to C2.

Current C (C1)

Future C (C2)

O Q1

Q2A

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2A The Present Value of

Income:

Q1 + Q2/(1+r) = I

Current C (C1)

Future C (C2)

Budget Line:

C1 + C2/(1+r) = Q1 + Q2/(1+r) = I

(Slope = - (1+r))

O I

I (1+r)

Q1

Q2A

• As in standard choice problems, we assume that agents in this economy have well defined preferences on consumption today versus consumption tomorrow.

C1

C2

O C1

C2 B

A

Q1

Q2

Equilibrium in Small Economy

The Current Account

• The current account is defined as the change in international wealth. So, in period 1,

CA1 = B1 – B0

• But, recall that C1 + B1 = (1+r)B0 + Q1, so

CA1 = rB0 + Q1 – C1

= Y1 – C1

= S

C1

C2

O C1

C2 B

A

Q1

Q2

C1

C2

O C1

C2 B

A

Q1

Q2

CA Deficit in Period 1

• Note that the consumption choice depends on the present value of income, not on its timing.

• In contrast, savings and the current account do depend on the timing of income.

C1

C2

O C1

C2 B

A

Q1

Q2

CA Deficit in Period 1

C1

C2

O C1

C2 B

A

Q1’

Q2’

CA Surplus in Period 1

A’

If the Endowment Point is A’ instead of Athe economy runs a CA surplus

Welfare Implications

• International capital markets improve welfare.

• The benefits from access to international markets are bigger the bigger the resulting CA imbalance (relative to autarky)

Capital Controls

• Suppose that residents of this economy are not allowed to borrow abroad.

C1

C2

O C1

C2 B

A

Q1

Q2

Suppose that this is the outcomeunder free capital mobility

C1

C2

O

A

Q1

Q2

Capital controls mean that agents cannotborrow in the world market, that is, points in the budget set for which C1 > Q1 are not available.

C1

C2

O

A

Q1

Q2

The resulting budget set is below and to the left of the red line.

C1

C2

O

A

Q1

Q2

The resulting budget set is below and to the left of the red line.

• The domestic interest rate must increase so that the domestic market for loans is in equilibrium.

C1

C2

O

A

Q1

Q2

Slope: - (1+rA )

The domestic interest rate must increaseto rA so that home residents are happy consuming their endowments.

• Summarizing: if the economy is a net borrower from the rest of the world, capital controls (no foreign borrowing allowed) eliminate CA deficits and result in high interest rates at home.

• If the economy is a net lender to the rest of the world, capital controls are irrelevant.

C1

C2

O C1

C2 B

A

Q1

Q2

Suppose instead that this is the outcomeunder free capital mobility

C1

C2

O C1

C2 B

A

Q1

Q2

A prohibition on foreign borrowing does not affect agents’ choices here.

Some Comparative Statics

A Fall in Current Income

• Suppose that Q1 (initial endowment) falls by some quantity Δ.

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

Q1 - Δ

Current C (C1)

Future C (C2)

O I’

I (1+r)

Q1

Q2 A

Q1 - Δ

This is the

new budget line

A’

I

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

Q1 - Δ

A’

C

Suppose the CA was originally zero.Although A’ is now feasible,C is the new consumptionpoint.

I’

Current C (C1)

Future C (C2)

O I

I (1+r)

C1

Q2 A

Q1 - Δ

A’

C

Suppose the CA was originally zero.Although A’ is now feasible,C is the new consumptionpoint.

CA deficit

• The result is that the country runs a CA deficit.

• Intuition: access to international capital markets allow countries to smooth out temporary shortfalls in income.

• A possible explanation for current US CA deficits?

• A fall in future income has the opposite effect: it induces international lending and, therefore, a current account surplus.

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

Q2 - Δ A’

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

Q2 - Δ A’

I’

Current C (C1)

Future C (C2)

O I

I (1+r)

C1

Q2 A

Q1

A’

C

Suppose again the CA was originally zero.C is the new consumptionpoint : the CA is now insurplus.

CA surplus

Q2 - Δ

I’

Transitory vs permanent changes in income

• Suppose that both Q1 and Q2 fall by the same amount.

• By itself, the fall in Q1 would tend to induce a CA deficits

• But the fall in Q2 acts in the opposite direction

• Hence the CA will move little.

• The lesson: transitory changes in income are strongly accommodated by CA surpluses or deficits; the CA is, in contrast, unresponsive to permanent income changes.

An Increase in the World Interest Rate

• Consider an interest rate increase from r to r’ > r.

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

I’

I’ (1+r’) r’ > r

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2 A

I’

I’ (1+r’) r’ > r

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2A

I’

I’ (1+r’) r’ > r

C

C1

CA surplus

If the CA was initially zero, and the interest rate increases, the current account must go into surplus.

(Exercise: How do we know that consumption does not go to a point like C’ in the next slide?)

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2A

I’

I’ (1+r’) r’ > r

C’

• Here we have assumed that the economy was originally neither lending nor borrowing.

• One consequence is that the economy is always better off if the interest rate changes.

• This is not the case, however, if the economy was a net lender or borrower at the original interest rate.

• If the economy was a lender at r, an increase in r causes a beneficial wealth effect that reinforces the previous effects.

• But if the economy was a borrower before the interest rate increase, the increase in r makes it poorer and can cause a welfare loss.

Current C (C1)

Future C (C2)

O I

I (1+r)

Q1

Q2A

I’

I’ (1+r’) r’ > r

CC’

Net Wealth and Trade Surpluses

• Recall that C1 + B1 = (1+r)B0 + Q1

C2 = (1+r)B1 + Q2 B1 = – (Q2 – C2)/(1+r)

• It follows that:

(1+r)B0 = B1 – (Q1 – C1) = - (Q1 – C1) – (Q2 – C2)/(1+r)

• Recall that Qt – Ct = Trade Surplus at t = TBt

(1+r)B0 = - TB1 – TB2/(1+r)

This says that initial foreign net wealth must equal the discounted value of trade deficits.

Algebraic Example

• Assume U(C1,C2) =log C1 + log C2

• Recall that optimal consumption then requires that

(∂U/∂C1)/ (∂U/∂C2) = 1+r, i.e. (1/C1)/(1/C2) = (1+r), or

C2 = (1+r)C1

• Combine the last expression [C2 = (1+r)C1 ] with the (present value) budget constraint:

C1 + C2/(1+r) = Q1 + Q2/(1+r) = IC1 + (1+r)C1/(1+r) = I

C1 = I/2

• Savings, or the current account, in period 1 are given by:

TB1 = Q1 – C1 = Q1 – (I/2)

But: Q1 + Q2/(1+r) = I, so:

TB = [Q1 – Q2/(1+r) ] / 2

TB = [Q1 – Q2/(1+r) ] / 2

• As expected, the trade balance tends to be positive if Q1 is large, negative if Q2 is large. Why?

• Here, an increase in the world interest rate r causes an improvement in TB

• If the trade balance is initially zero, it continues to be zero if Q1 and Q2 change in the same proportion (“permanent shocks have small effects on the trade balance”)