chapter 2 the inter-temporal approach to external balance · george alogoskoufis, international...

George Alogoskoufis, International Macroeconomics

Chapter 2 The Inter-temporal Approach to External Balance

The inter-temporal approach is based on the assumption that households and firms maximize their inter-temporal utility and the present value of their profits respectively (Ramsey 1928, Fisher 1930).

It begins by identifying the technological and market possibilities of an economy to choose the optimal inter-temporal path of consumption and investment. These possibilities are described by the inter-temporal budget constraint, which describes the conditions under which the economy can borrow and lend internationally, and by the domestic investment technology. Separate analysis of the inter-temporal budget constraints of the private and the public sector illuminates the relationship between public finance and external balance.

This approach is currently the dominant approach to international macroeconomics because of its theoretical consistency as well as its compatibility with the experience of open economies. 1

In order to examine the main characteristics of this approach we start with models of an economy lasting for two periods (Fisher 1930). This is the simplest possible inter-temporal model, and we use it to investigate some of the main characteristics of the inter-temporal approach. Most of the important properties of the two period model carry over to models where economies last for more than two periods. In the last part of this chapter we analyze a simple representative household economy lasting for an infinite number of periods.

2.1 The Path of Optimal Consumption in a Economy without Capital

We assume an economy which lasts for two periods, the present, period 1, and the future period 2. Income per household in period 1 is equal to y1, and income per household in period 2 is equal to y2. We shall assume that income per household is exogenous taking the form of a perishable commodity (manna from heaven) and that there is no capital. 2

There are L identical households. The representative household chooses the inter-temporal path of its consumption, in order to maximize a utility function which depends on consumption in the two periods.

See Sachs (1981, 1982) for one of the first introductions and empirical investigations of this approach. See also 1

Obstfeld and Rogoff (1996) and Obstfeld (1998) general surveys of this approach.

Such a economy, where income is exogenous, is referred to as a pure endowment economy.2

George Alogoskoufis, International Macroeconomics Ch. 2

(2.1)

where U is inter-temporal household utility, u the per period utility function of the household, c1 and c2 the consumption of periods 1 and 2 respectively, and ρ >0, the pure rate of time preference of the household. The per period utility function is twice differentiable and concave, satisfying u΄>0, u΄΄<0.

Inter-temporal utility is maximized subject to the inter-temporal budget constraint,

! (2.2)

where r is the real interest rate, at which households can borrow and lend in a competitive capital market. From (2.2), it follows that the present value of consumption cannot exceed the present value of income.

There is perfect certainty about future income, and the representative household passes away at the end of period 2. Thus, from the budget constraint (2.2), the representative household will consume all the income of period 2, plus its savings. From (2.2) this implies that,

! (2.3)

Substituting (2.3) in (2.1), the problem of the representative household is to select consumption in period 1, in order to maximize,

! (2.4)

From the first order condition for a maximum, i.e the condition that the first derivative of U with respect to c1 is equal to zero, we get,

! (2.5)

(2.5) is known as the Euler equation for consumption, and can be written as,

! (2.6)

The left hand side of (2.6) is the marginal rate of substitution between present and future consumption. At the optimum this is equated to the right hand side, which is the relative price of future and current consumption. The representative household cannot thus improve her lifetime utility by further substituting present for future consumption.

U = u(c1) +1

1+ ρu(c2 )

c1 +11+ r

c2 = y1 +11+ r

y2

c2 = (1+ r)(y1 − c1) + y2

U = u(c1) +1

1+ ρu (1+ r)(y1 − c1) + y2( )

′u (c1) =1+ r1+ ρ

′u (c2 )

11+ ρ

′u (c2 )′u (c1)

= 11+ r

!2


From (2.6), it is clear that the ratio of the marginal utilities of consumption, and thus the ratio of period 1 and period 2 consumption depends on the relation between the pure rate of time preference of the representative household and the real interest rate. This can be seen by rewriting (2.6) as,

! (2.7)

If the real interest rate is higher than the pure rate of time preference, then consumption in the second period is greater than consumption in the first period (the marginal utility is smaller), as the rate of return on savings exceeds the pure rate of time preference of the household. From (2.7) it follows that,

! The opposite happens if the real interest rate is lower than the pure rate of time preference. Then, second period consumption is lower than first period consumption, i.e,

!

If the real interest rate is equal to the pure rate of time preference, then, it follows that consumption in the two periods is the same. Thus, it follows that,

! (2.8)

Thus, in the special case where r=ρ, we end up with absolute consumption smoothing between the two periods.

2.2 General Equilibrium Under Financial Autarky and Financial Openness

Up to now we have only looked at the problem of the representative household that takes the interest rate as given. We shall now investigate the determination of the real interest rate under two alternative assumptions. The assumption of a closed economy, that cannot borrow and lend from the rest of the world, and the assumption of a financially open economy that can borrow and lend from the rest of the world at a given international real interest rate r.

Since we have assumed that there are L identical households, aggregate consumption and income will be given by,

! ! ! (2.9)

2.2.1 General Equilibrium under Financial Autarky

If the economy is a closed economy, a regime that we shall term financial autarky, the real interest rate will be determined by the equality of aggregate consumption and current income, since current income is in the form of a perishable commodity that cannot be transferred from period to period. Thus, in the market for goods, the equilibrium conditions in the two periods are,

′u (c2 )′u (c1)

= 1+ ρ1+ r

r > ρ ⇒ ′u (c2 ) < ′u (c1)⇒ c2 > c1

r < ρ ⇒ ′u (c2 ) > ′u (c1)⇒ c2 < c1

r = ρ ⇒ ′u (c2 ) = ′u (c1)⇒ c1 = c2 =(1+ r)y1 + y2

2 + r

Ct = Lct Yt = Lyt t = 1,2

!3


! , ! (2.10)

The equilibrium conditions (2.10) imply that,

! , ! (2.11)

Since aggregate consumption must be equal to aggregate income, and all households have identical preferences and income, each household will be consuming its current income in each period.

Combining (2.11) with the Euler equation (2.7), the equilibrium real interest rate under autarky, rA, will thus be given by,

! (2.12)

The equilibrium real interest rate under autarky is such as to make all households wish to consume their current income in every period. No household will borrow or lend in this autarkic equilibrium.

If the real interest rate was lower than rA, all households would want to consume more than their current income, they would all simultaneously try to borrow in order to consume more than their current income, and this would drive the real interest rate upwards.

On the other hand, if the real interest rate was higher than rA, all households would want to consume less than their current income, they would all simultaneously try to save and lend their savings to the other households, but since no household would want to borrow, this would drive the real interest rate downwards.

Thus, rA is the only real interest rate that ensures that there is neither borrowing nor lending under financial autarky.

The equilibrium under autarky is depicted in Figure 2.1, under the assumption that y1<y2. Since all households must consume their current income, the equilibrium is at point A, and the level of welfare is measured by the position of the highest possible indifference curve which is tangent to the budget constraint Y1AY2. The equilibrium real interest rate under financial autarky is determined by the slope of that indifference curve at point A. 2.2.2 General Equilibrium under Financial Openness

Let us now assume that the economy can participate in international financial markets, and that all domestic households can borrow and lend freely at the international real interest rate r, which is different that the autarky real interest rate rA. International borrowing and lending takes place through a real international bond F. We shall also assume that the domestic economy is small, in the sense that it cannot affect the international real interest rate.

Since there is no capital, the only outlet for savings is international bonds, while if domestic households want to borrow, they must issue international bonds themselves. From the national income identities we know that the current account of a financially open economy will be determined by,

C1 = Y1 C2 = Y2

c1 = y1 c2 = y2

1+ rA = (1+ ρ) ′u (y1)′u (y2 )

!4


! (2.13)

where CA is the current account in period t, and F the stock of international bonds held by domestic residents in period t.

For the economy of our example, in period 1 international bond holdings are equal to zero. Hence, the current account in period 1 is given by,

! (2.14)

Unlike the case of financial autarky, first period consumption can differ from first period income, since the household can finance any excess of consumption over income by borrowing from the rest of the world, or invest any excess of income over consumption in international bonds, that yield a real rate of return r.

The current account in the second period is given by,

! (2.15)

From the inter-temporal budget constraint of the representative household, it follows that,

! (2.16)

Using (2.16) to substitute for C2 in (2.15), we get,

! (2.17)

The inter-temporal budget constraint of the representative household implies that the current account in the second period will be equal to minus the current account in the first period.

If first period aggregate consumption is higher than first period income, there is a current account deficit in the first period. In the second period, the opposite will happen. There must be a current account surplus, in order to pay the international bond holders who lent the country in the first period.

If first period consumption was lower than first period income, and there is a current account surplus invested in international bonds, in the second period consumption will be greater than income, and there will be a current account deficit, as the country will consume more than its current income, selling its international bonds, and consuming the proceeds.

A central advantage of the inter-temporal approach to external balance is that it forces us to monitor the inter-temporal effects of external imbalances, through the inter-temporal budget constraint. Current deficits must be converted into future surpluses and vice versa if the country is to respect its inter-temporal budget constraint.

What will happen to the economy of our example when it gains access to international financial markets at an international real interest rate which lower than the autarky real interest rate?

CAt = Ft+1 − Ft = Yt + rtBt −Ct

CA1 = F2 = Y1 −C1

CA2 = Y2 + rF2 −C2 = Y2 + r(Y1 −C1)−C2

C2 = Y2 + (1+ r)(Y1 −C1)

CA2 = Y2 + r(Y1 −C1)−C2 = −(Y1 −C1) = −F2 = −CA1

!5


At the lower international real interest rate, all domestic households will want to consume more than their current income. Thus, aggregate consumption in the first period will be higher than aggregate income, and the country will run a current account deficit. In the second period, aggregate consumption will be lower than aggregate income, and the country will run a current account surplus, equal to the deficit of the first period.

Where will the equilibrium be determined? Under financial openness, the inter-temporal budget constraint (2.16) is depicted by the negatively sloped straight line in Figure 2.1. The path of consumption will be determined at the point where the inter-temporal budget constraint is tangent to the highest possible indifference curve of domestic households. This is at point B. At point B, first period consumption is higher than first period income, and second period consumption is lower than second period income.

Thus, under financial openness, domestic households can engage in an inter-temporal reallocation of their consumption, in a way that maximizes their welfare, and which was not possible under autarky. As a result, they can achieve a higher level of welfare. The reason is that financial openness allows domestic households to choose their consumption on the basis of the inter-temporal budget constraint (2.16) and not the much tighter budget constraint (2.10).

In Figure 2.1 the current account of period 1 is measured by the difference between income and consumption on the horizontal axis and the current account in period 2 by the difference between income and consumption on the vertical axis. In our specific example, there is a deficit in period 1 and an equivalent surplus in period 2.

If the international real interest rate was higher than the real interest rate under autarky, the opposite would happen. There would be a current account surplus in the first period and a current account deficit in the second period. The welfare of domestic households would also be higher than in the case of financial autarky.

We have thus demonstrated that in the simple endowment economy that we have examined, financial openness leads to an increase in the welfare of domestic households relative to autarky, as it allows them to engage in beneficial inter-temporal reallocations of their consumption that are not possible under financial autarky. These are the benefits of inter-temporal trade made possible by financial openness. Second, we have shown that financial openness results in current account deficits (or surpluses) in some periods, matched by countervailing current account surpluses (or deficits) in future periods. Thus, external balance is defined as a path of the current account of the balance of payments that satisfies the country’s inter-temporal budget constraint.

2.2.3 A Two Period Endowment Economy with a Non-Perishable Commodity

Up to now, we have been assuming that the exogenous income in the two periods took the form of a perishable commodity, that could not serve as a store of value from period to period. Let us now assume that period 1 income is non perishable, and can thus be saved for consumption in the second period. However, the commodity, although non perishable is assumed to depreciate at at rate δ, where 0<δ<1.

Under financial autarky, the product market equilibrium conditions (2.10) are now replaced by the budget constraint,

!6


! , ! (2.18)

Income no longer has to be equal to consumption in every period, as the commodity can by stored, but first period consumption cannot be higher than first period income.

One can show, using the Euler equations for consumption, that if first period income is lower than or equal to second period income, all households will end up consuming their current income in both periods. The economy will save in the first period, only if,

! (2.19)

Thus, first period income must be sufficiently higher than second period income to make it worthwhile for households to save in the first period. In fact it has to be so much higher, as to make the autarky real interest rate negative, and equal to δ, the depreciation rate.

Thus, with positive autarky real interest rates, the results of our previous analysis go through even for non perishable exogenous incomes. If income in period 1 is sufficiently higher than income in period 2, so as to make it worthwhile for domestic households to save in the form of a depreciating commodity, there will be positive savings in the first period, and the autarky real interest rate will be equal to -δ.

The two cases, with zero and positive savings in the first period are depicted in Figure 2.2. In the case of the equilibrium at A, first period income is lower than second period income. The equilibrium under autarky implies that households will consume their current income in each period. In the case of the equilibrium at B, first period income is sufficiently higher than second period income, to make it worthwhile for households to save part of their first period income and consume it in the second period.

Under financial openness, if the international real interest rate differs from -δ, the real interest rate under autarky, households will be better off, as they will be able to reach a higher indifference curve than in the case of autarky.

2.3. Optimal Consumption and Investment in a Small Two Period Economy

We now generalize the model by assuming a two period economy with a production sector, in which households, combine labor and capital in order to produce output. The main results of the inter-temporal approach survive this generalization.

2.3.1 A Two Period Production Economy with Labor and Capital

Let us assume a two period competitive economy, where households produce output using a production technology that requires labor and capital.

Output and income are produced using a neoclassical production function with constant returns to scale. This takes the form,

C2 = Y2 + (1−δ )(Y1 −C1) Y1 ≥C1

′u (Y2 ) ≥1+ ρ1−δ

′u (Y1)

!7


! (2.20)

where,

! and ! .

In the beginning of period 1, each household is endowed with initial capital k1, and during each period each household supplies one unit of labor. As we have L identical households, it follows that,

! , ! , where! , ! (2.21)

Aggregate capital in period 2 is equal to total capital in period 1, plus investment in period 1. It thus follows that,

! (2.22)

I1 is gross investment in period 1. δ is the depreciation rate, assumed to satisfy 0<δ<1.

In period 2 gross investment is negative, as the household consumes all its remaining capital, since there is no period 3. It thus follows that,

! (2.23)

The representative household selects the path of consumption and investment in order to maximize the inter-temporal utility function,

! (2.24)

under the inter-temporal budget constraint that the present value of consumption plus investment is equal to the present value of output and income. 3

! (2.25)

Investment in the two periods, and the capital stock in period 2 are determined by (2.22) and (2.23) and r is the real interest rate, determined in a competitive capital market.

Substituting (2.22) and (2.23) in the inter-temporal budget constraint (2.25), we get,

! (2.26)

From the first order conditions for a maximum of (2.24) subject to (22.6), we get,

Y = F(K ,L)

FK ,FL > 0 FKK ,FLL < 0,FKL ,FLK > 0

Y1 = F(K1,L) Y2 = F(K2 ,L) K1 = Lk1 K2 = Lk2

K2 = (1−δ )K1 + I1

I2 = −(1−δ )K2 = −(1−δ ) (1−δ )K1 − I1( )

U = u(C1) +1

1+ ρu(C2 )

C1 + I1 +C2 + I21+ r

= F K1,L( )+ F K2,L( )1+ r

C1 + I1 +C2 − (1−δ ) (1−δ )K1 + I1( )

1+ r= F K1,L( )+ F (1−δ ) (1−δ )K1 − I1( ),L( )

1+ r

In what follows we ignore the distinction between individual and aggregate consumption, assuming that the 3

representative household selects aggregate consumption and investment. !8


! (2.27)

! (2.28)

(2.27) is the usual Euler equation for inter-temporal efficiency in consumption. It implies that the marginal rate of substitution between present and future consumption is equated to the relative price of future and current consumption. The representative household cannot thus improve her lifetime utility by further substituting present for future consumption.

(2.27) is the usual condition for productive efficiency. The household will invest in physical capital up to the point where the marginal product of capital (net of the depreciation rate) is equal to the real interest rate.

2.3.2 General Equilibrium under Financial Autarky

Under financial autarky, the only outlet for private savings is investment in physical capital. In a closed economy, savings will be equal to gross investment in each period. Thus, in period 1 we shall have that,

! (2.29)

Substituting (2.29) in the inter-temporal budget constraint (2.26), after some re-arrangement, we get,

! (2.30)

(2.30) is the transformation curve between current and future consumption in a financially autarkic (closed) economy.

From (2.30), it follows that if households selected the lowest possible level of investment in period 1, and consumed all their capital endowment in the first period, they could enjoy the highest possible consumption level in period 1, equal to,

!

In such a case, consumption in period 2 would be equal to zero. At the other extreme, if consumption in period 1 was equal to zero, consumption in period 2 would be at its maximum, given by,

!

In between, the slope of the transformation curve is given by,

! (2.31)

′u (C2 )′u (C1)

= 1+ ρ1+ r

FK (1−δ ) (1−δ )K1 − I1( ),L( )−δ = r

F(K1,L)−C1 = I1

C2 = F (1−δ )K1 + F(K1,L)−C1,L( )+ (1−δ ) (1−δ )K1 + F(K1,L)−C1( )

C1 = F(K1,L)+ (1−δ )K1

C2 = F (1−δ )K1 + F(K1,L)( ),L⎡⎣ ⎤⎦ + (1−δ ) (1−δ )K1 + F(K1,L)( )

dC2

dC1= −[1−δ + FK (1−δ )K1 + F(K1,L)−C1( ),L( )]

!9


The slope of the transformation curve depicts the marginal rate of transformation between current and future consumption. The transformation curve has an increasing slope in C1, because of the diminishing marginal productivity of capital. Thus, the transformation curve is negatively sloped and concave, as depicted in Figure 2.3.

Note from (2.31) and (2.28), that under production efficiency, the slope of the transformation curve will be equal to the real interest rate. Also note from (2.27) that under consumption efficiency, the slope of indifference curves in Figure 2.3 will also be equal to the real interest rate. The real interest rate under financial autarky is thus determined at the point where both production and consumption efficiency hold. This is point A in Figure 2.3.

At point A, households choose consumption and investment in order to maximize their inter-temporal utility. In a competitive equilibrium, the real interest rate rA will be determined at the point where the marginal rate of substitution between current and future consumption equals the marginal rate of transformation between current and future consumption, as determined by the technology of production and the choice of consumption and investment by households.

Although the economy is closed, households can engage in inter-temporal trade through investment in physical capital, that allows them to shift output and income from period to period. The autarky real interest rate is determined at the point where the marginal rate of transformation between current and future consumption is equal to the marginal rate of substitution between current and future consumption in the preferences of domestic households. In this sense, the real interest rate depends on both the technology of production and the preferences of consumers.

2.3.3 General Equilibrium under Financial Openness

Under financial openness savings need no longer be equal to aggregate investment in physical capital. We shall assume that households can finance consumption and investment through a competitive international capital market, at the international real interest rate r. We shall also assume that the domestic economy is small, in the sense that it cannot affect the international real interest rate.

The current account in period 1 is determined by the excess of savings over domestic investment in period 1, as,

! (2.32)

The current account in period 2 is determined by the excess of savings over domestic investment in period 2, as,

! (2.33)

Maximization of the inter-temporal utility function of the representative household yields the first order conditions (2.27) and (2.28). However, under financial openness the real interest rate is the international real interest r. Thus, for (2.28) to be satisfied, investment in the first period must be such that the marginal product of capital is equal to the international real interest rate. Households must be indifferent in equilibrium between investing in domestic capital and international bonds. On the other hand, savings will be determined so that the Euler equation (2.27) for consumption is satisfied at the international real interest rate.

CA1 = F2 = Y1 −C1 − I1

CA2 = −F2 = Y2 + rF2 −C2 − I2

!10


A critical characteristic of (2.28) is that in a small open economy, investment and the capital stock are independent of consumer preferences. A country that can borrow abroad at an international real interest rate r would never want to ignore investment opportunities that yield more than r. To put it otherwise, investment is independent of national savings, because the country can borrow freely from the rest of the world.

This characteristic is due to specific assumption which we have made. First, that the economy is small and does not affect international real interest rates. A sizable economy would affect international real interest rates because it would affect the international balance of savings and investment. Second, we assume that the economy produces an internationally tradable commodity. If the economy produces internationally non-tradable goods as well, the result does not follow. Thirdly, we have assumed that international capital markets operate without imperfections that can limit the access of a solvent country to them. When there are factors such as the risk of bankruptcy, limiting access to international capital markets, national savings may affect domestic investment through a risk premium on real interest rates.

The comparison between financial autarky and financial openness is presented in Figure 2.3, under the assumption that the international real interest rate is lower than the domestic equilibrium real interest rate under autarky. Because of a lower real interest rate, both consumption and investment in period 1 are higher relative to autarky. Consumption is determined at point B, and the difference between output and investment at point B΄. The difference between the two is the current account. In period 1 there is a current account deficit, which is transformed into a current account surplus in period 2.

Under financial openness the economy is able to reach a higher level of welfare for domestic consumers, because its expanded possibilities to inter-temporally substitute consumption, through its participation in international capital markets.

It is straightforward to show that an increase in welfare would follow even if the international real interest rate was higher than the domestic autarky rate. Then, the economy would experience a fall in consumption and investment in the first period, and a rise in consumption in the second period. The country would experience a current account surplus in the first period, which, because of the higher world real interest rate, would allow it to have higher income and consumption in the second period.

Consequently, whenever the international real interest rate differs from the domestic autarky rate, the extra possibilities of inter-temporal trade made possible by a country’s participation in global capital markets, lead to increased welfare. Apart from the static benefits highlighted by the theories of international trade, participation in the global economy also entails dynamic benefits arising from the decoupling of domestic investment from domestic savings.

2.4 Optimal Determination of External Balance in a Multi-Period Economy

We now abandon the two period assumption, and assume, generalizing our model, that the economy lasts for ever. Instead of two periods, we have an infinite number of periods. 4

The model in this section is based on Sachs (1982).4

!11


The representative household determines a path of consumption to maximize the inter-temporal utility function,

! (2.34)

The maximization takes place under the constraint,

! (2.35)

where,

! (2.36)

Υ is aggregate output and income, Ι aggregate investment, and Cg aggregate government expenditure. F denotes net foreign assets. To keep the analysis simple we assume that aggregate output, investment and government expenditure are exogenous variables for the representative household.

(2.35) is the inter-temporal budget constraint of both the representative household and the country. On the left hand side is the present value of aggregate consumption and on the right hand the present value of the disposable income of the household, plus its net foreign assets.

(2.35) implies that,

! (2.37)

The current account determines the accumulation of net foreign assets, and is equal to the difference between aggregate national savings and domestic investment.

In order for (2.37) to satisfy (2.35), the following transversality condition must also hold,

! (2.38)

(2.38) suggests that the present value of net foreign assets, as time tends to infinity, should tend towards zero. This condition precludes the rapid accumulation of net foreign assets or net foreign debt. It requires that net foreign assets, or debt, should not cumulate at a rate that exceeds the real interest rate r. Thus, an economy with an infinite time horizon can continuously have positive or negative net foreign assets, on the condition that they do not cumulate at a rate higher than the real interest rate. It is only in this case that the transversality condition (2.38) and the inter-temporal budget constraint (2.35) are satisfied.

(2.35) can be re-written as,

Ut =1

1+ ρ⎛⎝⎜

⎞⎠⎟s=t

∞∑s−t

u(Cs )

11+ r

⎛⎝⎜

⎞⎠⎟s−t

s=t

∞∑ Cs = (1+ r)Ft +11+ r

⎛⎝⎜

⎞⎠⎟s−t

s=t

∞∑ Ys −Csg − Is( )

Is = Ks+1 − Ks

CAt = Ft+1 − Ft = Yt + rFt −Ct −Ctg − It

limT→∞

11+ r

⎛⎝⎜

⎞⎠⎟T

Ft+T +1 = 0

!12


! (2.39)

(2.39) suggests that the present value of future trade surpluses should equal the initial external debt (negative net foreign assets) of a country. It is only in this case that external debt is sustainable and the path of the balance of payments satisfies the inter-temporal budget constraint of the country. On the other hand, if the present value of future trade deficits is greater than the country’s initial external debt -(1+r)F, the country could increase its consumption, increasing the welfare of the representative household, without violating its inter-temporal budget constraint.

From the first order conditions for the maximization of (2.34) subject to (2.35), we get the familiar Euler equation,

! (2.40)

The interpretation of (2.40) is analogous to the interpretation of (2.7) in the two period model. If the real interest rate exceeds the pure rate of time preference, then future consumption is higher than current consumption, since the marginal utility of future consumption must be lower than the marginal utility of current consumption. The opposite applies if the real interest rate is lower than the pure rate of time preference. Then future consumption must be lower than current consumption.

In the case where the real interest rate is equal to the pure rate of time preference, a natural steady state condition in this model, then, from (2.40), the ratio of marginal utilities of consumption between periods will be equal to unity, hence optimal consumption will be constant. Thus, from (2.40), if r=ρ, it follows that,

! (2.41)

Substituting (2.41) in (2.35) and solving for optimal consumption, we get that,

! (2.42)

According to (2.42) the household consumes in every period the value of the current return of its total wealth, which consists of its net foreign assets, plus the net present value of its disposable income, when primary government expenditure and investment expenditure are subtracted from gross domestic product (GDP). This idea is linked to the permanent income hypothesis of Friedman (1957). The net current return is defined as the amount that can be consumed, leaving the net present value of the wealth of the household unchanged.

For a constant real interest rate we can define as the permanent level of a variable XP, as,

! (2.43)

−(1+ r)Ft =11+ r

⎛⎝⎜

⎞⎠⎟s−t

s=t

∞∑ Ys −Cs −Csg − Is( )

′u (Ct+1)′u (Ct )

= 1+ ρ1+ r

Ct = Ct+1 = Ct+2 = ...

Ct =r1+ r

(1+ r)Ft +11+ r

⎛⎝⎜

⎞⎠⎟s−t

Ys −Csg − Is( )s=t

∞∑⎡

⎣⎢

⎤

⎦⎥

11+ r

⎛⎝⎜

⎞⎠⎟s−t

XtP

s=t

∞∑ = 11+ r

⎛⎝⎜

⎞⎠⎟s−t

Xss=t

∞∑

!13


so that,

! (2.44)

The permanent value of a variable is the value of its current return at the going real interest rate, i.e. the hypothetical constant value of the variable with the same present value as the variable itself.

Substituting (2.44) for the present value of the exogenous real income Y, the exogenous real primary expenditure Cg and exogenous investment Ι, we get that,

! (2.45)

(2.45) implies that, if ρ=r, then the household consumes in steady state the return of its net foreign assets, plus the permanent value of its net disposable income, when exogenous investment and primary government expenditure are subtracted from permanent gross domestic product.

Substituting the consumption function (2.45) in the current account equation (2.37), we get,

! (2.46)

(2.46) incorporates the key predictions of the inter-temporal approach to the current account.

(2.46) suggests that optimal consumption behavior implies surpluses in the current account when there are temporary positive deviations of output from its permanent level, and deficits when there are temporary positive deviations of investment or public expenditure from their permanent levels.

If income is temporarily higher than than its permanent level, this does not lead to an increase in consumption in this model. Consequently, national savings rise temporarily above investment, and the current account shows a temporary surplus, which leads to accumulation of assets (bonds) from the rest of the world.

If public expenditure is temporarily higher than its permanent level, this does not affect private consumption. Consequently, national savings are reduced below aggregate domestic investment, and the current account moves into deficit. The deficit leads to a de-cumulation of net foreign assets or an increase in external debt.

Finally, if aggregate domestic investment is temporarily higher than its permanent level, this also leads to a deficit in the current account, because a temporary change in the investment does not affect permanent disposable income, private consumption and savings. Aggregate domestic investment temporarily rises above national savings and the country moves into a current account deficit and de-cumulation of net foreign assets or an increase in external debt.

XtP = r

1+ r11+ r

⎛⎝⎜

⎞⎠⎟s−t

Xss=t

∞∑

Ct = rFt +YtP −Ct

gP − ItP

CAt = Ft+1 − Ft = Yt −YtP( )− Ct

g −CtgP( )− It − It

P( )

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More complex inter-temporal models, in which aggregate domestic investment and income are endogenous variables, and the real interest rate is not always equal to its steady state value, have similar, if more complicated properties.

In the remainder of this book we shall return to models based on the inter-temporal approach, combining the inter-temporal approach with models that allow for long run economic growth, the role of money, wage and price stickiness, unemployment and other distortions.

2.5 Conclusions

In this chapter we have introduced the inter-temporal approach to the balance of payments. This approach takes as its starting point the technological and market possibilities of an economy to choose the optimal path of private consumption over time. These possibilities are incorporated in the inter-temporal budget constraint, which describes the conditions under which the economy can borrow and lend in international capital markets, as well as the domestic investment technology.

This approach is currently the dominant approach to international macroeconomics and is widely used for the analysis of current account imbalances, policy coordination, external debt and the operation of international capital markets.

In order to introduce the key features of the inter-temporal approach, we first analyzed models of economies that last for two periods (Fisher 1930), and then a representative household multi-period model.

We have demonstrated that participation in global capital markets enables an economy to achieve higher levels of prosperity through increased opportunities for inter-temporal trade. When the real interest rate under autarky differs from the international real interest rate, an economy can smooth the inter-temporal path of consumption more effectively, achieving a higher level of welfare.

In a multi-period representative household model, we have also demonstrated that when real output and income are temporarily high, the economy experiences a current account surplus, as national savings exceed domestic investment. On the other hand, when aggregate domestic investment, or real government expenditure, are temporarily high, the current account moves into deficit, as domestic investment exceed national savings.

These properties of the inter-temporal approach to the current account carry over to more general models of the inter-temporal approach, that allow for long run economic growth, the role of money and financial markets, wage and price rigidities and other distortions.

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Figure 2.1 The Welfare Benefits of Financial Openness in an Economy without Capital

!16


Figure 2.2 Equilibrium under Autarky in an Economy with a Non Perishable Endowment Income

!17


Figure 2.3 The Benefits of Financial Openness in an Economy with Capital

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References

Fisher I. (1930), The Theory of Interest, New York, Macmillan. Obstfeld M. and K. Rogoff (1996), Foundations of International Macroeconomics, Cambridge Mass.,

MIT Press. Obstfeld M. (1998), “International Finance”, in Eatwell J., M. Millgate and P. Newman (eds) The New

Palgrave: A Dictionary of Economics, London, Palgrave Publishers. Ramsey F. (1928), “A Mathematical Theory of Saving”, The Economic Journal, 38, pp. 543-559. Sachs J. (1981), “The Current Account and Macroeconomic Adjustment in the 1970s”, Brookings

Papers on Economic Activity, 1, pp. 201-282. Sachs J. (1982), “The Current Account in the Macroeconomic Adjustment Process”, Scandinavian

Journal of Economics, 84, pp. 147-159.

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chapter 2 the inter-temporal approach to external balance · george alogoskoufis, international...

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