1. Specification

Consider a small open economy producing both traded and nontraded goods. Agents in this economy derive utility from the consumption of both kinds of goods and from the holding of real cash balances. We denote these respectively as c_T, c_N, and m, where m is the nominal money stock (M) divided by a price index (P) with the latter defined as:

where α is a parameter whose economic interpretation will be given below, and P_T and P_N are, in turn, the domestic-currency prices of traded and nontraded goods. The instantaneous utility function u is therefore given by:

The representative agent is assumed to choose paths for c_T, c_N, and m by maximizing a functional of the form:

where Δ(t) is the discount factor which applies to utility received at instant t. Following Obstfeld (1981) and Uzawa (1968), the rate of time preference is endogenous in our model. The discount factor Δ(t) is given by:

where the rate of time preference δ at time s depends on the instantaneous rate of utility received by the household at that moment. ^5/ δ is assumed to have the properties δ’ > 0, δ’’ > 0, and δ-uδ’ > 0—i.e., the rate of time preference is an increasing function of the flow of utility (see Obstfeld 1981)).

In maximizing the functional U, the household has to respect a number of constraints. Let a denote the household’s financial wealth measured in units of traded goods. Financial wealth is the sum of the household’s stocks of money and bonds, so a is given by:

where e is the real exchange rate, defined as e=P_T/P_N, and b_p denotes the households’ bond holdings, measured in units of traded goods. Bonds are assumed to be internationally traded, and to yield the interest rate r. The rate r is both a nominal and real rate, since we shall assume that there is no ongoing traded goods inflation. The evolution of financial wealth over time is governed by:

when y denotes the value of real output in the economy and τ is the real value of the household’s tax liabilities. Taxes are taken to be lump-sum and measured in units of the consumption bundle. Equation (5) is the household’s instantaneous budget constraint. Its intertemporal budget constraint is given by:

This condition, together with the flow budget constraint (5), ensures that the present value of the household’s consumption (including liquidity services of money) does not exceed the present value of its resources. A final constraint that operates on the household is that its solution paths for both types of consumption as well as for money must be non-negative:

In solving the household’s problem, it is convenient to once again follow Obstfeld and Uzawa in using the change of variables Δ=Δ(t) to express the problem in terms of Δ, rather than t. ^6/ After making this transformation, the household’s maximization problem becomes:

subject to:

By applying the Maximum Principle and reversing the change of variables, we derive the first-order conditions for an interior maximum:

where λ is a costate variable and numerical subscripts denote partial derivatives. These equations characterize the solution to the household’s optimization problem for the general utility function given by equation (2).

It is much more convenient, however, to work with a specific utility function, and for this purpose we adopt the logarithmic version:

where β is a parameter with 0 < β < 1. Since this function has the familiar property that consumption of each good is proportional to total consumption expenditure, it simplifies matters to conduct the analysis below in terms of the latter variable. Let Ƶ denote total consumption expenditure measured in terms of traded goods, defined as:

By using the specific utility function (13) in equations (8)-(10) and making use of the definition (14), we can now rewrite the necessary conditions for the solution to the household’s problem as:

Equation (20) is derived by substituting equations (14) and (17) in (12). We have also defined the function V(Ƶ,e), given by:

where H = β[αlnα+(l-α)ln(l-α)] + (l-β)ln[(l-β)/rβ], and V( ) is obtained by substituting equations (15)-(17) in (13) and simplifying.

In addition to the household sector, the economy also contains a government and a central bank. The government purchases both traded (gT) and nontraded (gN) goods, collects taxes, and receives interest from both its own stock of bonds (b_G) and that of the central bank (b_c). Any excess of receipts over expenditures is devoted to bond accumulation. Thus the government’s budget constraint is given by:

We will suppose that initially the composition of the government’s expenditures mirrors that of the private sector. That is,

where:

In addition, we will assume that b_c, which represents the central bank’s stock of foreign exchange reserves, is positive, but that b_G is zero, so the government is neither a net creditor nor debtor.

Turning to the central bank, its only role is to maintain the exchange rate. It does so by exchanging domestic currency for foreign-exchange denominated bonds at an announced par value. Its balance sheet is given by:

where ${\mathrm{P}}_{\mathrm{T}}^{\mathrm{*}}$ is the foreign-currency price of traded goods. The second term, n, represents the cumulative value of the devaluation profits of the central bank and constitutes the bank’s net worth, since all operating profits (interest receipts on bonds) are, as indicated previously, transferred to the government. For simplicity, we assume that there was no previous devaluation; therefore n is zero initially.

Our model is closed by the specification of the conditions of production. We assume that the economy is continually at full employment and possesses a concave transformation frontier between traded and non-traded goods. It operates at the point of tangency between that frontier and a straight line with slope ^-_e^-1. This implies that output of traded and nontraded goods is given by:

with ${\mathrm{Y}}_{\mathrm{T}}^{\mathrm{\prime }}\mathrm{+}{\mathrm{e}}^{\mathrm{-}\mathrm{1}}Y\stackrel{\prime }{N}\mathrm{=}\mathrm{0}$. Since all income from production is received by the private sector, we also have:

with ${\mathrm{Y}}^{\mathrm{\prime }}\mathrm{=}{\mathrm{e}}^{\mathrm{\alpha }\mathrm{-}\mathrm{1}}\mathrm{(}{\mathrm{\alpha }\mathrm{Y}}_{\mathrm{T}}\mathrm{-}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\alpha }\mathrm{)}{\mathrm{Y}}_{\mathrm{N}}{\mathrm{e}}^{\mathrm{-}\mathrm{1}}\mathrm{)}{}_{<}{}^{>}0$. Output of nontraded goods must clear the market for such goods:

In the case of traded goods, however, any excess of production over domestic consumption can be sold abroad, implying a trade balance of y_T - c_T - g_T and a current account balance (ca), measured in terms of traded goods, of:

where:

is the economy’s net international creditor position.

2. Solution

To solve this model, it is necessary to specify the nature of fiscal policy. We assume initially that τ is set exogenously, and that g is set such that, given the initial value of b_c + b_G, the budget is in balance—i.e., ${\dot{b}}_G=0$. The composition of g is determined by equations (23) in the initial steady state. In response to shocks, however, we assume—unless otherwise stated—that g_N remains fixed at its initial value while g_T is adjusted to maintain budget balance. As we show below, this is an important assumption.

Proceeding to the solution of the model, we first find the value of e that clears the market for nontraded goods, given Ƶ and g_N:

Substituting equation (27) in the household’s saving equation (20) and then using (31), we can express private asset accumulation as a function of private expenditure, real private financial wealth, and the exogenous fiscal policy variables:

Next we use equation (18) to express λ as a function of private expenditure Ƶ, real private financial wealth a and the fiscal variables τ and g_N:

By differentiating equation (33) with respect to time and solving for $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}$ we have:

with:

Equations (32) and (34) represent a two-equation system of differential equations with a single predetermined variable (a). The system has a determinant Ω given by:

Thus the steady-state equilibrium defined by $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ is saddlepoint stable. The system is depicted graphically in Figure 1. The condition $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\mathrm{0}$ traces out a locus in a-Ƶ space with slope:

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Steady-State Equilibrium and Devaluation Dynamics

Citation: IMF Working Papers 1989, 001; 10.5089/9781451931235.001.A001

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Since Ƶ₁ = 0, however, the condition $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ is only satisfied at a unique value of a—i.e., the locus $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ is vertical. The stable arm of this system, depicted as SS in Figure 1, has slope given by:

where q is the stable (negative) root of the system. As can be verified from the direction of the arrows in Figure 1, the slope of SS must exceed that of the $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\mathrm{0}$ locus. It can readily be seen that for any given value of private wealth, say a₀, the solution to the household’s maximization problem must place private consumption on the stable path SS (say at point B, where consumption is Ƶ₀). This is because values of Ƶ above SS would place the household on a divergent path that eventually moves into the second quadrant, where a_t is negative and constraint (6) is violated. ^7/ Such paths are obviously not feasible. Paths that begin below SS, on the other hand, cannot be optimal, since such paths must always remain below SS, implying permanently lower consumption than is attainable along SS.

With the solution to the model in hand, we can now proceed to study the effects of devaluation as well as of fiscal policy in this model.

1. The standard case

Devaluation operates through familiar monetary channels in this model. Using equation (4) and the definition of the real exchange rate, private financial wealth can be expressed as:

Since a devaluation raises P_T in the same proportion as the change in the exchange rate, we have:

Devaluation lowers the real value of the pre-devaluation money stock and thus reduces private wealth. In terms of Figure 1, private wealth falls from a* to a⁰. This reduction in private wealth reduces household expenditure from Ƶ* to Ƶ₀, located on the saddle path SS. The dynamics of adjustment to the higher price of foreign exchange are straightforward. The reduction of household spending means that households begin to save. As they do so, household wealth and spending both rise over time along SS, until they regain their original values (a*,Ƶ*) at A. At point A, all real variables in the system have returned to their original values, except for certain items in the central bank’s balance sheet and the government’s budget. In terms of equation (25), the central bank’s balance sheet now shows a higher value of b_c, offset on the right-hand side by a larger value of net worth in the form of devaluation profits. In the government’s budget, transfers from the central bank earned in the form of interest receipts on the higher value of b_c finance a permanently higher value of g_T.

We now examine the dynamics of adjustment in more detail by investigating the responses of other macroeconomic variables.

By substituting equation (17) into (25) and differentiating with respect to time, the evolution of foreign exchange reserves over time can be shown to be given by:

Since $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}$ is positive during the period following devaluation, devaluation therefore induces a balance of payments surplus during the transition to the new steady state. By summing the budget constraints of the private sector (20) and the government (22), using the definition of the current account given by (29) and imposing the market-clearing condition for non-traded goods (28), we have:

since $\stackrel{\mathrm{\cdot }}{{\mathrm{b}}_{\mathrm{G}}}\mathrm{=}\mathrm{0}$ by assumption. Since a is rising over the segment BA in Figure 1, this means that devaluation induces a current account as well as a balance of payments surplus. Whether the capital account behaves similarly is more problematic. Using (36) and (37) the capital account surplus can be expressed as:

Since $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{>}\mathrm{0}$, the capital account will be in surplus $\mathrm{(}\stackrel{\mathrm{\cdot }}{{\mathrm{b}}_{\mathrm{p}}}\mathrm{<}\mathrm{0}\mathrm{)}$ if the saddle path SS is sufficiently steep. While it is possible to derive a set of sufficient conditions on the parameters of the model that ensure this result, devaluation need not in general induce a capital account surplus in this model.

From equation (31), we know that the nominal devaluation will result in a real exchange rate depreciation on impact. Since this depreciation must be reversed over time, the path BA must be characterized by continuous inflation. However, because $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}$ decreases as A is approached, the rate of inflation is highest immediately following the devaluation, and then decreases over time. With the nominal interest rate fixed by the assumed perfect mobility of capital, the domestic real interest rate must initially fall following the devaluation, then gradually recover its original level as inflation subsides during the return to the steady state.

Consumption of traded goods is proportional to total consumption (equation (15)), so the reduction in total consumption is matched by a similar reduction in consumption of traded goods, followed by gradual recovery. From equation (31), the elasticity of e with respect to Ƶ is given by -eƵe^-1Ƶ < 1, so consumption of nontraded goods must fall as well, though less than in proportion to Ƶ (equation (16)). Thus consumers shift the composition of their spending from traded to nontraded goods, but reduce total spending on both. Since private consumption of nontraded goods has fallen while government purchases of such goods are unchanged, output of nontraded goods must fall as well. This is at least partially offset, however, by increased output of traded goods due to the real exchange depreciation (equation (26a)).

In spite of the reduction in output of nontraded goods and increase in output of traded goods, it is possible to derive the effects of devaluation on total real output measured in terms of the consumption bundle. After some manipulation, the derivative of the function y in equation (27) can be reduced to:

Since b = b_p+b_c = a + (b_c-me^-α), where (b_c -me^-α) is the central bank’s net worth, and since both of these terms have been taken to be positive, b is also positive (i.e., the country in question is a net international creditor). Thus y’ is negative. That is, a real depreciation reduces the value of real output measured in terms of the consumption bundle. Since the real exchange rate e depreciates on impact, real output must fall. In this sense, then, devaluation is contractionary on impact. As the real exchange rate appreciates back to its original level, real output recovers its initial value over time. The basic results of this section are summarized in Figure 2, which illustrates the dynamic behavior of Ƶ, b_c+b_G, and e in response to a devaluation. The paths of both types of consumption as well as of real output can be inferred from the third panel of Figure 2, since these variables can all be expressed as functions of the real exchange rate.

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Dynamic Effects of Devaluation on Private Spending, Reserves, and the Real Exchange Rate

Citation: IMF Working Papers 1989, 001; 10.5089/9781451931235.001.A001

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2. The role of fiscal policy rules

What is the role of fiscal policy in these results? Notice that the conditions for Ricardian equivalence are present in our model. Future taxes are perfectly anticipated over an infinite horizon by consumers who can borrow and lend at the same rate as the government. In spite of this, devaluation is effective; an apparently paradoxical result, since devaluation simply transfers real wealth from the private to the public sectors. Presumably Ricardian consumers should pierce the government veil, leaving their behavior unaffected. In particular, though the private sector is poorer, the enhanced resources of the public sector should reduce future tax liabilities by an equivalent amount, leaving households no worse off than before. The reason this is not the case is the nature of the fiscal policy rule in effect. The government is assumed to devote its increased resources to the purchase of traded goods, not to tax reductions. Thus the private sector is indeed poorer after the devaluation. Suppose we had assumed instead that the government fixed its total spending measured in terms of traded goods (g) and altered taxes so as to keep its budget in balance. Then, setting $\stackrel{\mathrm{\cdot }}{{\mathrm{b}}_{\mathrm{G}}}\mathrm{=}\mathrm{0}$ in (22), using the resulting equation to solve for τe^-α, substituting in equation (20), and using (17) the private sector’s budget constraint would become:

which is unaffected by changes in a. Moreover, since $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\stackrel{\mathrm{\cdot }}{\mathrm{b}}$ in this case, the intertemporal budget constraint (6) now becomes:

which is also unaffected by changes in a. Thus household behavior is unaffected by changes in a and the economy never leaves its steady-state configuration. Devaluation is effective in the sense that bonds are transferred from the private to the public sector, but the usual monetary dynamics are absent.

1. Temporary reduction in government spending on nontraded goods

We continue to assume, as before, that the reduction in spending is motivated by a desire on the part of the public sector to increase its steady-state stock of bonds. In this sub-section, the government is assumed to achieve this end by temporarily reducing spending on nontraded goods, keeping spending on traded goods constant. When the original level of g_N is restored, g_T is raised to consume the additional interest receipts and maintain budget balance. Notice that, since the target is (b_c + b_G), g_N may be returned to its original value before this target is reached, if b_c remains short of its steady-state value. This will become clear below. There is, of course, no unique path of g_N that will achieve the desired result. The authorities presumably face a choice between large reductions in g_N lasting for a short time and smaller but more prolonged budget cuts. The analysis below will be for an arbitrary path of g_N that achieves the steady-state outcome.

A reduction in g_N leaves the $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ locus unaffected (since Ƶ_gN = 0), but causes the $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\mathrm{0}$ locus to shift downwards by the amount:

The new configuration is depicted in Figure 3. Although, as we shall show below, the reduction in g_N lowers the domestic price level, it has no effect on a, which is measured in units of traded goods. In the absence of devaluation, the price of these goods is unchanged. The path followed by Ƶ is governed by the following considerations: since private agents have perfect foresight, the reduction in g_N is known to be temporary. Thus it will be taken into account when agents formulate their optimal plans. One characteristic of such plans is that the costate variable λ not be anticipated to undergo discrete changes. ^10/ Since λ is a function of both g_N and Ƶ (equation (33)), this means that when g_N returns to its original value, agents must plan compensating changes in Ƶ that leave λ unchanged. From equation (33):

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Effects of a Temporary Reduction in Spending on Nontraded Goods

Citation: IMF Working Papers 1989, 001; 10.5089/9781451931235.001.A001

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Thus an increase in Ƶ must accompany the restoration of g_N to its original level. It follows, in terms of Figure 3, that the initial value of Ƶ must place it on a divergent path with respect to the steady state at B with the property that when g_N is restored to its original value, Ƶ can jump upwards to the saddle path SS. This rules out paths that begin above point A (such as that at G) and implies that Ƶ must fall on impact. Furthermore, comparison of equations (39) and (40) reveals that the upward displacement of Ƶ that must accompany the restoration of spending to its original level must fall short of the downward displacement of the $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\mathrm{0}$ locus. This rules out paths that begin below B and implies that Ƶ falls initially to a point on the segment AB such as C. From then on, it follows the divergent path CD and jumps to the point E on SS when the spending cut is rescinded. Finally, the economy returns to the original steady state along EA.

The dynamics of the balance of payments are quite different in this case from the devaluation case. First, the reduction in g_N leads to a discrete capital outflow as the lower value of Ƶ causes the private sector to shift its portfolio from money to foreign bonds. After this once-for-all portfolio adjustment, the balance of payments is in equilibrium at C, from which it gradually moves into surplus as Ƶ rises. The increase in gjq is accompanied by a discrete capital inflow in response to the increase in Ƶ from D to E in Figure 3. Finally, over the range EA the overall balance of payments exhibits a gradually decreasing surplus. Notice that, since $\stackrel{\mathrm{\cdot }}{\mathrm{a}}$ is negative over the range CD, the private sector runs a current account deficit over this range. Since its overall balance of payments is positive, however, its capital account must be in surplus. The government, on the other hand, runs offsetting current account surpluses and capital account deficits, with a balanced overall account, over this range. As the economy moves to the segment EA, the behavior of its external accounts resembles the devaluation case.

Turning to domestic variables, notice that, as in the devaluation case, equation (31) implies that the real exchange rate will depreciate on impact (this time because both Ƶ and g_N have fallen). The range CD is characterized by accelerating domestic inflation as Ƶ moves away from the $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ locus. The domestic price level jumps discretely (and e appreciates) when government spending is restored at point D. Over the range EA domestic inflation continues, but at a gradually decreasing rate. This behavior of the price level implies that the domestic real interest rate begins to fall gradually on impact, reaches a minimum at point D when government spending rises, then gradually returns to its original level over the range EA.

Qualitatively, the behavior of private consumption of both kinds of goods, as well as of real output, resemble the devaluation case, except for the discontinuous increase in consumption—and also in real output—when the government spending cut is rescinded. The key to this is the behavior of the real exchange rate, which determines the behavior of c_T, c_N and y.

2. Temporary tax increase

The dynamics of a temporary tax increase prove to be quite different. Notice first from equations (32) and (34) that an increase in τ shifts both the $\stackrel{\mathrm{\cdot }}{\mathrm{a}}\mathrm{=}\mathrm{0}$ and $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ loci by:

Also, from equation (33), when the tax increase expires, the costate variable λ will avoid a discontinuous jump if Ƶ responds according to:

Thus, Ƶ and τ must move in the same direction when the tax increase is removed.

These considerations govern the dynamics of the economy in response to a temporary tax increase. The paths of a and Ƶ are depicted in Figure 4. If the tax increase is sufficiently prolonged, consumer spending Ƶ will decrease on impact to a point like D on the segment AC. ^11/ C is the point at which $\stackrel{\mathrm{\cdot }}{\mathrm{Ƶ}}\mathrm{=}\mathrm{0}$ intersects the saddle path S’S’ through the steady-state B that would be associated with a permanent tax increase. Points below C are ruled out because Ƶ would remain below S’ S’ from such an initial point, requiring a later increase to reach SS. This is inconsistent with dƵ/dτ|λ = λ > 0. From point D, private consumption rises temporarily, eventually overshooting its steady-state value Ƶ*. Since consumption is initially depressed, private wealth increases during the initial portion of this path. As spending continues to rise, however, the private sector begins to dissave, eventually more than depleting the additional assets accumulated. Finally, when the tax increase is revoked, spending undergoes a discrete fall, to the point G on SS. ^12/ Over the remainder of the trajectory, wealth and spending rise together.

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Effects of a Temporary Tax Increase

Citation: IMF Working Papers 1989, 001; 10.5089/9781451931235.001.A001

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In contrast to both devaluation and a spending cut, in the case of a tax increase the paths of private expenditure and wealth can be nonmonotonic. It is not surprising, therefore, that the behavior of other macroeconomic variables is also more complicated in this case. The balance of payments, for example, exhibits a gradually increasing surplus over the range DE, in which the current account is in surplus while the capital account may be in surplus or deficit. Over the range EF, the private sector begins to run a current account deficit (due to a rising level of consumption), though one which falls short of its capital account surplus, while the public sector runs offsetting current account surpluses and capital account deficits. This situation comes to an abrupt end at point F when the tax increase is rescinded. Finally, over the range GA the overall balance is again in surplus, though this surplus gradually declines as the steady state is approached at A.

Unlike in the previous two exercises, in the case of a tax increase the real exchange rate overshoots its equilibrium value after an initial discrete depreciation. It appreciates continuously over the range DF as the domestic rate of inflation gradually increases, peaking at point F before undergoing a second discrete depreciation at that point. From G on, it returns to its original value through a gradual appreciation. The behavior of consumption of both types of goods, as well as of real output, can be inferred directly from that of the real exchange rate.

The results of this section are summarized in Figure 5. Notice that both the relative positions of the times t₁ and t₂ at which the expenditure cut and tax increase respectively are rescinded, as well as the relative initial (time t₀) values of the variables illustrated in the figure, are arbitrary, since both depend on the size of the spending cut or tax increase.

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Dynamic Effects of Temporary Fiscal Measures on Private Spending, Reserves, and the Real Exchange Rate

Citation: IMF Working Papers 1989, 001; 10.5089/9781451931235.001.A001

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