This chapter examines the dynamics of adjustment to changes in aid inflows in a stochastic general equilibrium model of a small and open economy. In particular, it explores the dynamic response to changes in aid under various combinations of spending and absorption policies.

The experiments exploit a theoretical model of a small and open economy that features monopolistic competition (Blanchard and Kiyotaki, 1987), sluggish price adjustment (Calvo, 1983), and the presence of traded and nontraded goods. When applied to open economies, this class of models is also known as the new open economy macroeconomics (NOEM) literature (Obstfeld and Rogoff, 1995).

The model reflects key features of low-income countries, in contrast to the usual application to emerging or developed economies. Low-income countries typically have less access to foreign borrowing and lower levels of foreign direct investment. To capture this feature, a closed capital account is assumed. As a result, the exchange rate is mostly determined by external trade and central bank actions, rather than by international capital flows, and the only source of financing for the trade deficit is the sale of aid dollars by the central bank. The authorities are assumed to target a constant nominal money supply, except insofar as spending differs from absorption. The exchange rate regime is thus a float. Prior to the aid shock, inflation and the exchange rate are stable.

The model is designed to be as simple as possible while still capturing the basic intuition of Chapter II in a micro-founded general equilibrium dynamic framework. As a result, much of potential interest is not modeled. First, there are no bonds, so that the option of sterilizing cannot be discussed. More broadly, the set of policy choices open to the authorities is limited. Second, given the short-term focus of the model, the stock of physical capital is fixed. Third, productivity is also fixed. In particular, a higher level of exports does not raise aggregate productivity through positive spillovers of “know-how” or through a dynamic “learning by doing” process. Fourth, the social value of government spending is ignored. In reality, foreign aid is often provided with the hope that government expenditures will lead to higher private sector productivity. To the extent that these gains are realized with lags, they should not affect our short-term analysis. Fifth, foreign currency is only used as a medium of exchange (to purchase imports) and not as a means of saving.^{1} Finally, the model is not calibrated to any particular country but with broadly sensible parameter values.

This model is a first effort toward a more complete characterization of the monetary policy challenge facing aid-dependent low-income countries. The list of omissions thus also represents an agenda for future work. The model is nonetheless sufficiently rich to confirm and illustrate many of the basic points from the previous chapters, as well as assess some questions that the intuition provided there cannot readily address:

Spending without absorption must be financed domestically, here by seigniorage. It raises government spending and lowers private consumption. Aid that is associated with this response results in macroeconomic instability, specifically higher inflation. The real exchange rate tends to appreciate in the long run but is more likely to depreciate in the short run (but neither of these exchange rate results are robust to plausible alternative parameter values).

An absorb-and-don’t-spend response appreciates the real exchange rate and raises private consumption. The model is too simple to capture the channel by which lower public debt (or lower inflation) encourages private investment and thus growth.

An absorb-and-spend response increases net imports. The real exchange rate is likely to appreciate, in the short and long run, by more than the spend-and-don’t-absorb response.

The next section describes the model and policy choices, while the rest of the chapter presents the implications of aid shocks and how these depend on different policy choices with respect to absorption and spending.

## The Model

The economy has two sectors, producing nontradable and exportable goods.^{2} Nontradable goods producers are monopolistic competitors. Prices in this sector exhibit sluggish adjustment as in Calvo (1983).^{3} Exportable goods producers, on the other hand, take the foreign currency value of their price as given by the world market. Hence, in the absence of changes in world prices, the domestic currency value of traded goods prices is entirely determined by the nominal exchange rate. Foreign demand for domestic exports is infinitely elastic, i.e., exporters can sell any amount at the given international price. However, their output is restricted by the diminishing marginal product of labor.

Domestic households (consumers) and the government purchase nontraded goods, domestically-produced exportables, and imported goods, although their preferences over various types of goods may differ. Household consumption is financed by their labor income, dividends from firm ownership, and returns on financial assets.^{4} Consumers hold two financial assets: domestic currency and nominal nonstate-contingent bonds that are traded only among consumers. Because all households are identical, the equilibrium bond holding is always zero. The (negative) return on money holdings is equal to price inflation.

The capital account is closed, i.e., the country’s residents and the government neither lend nor borrow from abroad. Therefore, in the absence of foreign exchange interventions by the central bank, the country’s trade balance must always be zero for the foreign exchange market to clear.^{5} Alternatively, in any period imports can only exceed exports by the amount of foreign exchange sold by the central bank.

Government spending is financed by labor income taxes and by foreign aid in the form of grants.^{6} Upon the receipt of aid, the government places aid dollars in the central bank and receives a domestic currency equivalent on its account. Next, it decides on the fraction of the aid money to be spent, while the central bank determines whether to add the aid dollars to foreign exchange reserves or whether to sell them on the market. The government’s budget constraint can be expressed as follows:

where *P*_{gt} is the price of a unit of the government’s consumption basket, *G*_{t} is the number of baskets purchased by the government, or the real value of government spending, τ is the tax rate, *W*_{t} is nominal hourly wage, *L*_{t} stands for aggregate labor hours, or aggregate employment, *t*, *S*_{t} is the nominal exchange rate, and γ is one of our key policy variables, measuring the fraction of aid spent by the government. The nominal money supply evolves as follows:

where *M*_{t} is the nominal money supply in period *t*, and θ is the fraction of aid dollars sold on the market by the central bank. Equation (2) states that the net effect of aid on the money supply is jointly determined by the government’s decision on spending (through γ) and the central bank’s decision on absorption/sterilization (θ). In the absence of absorption (θ = 0), higher government spending is entirely financed by money creation and, ultimately, inflation. On the other hand, full foreign exchange sterilization (θ = 1) could generate a real appreciation that would accommodate higher domestic aggregate demand through an increase in net imports.^{7} The experiments presented in the next section study the effects of random changes in

The exact parameter values used in the experiments are presented in Appendix 8.1.^{8} The assumed values imply that in the steady-state government spending is roughly 20 percent of GDP; nontraded goods firms’ prices are set with a 20 percent markup over marginal costs (an elasticity of substitution between goods of 6); consumers treat tradables and nontradables as equally important, whereas the government in its spending basket places a 90 percent weight on nontradables and a 10 percent weight on domestically-produced tradables and does not purchase imports; and aid finances about 8 percent of government spending. In dynamic simulations, each nontraded goods firm is assumed to face a 50 percent probability of adjusting its price each period. Finally, the experiments examine the effects of a 30 percent increase in aid (0.5 percent of the steady-state GDP). The possible effects of different parameter values are discussed in the context of each experiment.

## Anatomy of Macroeconomic Adjustment to Aid Inflows

### Spend and Don’t Absorb

A well-known property of New Keynesian models is that the real effects of permanent increases in the nominal money supply are short-lived because they are driven by nominal rigidities that disappear in the long run. In contrast, a permanent shock to aid inflows, when aid is spent but not absorbed, is in essence an amalgam of a pure monetary expansion because higher government spending is financed by liquidity injection, and a shock to the structure of aggregate demand when the government’s spending habits differ from those of the private sector.^{9} When, as here, the government strongly favors nontraded goods over tradables, such an increase in spending would have two opposing effects on the real exchange rate. A monetary expansion tends to generate nominal and, in the periods of incomplete price adjustment, real depreciation. On the other hand, a higher relative demand for nontraded goods would tend to raise their prices relative to prices of traded goods and result in real appreciation. If the aid shock is permanent, then these two effects compete in the short run, but only the second effect persists. A slow pace of domestic price adjustment may thus result in a short-run depreciation followed by a permanent real appreciation. Such overshooting is likely to lead to overshooting of other variables and implies that aid inflows may generate larger short-run volatility than is implied by the new long-run equilibrium. Therefore, distinguishing between the short-term and long-term effects of shocks to aid is important in this scenario.^{10}

Figures 8.1 and 8.2 present the effects of, respectively, permanent and temporary increases in the level of aid in the model. The first panel displays the different short- and long-run effects on the real exchange rate discussed above. In the short run, while nontraded goods prices have not completely adjusted to higher demand, the real exchange rate depreciates. As prices adjust, the effects of higher government spending push the real exchange rate toward appreciation. An interesting implication illustrated by Figure 8.2 is that if price rigidities are strong enough and the aid increase is temporary and not very persistent, then real appreciation may be very small or may never materialize, since it would happen late in the adjustment process. This may in part explain the real depreciation observed in some of the case studies.^{11}

**Spend and Don’t Absorb, Temporary Shock**

**Spend and Don’t Absorb, Temporary Shock**

**Spend and Don’t Absorb, Temporary Shock**

The real value of government spending in Figure 8.1 also overshoots its long-run equilibrium value due to sluggish price adjustment. Inflation rises rapidly due to the instant adjustment of the nominal exchange rate and the prices of traded goods and gradually returns to a new higher level. The effects on private consumption can be explained by the behavior of the three components of household income. First, a permanent increase in the inflation tax, which finances government spending, permanently lowers household income. Second, wage income rises as consumers try to compensate for a higher inflation tax and increased labor supply. Wage income rises more in the short run when real wages are higher due to price rigidity. Third, firms’ profits (households’ dividend income) fall in the short run, driven by lower profits of firms with unadjusted prices, but rise in the long run due to higher aggregate demand. The offsetting effects of higher wage and dividend income imply that consumption^{12} falls gradually due to the overshooting of real wages, and that in the long run it falls by less than the amount of inflation tax. This results in a positive net long-term effect on domestic aggregate demand.^{13}

The short-run shift of demand toward nontraded goods combined with real depreciation depresses aggregate demand for traded goods. On the other hand, as the real exchange rate appreciates over time, households increasingly substitute tradable goods for nontraded goods produced domestically and imported. Despite lower aggregate private consumption, the effect on the net demand for tradable goods is ambiguous, and depends on the relative strength of the substitution and income effects in consumer demand. Our particular parameter values lead to an unchanged net demand (and supply) for tradables as shown in Figure 8.1. Note that if the elasticity of substitution between the tradables and nontraded goods and labor supply elasticity were very low, then output in the traded goods sector would most likely decline in the long run.

Finally, note that in the absence of aid absorption the trade balance is unchanged, i.e., aid does not lead to any resource transfer from abroad. The fiscal expansion examined here could in principle be undertaken even without aid by borrowing directly from the central bank. The only difference is that central bank reserves are higher in the scenario with aid.

### Absorb and Don’t Spend

A sale of aid dollars (θ = 1) without an associated increase in government spending (γ = 0) generates a real resource transfer through higher net imports, but produces a monetary and real tightening. The pattern of adjustment to a permanent shock to aid is displayed in Figure 8.3.^{14} A monetary contraction reduces inflation, increasing consumers’ real income. A real appreciation in this case is unambiguous, although it is larger in the short run when some of the nontraded goods prices are fixed. Higher income provides an incentive for consumers to increase consumption and to supply less labor.^{15} Aggregate output and government revenues therefore decline, and more so in the short run. Note that this result differs from the analysis in Chapter I, which argues that an absorb-and-don’t-spend policy can be used to crowd in private sector investment (for example, by using aid to retire domestic government debt) and thereby stimulate growth. The model presented here, for considerations of tractability, does not model government debt or private investment behavior, both of which are important considerations for an absorb-and-don’t-spend strategy.

**Absorb and Don’t Spend, Permanent Shock**

**Absorb and Don’t Spend, Permanent Shock**

**Absorb and Don’t Spend, Permanent Shock**

Real exchange rate overshooting generates quite different responses of sectoral output. In the short run, when the appreciation is large and government spending is low, nontraded output declines while traded goods output expands. On the other hand, as the real exchange rate returns to the long-run level, output in both sectors declines. Nevertheless, household consumption of all types of goods rises at the expense of lower government spending and lower exports.

### Absorb and Spend

When aid is spent and absorbed, the nominal money supply remains constant (see equation 2). Hence, the main effect on the real exchange rate comes from higher government spending, which explains the absence of exchange rate overshooting in the first panel of Figure 8.4; instead, the real exchange rate gradually appreciates as nontraded goods prices adjust to the new demand structure. The long-run real appreciation is higher than in the spend-and-don’t-absorb case.

In the absence of an inflation tax, household income is affected by wage income and profits. Profits are mostly driven by nontraded goods producers, whose profits rise significantly due to higher government demand and firms’ monopoly power. Wages rise in the short run, due to a large output response in the non-traded sector, but fall in the long run. Lower employment in the long run results from the assumption of identically diminishing marginal products of labor in both sectors, combined with monopolistic competition in the nontraded goods sector. These assumptions imply that in the steady state, output is lower and marginal productivity of labor is higher in the nontraded sector. Therefore, a given expansion in the nontraded sector would require less labor resources than the same contraction in the traded sector. Alternatively, a given amount of labor resources freed as a result of a contraction of traded goods output would generate higher output in the nontraded goods sector. In the new long-run equilibrium, a mixture of these two effects is observed: lower aggregate employment and higher aggregate output. This outcome is not robust to various plausible alternative parameter assumptions. A lower assumed monopoly power in the nontraded sector, absence of perfect competition in the traded sector, and different sectoral technologies can nullify the implication that a demand-induced reallocation of resources toward non-traded goods improves aggregate efficiency.^{16} In any case, the increase in output in Figure 8.4 (about 0.02 percent of GDP) is negligible compared with the size of the aid shock (0.5 percent of GDP).

## Discussion of the Appropriate Policy Response

Even in this simple framework, countries may face difficult choices in responding to aid inflows, because each policy scenario has potentially negative effects on at least some variables. The model can be used to show how the expected utility of consumers depends on the nature of the policy response to aid shocks. However, it lacks too many features, notably investment and any explicit benefit to government spending, to provide much direct insight on this basis.^{17} But in anticipation of a more fully worked-out model, the results presented so far, applied somewhat loosely, can shed light on the relative merits of the different choices with respect to absorbing and spending.

It is useful to compare two responses: that which would be optimal given overall policy preferences of the authorities and that which would be the outcome of separate decisions by the government (for fiscal policy) and an independent central bank (for monetary policy). A useful starting point is to note that a spend-and-don’t absorb response would not likely be the optimal choice of a single authority. This reflects the idea that any benefits of domestically-financed fiscal expansions have already been exploited or do not arrive with aid shocks, not that they do not exist in general.^{18} To absorb and not spend is likely to be suboptimal when the initial rate of inflation is appropriate, but may be desirable in a highly inflationary environment. Thus, in the absence of high initial inflation, the two optimal long-run responses are likely to be spend and absorb or neither spend nor absorb. The long-run trade-off is between the potentially negative effects on the export sector and the stimulus to long-run growth provided by government investments.^{19}

It is possible that separate, uncoordinated decisions by the central bank and the government would result in a suboptimal outcome. The objectives of central banks typically involve maintaining macroeconomic stability through low and stable inflation. Another common objective is to maintain competitiveness by avoiding real exchange rate appreciation. The fiscal authorities share similar objectives, but it is reasonable to suppose that they place more weight on the benefits associated with government spending, less on macroeconomic stability, and less on the costs of exchange rate appreciation.

Consider first the decision of the ministry of finance. In general, its choice may depend on whether it expects the central bank to absorb or not. However, it is plausible that the ministry will choose to spend whatever the decision of the central bank with respect to absorption, as long as it considers that the benefits of spending outweigh either the costs of appreciation in the case of absorption or the costs of higher inflation in the case of nonabsorption.

The central bank thus may have to decide how to respond to the decision to spend. The central bank faces a trade-off. Absorption (i.e., the sale of aid dollars) avoids a rise in inflation and results in much smaller short-term volatility of other variables; however, it also results in a more appreciated real exchange rate (Figures 8.1 and 8.4). When it is not too concerned about real appreciation, the central bank will choose to absorb. If the central bank sufficiently dislikes real exchange rate appreciations, however, then it will choose not to absorb.

In sum, a central bank mostly concerned about real appreciation and a ministry of finance mostly concerned about spending might separately choose policies that result in a spend-and-don’t-absorb response, despite the fact that together they would not choose such an outcome. The case studies suggest that this situation may be common.

This analysis of policy responses is only suggestive. A next step would be to write down well-defined preferences for the two institutions and derive the noncooperative and coordinated equilibria. More generally, the model presented here abstracts from several important considerations. A more complete version of the model, calibrated to a specific country case, is the subject of ongoing work.

## Appendix 8.1. Model Description

This appendix presents the details of the model used in the experiments.

### Consumers

Each consumer maximizes life-time utility that depends on consumption, real money balances, and labor:^{20}

where *C*_{t} is private consumption expressed in units of domestic consumption baskets (to be defined below), *L*_{t} is their labor effort. The consumers’ budget constraint is given by

where *W*_{t} is the nominal wage, and *P*_{ct} and employing smaller case letters to denote real values of nominal variables, we can rewrite the budget constraint in real terms^{21} as follows:

Maximizing (A.1) subject to (A.2) yields the following first order conditions:

where *u*_{c}, *u*_{m}, *u*_{L} denote partial derivatives of the utility function with respect to consumption, real money balances, and labor, respectively.

### Consumption Baskets and Demands and Price Indices

The three types of goods traded in the economy are indexed by superscript (^{N}) for nontradables, (^{e}) for exportables, and (^{F}) for foreign imported goods. Consumption sub-baskets of each type of goods aggregate consumption of individual products using a constant elasticity of substitution (CES) function:

The total consumption basket of the consumers combines the three sub-baskets:

where *n*_{1} + *n*_{2} + *n*_{3} = 1. Here, λ and ω represent elasticity of substitution between various types of goods.^{22} The government also spends money on goods, but has a different consumption basket:

where *v*_{1} + *v*_{2} + *v*_{3} = 1. Varying *v*_{i}’s stresses differences in government’s preferences over different varieties of goods.

Price indices associated with each basket of goods follow similar notation and are defined as the minimum nominal cost of buying one unit of each (sub-) basket. They are given by

The law of one price (LOP) holds in the tradables sector. Furthermore, the foreign price is normalized to unity *S*_{t} is the nominal exchange rate and ^{23}

Price indices associated with the baskets of consumers and the government, respectively, are

Dividing both sides of these equations by *P*_{ct} provides an expression for “real” prices, or prices in terms of the private consumption basket. Using the LOP assumption

Demand functions associated with an individual good *i* within each category can be expressed as follows:

Finally, demands for aggregate sub-baskets are given by:

### Nontraded Goods Sector

The production function of the nontraded goods sector producers depends on labor (capital is assumed to be fixed) and is assumed to exhibit diminishing marginal returns.

where *Q*_{n} is aggregate productivity in the nontraded sector (held fixed in the baseline model.) Firms follow Calvo pricing; i.e., in each period every firm faces a fixed probability (1 – *q*) of changing its price. By the law of large numbers in equilibrium, *q* fraction of firms keep their prices fixed, whereas (1 – *q*) fraction of the prices adjust.

### Price-Setting Problem

Every individual firm *i* faces the following demand:

Combining the last two equations, the firm’s labor demand can be expressed as follows:

When provided a chance to re-set its (nominal) price, firm *i* seeks to maximize expected profits (in real terms):

where *J*_{t+j} is the firm’s discount factor between dates *t* and *t* + *j*.^{24} The first order condition, describing the optimal price is given by:

Using *V*_{1t} and *V*_{2t} to denote the numerator and the denominator of the right-hand side of the previous equation is useful for the purpose of solving the model. Note that *V*_{1t} and *V*_{2t} can be expressed recursively as follows:

### Aggregate Prices, Output, and Labor Demand in the Nontraded Sector

Since only a (1–*q*) fraction of the firms change their prices every period, the nominal price of the nontraded goods sub-basket follows the following law of motion:

where *P*_{ct} yields the real price of the nontraded goods basket:

Aggregate output in the nontraded goods sector (in units of the total consumption basket) is given by aggregate demand for nontradables:

Finally, aggregate labor demand of the nontraded goods producers can be expressed as follows:

### Exportable Goods Sector

Exportable goods producers have a similar production function: ^{25}

which implies that optimal output and employment levels are given by

Before exporting, the exportable goods producer must satisfy domestic demand, which from (A.11) and (A.12) can be expressed as

Thus, exports are given by

### Government and the Central Bank

The government can finance its spending by taxing labor income or by using its aid proceeds:

where γ is the key variable measuring the fraction of aid spent by the government and ^{26}

The central bank’s budget constraint is given by

where θ is the fraction of aid dollars sold on the market by the central bank. Note that in the absence of alternative sterilization measures, the net effect of aid on the money supply is jointly determined by the government’s decision on spending (through γ) and the central bank’s decision on absorption (θ). Dividing both sides by *P _{ct}* yields the law of motion for the real money supply:

The unused portion of the aid is added to the forex reserves.

### Market Clearing

The labor market clearing condition is

Combining equations (A.2), (A.26), and (A.27), one can obtain the economy-wide resource constraint, or the current account equation:

The equation highlights that in the absence of foreign borrowing, the only way the economy can consume more than it produces is through the central bank’s sale of foreign exchange. It can also be interpreted as the foreign exchange market equilibrium. The left-hand side represents net imports (or demand for foreign exchange in excess of export revenues), whereas the right-hand side represents the supply of additional foreign exchange by the central bank.^{27}

### Equilibrium

For the purpose of convenience it is useful to summarize the system of equations that describes equilibrium at any point in time.

where

In the model discussed, the uncertainty comes from aid inflows, which follow an AR(1) process:

Parameter values, used in the experiments are as follows:

α | λ | q | χ | ε | κ | τ | ρ |

0.8 | 5 | 0.3 | 0.1 | 3 | 2 | 0.2 | 0.7 |

n_{1} | n_{2} | v_{1} | v_{2} | A¯ | β | σ | Q_{n} = Q_{e}. |

0.5 | 0.25 | 0.9 | 0.1 | 0.012 | 0.96 | 0.001 | 1 |

α | λ | q | χ | ε | κ | τ | ρ |

0.8 | 5 | 0.3 | 0.1 | 3 | 2 | 0.2 | 0.7 |

n_{1} | n_{2} | v_{1} | v_{2} | A¯ | β | σ | Q_{n} = Q_{e}. |

0.5 | 0.25 | 0.9 | 0.1 | 0.012 | 0.96 | 0.001 | 1 |

## Appendix 8.2. Solution Method

The system of equations from (A.30) to (A.44) describes a dynamic nonlinear system. The method of collocation is applied to solve the system.^{28} Commonly used linearization-based techniques can result in very inaccurate calculations of the solution to a nonlinear model, particularly when shocks are large.^{29} The method of collocation helps assess and ensure reasonable accuracy over the desired part of the state space.^{30}

Solving the above system is complicated by (1) the nonlinear structure of the model, and (2) uncertainty created by aid shocks. The first step in solving the model is to separate variables into three groups: actions, states, and expectational variables.

The *actions* are the contemporaneous endogenous variables, i.e., agents’ decisions and equilibrium outcomes. In our model these include consumption (C_{t}), interest rate (i_{t}), prices *m _{t}*), labor supply (

*L*), wage rate (

_{t}*w*), inflation (π

_{t}_{ct}), and government spending (

*G*):

_{t}*State* variables are those that are predetermined and are therefore taken by agents as given at the beginning of each period. Typically they include all exogenous variables and past values of endogenous variables. In the simplified version of the model, there are three such variables: level of aid, previous period money holding, and past price of nontraded goods:

The state vector describes the state of nature that agents face each period and on which they base their actions. It follows a law of motion:^{31}

The *expectational variables* are expressions under the expectation sign. They appear in three places in the system: the Euler equation (A.30), and the expressions for *V*_{1t} and *V*_{2t} which enter the pricing equation (A.33). Thus, the vector of expectational variables is three-dimensional:

Bringing all the variables in equations (A.30) to (A.44) to the left-hand side, the system can be represented as^{32}

The next step is to note that the solution of the model relates actions to states via some nonlinear function:

Also note that the expectational variables are functions of the next period’s states and actions. Hence, the current value of *z _{t}* is also a function of the states:

Then, using the law of motion for *sv _{t}*,

*z*

_{t+1}must be given by

Plugging this into (A.49):

The main element of the collocation method is to approximate *E*_{t}*f*(*g*(*x*_{t}, *sv*_{t}, *e*_{t+1})) by discretizing the state space and the stochastic distribution of the shocks. The procedure goes as follows. First, the true continuous distribution of *e*_{t} (using Gaussian Quadrature) is approximated with a discrete probability distribution with *m* realizations {*e*_{1}, …, *e*_{m}} and associated probabilities {*prob*_{1}, …, *prob*_{m}}. Second, the function *f*(·) is approximated with a linear combination of *n*-degree Chebyshev polynomials:^{33}

where φ_{j} (*sv _{t}*) are polynomial basis functions and

*c*

_{j}are unknown coefficients. Thus, for any the system of equations (A.53) can be written as

which is a deterministic system and can be solved for *x*_{t} given *sv*_{t} given the polynomial coefficients *c*_{j}, *j* = 1, …, *n*. The coefficients are solved for by discretizing the state space, i.e., by adopting *n* values (nodes) of the state vector along the state space, denote {*sv*_{1}, …, *sv*_{n}}. Typically the nodes are chosen in the neighborhood of the steady state where the system is expected to evolve. The solution algorithm is as follows:

First, guess the values of

*c*_{j},*j*= 1, …,*n*and solve the system (A.55) at all nodes {*sv*_{1}, …,*sv*_{n}} given*c*_{j}’s.Second, given the first step solution, record the contemporaneous values of

*z*_{t}at each of the*n*nodes.Third, update the values of

*c*_{j},*j*= 1, …,*n*by imposing the adopted approximation:${z}_{k}=f\left({\mathit{sv}}_{k}\right)=\underset{j=1}{\overset{n}{\mathrm{\Sigma}}}{c}_{j}{\phi}_{j}\left({\mathit{sv}}_{k}\right),\forall k=1,\mathrm{.}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\mathrm{.}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\mathrm{.},n$(note that these are

*n*linear equations in*n*unknown coefficients*c*_{j}, so updating*c*_{j}’s is straightforward.)Finally, return to step 1 and iterate until convergence in

*c*_{j}’s is achieved.

^{}1

Buffie and others (2004) and O’Connell and others (2006) examine the effects of aid inflows in a model with currency substitution with a focus on various monetary and exchange rate policies. The model presented in this chapter is more explicit in accounting for the consumption-labor choice and for the effects of profitability of domestic firms on household income and focuses more closely on the interactions of monetary and fiscal policy.

^{}2

A detailed description of the model and the solution method is presented in Appendices 8.1 and 8.2.

^{}3

Monopolistic competition is a standard device to allow a role for aggregate demand in output determination and for price-setting behavior, which in turn generates nominal rigidities. The empirical relevance of this assumption will presumably depend on the country in question, and parameters can be chosen to yield any desired degree of monopolistic competition.

^{}4

All domestic firms are assumed to be owned by domestic consumers. Consumers receive dividends from firms, but take them as given, i.e., they do not actively participate in firms’ decision-making.

^{}5

This assumption is less appealing for countries with market access and substantial levels of foreign direct investment. However, we view it as a useful first-step approximation in keeping the analysis simple and tractable.

^{}7

This is likely to be true even when the government only spends aid proceeds on nontradables. A real appreciation would shift private demand toward imports, allowing the government to consume more nontraded goods.

^{}8

Instead of a linearization-based solution, we used the collocation method to solve this model (Appendix 8.2).

^{}9

Note that an increase in government spending in the absence of aid would be a similar amalgam of a pure monetary shock and a change in the structure of aggregate demand.

^{}10

For simplicity, we focus on the extreme cases when θ and γ are either 0 or 1. Intermediate cases can be interpreted as more moderate versions of the scenarios presented here. For example, when γ > θ, the interpretation would be similar to the case of spending without absorption.

^{}11

Other parameter values, such as those creating differences in sectoral productivity, high elasticity of substitution between traded and nontraded goods, and a higher share of traded goods in the government’s consumption basket may also result in real depreciation both in the short and long run.

^{}13

Output increases in the long run mainly because (1) the increase in government consumption is greater than the decline in private consumption, and (2) the government consumes domestic goods disproportionately, requiring higher domestic output. Even if private consumption demand fully offset government consumption demand in (1), the effect of (2) would still increase long-run output.

^{}15

This is related to the concept of the backward-bending labor supply curve. An early reference for this possibility in the developing country context is Berg (1961), who discusses conditions under which labor supply elasticity turns negative beyond some target level of income.

^{}16

For example, a high elasticity of substitution between traded and nontraded goods and/or higher labor productivity in the traded sector, among other factors, could reverse the output outcome.

^{}17

For similar reasons, the long-run effects on output do not allow a useful ranking of the merits of the different policy responses.

^{}18

It is possible that optimal domestically-financed spending would be greater with higher reserves, because the higher reserves might lower the country risk premium. However, this is likely to be a small effect. Moreover, borrowing domestically while building reserves would imply losses on the spread between the cost of borrowing and the yield on reserves. These considerations are outside the scope of the model presented here.

^{}19

As noted earlier, this trade-off cannot be formally addressed with the model presented here. See Prati and Tressel (2005).

^{}20

The benchmark model employs the following period utility function:

^{}21

With very few exceptions, “real” denotes variables expressed in units of household consumption baskets.

^{}24

Because firms are owned by consumers, the discount factor is set to equal the consumers’ intertemporal marginal rate of substitution.

^{}25

The optimal level of output is well defined as long as there is diminishing marginal product of labor (i.e., α < 1).

^{}26

Recall that the (consumer price index-based) real exchange rate *s _{t}* also equals the real price of tradables

^{}27

If consumers were allowed to hold foreign currency, the supply side would also be affected by the private sector’s decisions regarding currency composition of its asset portfolio. This case (dollarization) may be an interesting extension to the present set-up.

^{}28

Miranda and Fackler (2002) provide an extensive exposition of the method.

^{}29

Carroll (2001) and Kim and Kim (2003) illustrate the inaccuracies that could result from linearization-based solutions.

^{}30

See, for example, Gapen and Cosimano (2005).

^{}31

The law of motion in this case can be represented as

^{}32

Note that if we knew how *E*_{t}*z*_{t+1} is determined, then given the values of the state variables, we could determine agents’ actions by solving a deterministic system above. The difficulty presented by the uncertainty is that *E*_{t}*z*_{t+1} and *x*_{t} are jointly determined and need to be consistent with rational behavior.