A. The Representative Agent

For simplicity, there is only one private agent in each interest group. The representative agent of interest group h consumes one tradable good c_h,_t which is taken as the numeraire. The private agent can hold her wealth in the form of an internationally traded bond, whose real value is denoted by f_h,t, or in the form of domestic currency. The nominal stock of domestic currency, M_h,t, is chosen at the end of period t and carried over to period t + 1. Assuming PPP and normalizing the foreign price level to be constant at one yield the familiar relation between the domestic price level and nominal exchange rate: P_t = E_t. So inflation and nominal devaluation rates are the same, which are defined as:

Let r be the exogenous world real interest rate. For simplicity, it is assumed that the private agent’s subjective discount rate is the same as the real interest rate. The domestic nominal interest rate is defined as i_t = r + π_t. The private agent of each group has an initial stock of real bonds f_h,0 and an initial holding of nominal money M_h,0. In period 1, she receives a constant income y, a lump-sum transfer g_h,1 and pays income taxes τy. Then she chooses her private consumption c_h,1, adjusts her holdings of real bonds and real money to f_h,1 and $m_{h,1}\equiv \frac{M_{h,1}}{P_1}$, respectively. In the second period, she receives g_h,2 and y. She uses up all her income (g_h,2 and y) and accumulated wealth (f_h,1 and m_h,1) to pay for the income tax, inflation tax and consumption c_h,2.

The private agent’s utility function of group h is given by

where є is assumed to lie between 0 and 1 to ensure that the economy is always on the upward-sloping side of the Laffer curve. Notice that m_h,t–1 is the real balance chosen by private agent h at the end of period t-1 and carried over to period t.

The private agent’s budget constraint of group h for period 1 is given by:

and for period 2:

The consolidated budget constraint of both periods for private agent h is thus:

Summing up the consolidated budget constraints across all interest groups gives the intertemporal budget constraint of the private sector.

B. The Government

The government consists of a number of fiscal authorities (FA) and one central bank (CB). Each fiscal authority decides the optimal transfer for its own interest group. One way to interpret this assumption is that finance minister is weak and many spending ministers can influence his decision. Fiscal authority of interest group h is assumed to be benevolent in the sense that it chooses g_h,1, g_h,2 to maximize its private agent’s preference, taking other fiscal authorities’ decisions as given.

The initial government liabilities include a stock of net foreign debt nb₀ (b₀ is the debt per capita) and domestic nominal monetary liability ${\displaystyle {\sum }_{h=1}^n{m}_{h,0}}$. In period 1 the government makes the lump sum transfer g_h,₁ to the representative agent of interest group h (h = 1, 2, ..n), pays interest on its net debt, and changes its net debt holdings. It collects both income tax revenue and monetary revenue, including seigniorage and inflation tax. During period 2 the government makes another transfer g_h,2 and has to pay back both monetary and real debts. Its revenue comes from the inflation tax and income tax. It follows that the budget constraint of each period is:

Consolidating Eq. (7) and (8) yields:

The economy-wide resource constraint is obtained by combining Eq.(6) and (9):

C. Solving the Representative Agent’s Problem

The representative agent of group h chooses c_h,1, c_h,2, m_h,1, m_h,2 to maximize Eq.(2) subject to (5), taking g_h,1 and g_h,2 as given. The optimal conditions are:

The optimal condition of consumption implies that consumption is constant over time: $c_{h,1}^{*}=c_{h,2}^{*}={\overline{c}}_h,$ where

Symmetry condition indicates that ${\overline{c}}_h=\overline{c}$ for all h = 1, 2…n. Therefore, $\overline{c}$ can be determined by the economy-wide budget constraint.

Consumption is the scale factor in the money demand function. Let us define $a\equiv \frac{1}{u\prime (\overline{c})}$. The real balance demand function can be rewritten as:

Since only one inflation rate is prevalent in this economy, the real balance demand of each interest group is the same. The subscript h in m₀(M₀) and m₁(M₁) can be omitted.

D. The Central Bank’s Policy Rules

In line with the Tornell-Velasco model, it is assumed that the central bank’s rules are exogenous and are not the result of any optimization problem. In order to carry out a meaningful comparison between fixed and floating regimes, the CB is assumed to be able to commit to its policy rules only in period 1 and has to monetize the budget deficits in period 2 to remain solvent. The reason for this assumption will become clear later in this paper. Under a fixed exchange rate regime, the policy variable of CB is the devaluation rate in period 1:${\mathit{\pi }}_1^{fix}$. Under a flexible rate regime, the policy variable is the nominal money growth rate of period 1:

Right after the policy announcement of a fixed rate regime at the end of period 0, private agents can rearrange their portfolios by asset swaps:

${\displaystyle {\sum }_{h=1}^n{m}_0^h-{\displaystyle {\sum }_{h=1}^n{m}_{0-}^h}}\equiv n{m}_0-n{m}_{0-}=-(n{b}_0-n{b}_{0-}),$ where b_0– and m_0– are the levels of net foreign debt per capita and real balance before the announcement is made. There is no capital gain or loss. In contrast, due to the exchange rate movement at the end of period 0, there is a capital loss (gain) for the private agents under a flexible regime. To carry out a consistent comparison of two exchange rate regimes, Tornell-Velasco’s assumption that the government gives a rebate to private agents equal to the loss (gain) is adopted. This ensures that the government faces the same intertemporal budget constraint under different regimes.⁹ Based on the real balance demand condition of Eq.(12′), the government budget constraints (7), (8), and (9) can be rewritten as follows.

The intertemporal government budget constraint is obtained by combining (7′) and (8′):

The timing of the game is as follows. At the end of period 0, the CB makes its monetary policy announcement $({\pi }_1^{fix}\text{or}{\nu }_1)$. Each fiscal authority chooses its transfer in period 1: g_h,1 (h = 1, 2, ..n). Given the government decisions, the private agent of each interest group chooses her real balance for time 1: m_h,0. During period 1 the private agent h consumes c_h,1. At the end of period 1, each fiscal authority makes a decision on g_h,2 and then the private agent selects her desired real balance of period 2: m_h,l. During period 2, the government pays back both real and monetary debts, and delivers the transfer g_h,2. The private agent consumes all the remaining wealth and the game ends.

A. Tragedy of the Commons

If the CB is assumed to be able to precommit to its policy rules in both periods, the present value of government budget revenue is then fixed. It is equal to income tax revenue plus monetary revenue minus the government’s initial liability, regardless of the choice of exchange rate regime. In this case, a meaningful comparison between the two regimes can not be carried out. This model is then reduced to Velasco’s dynamic model of tragedy of the commons (TC), in which the strategic interactions among interest groups determines how government transfers are allocated intertemporally. Given $\overline{c},m_0({\pi }_1)$, and m₁(π₂), fiscal authority of interest group h simply maximizes the following objective function

subject to Eq.(7′) and (8′). Backward induction method to solve for the subgame perfect equilibrium yields optimal g_h,1, g_h,2 (see Appendix I for details). Since each interest group makes the same choice in equilibrium, the subscript h can be omitted. This will apply to all the following sections.

This model generates a “deficit bias” in the sense that fiscal authorities tend to spend more in the first period and leave a debt, although there is no reason for intertemporal smoothing. The socially optimal transfer is achieved when n = 1. The transfers in both periods are the same and there is no debt. When fiscal authorities independently choose their optimal spending, it’s clear that both $g_{TC,1}^{*}$ and $g_{TC,2}^{*}$ are decreasing with the number of fiscal authorities, n. However, the total transfers of period one $n{g}_{TC,1}^{*}$ and the debt b₁ can be easily shown to be increasing in n. Hence, the larger the number of interest groups, the larger the debts, and the stronger the deficit bias. This intertemporal distortion occurs because all fiscal authorities share one common property: government net assets. Each fiscal authority thinks that if it does not take more today, there will be less left for it tomorrow because other groups will take more today. So every interest group ends up spending more in the first period and a debt is left for period 2, although there is no reason for intertemporal smoothing.

B. Competitive Externality

In this part, it is assumed that the CB can only precommit in period 1, but n fiscal authorities could precommit in both periods. At the end of period 0, the fiscal authorities decide their transfer plans for both periods g_h,1, g_h,2 (h = 1, 2, ..n) and stick to their plans during the game. No decision is made at the end of period 1. This way, the dynamic role of government debt is assumed away since fiscal authorities can not reoptimize based on the debt level at the end of period 1. In this sense, the two-period dynamic model actually becomes a static one-period model.

Under a fixed exchange taking ${\pi }_1^{fix}(m_0)$ as given, each fiscal authority chooses g_h,1, g_h,2 to maximize its private agent’s utility function (2) subject to Eq.(9′). Under a floating rate regime, π₁ (m₀) is endogenously determined with nominal money growth rate ν₁ being the policy tool. From the definition of υ₁ and π₁, the following identity is obtained:

In Appendix II, it is shown that the equilibrium transfer paths under two exchange rate regimes are as follows:

This model exhibits the property of “competitive externality” (CE), which gives the economy a spending bias and an inflationary bias. The socially optimal transfer of $a(\frac{1}{\mathit{є}}-1)$is achieved when n = 1. With a weak CB and strong spending ministers, the benefits from spending accrue entirely to each interest group while the inflation costs are shared by all the groups. This distorts the fiscal authorities’ incentives and leads to higher government transfer. Notice that the distortion exists even in the first period under a fixed exchange rate regime. As long as the CB gives in by monetizing budget deficits in the second period, the fiscal authorities tend to overspend in the first period without worrying about inflationary consequences.

The distortion caused by “competitive externality” is intratemporal, not intertemporal. This model shows that different exchange rate regimes yield identical spending paths because fiscal authorities discount the future at the rate of world real interest. This is the result derived in Tornell-Velasco’s static model. In order to derive an equilibrium in which different regimes give different paths, an intertemporal distortion must be introduced. Tornell and Velasco (1995, 1998) assume that the fiscal authority discounts the future at a rate lower than the world real interest rate, which gives the policymaker a greater incentive to spend more under fixed rates. However, as shown later in this paper, the dynamics itself of the general model will generate the difference between fixed rates and flexible rates, and no intertemporal assumption is needed.

A. The Equilibrium Under Fixed Rates

Under a fixed exchange rate regime, π₁(m₀) is exogenously given. The unknowns are g_h,1, g_h,2, π₂, and m₁. The subscript h will be omitted because the decision of each fiscal authority is the same in equilibrium.

Period 2:

At the beginning of period 2, objective function of each fiscal authority changes. It no longer cares about its utility of period 1. Given the debt level of nb₁, fiscal authority of interest group h now chooses g_h,2 to maximize

subject to Eq.(8′), Eq.(12′) and $c_{h,2}=\overline{c}$. Simple first-order condition yields the optimal transfer as an increasing function of m₁:

Inserting $g_2^{fix}$ into the government budget constraint of period 2, we can express b₁ as a function of m₁ and use this expression to substitute b₁ in Eq.(7′). Then the government budget constraint in period 1 can be rewritten as:

Period 1:

Fiscal authority h will choose g_h,1 to maximize Eq.(2), its representative agent’s utility, subject to Eq.(17) and (18). Notice that π₁ and m₀ are exogenously given under a fixed exchange rate regime. The optimal g_h,1 is an increasing function of m₁.

Replacing $g_1^{fix}$ and $g_2^{fix}$ back to Eq.(9′) and dividing both side by n yield the intertemporal government budget constraint in terms of each interest group, from which equilibrium $m_1^{fix*}$ can be solved for. Let us define $k\equiv \tau y(\frac{2+r}{1+r})-(1+r)(b_{0-}+m_{0-})$,

As long as $m_1^{fix*}$ is solved, $g_1^{fix*}$ and $g_1^{fix*}$ can be easily derived. The fiscal authority’s problem under fixed exchange rates is fully characterized.

In contrast to the result of Tornell and Velasco (1995, 1998) that government spending is independent of the devaluation rate under fixed rates in the case of C.E.S. utility functions, both $g_1^{fix*}$ and $g_2^{fix*}$ are increasing functions of ${\pi }_1^{fix}$, the exogenous policy variable. In other words, the tightness of the exchange rate policy under a fixed regime will affect fiscal authorities’ decisions in this paper. This occurs because fiscal authorities can not precommit in the model. At the end of period 1, each fiscal authority would reoptimize and choose its optimal transfer for period 2 based on the debt level. The choice of ${\pi }_1^{fix}$ under fixed rates will determine how much monetary revenue the government can collect in period 1 and thereby affect how much debts the government needs to run in period 1. Therefore, ${\pi }_1^{fix}$ will affect the choice of fiscal authorities in period 2 and in turn the choice in period 1. In the Tornell-Velasco model, the fiscal authority is assumed to be able to commit to a certain path of spending and it does not reoptimize at the end of period 1. In the case of C.E.S utility functions, difference between the marginal cost and the marginal benefit of increasing government spending is independent of m₁.

B. The Equilibrium Under Flexible Rates

Since the maximization problem of each fiscal authority in period 2 is the same under different regimes, the optimal transfer at the second period is also given by:

Period 1:

Fiscal authority h chooses g_h,1 to maximize Eq.(2) subject to (15),(18), and (21).

The optimal g_h,1 is given as a function of m₁ and m₀. According to Eq.(15), m₀ can be expressed as an increasing function of m₁. Therefore, $g_1^{fle}$can be viewed as a function of m₁ alone.

Substituting $g_1^{fle}$ and $g_2^{fle}$ into Eq.(9′) and dividing both sides by n yields the intertemporal government budget constraint in terms of each interest group.

Unlike the case of a fixed exchange rate regime, m₀ is unknown under a floating exchange rate regime. In order to fully characterize the equilibrium, we need one more equation, which is given by Eq.(15). Equilibrium $m_0^{fle*}$ and $m_1^{fle*}$ can be solved from Eq.(23) and Eq.(15). As long as $m_0^{fle*}$ and $m_1^{fle*}$ are solved, $g_1^{fle*}$ and $g_2^{fle*}$ can be obtained according to Eq.(21) and (22). The subgame perfect equilibrium under flexible exchange rates is then fully characterized.

The above discussions show that optimal transfers under both regimes can all be expressed as functions of m₁. There are some interesting properties of these four functions, $F_1^{fix}(m_1),F_2^{fix}(m_1),F_1^{fle}(m_1)$, and $F_2^{fle}(m_1)$. Since fiscal authorities face the same maximization problem in period 2, $F_2^{fix}(m_1)$ and $F_2^{fle}(m_1)$ are actually the same function. Appendix III shows that all four functions are increasing with m₁, that is, government transfers decrease with the future inflation rate. This generates the punishment mechanism in this model. Future inflation or currency depreciation serves as punishment against fiscal laxity in this model. The higher the future inflation (depreciation), the less the government transfer. In addition, as long as υ ≠ 1, given the same value of m₁, the following relation is obtained:

Eq.(24) reveals an important difference between fixed and flexible exchange rate regimes: the greater incentive to spend more in period 1 under fixed rates. This occurs because the fiscal authorities will take into account their optimal behavior in period 2 making decisions in period 1. They understand that, under fixed rates, unsound fiscal policy will not cause any inflation in period 1 and inflation only happens in period 2. In contrast, under flexible rates, fiscal laxity will have an immediate inflationary impact in period 1. Therefore, fiscal authorities have a greater incentive spend more in the first period under fixed rates than under flexible rates, although they discount the future at the world real interest rate. This is the basis on which Tornell and Velasco (1995) build their critique about conventional wisdom. In their model, due to the absence of explicit dynamic analysis, they need the assumption that the fiscal authority discounts the future at the rate lower than the world real interest rate. Actually, the dynamics itself of the general model is sufficient to generate the greater incentive to spend more today under fixed rates and no special assumption on the discount rate is needed.

Figure 1 illustrates the above points clearly. The curve of $F_1^{fix}$ is always above the curve of $F_1^{fle}$. Whether the curve of $F_2^{fix}$ and $F_2^{fle}$ is above, below, or in-between the curves of $F_1^{fix}$ and $F_1^{fle}$ depends on n, number of interest groups. In all graphs of this paper, $F_2^{fix}(F_2^{fle})$ is drawn to be below $F_1^{fix}$ and $F_1^{fle}$.

Optimal government transfers in terms of m₁

Citation: IMF Working Papers 2003, 078; 10.5089/9781451850123.001.A001

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Optimal government transfers in terms of m₁

Citation: IMF Working Papers 2003, 078; 10.5089/9781451850123.001.A001

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Optimal government transfers in terms of m₁

Citation: IMF Working Papers 2003, 078; 10.5089/9781451850123.001.A001

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