2.1 Households

Households derive utility from consuming private and public goods according to the following lifetime utility function:

where β is the discount factor, c_t denotes consumption of the private (tradable) good and g_t denotes the public good. The utility function satisfies the usual concavity properties and it is separable in consumption of private and public goods:

where u_i denotes $\frac{du(i,j)}{di}$ and u_ij denotes $\frac{d^{2}u(i,j)}{didj}$. Households are endowed with one unit of labor, which they supply inelastically in domestic labor markets. They have access to external capital markets where they face a perfectly elastic supply of funds, and they can also save by holding cash balances. The household budget constraint is given by:

where $k_t^h$ denotes time t holdings of foreign bonds, which yield an interest r in each period; c_t denotes consumption of the private good, $M_t^h$ denotes time t nominal cash balances, w_t denotes wage income on which taxes (τw) are levied, π denotes dividends from firms and finally, ${\tau }_t^{MA}$ denotes transfers from the government (monetary authority).

Define $m_t^h\equiv \frac{M_t^h}{E_t}$ as real cash balances held by the households. Then households’ financial wealth is given by:

and the budget constraint can be re-written as :

Finally, money is introduced in the model by assuming that households faces a cash in advance (CIA) constraint for the purchase of consumption goods:⁴

Since the CIA constraint binds for any positive nominal interest rate, solving Equation (8) recursively yields the following household intertemporal budget constraint:

where (1 + αi_t) denotes the time t effective price of consumption.

The household optimization problem consists therefore in maximizing Equation (2) subject to (10) and the usual No Ponzi game condition $\underset{t\to \infty }{lim}\frac{a_t}{{(1+r)}^t}\ge 0$, taking {i_t, ${\tau }_t^{MA}$, ${\pi }_t^f$} and a₀ as given. Assuming the discount factor satisfies β (1 + r) = 1 (this ensures the existence of a steady state) the first order conditions that characterize the solution to the household’s problem are simply given by:

where λis the Lagrange multiplier associated to the intertemporal budget constraint (10). Equation (11) determines the optimal path of consumption. From the assumption of separability (u_cg = 0) follows that independent of the path of government expenditure, private consumption is constant over time as long as the nominal interest rate is constant over time.⁵ Equation (12), on the other hand, simply represents the usual transversality condition which follows from imposing the No Ponzi game condition in the optimization problem.

Finally, using Equation (12) the household intertemporal budget constraint can be written as:

2.2 Firms

Firms are modeled in a simple fashion. They produce the homogeneous tradable good using labor as the only input according to the following production function:

where 0 < σ < 1, l_t denotes labor and A_t > 0 denotes a productivity parameter. Firms are assumed to pay taxes on profits, defined as π_t = y_t – w_tl_t. Thus, the firms’ flow constraint is given by:

Whare w_t denotes wage per unit of labor, as mentioned before, ${\pi }_t^f$ are dividends paid to the shareholders (households); $k_t^f$ are firm’s holdings of foreign bonds; and τ is the tax rate on profits. Solving Equation (15) recursively and using Equation (14) yields:

where the LHS represents the firm’s market value, given by the discounted present value of dividends paid to the shareholders.

The firm’s optimization problem consists therefore in maximizing (16) subject the No Ponzi game condition $\underset{t\to \infty }{lim}\frac{k_t^f}{{(1+r)}^t}\ge 0$, taking {w_t} and $k_0^f$ as given. The first order conditions of this problem are simply given by:

Equation (17) implicitly defines the demand for labor as a function of the wage rate, while Equation (18) is the transversality condition for the firm, which again follows from imposing the No Ponzi game condition in the optimization problem.

2.3 Government

The government comprises both the fiscal and the monetary authority. The monetary authority sets monetary policy under a flexible exchange rate regime,⁶ by setting the rate of money growth in each period. The fiscal authority, on the other hand, levies nondistortionary taxes on firms’ profits and labor income.⁷ Government expenditure takes the form of a public good that provides direct utility to the households as shown in Equation (2). In order to finance (temporarily high) spending, the government can borrow from external markets both in foreign currency (at an interest rate r) or in domestic currency. As mentioned before, while the supply of funds in foreign currency is perfectly elastic (for Incentive Compatible levels of debt) the demand for local currency debt is not. As mentioned before, the demand for local currency instruments (d) is assumed to be increasing in the nominal interest rate of these bonds $(i_t^g)$ and decreasing in the expected depreciation rate $({\epsilon }_t^e)$:

This specification departs from open economy models like in Lahiri and Vegh (2003), where domestic debt is held only by domestic agents who derive liquidity services from these securities. In this case, nominal debt is held by foreigners. A set up a la Lahiri and Vegh (2003) would deliver the same qualitative results but it would complicate the solution of the model since it would introduce a link between the household problem and the government taxation problem, which is shut down with this specification. This will become clear in the next section.

As mentioned before, since sovereign debt contracts are not enforceable, the government can borrow from external sources (either in foreign or local currency) only as long as it has no incentives to repudiate on these obligations. In other words, the government’s Ramsey problem, which is described in detail in the next section, must satisfy an Incentive Compatibility (IC) constraint of the form:

where $V_t^R$ denotes welfare under sovereign debt repayment and $V_t^D$ denotes welfare under default. As it is shown below, this IC constraint is equivalent to an endogenous upper bound on total public debt.

In the event of a default, the government loses its access to capital markets, forcing the fiscal budget to be balanced every period.⁸ In addition, it is assumed that a default also results in a fiscal cost of θ units of consumption good in each period.⁹ It follows that in the event of a default government expenditure is given by:

which is clearly driven by the endogenous behavior of tax revenues (Г_t). Provided that total public debt is below its upper bound level (the IC constraint holds) the government’s budget constraint is given by:

where b_t denotes foreign currency denominated debt, g_t is government expenditure in the form of the public good, Г_t denotes tax revenues; D_t–1 denotes the nominal value of previous period domestic currency debt, d_t denotes its real value, and ${\tau }_t^{MA}$ denotes central bank’s transfers to the household.

Money is introduced in the economy as a transfer to the household (“helicopter drop”), which implies that ${\tau }_t^{MA}=\frac{M_t-M_{t-1}}{E_t}$.

Notice also that tax revenues are given by

so fiscal revenues are proportional to output in this set up. The tax rate τ_t is assumed to be constant over time, so we focus on government expenditure as the fiscal policy instrument.

Finally, the following timing of policy actions is assumed: at the beginning of each period, before resorting to capital markets to finance any fiscal imbalance, the MA announces a target for nominal money growth:

This introduces, in a simple way, a potential problem of commitment since the government would have incentives to deviate from the announced policy in order to inflate nominal debt away. This commitment problem has been largely studied in the literature and it is the central argument to explain why governments in emerging markets are unable to borrow in their own currencies. The contribution of this paper is precisely to show that this same problem can explain the differences in cyclical patterns of fiscal policy between developed and developing countries.

Finally it is assumed, without loss of generality, that the nominal interest rate on domestic debt is constant over time ($i_t^g=i^g$ for all t), so that the real domestic interest rate is simply determined by the depreciation rate (ε_t).

2.4 Equilibrium Path

A competitive equilibrium is a sequence for quantities {c_t,$k_t^h$,b_t,d_t,$m_t^h$,g_t,$k_t^f$,l_t,${\pi }_t^f$} and prices {E_t, i_t, $i_t^g$, w_t} such that: i) {c_t, $m_t^h$, $k_t^h$}solve the households’ maximization problem given {E_t, i_t, w_t} and{g_t, ${\pi }_t^f$}; ii) {l_t} solves firms’ maximization problem given {w_t}; iii) the government intertemporal budget constraint holds; and iv) markets clear: l_t= 1, $m_t^h=m_t$.

As agents are rational and there is no uncertainty in the economy, it follows that they perfectly anticipate the depreciation rate ${\epsilon }_t^e={\epsilon }_t$. Then the nominal interest rate is given by:

Furthermore, it can be shown that in equilibrium the depreciation rate ε_t must equal the rate of money growth (ε_t = μ_t), as it is standard in models with flexible exchange rates.¹⁰

From the equilibrium condition for labor markets follows that l_t = 1, thus output is given by y_t = A_t. Combining the household’s budget constraint (8) and the firm’s budget constraint (15), and using the fact that ${\tau }_t^{MA}=\frac{M_t-M_{t-1}}{E_t}$, the household intertemporal budget constraint (10) can be written as:

where $k_0^p\equiv (k_0^h+k_0^f)$ denotes private sector holdings of foreign assets. It is clear from this equation, Equation (11) and the assumption of separability of the utility function, that the optimal path of private consumption can only be affected by an intertemporal distortion due to an irregular path of nominal interest rates.¹¹

On the other hand, using ε_t = μ_t and ${\tau }_t^{MA}=\frac{M_t-M_{t-1}}{E_t}$ to substitute in the government budget constraint yields the fiscal constraint:

Where $k_t^g\equiv -(b_t+d_t)$ denotes government net external position. Notice that at time t, the stock of nominal debt d_t–1and the interest rate i^g are given. This implies that, by increasing the rate of money growth μ_t (and thus depreciating the currency) the government can reduce the real value of public liabilities which implies a transfer of resources from foreigners to domestic households. This creates the potential commitment problem that, in equilibrium, will affect the currency composition of public debt.

Finally, define $k_t\equiv k_t^p+k_t^g$ as the country net external position. Then combining Equations (27), and (28) yields the resource constraint of the economy:

Notice that this resource constraint differs from the classical case only by its last term, which represents the social benefit (cost) of borrowing from abroad in domestic currency.

Finally, note that the equilibrium path is not yet fully specified since we have not determined the actual paths of the rate of money growth {μ_t} and government expenditure {g_t}, which are in fact policy instruments that affect the path of consumption as it is clear from Equations (11) and (25), as well as the path of government debt b_t and d_t. In the next section, we study the how the path of μ_t and g_t are set.

3.1 The Commitment Case

Consider the case of a monetary authority that announces a rate of money growth ${\mu }_t={\mu }_t^A$ (before borrowing) and commit to this policy. Since in equilibrium ε_t = μ_t, the announcement about money growth rate is credible then ${\epsilon }_t^e={\mu }_t^A$, $i_t=(1+r)(1+{\mu }_t^A)-1$ and ${}^e{I}_t^g=(1+r)-\frac{1+i^g}{1+{\mu }_t^A}$. Then the Ramsey Problem of the government can be written as a simple dynamic programming problem in the following way:

Equation (31) is the household budget constraint once we take into account the solution to the optimization problem of the firm and the central bank transfer policy ${\tau }_t^{MA}=\frac{M_t-M_t}{E_t}$. From this equation and Equation (33) follows that the government does not collect inflation tax precisely because of the central bank’s transfer policy. This allows us to focus solely on the effect that a currency depreciation has on the real value of domestic debt, as a source of income for the government. Equation (32) follows from the household’s first order condition and ensures that the optimal path of consumption in the government problem is implementable in a decentralized equilibrium. Equation (33) is the fiscal constraint, following from (22). Equation (34) simply denotes the opportunity cost of holding domestic debt, as explained before. Finally, Equation (35) is the Incentive Compatibility constraint necessary to ensure that the government does not default on external debt. Notice that V^D does not depend on the level of debt of the government or the announcement of the depreciation rate precisely because in the event of a default, all public debt is defaulted. However, as we have assumed that debt contracts are enforceable in the private sector, even in the case of a sovereign default welfare is a function of the private sector foreign position K^p’. Furthermore, as we have assumed that the tax rate τ is constant over time and since the utility function is separable in private and public good consumption (u_cg = 0), it follows from Equations (31) and (32) that the optimal path of private consumption can only be affected by monetary policy to the extent that it distorts the path of nominal interest rates. If the optimal path of inflation is constant, then the private sector problem is completely independent of the government problem and, as a result, it can be shown that $V_{k^{p\prime }}^D(k^{p\prime })=V_{k^{p\prime }}^R(k^{p\prime },k^{g\prime },{\mu }^{A\prime })$.

The first order conditions that characterize the solution to this optimization problem, in a stationary economy, are given by:

Where ${\eta }_{d_t,I_t^g}\equiv \frac{\partial d_t/d_t}{\partial I_t^g/I_t^g}$ denotes the elasticity of the demand for domestic debt with respect to the nominal interest rate. Equations (36) and (37) define the optimal path of consumption of the private and the public good, respectively. It is clear from these Equations that c_t is always constant over time, while the path of g_t depends on whether the IC constraint binds or not. The former implies, as follows from Equations (32), (25) and (26) that the rate of money growth (μ_t), the depreciation rate (ε_t), and the nominal interest rate (i_t) are constant over time. Equation (38) on the other hand, defines the actual level for the rate of money growth. Given i^g, the optimal level of money growth is the one that maximizes the benefits from borrowing in domestic currency $(d_t{I}_t^g)$, which is exactly the level at which the elasticity of demand is equal to one (the top of the Laffer curve).

Thus, in equilibrium, there exist an optimal positive level of local currency denominated debt and therefore an optimal currency composition of public debt. Define ${\rho }_t\equiv -d_t/k_t^g$ as the proportion of public debt denominated in domestic currency, it follows that the steady state value of ρ_ss is given by:

From the assumption of separability of the utility function, we know that $V_t={\displaystyle {\sum }_{j=0}^{\infty }{\beta }^{j}u(c_{t+j})}+{\displaystyle {\sum }_{j=0}^{\infty }{\beta }^j\upsilon (g_{t+j})}$; and, since the path of private consumption is the same under repayment or default, it follows that the IC constraint can simply be written as:

where ${\displaystyle {\sum }_{j=0}^{\infty }{\beta }^j\upsilon (g_{t+j}^D)}={\displaystyle {\sum }_{j=0}^{\infty }{\beta }^j\upsilon (A_t\tau -\theta )}$ as follows from Equation (21). That is, the IC constraint only depends on fiscal policy (the path of government expenditure) precisely because we have separated the private sector from the public sector problem.

3.2 No Commitment

Consider now the case of a monetary authority that cannot credibly commit to a certain policy. In this case, the announcement μ^A’ is irrelevant and it does not affect the demand for domestic currency debt instruments.

Notice that the problem is exactly as before except for the fact that now d(^eI^g) does not depend on the previous announcement μ^A but on the expectation of the depreciation rate μ^e. Disregarding reputational equilibria, the latter implies that the government takes d(^eI^g) as given when choosing the depreciation rate at time t.

The first order conditions that characterize the solution to this optimization problem are given by: