Positive and Normative Implications of Liability Dollarization for Sudden Stops Models of Macroprudential Policy

Abstract

“Liability dollarization,” namely intermediation of capital inflows in units of tradables into domestic loans in units of aggregate consumption, adds three important effects driven by real exchange rate fluctuations that alter standard models of Sudden Stops significantly: changes on the debt repayment burden, on the price of new debt, and on a risk-taking incentive (i.e., a negative premium on domestic debt). Under perfect foresight, the first effect makes Sudden Stops milder and multiple equilibria harder to obtain. The three effects add an “intermediation externality” to the macroprudential externality of standard models, which is present even without credit constraints. Optimal policy under commitment can be decentralized equally by taxing domestic credit or capital inflows, and hence capital controls as a separate instrument are not justified. This optimal policy is time inconsistent and follows a complex, nonlinear schedule. Quantitatively, an optimized pair of constant taxes on domestic debt and capital inflows makes crises slightly less likely and yields a small welfare gain, but other pairs reduce welfare sharply. For high effective debt taxes, capital controls and domestic debt taxes are again equivalent, and for low ones, welfare is higher with higher taxes on domestic debt than on capital inflows.

Appendix: Properties of the \({\text {BB}}^{\mathrm{SSLD}}\) Curve

The ${\text {BB}}^{\mathrm{SSLD}}$ curve determines the value of $p^N$ that corresponds to a value of $c^T$ such that both the resource and the collateral constraint hold with equality [see Eq. (13) in the text]. Formally, the ${\text {BB}}^{\mathrm{SSLD}}$ curves is given by the following function:

$$\begin{aligned} p^N(c^T)=\frac{c^{T}-(1+\kappa )y^T-p^c(c^T) b^c}{\kappa {\overline{y}}^N }. \end{aligned}$$

We omit time subscripts for simplicity, but notice $b^c$ corresponds to the outstanding debt at the beginning of the period. Given the parametric restrictions on $c^T$, $\kappa$, $y^T$, ${\bar{y}}^N$ , and $b^c$, the fact that $p^c(c^T)$ is continuous implies that the ${\text {BB}}^{\mathrm{SSLD}}$ curve is continuous.

The horizontal intercept of the ${\text {BB}}^{\mathrm{SSLD}}$ curve is found by evaluating the above expression when $p^N=0$. Using Eq. (3) to determine the consumption price index, when $p^N=0$, it follows that the intercept is the value of tradables consumption such that $c^T=(1+\kappa )y^T+\omega ^\frac{1}{\eta }b^c$.

To obtain the slope of the ${\text {BB}}^{\mathrm{SSLD}}$ curve, we take the first derivative of the above expression, which yields $\frac{\partial p^N}{\partial c^T}=\frac{1-\frac{\partial p^c}{\partial c^T} b^c}{\kappa {\overline{y}}^N}$. The sign of the slope depends on the signs of $\frac{\partial p^c}{\partial c^T}$ and $b^c$. Using Eq. (3), it follows that because of the CES structure of preferences, $\frac{\partial p^c}{\partial c^T}=(1+\eta )\frac{1-\omega }{\omega }\left[ \omega +(1-\omega )(c^T)^{\eta }\right] ^{\frac{1}{\eta }}(c^T)^{\eta -1}>0$. Moreover, since we are interested in economies with debt ($b^c<0$), it follows that $\frac{\partial p^N}{\partial c^T}>0$. Hence, the ${\text {BB}}^{\mathrm{SSLD}}$ curve is increasing in $c^T$

To determine whether the ${\text {BB}}^{\mathrm{SSLD}}$ curve is concave or convex, we analyze its second derivative, which is the following:

$$\begin{aligned} \frac{\partial ^2 {p^N}}{\partial {c^T}^2}=-\frac{\frac{\partial ^2 p^c}{\partial {c^T}^2}b^c}{\kappa {\overline{y}}^N}. \end{aligned}$$

The sign of this derivative is the same as the sign of the second derivative of $p^c$ with respect to $c^T$. This derivative can be expressed as:

$$\begin{aligned} \frac{\partial ^2 p^c}{\partial {c^T}^2}=(1+\eta )\frac{1-\omega }{\omega }\left( \omega +(1-\omega ) (c^T)^\eta \right) ^{\frac{1}{\eta }}(c^T)^{2(\eta -1)}\left[ \frac{1-\omega }{\omega +(1-\omega )(c^T)^\eta }+(\eta -1)(c^T)^{-\eta }\right]. \end{aligned}$$

All the terms in the right-hand side of this expression are positive, except for the last term in square brackets, which has an ambiguous sign. Hence, the sign of this derivative is determined by the sign of the term $\left[ \frac{1-\omega }{\omega +(1-\omega )(c^T)^\eta }+(\eta -1)(c_T)^{-\eta }\right]$. We can characterize the conditions determining the sign of this term by first reducing it to this expression:

$$\begin{aligned} \left[ \frac{1}{\tilde{\omega }+(c^T)^\eta }+\frac{\eta -1}{(c^T)^{\eta }}\right] \end{aligned}$$

(30)

where we used the definition $\tilde{\omega }\equiv \omega / (1-\omega )$. Analyzing this expression, it follows that:

$$\begin{aligned} \left[ \frac{1}{\tilde{\omega }+(c^T)^\eta }+\frac{\eta -1}{(c^T)^{\eta }} \right] \gtreqqless 0 \Leftrightarrow \eta \gtreqqless \frac{\tilde{\omega }}{\tilde{\omega }+(c^T)^\eta }. \end{aligned}$$

(31)

Notice the expression in the right-hand side of the last inequality is always a positive fraction, but its magnitude varies with $c^T, \omega$ and $\eta$, which is what makes the direction of the inequality ambiguous. Still, given that $\eta >-1$ from the CES functional form and that $c^T>0$ and $0<\omega <1$, we can establish the following three results:

• 1. The$p^c$and${\text {BB}}^{\mathrm{SSLD}}$functions are strictly concave when the elasticity of substitution between tradables and nontradables is greater or equal to 1: If $-1<\eta \le 0$ (i.e., $1/(1+\eta )\ge 1$), then $\eta < \frac{\tilde{\omega }}{\tilde{\omega }+(c^T)^\eta }$, and hence $\left[ \frac{1}{\tilde{\omega }+(c^T)^\eta }+\frac{\eta -1}{(c^T)^{\eta }}\right] <0$ and thus both $p^c$ and ${\text {BB}}^{\mathrm{SSLD}}$ are strictly concave, for any positive $c^T, \omega$.

• 2. The$p^c$and${\text {BB}}^{\mathrm{SSLD}}$functions are strictly convex when the elasticity of substitution between tradables and nontradables is less or equal than 1/2: If $\eta \ge 1$ (i.e., $1/(1+\eta )\le 1/2$) then $\eta > \frac{\tilde{\omega }}{\tilde{\omega }+(c^T)^\eta }$, and hence $\left[ \frac{1}{\tilde{\omega }+(c^T)^\eta }+\frac{\eta -1}{(c^T)^{\eta }}\right] >0$ and both $p^c$ and ${\text {BB}}^{\mathrm{SSLD}}$ are strictly convex, for any positive $c^T, \omega$.

• 3. The$p^c$and${\text {BB}}^{\mathrm{SSLD}}$functions are concave (convex) for sufficiently low (high)$c^T$when the elasticity of substitution is between 1/2 and 1: If $0<\eta <1$ (so that $1/2< 1/(1+\eta ) <1$), the sign of $\left[ \frac{1}{\tilde{\omega }+(c^T)^\eta }+\frac{\eta -1}{(c^T)^{\eta }}\right]$ changes from negative to positive as $c^T$ rises. Around $c^T=0$ we obtain $\eta < \frac{\tilde{\omega }}{\tilde{\omega }+(c^T)^\eta }=1$ and as $c^T$ increases $\frac{\tilde{\omega }}{\tilde{\omega }+(c^T)^\eta }$ falls. Hence, near $c^T=0$ the derivatives are negative and the curves are concave. By continuity, for sufficiently low $c^T$ the curves are concave, and for sufficiently high $c^T$ the curves turn convex.

Bianchi (2011) notes that estimates of the elasticity of substitution in tradables and nontradables consumption for emerging markets are in the [.4, .83] interval. Hence, most of this range falls in the region where the third result above applies. Quantitatively, however, in our baseline calibration for the perfect-foresight experiments taken from Bianchi’s work (which uses $\eta =0.205$, $1/(1+\eta )=0.83$) and for other exercises using reasonable values of $b^c$ and $y^T$ and any $0< \eta <1$, we found that ${\text {BB}}^{\mathrm{SSLD}}$ is either convex or nearly linear, except for a slightly concave segment for very low $c^T$. In all of these experiments, the concavity is visible only for $c^T<0.05$ compared with a perfect-foresight unconstrained equilibrium of $c^T=0.92$ using our baseline calibration, or a Sudden Stop outcome of $c^T=0.83$ for a wealth-neutral negative shock of nearly 20% from an initial income of $y^T=1$. Hence, assuming convex or nearly linear ${\text {BB}}^{\mathrm{SSLD}}$ curves when the elasticity of substitutions is less than unitary is innocuous.