Terms of trade volatility, exports, and GDP

Abstract

This paper relates terms of trade volatility to exports and output in a two-sector model where entrepreneurs can produce non-tradable goods or pay a fixed cost in order to export. In order to compensate exporters for the fixed entry cost, the expected return to exporting must exceed the expected return to non-tradable production. As a result, exporters are more risk-exposed and trade volatility decreases entry into the export sector. However, terms of trade insurance and hedging strategies can increase exports, GDP, and welfare.

Appendix

1.1 Proof of Proposition 2

• (i) When $f = 0$, (14) implies that $\delta = 1 - \beta$. Thus, the labor force share of the tradable sector ($\delta$) is constant and independent of terms of trade volatility. Since $\delta$ is in independent of terms of trade volatility, (9) implies that $\partial E(G)/\partial \sigma_{{p_{x} }} = 0$. On the other hand, since (10) is concave, in the terms of trade, $\partial^{2} \hat{G}/\partial p_{x}^{2} < 0$, we have $\partial E(\hat{G})/\partial \sigma_{{p_{x} }}^{2} < 0$. Since $\delta$ is constant, (11) implies that $\partial E(W)/\partial \sigma_{{p_{x} }}^{2} = K\partial E(\hat{G})/\partial \sigma_{{p_{x} }}^{2} < 0$.

• (ii) Using (14), define

  $$H(\delta (\sigma_{{p_{x} }}^{2} ),\sigma_{{p_{x} }}^{2} ) \equiv \left\{ {\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } - \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{1 - \beta } } \right\}E\left( {p_{x}^{1 - \beta } } \right) - f/K = 0.$$

  Then, the implicit function implies that

  $$\begin{aligned} \frac{\partial \delta }{{\partial \sigma_{{p_{x} }}^{2} }} = \frac{{ - \partial H/\partial \sigma_{{p_{x} }}^{2} }}{\partial H/\partial \delta } & = \frac{{ - \left\{ {\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } - \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{1 - \beta } } \right\}\partial E(p_{x}^{1 - \beta } )/\partial \sigma_{{p_{x} }}^{2} }}{{ - \frac{\beta }{{\delta (1 - \delta )^{2} }}\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } E(p_{x}^{1 - \beta } )}} \\ & = \frac{{\left\{ {1 - \frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right\}\partial E(p_{x}^{1 - \beta } )/\partial \sigma_{{p_{x} }}^{2} }}{{\frac{\beta }{{\delta (1 - \delta )^{2} }}E(p_{x}^{1 - \beta } )}} = \frac{{\left\{ {\frac{1 - \delta - \beta }{1 - \delta - \beta + \beta \delta }} \right\}\overbrace {{\partial E(p_{x}^{1 - \beta } )/\partial \sigma_{{p_{x} }}^{2} }}^{ < 0}}}{{\frac{\beta }{{\delta (1 - \delta )^{2} }}E(p_{x}^{1 - \beta } )}} < 0. \\ \end{aligned}$$

  (15)

Terms of trade volatility therefore decreases the labor force share of the tradable sector. The decline in tradable production decreases GDP in (9) and (10) since

$$\frac{\partial EG}{{\partial \sigma_{{p_{x} }}^{2} }} = \overbrace {{\frac{{\partial \delta /\partial \sigma_{{p_{x} }}^{2} }}{1 - \beta }}}^{ < 0}E(p_{x} ) < 0.$$

(16)

$$\begin{aligned} \frac{{\partial E(\hat{G})}}{{\partial \sigma_{{p_{x} }}^{2} }} & = \frac{\partial }{{\partial \sigma_{{p_{x} }}^{2} }}E\left( {\frac{{\delta^{1 - \beta } (1 - \delta )^{\beta } }}{{(1 - \beta )^{1 - \beta } \beta^{\beta } }}p_{x}^{1 - \beta } } \right) \\ & = \left( {\frac{{\delta^{1 - \beta } (1 - \delta )^{\beta } }}{{(1 - \beta )^{1 - \beta } \beta^{\beta } }}} \right)\overbrace {{\frac{{\partial E(p_{x}^{1 - \beta } )}}{{\partial \sigma_{{p_{x} }}^{2} }}}}^{ < 0} + \frac{1}{{(1 - \beta )^{1 - \beta } \beta^{\beta } }}\frac{{\partial \left( {\delta^{1 - \beta } (1 - \delta )^{\beta } } \right)}}{\partial \delta }\overbrace {{\frac{\partial \delta }{{\partial \sigma_{{p_{x} }}^{2} }}}}^{ < 0}E\left( {p_{x}^{1 - \beta } } \right) < 0 \\ & \Leftarrow \, \frac{{\partial \left( {\delta^{1 - \beta } (1 - \delta )^{\beta } } \right)}}{\partial \delta } < 0 \Leftrightarrow (1 - \beta )\delta^{ - \beta } (1 - \delta )^{\beta } + \delta^{1 - \beta } \beta (1 - \delta )^{\beta - 1} > 0 \Leftrightarrow 1 - \delta > \beta { ,} \\ \end{aligned}$$

(17)

(where $\Leftarrow$ denotes a sufficient condition).In order to show that terms of trade volatility decreases welfare, we differentiate (11),$W = \delta u_{x} + (1 - \delta )u_{n} - \delta f = K\hat{G} - \delta f$,giving

$$\frac{dE(W)}{{d\sigma_{{p_{x} }}^{{}} }} = \left[ {\frac{\partial E(W)}{{\partial \sigma_{{p_{x} }}^{{}} }}} \right] + \left[ {\frac{\partial E(W)}{\partial \delta }\frac{\partial \delta }{{\partial \sigma_{{p_{x} }}^{{}} }}} \right]$$

(18)

$$= \left[ {\delta \frac{{\partial E(u_{x} )}}{{\partial \sigma_{{p_{x} }}^{{}} }} + (1 - \delta )\frac{{\partial E(u_{n} )}}{{\partial \sigma_{{p_{x} }}^{{}} }}} \right] + \left[ {\frac{\partial }{\partial \delta }\left( {\delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta f} \right)} \right]\overbrace {{\frac{\partial \delta }{{\partial \sigma_{{p_{x} }}^{{}} }}}}^{ < 0} ,$$

(19)

where Proposition 2(ii) implies that $\partial \delta /\partial \sigma_{{p_{x} }}^{{}} < 0$. The first bracketed expression in (19) is the welfare effect of increasing income volatility conditional on the allocation of labor across sectors. The second bracketed expression is the welfare effect of labor reallocation.

If we only consider a marginal volatility increase starting from the equilibrium labor allocation in (13), however, the second brackets are zero: Although the market cannot eliminate the terms of trade risk, the labor allocation has already adapted to the risk. Formally,

$$\frac{\partial }{\partial \delta }\left( {\delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta f} \right) = \underbrace {{\left[ {E(u_{x} ) - E(u_{n} ) - f} \right]}}_{ = 0} + \left[ {\delta \frac{{\partial E(u_{x} )}}{\partial \delta } + (1 - \delta )\frac{{\partial E(u_{n} )}}{\partial \delta }} \right] = 0,$$

(20)

where the equality follows because

• (i) Eq. (13) implies that the first brackets are zero.

• (ii) Substituting the utility expressions in (14) into the second brackets shows that

$$\delta \frac{{\partial E(u_{x} )}}{\partial \delta } + (1 - \delta )\frac{{\partial E(u_{n} )}}{\partial \delta } = 0,$$

(21)

$$\Leftrightarrow \left\{ {\delta \frac{\partial }{\partial \delta }\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } + (1 - \delta )\frac{\partial }{\partial \delta }\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{1 - \beta } } \right\}KE\left( {p_{x}^{1 - \beta } } \right) = 0,$$

$$\Leftrightarrow \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } \frac{(1 - \delta - \beta + \beta \delta )\beta + \beta \delta (1 - \beta )}{{(1 - \delta - \beta + \beta \delta )^{2} }}\left\{ { - \delta \beta \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - 1} + (1 - \delta )(1 - \beta )} \right\} = 0,$$

$$\Leftrightarrow - \delta \beta \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - 1} + (1 - \delta )(1 - \beta ) = 0 \Leftrightarrow - (1 - \delta - \beta + \beta \delta ) + (1 - \delta )(1 - \beta ) = 0,$$

$$\Leftrightarrow (1 - \delta )(1 - \beta ) = (1 - \delta )(1 - \beta ).$$

The welfare effect (19) therefore simplifies to

$$\begin{aligned} \frac{dE(W)}{{d\sigma_{{p_{x} }}^{{}} }} & = \left[ {\delta \frac{{\partial E(u_{x} )}}{{\partial \sigma_{{p_{x} }}^{{}} }} + (1 - \delta )\frac{{\partial E(u_{n} )}}{{\partial \sigma_{{p_{x} }}^{{}} }}} \right] \\ & = K\left[ {\delta \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{ - \beta } + (1 - \delta )\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }} \right)^{1 - \beta } } \right]\underbrace {{\frac{{\partial E(p_{x}^{1 - \beta } )}}{{\partial \sigma_{{p_{x} }}^{{}} }}}}_{ < 0} < 0, \\ \end{aligned}$$

where $\frac{{\partial E(p_{x}^{1 - \beta } )}}{{\partial \sigma_{{p_{x} }}^{{}} }} < 0$ by Jensen’s inequality. □

1.2 The welfare effects of exchange rate management and entry subsidies

In this section, we show that the nominal exchange rate does not have any real effects in the model and that entry subsidies cannot increase welfare holding terms of trade volatility constant.

The effects of the nominal exchange rate We can choose the units so the price of the import good is the price of a foreign currency unit. Then, the domestic currency price of the export good is $p_{x}^{d} = p_{x} /e,$ where $e$ is the nominal exchange rate (foreign currency per unit of domestic currency). Domestic currency GDP is

$$G^{d} = \delta p_{x}^{d} + (1 - \delta )p_{n}^{d} .$$

where the domestic currency price of non-tradables solves

$$1 - \delta = \beta G^{d} /p_{n}^{d} \Leftrightarrow ,$$

$$p_{n}^{d} = \frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x}^{d} = \frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x} /e.$$

The price level in domestic currency is $P^{d} = (p_{n}^{d} )^{\beta } (1/e)^{1 - \beta } = \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x}^{d} } \right)^{\beta } (1/e)^{1 - \beta }$since an imported good costs $1/e$ domestic currency units.

In terms of foreign currency or, equivalently, imported goods, however, for each $\delta$, all of the variable values remain the same as in Eqs. (7–10) in the paper:

$$G = eG^{d} = \delta p_{x} + (1 - \delta )\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x} = \delta p_{x} + (1 - \delta )\frac{\beta \delta }{(1 - \delta )(1 - \beta )}p_{x} = \frac{\delta }{1 - \beta }p_{x} ,$$

$$p_{n} = ep_{n}^{d} = \frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x} ,$$

$$P = eP^{d} = \left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x} } \right)^{\beta } ,$$

$$\hat{G} = G^{d} /P^{d} = \frac{{\frac{\delta }{1 - \beta }ep_{x}^{d} }}{{\left( {\frac{\beta \delta }{1 - \delta - \beta + \beta \delta }p_{x}^{d} } \right)^{\beta } (1/e)^{1 - \beta } }} = \frac{{\delta^{1 - \beta } (1 - \delta )^{\beta } }}{{(1 - \beta )^{1 - \beta } \beta^{\beta } }}(ep_{x}^{d} )^{1 - \beta } = \frac{{\delta^{1 - \beta } (1 - \delta )^{\beta } }}{{(1 - \beta )^{1 - \beta } \beta^{\beta } }}p_{x}^{1 - \beta } = G/P.$$

Agent incomes in terms of the import good remain $I_{x} = p_{x}$ and $I_{n} = p_{n}$. Finally, the labor allocation $\delta$ is the same since the utility expressions are unchanged from (12):

$$u_{j} = (\beta I_{j}^{d} /p_{n}^{d} )^{\beta } ((1 - \beta )I_{j}^{d} /e)^{1 - \beta } = (\beta eI_{j} /ep_{n} )^{\beta } ((1 - \beta )eI_{j} /e)^{1 - \beta } ,$$

$$= (\beta I_{j} /p_{n} )^{\beta } ((1 - \beta )I_{j} )^{1 - \beta } = KI_{j} /P,$$

and the income ($I_{j}$, $j = x,n$), and price levels ($P$) remain unchanged.

The effects of entry subsidies Assume that the government can pay an entry subsidy $s$ using reserves or lump-sum taxes. Rewriting Eq. (14), which defines the labor allocation, as

$$J(\delta (s),s) = K\left\{ {\left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{ - \beta } - \left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{1 - \beta } } \right\}E\left( {p_{x}^{1 - \beta } } \right) - (f - s) = 0,$$

the implicit function theorem implies that

$$\frac{\partial \delta }{\partial s} = - \frac{\partial J/\partial f}{\partial J/\partial \delta } = - \frac{1}{\partial J/\partial \delta } > 0$$

so entry increases. The welfare expression is unchanged from (11) in the main paper because the government pays the subsidy cost, that is,

$$W = \delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta (f - s) - \delta s = \delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta f.$$

The welfare effect of a marginal subsidy is

$$\frac{dE(W)}{df} = \frac{\partial E(W)}{\partial \delta }\frac{\partial \delta }{\partial s} = \left[ {\frac{\partial }{\partial \delta }\left( {\delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta f} \right)} \right]\overbrace {{\frac{\partial \delta }{\partial s}}}^{ > 0}.$$

However, equation (20) shows that, starting from the market equilibrium, the bracketed term is zero. More generally, given any fixed labor allocation $\delta$

$$\frac{dE(W)}{df} = \frac{\partial E(W)}{\partial \delta }\frac{\partial \delta }{\partial s} = \frac{\partial }{\partial \delta }\left( {\delta E(u_{x} ) + (1 - \delta )E(u_{n} ) - \delta f} \right),$$

$$= \left[ {E(u_{x} ) - E(u_{n} ) - f} \right] + \underbrace {{\left[ {\delta \frac{{\partial E(u_{x} )}}{\partial \delta } + (1 - \delta )\frac{{\partial E(u_{n} )}}{\partial \delta }} \right]}}_{ = 0},$$

where (21) shows that the second brackets are zero. The first brackets are not zero outside of the market equilibrium (13). However, substituting (14) into the first brackets shows that

$$E(u_{x} ) - E(u_{n} ) - f = K\left\{ {\left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{ - \beta } - \left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{1 - \beta } } \right\}E\left( {p_{x}^{1 - \beta } } \right) - f.$$

This expression—the marginal welfare gain from labor reallocation—is strictly decreasing in $\delta$ since $\frac{\partial }{\partial \delta }\left\{ {\left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{ - \beta } - \left( {\frac{\beta \delta }{(1 - \delta )(1 - \beta )}} \right)^{1 - \beta } } \right\}KE\left( {p_{x}^{1 - \beta } } \right) < 0$. It equals zero in the market equilibrium allocation defined by (14). Thus, increasing $\delta$ to any point exceeding its equilibrium value decreases welfare.