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Let the real exchange rate $Q=\frac{P^*e}{P}$, where $P^*$ is the 'world' price level (everything with '*' is of the 'world'), $P$ is the domestic price level, and $e =\frac{\text{Home Currency}}{\text{Foreign Currency}}$ the nominal exchange rate. The domestic country doesn't influence economic world fundamentals.

By depreciation, I mean an increase in $Q$.

In real terms, I'm going to define:

  • Exports: $X=\sigma(Q)y^*$, with $\sigma(Q)$ being a function of $Q$ that gives the share of the domestic country in world trade. With an increase in Q, foreign goods are more expensive relative to home goods, so home share of world trade increases.
  • Imports: $Y=Q m(Q)y$, with $m(Q)$ being the marginal propensity to import (increase in Q $\rightarrow$ decrease in $m(Q)$), and $y$ domestic output.

From the above definitions for exports and imports, we're able to define the trade balance equilibrium output $y_{BT}=\frac{\sigma(Q)}{Qm(Q)}y^*$. So, assuming the Marshall-Lerner condition, we know that a depreciation in $Q$ will make the $y_{BT}$ to increase.

I'm reading a textbook that states: assuming the world real interest rate $r^*$(=domestic real interest rate $r$) is constant, when the $Q$ increases, $y_{BT}$ will increase by more than the domestic output $y$.

Well,I'm trying to check this assertion.

For that I define IS (with keynesian consumption function) as $y=\frac{1}{1-c_1(1-t)+Qm(Q)}(c_0+I(r)+G+\sigma(Q)y^*)$. In this definition I'm trying to keep everything simple, i.e. no lags for the interest and exchange rate. So, when there's a depreciation in $Q$ from $Q_1 \rightarrow Q_2$, we have

  • $\Delta y_{BT}=\left(\frac{\sigma(Q_1)}{Q_1m(Q_1)}-\frac{\sigma(Q_2)}{Q_2m(Q_2)}\right) y^*$
  • $\Delta y=D(\frac{1}{1-c_1(1-t)+Q_2m(Q_2)}-\frac{1}{1-c_1(1-t)+Q_1m(Q_1)})+(\frac{\sigma(Q_2)}{1-c_1(1-t)+Q_2m(Q_2)}-\frac{\sigma(Q_1)}{1-c_1(1-t)+Q_1m(Q_1)})y^*$

where $D$ is the spending independent of $Q$.

From this, how we can conclude $\Delta y < \Delta y_{BT}$?

Any help help would be appreciated.

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