Marshall-Lerner Condition

This condition says that, for a currency devaluation to have a positive impact in trade balance, the sum of price elasticity of exports and imports (in absolute value) must be greater than 1. The principle is named for economists Alfred Marshall and Abba Lerner.

As a devaluation of the exchange rate means a reduction on price of exports, quantity demanded for these will increase. At the same time, price of imports will rise and their quantity demanded will diminish.

The net effect on the trade balance will depend on price elasticities. If goods exported are elastic to price, their quantity demanded will increase proportionately more than the decrease in price, and total export revenue will increase. Similarly, if goods imported are elastic, total import expenditure will decrease. Both will improve the trade balance.

Empirically, it has been found that goods tend to be inelastic in the short term, as it takes time to change consuming patterns. Thus, the Marshall-Lerner condition is not met, and a devaluation is likely to worsen the trade balance initially. In the long term, consumers will adjust to the new prices, and trade balance will improve. This effect is called J-Curve effect.

Mathematical Derivation

Here 'e' is defined as the price of one unit of foreign currency in terms of the domestic currency.

Using this definition, the trade balance is given by:

$N\_x\; =\; X\; -\; Qe$

where X denotes exports, and Q imports.

Differentiating with respect to e gives:

$frac\{partial\; N\_x\}\{partial\; e\}\; =\; frac\{partial\; X\}\{partial\; e\}\; -\; efrac\{partial\; Q\}\{partial\; e\}\; -\; Q$

Dividing through by X:

$frac\{partial\; N\_x\}\{partial\; e\}frac\{1\}\{X\}\; =\; frac\{partial\; X\}\{partial\; e\}frac\{1\}\{X\}\; -\; frac\{e\}\{X\}frac\{partial\; Q\}\{partial\; e\}\; -\; frac\{Q\}\{X\}$

At equilibrium, $X\; =\; eQ$. Therefore:

$frac\{partial\; N\_x\}\{partial\; e\}frac\{1\}\{X\}\; =\; frac\{partial\; X\}\{partial\; e\}frac\{1\}\{X\}\; -\; frac\{1\}\{Q\}frac\{partial\; Q\}\{partial\; e\}\; -\; frac\{1\}\{e\}$

Multiplying through by e:

$frac\{partial\; N\_x\}\{partial\; e\}frac\{e\}\{X\}\; =\; frac\{partial\; X\}\{partial\; e\}frac\{e\}\{X\}\; -\; frac\{partial\; Q\}\{partial\; e\}frac\{e\}\{Q\}\; -\; 1$

Which can be expressed as $frac\{partial\; N\_x\}\{partial\; e\}frac\{e\}\{X\}\; =\; eta\_\{Xe\}\; -\; eta\_\{Qe\}\; -\; 1$

where $eta\_\{Xe\}$ and $eta\_\{Qe\}$ are common notation for the elasticity of exports and imports with respect to the exchange rate respectively.

In order for a fall in the relative value of a country's currency (i.e. a rise in e using the above definition) to have a positive effect on that country's trade balance, the left hand side of the equation must be positive (i.e. for a rise in e to cause a rise in $N\_x$)

Therefore:

$eta\_\{Xe\}\; -\; eta\_\{Qe\}\; -\; 1\; >\; 0\; Rightarrow\; eta\_\{Xe\}\; -\; eta\_\{Qe\}\; >\; 1$

Which can be written as:

$eta\_\{Xe\}\; +\; left|eta\_\{Qe\}\; ight|\; >\; 1$