Marshall–Lerner condition

The Marshall–Lerner condition (after Alfred Marshall and Abba P. Lerner) has been cited as a technical reason why a reduction in value of a nation's currency need not immediately improve its balance of payments.^[1] The condition states that, for a currency devaluation to have a positive impact on trade balance, the sum of price elasticity of exports and imports (in absolute value) must be greater than 1.

As a devaluation of the exchange rate means a reduction in the 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.^{[citation needed]} 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. For example, assume a country is a net importer of oil and a net producer of ships. Initially, the devaluation immediately increases the price of oil, and as consumption patterns remain the same in the short term, an increased sum is spent on imported oil, worsening the deficit on the import side. Meanwhile, it takes some time for the shipbuilder's sales department to exploit the lower price and secure new contracts. Only the funds acquired from previously agreed contracts, now devalued by the currency devaluation, are immediately available, again worsening the deficit on the export side.

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 denominated in domestic currency, with domestic and foreign prices normalized to one, is given by:

N_x = X − Qe

where X denotes exports, and Q imports.

Differentiating with respect to e gives:

$\frac{\partial N_x}{\partial e} = \frac{\partial X}{\partial e} - e\frac{\partial Q}{\partial e} - Q$

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 η_Xe and η_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:

$\eta_{Xe} + \left|\eta_{Qe}\right| > 1$

References

1. ^ Davidson, Paul (2009), The Keynes Solution: The Path to Global Economic Prosperity, New York: Palgrave Macmillan, p. 125, ISBN 9780230619203 .

Further reading

• Rose, Andrew K. (1991), "The role of exchange rates in a popular model of international trade: Does the ‘Marshall–Lerner’ condition hold?", Journal of International Economics 30 (3–4): 301–316, doi:10.1016/0022-1996(91)90024-Z .