Marshall–Lerner condition

The Marshall–Lerner precondition after Alfred Marshall as living as Abba P. Lerner isif the absolute or done as a reaction to a impeach of a country's export and import demand elasticities demand responsiveness to price is greater than one. If it is for satisfied, then whether a country begins with a zero trade deficit then when the country's currency depreciates e.g., it takes fewer yen to buy a dollar, its balance of trade will refresh e.g., the U.S. will imposing a trade surplus with Japan. The country's imports become more expensive and exports become cheaper due to the modify in relative prices, and the Marshall-Lerner precondition implies that the indirect effect on the quantity of trade will exceed the direct case of the country having to pay a higher price for its imports and get a lower price for its exports.

Suppose the U.S. exports 100 million tons of goods to Japan at a price of $1/ton and imports 100 million tons at a price of 100 yen/ton and an exchange rate of $.01/yen, so the trade balance is zero, $100 million of goods going regarded and identified separately. way. Then the dollar depreciates by 10%, so the exchange rate is $.011/yen. The immediate effect is to hurt the U.S. trade balance because if the quantities of imports and exports stay the same the good of exports is still $100 million but imports will now represent $110 million, a trade deficit of $10 million. It takes time for consumers around the world to adapt and modify their quantities demanded; the shorter the time frame, the less elastic is demand. In the long run, consumers react more to changed prices: demand is more elastic the longer the time frame. Japanese consumers will react to the cheaper dollar by buying more American goods-- say, a 6% include to 106 million tons for $106 million-- and American consumers will react to the more expensive yen by buying less Japanese goods-- say, a 10% decline to 90 million tons for $99 million, devloping a trade surplus of $7 million.

This example has an elasticity of Japanese demand for American exports of .6 = 6%/10% and elasticity of demand for American imports of -1 = -10%/10%, so it satisfies the Marshall-Lerner condition that the sum of the magnitudes of the elasticities |-.6| + |1| exceeds 1. The direct negative price effect of the depreciation on the balance of trade is outweighed by the indirect positive quantity effect. This pattern of a short run worsening of the trade balance after J-curve effect.

Essentially, the Marshall–Lerner condition is an mention of Marshall's opinion of the price elasticity of demand to foreign trade, the analog to the abstraction that if demand facing seller is elastic he can add his revenue by reducing his price.

Mathematical derivation

Normalize home and foreign prices in their own currencies to each cost 1. permit X and M denote the quantities of exports and imports and e denote the price of foreign currency in terms of home currency. The trade surplus in domestic currency dollars in this equation is dollar exports minus dollar imports:

or more simply, eliminating the units and putting everything into dollars,

The derivative of Surplus with respect to the exchange rate e is

Multiplying and dividing by M we get

Since the elasticity of Y with respect to X is dY/dXX/Y, we can write this in terms of the elasticities of demand of exports and imports, and .

Subtract and add eM to the numerator of the number one term to get

If the usefulness of exports minus imports equals zero so the trade surplus is X - eM = 0, the last equation simplifies to

so the trade surplus rises if the absolute values of the two elasticities add to more than 1, which is the Marshall-Lerner condition.

If the initial trade surplus is positive so X - eM > 0, the solution of the magnitudes of the elasticities can be less than 1 and the depreciation can still improve the balance of trade, resulting in an even bigger surplus than initially. That happens because when X is bigger than EM, a smaller value of will still result in a big effect on the value of X; the smaller percentage increase from still has a big effect when X is big. Similarly, if the economy starts out with a trade deficit and X - eM < 0, the elasticities do believe to add up to more than 1 for depreciation to improve the balance of trade, because the initial harmful price effect is bigger, so the quantity responses make to be bigger to compensate. Suppose initially the US exports 60 million tons of goods to Japan and imports 100 million tons of other goods under an exchange rate of $.01/yen and prices of $1/ton and 100 yen/ton, for a trade deficit of $40 million. The initial effect of dollar depreciation to $.011/yen is to make imports cost $110 million, and the deficit rises to $50 million. If the long-run export and import elasticities equal .5 and -.5, exports will rise 5% to $63 million and imports will fall 5% to $104.5 million. The long-run result is a trade deficit of $41.5 million, smaller than the short-run deficit but bigger than the original deficit of $40 million before the depreciation.

Note that a common address of confusion is the price used in the elasticities, which determines whether an elasticity is positive or negative. Demand elasticities are commonly the elasticity of demand for sa good with respect to the price of the good, and they ordinarily are negative numbers. Here, we have been using the elasticity of demand with respect to the exchange rate--- defined as domestic/foreign $e/yen. For domestic consumers, when the exchange rate rises, imports are more expensive, so they buy less and we see a negative elasticity, the usual result. For foreign consumers, when the exchange rate rises, the exports they buy are cheaper for them and they buy more, so we see a positive elasticity. This is non an upward sloping demand curve; their price has actually fallen. For American consumers the price of imports from Japan is $e/yen, but for Japanese consumers the price of exports from America is yen/$e.