• Charles Bromley-Davenport

The Most Beautiful Proof in Econometrics: A Mathematical Derivation of the Marshall-Lerner Condition

Like all second year A-Level Economics classes, we were introduced to the Marshall-Lerner Condition while studying the key macroeconomic theme ‘Balance of Payments’.

Briefly explained, the Marshall-Lerner Condition (MLC) states that for an improvement in the Balance of Payment’s Current Account following a currency devaluation, the total sum of demand elasticity for imports and exports must be greater than 1.

This finding therefore acts as the logical premise behind the distinct shape of the ‘J-Curve’, which models the effect of the Balance of Payment initially worsening in the short-run, before an improvement will be noticed (as modelled below).

The initial worsening can be explained through the ‘lag period’ where demand for imports are inelastic due to firms being trapped in now more expensive pre-existing contracts, or as a consequence of inelastic supply in the short-run. This downwards pressure accumulates further through demand for exports likely being inelastic during a similar period as a result of foreign consumers having to adjust to price changes over time – and so the volume of exports is largely rigid. Thus causing an initial deterioration.

In the medium to long-run however, as the demand for imports and exports become more elastic (PED > 1), firms are able to negotiate more price competitive export contracts, causing an increase in the volume of exports, as well as imports being expected to conversely reduce due to the prices having relatively appreciated in value. This therefore indicates the gradual improvement in the Current Account.

I have modelled the mathematical logic behind this idea below.