A devaluation is supposed to improve the trade balance. Whether it does depends on a race between price and quantity, and the condition that settles the race is simpler than it looks.
Set up the accounting first, because most confusion here is accounting confusion. Export prices are sticky in the home currency. A devaluation therefore lowers the foreign-currency price of exports by the full amount, and foreign buyers respond with elasticity η_X. Import prices are sticky in the foreign currency. A devaluation raises the home-currency price of imports by the full amount, and domestic buyers cut volume with elasticity η_M. For a devaluation of size d from balanced trade with exports and imports both equal to X₀:
- Home-currency export value rises by
X₀ × η_X × d, because volume rises and the home price did not change. - Home-currency import value changes by
−M₀ × (1 − η_M) × d, because the price rose by the full d while volume fell by η_M × d.
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Marshall-Lerner condition formula
Add them and the whole condition falls out:
ΔTB = X₀ × d × (η_X + η_M − 1)
The trade balance improves if and only if η_X + η_M > 1. That is the Marshall-Lerner condition, and the reason it is a sum rather than a product is now visible: each elasticity has to earn its keep against the single unit of adverse price effect on the import bill.
Test it on stated numbers. With η_X = 0.45 and η_M = 0.40, the sum is 0.85 and a 15% devaluation on a balanced ₹100 billion of trade worsens the balance by ₹2.25 billion. With η_X = 1.00 and η_M = 0.80, the sum is 1.80 and the same devaluation improves it by ₹12.00 billion.
What if trade is not balanced to start with?
Trade is rarely balanced to start with, and the condition generalises. With exports of 90 and imports of 120, the change per unit of devaluation is X₀ η_X + M₀ η_M − M₀, and at η_X = 0.45, η_M = 0.40 that is −31.500000 per unit. Break-even now requires X₀ η_X + M₀ η_M = M₀, which no longer collapses to a condition on the bare sum, so say which elasticity you are moving: hold η_M at 0.40 and η_X has to reach 0.800000, taking the sum to 1.200000 against 1.00 on balanced trade. The adverse price effect applies to the whole import base while the favourable volume effect applies to the smaller export base. A country with a big trade deficit needs more elasticity, not less, to devalue its way out.
What is the J-curve?
Now the timing, which is where the J-curve lives. Elasticities are small on impact and grow, because contracts are already signed and substituting a supplier takes quarters, so the sum starts below 1 and rises through it. Worked Example 6 tabulates the path for a 15% devaluation: the trough is −₹11.25 billion in the first quarter, 11.25% of pre-devaluation exports, and the balance crosses back through zero at quarter 3.400000. Draw it and you get the letter J. Do not confuse it with the private-equity J-curve, which describes a fund's net cash flows to investors and shares nothing with this except a shape.
The J-curve is the single most useful piece of forecasting discipline in the trade-balance material, because it tells you what a correct policy looks like while it is failing. A finance ministry that devalues and sees the trade deficit widen for three quarters is not necessarily wrong. It may be at quarter two. Judging the policy at the trough is how a correct decision gets reversed.