Nov 25 2013

## A mathematical proof of the Marshall Lerner Condition

One of the toughest topics to teach in IB higher level Economics is the Marshall Lerner Condition, which is an International Economics concept which states the following:

If the combined price elasticities of demand of a nation’s imports and exports is greater than one (PEDx + PEDm > 1), then a depreciation or a devaluation of the nation’s currency will move its current account balance towards surplus.

This is a concept I have been teaching for eight years now, and I have even written about it in my textbook and produced a YouTube video lecture explaining it to students, but one thing I’ve never done is attempted a mathematical proof of the concept (needless to say, I avoid using math as much as possible, and the prospect of “proving” the MLC was always too daunting).

But this evening I received an email from an Economics teacher in Paris asking for just such a proof. So I buckled down and worked it out. In her email, the teacher said:

The Marshall Lerner Condition states that if the PEDx + PEDm > 1 then a depreciation in a country’s currency will reduce a current account deficit.

Suppose the PED for exports = .6 and the PED for imports = .5. The sum is greater than 1, therefore the MLC is met. A depreciation of this country’s currency should therefore improve its current account balance.

But based on my analysis, this country’s current account should be getting worse, not better.

For Exports: price is decreasing but the quantity demanded is increasing by proportionally less (since PEDx = 0.6) so the country’s total export revenue is decreasing

For Imports: price is increasing and quantity demanded is decreasing by proportionally less (since PEDm = 0.5) so the country’s total spending on imports is increasing

The country’s revenues from exports are decreasing while the country’s spending on imports are increasing, so overall the trade balance is getting worse (moving deeper into deficit) not improving.

What am I doing wrong?

This teacher’s email really stumped me at first, because her logic is totally sound. I figured the only way I was going to be satisfied was if I worked it mathematically. So here’s the result and the reply I sent to the teacher:

Hello,

Your email really got me thinking about this. Your logic stumped me at first, but then inspired me to go work it out with numbers. So, hopefully my “proof” of the MLC below will clarify your confusion.

To simplify the analysis we will use easy numbers. I will use your values of PEDx = 0.6 and PEDm = 0.5

**Assumptions:**

- The US and Canada are trading partners
- Current exchange rate: $1 US = $1 CA
- US exports 10 widgets at $1 US apiece for a total export revenue of $10 US
- US imports 10 wingdings at $1 CA apiece for a total import expenditure of $10 US
- US trade balance: $10 – $10 = 0
- PEDx = 0.6 and PEDm = 0.5

Next, assume the US $ depreciates by 10% against the CA $. Now,

- $1 US = $0.90 CA
- $1 CA = $1.11 US

Impact on imports:

- Price to Americans of Canadian wingdings rises to $1.11 US
- Quantity demanded falls by 5.5% to 9.45
- Total expenditures on Canadian imports expressed in US $: $1.11 x 9.45 = $10.49

In order for the US trade balance to improve *US export revenues must increase by more than $0.49 US*.

Impact on exports:

- Price to Canadians of US widgets falls by 10% to $0.90 CA
- Quantity demanded increases by 6% to 10.6
- Total revenue from exports to Canada expressed in CA $: $0.90 x 10.6 = $9.54 CA.
- Since $1 CA = $1.11 US, the value of US exports to Canada expressed in US $ is $9.54 x $1.11 = $10.59

Expressed in US $, exports increased by $0.59 and imports increased by $0.49.

*Therefore, US net exports are now $10.59 – $10.49 = $0.1. The MLC is met and the US trade balance moves into surplus.*

I think the only mistake with the logic you applied in your email was that you were not considering that a country’s balance of trade is measured in *its own home currency*. As you can see, if we measured the value of US exports following the depreciation in *Canadian dollars*, the export revenues actually decreased following the depreciation of the US $, moving the US into a current account deficit. But even though Canadians are spending less of their own dollars on US goods, the Canadian dollar has now appreciated by 11%, therefore the value of US exports expressed in US $ actually *increases* (due to the now weaker US $)!

I hope this all makes sense! Thanks for inspiring me to buckle down and tackle this analysis! I’ve been teaching this concept for eight years and have never actually taken the time to walk through a proof like this.

Best,

Jason