marshall lerner condition

i don't get where this comes from?
i went to wiki, but the answer wasn't clear, they said:

Trade (B)alance = X*P - M
where X is the amount exported
P is the export price/amount
M is the import expenses

then they differenciated wrt P giving:

dB/dP = X*Ex + X -M/P*Em

where Ex & Em are the export & import elasticities
then dividing by X, noting X*P = M (as initially in balance), noting dB/dP<0
then 1< |Em| +|Ex|

but in this derivation they seem to mix their definition of P - is it international price i.e. $$'s or local price i.e pounds?? what exactly are they differenting by?

does anyone have a greater understanding of where this condition comes from?

 

Some of the assumptions for the Marshall - Lerner Condition to hold:

i) Domestic and foreign prices remain constant, therefore one can use the nominal exchange rates as the real exchange rates

ii ) Supply elasticities are infinite, therefore changes in the trade balance is affected only by changes in the demand

iii ) At the time of depreciation (or appreciation) concerned, the trade balance is zero

iv) There is no 'cross price' elasticities between exports and imports, therefore imports and exports are independent of each other

v) Income levels between exporters and importers are kept constant, therefore demand is only affected by the exchange rate

(Reference: Salasevecius R. & Vaicius P. (2003). Exchange Rate-Trade Balance Relationship: Testing the Marshall - Lerner Condition in the baltic states, www2.sseriga.edu.lv/library/working_papers/FT_2003_13.pdf )


The trade balance stands for "...the difference between the monetary value of exports and imports in an economy over a certain period of time.."
( http://en.wikipedia.org/wiki/Trade_balance )

Letting the local currency be pounds and the international currency be dollars
According to your definitions I think the p should stand for the amount of dollars for one pound i.e. the $ : pound exchange rate
It is just a variable, however, I think the elasticity is usually defined in terms of the pound:$ exchange rate
There B is defined in the foreign currency and not correctly
it should be B = X/p - M, i think

Note that if the dollar to pound exchange rate is 1 pound = 2.03 dollars then p is taken as 2.03

Now if we want a devaluation in the local currency
then the exchange rate weakens, thus more dollars can buy 1 pound and p would increase
therefore to have a positive impact on the trade balance, which is what the M-L condition is looking for, the slope of the function needs to be positive (not negative)
thus we require dB/dp to be greater (not less than) 0
and the proof does not work out

Therefore you are 'technically' correct, we need dB/dp to be greater than 0, They don’t have an sensible definition of p
According to their definition of p, I don't think the proof works out!


Here is a better proof?

Marshall – Lerner Condition Proof:
Let B = Trade Balance
Let Xv = Real volume of Exports
Let pd = Price of domestic goods
Let Iv = Real volume of Inports
Let pi = Price of foreign goods
Let E = Nominal exchange rate (i.e. amount of domestic currency per unit of foreign currency)

Therefore
B = (Xv).(pd) - (Iv).(pi).(E)

(note B is defined in the local currency)

Now by assumption i) i.e. Domestic and foreign prices remain constant, therefore one can use the nominal exchange rates as the real exchange rates

We get:

B = X - (M). (E)

Where X and M are now the nominal prices of exports and imports respectively

Note X is a function of E

and E stands for the amount of pounds per one dollar

Differentiating w.r.t E
dB/dE = (dX)/(dE) - (dM/dE).(E) - (M).(dE/dE) ... (eqn a)

Now if a devaluation in the local currency occurs (i.e. E increases since more pounds are needed for one dollar), this should lead to an increase in the balance of trade therefore dB/dE > 0
i.e. the slope of the function is positive and B and E are positively related

Now from eqn a

(dX)/(dE) - (dM/dE).(E) - (M).(dE/dE) > 0

which implies
[(dX)/(dE)].[(E/X).(X/E)] - [(dM/dE).(E)].[M/M] - (M).(1) > 0

then
[(dX/dp).(E/X)][X/E] - [(dM/dE).(E/M)].[M] - M > 0 ... (eqn b)

Now define the elasticity’s of export and import as:
Ex = (dX/dE).(E/X)
and Em = (dM/dE).(E/M)

then substituting Ex and Em into eqn b implies:

(Ex).(X/E) – (Em).(M) – M > 0

Dividing by M gives:

[Ex]. [(X)/(ME)] – Em -1 > 0 ... (eqn c)


Now by assumption iii) i.e. At the time of depreciation (or appreciation) concerned, the trade balance is zero
implies that

B at outset = 0
therefore X - (M).(E) = 0
and therefore X = (M).(E)

therefore eqn c becomes:

Ex – Em -1 > 0

therefore

Ex – Em > 1

therefore

| Ex – Em | > |1|

(since 1 > 0 and Ex-Em > 1 > 0 )

Now from absolute value algebra:

| Ex + (-Em) | <= |Ex | + |-Em|

and |-Em| = |Em|

therefore

|1| < |Ex - Em| <= |Ex| + |Em|

which implies that:

|1| <= |Ex| + |Em|

and thats the proof!


Alternatively we could define B in the foreign currency
then using the same notation
B = X/E - M and dB/dE > 0
The proof is similar to the above, however, a few more tricks are required

i.e. i) differentiate w.r.t. E,
ii) then set > 0,
iii) multiply by E^2,
iv) divide by X,
v) define Ex and Em as before and substitute into the equation
vi) set X/E = M and cancel out a factor by setting it equal to 1
vii) It leaves Ex-1-Em >0
viii) Use absolute-value algebra to get 1 <= |Ex| and |Em| as shown in the proof above


This was a very tricky and challenging question!

I hope this helps?

 

nice introduction of E, it has also clarified my definitions of elasticity.
thanks