Question

The following function describes how the value of import of country i from country j, Fij is determined:

ln Fij = −14.44 + 0.852 ln Gi + 0.178 ln Gj − 1.119 ln Dij ,

where Gi is GDP in country i, Gj GDP in country j, and Dij the distance between the two countries.

Economists usually convert many aggregate measures of an economy in nature logs. According to the chain-rule d ln x = dx x , which measures the percentage change of x. Hence d ln y d ln x approximates the impact of 1% change of x on y, where the impact is also measured by percentage.

For example, let y be student grade and x study time measured in hours. If y = 0.5x, then dy dx = 0.5 and one additional hours of study will increase grade by 0.5 point. If ln y = 0.5 ln x, then d ln y d ln x = 0.5 and 1% increase of study time will increase grade by 0.5%.

Using derivatives to answer the following questions.

(a) If country i’s GDP increases by 5% , what is the percentage change of its import from country j approximately?

(b) If country j’s GDP decreases by 4% , what is the percentage change of country i’s import from country j approximately?

(c) If the distance between country i and j increases by 3% , what is the percentage change of country i’s import from country j approximately?

I believe that I'm meant to take partial derivatives for each question, but after viewing another chegg response to this question I became a bit confused. For (a) I calculated a derivative of d(-14.44+0.852*lnGi), or 0.852/Gi. That said, I've never been very keen on logs and am a little bit thrown off by the ln on the left side of the equation (ln Fij). Thanks for any help you can provide!

Answer 1

Actually you are absolutely right, here you need to take partial differentiation in each cases.

Let’s assume that “dY/dX = b > 0”, => if “X” increases by “1” unit, => “Y” the dependent variable will increase by “b” units, on the hand if we have “dlogY/dlogX = b > 0”, here the differentiation measure the % change in Y due to 1% change in “X”, => if “X” increases by 1%, => “Y” will increase by “b”%.

Consider the given model given below.

InFij = (-14.44) + 0.852*InGi + 0.178*InGj – 1.119*InDij, where “Fij = import of country “i” from country “j”.

a).

Now, if GDP of “country i” increases by “5%”, => dInGi = 5, => given other factors effecting “Fij” remain same, the “% change in Fij” is “(0.852*5) = 4.26”, => dInFij=4.26. So, if “Gi” increases by “5%” implied the import from “country j” will increase by “4.26%”.

b).

Now, if GDP of “country j” decreases by “4%”, => dInGj = (-4), => given other factors effecting “Fij” remain same, the % change in Fij” is “0.178*(-4) = (-0.712) < 0”, => dInFij = (-0.712). So, if “Gj” decreases by “4%” implied the import from “country j” will also decrease by “(0.712)%”.

c).

Similarly, if the distance between the two country increases by 3%, => the % change in import will be, “dIn Fij = (-1.119)*3 = (-3.357) < 0”, So, the import will decreases by “3.357”%.