Intermediate Macroeconomics 2 Coursework

Question 1

a)

The production function is Y=AKαL1-α as it is a constant-returns-to-scale production function. Computing the Solow residual giver you ∆AA , which is the rate of technological progress.

From the question: α=0.2 ; ∆LL=n=0.01; ∆KK=0.07 ; ∆YY=0.05

Taking natural logarithms of the production function gives: ln⁡(Y)=ln⁡(A)+αln⁡(K)+1-αln⁡(L)

Taking the first difference of both sides yields:

ln⁡(Y)-ln⁡(Y-1)=ln⁡(A)-ln⁡(A-1)+α(lnK-lnK-1)+1-α(ln⁡(L)-ln(L-1))

From here we can use the approximation: lnX-lnX-1≅X-X-1X-1=∆XX
Giving:

∆YY=∆AA+α∆KK+1-α∆LL

This is the growth accounting equation and can be rearranged to give the Solow residual:

∆AA=∆YY-α∆KK-1-α∆YY

∆AA=0.05-0.20.07-0.80.01=2.8%

b)

The new production function is: Y=Kα(AL)1-α

I will compute the Solow residual in the same way as before. Starting with taking natural logarithms:

ln⁡(Y)=αln⁡(K)+(1-α)ln⁡(A)+1-αln⁡(L)

Taking the first difference:

ln⁡(Y)-ln(Y-1)=α(ln⁡(K)-ln⁡(K-1))+(1-α)(ln⁡(A)-ln(A-1))+1-α(ln⁡(L)-ln(L-1))

Using the approximation: : lnX-lnX-1≅X-X-1X-1=∆XX to give the growth accounting equation:

∆YY=α∆KK+(1-α)∆AA+1-α∆LL

And rearranging to give the Solow residual and plugging the numbers in:

∆AA=11-α∙∆YY-α1-α∙∆KK-∆LL=3.5%

Note that the growth rate of technology when the labour-augmented production function is used, of 3.5% is greater than that of the Standard Cobb-Douglas, of 2.8%. Your boss was hoping it would fall, but it has risen.

c)

The Long run growth rate of human capital is ∆HH=2.5%

Our new production function is: Y=Kα(AHL)1-α

Again, computing the Solow residual in the same was as before. Taking natural logarithms:

ln⁡(Y)=αln⁡(K)+1-αln⁡(A)+1-αln⁡(H)+1-αln⁡(L)

After taking the first differences and making the approximation: lnX-lnX-1≅X-X-1X-1=∆XX
We get: