Question

A) Considerian economy with the following production function: Y =18K1/3L 2/3 and factor supplies: L = 27, and K = 27. Under the assumptions of the Neoclassical model, compute the equilibrium value of the real wage.

B) Suppose that GDP (Y) initially is 5,000. Investment (I) is given by the equation I = 1500 – 10,000r, where r is the real interest rate. Consumption (C) is given by the equation C = 500 + 0.75(Y – T) . Government spending (G) and taxes (T) are exogenous, G=1000 and T=1000. In this economy, a fall in GDP by 100 will cause the real interest rate to ____ and investment to____. (No computation required for this problem.)

Answer 1

A) The equilibrium real wage under neoclassical model is simply equal to the marginal product of labor.

The production function is given to us as,

Y = 18 (K)^1/3 (L)^2/3

The marginal product of labor will be calculated as,

MPL = Y/L

MPL = (18 × (K)^1/3 (L)^2/3)/L

MPL = 18 (K)^1/3 × 2/3 × (L)^2/3 - 1

MPL = 18×(K)^1/3 × 2/3 × (L)^-1/3

MPL = 18 × 2/3 × (K/L)^1/3

MPL = 12 × (K/L)^1/3

Now putting K = 27 and L = 27 we get,

MPL = 12 × (27/27)^1/3

MPL = 12 × 1

MPL = 12

So the value of real wage here is equal to 12.

B.

Y = C + I + G

Y = 500 + 0.75(Y - T) + 1500 - 10,000r + G

Now putting Y = 5,000 and G = T = 1,000

5,000 = 500 + 0.75(5,000 - 1,000) + 1500 - 10,000r + 1,000

5,000 = 500 + 0.75×3,000 + 1500 - 10,000r + 1000

5,000 = 500 + 3,000 + 1500 + 1,000 - 10,000r

5,000 = 6,000 - 10,000r

10,000r = 6,000 - 5,000

10,000r = 1,000

r = 1,000/10,000

r = 1/10

r = 0.1

Now putting r = 0.1 in investment function we get,

I = 1500 - 10,000r

I = 1500 - 10,000×0.1

I = 1500 - 1000

I = 500

Now GDP Y falls by 100, so new Y = 100.

Y = 500 + 0.75(Y - T) + 1500 - 10,000r + G

4900 = 500 + 0.75(4900 - 1000) + 1500 - 10,000r + 1000

4900 = 500 + 0.75×3900 + 1500 - 10,000r + 1000

4900 = 500 + 2925 + 1500 - 10,000r + 1000

4900 = 5925 - 10,000r

10,000r = 5925 - 4900

10,000r = 1025

r = 1025/10,000

r = 0.1025

Now putting new equilibrium interest rate in investment function we get,

I = 1500 - 10,000r

I = 1500 - 10,000 × 0.1025

I = 1500 - 1025

I = 475.

So due to fall in equilibrium GDP Y by 100 the real interest rate to increases and the investment to fall.

I did the mathematical derivation for your ease to understand, we could have directly said that investment and interest rate are negatively related. And output and investment are directly related. So it the equilibrium output falls then we know it must have been the case that investment must have fallen too. And the relationship between real interest rate and investment being negative which means a fall in investment is associated with an increase in interest rate.