Question

A6-9. Imagine an economy with an average level of human capital (H) equal to 5, an average level of technology (T) equal to 5, and with a labour force (L) and capital stock (K) both equal to 64. Also suppose that the long-run output in this economy is given by Y = (T+H)K12L1/2, (a) Given the information above, calculate Y and Y/L. [3]. (b) Suppose both the K and L increase to 100. What happens to Y and Y/L? Does this aggregate production function exhibit “constant returns to scale” (CRS)? [4]. (c) Imagine that only L increases from 64 to 100. Calculate Y and Y/L. [3] (d) Imagine that only K increases from 64 to 100. Calculate Y and Y/L. [3]. (e) Now suppose that growth is driven not by any increase in the factors of production, but rather by increases in both T and H from 5 to 6.25. What happens to both Y* and Y*/L? [3]. (f) Assuming that the population grows at the same rate as the labour force, so that we can interpret any changes in average product (Y/L) as changes in per capita GDP, rank the growth scenarios described above in terms of which would be most desirable. [4]

Answer 1

(a)

Plugging in given values,

(i) Y = (5 + 5) x (64)1/2 x (64)1/2 = 10 x 8 x 8 = 640

(ii) Y/L = 640 / 64 = 10

(b)

Plugging in new values,

Y = (5 + 5) x (100)1/2 x (100)1/2 = 10 x 10 x 10 = 1000

Y/L = 1000 / 100 = 10

When both K and L are increased by (100/64) times, aggregate Y has increased by exactly (1000/640) = (100/64) times. Hence it exhibits constant returns to scale.

(c)

Plugging in new values,

Y = (5 + 5) x (64)1/2 x (100)1/2 = 10 x 8 x 10 = 800

Y/L = 800 / 100 = 8

(d)

Plugging in new values,

Y = (5 + 5) x (100)1/2 x (64)1/2 = 10 x 10 x 8 = 800

Y/L = 800 / 100 = 8