Question

Consider an economy described by the aggregate production function
Y = F(K,L) = K^0.3L^0.7
(a) Does this production function have constant returns to scale (CRS)? Why?
(b) What is the marginal product of capital (MPK)?
(c) What is the marginal product of labor (MPL)?
(d) Use your answers in (b) and (c) above to show that this production function displays
diminishing marginal product (DMP).
(e) Are K and L "complements"? Explain in some detail.
(f) What is the capital share?
(g) When capital per worker rises, what happen to labor share of national income?

Answer 1

Answer on Question #66850 – Economics - Macroeconomics

Question

Consider an economy described by the aggregate production function $Y = F(K,L) = K^0.3L^0.7$.

(a) Does this production function have constant returns to scale (CRS)? Why?

(b) What is the marginal product of capital (MPK)?

(c) What is the marginal product of labor (MPL)?

(d) Use your answers in (b) and (c) above to show that this production function displays diminishing marginal product (DMP).

(e) Are K and L "complements"? Explain in some detail.

(f) What is the capital share?

(g) When capital per worker rises, what happen to labor share of national income?

Answer

(a) Does this production function have constant returns to scale (CRS)? Why?

Yes, because

$\alpha + \beta = 1$

$0.3 + 0.7 = 1$

(b) What is the marginal product of capital (MPK)?

Marginal product of capital shows how much extra output an extra unit of capital can produce

$MPK = 0.3K^{(-0.7)}L^{0.7}$

(c) What is the marginal product of labor (MPL)?

$MPL = 0.7K^{0.3}L^{(-0.3)}$

(d) Use your answers in (b) and (c) above to show that this production function displays diminishing marginal product (DMP).

$(-0.7) < 0$ and $(-0.3) < 0$ this means, holding other variables constant, that the increase in K decreases MPK and that the increase in L decreases MPL.

(e) Are K and L "complements"? Explain in some detail.

Yes they are, because when L increases, MPK grows and vice versa, when K enlarges, MPL rises.

(f) What is the capital share?

Capital share $(\alpha) = 0.3$

(g) When capital per worker rises, what happen to labor share of national income?

Labor share declines

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