Question

An economy has a Cobb–Douglas production function:

Y=Kα(LE)1−αY=Kα(LE)1−α

The economy has a capital share of 0.30, a saving rate of 42 percent, a depreciation rate of 5.00 percent, a rate of population growth of 2.50 percent, and a rate of labor-augmenting technological change of 4.0 percent. It is in steady state.

Solve for capital per effective worker (k∗)(k∗), output per effective worker (y∗)(y∗), and the marginal product of capital.

k∗=k∗=

y∗=y∗=

marginal product of capital =

Answer 1

Production function is given by

Y=K^α(LE)^1−α => Y/(EL) = (K^α(LE)^1−α)/(EL) = (K/(LE))^α

Now, y = Y/(EL) and k = K/(EL)

=> y = k^α

It is given that capital share = 0.30 => α = 0.30

=> y = k^0.3

Steady state occurs when k = 0.

As k = sy - (d + n + g)k

where s = saving rate = 42% = 0.42, d = depreciation rate = 5% = 0.05, n = population growth rate = 2.5% = 0.025, g = rate of labor-augmenting technological change = 4% = 0.04.

Hence, as at steady state k = 0 => k = sy - (d + n + g)k = 0 => 0.42k^0.3 - (0.05 + 0.025 + 0.04)k => k^0.7 = 0.42/0.115

=> k* = 6.36

Hence, Steady state capital per effective worker (k*) = 6.36

Thus y* = k*^0.3 = 6.36^0.3 = 1.74

Hence, Steady state output per effective worker (y*) = 1.74.

Marginal product of capital = dy/dk = 0.3k^-0.7 => 0.36.36^-0.7  = 0.082