Question

Growth and capital over-accumulation

Suppose two countries, A and B, with the same production function Y = KαL1−α.
The value of α is 0.30, the growth rate of population is 2% and the depreciation rate is 5%.
a) Show that with price-taking firms the share of labor must be 1 − α.
b) Compute the stock of capital, output and consumption per unit of labor in the steady
state if the savings rates were 25% for country A and 35% for country B.
c) Compare both economies to the Golden Rule.
d) Explain what would happen to both countries if suddenly their savings rate became
the Golden Rule savings rate.

Answer 1

The labor share with price-taking firms would be 1 - α. The stock of capital, output, and consumption per unit of labor can be calculated using the savings rates. Comparing to the Golden Rule requires finding the savings rate that maximizes consumption per unit of labor. A sudden change to the Golden Rule savings rate would impact the levels of capital, output, and consumption per unit of labor in both countries.

Step-by-step explanation:

With price-taking firms, the share of labor is determined by the value of α in the production function Y = KαL1−α. Since α is 0.30, the share of labor would be 1 - 0.30 = 0.70.

To compute the stock of capital, output, and consumption per unit of labor in the steady state, we need to use the savings rates for country A and B. The formula for capital per unit of labor is K/L = (s * Y) / (n + d + g). Using the savings rates of 25% for country A and 35% for country B, we can calculate the values. Note that n is the growth rate of population, d is the depreciation rate, and g is the growth rate of technology.

Comparing both economies to the Golden Rule requires calculating the levels of capital, output, and consumption per unit of labor that maximize utility. This can be done by finding the savings rate that leads to the highest level of consumption per unit of labor in the steady state.

If the savings rate suddenly became the Golden Rule savings rate, it would lead to changes in the levels of capital, output, and consumption per unit of labor in both countries. The exact changes would depend on the initial savings rates and the difference between the previous savings rates and the Golden Rule savings rate.