Answer to Question #174955 in Macroeconomics for George

Question

Real GDP in Canada is produced using the following production function: Y = AK^0.5 N^0.5 Where A is productivity, K is the capital stock and N is the size of the labour force. Households save a constant fraction of output: S=0.3Y The labour force grows at a rate of 5% and capital depreciates at a rate of 5%, therefore: n=0.05 and d=0.05

a) Suppose that productivity, A, is constant at A=5. What is the steady state capital per capita, output per capita and consumption per capita?

b) In order for output per worker to double in the steady state, what must be the increase in A if nothing else changes? Solve for the new level of A.

c) Suppose that the savings rate increase to 0.5. What is the new steady state capital, output and consumption per capita?

d) Based on your result in part c), do you think the initial savings rate of 0.3 was the golden rule level of saving?

Answer 1

a) $Y =5K ^{0,5 }N ^{0,5}$

α+β=1

$Y=\sqrt{(KN)}$

$y=\frac{Y}{L}$

k=K/L

$\frac{Y}{L}=\frac{\sqrt{(KL)}}{L}$

$y=\sqrt{(\frac{K}{L})}=\sqrt{k}$

b) A=2

c)

Let's assume that the economy is in a stable state: with a stable level of capital strength kx and a corresponding savings rate Sy. Suppose that under the influence of external changes, the savings rate increased to s2. This will lead to an increase in the sustainable level of capital strength to

k*2, since the investment at k* will exceed the level necessary to maintain k at the same level and capital strength will begin to grow until it reaches k2.

Labor productivity = Ef (k) will increase due to the growth of A; and with the growth of labor efficiency E, so in the transition period, the rate of labor productivity growth will exceed g. As soon as k reaches k2, the rate of labor productivity growth will fall to g. Thus, an increase in the savings rate will lead to a temporary increase in the rate of labor productivity growth.

This change affects the level of capital and productivity, rather than the rate of their growth in a stable state.

d)According to the Solow model, each level of the savings rate corresponds to a certain stable state. Therefore, there is a problem of choosing the optimal savings rate.

When making a choice in favor of a particular sustainable state, a politician with the goal of maximizing the economic well-being of society will want to choose a sustainable state with the highest level of consumption.

The level of capital accumulation that provides a stable state with the highest level of consumption is called the Golden Level of Capital Accumulation.

When the initial capital ratio is higher than the Golden Rule, the achievement of a stable state with a maximum consumption is accompanied by a higher level of consumption.

When the initial capital ratio is lower than the Golden Rule, achieving a stable state under the Golden Rule requires an immediate reduction in consumption in the present in order to increase it in the future.