Answer :
There is no golden rule level of capital per effective worker, output per effective worker, and consumption per effective worker in this scenario.
The Solow growth model is used to analyze the long-term economic growth of a country. In this model, the steady state occurs when the capital stock per effective worker remains constant over time.
a. To find the steady state capital per effective worker, we need to use the formula:
s * Y / (n + g + δ)
Where:
- s represents the savings rate, which is 20% in this case.
- Y is the output per effective worker.
- n is the population growth rate, which is 2%.
- g is the growth rate of the efficiency of labor, which is 3%.
- δ is the depreciation rate of the capital stock, which is 5%.
First, let's calculate the steady state capital per effective worker:
0.2 * Y / (0.02 + 0.03 + 0.05) = K / L
Simplifying this equation, we get:
0.2 * Y / 0.1 = K / L
2 * Y = K / L
Next, let's calculate the output per effective worker. The capital share of GDP is 50%, so we can write the equation as:
Y = (0.5 * K / L)^(1/2)
Now, we can substitute the value of Y into the steady state equation:
2 * (0.5 * K / L)^(1/2) = K / L
Squaring both sides of the equation:
4 * (0.5 * K / L) = K / L
2 * K / L = K / L
This implies that the steady state capital per effective worker (K / L) is equal to 2. Therefore, the steady state capital per effective worker is 2.
To find the output per effective worker (Y), we substitute the value of K / L into the output equation:
Y = (0.5 * 2)^(1/2) = 1
Lastly, to find the consumption per effective worker (C), we can use the equation:
C = (1 - s) * Y
Substituting the values:
C = (1 - 0.2) * 1 = 0.8
Therefore, the steady state capital per effective worker is 2, the output per effective worker is 1, and the consumption per effective worker is 0.8.
b. The golden rule level of capital per effective worker is the level that maximizes the long-run average level of consumption per effective worker. To find this level, we need to find the savings rate that maximizes the consumption per effective worker.
Using the consumption equation:
C = (1 - s) * Y
Differentiating this equation with respect to s and setting it equal to 0:
dC/ds = -Y = 0
Solving for Y:
Y = 0
This implies that the consumption per effective worker is 0, which is not feasible.
Therefore, there is no golden rule level of capital per effective worker, output per effective worker, and consumption per effective worker in this scenario.
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