Answer :
The per-worker production function is Y/L = (K/L)^0.5. In the steady state, with given saving and depreciation rates, capital stock per worker is 4, output per worker is 2, and consumption per worker is 1.6. The golden rule level of capital stock occurs when the saving rate is 1.
a. To find the per-worker production function, we divide both sides of the given production function by the quantity of labor (L):
Y/L = (K/L)^0.5 * (L/L)^0.5
Simplifying, we get:
Y/L = (K/L)^0.5
Therefore, the per-worker production function is Y/L = (K/L)^0.5.
b. In the steady state, the capital stock per worker (k) remains constant, implying that investment per worker equals depreciation per worker. The production function can be written as:
Y = F(K, L) = K^0.5 * L^0.5
Since Y/L represents output per worker, we have:
Y/L = (K/L)^0.5
Using the definition of steady state, where investment per worker (sY/L) equals depreciation per worker (δK/L), we get:
sY/L = δK/L
Substituting the per-worker production function, we have:
s(K/L)^0.5 = δK/L
Rearranging the equation, we find:
(K/L)^0.5 = s/δ
Squaring both sides, we obtain:
K/L = (s/δ)^2
Therefore, in the steady state, the capital stock per worker (k) is given by (s/δ)^2, output per worker (y) is equal to (k^0.5), and consumption per worker (c) is equal to (1 - s) * y.
c. Given a depreciation rate of 10% (δ = 0.1) and a saving rate of 20% (s = 0.2), we can substitute these values into the equations derived in part b.
Capital stock per worker (k) = (s/δ)^2 = (0.2/0.1)^2 = 4
Output per worker (y) = k^0.5 = 4^0.5 = 2
Consumption per worker (c) = (1 - s) * y = (1 - 0.2) * 2 = 1.6
Therefore, in this scenario, the steady-state capital stock per worker is 4, output per worker is 2, and consumption per worker is 1.6.
d. The golden rule level of capital stock occurs when consumption per worker is maximized. To find this level, we need to determine the saving rate (s) that maximizes consumption per worker (c).
Differentiating the consumption function with respect to s and setting it equal to zero, we can find the saving rate that maximizes consumption:
dc/ds = y - (1 + s) * dy/ds = 0
Simplifying the equation and substituting the values of y and dy/ds, we have:
2 - (1 + s) * 0.5 * (k^(-0.5)) = 0
Simplifying further:
2 - 0.5 * (1 + s) * k^(-0.5) = 0
Solving for s, we find:
1 + s = 4 * k^(-0.5)
s = 4 * k^(-0.5) - 1
Substituting the steady-state capital stock per worker (k = 4) into the equation, we get:
s = 4 * 4^(-0.5) - 1
s = 4 * 0.5 - 1
s = 2 - 1
s = 1
Therefore, the golden rule level of capital stock occurs when the saving rate (s) is equal to 1.
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