A. To derive the equation for output per effective worker, we need to divide both sides of the production function by EN:
Y = 100(K^3/10)(EN^7/10)
Divide both sides by EN:
Y/EN = 100(K^3/10)(1)
So, the equation for output per effective worker is:
y = 100 * k^3/10, where k = K/EN.
B. To calculate the steady state levels, we first need to know the depreciation rate (δ) of capital. Since capital lasts for an average of fifteen years, we can assume the depreciation rate is 1/15 = 0.0667.
Next, we use the Solow growth model to find the steady-state levels of capital per effective worker (k*), output per effective worker (y*), and consumption per effective worker (c*):
k* = [sA / (n + g + δ)]^(10/7), where s is the saving rate, A is the technology level, n is the population growth rate, and g is the technological progress rate.
In our case, s = 0.05, A = 1 (since we have not been given any specific value for technology level), n = 0.005, g = 0.025, and δ = 0.0667.
k* = [(0.05*1) / (0.005 + 0.025 + 0.0667)]^(10/7)
k* = (0.05 / 0.0967)^(10/7)
k* = 0.51685^1.42857
k* ≈ 0.756 (rounded to 3 decimal places)
Now, we can calculate y* using the derived output per effective worker equation:
y* = 100 * k*^3/10
y* = 100 * (0.756)^3/10
y* ≈ 14.583 (rounded to 3 decimal places)
Finally, we calculate c* using the formula c* = (1 - s)y*:
c* = (1 - 0.05) * 14.583
c* = 0.95 * 14.583
c* ≈ 13.854 (rounded to 3 decimal places)
So, the steady state levels for capital per effective worker, output per effective worker, and consumption per effective worker are approximately 0.756, 14.583, and 13.854, respectively.
Suppose the production function is Y = 100(K3/10)(EN)7/10 and capital lasts an average of fifteen years. The rate of population growth is 0.5%. The rate of technological progress is 2.5%. The saving rate is 5%.
A. Derive the equation for output per effective worker y = Y/EN = f(k), where k equals the amount of capital per effective worker.
B. Calculate the steady state levels for each of the following: (1) capital per effective worker, (2) output per effective worker, and (3) consumption per effective worker.
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