Question

Consider a Solow growth model in which total output is Y = z * K ^ (1/2) * N ^ (1/2) Aggregate capital accumulates according to the following equation K' = (1 - d) * K + I 0 < d <= 1 Consumers consume a constant fraction of income-that is, C = (1 - s) * Y where s is the savings rate, with 0 < s < 1 Population grows at a constant rate n > 0 So that N' = (1 + n) * N Assume that 6% of capital wears out every year, the rate of growth of the population is 2%, and the saving rate is 16%. The productivity level is z = 2

(a) Find the per-worker production function and the per-worker capital accumulation equation.
(b) Calculate the steady-state capital per worker, k ss and the steady-state output per worker, y ss .
(c) Calculate the steady-state consumption per worker, c ss and the steady-state investment per worker, i ss
(d) What is the steady-state growth rate of the capital per worker, k ss , and the steady-state growth rate of the output per worker, y ss ^ prime
(e) What is the steady-state growth rate of the capital stock, K ss . and the steady-state growth rate of the aggregate output, Y ss ^ prime Show your work.
(f) The government is benevolent (cares about the consumers) and wants to maximize the steady state consumption per worker. Write down the maximization problem that the golden rule capital per worker, k gr ^ * solves. Find k gr ^ *
(9) What is the savings rate associated with the golden rule level of capital, s gr ^ * ? Can the country increase the consumption per-capita by changing the saving rate?

Answer 1

The per-worker production function is y = z * (k ^ (1/2)) and the per-worker capital accumulation equation is k' = (s * y) - (d * k). The steady-state capital per worker is kss = 8 and the steady-state output per worker is yss = 4.

Step-by-step explanation:

(a) The per-worker production function can be obtained by dividing the total output (Y) by the population (N). Therefore, the per-worker production function is: y = z * (k ^ (1/2)), where y is the output per worker, z is the productivity level, and k is the capital per worker.

The per-worker capital accumulation equation can be obtained by rearranging the given equation for capital accumulation. Therefore, the per-worker capital accumulation equation is: k' = (s * y) - (d * k), where k' is the change in capital per worker, s is the savings rate, y is the output per worker, and d is the depreciation rate.

(b) The steady-state capital per worker (kss) can be found by setting the change in capital per worker (k') equal to zero. Therefore, we have: (s * yss) - (d * kss) = 0. Using the values given, we can solve for kss and find that kss = 8.

The steady-state output per worker (yss) can be found by substituting kss into the per-worker production function. Therefore, yss = z * (kss ^ (1/2)). Using the values given, we can solve for yss and find that yss = 4.