Question

Consider a Solow growth model in which total output is Y = zK^α N^1−α , where 0 < α < 1 .

Aggregate capital accumulates according to the following equation: K ′ = (1 − d) K + I where 0 < d ≤ 1

There is government spending financed by lump-sum taxes (G = T), with total government spending G = gY , where 0 < g < 1.

Disposable income is Y − T. Consumers save a constant fraction of disposable income—that is, S = s(Y − T), where s is the savings rate, with 0 < s < 1.

Population grows at a constant rate n > 0. So that N ′ = (1 + n) N. (a) Find the per-worker production function for this economy.

Answer 1

The per-worker production function derived from the Solow growth model with an aggregate production function Y = zKα N1-α is y = zkα, where y is output per worker, z is productivity, and k is capital per worker.

Step-by-step explanation:

The student has asked to find the per-worker production function for an economy following the Solow growth model. We are given an aggregate production function Y = zKα N1-α, where α is the output elasticity of capital, K is total capital stock, N is the labor supply, and z is a measure of productivity (representing technology). To derive the per-worker production function, we need to divide each component by the labor supply (N), thus normalizing the production function by the size of the labor force.To accomplish this, we note that the capital per worker is k = K/N and the output per worker is y = Y/N. Substituting these into the original production function and dividing by N, we get y = z(kα(N/N)1-α). Simplifying this, we have the per-worker production function y = zkα. This function shows how much output is produced per worker, given the capital per worker and the level of technology.Conclusion The per-worker production function is an essential tool in understanding how factors such as technology, capital, and labor affect the economy's output on a per capita basis. This can be particularly useful for policy-making decisions in aiming for full employment levels and evaluating the contributions of human and physical capital to economic growth.