Question

Consider an economy with a production function: Y = F(K, L) = K^α L^(1−α). The economy has a constant savings rate s, capital depreciates at rate δ, and the population growth at rate n.

Let α = 0.36, s = 0.2, δ = 0.05, and n = 0.05.
What is the steady-state capital per worker, output per worker, and consumption per worker in this economy?
Assume in period one, the total workforce is L1 = 100 and the capital stock be K1 = 30.
Solve for output Y1 and for the following variables in per worker terms: k1, y1, c1, sy1, and (δ + n)k1.

Answer 1

The steady-state quantities of capital, output, and consumption per worker in an economy can be determined using the per-worker production function, accounting for the constants of savings, depreciation, and population growth. Using the given values, specific variables such as k1, y1, c1, sy1, and (δ + n)k1 can be calculated.

Step-by-step explanation:

To analyze the steady-state capital per worker, output per worker, and consumption per worker in an economy with a production function Y = K^0.36 L^0.64, given the parameters α = 0.36, s = 0.2, δ = 0.05, and n = 0.05, we begin by calculating per worker terms from the total output and capital.

The total output in period one is calculated as Y1 = K1^α * L1^(1-α) = 30^0.36 * 100^0.64. Then, we derive per worker terms by dividing by the workforce (L1): k1 (capital per worker) = K1 / L1 = 30 / 100, y1 (output per worker) = Y1 / L1. The consumption per worker is given by the formula c1 = (1 - s) * y1.

Savings per worker would be sy1 = s * y1, and the depreciation plus population growth times capital per worker is (δ + n)k1.

Using these formulas and the given values, we can solve for the steady state of each variable:

• k1 = 30 / 100 = 0.3
• Y1 = 30^0.36 * 100^0.64
• y1 = Y1 / 100
• c1 = (1 - 0.2) * y1
• sy1 = 0.2 * y1
• (δ + n)k1 = (0.05 + 0.05) * 0.3