# Question: consider the solow growth model output at time t is...

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Consider the Solow growth model. Output at time t is given by the production function Yt = AK 1 3 t L 2 3 where Kt is total capital at time t, L is the labour force and A is total factor productivity. The labour force and total factor productivity are constant over time and capital evolves according the transition equation Kt+1 = (1 − d) ∗ Kt + It , where d is the depreciation rate. Every person saves share s of his income and, therefore, aggregate saving is St = s ∗ Yt and aggregate consumption is Ct = (1 − s) ∗ Yt . Each period, savings equal investment: It = St . Let yt and kt denote output per worker and capital per worker, respectively.

1. Derive the level of output per worker as a function of capital per worker and model parameters (A, d and s). Derive the transition equation for capital per worker, i.e. express kt+1 as a function of kt and model parameters.

2. Suppose that A = 2, d = 0.05 and s = 0.2 and the initial level of capital per worker is k1 = 15. Calculate the economy’s output per worker, consumption per worker and the capital-output ratio for period 1.

3. What is the level of investment per worker in period 1? How much capital per worker will depreciation in period 1? Will output per worker be higher or lower in period 2? Explain (you do not need to calculate the next period’s capital per worker).

4. Draw a graph with capital per worker in period t on the horizontal axis and capital per worker in t + 1 on the vertical axis and denote k1, k2 and k3 on the graph (the graph does not need to be drawn to scale). Explain what will eventually happen to the evolution of capital per worker. Describe the implication of the evolution of capital per worker for the evolution of output per worker. Suppose that in period T (where T is a large number) the economy has reached its steady state.

5. Use the transition equation to calculate the steady state level of capital per worker k ∗ as a function of model parameters (do not calculate the exact numerical value). [Hint: recall that the economy is in steady state when kt = kt+1 = k ∗ .]