9. In the Solow growth model, output Y is produced using capital K and labour L. Assume the production function is Y = K L, which has total factor productivity constant over time. The change over time in the capital stock is K = IK, where I is investment and is the depreciation rate. The population and labour force L are constant over time. Investment I is equal to saving S, which is a fraction s of income. Let k = K/L and y = Y/L denote capital per worker and output per worker. (a) [5 marks] Show that y = k and k = s kk, and solve for the steady-state values of k and y where k = 0. (b) [2 marks] Using a diagram, explain intuitively why there is a steady state where growth of output per worker is zero, and why the economy converges to this steady state in the long run. Suppose that around the world, some countries choose to save a higher fraction s of income. In all other respects, assume countries are identical. (c) [3 marks] Explain what is meant by the terms absolute convergence and conditional convergence. Explain which one is predicted by the Solow model. (d) [1 mark] Comparing two countries where one has double the saving rate of the other, what is the Solow model's prediction for their relative levels of income per worker in the long run? Now suppose that the saving rate s is not constant in a country. When income is low, households must spend most of their incomes on essential goods and services, while if income were higher, households would be able to save a greater fraction of income. Specifically, assume that the saving rate is s = 0.05 if y < 1, and s = 0.35 (0.3/y) if y 1. This behaviour of the saving rate s is common to all countries. The depreciation rate in all countries is = 0.1. (e) [1 mark] Assuming y < 1, find the steady-state level of y. (f) [3 marks] Show that there are two steady-state values of y greater than 1, and find these values of y. [Hint: One steady state is y = 1.5.] (g) [3 marks] Illustrate the new 'saving line' sy in your diagram from part (b) and use it to explain which steady state the economy will converge to if it starts from a level of k where y < 1.5. Which steady state will the economy converge to if y > 1.5 initially? (h) [2 marks] Empirically, saving rates and levels of income per worker are positively correlated across countries. In light of your findings above, can this be taken as evidence that exogenous differences in saving behaviour across countries are the cause of some of the income differences? Explain.