Consider the following production function: what is the expression for rK and wL?

1. Consider the following production function: Y = AKl-YLY, where Y is production, A is productivity, K is capital, L is labor, and y E (0, 1) is a parameter. Let w denote the wage rate and r denote the rental rate of capital. Solve the profit maximization problem of the firm: max AKI-YLY -wL-rK. What K , L is the expression for rk and wL? Comment on your results. Show every step of all calculations. 2. Suppose GDP per capita is given by y = Ak, where A is productivity and & is capital per person. The following table reports values for observed capital per capita (pc), k, and observed GDP per capita, y, for Japan, Brazil, and India, where all values are reported relative to Japan (that is, k and y in Japan have been normalized to a value of 1.000). (a) Fill out the table. That is, compute the value for predicted GDP per capita in Brazil and India if productivity, A, is equal to 1, and compute the model-implied productivity (also known as TFP) value in Brazil and India (that is, the value of A required for the model to match the data). Show every step of all calculations. (b) Decompose the differences in GDP per capita between Japan and Brazil into two components: differences in productivity and differences in capital per capita. That is, what share of the difference in GDP per capita between Japan and Brazil is explained by differences in productivity between the two countries, and what share of the difference is explained by differences in capital per capita between the two countries? (Hint: use that y = Ak? and write out the expression for y(Brazil) ) y(Japan ). Run the same decomposition exercise for Japan and India. Relative to Japan (Japan = 1.000) Country Observed capital/pc (k) Observed GDP/pc (y) Predicted GDP/pc if A = 1 Implied TFP (A Japan 1.000 1.000 1.000 1.000 Brazil 0.449 0.491 ? ? India 0.083 0.153