Question

Consider a country with two cities A and B that produce output according to Yi = (N) (K), j = A, B, where Ni and Ki represent the number of workers and the stock of capital used in production, respectively. Assume that each city-firm faces diminishing returns to each input but increasing returns to scale, i.e. 0 < a <1, 0<B<1, and a +8 > 1. Throughout this problem we consider a benevolent social planner who tries to maximize the country's total welfare. The planner realizes this is a complicated problem because the capital goods make the problem dynamic. But she is very clever and realizes she can split the problem in two. First, she maximizes total output in the country subject to two constraints: one, the sum of labor in the two cities must be equal to N, which is assumed to be constant. The second constraint is that output in each city is given by the production functions introduced above. Capital in each city is taken as given in this first stage. a. Write down the planner's first stage problem. b. Find the optimal NÅ ratio, which should be a function of K. c. Does this optimal rule imply that the city with more capital ends up with more labor too? Provide economic intuition for your answer. It can be shown that this optimal rule implies that total output per capita in the economy is given by f(k4, kB,N) = Nº+8-1 D=A.B (k) 1- d. Show that, if kA > kB, then fa > fb, where f, is the partial derivative of f with respect to k, i.e, the MPK of city j. What is the economic intuition behind this result?

Answer 1

Ni represents Labour used in the production of output of the two cities A and B

Ki represents capital used in the production of output of the two cities A and B.