Consider the Ramsey model studied in class with two goods in the economy, consumption and investment. However, these two goods are now produced in separate sectors. In the consumption sector the production function is C_t = k (A_ctn_ct)¹ while in the investment sector I_t = k (A_itn_it)¹. Assume that the production factors can move freely across the economy between the two sectors.This waythe total capital stock is K_t = k_ct + k_itand aggregate labor is 1 = n_ct + n_it. All this implies that the prices of consumption and investment are no necessarily equal. Let P_t be the price of the consumption good in period t in terms of the investment good and let R_t and W_t be the rental prices of capital and labor. Finally, notice that A_ct and A_it represent exogenous technological progress in each sector and grow at exogenous rates _c and _i.

a)Write down the maximization problem of the two firms in the economy and compute the equilibrium prices for capital and labor. Are the same in both sectors? Why?

b)Using the first order conditions obtained in the previous section compute the optimal capital-labor ratios in each sector. What can you say about the aggregate capital- labor ratio?

c)Use the results obtained in the two previous sections and compute the price of con- sumption as a function of A_it and A_ct (give detailed economic intuition of this result). Then, derive the value of P_tC_t as a function of K_t, A_it and n_ct.

d)Show that aggregate production Y_t = I_t + P_tC_t can be expressed as an aggregate

production function Y_t = KA¹.