Suppose the stochastic version of the neoclassical growth model. The preferences of the representative household are given by 00 EOZBt (inct (977.15), t:0 with 0 0, where ct is consumption and nt is working time / employment. The aggregate production function is a 13 1/2: = Ztkt 71,: a with 0 0 is i.i.d., with mean ,LL and variance 02. Physical capital is accumulated according to kt+1 = it + (1 5) kt: where it is gross investment and 6 E [03 1] is the rate of physical depreciation. (a) Formulate the social planner's problem recursively. Explain your choice of state vari ables. (b) Derive and interpret the optimality conditions. (c) Suppose capital fully depreciates during production. Verify, using the planner's optimal ity conditions, that the optimal decision rule for investment is of the type it = Byt. (d) Explain the economics behind the decision rules for saving and employment. Do you nd they depend on the marginal products of capital and labor in a plausible way? (6) (1\") Find the equilibrium wage rate and real interest rate, as a function of the current state of the economy. Explain your reasoning in detail. According to the data, the following tends to happen during business cycles: (1) con sumption, investment, and employment all comove positively with output (they are procyclical), (ii) consumption is less volatile than output, employment is as volatile as output, and investment is more volatile than output. The evidence also suggests that real wages are procyclical, whereas real interest rates are countercyclical. Is this ver sion of the neoclassical growth model consistent with these facts? Explain the economics behind your nding for how prices comove with output