economics only please

Question 7 An economy is populated by identical agents with expected lifetime utility given by: Eo Each period, agents rent their beginning-of-period capital stock (k,) to (identical) firms (the rental rate of capital is denoted r, ) and also supply inelastically one unit of labor to firms (the wage rate is denoted w;). The income generated by these factor supplies are used to acquire consumption (c) and new capital. The depreciation rate of capital is assumed to be 100%. Firms choose labor and capital every period in order to maximize profits where output is given by the technology: where the law of motion for =, is = = 3,_18, where 0 , (2) where y, is the individual's stochastic endowment in period , and y' = (vo, VI, . . . . V) is the corresponding history. Assume #(1 + r) = 1. a. Let yi+1 = pyr + er, where a, is a zero mean random shock. What is the Bellman equation corresponding to (1)-(2) above? (Feel free to use the expectations operator instead of summing/integrating over possible events.) b. In the problem-set, we showed that if u(c) = ac - be, the solution took the form " =[a(1 +r) + EZ7 (3) $=0 That is, consumption in period / equals the annuity value of total assets plus "perma- nent income" (a term coined by Milton Friedman). Find the consumption policy function that satisfies your Bellman equation in (a), and show that your (recursive) solution coincides with (3) (Hint: Use the (recur- sive) Euler equation to derive the policy rule - the value function is very difficult to recover.). c. What is the marginal propensity to consume,' , 24? How does the MPC change with the parameter p? Interpret