summarise the work for easy revision please thankyou

Question 7 An economy is populated by identical agents with expected lifetime utility given by: Bo [ [A Inc. ] 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,_16, where 0 Bucci(y')) (1) 1=0 subject to "( ) +a+() = (1+ra( )+x (2) where y, is the individual's stochastic endowment in period ?, and y' = (yo, VI, . .., Vr) is the corresponding history. Assume B(1 + r) = 1. a. Let yr+1 = py, + 8,, where &, 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) + E, > (1 trys )its] (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, y Ac? How does the MPC change with the parameter p? Interpret.Question 1 Consider the sequence of function {on (x))2, defined by Pati(x) = max fu(x - x') + Bon(x) ) Y'E[0,x] with Do(x) = 0. Under which conditions on u and / do we know that o (x) = lim, -+Do Dn(x) exists and is unique and continuous? Explain briefly. Question 2 Consider a representative agent, optimal growth model with no popula- tion growth in which agents have logarithmic preferences and assume that agents' discount factor () is equal to 0.97. If the economy is growing at 3%, what will the equilibrium one-period real interest rate be in this economy? (All interest rates and growth rates are expressed on an annual basis.) Suppose consumption uncertainty is introduced into this economy. How will this affect the (average) equilibrium real interest rate? Question 3 Let +1 denote the realized gross one-period return on a risky asset pur- chased at time / and ry denote the gross one-period return on a risk-free asset (also pur- chased at time /). Assuming that consumption growth and returns are serially uncorrelated, then the risk premium associated with the risky asset can be written as: E (Fitl)-ryt = -Brys [p (m,+1, F+1)o (m+1)o (+1)] where / is the discount factor, m, represents agents' stochastic discount factor and p (.) and (.) represent correlation and standard deviation respectively, Derive the above expression and discuss its implications for the equity premium puzzle