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(10) Consider a consumer with preferences U = Z 1 0 e tu(ct)dt; where is the subjective discount rate, c is consumption, and u(c) = ln c. She receives an exogenous ow of income y and can borrow or lend freely at a constant interest rate r; subject to a no-Ponzi-game condition that rules out innite debt. Her ow budget constraint is a_ = ra + y c; where a represents nancial wealth and a0 is the initial value. (a) Derive the rst-order conditions for optimal consumption. (b) At what rate does consumption change? Interpret its sign. (c) Derive the optimal decision rule for consumption. Provide an interpretation for the case = r: 2. (20) Consider the following optimal growth model. Agentspreferences are given by E0 "X1 t=0 tAt ln ct # The term At denotes an i:i:d: demand shock. Output is a concave function of beginning of period capital and a technology shock. That is, yt = ztkt It is assumed that zt is i:i:d:both over time and with respect to At . (That is, the two shocks are independently distributed.) The depreciation rate of capital is 100%. Within this environment, do the following (a) Set up the social planner problem as a dynamic programming problem and derive the necessary conditions. (b) Conjecture a solution to the policy functions for consumption and savings. Derive the equations that determine these optimal policy functions and characterize their qualitative behavior. (c) Suppose one period (real) bonds were introduced into this economy. How do demand shocks aect interest rates in this economy? Explain.

You decide to short 100 shares of GME at $300 as you believe the stock price is overvalued. The initial margin and maintenance margin requirement is 50% and 30% respectively. Ignoring margin interests.

a) Construct the account balance sheet once the short selling position is set.

b) Calculate the annual percentage return (APR) and the effective annual return (EAR) of your trading strategy if you settle your position at the price of $270 in 20 days later, assuming

that a year has 360 days. c) Calculate the price level that you will receive margin call. d) Explain why short selling may not be a good 'investment'.

(20) Consider the basic real business cycle model in which the representative agent maximizes the expected value of discounted utility given by: E0 "X1 t=0 t (ln ct ht) # Technology is given by a standard Cobb-Douglas production function, i.e. yt = ztk t h 1 t in which zt is an i:i:d: technology shock with c:d:f: given by G (zt): The depreciation rate of capital is 100%. Do the following: (a) Solve the model as a social planner problem and derive the optimal policy functions. (b) Discuss how well the implied time series characteristics of the model match those seen in the data. (Restrict your discussion to those features discussed in class.) Discuss a minimal set of changes in the model environment that could produce better consistency between the model and business cycle da

(20) Consider the standard growth model in discrete time. There is a large number of identical households (normalized to 1). Each household wants to maximize life-time discounted utility U(fctg 1 t=0) = X1 t=0 tu(ct): Each household has the following endowments: a) an initial capital stock x0 at time 0, b) one unit of productive time per period that can be devoted to work, and c) one unit of land. Final output is produced according to yt = F(kt ; nt ; lt); where F is a CRS production function, and kt ; nt ; lt denote capital, labor, and land services, respectively. This technology is owned by rms whose number will be determined in equilibrium. Output can be consumed (ct) or invested (it). We assume that households own the capital stock (so they make the investment decision) and the land, and they rent out capital, labor, and land services to the rms. The depreciation rate of the capital stock (xt) is denoted by . 1 The land does not depreciate, i.e. it is in xed supply. Finally, we assume that households own the rms, i.e. they are claimants to the rmsprots. The functions u and F have the usual properties that we have described in class.2 (a) Consider an Arrow-Debreu world. Describe the householdsand rmsproblems and carefully dene an AD equilibrium. How many rms operate in this equilibrium? (b) In this economy, why is it a good idea to describe the AD equilibrium capital stock allocation by solving the (easier) Social Planners Problem? From now on assume that F(kt ; nt ; lt) = A k1 t n 2 t l 112 t and = 1, with A > 0 and 1; 2 2 (0; 1). The households instantaneous utility function is given by u(c) = ln(c). (c) Fully characterize (i.e. nd a closed form solution for) the equilibrium allocation of the capital stock. (Hint: Guess and verify a policy rule" of the form kt+1 = gk t , where g is an unknown to be determined.) In the remaining questions, for full credit, your answers should be functions only of the parameters of the model, i.e. 1; 2; ; A; x0 etc. (d) What is the ADE value of the rental rate of capital as t ! 1? (e) What is the ADE price of the consumption good in t = 1? (f) What is the ADE price of land services in t = 2