. Assume an individual faces an instantaneous quadratic utility function, u(c,) = (o (-Z' ef) so that u'(e;) > 0. Continue to assume, for simplicity reasons, that both the interest rate and the discount rate are zero. In particular, suppose there is uncertainty about the individual's labour income (y{s) each period (future income). The individual maximizes expected lifetime utility E[U/] = E,[Z]_;( , 7rfjj.a = 0 subject to the lifetime budget constraint: Iiice = Ay + Ea a. Using the random walk hypothesis, will uncertainty about future income affect consumption! b. Show whether or not this uncertainty about future income will affect expected lifetime utility? c. Will the result in {b) change if u(c;) = log(1 ac;)? d. The random-walk hypothesis argues that change in consumption is unpredictable. Discuss this with the necessary derivations. e Using the instantaneous utility above, also show that the utility function exhibits increasing relative risk aversion (IRRA). and which individuals is it consistent with in the econemy as wealth increases? f Using the Consumption Capital-Asset Pricing Model, show that a marginal increase in holdings of an asset that is risky does not increase the variance of the individual's consumption g Further show that the expected-return premium that an asset must offer relative to the risk-free rate is proportional to the covariance of its return with censumption. h. Derive the equity premium puzzle and discuss the intuition behind your derivation. How can the government of Ghana benefit from the equity premium puzzle in terms of its debt and capital investment! i. With derivation explain why it is difficult to reconcile returns on stocks and bonds j. Some developed countries have debt-GDP ratio of as high as berween 200% and 400% whiles developing countries with debt-GDP ratio of less than 100% have become big national issues. Is this really a problem? 2. Consider a Cobb Douglas production function F(K, L, A) = AK\"L*% 0 0; and all variables in dme a. Set up the present value Hamiltonian by stating clearly the problem of the firm. Find the first order conditions and solve the model for K as a function of 7, L, T and A. b. Set up the current value Hamiltenian. Find the first order conditions and solve the model for Kas a function of 7, L, 7 and A. Compare the results to (b) c. Find equations for the dynamics of K and g and show their graphs at the steady state. d. Consider an unanticipated and permanent increase in . How do capital and investment behave over time? e. Consider an unanticipated and permanent increase in 7. How do capital and investment behave over time? f. Consider an unanticipated and permanent increase in . How do capital and investment behave over time? g Imagine that at time 0, we learn that T will increase at some future time T. How do capital and investment behave over time! h. A temporary decrease in i. The AGI has complained about recent high taxes on investment in Ghana. Suppose that a firm is taxed at rate T, as above, on profits such that the marginal benefic of a ene dellar increase in investment is MPe (1 7). Bur the firm is also entitled to various tax credivbenefits on the investment. If the tax-saving opportunities are represented as , the cost of borrowing is v, and the true rate of depreciation is &. As the Chief Economist, what will be your advice to the Trade and Industry Minister, seeking your competent advice on how to create incentive for investment! Given tand @ what should be the investment decisien (equilibrium condition) of the firm? Be clear with your derivations and intuition. 4 Critically review the literature on government debt and the government risk premium puzzle. 5. Answer the following questions: a. WWhatare the stylized facts about Business Cycles? b. Consider a Real Business Cycles model where the agent maximizes max, ., Yo Bt [Ing, + In(1 = n,)) subject to k,,, = A kEn!% , Wt; where n, is employment, , is consumption, k, is capital and the productivity/technology shock A, follows an AR process InA, = pind,_, + &; p is the persistence parameter, and &, is independently and identically distributed (ii.d) random variable with mean 0 and variance 5. Using your equilibrium trace how one standard deviation of positive technology shock will impact real wvariables such as output, consumption, capital accumulation, employment etc. Describe your impulse response functions, i. Howdoesp=099anda = % affect the performance of the model? ii. Ifp =0 whatwill be the response of capital, output, employment and consumption to a positive productivity shock? What is the nature of your impulse response functions! iv. Now consider the effect of a one time (temporary) positive technology shock. Assume the share of capital, @ = 3 /4. Let y; be the output at time 1. Suppose that at time T technology shock is realized and let = 1 . Suppose that the technology shock is entirely temporary, ie, & =0vt > T. 'What would be the response of output. capital, labour and consumption