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ECO4145A00 Mathematical Economics || Professor: N. V, guy/$9 Final Exam April 21, 2020 Du ration: 7:00PM-10:30PM The exam is open book, and laptops are allowed. It contains four problems each of which is worth 25 points. 1. Consider an individual, assumed to live forever, whose wealth in period t) is MD > 0. This individual does not work, and his consumption in each period t : 0, 1, is financed from his wealth. Let cD denote his consumption in period 0. The amount of wealth left after paying forthe consumption in period 0 is thus m\" ecu, which is then invested. At the beginning of period 1, the wealth of this individual is given by Lugrluuu ecu), where r1 is the rate of return to the investment made in period 0. Assume that r1 is random and that the uncertainty concerning r] is resolved at the beginning of period 1. In period 1, the individual has to choose a consumption level c]. The wealth of the individual at the beginning of period 2 is then all: rz(w, ecj), where r; is the random rate of return tothe investment made in period 1. In general, if w: is the wealth of the individual at the beginning of period t, and cl is his consumption in that period, then the wealth at his disposal at the beginning of period H l is wt.) =n.1(w, 7 q), where rm is the rate of return to the investment made in period I. Assume that rt, 1' = 0, l, 2, ..., are independent and identically distributed random va riables. Find a consumption function , say ct=c[xt], that gives the consumption in each period as a function of the wealth available at the beginning of that period, to maximize the following expected discounted utility: I 1., .E 2206' 1 c, '7], where E denote the expectation operator; 6, 0 1, is the intertemporal elasticity of substitution. In answering this question, assume Em]: F1 is the intertemporal elasticity of substitution. Furthermore, note that in the ,__ objective function, p > 0 is the rate used to discount social welfare. (a) What is the differential equation that governs the motion of the capital stock? What is the differential equation that governs the motion of the current shadow price of capital? Illustrate the convergence of the economy to the longrrun equilibrium with the help of a phase diagram. (b) Show that the value of the current Hamiltonian at a point on the optimal trajectory represents the national income of the economy atthat instant