3. Consider a stochastic DGE model with uncertainty about future income and interest rates. The representative consumer has the utility function U₁=EB'u(c), faces the budget constraint c, +a, = y₁+(1+r),,, for 1-0 120, and has an initial assets position given by a.. a) Derive analytically the Euler equation for the optimal allocation of consumption. b) Use your result to explain why the covariance between consumption and assets' return is a measure of macroeconomic risk and how this affects consumption. c) Show how your result can be used to derive a formula for pricing risky assets and provide an economic interpretation.

3. Consider a stochastic DGE model with uncertainty about future income and interest rates. The representative consumer has the utility function U₁=EB'u(c), faces the budget constraint c, +a, = y₁+(1+r),,, for 1-0 120, and has an initial assets position given by a.. a) Derive analytically the Euler equation for the optimal allocation of consumption. b) Use your result to explain why the covariance between consumption and assets' return is a measure of macroeconomic risk and how this affects consumption. c) Show how your result can be used to derive a formula for pricing risky assets and provide an economic interpretation.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Consider the following consumer’s problem: max u(c1 ) + βu(c2) c1c2 subject to c1 + s1 = y1 c2 = y2 + (1 + r)s1 u(c)={1/(1-1/σ}c1-1/σ a) Describe all equations using economic terminology.b) Derive the intertemporal budget constraint for this consumer.c) Set the Langrangean for this problem.d) Find the first order conditions for a maximum.e) What is the effect of an increase in the interest rate on savings? Show in equation and explain.
Review Appendix 2A of Griffin (on Lagrangian Method). Especially go over Box 2A.1. Using the utility function U = (x^1)(5w^1)(5), income = 40k, p_x = 1, find the demand function for water. (Follow the same steps.).
Consider the model where an individual has wealth k which they can either save or consume. If they save it, they receive a fixed and exogenous return r. The instantaneous utility function is given by: u(c, k) = c + a(k) where c is consumption, k is wealth, and a(k) is a function that defines the utility that an individual gets from holding wealth. The growth in wealth is given as the returns on wealth rk, plus income from working z(t), minus consumption c(t). a. Write out the differential equation for wealth. b. For an infinite time model, set up the optimal control problem with discounting at a rate 8. c. Write the current-valued Hamiltonian of this problem. d. Derive the steady-state level of consumption.
Please solve the last 3 parts
Let's incorporate the labor-leisure trade-off and capital income taxes in the two-period model. Let c₁, c₂ be consumption in two periods, I the number of hours worked, Te Te the proportional taxes on consumption in 2 periods, s the saving rate, w the wage rate, b pension in the 2nd period, and 7, the tax on savings (capital income tax). The household's maximization problem in this case is: given by maxe₁,e2,8,1-1 log(c₁) + log (1-1)+5log (c₂) such that (1+T₂) C₁+8 = (1-7)wl and (1+T₂)C₂ = [1+r(1-Ts)]s+b, where measures how the household values leisure vis-a-vis consumption.
3. Consider a parent who is altruistic towards her child, but also cares about her own consumption. The parent's utility over her own consumption and that of her child is up = log(co) +a log(ci) where c is the child's consumption, and a > 0 is the degree of parental altruism. Suppose that the parent can invest in the child's human capital by spending money (e) on her education; education generates human capital h /() and human capital is paid at rate w. The parent has a total income of (a) Write down an expression for the child's future consumption in terms of the parent's choice of e. (b) Now write down the Lagrangian for the parent's decision problem.
6. If intertemporal preferences are consistent and the lifetime utility function is additive, then the discount function 8(t) must be (a) bounded (b) exponential (c) hyperbolic (d) linear (e) logarithmic
question ii
Solve all this question......you will not solve all questions then I will give you down?? upvote...
A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
A decision maker allocates an endowment of W > 0 dollars across two periodst = 1, 2. He discounts the future by β ∈ (0, 1) while facing a gross interest rateof R > 1. His utility is the same as studied in class. Solve for the intertemporalchoice problem. Show that the optimal consumption is decreasing over time ifβR < 1, constant over time if βR = 1, and increasing over time if βR > 1.
Thank you for your solutions of questions 1-3. Please answer questions 4-5.