2. (CES felicity) Consider the following utility maximization of some consumer over time. /or(e(t)/σ e-ptdt subject...
3. Stone-Geary felicity) Consider the following utility maximization of some consumer over time. cr(e(t)-c)사-l eMt subject to k(0)=k(T)=0, where w(t) and r(t) are given paths of real wages and the real rate of return of the single asset and σ ël is a positive constant. The parameter c represents the minimum subsistence level of consumption. Compute the Euler equation.
1. (logarithmic felicity) Consider the following utility maximization of some consumer over time. Mr,( .1。[loge(t) e--dt subject to k(0)=k(T)=0, a(t)-w(t)x1+r(t)xa(t)-c(t). where w(t) and r(t) are given paths of real wages and the real rate of return of the single asset. Compute the Euler equation.
6.Consider the following utility maximization of some consumer over time. Max "dt subject to a(0)a(T)-0 a(t)-w(t)x1+r(t)xa(t)-ct) where w(t) and r(t) are given paths of real wages and the real rate of return of the single asset and σ # 1 is a positive constant. The parameter c represents the minimum a(t)- w(t)x1+r(t)xa(t)-c(t), subsistence level of consumption. Compute the Euler equation.
Solve the following Utility Maximization Problem for x* and y* that Max U(x,y)= ln(x) subject to Pxx + pyy = I ----.-.(2) where In denotes the natural logarithm (base e) and x and y>0. a) (25 points) by Substitution and show that your values of x* and y* max U (x*,y*). Problem 1. b) (20 points) by the Lagrange Multiplier Method
3. Consider a representative consumer who has preferences over an aggregate consumption good e and leisure. Her preferences are described by the uility function: U(c,l) In(e) +In(l) The consumer has a time endowment of h hours which can be used to work at the market or enjoyed as leisure. The real wage rate is w per hour. The worker pays a proportional wage tax of rate t, so the worker's after-tax wage is (1 t). The consumer also has dividend...
Problem 1 Consider the following two-period utility maximization problem. This utility function belongs to the CRRA (Constant Relative Risk Aversion) class of functions which can be thought of as generalized logarithmic functions. An agent lives for two periods and in both receives some positive income. subject to +6+1 4+1 = 3+1 + (1 + r) ar+1 where a > 0,13 € (0, 1) and r>-1. (a) Rewrite the budget constraints into a single lifetime budget constraint and set up the...
Consider an economy occupied by two households (i- A, B) who are facing the two-period consumption problem. Each household i - A, B is facing the following utility maximization problem: max subject to ci +biy(1+r)bo where Vi and US are household i's exogenous income in period t 1.2. cỈ and c are household i's consumption in period t 1,2. bo,bi is household i's bond holdings of which bo is exogenously given, r is the real interest rate, and 0 <...
2. Consider the following four consumers (C1,C2,C3,C4) with the following utility functions: Consumer Utility Function C1 u(x,y) = 2x+2y C2 u(x,y) = x^3/4y^1/4 C3 u(x,y) = min(x,y) C4 u(x,y) = min(4x,3y) On the appropriate graph, draw each consumer’s indifference curves through the following points: (2,2), (4,4), (6,6) and (8,8), AND label the utility level of each curve. Hint: Each grid should have 4 curves on it representing the same preferences but with different utility levels. 3. In the following parts,...
Suppose a consumer has quasi-linear utility: u(x1, x2) = 3.01 + x2. The marginal utilities are MU(X) = 2x7"! and MU2:) = 1. Throughout this problem, assume P2 = 1. (a) Sketch an indifference curve for these preferences (label axes and intercepts). (b) Compute the marginal rate of substitution. (c) Assume w> . Find the optimal bundle (this will be a function of pı and w). Why do we need the assumption w> (d) Sketch the demand function for good...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...