The interest rate is 10%, and y = (600, 200). The total utility function is max V (C₁, C₂) = ln(c₁) + In(c₂) C1,C2 The MRS at c = (2,1) is closest to: a. -1.10 b. -2 c. -1/2 d. -4
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- 103. Consider a home energy storage (battery) system that can store up to 2 units of energy. At every time step, there is a demand for energy in the home which is drawn from 0,1,2 units with equal probability independent of the demands in the previous time steps. At every point in time, you have to satisfy the demand either by discharging the needed energy from the battery or purchasing power from the grid (or a combination of the two). You could also choose to purchase power from the grid to charge your battery. The grid energy price is either H(igh) or L(ow) according to a Markov chain (Price moves from H to L with probability p, and from L to H with probability q). (a) Model the decision making as an infinite horizon MDP where the objective is to minimize the discounted cost of energy purchased over an infinite horizon. (b) Write down a policy a of your choosing. Perform two steps of the operator T, for your policy, followed by one step of T..Suppose a household has the following lifetime utility function: U=c1/2 + ẞc¹/2 12tt+1 A) Find expressions for the partial derivatives of lifetime utility, U, with respect to period t and period t + 1 consumption. Is marginal utility of consumption in both periods always positive? B) Find expressions for the second derivatives of lifetime utility with respect to period t and t+1 consumption, i.e., 2U and a 20_Are these second derivatives always negative for ac²²+1 any positive values of period t and t+1 consumption? C) Derive an expression for the indifference curve associated with lifetime utility level Uo (i.e., derive an expression for C++₁ as a function of U₁ and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of c?
- Problem 1 Jane has the following utility function: u(w) w^.5 or the square root. function Which of the following prospects would she prefer? P1(.4, 3,000, 600) P2(.7, 1,200, 1,000) P3(.6, 2,000, 100)2 3) For each utility function, determine the marginal utility with respect to x, marginal utility with respect to y, and the MRS. (3pts each) a) U(x, y)-3x+2y b) U(x, y) 10x¹/5 y c) U(x, y)=x+ y²6. 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
- A person's utility function is given by U(x, y, z) = 7xyz, where x, y, z denote then number of units of three commodities X,Y, and Z that the person consumes. The pricesr per unit of X,Y, and Z are 5 euro, 1 euro, and 3 euro respectively. If ther person has a budget of 270 euro, the person's utility is maximised when they consume units of X, units of Y, and units of Z. The person's maximum utility is If the person's budget is increased to 271 euro, then using the method of lagrange multipliers we find that the maximum utility is approximately1. A standard model of choice under risk is Expected Utility Theory (EUT) in which preferences over lotteries that pay monetary prizes (x₁, x2, ..., xs) with probabilities (P1, P2, ..., Ps) with Eps = 1 are represented by the function L S (a) What does it mean to say that a function represents the consumer's prefer- ences? Σpsu(xs) Choice 1 8=1 (b) State and briefly comment on the axioms required for the EUT representation. (c) Consider the following experiment of decision making under risk in which sub- jects are asked which lottery they prefer in each of the following two choices: Lottery B 0 with prob. 0.01 10 with prob. 0.89 50 with prob. 0.10 Lottery D Choice 2 Lottery A 0 with prob. 0 10 with prob. 1 50 with prob. 0 Lottery C 0 with prob. 0.90 10 with prob. 0 50 with prob. 0.10 Suppose that the modal responses are Lottery A in Choice 1 and Lottery D in Choice 2. Assume that utility of zero is equal to zero and illustrate why it is not possible to reconcile these experimental…1. Bob has a utility function U (x, y) Vai +0.8 /T2 + 0,64/x3 over his incomes *1, E2, X3 in the next three years. This function is an example of (A) expected utility; (B) quasi-hyperbolic utility function; (C) discounted utility; (D) none of the above.