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
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Given : intertemporal preferences are consistent and lifetime utility function is additive.
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- 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?Question 4 A consumer is maximising her utility function: U (x, y) = (205 + y05)², subject to the budget constraint 4x + 2y 108. (a) Set up the Lagrangian function of this utility maximisation problem and derive the first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the Lagrange multiplier. (c) What are the utility maximising amounts of x and y if the budget constraint changes to x + 2y = 36? How would A change? Explain your reasoning. (Hint: You do not need to calculate A, rather comment on how it would change and why.)Consider an economy with two goods, consumption c and leisure 1, and a representative consumer. The consumer is endowed with 24 hours of time in a day. A consumer's daily leisure hours are equal to 1 = 24-h where h is the number of hours a day the consumer chooses to work. The price of consumption p is equal to 1 and the consumer's hourly wage is w. The consumer faces an ad valorem tax on their earnings of 7 percent. The con- sumer also receives some exogenous income Y that does not depend on how many hours she works (e.g. an inheritance). The consumer's preferences over consumption and hours of work can be represented by the utility function U(c, h) = c-3h¹+, where 3 > 0 and p > 0 are parameters. 1+p
- 1. 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…Jien is just bored all the time; no amount of success makes him happy, it seems. Below is a list of his income for the last several years and the utility he experienced per dollar of income: Year Yearly Income Utility per Dollar Earned 2017 $60,000 2 utils 2018 $70,000 1.8 2019 $100,000 1.5 2020 $120,000 1 2021 $145,000 0.40 From the above, we can say that Jien most likely is different from most people economists study in terms of risk attitudes is "risk loving" will not take a fair bet has a utility of wealth curve that is a straight lineExercise 1 A consumer's utility function is u(x, y) = x²y³ and the budget constraint is prx +Pyy ≤ w, where the parameters pr, Py and we are all strictly positive. (a) Solve the consumer's utility maximization problem. (b) Use the envelope theorem to estimate the change in the indirect utility function (i.e., the problem's value function) when the price py is changed to py + e, with e > 0.
- A consumer is maximising her utility function: U(r. y) = (r05 +y0.5)2, subject to the budget constraint 4x + 2y = 108. (a) Set up the Lagrangian function of this utility maximisation problem and first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the multiplier. (c) What are the utility maximising amounts of x and y if the budget constrain to x+ 2y = 36? How would change? Explain your reasoning. (Hint: Yc need to calculate A, rather comment on how it would change and why.)What is a lifetime utility function, and in what sense does it exhibit diminish-ing returns?A 2015 report by the music industry estimated the revenue lost to the industry every yearfrom illegal downloading. In this problem we will derive some of the estimates that may havegone into their calculation (approximately).First, start with the individual consumer’s problem. Suppose a typical consumer has a yearlyentertainment budget of I that they can allocate between music downloads (D) and otherforms of entertainment (E). Consumer preferences are characterized by a utility functionU(D, E). a.) Write an expression for the consumer’s budget constraint as a function of their entertainment budget and the prices of music downloads (Pd ) and other entertainment (Pe). (b) Write the consumer’s constrained optimization problem in Lagrangian form. (Note: Youdo not need to solve it or derive first order conditions.)
- Mathemeticaly prove that slope of budget line is -P1/P2DEFINE Limit of consumption optionsGive U=XY-2Y and Px=4 ,Py=2 want to minimize budget that subject to is PX+PY=288 1).write out the Lagrangian function 2.Use First order condition to find X and Y 3.Test for the second order condition for minimum satisfied.