3. Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Suppose the investor has time-additive utility with uo = u and u = du for a common function u and discount factor 8. Suppose the investor has labor income Ỹ at the end of the period, so she chooses Co to maximize u (Co) + 6E [u (Wo – Co) R + Y)] . Suppose the investor has convex marginal utility (u" > 0) and suppose that E Y = 0. Show that the optimal Co is smaller than if_ Y = 0. Note: This illustrates the concept of precautionary savings the risk imposed by Ỹ results in higher savings Wo - Co.
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- 3. Consider an agent who evaluates utility delayed by k periods with a discount factor of B8k. Time is discrete and indexed by t E {0,1,2, ...}. This individual has to complete a project (which only takes one period to complete) before or during period T, where the (undiscounted) utility cost of completing the t project in period t is a) Explain the difference between an exponential discounter, a naïve hyperbolic discounter and a sophisticated hyperbolic discounter. b) Suppose the individual is an exponential discounter with B completed? = 1 and 8 = 1. When will the project be c) Now suppose the individual is a naïve hyperbolic discounter with B = and 8 = 1. Calculate when this individual will plan on completing the project, and when it will actually be completed. d) Now consider the behaviour of a sophisticated hyperbolic discounter with B = and & = 1. Prove that if T is even, then the individual will finish the project in period 0, whereas if T is odd the project will be completed in…1. A woman with current wealth X has the opportunity to bet an amount on the occurrence of an event that she knows will occur with probability P. If she wagers W, she will received 2W, if the event occur and o if it does not. Assume that the Bernoulli utility function takes the form u(x) = -e-rx with r>0. How much should she wager? Does her utility function exhibit CARA, DARA, IARA?The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica- tion of preferences in economics that captures risk aversion and intertemporal consump- tion smoothing. The CRRA utility function has the desirable property that the degree of risk aversion is constant and independent of the level of consumption. This means that as a household's consumption grows, its willingness to take risks remains the same. The coefficient of relative risk aversion (σ) measures the extent to which households are risk- averse and prefer a smooth consumption path over time. A higher value of σ indicates a greater degree of risk aversion and a stronger preference for consumption smoothing. Consider a two-period endowment economy with a large number of identical house- holds. Each household has the following lifetime utility function: U(j) = C+(j) 1-0 - 1 1-σ +ẞ C++1(j) 1-0 - 1 1-σ where C₁(j) and C++1(j) are consumption in periods t and t + 1 for household j, re- spectively, ẞ is…
- The consumer choice is not restricted to the choice of consumptiongoods. In fact, it can apply to all our decisions that involve trade-offs. Suppose Mary has awage per hour of 10 euros. With her earned income she consumes. That isC=wH per day.She also decides how many hours to work of take leisure time each day.H=24-N, whereHis work and N is leisure. Her utility is given by （picture） Solve for the optimal decision of labor/leisure. Plot the budget constraint and the indif-ferent curve. What is the labor supply function?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…1. A woman with current wealth X has the opportunity to bet an amount on the o ccurrence of an event that she knows will occur with probability P. If she wager s W, she will received 2W, if the event occur and if it does not. Assume that t he Bernoulli utility function takes the form u(x) = -e-TX with r> 0. How much should she wager? Does her utility function exhibit CARA, DARA, IARA?
- 2. A small college is trying to predict enrolment for the next academic year. Thevice president for business states that enrolment has tended to follow apattern described by E = 18,000 – 0.5P, where E denotes total enrolment andP is yearly tuition.a) If the school sets tuition at €20,000, how many students can it expect to enrol?b) If the school wants to maximize total tuition revenue, what tuition should itcharge?c) As the vice president for business, what tuition would you recommend? Explainbriefly.d) Due to a strong post-COVID-19 recovery, the income conditions in the regionimprove substantially. Explain in one sentence how this could affect the college’senrolment pattern and the enrolment level maximizing its tuition revenue.Betty is looking for a job. She considers job opportunities intwo cities. Bettyís utility is given by y- x, where y is the lifetime income andx is the amount spent on buying a house. The income from City 1 fluctuatesalthough the house price is stable. On the contrary, the income from City2 is stable while the house price fluctuates. If she moves to City 1, Bettycan earn a lifetime income y1 with probability alpha and 1 + y1 with probability1-alpha . The house price in City 1 is x1. Moving to City 2 means that Bettycan earn an income of y2. However, the house price is x2 with probabilitygamma and 1 + x2 with probability 1-gamma . Do the following: (a) Write down theexpected utilities associated with living in the two respective cities, i.e., V1and V2. (b) Derive the condition under which Betty chooses City 1.6. for a large company (she travels by car). She earns 900 euros a week in fixed salary, but if she reaches the set for a week sales target, she receives a bonus of 325 euros that week. The probability of getting the bonus is 0.2. A person is a traveling salesman of goods Her utility function over money is given by U=√W, where W is the wealth in SEK per week. 6a 6b What is her expected weekly bonus? Her boss offers her another job within the same company. If so, she would work in an office with a certain personnel responsibility and receive a fixed salary per week (no bonuses). What salary per week must her boss at least offer her in order for her to accept new work? Justify your answer based on expected utility theory.
- 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?A consumer whowill consume c1 in period 1 and c2 in period 2 exhibits pure time discounting if they evaluate their consumption using the utility functionU(c1; c2) = u(c1) + βu(c2)where 0 < β < 1, and β < 1 captures the idea that consumption inthe future is not worth as much as current consumption. Note thatU(·; ·) depends on two arguments, while u(·) depends on only oneargument. We assume that u0(·) > 0, u00(·) < 0, and suppose thatthe consumer can split their current wealth, W, between the twoperiods as they please. In this problem, you are supposed to findproperties of the the optimal savings, s, as the solution tomax0≤s≤W U(W - s; s) = u(W - s) + βu(s):1. Using the FOCs, show that W - s∗ is larger than s∗ for anyβ < 1.2. Show that s∗ is a weakly increasing function of W (this can bedone without FOCs).3. Show that s∗ is a weakly increasing function of β (this too canbe done without FOCs).4. If u(x) = log(x), give the optimal savings as a function of β andW.Q1. A farmer believes there is a 50-50 chance that the next growing season will be abnormally rainy. His expected utility function has the form Expected utility = 0.5lnYNR + 0.5lnYR Where and represent the farmers income in the state of ‘normal rain’ and ‘rainy’ respectively. Suppose the farmer must choose between two crops that promise the following income prospects Crop YNR YR Wheat $83,000 $10,000 Maize $83,000 $15000 What mix of wheat and maize would provide maximum expected utility to this farmer?