Suppose a consumer faces uncertainty over his income in period and he assumes following utility function max E,{u(c,)+ Bu(c,)} subject to c =1-a;c, =w, +(1+r)a, where income in period 2, w, is uncertain. w, is subject to two states with probability of 1+ɛ with probability p and 1- ɛ with probability 1- p. Given these conditions, derive the Euler condition under uncertainty.

Suppose a consumer faces uncertainty over his income in period and he assumes following utility function max E,{u(c,)+ Bu(c,)} subject to c =1-a;c, =w, +(1+r)a, where income in period 2, w, is uncertain. w, is subject to two states with probability of 1+ɛ with probability p and 1- ɛ with probability 1- p. Given these conditions, derive the Euler condition under uncertainty.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.1P

See similar textbooks

Similar questions

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?
Y5 Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort).a. In the absence of insurance, does Alfred exert effort to lower the probability of fire?HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs.b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the…
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.
a. Suppose that you took part in a lottery that has a chance to increase, decrease or have no effect on your level of income. With probability 0.5, your income remains at it original level K500; with 0.2 probability, your income increases to K700; and with probability 0.3, your income decreases to K400. The utility function is.u（1） =I^0.7where I denote income leveli.Using the utility function show that the consumer's risk preference is averse. (2marks)ii.Calculate both the EU and EV of the income. (4marks)iii.Using the results in (il) above, indicate the attitude to risk of this consumer. (2marks)
A person has wealth of $500,000. In case of a flood her wealth will be reduced to $50,000. The probability of flooding is 1/10. The person can buy flood insurance at a cost of $0.10 for each $1 worth of coverage. Suppose that the satisfaction she derives from c dollars of wealth (or consumption) is given by u(c) = √c. Let CF denote the contingent commodity dollars if there is a flood (horizontal axis) and CNF denote the contingent commodity dollars if there is no flood (vertical axis). (a) Determine the contingent consumption plan if she does not buy insurance. (b) Determine the contingent consumption plan if she buys insurance $K. (c) Use your answer in (b) to eliminate K and construct the budget constraint (BC) that gives the feasible contingent consumption plans for different amounts of insurance K. Determine the slope of budget line (both graphically and by forming the price ratio).
A person has wealth of $500,000. In case of a flood her wealth will be reduced to $50,000. The probability of flooding is 1/10. The person can buy flood insurance at a cost of $0.10 for each $1 worth of coverage. Suppose that the satisfaction she derives from c dollars of wealth (or consumption) is given by u(c) = √c. Let C denote the contingent commodity dollars if there is a flood (horizontal axis) and CNF denote the contingent commodity dollars if there is no flood (vertical axis). 1 Determine the contingent consumption plan if she does not buy insurance. 2 Assume that the person has von Neumann-Morgenstern utility function on the contingent consumption plans. Write down the expected utility U(CF, CNF) and derive the MRS. 3 Solve for optimal (CF, CNF). To this end, first use the tangency condition (TC) to find the relation between the two contingent commodities (CF, CNF). Next, use (BC) to solve for their values. What is the optimal amount of insurance K the person will buy? (Note:…
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?
[**] Simon has current wealth of $36, including $20 in cash. With probabil- ity Simon's money will be stolen (contingency 1), leaving him with only $16 to spend on consumption, and with probability his cash will not be stolen (contingency 2) so he can spend the entire $36 on consumption. Simon has utility function u (0₁.0₂) = √6 + 2√ where c is consumption spending in contingency i. Show Transcribed Text 1. What is the expected value of Simons contingent consumption? 2. What is the expected utility of Simons contingent consumption? 3. What is Simons MRS at his current contingent consumption bundle 4. An insurance company offers to fully insure Simon against the risk of theft for a premium (price) of $P. That is, if Simon pays $P to the insurance company, then the insurer will replace Simons $20 in contingency 1. If Simon buys the insurance his contingent consumption bundle is therefore (36 P; 36 P). i. What is Simons MRS at the fully insured contingent consumption bundle? ii. At what…
INV 1 5aiv Suppose that you have the following utility function: U=E(r) – ½ Aσ2 and A=3 Suppose that you have $10 million to invest for one year and you want to invest that money into ETFs tracking the S&P 500 (US) and S&P/TSX 60 (Canada) index, which are often used as proxies for the US and Canadian stock markets, respectively, and the Canadian one-year T-bill. Assume that the interest rate of the one-year T-bill is 0.35% per annum. You have found two ETFs that you are interested in. From a set of their historical data between 2001 and 2019, you have estimated the annual expected returns, standard deviations, and covariance as follows: ETFUS : E(r)= 0.070584 standard deviation = 0.173687 ETFCDA : E(r)= 0.073763 standard deviation = 0.16816 Covariance between ETFUS and ETFCDA = 0.02397 What is the standard deviation for ETFCDA?
Consider an individual who gets a utility of u(x) - x^1/2 from his total wealth x. Amsume that he has 160.000 AZN in the bank and owns a car with a value of 90,000 AZN. It is expected that will be stolen within the next year with 20% probability, whereas nothing will happen with. Your company tries to sell him an insurance package with the following properties; as an insurance premium now. (ii) if his car is stolen, your company will pay him a partial ation of 55,000 AZN. (iii) if his car is not stolen, there will be no paytent made by your .Should the individund buy this package, if the insurance premium in 12,500 AZN? Explain
4 Lata gets to consume 16 units of food (FS) if there is sunshine (probability 3/4) and 4 units of food (FH) if there is hurricane (probability 1/4). Will she accept a bundle (9,25) with same probabilities? Find out the equation of indifference curve that passes through the new bundle. Also calculate certainty equivalent and risk premium for the new bundle. Explain your answer with the help of appropriate diagrams if her benefit function for food is given by (i) W(F) = F^2 (ii) W(F) = F^1/2 (iii) W(F) = F
Give 1 problem solving example of Expect Value Maximization and its solutions using this formula.