Answer the following questions

Problem Set 1 A. Risk Aversion Consider a risk averse consumer with probability p of becoming sick. Let /, be the consumer's income if he becomes sick, and let Ins be his income if he does not become sick, with Is T}. eligibility for Medicaid (and the transfer ofC} is lost When hours of work are between L. and L2 hours the worker's total assets is less than when hours of work is less than L. or greater than L1. Thus. workers have no incentive to work between L. and L2 hours. 1. Suppose the state reduces the Medicaid income eligibility threshold, li a. Would this make it easier or harder for people to qualify for Medicaid? b. What would happen to the size ofthe region where workers have no incentive to work? c. Would L. increase. decrease or stag.r the same? What about L2? Problem #1: Problems from Phelps Phelps, Ch. 8-problems #1 and #4; Ch. 9-problem #2; Ch. 15-problem #3 Problem #2: Type I vs. Type II error Consider the model of drug testing by the FDA that we discussed in class. A drug could be either good or bad and you have a test to help determine which kind of drug it is. Good drugs tend to yield high test scores, bad drugs low test scores. Unfortunately, this is not always the case. Here is the probability distribution of test scores for good and bad drugs. You set a threshold level of the test such that if the test yields a score above the threshold, you approve the drug, and not otherwise. TestRes (1) For a drug with the above distributions of test scores for good and bad drugs, plot the probability of a type I error (approving a bad drug) on the x-axis against the probability of a type II error (failing to approve a good drug) on the y-axis as the threshold approval level moves from -4 to 4. Getting the exact numbers right is not as important as getting the general shape right. (Incidentally, this graph you are making is known as a receiver-operator curve). (2) What will be the shape of the receiver-operator curve if there is no overlap in the distribution of test results for good drugs and bad drugs (that is, good drugs always yield a test result above some number, say x; while bad drugs always yield test results below x)? (3) What will be the shape of the receiver-operator curve if the distribution of test results for good and bad drugs exactly overlap?Problem #3: Medisure It is the year 2005 and after graduating from Stanford, you have become the president of a small isolated island nation in the South Pacific. Your country has a health care system much like the one in the U.S., except much smaller. The government has a program "Medisure" designed to provide insurance for the old and the very old for free. Unlike the American Medicare system, the recipients of Medisure do not face any cost sharing - including premiums, deductibles, or copayments; that is, the elderly pay nothing. However, like Medicare, each young person pays $1,000 in taxes into Medisure to support their elders (and like Medicare, they have no choice). Also like Medicare, there is a trust fund, currently with $1 million dollars in it, which will reportedly go bankrupt any day now. At least, that's what your political opponents say. You are up for re- election soon and have the unenviable task of saving Medisure. Here is the current population of your island: Age Population Young 100 Old 90 Very Old 50 Here are the yearly expenditures on medical care per person on your island: Age Expenditures Young Old $1,000 Very Old $4,000 There are some unique features about your island: There is full employment for every member of the young generation, no matter how high taxes get; All of them are able to pay the full $1,000 for Medisure. Each year, the young become old, the old become very old, and all of the very old die. Here are the death rates in these life transitions: Age Death Rate Young > Old 0.00% Old > Very Old 44.44% Very Old -> Heaven 100% . Your political life will last long enough that you will face the consequences of your decisions. Each year, there are 100 young people born. The trust fund earns no interest and there is no inflation. In answering the following questions, it might be helpful to use a spreadsheet. (1) Are your political opponents right? That is, will the Medisure trust fund ever go bankrupt? If so, when? (2) As if things weren't bad enough, your scientists have come up with a breakthrough medical technology that will decrease death rates without affecting per-person medical costs for the elderly. The new death rates are: Age Death Rate Young > Old 5.00% Old > Very Old 36.84% Very Old > Heaven 100% Will the Medisure trust fund ever go bankrupt now? If so, when?Question 2 (Optimal Consumption-Labor Choice - Additional Problem in Homework) Consider a Robinson Crusoe simple economy without storage technology. The production function with only labor (() as factor of production is y = f (() = 6 2. The utility function from consuming c and exerting labor { is u(c, () = log (c) - 4. The market-clearing condition requires that y = c Crusoe seeks to maximize his utility subject to his production. a) Find the marginal product of labor. b) Find the equation of an indifference curve given a level of utility T. ") Find the slope of an indifference curve given a level of utility U. Does it hold that, for the same level of labor supply , the higher indifference curve has higher slope?' There are two ways to solve for the optimal consumption-labor choice. Method 1: Using the tangency condition (MRS = MPL), note that MRS = - (Ou/02) / (Ou/Bc) d) Graphically exhibit the optimal consumption-labor choice. e) Find the marginal rate of substitution between consumption and labor supply. f) Find the optimal consumption, c", and labor supply, (". Show the answers in graph drawn in d). (Hint: Remember that Crusoe eats, c, all he produces, y = (1/2.) Method 2: Substituting the market-clearing condition - since c = y = (2, one can substitute e into utility function to receive u (21/?, {) = log (21/?) - & The problem is now a single variable unconstrained maximiza- tion problem with & as choice variable. g) Find the optimal consumption, c', and labor supply. . Question 3 (Substitution and Wealth Effect from a Shift in Production Function - Problem 2.8 in Homework) Assume the production function is y = Av + B. What are the effects on household's work effort, f, output and consumption, c, from: a) An increase in the coefficient A b) An increase in the coefficient B (This is a good question to review 2c) above.) Question 4 (Productivity - Problem 2.11 in Homework) A popular measure of productivity is the ratio of output (e.g., real GDP) to employment (e.g., works-hours). that is, y/4. Consider the usual concave production function that takes the form of y = AP", 0