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PUBPOL 5210 – Problem Set #2 (Uncertainty and Adverse Selection)

[100 points]

Due October 16, 2025

1) Suppose during your final year at Cornell you are offered a job with a new startup or Accenture Global Health Consulting. Accenture pays a constant salary and no bonuses and would pay you $40,000. The start-up will pay you one of two salaries based on firm performance. You investigate past performance, and you see that 50% of the time people earn $90,000 but also 50% of the time people only earn $6,400.

a) (10 points) If you made your job choice based on the expected income which job would you choose?

b) (10 points) If you made your compensation choice based on expected utility and your utility function is given by U = √I  (utility is equal to the square root of Income) which job would you choose?

c) (10 points) Assuming you make your decision based on expected utility what is the lowest amount of income Accenture could pay you to accept the job over the start-up.

2) (10 points) Now that Donald Trump is in his second term he is relooking at investing in a land deal in Moscow (even with the going on). He is risk neutral and his utility function is U(I) = I where I is income.  He believes that there is an 80% chance that he will earn $100,000,000 from the investment and a 20% chance that he will earn $500,000,000.  Trump can choose to put his money in a safe asset that guarantees him $200,000,000 for sure. Michael Cohen (even though he served in prison and is no longer friends with Trump) is still the best expert on Russian issues and can tell President Trump for sure whether he will make the $500,000,000 on the land deal. Prior to Michael Cohen looking into the deal Trump believes that there is an 80% chance that he will report that the deal is not a good one. How much will he be willing to pay Michael Cohen to look into the deal so that he can have perfect information about whether the Moscow deal will give him the big payoff?

3) Suppose individuals have a utility function given by U = √W  (Utility is equal to the square root of Wealth) and is thinking about buying health insurance.  Individuals start with a wealth of $100 and face a .2 probability of getting ill.  If they do get ill they suffer losses $36. Suppose there is an insurance company that is willing to offer full insurance to the consumer (they will cover all the losses from illness if the consumer does gets ill).

3a) (10 points) What is the maximum amount of money the consumer would be willing to pay for full insurance?

3b) (10 points) If the insurance company can extract the full willingness to pay by the consumer how much money will they make on each consumer.

Now however there happens to exist high risk individuals in the community that the insurance company did not know existed. They have the same exact utility function as the previous consumers but have a much higher probability of an accident. These higher risk individuals have a .5 probability of having an accident. There is still a $36 loss to the consumer if an accident occurs.    

3c) (10 points) Would the high-risk person want to buy the contract and if they do will the insurance company lose money for each high-risk person buying it? (recall that they made money when low risk people purchased the contract).  

3d) (20 points) Use the graph structure we developed in class (with Wealth if there is an accident on the vertical axis and Wealth without an accident on the horizontal axis) to show why the high-risk person would be so attracted to this contract (that they get on a higher indifference curve when buying this insurance contract versus not buying insurance.

3e) (10 points) Now suppose the insurance company realizes that they are losing lots of money because there are two types of consumers (high risk and low risk) but still cannot determine which consumers are high risk and which consumers are low risk.  Explain in words how the insurance company is likely to respond to this situation.