Question Description

This assignment contains 3 problems on Employers surplus from the high skilled  . The first problem includes step-by-step directional questions that should lead you through the logical steps needed to solving these types of problems. Please also refer to the notes and try to see the general patterns of logical reasoning needed for these types of questions. The main difficulty in dealing with the new material is that you need to use different elements of methodology that were covered at different times during the semester and at the same time be able to analyze the text of a problem to pick up the necessary elements and put all of these together.

All 3 problems are set up as essay questions. Get used to showing your work and writing all the intermediary steps while also making sure your final numerical answers are correct.

ADVERSE SELECTION PROBLEMS 

Problem I: An insurance company can insure two types of drivers: risky drivers and safe drivers. The cost to insure a risky driver is $350, while the cost to insure a safe driver is only $50. A risky driver is willing to pay up to $500 for insurance, while a safe driver is willing to pay up to $100 for insurance. Explain and analyze the market outcomes under different informational scenarios (full information, asymmetric information, and incomplete but symmetric information). For each case, state who gets to purchase insurance, at what price, and what are the resulting surpluses (for buyers, seller, and in total). Assume there is only one insurance company in town (so it gets to price its insurance as high as it wants without any competition and consumers can either take it or leave it). Also assume there are 300 drivers in total – 100 of them are risky and 200 are safe. 

Case I. Full Information (everybody knows who is a risky/safe driver) – the market essentially functions as two separate mini-markets: the one for risky drivers and the one for safe drivers. 

  1. Given, the cost of insuring a risky driver and his willingness to pay, is trade possible between risky drivers and the insurance company? 
  2. If YES, how much should the insurance company charge a risky driver to maximize its profit and make sure these drivers will buy it? Remember there is no competition! 
  3. At this price, what is the surplus for a risky driver? What is the surplus (profit) for the insurance company from selling to a risky driver? 
  4. What is the total market surplus (keep in mind there are more than 1 risky driver on the market so you need to all the surpluses for each transaction)? 
  5. Repeat 1-4 for the safe drivers. 
  6. Summarize the market outcome for the case of full information. 

Case II. Asymmetric Information (buyers know their true type, but the insurance company cannot tell a risky driver from a safe driver) – so the insurance company has to “average out”. It can no longer price differently, it has to offer insurance at one single price. 

  1. Given the uncertainty, what is the expected cost of insuring a randomly picked client (or the average cost per client if you insure everyone)? This will be the minimum price the insurance company will be willing to sell insurance (nobody wants to price below cost). 
  2. At this minimum price, and given the individual willingness to pay of each buyer (keep in mind they know who they are), who will buy insurance? 
  3. What will be the profit of the insurance company in this case? 
  4. Can the insurance company do better than this by offering insurance at a different price? In other words, if the insurance company anticipates what could happen given your answers in point 2 and 3, would they even do what   you said they would in point 2? 

5. Summarize the market outcome (who buys and at what price) and compute the surpluses. Contrast with the full information case. 

Case III: Incomplete but Symmetric Information (nobody knows who is risky and who is safe) – you might say this is unrealistic, but it serves a purpose to pretend that this can actually happen. We want to see that is NOT the simple missing of information that creates the problem of adverse selection, but the ASYMMTERY of it. 

  1. If drivers also don’t now their true type (but they know the likelihood of being risky vs. safe), what is their expected willingness to pay for insurance? 
  2. We already know from Case II, what is the expected cost for the insurance company. But if we didn’t, we would need to calculate it here again, because in this case there is uncertainty on both sides of the market. 
  3. Given the expected cost, and expected willingness to pay, will there be trade possible on this market? If so, at what price? 
  4. Imagine now that after everyone buys, drivers realize their true types. Carefully calculate all the relevant individual surpluses and the total market surplus. For individual surpluses, you need to calculate the surplus of a risky driver, the surplus of a safe driver, the surplus of the insurance company when selling to a risky driver, and the surplus of the insurance company when selling to a safe driver. 
  5. Summarize and contrast with Case I and Case II. 

CASE IV: Alternative Scenario – redo Case II, under the assumption that safe drivers are now willing to pay up to $200 for insurance. 

Following these step-by-step questions, you should be able to solve this problem. It is important however that you start developing the intuition needed to solving these types of problems without having these types of directions. Refer to the lecture notes as well and try to see the “big patterns” of adverse selection problems so you can ask these intermediate questions yourselves and be able to solve problems such as the ones below. 

Problem II: There are M sellers of used cellphones on the market and many more potential buyers who compete against each other to buy these cellphones. Assume cellphones can either be of good quality or bad quality. The reservation seller prices are $150 for a good quality phone, and $20 for a bad quality phone. Buyers are willing to pay up to $200 for a good quality phone, and up to $50 for a bad quality phone. The ratio of good phones to bad phones among these M sellers is 1/2. That means that for every good phone there are two bad phones or that the probability of dealing with a good phone is 1/3 and the probability of dealing with a bad phone is 2/3. 

a) What will happen if buyers can determine the exact quality of the phones before buying? What goods get traded, what are the prices, and what is the total market surplus in this full information case? 

b) What happens under asymmetric information? Just as before, state which goods get traded, at what prices, and what is the total surplus on the market? 

c) What should be the seller’s reservation price for the high quality phone (instead of $150) such that we will not observe an adverse selection problem? 

Problem III: There are two people who apply for a job. One is highly skilled, the other one is not. Their productivities (profits that they bring to the employer) are $1000 and $300, respectively. Their reservation wages are $500 and $100, respectively. 

a) Assume the employer can distinguish who is highly skilled and who is not. Who will get hired and at what wage? Assume he can hire both or just one of them depending on which is more profitable. What will be the profit for the employer? 

b) Assume the employer cannot distinguish between workers, but the workers know who they are. What is the wage that he would offer, who will get hired and what will be the profit for the firm in this case? 

c) Propose a signaling contract (it doesn’t need to be optimal) based on education that could improve the outcome. Assume the costs for one year of education are $10 and $50, respectively for the high skill and low skill worker. You need to find wages for the two types, and a level of education that results in a separating equilibrium and that both workers will accept. 

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