Screening. Sandeep Baliga. March 29. Sandeep Baliga () Screening March 29 1 / 26

Transcription

1 Screening Sandeep Baliga March 29 Sandeep Baliga () Screening March 29 1 / 26

2 Screening So far we studied competitive equilibrium and benevolent government intervention when there is asymmetric information. But a monopolist may also face asymmetric information. How does asymmetric information affect the optimal solution that is offered by a monopolist? How about an oligopolist? We will assume the uninformed party moves first and offer a contract/mechanism. This is called screening. When informed party move first, it is called signaling. We will begin with screening ranging from monopoly, competitive to competitive with behavioral types. Sandeep Baliga () Screening March 29 2 / 2

3 Monopolistic Screening There is a risk-neutral firm that produces a unit of a good of quality q at cost c(q) where c (0) = 0, c ( ) = and c (0) > 0. Its profits from a sale of one unit at price t are V = t c(q). The agent has a unit demand and has a payoff function U = u(q, θ) t = θq t. His outside option is zero. Notice, for q > q u(q, θ) u(q, θ) is increasing in θ or u 12 > 0. This is called the Spence-Mirrlees condition or single-crossing property. Agents with higher θ s are willing to pay more for a given increase in quality. Diagram here. We assume there are just two types of consumers: θ > θ with a proportion p of high types. Sandeep Baliga () Screening March 29 3 / 26

4 The First-best: Complete Information The principal solves for each θ maxt c(q) such that q,t θq t 0. Therefore, we get c (q (θ)) = θ and t = q (θ)θ. Therefore, as θ > θ, we get q ( θ) > q (θ). This is first-degree price discrimination. Add diagram here Sandeep Baliga () Screening March 29 4 / 26

5 The Second-best: Incomplete Information. By the revelation principle, we restrict attention to contracts or menus of the form {(q, t), ( q, t)} such that type θ will choose the first option and type θ the second. Therefore, the principal s program is subject to max p[ t c( q)] + (1 p)[t c(q)] {(q,t),( q, t)} θ q t θq t (IC ) θq t θ q t (IC ) θ q t 0 (IR) θq t 0 (IR) Sandeep Baliga () Screening March 29 5 / 26

6 The two IC constraints can be combined to give Therefore, IC implies q q. We will now simplify the problem. θ ( q q ) t θ ( q q ). 1. First, from θ > θ and (IC ) and (IR), we get (IR). Therefore, we can omit it from the program as it holds automatically. 2. Second, (IC ) binds (i.e. holds as an equality at the optimum). Suppose not. Then, increase t. This increases the maximand while relaxing (IC ) and leaving (IR) unchanged. Contradiction. 3. Third, as (IC ) binds, (IC ) is slack when q q. Therefore, we can omit (IC ) too. 4. Finally, (IR) must bind: otherwise, increase t. Sandeep Baliga () Screening March 29 6 / 26

7 Therefore, we get Therefore, the principal s problem is t = θq θ q θq + θq = t = θ q θq. max p[ θ q θq c( q)] + (1 p)[θq c(q)] {(q,t),( q, t)} subject to q q. Sandeep Baliga () Screening March 29 7 / 26

8 Ignoring the monotonicity constraint. The first order conditions are θ = c ( q) θ = c (q) + p 1 p θ. Hence, q is set at the first-best level and q < q (θ). First, notice the monotonicity constraint is satisfied. Second, notice that for large p, the second equation cannot be satisfied for positive q and, therefore, it is then optimal to just exclude the low types and serve only the high types. When q > 0, the high type earns information rents. We can then substitute back to find prices etc. Sandeep Baliga () Screening March 29 8 / 26

9 The general properties are: 1. The high type gets an effi cient allocation: effi ciency at the top 2. The lowest type gets an ineffi cient allocation with ineffi ciently low sales: underproduction. 3. The IR constraint for the lowest type binds. The high type gets informational rent. These properties are true in general models. The monotonicity constraint we omitted may bind though This is second degree price discrimination. Sandeep Baliga () Screening March 29 9 / 26

10 Competitive Screening in Health Insurance Rothschild and Stiglitz (1976) extend the model of Akerlof to oligopoly. Consumers have private information about health risks Firms determine the set of policies available to consumers. Sandeep Baliga () Screening March / 26

11 The 2x2 Model There are two types of consumers: Good (G) and Bad (B) risks. There are two possible outcomes: Loss (i.e. Sickness), No Loss. Each consumer knows own risk level, p G or p B. Bad risks have greater probability of sickness/loss: p G < p B. Initial wealth W, possible loss of L < W. Sandeep Baliga () Screening March / 26

12 Competitive Screening with Symmetric Information. Suppose that a consumer has probability of loss p (could be good or bad but it is known). Consumer has utility function U. There are two insurance companies. They are rich and can transfer wealth between states. They demand a premium in return for a repayment if there is a loss. Insurance companies Bertrand compete and must make zero profits in equilibrium. Specifically, consumers go to the company that maximizes their utility. If both give them the same expected utility, they divide equally. Sandeep Baliga () Screening March / 26

13 The Actuarially Fair Line Define: W 1 = Consumption in State 1 (No Loss) and W 2 = Consumption in State 2 (Loss). Note W 1 = W premium W 2 = W L premium + repayment. Key point: Since insurance companies make zero profits, they cannot be getting any net wealth in equilibrium. Consumers get it all This gives actuarially fair line/fair odds line/zero profit line: W 2 = W pl p (1 p) W 1. p Sandeep Baliga () Screening March / 26

14 Graphical Analysis of Equilibrium with Symmetric Information Sandeep Baliga () Screening March / 26

15 Indifference Curves for Two Types of Consumers Suppose that there are two types of consumers with same utility function for wealth, but different risk levels. Then the indifference curves for the bad type are relatively flat. Bad type puts more weight on State 2 than does Good type. This means that the indifference curves for the two types satisfy the single-crossing property companies can screen the types. Sandeep Baliga () Screening March / 26

16 Equilibria and Feasible Contracts A fraction λ of consumers are Good and 1 λ are Bad. Each firm f offers a pair of contracts (W 1B, W 2B ) and (W 1G, W 2G ). They still have to make zero expected profits in equilibrium. There are two potential types of equilibria. In pooling equilibria (W 1B, W 2B ) = (W 1G, W 2G ). All types get the same contract. In separating equilibria (W 1B, W 2B ) = (W 1G, W 2G ). Different types choose different contracts. Sandeep Baliga () Screening March / 26

17 Pooling Equilibria Do Not Exist We will show cream-skimming destroys pooling equilibria Let p = (1 λ) p B + λp G. Fair odds line in any pooling equilibrium is Draw picture. W 2 = W pl p (1 p) W 1. p Sandeep Baliga () Screening March / 26

18 Separating Equilibrium: Incentive Compatibility Constraints and Zero Profit Conditions Each insurance company offers a pair of contracts {(W 1B, W 2B ), (W 1G, W 2G )} where (W 1B, W 2B ) = (W 1G, W 2G ). For types to separate we must have p B U(W 2B ) + (1 p B )U(W 1B ) p B U(W 2G ) + (1 p B )U(W 1G ) p G U(W 2G ) + (1 p G )U(W 1G ) p G U(W 2B ) + (1 p G )U(W 1B ) Firms learn agents types through their choice and hence must make zero profits type by type. Sandeep Baliga () Screening March / 26

19 Graphing the Separating Equilibrium. Step One: Incentive Condition for Good Types Sandeep Baliga () Screening March / 26

20 Graphing the Separating Equilibrium. Step Two: Unique Contract for Good Types Sandeep Baliga () Screening March / 26

21 Graphing the Separating Equilibrium. Step Three: Contract for Bad Types Must Be Effi cient Sandeep Baliga () Screening March / 26

22 Graphing the Separating Equilibrium. Step Four: Separating Equilibrium May Not Exist Either! Sandeep Baliga () Screening March / 26

23 Solutions and Obamacare Analogy Mixed strategy equilibrium exists. Mandate pooling by law. Force insurance companies to offer the same contract within some classes, Bronze, Silver etc. Does not welfare dominate separating equilibrium as low risks subsidize high risks. Sandeep Baliga () Screening March / 26

24 Why is there so much money in plastic? Link

25 Competition with Heterogeneous Consumers 32% 15% 15% 24% Source: Estimate from Brittain Associates 14% Dazed and Confused Schemers Credit Addicts Reward Addicts Bookkeepers

26 Let s define the terms. BOOKKEEPERS: Own credit cards as a way to conveniently keep track of their financial lives. REWARD ADDICTS: Behavior based on a very single minded accumulation of rewards CREDIT ADDICTS: Always carry balances and often live beyond their means. SCHEMERS: Motivated by a sense of beating the system. DAZED AND CONFUSED: Do not really understand the facts and features associated with the cards they own. They often do not know that their cards have fees, what the interest rate is, or what valueadded proposition their card holds.

27 Self control: To Drink or Not To Drink ME right now ME a couple of hours later at the bar 5 Go to the bar with my friends and drink responsibly 3 Go directly home; watch old sitcoms

28 With Self control, we can have it both ways -10 ME right now ME a couple of hours later at the bar 5 Go to the bar and indulge myself but feel sick the next day 3 Go to the bar with my friends and drink responsibly Go directly home; watch old sitcoms

29 With Self control: The Devil Inside Us But ME a couple of hours later at the bar is a different person (-10, 10) ME right now (5, 5) 3 (my utility now, utility of my future self)

30 2 of irrationality! The behavior considered above is irrational!! A traditional rational agent has just one self He behaves consistently And does not experience preference reversals But we know, of course, that people can be irrational; but irrationality comes in degrees A sophisticated consumer understands that she may be irrational in the future. She anticipates this and tries to mitigate the harmful effects. How?

31 A sophisticated agent is not necessarily rational but at least he anticipates his own irrationality (-10, 10) ME right now My future self 3 Commit not to go to the bar. My present self commits my future self not to indulge (5, 5)

32 A dazed and confused agent is irrational and he unaware that he is irrational! (-10, 10) My future self (5, 5) ME right now Forgo the commitment because of failure to anticipate the actions of my future self. I go to the bar naively thinking I will be able to exercise self-control 3

33 A Simple Model of Demand for Credit Cards Model due to Ausubel. Let s simplify by assuming three kinds of consumers: Convenience users: Bookkeepers and reward addicts and schemers. Credit addicts: Can t find a better source of credit than credit cards. Dazed and Confused: People who do not intend to borrow but do so anyway. They repay their loans and are good credit risks. Key assumption: An issuer does not now which group an applicant belongs to when they apply for a card. If you were an issuer which group would you like to target? Which groups are you going to want to avoid? How responsive are the three groups to interest-rate cuts?

34 The impact of an interest-rate cut I: Suppose your competitors are all charging high interest rates and that you are all making positive net profits. The standard story says that either an existing issuer should cut rates or someone may enter at a lower rate. Is this a good idea here?

35 A simple model of switching costs and adverse selection I Both credit addicts and dazed and confused consumers have search costs between $0 and $100 and switching costs are evenly distributed over this interval. The data suggests both groups are roughly the same size. A credit addict expects to roll over $1000 if he does not default, but he will default with 50% probability. A dazed and confused customer expects to have a balance of only $100 but actually ends up with a rolling debt of $500. He never defaults. Suppose Bank 1 is setting an interest rate of 20% and Bank 2 is thinking of cutting it to 18%.

36 A simple model of switching costs and adverse selection II A credit addict is looking at a potential gain of = $20 and compares this to his switching costs So, given switching costs are uniformly distributed between $1 and $100, about 20% of credit addicts will switch. A dazed and confused consumer only expects to roll over $100 and expects a saving of = $2. So only about 2% of these will switch. (Of course if they were not dazed and confused, more would switch!). So Bank 2 will end up with the high default risks and very few good risks! Raising rates may make more sense than lowering them.

37 Super FICO and the leveling of the playing field Credit card companies derive considerable advantage from their own proprietary models of default risk They supplement generic credit bureau information and credit scoring agencies with their own statistical models backed by their massive data sets. Generally, credit scores do a good job predicting default risk, but of course they are still imperfect Fair-Isaac is offering a new product that dramatically improves the ability to sort various segments by default risk and other relevant attributes What would that do to the industry?

38 Making into Suppose people who stay in If a hotel cuts the price of the hotels either just pay for the room to attract dazed and room ( cheapskates ) or pay for confused, it will also attract a the room and then lose selfcontrol and hit the expensive lot of cheap-skates and may end up making less money: mini-bar ( dazed and Room Prices and Mini-bar confused ). prices stay high despite competition. Other examples: car rental companies selling insurance and pre-paid gasoline; Best Buy pushing extended warranties

39 Know Thyself

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