Lecture 7 Asymmetric Information 2

Asymmetric information 2Lecture 7 – Tom Holden

Intermediate Microeconomics Semester 2

http://micro2.tholden.org/

ECO2051 – Intermediate Microeconomics 1

http://micro2.tholden.org/

Outline

• Moral hazard in insurance markets.

• Principal-agent models.

• Optimal contracts.

• If we have time, we’ll also cover:• The impact of AI on health.

• And current issues in health policy.


Reading

• Varian, Chapter 37

• MKR, Chapter 17 (second half)

• Additional reading (if desired):

• Frank Cowell, “Microeconomics: Principles and Analysis”• Chapter 11, especially 11.4

• Hugh Gravelle & Ray Rees, “Microeconomics”• Chapter 19, especially sub-section G

• Chapter 20, especially sub-section C

ECO2051 – Intermediate Microeconomics

Asymmetric information and moral hazard

• Our analysis of the signalling model of education was a model of hidden characteristics. The relevant characteristic was unknown but fixed.

• Moral hazard is hidden behaviour. Because an individual can’t be monitored, they may choose to act in a way which is undesirable for the other party.

• To overcome this incentives must be introduced to motivate individuals to act in the interest of the other party.

• This brings us into the world of principal-agent models.


Moral hazard in insurance markets (1/4)

• Suppose my risk depends on how much effort I put into being careful.• Probability of the bad state is 𝑝 𝑒 where 𝑒 is effort.

• Assume:• 𝑝 0 = 1 (if you make no effort the bad state will occur).

• 𝑝′ 𝑒 < 0 for all 𝑒 (i.e. effort always reduces the probability of the bad state).

• Utility is quasi-linear in consumption and effort i.e. 𝑈 𝑐, 𝑒 = 𝑢 𝑐 − 𝑒.

• Expected utility is:𝑝 𝑒 𝑢 𝑦𝐵 − 𝜋𝑥 + 𝑥 + 1 − 𝑝 𝑒 𝑢 𝑦𝐺 − 𝜋𝑥 − 𝑒

• 𝜋 is the premium.

• 𝑥 is how much insurance I take out.

• 𝑦𝐵 and 𝑦𝐺 are my incomes in the bad and good states, respectively.



• We want to maximise:

𝑝 𝑒 𝑢 𝑦𝐵 − 𝜋𝑥 + 𝑥 + 1 − 𝑝 𝑒 𝑢 𝑦𝐺 − 𝜋𝑥 − 𝑒

• If we are in an internal solution with 𝑒∗ > 0 and 𝑥∗ > 0:

• FOC 𝑥: 0 = 𝑝 𝑒∗ 1 − 𝜋 𝑢′ 𝑦𝐵 − 𝜋𝑥∗ + 𝑥∗ − 1 − 𝑝 𝑒∗ 𝜋𝑢′ 𝑦𝐺 − 𝜋𝑥∗

• So: 𝑢′ 𝑦𝐵−𝜋𝑥

∗+𝑥∗

𝑢′ 𝑦𝐺−𝜋𝑥∗ =

𝜋 1−𝑝 𝑒∗

𝑝 𝑒∗ 1−𝜋

• FOC 𝑒: 1 = 𝑝′ 𝑒∗ 𝑢 𝑦𝐵 − 𝜋𝑥∗ + 𝑥∗ − 𝑢 𝑦𝐺 − 𝜋𝑥∗

• Will people ever perfectly insure in an internal solution?



• Free entry of insurers means their profits are zero, so:

0 = 𝜋 − 𝑝 𝑒∗ 𝑥∗, i.e. 𝑝 𝑒∗ = 𝜋.• Premiums are set equal to the actuarially fair level given consumers level of

care.

• But 𝑝 𝑒∗ = 𝜋 means 𝑢′ 𝑦𝐵−𝜋𝑥

∗+𝑥∗

𝑢′ 𝑦𝐺−𝜋𝑥∗ = 1 if consumers insure, so:

𝑢 𝑦𝐵 − 𝜋𝑥∗ + 𝑥∗ = 𝑢 𝑦𝐺 − 𝜋𝑥∗

• But we already said this was impossible in an internal solution. Thus, either:• Consumers will not make any effort, so 𝑒∗ = 0, but 𝑒∗ = 0 implies 𝜋 =𝑝 𝑒∗ = 1, so consumers are indifferent about how much insurance they purchase. OR

• Consumers will not insure, and will choose the optimum level of effort given this.

• By revealed preference, the second option must be preferable.



• Summary:• Moral hazard leads the insured to take no care to avoid the bad outcome,

meaning it will certainly happen.

• Insurers realise this, and set premiums to 100%, so insurance provides no benefit.

• People thus choose not to insure at all.

• People would like to be able to commit to making an effort while insured.


Solutions to moral hazard in insurance market

• Does the solution we suggested for adverse selection work here?

• Another possibility is to prevent people from ever fully insuring.

• E.g. require a “deductible” or a “co-pay”.

• People are then worse off in the bad state, so have an incentive to provide effort.


Designing incentives

• You want someone to do something for you, but you cannot fully observe what they do, how do you pay them to get the best results?• You must ensure that their incentives are aligned with your objectives.

• Generally you’d want their pay to increase with their effort.

• This problem is often referred to as a principal-agent problem.• The principal proposes the contract.

• The agent decides whether to accept the contract and then performs under its terms.

• Principal = owner , Agent = worker


Incentive design: The principal

• Risk-neutral principal has profit function 𝜋 𝑥 − 𝑤, where:

• 𝜋 𝑥 is the (risky) gross profits,

• 𝑥 is the amount of effort the agent supplies, and

• 𝑤 is the wage the principal pays to the agent.

• We assume:

• 𝔼𝜋′ 𝑥 > 0 for all 𝑥 (increasing effort increases expected profits), and

• 𝔼𝜋 0 = 0 (with no effort, expected profits are zero).


Incentive design: The agent

• Agent has “mean-variance” utility (see Varian chapter 13)

𝑢 = 𝔼𝑤 − 𝜌𝔼 𝑤− 𝔼𝑤 2 − 𝑐 𝑥 , where:• 𝑤 is their income,

• 𝜌 measures their risk aversion, and

• 𝑐 𝑥 is the cost for them of supplying effort 𝑥.

• We assume:• 𝑐′ 𝑥 > 0 for all 𝑥 (increasing effort reduces utility).

• 𝑐 0 = 0 (exerting no effort costs nothing).

• Assume agent has an outside option that gives a utility of 𝑢.


Efficient outcome (1/2)

• Suppose that the principal could observe the agent’s effort.

• Let 𝑥∗ be the principal’s desired effort level.

• The principal could then give a contract giving zero wage unless the agent set their effort to 𝑥∗.

• How do we calculate the minimum wage the agent would accept when 𝑥 = 𝑥∗?


Efficient outcome (2/2)

• 𝑢 = 𝑤 − 𝑐 𝑥∗ . So 𝑤 = 𝑢 + 𝑐 𝑥∗ .

• Principal’s expected profits are thus: 𝔼𝜋 𝑥∗ − 𝑢 − 𝑐 𝑥∗ .

• FOC: 𝔼𝜋′ 𝑥∗ = 𝑐′ 𝑥∗ .

• I.e. expected marginal benefits equal marginal costs.


Asymmetric information case (1/5)

• Suppose now that the principal cannot see the agent’s effort.• If there is no risky component to the job, then from observing 𝜋 𝑥 the

principal can work out 𝑥. (So there is no problem.)

• Suppose also then that profits are risky, i.e. 𝜋 𝑥 has a positive variance.• For simplicity we will assume the variance of profits is equal to 𝑣𝜋 𝑥 .

• If no effort is undertaken, profit is zero, but also has zero variance.

• Also for simplicity, we restrict the principal to offering contracts of the 𝑤 = 𝑎 + 𝑏𝜋.



• Suppose the principal would like to induce the agent to provide an effort of 𝑥⋄.• Perhaps the principal wants the agent to exert the efficient level in which

case 𝑥⋄ = 𝑥∗. May not be optimal though.

• Two conditions:• Participation constraint:

• The agent must prefer providing 𝑥⋄ to her outside option.

• I.e. 𝑢 ≤ 𝑎 + 𝑏𝔼𝜋 𝑥⋄ − 𝜌𝑏2𝑣𝔼𝜋 𝑥⋄ − 𝑐 𝑥⋄ = 𝑎 + 𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋 𝑥⋄ − 𝑐 𝑥⋄

• Always optimal for the principal to set 𝑎 = 𝑢 − 𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋 𝑥⋄ + 𝑐 𝑥⋄ .

• Incentive compatibility constraint:• The agent must prefer providing 𝑥⋄ to any other level.

• I.e. 𝑥⋄ satisfies the agent’s FOC: 0 = 𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋′ 𝑥⋄ − 𝑐′ 𝑥⋄ . May be solved for 𝑏.



• Principal’s expected profits, with optimal 𝑎 are:

𝔼𝜋 𝑥⋄ − 𝑢 + 𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋 𝑥⋄ − 𝑐 𝑥⋄ − 𝑏𝔼𝜋 𝑥⋄

= 1 − 𝜌𝑏2𝑣 𝔼𝜋 𝑥⋄ − 𝑢 − 𝑐 𝑥⋄

• If 𝑏 = 1 then principal’s objective and agent’s objective are identical (other than a vertical shift).• So when 𝑏 = 1 the incentive compatibility condition is satisfied.

• In this case we have a “residual claimant” scheme. The principal keeps −𝑎 (𝑎will be negative here) and the agent gets the rest.

• Agent bears all risk, but also all marginal rewards from their effort.

• Only efficient however if 𝜌𝑣 = 0, so either the agent is risk neutral, or the task’s variance is constant.


Residual claimant example problem

• An owner wishes to incentivise his risk neutral manager. The manager attends for a full working week, but the hours of effort he puts in are unobservable to the owner.• The cost of each hour’s effort for the manager is equivalent to £30.

• The efficient solution has gross profits of £10000 and 30 hours of effort.

• The manager has an outside option which enables him to earn £3000, but he must put in 40 hours of effort.

• What is the optimal compensation scheme?

• Can you think of any examples of residual claimant compensation?

• Why aren’t residual claimant contracts more common?



• 𝑏 = 1 is not necessarily optimal from the principal’s perspective though.

• Principal should maximise his profits subject to the agent’s incentive compatibility constraint.

• Lagrangian:ℒ = 1 − 𝜌𝑏2𝑣 𝔼𝜋 𝑥⋄ − 𝑢 − 𝑐 𝑥⋄ + 𝜆 𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋′ 𝑥⋄ − 𝑐′ 𝑥⋄

• FOC 𝑥⋄: 1 − 𝜌𝑏2𝑣 𝔼𝜋′ 𝑥⋄ + 𝜆𝑏 1 − 𝜌𝑏𝑣 𝔼𝜋′′ 𝑥⋄ = 𝑐′ 𝑥⋄ + 𝜆𝑐′′ 𝑥⋄

• FOC 𝑏: 2𝜌𝑏𝑣𝔼𝜋 𝑥⋄ = 𝜆 1 − 2𝜌𝑏𝑣 𝔼𝜋′ 𝑥⋄



• Suppose that profits and costs are both linear, with 𝜋 𝑥 = 𝜋1𝑥, 𝑐 𝑥 = 𝑐1𝑥. Then we can solve the FOC 𝑥⋄ equation to get:

𝑏 =1

𝜌𝑣1 −

𝑐1𝜋1

• As the agent gets more risk averse (i.e. 𝜌 increases), or the variance of profits increases (i.e. 𝑣 increases), the optimal choice of 𝑏 decreases, and the firm will take on more and more of the risk.

• We will also move further from the efficient effort level.

• Principal will usually want to set 𝑎 > 0 and 𝑏 > 0, in order to both reduce the agent’s risk (encouraging participation), and to incentivise them to perform effort.


Reading on health

• An interesting letter from Economists to Obama can be found at:

• http://economix.blogs.nytimes.com/2009/11/17/economists-letter-to-obama-on-health-care-reform/

• Also http://www.washingtonpost.com/wp-dyn/content/article/2009/07/27/AR2009072701905.html

• Analysis of NHS reforms (magazine article) http://cep.lse.ac.uk/pubs/download/cp323.pdf


http://economix.blogs.nytimes.com/2009/11/17/economists-letter-to-obama-on-health-care-reform/

http://www.washingtonpost.com/wp-dyn/content/article/2009/07/27/AR2009072701905.html

http://cep.lse.ac.uk/pubs/download/cp323.pdf

Market failures in health care markets

• Both moral hazard and adverse selection have impacts on healthcare markets.

• There are a number of types of asymmetric information.

• Individuals would like to be insured, but have hidden information about how likely they are to be sick.• Those who are most likely to be sick will demand more insurance.• Adverse selection.

• Individual’s behaviour influences how likely they are to be sick, insurers cannot observe this.• Moral hazard.

• Individuals do not have full information about the type of treatment that would be best for them.


Asymmetric information and insurance

• For insurance to work it must be the case that the insurer and the potential insured have the same info about the probability of bad outcomes.

• The failure of this assumption leads to adverse selection.

• It must also be the case that any actions an individual (or his doctor) takes to change his risk are fully observable to the insurer.

• The failure of this assumption leads to moral hazard.


Solutions to adverse selection in health insurance markets

• Medical examinations, to identify type. But this may just mean that some individuals are not insured.

• Group health plans. Insurance which operates through the employer means that a wider range of risks are pooled and avoids adverse selection.

• Indirect targeting. Premiums are based on the average risk for the group. Again this can lead to high premiums and non-insurance for some groups.

• Co-payment and deductibles, but these reduce the welfare gains from being fully insured.


Moral hazard

• Insured individuals – May not be as careful as they could be with their health. However, this is less likely than in other insurance markets as individuals experience discomfort if they are ill. Co-payment of costs could help to reduce this problem.

• Insured individuals and doctors have no incentive to consider the costs of medical care if they are being paid by the insurance company.• Evidence of a massive amount of price dispersion in the pharmacy market

(Sorenson 2001, Ohlen & Smith 2011). The latter paper finds a coefficient of variation for the price of some drugs of 64%.

• This can be overcome if insurers, doctors and pharmacies are combined into the same organisation.• Risk that incentives could move too far into the other direction, so that the

doctor would have an incentive to under-provide care?


International comparisons

• At least prior to Obama’s reforms the private US system had some predictable problems.• Large costs, because doctors have no incentive to keep them down.

• Some individuals could not afford insurance at this high cost.

• Huge diversity of provision, many are getting great care, others none at all. Life expectancy is highly unequal.

• Despite being a private system it is still very costly to the tax payer, almost half of spending is public money. This is because the public sector is only insuring the highest risks (Medicare).


Comparative spending and outcomes

Health Spending Over Time

1970 1980 1990 2000

GDP PC GDP PC GDP PC GDP PC

Canada 7 294 7.1 783 9 1737 9.9 3003

Germany 6.2 270 8.7 965 8.5 1748 11.1 2996

Sweden 6.9 309 9.1 936 8.4 1579 9.2 2594

USA 6.9 347 8.7 1055 11.9 2738 15 5635

UK 4.5 164 5.6 482 6 986 7.7 2231

GDP: Gives health spending as a proportion of GDP.

PC: Stands for health spending per capita in US dollars.

Health Outcomes over Time

1970 1980 1990 2000

LE IM LE IM LE IM LE IM

Canada 72.9 18.8 75.3 10.4 77.6 6.8 79.7 5.4

Germany 70.4 22.5 72.9 12.4 75.2 7 78.4 4.2

Sweden 74.7 11 75.8 6.9 77.6 6 80.2 3.1

USA 70.9 20 73.7 12.6 75.3 9.2 77.2 7

UK 71.9 18.5 73.2 13.9 75.7 7.9 78.5 5.3

LE is life expectancy. IM is infant mortality, deaths per 1000 live births.


Obamacare (original plan)

• Obama wished to overcome the problems of the uninsured and lower the high cost to individuals.

• Features:• Expanding Medicaid (for low incomes).

• Forcing insurance companies to offer the same coverage and premium to people independent of their current health and/or location.

• Forcing individuals to have health insurance.

• More money for research etc.

• Stopping insurers using certain pieces of available information was bad for them. But forcing all individuals to have health insurance was good for them. (Even low risk people must purchase the same mid-premium policy.)

• Should substantially help with the adverse selection problem.

• Bad for moral hazard though?


What about the NHS?

• Recently, the reform agenda has shifted to the UK.

• GPs are now in charge of commissioning services; they hold budgets for their patients and decide how they should be spent.• They could profit from this, so should have an incentive to keep costs down.

• But will they act in the best interest of patients?

• A constant complaint against the NHS is that a lack of competition reduces quality and inflates costs.• In fact the UK looks quite good on costs, although could do better.

• Evidence suggest that previous market-based reforms have been beneficial and more could be done (ending national pay-deals).


Summary

• We’ve observed several important models of AI with striking real world implications for:• Quality choice

• Price setting

• Education

• Compensation schemes

• Insurance markets and health reform

• There are many others.

• Hopefully the tools we have learned will help you to think about how other similar problems are resolved.


Documents

Lecture 7 Asymmetric Information 2