Asymmetric information

The previous lesson broke the efficiency result by letting costs fall on people outside the trade, and this one breaks it without leaving the trade at all.

The competitive model assumes buyers and sellers know what is being sold. Drop that, let one side know something the other does not, and markets fail in a way that is harder to fix than an externality, because the missing thing is not a price but a fact. George Akerlof, Michael Spence and Joseph Stiglitz shared the 2001 Nobel prize for working out the consequences, and the machinery they built is now most of what applied microeconomics does.

Risk aversion is concavity

Insurance is the cleanest setting for information problems, and it needs one piece of apparatus first: a way to talk about choices whose outcome is uncertain.

Suppose a person's utility depends on wealth through a function $u (w)$ , and they evaluate a gamble by its expected utility, the probability-weighted average of the utility of each outcome. If $u$ is concave, so $u'' < 0$ , then the utility of the average exceeds the average of the utilities, and the person prefers a certain sum to a gamble with the same mean. That is risk aversion, and it is nothing more than diminishing marginal utility of wealth: the pound gained in the good state is worth less than the pound lost in the bad one.

Two numbers make this operational. The certainty equivalent is the sure amount giving the same utility as the gamble, and the risk premium is the gap between the gamble's expected value and its certainty equivalent, which is the most the person would pay to shed the risk beyond its actuarial cost.

Example. Someone with $u (w) = \sqrt{w}$ has wealth £10,000 and faces a 25 per cent chance of losing £3,600. Find the expected wealth, the certainty equivalent, the risk premium and the most they would pay for full insurance.

Expected wealth is $0.75 \times 10000 + 0.25 \times 6400 = £ 9100$ . Expected utility is $0.75 \times 100 + 0.25 \times 80 = 95$ , so the certainty equivalent is $95^{2} = £ 9025$ . The risk premium is $9100 - 9025 = £ 75$ . Full insurance leaves them with $10000$ minus the premium for certain, and they accept any premium up to $10000 - 9025 = £ 975$ . The actuarially fair premium is $0.25 \times 3600 = £ 900$ , so there is £75 of room for the insurer's costs and profit.

Now you. The same person faces a 20 per cent chance of losing £5,100. Find the certainty equivalent, the risk premium, the fair premium and the maximum premium.

Answer

Expected wealth is $0.8 \times 10000 + 0.2 \times 4900 = £ 8980$ . Expected utility is $0.8 \times 100 + 0.2 \times 70 = 94$ , so the certainty equivalent is $94^{2} = £ 8836$ . The risk premium is £144, the fair premium is $0.2 \times 5100 = £ 1020$ , and the maximum premium is $10000 - 8836 = £ 1164$ .

That gap is why insurance markets exist. A risk-averse individual and a risk-neutral insurer pooling thousands of independent risks both gain from the trade, and with full information the market works. Everything that follows is about what happens when the insurer cannot tell one customer from another.

Adverse selection: the market for lemons

Akerlof's 1970 paper, rejected by three journals before it was published, made the argument with used cars. Sellers know whether their car is sound; buyers do not.

Example. Half of used cars are good and half are lemons. A good car is worth £8,000 to its owner and £10,000 to a buyer; a lemon is worth £3,000 to its owner and £4,000 to a buyer. Show what happens with and without information.

With information, every car trades: buyers value each type above its owner does, so there are gains from trade on both. Without it, a buyer facing an unknown car values it at $0.5 \times 10000 + 0.5 \times 4000 = £ 7000$ and will pay no more. But no owner of a good car will accept £7,000 for something worth £8,000 to them, so good cars are withdrawn. Only lemons remain on offer, buyers work this out, and the price falls to £4,000. The market for good used cars has disappeared, even though every one of those trades would have made both parties better off.

Now you. Suppose instead that 60 per cent of cars are good, that a good car is worth £8,000 to its owner and £12,000 to a buyer, and lemons are unchanged. What happens?

Answer

The expected value of an unknown car is $0.6 \times 12000 + 0.4 \times 4000 = £ 8800$ , which exceeds the £8,000 reservation price of a good car's owner. So good cars stay on the market and everything trades at £8,800, with owners of good cars gaining £800 and owners of lemons gaining £5,800. The market survives when the quality gap is small enough and good cars are common enough relative to the buyers' valuation of them.

This is adverse selection: the price that clears the market attracts exactly the participants it should not.

Unravelling in insurance

The mechanism is general, and insurance is where it bites hardest, because there the hidden characteristic is the very thing being priced.

A premium set for average risk is a bargain for high risks and poor value for low risks, so low risks drop out. The average risk of those remaining rises, the premium rises with it, and more of the healthy leave. The process feeds on itself, which is why it is called a death spiral rather than a shift to a new equilibrium. David Cutler and Sarah Reber documented a complete one at Harvard, which changed its employee health contributions in 1995: the more generous plan attracted sicker enrollees, its premium climbed, healthier staff left it, and it was withdrawn entirely within two years.

Michael Rothschild and Joseph Stiglitz showed in 1976 that this can leave a competitive insurance market with no equilibrium at all in the pooling sense: any single contract attracting both types can be undercut by one designed to attract only the low risks. The standard responses are mandatory coverage, which removes the option to leave, and community rating with an individual mandate attached, which is why almost every health financing system in the world contains some form of compulsion.

Signalling and screening

If the informed side can prove something, the market can be rescued. Signalling is the informed party taking a costly action to reveal what they know; screening is the uninformed party offering a menu designed to make the other side reveal it.

Michael Spence's 1973 model is the sharpest illustration, precisely because it assumes education is useless. Suppose high-productivity workers are worth £60,000 a year and low-productivity ones £40,000, employers cannot tell them apart, and a year of education costs a low-productivity worker £8,000 in effort and a high-productivity one only £4,000, while adding nothing whatever to what either can do.

Let firms pay £60,000 to anyone with at least $y^{*}$ years and £40,000 otherwise. A low type stays uneducated if $40 \geq 60 - 8 y^{*}$ , which needs $y^{*} \geq 2.5$ . A high type gets educated if $60 - 4 y^{*} \geq 40$ , which needs $y^{*} \leq 5$ . So any requirement between 2.5 and 5 years separates the types, and the market reaches the full-information wages by burning real resources on a qualification that teaches nothing. At three years the high type spends £12,000 to earn £20,000 more: privately worth doing, socially pure waste.

Example. Take high productivity at £80,000 and low at £50,000, with education costing £10,000 a year for low types and £5,000 for high types. Find the range of separating requirements.

The low type must prefer no education: $50 \geq 80 - 10 y^{*}$ needs $y^{*} \geq 3$ . The high type must prefer education: $80 - 5 y^{*} \geq 50$ needs $y^{*} \leq 6$ . So any requirement from three to six years separates them.

Now you. Suppose the cost per year for the high type rises to £6,000, with everything else as in the previous case. What is the new range?

Answer

The low type's condition is unchanged, so $y^{*} \geq 3$ . The high type now needs $80 - 6 y^{*} \geq 50$ , so $y^{*} \leq 5$ . The range narrows to three to five years, and if the high type's cost rose above £10,000 a year the range would vanish and no separating requirement would exist.

Two conditions make signalling work, and both are visible in the algebra. The signal has to be costly, or everyone sends it. And it has to be differentially costly, cheaper for the type it is meant to identify, or it separates nobody. That is the test to apply to any claimed signal: warranties are credible because they cost a maker of bad products more, and an advertising campaign signals confidence in a product for the same reason.

The uncomfortable implication is that a signalling equilibrium can be socially wasteful while every individual in it behaves sensibly. How much of the return to education is signalling and how much is genuine skill remains one of the most argued-over empirical questions in economics, and the honest answer is that both are present and the split is not settled.

Moral hazard

Adverse selection is about hidden characteristics, known before the contract. Moral hazard is about hidden actions, taken after it. An insured driver takes marginally less care; a salaried worker exerts marginally less effort. Neither is dishonesty. It is the ordinary marginal calculation from the first lesson, applied to a person who no longer bears the full cost of their own choices.

The arithmetic is a first-order condition. Suppose an agent choosing effort $e$ produces output $20 e$ at a personal cost $e^{2}$ . If the agent keeps all the output, they maximise $20 e - e^{2}$ , giving $e = 10$ and output of 200. If instead they receive a share $s$ of output, they maximise $20 s e - e^{2}$ , giving $e = 10 s$ . At a half share effort falls to 5 and output halves to 100; at a quarter share effort falls to 2.5 and output to 50. Effort tracks the share of the marginal return the agent captures, which is the whole of contract design in one line.

That creates the central tension. A risk-averse agent wants insurance against outcomes they cannot control, which means a fixed wage, and a fixed wage sets $s = 0$ and destroys effort. A pure output contract gives perfect incentives and loads all the risk onto the person least able to bear it. Real contracts sit between: a base salary plus a bonus, an insurance policy with a deductible, a franchise arrangement, a tenancy that splits the crop.

The empirical magnitude is well established for health care. The RAND Health Insurance Experiment, which randomly assigned American families to plans with different cost-sharing between 1974 and 1982, found that those given free care used about a third more of it than those facing substantial deductibles, with no detectable difference in health outcomes for most participants, though the poorest and sickest did worse under cost sharing. That is moral hazard measured under randomisation, and it is also a warning that the efficiency loss and the health loss are different quantities.

What the theory gets wrong

The models in this lesson are unusually clean, and the evidence is messier in ways worth knowing.

The signature prediction of adverse selection is a positive correlation between how much coverage someone buys and how much they claim. Pierre-André Chiappori and Bernard Salanié tested this carefully on French motor insurance in 2000 and found no such correlation among new drivers. The likely explanation is that two effects run in opposite directions: high risks buy more cover, but cautious people also buy more cover, and caution and risk pull the correlation apart. Testing for private information is much harder than the theory suggests.

Signalling models generally have many equilibria, since any $y^{*}$ in the separating range works, and the theory has nothing to say about which one occurs. That is not a small gap when the practical question is how many years of schooling an economy should be buying.

And the whole apparatus assumes the informed party knows their own type, which is often false. Most drivers believe they are above average, most entrepreneurs overestimate their prospects, and adverse selection built on accurate self-knowledge is then modelling something that is not happening.

What survives all of that is the core insight, and it is robust: a market can be perfectly competitive, free of externalities, and still fail, because the thing being traded cannot be verified. That completes the catalogue of failures. The last lesson steps back to all markets at once and asks exactly what the competitive result claims when it does hold, and how much of the framework survives the evidence against it.