Choice under risk and uncertainty

What is uncertainty?

• Uncertainty concerns missing information, where the link between cause and consequence, choices and outcomes, is not clear

Risk

• This is where an agent knows the achievable outcomes and probabilities of achieving those outcomes rather than simple uncertainty

Expected outcomes (and utility)

• Involves multiplying the outcome of a situation by the probability that it happens 

• Expected utility involves attaching a utility weight to an outcome; in the case of a lottery, it might look like EU(w) = p₁ux₁ + p₂ux₂, where u = utility, p = probability, EU(w) is expected utility of wealth, and x is an outcome 

Certain Equivalent 

• This is the concept of the certain lottery which is equivalent to the risky one 

• This is the amount of money it would take for a person to be indifferent between the certain amount and the uncertain lottery (the ‘riskier’ choice)

• u(CE) = EU(w) = pu(w1) + (1 − p)u(w2)

Risk attitudes

• Risk averse | CE < E(w), EU(w) < u(E(w)) — This basically means that the expected utility of the wealth is less than the utility of actually having it; they have more utility by not taking the risky lottery

• Risk neutral | CE = E(w), EU(w) = u(E(w))

• Risk lover | CE > E(w), EU(w) > u(E(w))

What is Insurance?

• Insurance is effectively trading risk; a person would prefer to pay some amount in the eventuality that they are faced with a bad state (lose their car, etc)

Optimal Insurance Contract

• k is the amount ensured, w is the value of the insured object, A is the potential damage, á is the price per pound insured, p is the probability of the damage-causing event 

• Thus, expected utility maximising equation is max_kEU(k) = (1 − p) u(w − αk) + pu(w − A + (1 − α)k)

Insurance FOC

• dEU(k)/dk = α(1 − p)u′(wg ) − (1 − α)pu′(wb) = 0

• (1 − p) u′(wg) / pu′(wb) = (1 − α) / α

• Full insurance, or constant wealth, only occurs when k = A

• To make insurance optimal, we must have that p = α, to ensure a ‘fair price’

Willingness to pay and risk premium

• An agent is only willing to pay the difference between wealth in the good outcome and the Certain Equivalent

• The risk premium is the amount an agent is willing to pay with respect to a risk neutral person, quantified by the difference between E(w) and the Certain Equivalent

Adverse Selection

• Involves the situation when two people enter a contract with asymmetric information; a risky driver who should pay a higher insurance premium tries to enjoy lower insurance prices by hiding their behaviour

• This is also called a ‘hidden characteristic’ or ‘hidden information’ problem

• This could lead to non-efficient outcomes or even stop trading in some markets

Moral Hazard 

• Occurs when someone has no incentive to mitigate risk because they’re completely insured

• In the case of bicycle insurance, a person might not take any care at all if they are reimbursed completely for losing it because they have already made the investment in insurance (as opposed to investing in a really heavy and secure bike-lock)

• Occurs when one side of the market can’t observe the actions of the other — a hidden action problem 

Signalling

• A way to try and solve the ‘lemon market’ problem; sellers can offer warranties to insure a consumer against the risk of their car being a lemon

• Owners of the good cars can afford this whereas the owners of the bad cars can’t

The Incentive Compatibility Constraint

• Simply states that the utility to the worker of some effort level x* must be greater than any other effort level; this effort level is the one which maximises profit for the incentiviser

• Marginal cost of effort must be equal to the marginal product of effort

• The incentiviser can aim to do this through many ways:

How might a landowner incentivise a worker in line with the incentive compatibility constraint?

1. Rent (rent the land to the worker, they keep all the profit over the amount they pay back to the owner)

2. Wage labour pays a person an hourly rate

3. Take-it-or-leave-it: the worker is paid if they work at x* and nothing else

4. Sharecropping is where the owner and worker both take a percentage of the product of labour