Expected Utility Theory - EC-2065 Lecture 6 PDF

Expected Utility theory  Labour supply again We considered labour supply and spent some time thinking about corner solutions in the last part. A corner solution occurred where the slope of the IC was greater than the slope of the budget constraint. Or MRS>w (setting the prices...

Expected Utility theory  Labour supply again We considered labour supply and spent some time thinking about corner solutions in the last part. A corner solution occurred where the slope of the IC was greater than the slope of the budget constraint. Or MRS>w (setting the prices of consumption goods=1)  The reservation wage We can use this condition to determine the wage at which the individual would be indifferent between working and not. This is known as the reservation wage, the minimum wage that the individual would take a job offer MRS=wres This gives us a clue as to what causes (at least some) unemployment.  An individual may be searching for work and, given job characteristics etc, will accept a job if w> wres What is the reservation wage? Economic agents operate in an uncertain environment. There is likely to be a distribution of wages that could be offered  The individual will make his/her decision on the basis of the wage that was expected. These expectations may need to be updated in the light of new information (Bayes rule, Bayesian decision theory) We need to model decisions under uncertainty; this is expected utility theory  Risk versus Uncertainty We are going to differentiate between two different ways in which the future may not be known. Horse races or Roulette wheels What is the difference?  When playing a roulette wheel the probabilities are known Everyone agrees on the likelihood of black So we (the researcher) can treat this as something we can observe Probabilities are objective This is a situation of risk  When betting on a horse race the probabilities are unknown Different people may apply different probabilities to a horse winning We cannot directly observe a person’s beliefs Probabilities are subjective This is a situation of uncertainty (or ambiguity)  Attitudes to risk Would you accept the following lottery; – win £1 with probability.5 or lose £1 with probability.5? This is known as a fair bet. It is a fair bet because the expected value is equal to zero. The expected value is given by Where is the probability of outcome 1 and is the return in outcome 1 and is the probability of outcome 2 and is the return in outcome 2.  Does (should) size matter? Which would you prefer of the following? – A lottery ticket that pays out £10 with probability.5 and £0 otherwise, or – A lottery ticket that pays out £3 with probability 1 – =0.5*10=£5>£3 How about: – A lottery ticket that pays out £100,000,000 with probability.5 and £0 otherwise, or – A lottery ticket that pays out £30,000,000 with probability 1 – =0.5*100000000=50000000>30000000  Inconsistency Your answer to these questions should not change if your utility function is stable in some predictable way, for example strictly concave (or convex). In both cases the expected value of the lottery is greater than the value of the certain outcome. The option you chose in case 1 should be the same as that chosen in case 2, or vice versa. BUT this is not the case experimentally.Why?  Expected value or not Usually, people do not simply go by expected value but have attitudes towards risk An agent is risk-neutral if she only cares about the expected value of the lottery ticket An agent is risk-averse if she always prefers the expected value of the lottery ticket to the lottery ticket – Most people are like this An agent is risk-seeking if she always prefers the lottery ticket to the expected value of the lottery ticket  The expected utility function The expected utility function is a function that allows us to rank alternatives in terms of expected utility, rather than expected value There are some assumptions about preferences that need to be satisfied (and sometimes these are violated) but if so then we can use our expected utility function to determine optimal choices. Expected value versus expected utility The expected value of a risky prospect is Where is the probability of state 1, is the probability of state 2, and is the return in state i. Expected utility is In this case we are interested in the weighted sum of the UTILITIES rather than the values.  Decreasing marginal utility Typically, at some point, having extra money does not make people much happier (decreasing marginal utility) utility buy a nicer car (utility = 3) buy a car (utility = 2) buy a bike (utility = 1) $200 $1500 $5000 money  Different possible risk attitudes under expected utility maximization utility money Green has decreasing marginal utility → risk-averse Blue has constant marginal utility → risk-neutral Red has increasing marginal utility → risk-seeking Grey’s marginal utility is sometimes increasing, sometimes decreasing → neither risk-averse (everywhere) nor risk-seeking (everywhere)  A fair bet U(w) EU=0.5*u(w0-£1)+ 0.5*u(w0+£1) u(w0+£1) Note that the utility u(w0) from turning down the lottery is greater than u(w0-£1) the expected utility from the lottery. w0-£1 w0 w0+£1  Certainty equivalent and risk premium rp U(w) EU=0.5*u(w0-£1)+ 0.5*u(w0+£1) The certainty equivalent u(w0+£1) income (wc) is the u(w0) certain income that u(w0-£1) would make the individual as well off as the expected utility of the gamble w0-£1 wc w0 w0+£1 The risk premium is how much the individual would pay to avoid the risk. This is given by rp= w0-wc  Attitudes to risk Risk averse: w0>wc, rp>0, diminishing marginal utility of income Risk neutral: w0=wc, rp=0, constant marginal utility of income Risk averse: w0