How do we cope with uncertainty?

Transcription

1 Topic 3: Choice under uncertainty (K&R Ch. 6) In 1965, a Frenchman named Raffray thought that he had found a great deal: He would pay a 90-year-old woman $500 a month until she died, then move into her grand apartment...but Raffray died at the age 77, having flocked over $184,000 for an apartment he never got to live in... This reminds us that life is full of uncertainty Agents face uncertainty both as demanders of goods but also as suppliers of inputs When buy car, it might be a lemon! (bad quality) When you supply labour you may suffer serious accident while on the job. But outcomes are not always bad: You might win the lotto (it could be you!), You invested in housing and the market exceeded your expectations. The bottom line is that consumers make decisions without knowing for certain what the consequences will be! In this topic we use the basic tools we have learned so far to study choice under uncertainty! 1 2 How do we cope with uncertainty? To study uncertainty we look at an individual s decision whether or not to accept a gamble. Suppose M= and offered a gamble For each 1 you bet you lose 1 if a heart is drawn but receive 40p if club, spade or diamond. How much do you bet? To answer this we use the procedure developed so far. But need to say something on what goods/commodities you are choosing when deciding how much to bet. But with one difference now we have contingent commodities and states of the world. consumption 140 if a not heart slope: e -1 1 endowment: no gambling? 3 states of the world 99 consumption if a heart 4 Probability and expected value What about the likelihood that different states of the world occur? This likelihood is measured by probability (if event cannot occur the probability is zero; if it occurs with certainty the probability is 1). Probability of a heart drawn from a fair deck of cards is 13/52=1/4. Probability of not a heart is drawn is ¾. For a given random process the probabilities of all the states of the world must sum up to 1. What is then the expected winnings of previous gamble? ¾ *.40 + ¼ * (-1) =.05 ; this is the expected value of the gamble (what you expect on average to happen). Suppose now that the gamble is as follows: if a heart appears you win a 1 if not then you lose 0.33, what is the expected value of the gamble? ¼ * 1 ¾ * 0.33 = =0 a gamble with 0 expected monetary gain is called actuarially fair (on average you win or lose nothing) but what is the slope of the budget constrain associated with this bet? it is simply ρ * W + (1 ρ)* L = 0 L ρ = W 1 ρ 5 6 1

2 a not heart Example: Bet 20, with fair odds (win 20 with prob. ¼, lose 20/3 with prob. ¾) e ¼*(+20)+¾*(-20/3) =1200/12=. odds: the ratio of probabilities of two events ρ * W + (1 ρ)* L = 0 L ρ = W 1 ρ fair odds line: The expected value of consumption is equal at every bundle on fair odds line a heart 7 Just as individuals have preferences between different goods, they have preferences between consumption in different states of the world (i.e., different combinations of contingent commodities). The normal assumptions still apply here but we need one additional: the marginal utility of consumption is independent of the state of the world. Preferences between contingent commodities reflect an individual s attitude towards risk. 8 Risk averse consumers are those who do not like risk. They always prefer a sure thing and will not accept an actuarially fair gamble (this is the definition). Risk lovers prefer uncertainty so they prefer a gamble to a sure thing. Risk neutral people are indifferent among all alternatives with the same expected value. Preferences (risk loving) other Now we need to rank the various alternatives. You care about the amount of consumption in both states of the world certainty line connects all possible endowment points (it is the locus of all possible certain consumption levels slope ρ * W + (1 ρ)* L = 0 L ρ = W 1 ρ fair odds line 9 heart 10 Preferences (risk neutral) Preferences (risk averse) other certainty line other certainty line fair odds line= indifference curve fair odds line heart 11 heart 12 2

3 Equilibrium Equilibrium for a risk averse (how much does she bet?) Equilibrium occurs where the highest indifference curve possible is reached on the budget constraint. Will a risk averse consumer accept an actuarially fair gamble? No! Why? Because ends up with same level of wealth (equal to her endowment!) without facing the uncertainty of the gamble! other a bet e though on average there is a gain she does not bet a lot of money budget constraint 13 heart 14 Equilibrium for a risk lover (how much does she bet?) heads a e though on average there is a loss she bets all of her income (corner solution) Which type of preference is more relevant for understanding behaviour? There is strong evidence that for the important decisions most people are risk averse (most people buy insurance at a high price). Lotteries (and casinos!) suggest that many people are risk lovers for small bets. tails Risk premia Risk premium: extra return on an investment to compensate for risk. Example. return in bad state certainty line e b return in good state 18 3

4 Should we keep all eggs in one basket? Diversification: the process of buying several assets in order to reduce risk. Example: 2 assets, both same expected return, but (perfectly) negatively correlated. Then split investment! (Diversify). Application: Tax evasion Tax evasion: the failure to pay taxes that are legally due. Optimal policy? If risk averse penalty should be such that she faces an actuarially fair gamble. The higher the tax the higher should the penalty be! (See example) Example Consider Elena who is risk averse and amoral citizen. Her before tax income is 1,500. She faces a tax system t=1/3 (if declares 1,500 then net income is 1,500(1- t)= 1,000. If she decides to evade taxes and caught then is is fined f=0.8 for every evaded. not (1-r) 1,500 C.na 1,000 b e certainty line 21 1,000-C.a=taxes not paid C.a 1, audited (r) 22 Optimal policy Recall that Elena is risk averse. What it then the optimal policy (i.e, policy that reduces tax evasion). (1 r) t rf = 0 f * 1 r = t r Equation says: The penalty rate should be set so that it at least equals the odds of not being audited time the tax rate. Suppose r=0.1 then (1-r)/r=9 so a high penalty is required (9*t=3 for each evaded the evader should be fined 3.) Also the higher the tax, t, the higher the fine, f. Why?

5 Insurance Previously we asked whether consumers would like to gamble. Typically consumers have no choice (life is uncertain). Fair insurance Premium: the price of obtaining insurance coverage. Actuarially fair insurance: the premium paid is equal to the expected payout of the insurance provider. Example: Scarlet is obstetrician facing probability p of being sued and losing. She wants to buy insurance with premium k Preferences (full insurance) Actuarially fair is when p=k. endowment C-ns a slope: -p/(1-p) if sued the she has 0 income C-ns a insurance expenditure C*ns sl: -p/1-p in the sense that C*s=C*ns b b C-s 27 C*s C-s 28 Demand for unfair insurance Equilibrium with actuarially unfair insurance If the premium equals expected benefit then firms make no profits (possibly negative if consider labour costs, i.e., operating costs). So typically insurance policies are unfair (expected payments are less than premium paid---on average). Think of two different people (that belong two different risk classes) but yet pay more or less on same risk premia. 29 C-ns 45 certainty line C-s 30 5

6 When the insurance policy is actuarially unfair even a risk-averse individual purchases less than full insurance. Intuition: When the premium is higher than the expected payoff it is rational to assume some risk in return for reducing the payments to the insurance. Decision making with many uncertain outcomes: Von Neumann-Morgenstern utility So far the focus was on only two possible outcomes. But in many realistic situations there are more than two possible outcomes. We now introduce decision trees A decision tree Utility functions for uncertain outcomes decision node economics pass fail pass ,000 3,000 7,000 U c, c, c,..., c ; ρ, ρ, ρ,..., ρ ) = ρ u( c ) ρ u( c ) ( n n 1 1 n n Von Neumann-Morgenstern utility function: A utility function where the utility associated with some uncertain event is the expected value of the utilities of each of the possible outcomes politics fail 0.6 2,000 Now, what about sequential decisions? Asymmetric Information In markets buyers and sellers have different information Example 1: Seller and buyer of second hand car; seller knows quality but buyer does not (car may be lemon or plum ). Example 2: Labour markets: Employees know ability of themselves but employer does not. Example 3: Incumbents know ability but voters do not. Information about quality, ability may be costly to obtain (sometimes impossible: may be impossible or costly for a firm to determine how productive employees are). All these problems of asymmetric information may cause problems with the efficient functioning of a market. Terminology: Hidden action: action taken by a party not observed (but affected) by another

7 Adverse selection: Selection of inappropriate outcomes due to asymmetric information (e.g car market). Moral hazard: inappropriately taken actions (e.g insurance). (Aside: Akerlof, Spence and Stiglitz Nobel Prize winners in economics for their work on asymmetric information). The market for Lemons (interesting example) people want to sell their second hand cars. people want to buy a second hand car. Everyone knows that 50 are lemons and 50 are plums. Asymmetric information: owner knows quality but buyers does not. The owner of a lemon wants 1, The owner of a plum wants 2,000. The buyers are willing to pay 2,400 for a plum and 1,200 for a lemon. If information is perfect then market works Problems arises when quality is not observable. Assume that, when quality is not observed a buyers is willing to pay the expected quality of a second hand car 1 1 1,200+ 2,400 = 1, In this case no plum will be sold only lemons! Why? Because the owners of plums would not sell because the price offered by the buyers is below their reservation price. Only sellers of lemons would be in the market Quality choice Variation of previous idea quality now is determined by producers Suppose consumers buy umbrellas (U) for high quality they are willing to pay 14 and for low 8 (and is impossible to tell quality without trying Us) Suppose that cost of of both qualities of U is What is the equilibrium quality of Us? Consumers are uncertain about quality so expect average quality sold p = 14q + 8(1 q) 3 cases to consider 1. Only low quality Us will be produced 2. Only high quality Us will be produced 3. Both qualities will be produced

8 Case 1: Consumer will only pay low-quality price (but because it costs no Us will be produced) Case 2: Consumers are willing to pay 14 but price is (so consumer surplus is 2.50) Case 3: Competition makes sure that price is Average quality must have a value (for consumers to buy) of at lease the cost of Us that is, 14q + 8(1 q) Equilibrium Price, p q + 8(1 q) q=1 Demand Many equilibria!! Supply 7/12 Fraction of high quality Us, q Adverse selection Example described adverse selection problem (low quality crowd out high quality products). Why? Because information is costly to acquire. Another example: Insurance Insurance company wants to design an insurance scheme for bicycles based on average theft rate. What will happen? Insurance company will go bankrupt! Why? Only high risk consumers would buy insurance! (they are the ones who need it!)--adverse selection again! It then follows that insurance should base scheme on risky customers. Similar problem with health insurance Moral hazard Moral hazard arises when probability of event is affected by the action taken: Example: insurance of bicycle but also health insurance. If consumer is completely insured against the bad outcome then the incentive to take appropriate care is missing: This lack of incentive is called moral hazard. Problem arise because effort is unobservable! Moral hazard and adverse selection Moral hazard refers to situations where one side of the market cannot observe actions of the other (also called hidden action). Adverse selection refers to situations where one side of the market cannot observe the type or quality of the goods on other side of market (also called hidden information)

9 Equilibrium characterisation Equilibrium in a market involving hidden actions is characterised by some form of rationing. Equilibrium in a market involving hidden information is characterised by too little trade (due to negative externalities the bad quality imposes on good quality good). Equilibrium in both instances is inefficient. Signalling In car market example asymmetric information caused problems But the owners of good cars have incentive to signal the quality of car to potential buyers (warranty!) Signalling makes market perform better Spence s model of education 2 types of workers, able and unable. The able has marginal product a and the unable b, with a>b. In total there are L able and K unable workers. Output is al+bk. If ability was known the firm would offer wage equal to marginal productivity of types. But ability is not observable so firm offers expected wage w=la+kb. Workers (in particular the able ones) have incentive to distinguish themselves from the unable ones. How? By choosing some level of education not attainable by the unable ones Equilibria In this example there are 2 possible (for this course) equilibria. Pooling: meaning that both types choose the same level of education. This is an equilibrium in which able and unable workers pool on the same policy. In this case firm cannot distinguish type and therefore offers expected wage. Separating: Both types have an incentive to choose a level of education such that the types of workers are revealed. In this equilibrium firm knows types and sets wages according to marginal productivity of workers. Both equilibria have interesting features The separating is inefficient from a social point of view. Able workers invest in signals that confer no social benefit (only private)

10 In the pooling the wage set, in the presence of unable workers, is depressed (externality), giving the able workers the incentive to invest in costly signals In general signalling is not bad since, as in example with types of cars, it facilitates trade. END OF TOPIC