ECN6112 ADVANCED MICROECONOMIC

1. ECN6112 ADVANCED MICROECONOMIC Topics: 1) Risk Choices and Risk Aversion 2) Intertemporal Choices and Capital Decisions 3) Asymmetric Information

2. Risky Choices and Risk Aversion Outline: • Probability • Expected Utility Function • Risk Preference • Risk Aversion Coefficient • State Dependent Utility

3. Introduction • Aim: to introduce concepts of risk and uncertainty • Investigate impacts they have on agents’ decisions • Expected utility is extensively used by applied economists as a tool for investigating this impact • We define households’ risk preferences in terms of preferences • Risk-averse • Risk-seeking • Risk-neutral

4. Probability • Probability is a measure of likelihood an outcome will occur • An outcome could be some level of income, amount or quality of a commodity, or a change in price • Probabilities of all outcomes were either 0 or 1 • Can be classified as subjective or objective

5. SUBJECTIVE PROBABILITY • Where a household has a perception that an event will occur • Household’s perception is based on • Conditions of current market prices • Its preferences • Its income • Any changes in these conditions will alter household’s perceptions • Households with different preferences will have different perceptions • Results in different subjective probabilities by individual households through time and across various households

6. OBJECTIVE PROBABILITY • Frequency with which certain outcomes will occur • Objective probability of an outcome can be measured from past experiences • For example, if it is observed over some time interval that a household goes to movies once every 4 weeks • Probability of going to movie on any weekend is ¼ or 25% • An objective probability, measured by frequency of occurrence • Depends on past household behavior

7. Mutually Exclusive Probabilities • Both subjective and objective probabilities are mutually exclusive • Only one outcome will occur • For example, a household will either go to the movies this weekend or not • Probabilities associated with each possible outcome always add up to 1 • If ¼ is probability household goes to the movies, then ¾ is probability it will not • In general, assuming a finite number of outcomes, k where j= probability of outcome j occurring

8. States of Nature • In making future consumption plans, a household will consider probability of possible outcomes • In determining these probabilities, household may use a combination of subjective and objective probabilities • Household is faced with choosing alternatives with uncertain outcomes by means of known probabilities • Each risky alternative is called a state of nature or lottery, N • For example, If probability of going to movies was ¾ and not going was ¼, state of nature would be reversed, (¾, ¼)

9. States of nature • In general, a state of nature is a set of probabilities associated with all k outcomes • Assumption: Households are rational when facing these states of nature with a preference relation • Outcomes could be in any form • Including consumption bundles, money, or an opportunity to participate in another state of nature • Fundamental difference between commodities and states of nature • Commodities can and generally are consumed jointly • States of nature, by their definition of being mutually exclusive, cannot be consumed jointly • Either one state of nature exists or another • For example, a household cannot have a 25% probability of going to the movies and a 75% probability of going to the movies at the same time

10. Expected Utility Function • Specifically, for two possible states of nature, expected utility function is • Where U is utility function associated with contingent commodity bundles x1 and x2 consumed in states of nature 1 and 2, respectively • 1 and 2 are respective probabilities of states of nature occurring • 1 + 2 = 1

11. Expected Utility Function • A contingent commodity is a commodity whose level of consumption depends on which state of nature occurs • Expected utility function is weighted (probability) sum of utility from consumption in the states of nature • If only one of the states of nature occurs, say state 1, then 1 = 1 and 2 = 0 • Utility function reduces to • Standard utility function, with certainty assumed • With uncertainty, probabilities are 0 < (1, 2) < 1 • Utility function represents average or expected utility given alternative possible states of nature

12. Risk Preference • Variability in outcomes of some state of nature when probability of each outcome is known is called risk • Generally, statistical concept of variance is employed as a measure of risk • Variance is a measure of spread of a probability distribution -measures variability in outcomes • The greater the variance and thus the variability in outcomes, the greater is the risk • For example, for a salaried employee the state of nature associated with receiving a fixed salary per month is risk free • It has no variability, no risk, and a variance of 0 • In contrast, an employee working on commission has some risk

13. Risk Preference • In general, unless compensated for risk exposure • Households appear to have an aversion to risk • In an effort to reduce this risk exposure, they will attempt to shift risk onto another agent • Purchasing insurance is a form of shifting risk • Its prevalence suggests risk aversion is common not only among households but also among firms

14. Risk Aversion • In a game of tossing a coin, if it is heads a household will receive $5 and if it is tails it must pay $5 • Expected value of game is ½(5) + ½(-5) = 0 • Called actuarially fair • Actuarially fair games • Games in which expected values are zero • Cost of playing a game is equivalent to game’s expected value • If instead payoff for heads is increased to $15, expected value would be • ½(15) + ½(-5) = 5 • A $5 cost of playing the game would reduce expected payoff to zero, which is actuarially fair

15. Risk Aversion • In terms of risk preferences, a household has risk-averse preferences if it refuses to play actuarially fair games • Unless there is some utility in actual process of playing and potential loss from playing is relatively small • It would prefer its current certain state to a risky state with a possible loss in happiness • Some individuals play actuarially unfair games • Expected value is negative • For some individuals who play slot machines or state lotteries, fun of playing and dreams of winning yield utility in excess of small cost of playing

16. Concave Utility Function • Risk-averse households would generally prefer a certain salary • Versus a variable salary based on sales with an expected value equivalent to certainty salary • A risk-averse household will not play actuarially fair games • For this household winning the game increases utility by a lesser degree than losing reduces it • Such a household prefers its current level of wealth over variance in wealth associated with an actuarially fair game

17. Concave Utility Function • Mathematically, for two possible states of nature, risk aversion may be stated as • U(1W1 + 2W2) > 1U(W1) + 2U(W2) • Where W1 and W2 are levels of wealth resulting from possible outcomes 1 and 2, respectively • Level of utility associated with expected wealth, U(1W1 + 2W2), is greater than state of nature resulting in expected utility of wealth, 1U(W1) + 2U(W2) • If outcomes were an employee’s salary, a risk-averse employee would prefer state of nature with certain salary resulting in U(1W1 + 2W2) • Over uncertain salary with an expected utility level of 1U(W1) + 2U(W2) • Inequality is called Jensen’s inequality • Defines concave function illustrated in Figure 1

18. Figure 1 Risk-averse preferences

19. Concave Utility Function • A zero level of wealth corresponding with U = 0 results from a positive linear transformation of expected utility function • Due to concave nature of utility function • Weighted average of wealth levels, W1 and W2, yield lower expected utility of wealth, 1U(W1) + 2U(W2), than utility of expected wealth, U(1W1 + 2W2) • Risk aversion is equivalent to a concave expected utility function • Implies diminishing marginal utility of wealth • ∂2U/∂W2 < 0 • As wealth increases, marginal utility of wealth declines

20. Risk seeking • Some individuals may have risk-seeking preferences • Also called risk-loving preferences • Prefer a random distribution of wealth over its expected value • Gravitate toward occupations with more variable income streams • Examples are being self-employed, day-trading stocks, or working as a commodities trader • Some individuals are not only risk seeking in terms of their wealth, but also in life itself • Examples are race-car drivers, soldiers in special military forces, and criminals • Most individuals do not have such extreme risk-seeking preferences • However, at least for small potential losses in income, many households have some risk-seeking characteristics • For example, many households engage in gambling • Overall, households are generally risk averse when it comes to relatively large potential losses

21. Risk seeking • Mathematically, risk seeking reverses inequality for risk aversion • For two possible states of nature, risk-seeking preferences are defined as • U(1W1 +2W2) < 1U(W1) + 2U(W2) • As illustrated in Figure 2, for risk-seeking preferences expected utility function is now convex • Weighted average of wealth levels, W1 and W2, yield a higher expected utility of wealth, 1U(W1) + 2U(W2), than utility of expected wealth, U(1W1 + 2W2) • Risk seeking is equivalent to a convex expected utility function • Implies increasing marginal utility of wealth, ∂2U/∂W2 > 0 • As wealth increases, marginal utility of wealth increases

22. Figure 2 Risk-seeking preferences

23. Risk neutrality • Intermediate between risk aversion and risk seeking is linear expected utility function representing risk neutral preference • Households with risk-neutral preferences are not concerned with variation in wealth • Only concern is expected value of wealth • A risk-neutral household is indifferent between receiving a certain income versus an uncertain income • Provided expected income from uncertain outcome is equivalent to certain income • Considering two outcomes, a household is considered risk neutral when utility of expected wealth is equal to expected utility of wealth • U(1W1 +2W2) = 1U(W1) + 2U(W2)

24. Risk neutrality • Risk-neutral preferences are depicted in Figure 3 • Expected utility function is now linear • Weighted average of wealth levels, W1 and W2, yield same expected utility of wealth, 1U(W1) + 2U(W2), as utility of expected wealth, U(1W1 + 2W2) • Equivalent to a linear expected utility function • Implies constant marginal utility of wealth, ∂2U/∂W2 = 0 • Marginal utility is constant at all levels of wealth

25. Figure 3 Risk-neutral preferences

26. Certainty Equivalence • Amount of return a household would receive from a certain outcome so it is indifferent between a risky outcome and this certain outcome • Specifically, given two uncertain outcomes • U(C) = 1U(W1) + 2(W2) • Illustrated in Figure 4 • At a level of utility 1U(W1) + 2U(W2), household is indifferent between receiving C with certainty or receiving risky outcome, 1W1 + 2W2 • Certain return is less than risky outcome • Implies a risk-averse household is willing to trade some expected return for certainty • Definition of risk aversion can be equivalently defined as C < 1W1 + 2W2 • Inequality sign is reversed for risk-seeking preferences • Inequality becomes an equality for risk-neutral preferences

27. Figure 4 Risk aversion and certainty equivalence

28. Insurance • Insurance companies have developed to where one can acquire an insurance policy against almost any risky outcome • Certainty equivalence underlies functioning of insurance • Households are willing to pay to avoid risky outcomes • Maximum amount they are willing to pay is difference in expected wealth and certainty equivalence of wealth, (1W1 + 2W2) - C • Paying more than this difference would result in a loss of utility • They will be willing to pay less than this amount in the form of insurance premiums

29. Actuarially Favorable Outcomes • If an uncertain state is actuarially favorable, a risk-averse household will also accept some risk • Actuarially favorable outcomes • Assets where expected value is positive or cost of asset is less than asset’s expected value • An asset is the title to receive commodities or monetary returns at some period in time • If returns are in form of commodities, asset is called a real asset • Assets yielding monetary returns are called financial assets

30. Actuarially Favorable Outcomes • Can demonstrate willingness of households to take on some risk by considering two assets, one certain, or risk free, and the other risky • Let rC be rate of return for certain asset and let r1 and r2 be two possible rates of return on risky asset • Probability of r1 occurring is , so (1 - ) is probability of outcome r2 • Assume mean return of risky asset exceeds return of certain asset • r1 + (1 - )r2 > rC • Otherwise, risk-averse households would never invest in risky asset • Let R and C denote proportion of wealth invested in risky and certain assets, respectively • Assume all wealth is invested in these two assets, so R + C = 1 • Objective of a household is to determine optimal amounts of wealth to invest in the two assets • Household has a probability  of earning a (RWr1 + CWrC) return and a probability of (1 - ) of earning a (RWr2 + CWrC) return

31. Actuarially Favorable Outcomes • Assuming a logarithmic expected utility function, a household is interested in maximizing its return for a given level of wealth • Taking a linear transformation by subtracting ln W yields • Incorporating constraint into expected utility function yields • F.O.C. is

32. Actuarially Favorable Outcomes • Assume household is unwilling to take any risk, so R = 0 • F.O.C. reduces to

33. Actuarially Favorable Outcomes • If expected rate of return on risky asset is the same as the certain return, a risk-averse household will not invest in risky asset • When expected rate of return is greater than certain return, R > 0, risk-averse households will accept some risk • At the point where returns are greater than the certain return for all possible outcomes, R = 1 • Households will invest only in risky asset • For two possible outcomes, this occurs where both r1 and r2 are greater than rC • This example of a risk-averse household willing to accept some risk is an exercise in diversification • In general, considering many assets, as long as assets are not perfectly correlated, there are some gains from diversification

34. Risk-Aversion Coefficient • Arrow-Pratt risk-aversion coefficient,  • A measure of degree of risk aversion • Defined in terms of expected utility function •  = -U"/U‘ • For a risk-neutral household, expected utility function is linear • U" = 0, which results in  = 0 • A risk-averse household will have a concave expected utility function indicated by U" < 0 • The more risk averse a household, the more concave is the function • Thus, the larger will be -U" and  • For an extremely risk-averse household,  =  • Such households will always fully insure against any risk regardless of the price • Examples are an individual with major depressive disorder who will not get out of bed or individuals with a particular phobia such as flying or driving

35. Risk-Aversion Coefficient • Similarly, a risk-seeking household will have a convex expected utility function with U" > 0 • For an extremely risk-seeking household,  = - • Such households would not insure against any risk • An example is a suicide bomber • Note that second derivative, U", is divided by U' • Risk-aversion coefficient will not vary by a linear transformation of utility function

36. Risk-Aversion Coefficient • Employing risk-aversion coefficient as a measure of risk aversion provides a comparison across household wealth levels • Generally, it is assumed that wealthier households are willing to take more risk than less-wealthy households • Thus, as a household’s wealth increases, the risk-aversion coefficient declines • ∂/∂W < 0 • An example of a utility function representing decreasing risk aversion is • U = ln W • Where risk-aversion coefficient is  = 1/W • As wealth increases, aversion to risk declines • ∂/∂W = -1/W2 < 0 • However, not all utility functions represent decreasing risk aversion

37. Risk-Aversion Coefficient • Risk-aversion coefficient can also be employed for a comparison of risk preferences across households • For example, given two households A and B • Household A would be more risk averse than household B if A > B for all levels of household wealth • Allows a partial ordering of households’ preferences from households who least prefer a risky outcome to those households who are less averse to risky outcome • However, it is not a complete ordering • If, for example, A > B for some but not all levels of wealth, then it is not possible to state household A is always more risk averse than household B

38. State-Dependent Utility • A household may derive utility from not only the monetary returns • But also from states of nature that underline them • Called state-dependent utility • Given this expected utility function for a household, we can determine utility-maximizing state of nature subject to a constraint on wealth • Assume a household has a choice of two states of nature • For example, state 1 could be running its own business and • State 2 could be having a position in a large international firm

39. State-Dependent Utility • Associated with each state is a set (bundle) of contingent commodities • Each set is available only if the particular state occurs • By representing these commodity bundles in terms of monetary values (W1 and W2) we can determine level of wealth required for obtaining each bundle • Like all commodities, contingent commodities have an associated price • This price is the cost of a particular uncertain state of nature occurring with certainty so that the associated contingent commodities may be received • Example: Political lobbyist’s expenditures on legislators to assure passage of some legislation

40. State-Dependent Utility • Let p1 and p2 be the price of receiving contingent commodities associated with states 1 and 2, respectively • In purchasing one of these contingent commodities, household is constrained by a given level of initial wealth, W • Thus, household’s contingent budget constraint is • p1W1 + p2W2 = W • Recall that only bundle W1 or bundle W2 can occur and be consumed • p1/p2 indicates market tradeoff between states 1 and 2 • Measures rate at which contingent commodities in state 2 can be substituted for contingent commodities in state 1, holding level of initial wealth constant

41. Actuarially Fair Prices • Assume prices p1 and p2 are actuarially fair and market for contingent commodities within alternative states is active with many buyers and sellers • Called well-developed markets • If state 1 occurs with probability  and state 2 with probability (1 - ) • Market will reveal these probabilities set p1 =  and p2 = (1 - ) • Assuming the household expects state 1 to occur with probability , expected utility associated with the two contingent commodity bundles is • U(W1, W2) = U(W1) + (1 - )u(W2) • Household will attempt to maximize this expected utility, given its budget constraint • Lagrangian is

42. Actuarially Fair Prices • F.O.C.s are • From these F.O.C.s we obtain • Assuming actuarially fair markets for contingent commodities • Results in U'(W*1) = U'(W*2), so W*1 = W*2

43. Actuarially Fair Prices • Thus, a risk-averse household facing actuarially fair markets will be willing to pay for a state with certainty • Where final level of wealth is the same regardless of which state occurs • An example is market for insurance • Where price for achieving a given state is the insurance premium and the contingent commodity is the contingent insurance claim

44. Actuarially Fair Prices • With actuarially fair insurance markets, risk-averse households will fully insure against possible losses • Will purchase insurance up to point where level of wealth is same regardless of whether state of nature involves a loss • Illustrated in Figure 5 • Axes measure wealth for the two alternative states • Certainty line is a 45 line from origin • Measures same level of wealth regardless of which state of nature occurs • Indifference curves measure equivalent level of expected utility to certainty level associated with certainty line • Points not on certainty line represent utility levels associated with some uncertainty in which state of nature will occur • A household’s budget constraint is represented by budget line W • Every point on budget line represents actuarially fair probability of the two states of nature occurring

45. Figure 5 Risk aversion in state-dependent utility

46. Actuarially Fair Prices • A risk-averse household is unwilling to play an actuarially fair game • Movements off certainty line on budget line will result in lower levels of utility • For a risk-averse household to be willing to accept some risk, they must be compensated in the form of additional wealth • A risk-averse household has indifference curves that are convex to origin • Point A is unobtainable with initial wealth W • However, at point A • Household can maintain same level of utility, but requiring less initial wealth • By moving down along indifference curve • Decrease in required initial wealth continues until point B is reached on certainty line • Where wealth levels of the states are equal

47. Actuarially Unfair Prices • If market is actuarially unfair, a risk-averse household may choose to accept some risk • For example, consider an unfair price ratio of p1/p2 < /(1 - ) • From F.O.C.s for expected utility maximization • Assuming actuarially unfair condition p1/p2 < /(1 - ) • Results in U'(W*1) < U'(W*2) • So, assuming diminishing marginal utility of income, W*1> W*2

48. Actuarially Unfair Prices • Thus, a risk-averse household facing actuarially unfair markets will be willing to accept some risk • Since relative price of W1 is low compared with actuarially fair price ratio • Illustrated in Figure 6 • Low price results in budget line tilting outward • Establishes a tangency with indifference curve to the right of certainty line • Results in risky outcome of W*1 > W*2

49. Figure 6 Actuarially unfair markets

50. Risk Seeking • A risky outcome will also result if a household is risk seeking • Illustrated in Figure 7 • Risk-seeking household’s indifference curves are concave from origin • Implies expected value for a certain outcome must be higher than any risky outcome before household would be indifferent between outcomes • Results in a corner solution, point A • Household prefers outcome with greater risk