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Lecture 8

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Lecture 8

Wednesday 2013/6/18, 1-3pm.

Price Theory

Risk and Uncertainty

Chapter 18

- States of the world can be considered as different goods
- States can be traded. How?
- As there is trade, there is a price. What’s the price?

- Key elements: a complete set of states, and a probability distribution across these states.
- Expected value: the average value of all the states, weighted by the corresponding probabilities.
- Graphically, we can use iso-expected value lines.

- Budget line and prices
- Fair odds
- Reflect true probabilities of various states of world
- Expected value of betting same as expected value of not betting

- Individual offered fair odds
- Budget line coincides with expected value line

- How to measure the risk of a stochastic variable (or a risk asset, or a gamble…)?
- You need a variation index, such as variance or standard deviation.
- Note that these measures are different from risk-attitude measures.

Section 18.1

- Define utility U as a function of wealth (W)
- If U is concave (U’’<0), the individual is considered risk-averse
- If linear (U’’=0) , then risk-neutral
- If convex (U’’>0), risk-loving

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- U($90) = 12, U($0) = 2.
- Expected utility is

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- U($90) = 12, U($0) = 2.
- Expected utility is

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- Expected money value of the lottery is

- EU = 7 and EM = $45.
- U($45) > 7 $45 for sure is preferred to the lottery risk-aversion.
- U($45) < 7 the lottery is preferred to $45 for sure risk-loving.
- U($45) = 7 the lottery is preferred equally to $45 for sure risk-neutrality.

12

EU=7

2

$0

$45

$90

Wealth

U($45) > EU risk-aversion.

12

U($45)

EU=7

2

$0

$45

$90

Wealth

U($45) > EU risk-aversion.

12

MU declines as wealth

rises.

U($45)

EU=7

2

$0

$45

$90

Wealth

12

EU=7

2

$0

$45

$90

Wealth

U($45) < EU risk-loving.

12

EU=7

U($45)

2

$0

$45

$90

Wealth

U($45) < EU risk-loving.

12

MU rises as wealth

rises.

EU=7

U($45)

2

$0

$45

$90

Wealth

12

EU=7

2

$0

$45

$90

Wealth

U($45) = EU risk-neutrality.

12

U($45)=EU=7

2

$0

$45

$90

Wealth

U($45) = EU risk-neutrality.

12

MU constant as wealth

rises.

U($45)=EU=7

2

$0

$45

$90

Wealth

- For risk-neutral individuals, state-contingent consumption plans that give equal expected utility are equally preferred.

Cna

Indifference curvesEU1 < EU2 < EU3

EU3

EU2

EU1

Ca

- What is the MRS of an indifference curve?
- Get consumption c1 with prob. 1 andc2 with prob. 2 (1 + 2 = 1).
- EU = 1U(c1) + 2U(c2).
- For constant EU, dEU = 0.

Cna

Indifference curvesEU1 < EU2 < EU3

EU3

EU2

EU1

Ca

- Q: How is a rational choice made under uncertainty?
- A: Choose the most preferred affordable state-contingent consumption plan.

- Absolute risk aversion
- Relative risk aversion

- Considering forming a portfolio with one risky asset and one risk-free asset.
- If the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollarsof the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing).
- Thus economists avoid using utility functions, such as the quadratic, which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication.

- Similarly, if the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (or constant, or increasing).

Risk attitude and wealth

- In general, individuals are more risk-averse than businesses
- Possibly because businesses know better about risk diversification.
- Budget constraint is less a concern for businesses.

- Societies’ desire for risk neutrality in some instances
- Individual entrepreneurial endeavors promote risk aversion
- Underinvest in risky projects
- Provides corporations with a good buffer

- Transfer of risk from one party to another – transferring risks from RA agents to RN or RL ones increases efficiency.
- Imperfect information
- Moral hazard
- Adverse selection

- The prevailing health insurance programs in Taiwan provide compensations to women who have a hysterectomy before age 45.
- This design leads to a typical phenomenon of moral hazard, creating a spike in the number of hysterectomies before insurants reaching age 45.

- Uninsurable risks
- A risk that cannot be diversified
- Eg. Group risk
- Eg. Health risk
- Eg. Profit or income

Section 18.3

- Futures contract
- Deliver specified good at specified future date at specified price

- Futures market
- Market for futures contracts

- Spot market
- Market for goods for immediate delivery

- Spot price
- Price in spot market

- Speculator
- Attempts to earn profits in futures market
- Predicts future changes in supply or demand

- Speculation and welfare
- Guess future correctly
- Earn profit
- Increase social welfare

- Guess future correctly

Section 18.4

- Returns
- Valued assets not for uses in consumption but for potential increase to owners’ wealth

- Expected returns
- Expected present value of those returns

- Standard deviation
- Measure of risk
- Spread across possible outcomes

- Investors

- Combination of risky assets
- Standard deviation of portfolio at most equal to average standard deviation of individual stocks
- Expected return to portfolio exactly equal to average expected returns of individual stocks

- Efficient set and portfolios

- Capital asset pricing model
- Assumes investor cares only about expected return and risk
- Measures risk by standard deviation

- Risk-free asset
- Market line and portfolios

- Rational investor
- Combines risk-free asset with market portfolio in some proportions in hold portfolio