Uncertainty and Public Policy

1
Uncertainty and Public Policy

• Expected Value and Expected Utility
• Risk-control and risk-shifting mechanisms
• Alternative models of individual behaviour under
  uncertainty
• Moral hazard and medical care
• (chap 7)

2
Uncertainty and Public Policy

• Most choices are made under uncertainty as to the
  possible outcomes these choices are essentially
  gambles
• The WTP for a risky commodity depends on the
  likelihood of possible outcomes, as perceived by
  the consumer.
• How do individuals respond to changes in these
  perceived probabilities?
• How can public policies affect these perceptions?

3
Uncertainty and Public Policy

• Objective to assess the economic costs of
  uncertainty and the benefits of its reduction
• Alternative models of individual behaviour under
  uncertainty
• The expected utility theorem individuals
  mazimize expected utility.
• Theory of games against persons the Slumlords
  Dilemma (a prisonners dilemma)
• Bounded Rationality

4
States of the World

• Set of alternative states of the world the
  different and mutually exclusive outcomes that
  may result from the process generating the
  uncertainty
• Examples
• Fliping a coin 2 states, heads or tails
• Throwing a die six possible outcomes may arise
• Notice that the definition of the relevant states
  of the world depends on the game that is being
  played for a student, in this course it may be
  Pass or Fail, while for another may be 18, 17,
  16, less than 15, and for another, 10, 11, 12,
  13, 14, more than 15...

5
Expected Value

• Payoff of each state of the world, Xi
• Expected value of a risky situation the sum of
  the payoff in each possible situation weighted by
  the probability that it will occur
• Suppose that there are N possible states, each
  with a payoff Xi and a probability of occurring
  of Pi, the expected value is,

6
Expected Value

• If all the possible states are considered it must
  always be true that
• Consider the following three different games
• Game 1 A coin is flipped. If heads turn out,
  you win (the payoff is) 100 and if tails appear,
  the payoff is -0,5.
• Game 2 if heads, you win 200 if tails, you lose
  100
• Game 3 If heads, you win 20,000 if tails, you
  lose 10,000 the losses may be paid in small
  amounts spread over 30 years

7
Expected Value

• Everybody would like to play game 1, many people
  (although less) would play game 2 and very few
  people would play game 3. Yet
• EV1(1/2)100 (1/2)(-0.5)49.75
• EV2(1/2)200 (1/2)(-100)50
• EV3(1/2)20000 (1/2)(-10000)5000
• Having a positive or a large expected value is
  not sufficient to make a game attractive in
  addition to the expected value, most people also
  consider how they feel about each possible outcome

8
Expected Value

• A fair game is a game whose entry price equals
  its EV. An entry price of 50 makes game 2 a fair
  game, EV20
• Risk-averse people are those who choose not to
  play fair games
• Bernoulli explained this pattern people value
  not the expected monetary outcome, but the
  expected utility (its moral value). If utility
  functions have decreasing marginal utility of
  income, an expected gain of amount X is less
  valued than an equal loss of the same amount.
• We consider that the sole relevant outcome of a
  gamble is the level of wealth to wich it
  corresponds

9
Expected Utility

• The formal economic theory of choice, formulated
  by Von Neumann, and Oskar Morgenstern, assumes
  the existence of a utility function U that
  assigns numerical values with the satisfaction
  associated with different outcomes.
• The expected utility is the sum of the utility
  associated with each possible state of the world,
  weighted by the probability that it will occur
  is the expected value of utility over all
  possible outcomes
• The Expected Utility Theorem, states that the
  individual chooses among alternatives in order to
  maximize expected utility

12th S, Oct 25
10
Utility function of von-Neumann-Morgenstern

• Suppose there are n possible outcomes, where X1
  is the least preferred and Xn the most preferred.
  Now assign arbitrary utility numbers to these two
  extreme results, for instance U(X1)0, U(Xn )1
• Given these two values, we can assign utility
  levels to the other possible outcomes
• Start with Xi. Ask the individual to state the
  probability pi that would make him indifferent
  between Xi with certainty and playing a game with
  probability pi of getting Xn and probability 1-
  pi of obtaining X1. (There should exist a
  sufficiently atractive game to make the
  individual accept to play it. The higher Xi, the
  higher must be pi for the individual to accept
  it)

11
Utility function of von-Neumann-Morgenstern

• This probability, pi, represents how desirable
  outcome Xi is the von Neumann - Morgenstern
  technique consists in defining the utility of Xi
  as the expected utility of the game that the
  individual considers equally desirable to Xi
• U(Xi) pi U(Xn)(1- pi )U(X1) or
• U(Xi) pi 10 pi
• The utility number of any outcome is the
  probability of winning the top prize in a game
  that the individual considers equivalent to that
  outcome
• Now, p10 and pn 1
• A rational individual will choose among gambles
  based on their expected utilities, ie, on the
  expected values of these Von-Neumann Morgenstern
  utility index numbers

12
Utility function of von-Neumann-Morgenstern

• Example a game has only 2 possible outcomes
  winning 50 or 0. Consider that the Utility of
  winning 50 with certainty is 1 and the utility of
  winning 0 with certainty is zero. Consider now
  the outcome 10 with certainty. Its utility must
  be less than 1 and larger than 0.

And there must (?) exist a lottery, with a
probability of wining between 0 and 1 that this
consumer considers indifferent to 10 with
certainty . Suppose it is 0.4
Utility
1
A
0.4
0
10
Wealth
50
13
Utility function of von-Neumann-Morgenstern

• The vertical axis shows the probability of
  winning 50 necessary to make the individual
  indifferent between playing that game and getting
  the wealth on the horizontal axis with certainty
  utility index of von-Neumann-Morgenstern

A
Utility
1
0.4
0
10
Wealth
50
14
Utility function of von-Neumann-Morgenstern

• Ray OA shows, for each possible probability, the
  expected utility (on the vertical axis), and the
  expected value of the lottery (on the horizontal
  axis). In this case the expected value of the
  lottery with a 0.4 chance of winning 50 is
  E(V)0.4500.6020.

The expected utility of the lottery is
E(U)0.4U(50)0.6U(0) 0.4100.4
Utility
A
1
U(W)
0.4
C
0
10
20
Wealth
50
15
Utility function of von-Neumann-Morgenstern

• Risk-lovers are individuals who prefer to accept
  fair games, (utility function is convex) and risk
  neutrals are indifferent to fair games
• Certain-wealth equivalent is the amount of
  certain wealth that gives the individual the same
  utility he has under the risky situation 10

• Pure risk cost the difference between the
  expected wealth of a risky situation and its
  certain equivalent 20-10

Utility
A
1
0.8
D
0.4
C
0
50
10
20
40
Wealth
16
Utility function of von-Neumann-Morgenstern

• The lottery indifferent to a certain-equivalent
  wealth of 10 is having a proabability of wining
  (50) of 0.4. And the expected value of that game
  is 20
• Point C represents the expected value and the
  expected utility from the gamble 0.4. Gambles
  with higher (lower) probabilities of winning have
  expected values and expected utilities on the ray
  to the right (left) of C

Utility
1
A
0.4
C
0
10
50
20
Wealth
17
Utility function of von-Neumann-Morgenstern

• The utility index ranks risky situations
• consider 20 with certainty the individual is in
  point D. Now propose the individual to play the
  fair game consisting of winning 50 with
  probability p0.4 at a price of 20 (the expected
  value of the game).

The expected utility of the gamble is less than
the utility of the certain 20 the individual
will not play this fair game he is risk-averse
Utility
A
1
D
0.4
C
0
20
10
Wealth
50
18
Utility function of von-Neumann-Morgenstern

• Consider the game that gives 50 with probability
  p0.8 and has entry price of 20. Its expected
  value is 40. Its expected utility, 0.8, is larger
  than the utility at D This risk-averse person
  will accept to play it

Risk aversion is due to the concavity of the
utility function. Its slope is decreasing,
meaning that the marginal utility of income is
decreasing
Utility
A
1
0.8
D
0.4
C
0
20
40
10
Wealth
50
19
Utility function of von-Neumann-Morgenstern

• Risk-averse persons will pay to reduce risk this
  is the reason for the existence of insurance
• Consider that W050, and the probability of being
  stolen is 0.6 (X1stolen0, Xn, not stolen50). C
  is now the expected value of this situation. The
  certain wealth equivalent is 10 for individual
  red, 14 for individual green

Utility
A
1
D
0.4
C
0
10
20
40
14
Wealth
50
20
Utility function of von-Neumann-Morgenstern

• The individual with the green Utility function
  will pay up to 30636 (the expected loss plus
  the pure risk cost) to get a full-coverage
  insurance against theft. His net wealth, after
  paying this insurance, would be 50-3614, whether
  he is stolen or not. The other will pay up to 40
  to avoid the risk

Utility
A
1
D
0.4
C
0
40
10
Wealth
50
20
14
21
Utility function of von-Neumann-Morgenstern

• How does the amount of risk affect this
  individuals choice? Suppose that W050, but only
  40 are at a risk of being stolen. The probability
  of being stolen is 0.6 again. The individual will
  be at point B (if stolen) or A (if not stolen).
  The straight line BA represents now the possible
  combinations of expected wealth and expected
  utility.

Utility
The risk is lower now
A
1
D
B
0.4
C
0
10
Wealth
50
20
22
Utility function of von-Neumann-Morgenstern

• The gamble is smaller. Then
• For any expected wealth, the espected utility of
  a smaller gamble is higher
• For any expected wealth, a smaller gamble has a
  lower pure risk cost

Example p of theft 0.75 EV0.25500.751020 EU
0.25U(50)0.75U(10) .55gt 0.4 Lower stakes
give higher utility
Utility
A
1
D
K
0.55
0.4
C
0
10
Wealth
50
20
23
Measures of risk

• The pure risk cost depends on the individuals
  preferences
• There are several measures that do not depend on
  preferences, like the variance and the
  standard-deviation of the outcomes
• Variance is
• it can be shown that the pure risk is
  aproximately proportional to the variance
• A general rule of thumb is that there is
  aproximately 90 of chances that the actual
  outcome will lie within 2 Standard-Deviations of
  the expected value

24
Risk-aversion and gambling

• Empirically, risk-averse behaviour is
  predominant everybody diversifies portfolios,
  and buys insurance at larger than fair prices.
• Basic risk aversion may be compatible with a
  risk-lover behaviour for other ranges of wealth

25
58
50
48
25
Risk-control and risk-shifting mechanisms

• Risk-pooling a group of individuals facing
  independent risks agree to share any losses (or
  gains) among them
• Ex n2, W0 50, and 5 has a p .2 of being
  stolen. For each individual there are only two
  possible outcomes, 50 with p .8 and 45 with p
  .2 gt EV 49
• Suppose they agree to pool their risks.
  Possibilities are
• 1. Neither is robbed W50, p .8(.8).64gt32
• 2. They are both subject to theft, W45, p
  .2(.2).04gt1,8
• 3. One is robbed and the other is not, W47.5, p
  .32gt15,2
• EV49 again, but the probability of ending up
  close to it is larger the risk has been reduced

26
Risk-pooling

• When the number of individuals joining the
  agreement increases, the risk costs decrease.
  This is the logic of insurance, where insurance
  companies bear the transaction costs of
  organizing the pool, keeping information about
  damages and making the required transfers. When
  these transaction costs are large relative to
  expected losses, people should self-insure
• Risk pooling also explains why firms in unrelated
  business merge to become conglomerates, or why
  people diverfy their portfolio

27
Risk-spreading

• Risk-spreading occurs when individuals share the
  returns of one risky situation
• Examples diversification of firm ownership on
  the stock market
• The proof that risk spreading reduces risk is
  that it must decrease the variance, which is, in
  turn, proportional to the risk cost. Xi is the
  ith outcome of the risky investment

28
Risk-spreading

• Other institutions that facilitate risk-spreading
  are futures markets in these markets, the
  supplier sells part of its future production at a
  price specified now certainty of income is
  achieved
• Im may happen that the total amount of risk in
  not exogenously given. Instead, total risk (or
  risk creating activities) may increase due to
  risk-cost reducing mechanisms

29
Social mechanisms designed to reduce the costs
of risk

• Social mechanisms aimed at decreasing or shifting
  risk-costs
• Limited liability (risk-shifting)
• Disaster insurance
• Quality certification like occupational licensing
  requirements the decrease in uncertainty is
  accompanied by entry restrictions that create
  market power (M. Friedman and K. Arrow)
• Consumer product safety restrictions
• Health standards on workplace
• Medical care insurance moral hazard and the
  increasing costs of medical insurance systems

30
Alternative models of behaviour under
uncertainty game theory

• When uncertainty concerns people behaviour,
  instead of states of nature, game theory helps.
• Strategic games and the prisonners dilemma the
  dominant strategy is for each party not to
  cooperate.
• Applications
• adjacent buildings in a slum. The solution must
  be the internalization of benefits and costs
• world free trade and GATT
• The extreme risk-averter and the maximin strategy

31
Alternative models of behaviour under
uncertainty bounded rationality

• There are limits to human rationality. In face of
  excessively complex games people develop
  satisficing (and not optimizing) routines
• Empirical evidence (disaster insurance in
  earthquake areas) suggests that in some
  circumstances compulsory insurance may be
  required
• Thaler suggested that two particular kinds of
  bounded rationality are frequent loss aversion
  and myopia
• Loss aversion. People weight more heavily losses
  than gaind
• Myopia people frame long run decisions in terms
  of their short-term consequences

32
Alternative models of behaviour under
uncertainty bounded rationality

• Kahneman and Tversky found that people (i) weight
  gains and losses separately and then add their
  separate values and (ii) attach more importance
  to losses than to gains.
• Suppose that at a moment, you are confronted with
  an unexpected gain of 100 and an unexpected loss
  of 80. Overall, wealth increases by 20. However,
  utility may decrease
• Their hypothesis is that people evaluate
  alternatives with a value function, defined over
  changes of wealth, and not with a conventional
  utility function

33
Alternative models of behaviour under
uncertainty bounded rationality

• Conventional utility function

• The Kahneman-Tversky Value function

V(100)
-80
100
V(-80)
W0
W020
34
Alternative models of behaviour under
uncertainty bounded rationality

• Consequences of the Value function the framing
  effect. According with the value function
  hypothesis, one should
• Segregate gains decomposing a large gain into
  smaller ones increases utility
• Combine losses two separate losses cause less
  pain if combined together in a single, larger one
• Combine small losses with a larger gain
• Segregate small gains from large losses a loss
  of 200 accompanied by a gain of 25 is better for
  most people than a loss of 175 (silver-lining
  effect)

35
Alternative models of behaviour under
uncertainty bounded rationality

• The equity premium puzzle the allocation of
  portfolio between stocks and bonds, (with a
  difference in their long-run average rate of
  return of 5-6 percent) can only be an equilibrium
  with an implausibly high risk-aversion. The
  hypothesis of bounded rationality can help
  explain this puzzle.
• More information may increase myopia
• If human behaviour is characterized by myopia and
  loss aversion, this must be taken into account on
  the discussion of (i) policies concerning
  individual control of retirement accounts and
  (ii) activities with negative environmental
  effects

36
Moral hazard and medical insurance

• Medical insurance is (partyly) responsible for
  the dramatic increase in the costs of medical
  care that has occurred worldwide.
• Medical insurance reduces the cost of risk,
  through risk-shifting and risk-pooling. However,
  the insurance changes the economic incentives
  faced by individuals and hence their behaviour
• Medical expenses are not random effects they
  depend on the occurrence of a random effect (the
  illness), but also on the consumers (and his
  doctors) preferences, incomes and on the prices
  of the services. And insurance changes these
  prices.

37
Moral hazard

• Moral hazard problems arise when in a deal one of
  the parties has incentives to undertake a hidden
  action (impossible to monitor) that harms the
  other contractor.
• It requires that there is an information
  asymmetry between the two parties envolved in the
  deal, and that this asymetry concerns the hidden
  action, (an action taken by one of the
  contractors after the deal and which affects the
  outcome (but not completely determines it)
• In a principal-agent relationship with hidden
  action there is always a moral hazard problem
• In the health-insurance problem the principal is
  the company and the agent is the insured. The
  hidden action is the excessive consumption of
  medical services if ill

38
Moral hazard and medical insurance

• With full-coverage insurance, there is no cost
  control of the medical care industry.
• In fact, full coverage private insurance can be
  offered only if demand in inelastic.
• Example 2 states, ill or healthy with pi0.5.
  If uninsured the individual will buy 50 units at
  a price 1 if illness occurs. The expected cost is
  25, and every body would purchase an insurance at
  this premium.

• Inelastic demand

Demand
1
50
Quantity of medical services
39
Moral hazard and medical insurance

• With a full-coverage insurance, and elastic
  demand EAG the individual will buy 100. The cost
  for the insurance company is now 0.510050 and
  it will offer the insurance at this price
• The benefit to the individual is 25 as before and
  hence he may not buy the insurance
• we have a prisonners dilemma everybody would be
  better if restraining from excessive consumption
  but no one will do it by himself.

• Elastic demand and p0

Demand
E
A
1
G
50
100
Quantity of medical services
40
Medical insurance with deductibles

• This problem may also affect actions that might
  prevent the illness
• Deductibles and coinsurance may help decrease the
  moral hazard problem in health insurance
• Suppose there is a deductible for the first 60
  units. Then the individual has 2 options
• A not file a claim and pay 50
• B file a claim, pay 60 and consume 100
• This person chooses B

• Elastic demand

Demand
E
A
1
G
50
Quantity of medical services
41
Medical insurance and deductibles

• Elastic demand

• If there is a demand curve for insurance for each
  illness, the existence of the fixed deductible
  leads people to file claims in the presence of
  serious diseases and not for less severe problems

Demand
E
A
1
G
50
Quantity of medical services
42
Medical insurance and coinsurance

• Suppose t0.1 the individual will purchase less
  than before, which means that the insurance will
  be less costly. The lower the elasticity, the
  lower will be the restrain in consumption
• Coinsurance partially shifts the risk it shifts
  the more expensive units, ie, prevents the
  losses where the marginal utility of wealth is
  gratest and decreases the consumption of the
  least-valued units

• Coinsurance

Demand
E
A
1
G
50
Quantity of medical services
43
Insurance and adverse selection

• Another relevant informational asymmetry is
  adverse selection it occurs when different
  individuals have different probabilities of
  having unfavourable outcomes. If insurance
  providers do not have accurate measures of
  expected loss, they will not be able to set
  equilibrium premiums and the insurance market
  will not function properly
• The market for lemons
• Signaling if the low-risk individuals could
  signal that, they would benefit. Signals, to be
  informative, must be difficult to fake (Sports
  car vs a peugeot 404)

44
Designing national health-care systems

• Usually, deductibles and coinsurance are income
  contigent
• How can the social cost be measured? Usually the
  market price conveys the information about the
  amount of resources given up by society in order
  to supply one further unit. But in this sector
  entry restrictions may make the market price
  higher than the social cost. If so, the optimal
  quantity of medical services will be larger than
  the one uninsured people would choose.

45
Designing national health-care systems

• What about measuring benefits?
• People seek health, not medical service, and the
  relation between the 2 is not well known
  (disagreement among doctors is common)
• The physician is supposed to be the patients
  agent, but in a fee-for-service system is also
  the seller of the services. There is an obvious
  conflict of interest. A solution is managed care
  systems, whereby suppliers have incentives to
  save. Other is hospital charges determined in
  accordance with a schedule of diagnostic related
  groups. Still another is a national health service