Topics in Health and Education Economics

1
Topics in Health and Education Economics class2

• Matilde P. Machado
• matilde.machado_at_uc3m.es

2
2.2. Adverse Selection/Risk SelectionRothschild
Stiglitz (QJE,1976)

• Summary
• Shows the impact of imperfect information on the
  equilibrium outcome of a competitive insurance
  market.
• Insurance companies offer insurance contracts
  that rely on a self-selection mechanism
• High risk individuals cause an externality on low
  risk individuals
• Everyone would be better off (or as well off) if
  risks were revealed ex-ante.

3
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Model I the case of a single type of consumer
• There are two states of nature no accident,
  accident.
• No insurance wealth W,W-d
• p probability of accident occurring
• a(a1,a2) is the insurance vector where a1 is the
  premium paid by the consumer and a2 is the net
  compensation in case of accident, i.e. a2 q- a1
  where q is the insurance coverage.
• With insurance wealth W- a1,W-d a2

4
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• 1. Supply side of the insurance market
• Insurers are risk-neutral, they maximize expected
  profits
• Perfect competition ? Expected profit 0

Expected profit of selling contract a to
individuals with probability of accident p
Competitive relation between the net compensation
and the premium
5
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• It is well-known that at actuarial fair premiums
  risk averse individuals want to hire full
  insurance.

6
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)
Full insurance
7
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

W2 Wealth after accident
W1W2 Full insurance combinations
W2gtW1
relevant
W1gtW2
45º
W1 Wealth with no accident
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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• All wealth combinations along EF are feasible
  (i.e. Expected profit0). F is the feasible point
  of Complete insurance

W2W1 (Completely insured)
W2 (with accident)
Possible wealth combinations with competitive
insurance, slope - (1-p)/p
W2gtW1
F
Wealth without insurance
W1gtW2
E
W-d
45º
W1 No accident
W
9
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Derivation of EF line

10
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Definition of Equilibrium The equilibrium in a
  competitive insurance markets is a set of
  contracts such that
• No contract in the equilibrium set makes negative
  profits
• There is no contract outside the set that, if
  offered, would make no-negative profits.
• We know that when the premium is actuarial fair,
  risk averse prefer F to the rest of the possible
  wealth combinations, i.e. complete insurance.

11
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• 2. Demand Side of the Insurance Market
• Individuals maximize their expected utility that
  depends only on their wealth
• Individuals know p
• Their utility function is state independent
• Their expected utility function is

12
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The slope of the indifference curve at F equals
  the slope of
• EF line. F is therefore the optimum point
  (highest i.c.) in EF.

The slope of the indifference curve at W1W2 is
independent of U(.) and the same as the EF line.
The tangency of the indifference curve to EF
shows that individuals maximize their expected
utility at F.
13
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The equilibrium

W2W1 (total insurance)
W2
F
Optimal net compensation
a2
E
W-d
45º
W1
W
Optimal premiuma1
14
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• For a(a1, a2) to be an equilibrium we need to
  check that it satisfies the two previous
  conditions which amount to
• BE0 (The wealth combination must be along the EF
  line)
• Any other insurance contract (a1,a2) that
  consumers may prefer has negative benefits lt0.
  Those would be contracts that lead to wealth
  combinations in higher indifference curves and
  obviously those are lower premium for the same
  (or higher) compensation or higher compensations
  for the same (or lower) premium which yield
  negative expected profits.

15
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Model II the case of two types of consumers
• Low Risk probability of accident pL
• High Risk probability of accident pH
• pH gt pL
• l proportion of high risks 0ltllt1
• Individuals know their types and their
  probability of accident
• Individuals only differ in their risk, the
  insurance company cannot distinguish them
  ex-ante, however the insurer knows the values of
  pH , pL and l.

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The insurance company knows that for the same
  premium, high risks would like to hire more
  coverage. It will use this information to devise
  self-selection mechanisms.
• Individuals cannot buy more than one insurance
  contract
• There can only be two types of equilibrium
• Pooling both types buy the same contract
• Separating Each type buys a different contract
• It can be shown that a Pooling Equilibrium never
  exists.

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Proof that a Pooling Equilibrium never exists.
• By contradiction, suppose a(a1, a2) is a pooling
  equilibrium, in that case, it can be shown that
  Exp. Profits are a function of the average prob.
  of accident

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The rate of substitution between the two states
  of nature can be derived for the high risks
• And similarly for the low risks

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• So the difference of the rate of substitution for
  high and risks only depends on the probabilities

The indifference curve of the Low risks (for the
equilibrium contract a) is steeper in absolute
terms.
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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)
Note that a must be in the EF curve so that the
expected profit is 0
W2W1
W2
a
F
Note that EF has now a slope that depends on the
average prob. of accident
UH
W-d
UL
45º
W1
W
21
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The existence of a contract b shows that a is not
  an eq.

W2W1
W2
a
F
b is preferred to a by the low risk and since it
is in EF or even slightly above can be offered in
the market (because only low risks would buy it).
UH
W-d
UL
45º
W1
W
22
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• If an equilibrium exists it MUST be separating.
  They assume that there is no cross-subsidization,
  i.e. competition forces the insurance company to
  break even for every contract. The zero expected
  profit conditions are now given by
• Which imply two different wealth combination
  lines from the initial E point.

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• For the same premium the insurance company could
  offer a much higher net compensation to the low
  risks.

W2W1
W2
L
Slope EL
H
E
45º
W1
Premium a
Measures competitive net compensation for high
and low risks for a premium a
Slope EH
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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• However H and L do not constitute an equilibrium

Of all wealth combinations along EH, aH is the
preferred one by the high risks. And of all those
along EL, q is the preferred by the low risks. We
know that Ep0 if aH is sold to the high risks
and q to the low risks. The problem is that q is
also preferred to aH by the high risks since it
means higher wealth in both states of nature.
UH
W2
L
q
UH
H
aH
E
45º
25
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• And the insurer will have negative expected
  profits if it sells q to everyone. That is if
  all individuals buy the contract q then

W2
L
q
UH
UH
aH
E
45º
26
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• The segment between EaL shows the set of wealth
  combinations that could be offered to the low
  risks at zero expected profit and are NOT
  preferred to aH by the high risks (incentive
  compatibility constraint)
• The set of candidates to an equilibrium is aH and
  contracts along EaL . Of all those along EaL , aL
  is the preferred by the low risk. So we will
  check under which conditions aH ,aL is an
  equilibrium.

UL
W2
L
q
UH
aL
aH
E
45º
27
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• To prove that aH, aL is indeed an equilibrium
  The first condition is satisfied because the
  insurer has expected zero profit in both
  contracts. The second condition is the difficult
  one. The existence of equilibrium depends on the
  percentage of high risks, l. It turns out that if
  l is high enough there is an equilibrium,
  otherwise there isnt.

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2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Suppose g is also offered. Then all individuals
  would prefer g to the eq. candidate. The question
  is can g be offered in the market? For aH,aL to
  be an equilibrium, the exp. profit from g should
  be lt0.

W2
L
q
UH
g
aL
H
aH
E
UL
45º
29
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• For a low level of l, the line of possible
  wealth combinations achieved with contracts that
  are sold to both types and breakeven is near EL
  for example EF2. If l is high then the line is
  somewhere close to EH e.g. EF1.

W2
q
F2
UH
g
F1
aH
E
UL
45º
For the high value of l (EF1) g cannot be offered
at a profit so aH,aL is an equilibrium. For a
low of l (EF2) there is no equilibrium in this
competitive market.
30
2.2. Adverse Selection/Risk Selection Rothschild
Stiglitz (QJE,1976)

• Conclusions
• the incomplete information may cause a
  competitive market to have no equilibrium
• The high risks are a negative externality on the
  low risks
• Everyone would be at least as well off if
  everyone revealed their type

31
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• This paper is nice because
• It is an application of RS
• It uses data!! (The British Household Panel
  Survey)
• It tests adverse selection in a market where
  there are private health insurers on top of a
  National Health Service (NHS).
• They find evidence of adverse selection in the
  private insurance market

32
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Setting The British Private Health Insurance
  Market.
• Whats especial?
• There is a public system that covers all
  expenditures (no copayments with few exceptions
  e.g. dental) and everyone.
• If people want to they can hire a private
  insurance. Private and Public are then
  substitutes for care. Everyone contributes to the
  public system through taxes.
• Other systems such as most of the American is
  purely private (except for Medicare/Medicaid)
• In other systems, the private is supplementary,
  people hire the private to cover for copayments
  and services not covered by the public (Belgium,
  France).

33
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Model
• Individuals decide whether to take private
  insurance after observing their type and before
  becoming ill.
• Individuals when become ill must choose where
  they want to be treated (private/public) and
  private and public services cannot be combined.
• If an individual chooses the private service, the
  private insurer must cover the full cost.
• Everyone contributes to the financing of the
  public health service regardless of whether
  he/she uses the public health services (PUB).
• There are a large set of insurers (similar to
  RS)
• Individuals can be of one of two types L,H,
  1gtPHgtPLgt0 and they know their type
• g (1-l) proportion of low risks, 0lt glt1

34
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Model (timing)

Individuals decide whether or not to take private
insurance, and which insurer, conditional on the
packages of all providers and their type
Health authority chooses package
Private insurers simultaneously make their
packages conditional on the package of the HA
35
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• If an individual buys private insurance, he has
  double coverage and in case he gets sick must
  decide if he wants the public (PUB) or private
  treatment (PRI)
• L0 loss if an individual gets sick and does not
  seek treatment
• (Lpri,q) private contract (q premium, Lpri
  Loss i.e. L0-Lpri coverage)
• (Lpub,0) is the outside option of the
  individual offered by the public system (premium
  is paid through taxes). Assumption LpubltL0 the
  public is effective.

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2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• W individuals initial wealth net of taxes
• If the individual does NOT take private
  insurance
• And he does not become ill W
• And he becomes ill W-Lpub
• If the individual TAKES private insurance
• And he does not become ill W-q
• And he becomes ill W-q-Lpri
• Note Private contracts where LprigtLpub are
  irrelevant because they are strictly dominated by
  the public package (the assumption implies that
  if ill the individual goes to the private
  system). The insurer commits to ensure the
  individual does not suffer a loss larger than Lpri

37
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• If the individual does NOT take private
  insurance
• Expected utility for probability p is
• If the individual TAKES private insurance
• Expected utility for probability p is
• The expected profit of a contract (L,q) is

Is the average probability of getting sick
coverage
38
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• In the case of no-illness for a contract (L,q)
  and wealth w
• In the case of illness
• The expected profit of a contract (L,q) is

39
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• We have 2 zero-isoprofit curves depending on the
  individuals type, with slopes
• The zero-isoprofit curves go through point E
  (w,w-L0) (i.e. no insurance)

40
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• 3.1. Symmetric Information
• If there is NO public system, then the
  equilibrium would be just as in RS with
  symmetric information, i.e. efficient contracts
  or complete coverage to both types i.e. aL,aH
  i.e. for JH,L

41
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• In the presence of a public system the status-quo
  is not E(w,w-L0) but P(w,w-Lpub)

W2a
Possible positions of the public package. If
LpubL0 then E is the status quo with public
coverage (i.e. it is as if the public system did
not existed)
L
aL
W-pLL0
UH
aH
W-pHL0
E
W-L0
45º
W-pHL0
W-pLL0
W1n
W
42
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Assumption 1 If all individuals of type J are
  indifferent between the public package P and the
  best private contract for them, all these
  individuals choose the public package.

43
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• In the presence of P(w,w-Lpub) some private
  contracts (in the symmetric information eq.) may
  not be attractive any more

an
a
P3
L
aL
W-pLL0
UH
aH
W-pHL0
P2
E
W-L0
P1
45º
n
W-pHL0
W-pLL0
If PP1 then private market not affected, the
public contract is NOT ACTIVE If PP2 aH not
attractive anymore If PP3 no private contract
is attractive, the private market is NOT ACTIVE
44
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• The equilibrium will depend on P

W2
L
aL
W-pLL0
UH
L0
aH
W-pHL0
H0
E
W-L0
45º
W-pHL0
W-pLL0
Proposition 1 In case P is between E-H0 the
equilibrium is aH,aL if P lies between H0
and L0 the equilibrium is P,aL, in case P lies
strictly above L0 then only P exists
45
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• So if both sectors are active (i.e. P is between
  H0 and L0) and there is no adverse selection the
  probability of illness among the ones with a
  private insurance is pL i.e. the low risk, lower
  than the average probability of illness.

46
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• 3.2. Asymmetric Information
• If there is NO public system, then the
  equilibrium would be just as in RS with
  asymmetric information, i.e. efficient contracts
  or complete coverage to the high risks and less
  than full coverage to the low risks
• We know that in RS we need the proportion of
  high risks to be high enough so that there is
  equilibrium i.e. gg
• Assumption 2 assume gg

47
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• The equilibrium will depend on P

W2
L
W-pLL0
UH
aH
H0
W-pHL0
E
W-L0
L1
45º
W-pHL0
W-pLL0
Again H0 is the public contract such that a high
risk is indifferent between the public contract
and the private contract aH. L1 is the public
contract such that the low-risk is indifferent
between P and . Note that H0 is above L1
(Lemma 2)
48
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Case 1 P lies below L1 ? eq is P is not
  active
• Case 2-3 P coincides with L1 or is between L1
  and H0 the equilibrium is aH,P (assumption 2
  is no longer necessary for the existence of the
  equilibrium.) Both Public and Private are ACTIVE.
• Case 4-5 P coincides with H0 or is above H0, in
  the equilibrium only the public system exists.
• If both sectors are active under adverse
  selection then the probability of illness of
  those who purchase private insurance is pH which
  is larger than the average probability of illness.

49
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Their Main Theoretical Result
• When both markets are active i.e. the private and
  the public then, with perfect information (i.e.
  no adverse selection) only the low risk would buy
  private insurance
• When both markets are active i.e. the private and
  the public then, with asymmetric information and
  therefore adverse selection only the high risks
  would buy private insurance.
• i.e. the sign of the correlation between the
  probability of buying private insurance and
  individuals risk depends on whether there is
  adverse selection. This prediction can be tested.

50
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Empirical Test
• In the UK the private system is substitute to the
  public one.
• Everyone is entitled to the public system, and
  people pay it through taxes regardless of
  utilization
• The private system offers better access, in
  particular negligible waiting times, in the
  models notation it is true that LpubgtLpri
• Data Bristish Household Panel Survey (BHPS)

51
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Empirical Test (cont.)
• In the UK, Private and Public systems are active
  so the model tells us that
• In the absence of adverse selection (perfect
  information) only the low risks buy private
  insurance
• In the presence of adverse selection only the
  high risks buy private insurance.
• We could therefore design a test by comparing the
  risk of requiring medical care of those that
  decide to buy medical insurance to those that
  decide not to buy it. However this would have
  problems
• One does not observe whether people require
  medical care, only observes whether people get
  medical care
• Access conditions are better in the private
  insurance so we could observe more people getting
  medical care under private insurance and wrongly
  conclude they were more needy of care.

52
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Empirical Test (cont.)
• To avoid the bias that may be introduced because
  of differences in access to care between private
  and public, they restrict the sample to
  individuals that have the same access to
  hospitalizations restrict to individuals that
  have private insurance and distinguish between
• Individuals who buy private insurance on their
  own
• Individuals who have private insurance as a
  fringe benefit from their employer (i.e. they get
  it for free)
• The test compares the probability of
  hospitalization of those who purchase private
  insurance to those who get it as a fringe
  benefit.

Note as we mention in the previous class
hospitalizations are likely to be free of moral
hazard.
53
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• They restrict the sample to England and to
  employees employed on a permanent basis.
• The identification assumption is that the
  individuals that get private insurance as a
  fringe benefit (conditional on covariates) are of
  the same risk, on average as the population of
  english permanently employed, i.e. the health
  insurance is orthogonal to health status
  conditional on covariates such as age, education,
  gender and income.

54
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Potential bias
• Employer driven some jobs are more likely to
  offer employer-paid health insurance then others.
  If health of individuals in these different jobs
  differ (conditional on covariates) then we may be
  biasing the results. For example, in agriculture
  the of employed paid individuals is only 15
  while in the banking, finance, insurance,
  business sector is of 66. By occupation, the
  of managers with employer-paid private insurance
  is of 63 while for operators of plants and
  machines it is only 38. To control for this
  possibility they include in the regressions
  industry and occupation among the covariates.
  This means that conditional on being of a certain
  occupation we compare the effect of buying
  private insurance.

55
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Potential bias (cont.)
• Employee driven If less healthy people search
  for jobs that offer private insurance.
• Some evidence (no proof) Most people that
  changed jobs offer reasons other than private
  insurance, for example, more money or better
  chances of promotion.
• In any case, this would imply a smaller
  difference in risks between the two groups
  (downwards bias) ? the true adverse selection
  would be even higher than what they find.

56
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• The idea is

Pool of permanently employed
Conditional on X, same average prob of
Hospitalization, pbar
Do not Get insurance as Fringe Benefit
Get insurance as Fringe Benefit
Different risks
Comparison in the data
Buy insurance
Dont Buy
57
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Note again that if the difference in
  probabilities of hospitalization between the two
  groups were statistically the same, this would
  not be evidence in favor of symmetric information
  (i.e no adverse selection) since in the absence
  of adverse selection the difference should be
  negative not zero.
• Probit model with the dependent variable as
  hospitalization

58
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)
IND1 if the individual pays for the health
insurance directly 0 if Fringe Benefit
This table shows the coefficients not the
marginal effects i.e. the effects on the
probabilities. The marginal effect of IND is
0.021 given that the average probability of
hospitalization is 0.064, this is quite HIGH!
59
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Probit model review

60
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Their test of adverse selection concludes that
  individuals that purchase private insurance have
  a higher probability of hospitalizations (are
  higher risk) than individuals that have private
  insurance as a fringe benefit. This constitutes
  evidence of adverse selection in the British
  private medical insurance market.

61
2.2. Adverse Selection/Risk Selection Olivella,
Vera-Hernández (WP06/02)

• Robustness
• What if what they are finding is just the result
  of the fact that individual purchased policies
  are more generous than firm purchased policies
  and this is what makes us observe more
  hospitalizations? They argue that if individual
  policies were more generous then the probability
  that conditional on being hospitalize individuals
  choose the public service (NHS) should be smaller
  for those having individual purchased policies.
  They run a probit (dependent variable is choose
  NHS, conditional on hospitalization) and find
  that the coefficient of IND is not statistically
  significantly different from zero.

62
Review Discrete choice Models Random Utility
framework (Train, chap2)

• A decision maker n faces J alternatives
• The decision maker chooses the alternative that
  gives him/her the greatest utility
• The researcher does not know U, it observes the
  decision and some attributes of the alternative j
  xnj and some attributes of the decision maker sn.
• Rewrite the utility function as a function of
  observables and an additive unobservable

63
Review Discrete choice Models Random Utility
framework (Train, chap2)

• The probability of choosing alternative i can be
  written as

This integral is of dimension J (i.e. the number
of alternatives)
This integral has a closed form for certain
distributions of e i.e f(.). For example if f()
is iid extreme value than it converts to logit.
Or when f() is generalized extreme value we
obtain nested logit.
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Review Discrete choice Models Random Utility
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• There are several things one must know about
  these models
• Only differences in utilities matter, its
  absolute value does not matter add a constant
  to the utility of every alternative and the
  decision maker keeps choosing the same
  alternative
• alternatively notice that the decision only
  depends on the difference of utilities

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Review Discrete choice Models Random Utility
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• There are several things one must know about
  these models
• When adding an alternative-specific constant, one
  must normalize one of them, i.e. they are
  identified up to a constant since all ki, kj for
  which the differenced are possible estimates.
• This is equivalent to normalize for example ki0
  and estimate kj-d

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Review Discrete choice Models Random Utility
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• Notice that the model cannot incorporate
  attributes of the decision maker because they do
  not vary by alternative, i.e. they cancel out.
  Imagine we could, take wn to be individual ns
  income. Then
• Only if income has a different impact on the
  different alternative utilities

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Review Discrete choice Models Random Utility
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• Another way to include socio-demographic
  characteristics of the individual is to interact
  them with the alternatives characteristics
• The scale of utility is irrelevant, that is if
  alternative i is preferred to alternative j then
  we can scale up or down utility that the result
  still holds true
• Normalizing the scale of utility means
  normalizing the variance of the error term. The
  coefficients should be interpreted accordingly,
  suppose var(e)s2

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Review Discrete choice Models Random Utility
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• Property 6 is important when comparing the
  results of the same model estimated in different
  samples. For example the model
• Was estimate for Chicago and Boston

Which imply a lower variance of the error term in
Boston, Time and Cost explain more in Boston
than in Chicago, i.e. unobservable factors are
less important in Boston
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Review Discrete choice Models Random Utility
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• The Logit Model is derived by assuming that the
  error terms are independently and identically
  distributed extreme value (varp2/6)
• The probability of choosing alternative i is
  given by

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Review Discrete choice Models Random Utility
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• The IIA (Independence of Irrelevant alternatives)
  for any two alternatives i and k the ratio of
  probabilities is given by
• i.e. independent of the rest of alternatives
  which is quite restrictive. No matter if there
  are 2 or 10 alternatives the odds ratio is
  exactly the same.

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Review Discrete choice Models Random Utility
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• GEV- Generalized Extreme Value
• Because IIA is so restrictive other models
  emerged to avoid or minimize its impact. The one
  that interests us now is the Nested Logit. The
  choice set is partitioned into subsets, called
  nests. The nested Logit does not completely relax
  the IIA assumption
• 1. IIA holds within the nest- i.e. for any two
  alternatives within a nest the odds ratio does
  not depend on any other alternatives.
• 2. IIA does not hold across nests- the odds
  ratio between two alternatives of different nests
  depend on all alternatives in those nests.

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Review Discrete choice Models Random Utility
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• Example, commuting to work, I may consider
• car, car pooling within the same nest
• Bus, train in a separate nest
• Therefore, suppose that our car breaks down, then
  the probability of all the other alternatives
  should rise BUT NOT proportionately i.e. the
  probability of car pooling should rise by more
  than the probability of train/bus.

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Review Discrete choice Models Random Utility
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• Suppose we have K nests, B1,.BK then the
  en(en1,enJ) has the following distribution
• The parameters lk measure the independence within
  each nest k. If lk 1 there is NO correlation in
  nest k, if this is so for all k we are back to
  LOGIT If lk lt1 then the error terms are
  correlated within nest k. Still the model assumes
  no correlation between the es across nests.

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Review Discrete choice Models Random Utility
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• The probabilities are given by

IIA holds within the nest
IIA does not hold but the odds ratio only depends
on alternatives in nests k and l. Independence of
irrelevant nests (IIN)
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Review Discrete choice Models Random Utility
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• The meaning of these lambdas. Suppose there are
  only 2 nests (nest1Portugal and nest2abroad)
  then we can understand the difference in the
  lambdas as a strictly preference for one of the
  nests