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Summary - Incentives & behavior
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- Incentives and behavior
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An-Sophie Baudoin 2019-2020INCENTIVES & BEHAVIOR
INTRODUCTION
Contrary to classic economic assumptions, evidence shows that humans do not decide rationally, their
preferences change over time and they can sometimes be interested in the well-being of others and not
only just in their own payoff.
GAME THEORY I
Game theory analyses strategic interaction, that is when the behavior of two or more parties influences
each party’s well-being. We consider the case of static games in which players choose their actions
simultaneously.
FUNDAMENTAL CONCEPTS
A game consists of:
- A number of players i ∈ N = {1, …, n}.
- A number of possible strategies for each player: si ∈ Si.
- A utility function that determines the player’s payoffs as a function of the chosen strategies:
ui : S → ℝ.
Each player maximizes his own payoff and knows the whole structure of the game. In a game with
complete information, each player’s payoff function is common knowledge among all players.
The game can be displayed in a payoff matrix when there are only two players:
1 2 LEFT RIGHT
UP (1,3) (0,1)
DOWN (2,1)* (1,0)
*NE
A dominant strategy is a strategy which is always optimal, regardless of the opponents’ strategies.
In the example, DOWN is dominant strategy for 1 because whatever 2 chooses, DOWN is always better
than UP. LEFT is a dominant strategy for 2 because whatever 1 chooses, LEFT is always better than
RIGHT. In this case, we get an equilibrium in dominant strategies.
Iterated elimination of strictly dominated strategies (IESDS):
In most games, there are no dominant strategies (and therefore no equilibrium in dominant strategies).
To analyze such games, we drop strictly dominated strategies from the payoff matrix:
1 2 LEFT MIDDLE RIGHT
UP (1,0) (1,2) (0,1)
DOWN (0,3) (0,1) (2,0)
A strategy α is strictly dominated by another strategy β, if strategy β yields a strictly larger payoff than
strategy α for all possible choices of the opponents. It is advisable to proceed by elimination and consider
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the remaining options available at each step, while keeping in mind that players are rational (= common
knowledge). Here, we firstly eliminate RIGHT, then DOWN, and then LEFT.
NASH EQUILIBRIUM
In most games, there are neither dominant, nor strictly dominated strategies. Therefore, we search for
the (mutual) best responses.
To find a Nash equilibrium, it is needed to mark for each strategy of player 2 the best response of player
1 by underlining the corresponding payoff of player 1, and then, mark for each strategy of player 1 the
best response of player 2 by underlining the corresponding payoff of player 2. The cell where both
payoffs are underlined corresponds to the Nash equilibrium. The corresponding strategies are the
mutual best responses.
In a Nash equilibrium, no player has an incentive to unilaterally deviate and expectations must be
correct.
- Players meet before the game: they agree on self-enforcing strategy profiles, in which each
player’s strategy must be a best response to the strategies of the other players.
- Players do not meet before the game: they have to believe that the opponents’ strategy are
mutually consistent.
In many games, there are multiple Nash equilibria. When it happens, it is not clear which one will be
played or whether a Nash equilibrium will be played at all. This can be solved with these concepts:
- Focal points: centers of interest, properties of the environment that can be used to anticipate the
opponent’s behavior; they show that people can coordinate without communication. This
include social norms (“ladies go first”).
- Pareto optimality: we may argue that a particular Nash equilibrium is played if it yields much
higher payoffs for the players than any other equilibrium (1). However, if players are uncertain
about their opponents’ strategy, it may make sense for them to choose another less risky strategy
(2).
(1) Payoff dominant NE (2) Risk dominant NE
1 2 LEFT RIGHT 1 2 LEFT RIGHT
UP (100,100) (0,0) UP (9,9) (0,8)
DOWN (0,0) (1,1) DOWN (8,0) (7,7)
- Coordination by communication
- Equilibrium refinements
MIXED STRATEGIES
Sometimes, there does not exist a Nash equilibrium in pure strategies. However, there exists a Nash
equilibrium in mixed strategies: each player chooses LEFT with a probability of ½ and RIGHT with a
probability of ½.
1 2 LEFT RIGHT
LEFT (-1,1) (1,-1)
RIGHT (1,-1) (-1,1)
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A mixed strategy for player x is a vector that assigns to each pure strategy a probability with which this
strategy will be chosen. A player would only use a mixed strategy when he is indifferent between several
pure strategies, and when keeping the opponent guessing is desirable (when the opponent can benefit
from knowing the next move).
To find the equilibrium in this mixed strategy, we suppose that player 1 expects player 2 to choose LEFT
with a probability q.
Expected payoff of player 1 from choosing LEFT:
E[u(LEFT)] = q x (-1) + (1-q) x 1 = 1-2q
Expected payoff of player 1 from choosing RIGHT:
E[u(RIGHT)] = q x 1 + (1-q) x (-1) = 2q-1
The player is indifferent between LEFT and RIGHT if:
E[u(LEFT)] = 1-2q = 2q-1 = E[u(RIGHT)]
→ In equilibrium, we must have q = ½
At a mixed strategy, a player is always indifferent between those pure strategies (= degenerate mixed
strategies) that he chooses with positive probability.
In generic game, there is an uneven number of Nash equilibria.
• Pure strategy: an unconditional, defined choice that a person makes, regardless of the other player’s
strategy; when each player plays one specific strategy and sticks with it (and no one has an incentive to
deviate)(compared to a mixed-strategy where at least one player randomizes his choice).
• Pure strategy Nash equilibrium: Nash equilibrium containing only pure strategies for each player.
• Symmetric Nash equilibrium: when all players use the same strategy (possibly mixed) in the
equilibrium.
• Simultaneous game: players have at least some information about the strategies chosen on others and
thus may contingent their pay on past moves.
• A game is one of complete information if all factors of the game are common knowledge (set of
strategies and payoffs).
GAME THEORY II
We consider dynamic games, in which players decide sequentially, by adding a time structure. We
display such games graphically in a game tree.
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Player B observes the choices of player A and then chooses his strategies.
- Player A has the strategies L and R.
- Player B has the strategies ll, lr, rl, rr.
Interpretation of lr: B plays l if A chooses L and r if A chooses R.
In a sequential game, a strategy specifies for each decision node the action that will be chosen if the
decision node is reached.
To solve the sequential game, it can be useful to translate it back into a payoff matrix.
A B ll lr rl rr
L (1,9) (1,9) (3,8) (3,8)
R (0,0) (2,1) (0,0) (2,1)
The equilibrium (L,ll) is not very plausible. Player B threatens to choose l if A chooses R. If A believes
that B will realize his threat, it is optimal for A to choose L. But the threat is not credible: if A chooses
R, it would be better for B to choose r. If A anticipates that B will choose r, it is better for him to choose
R instead of L. Therefore, (L,ll) is not a subgame-perfect Nash equilibrium.
• The extensive form representation of games contains the following information:
- The set of players {1, …, n}.
- The order of moves.
- What the players possible actions are when they move.
- What each player knows when he makes his decision.
- The players’ payoffs as a function of moves that were made.
- The probability distribution over exogenous events: indeed, exogenous events may be part of
the game. They are modeled as moves by the player Nature. Nature chooses events from a given
set of events according to some probability distribution.
The extensive form is thus a graphical representation of a sequential game that provides information
(mentioned above). This structure excludes cycles; a same node cannot be reached on different paths.
• The game tree can be used in sufficiently simple dynamic games, as a set of ordered and connected
nodes, which are points at which players can take actions. The game starts with exactly one initial node,
that represents the 1st decision to be made. “….” in a game tree means that the player who has the dots
at his level cannot observe the action of the other player.
• Decision nodes: exactly one player can choose an action and each action leads to a new decision or
terminal node.
• Terminal nodes: here the game ends and payoffs are realized.
• Information sets are used to describe the informational structure of the game (such as what the player
knows when he moves). For a player, it is a set of decision nodes satisfying the following properties:
- At all nodes of the information set, the same player moves.
- The player cannot distinguish between different nodes of the information set, this implies that
at every node of the information set, he faces the same set of actions.
- Every node is an element of exactly one information set.
- Players always recall their actions, unless specified otherwise.
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• If all information sets of a game are singletons (one point = one node), it is a game with perfect
information.
• If at least one information set contains more than one node, it is a game with imperfect information.
It also means that a player does not know what actions other players took up to that point (….).
• In games of incomplete information, players may not know the opponents’ preferences.
SUBGAME-PERFECT EQUILIBRIUM
The Nash equilibrium ignores the dynamic structure of an extensive game: NE may rely on non-credible
threats. For implausible threats to be excluded, the equilibrium concept has to be refined.
The idea behind the subgame-perfect equilibrium is that each player’s strategy must be optimal, given
the other players’ strategies not only at the start of the game, but in each subgame, even if it is not
reached in equilibrium.
A subgame in an extensive form game has the following properties:
- It begins at a decision node α that is a singleton information set.
- It includes all the decision and terminal nodes following α in the game (but no nodes that do not
follow α).
- It does not cut any information sets.
A subgame is a part of the game that remains to be played starting at a point at which the complete
history of the game thus far is common knowledge among all players. Each extensive form game has at
least one subgame (the subgame itself). When dots (“….”) are involved, there is no subgame at that
level because the player does not have information about the other player.
A Nash equilibrium is subgame-perfect if the players’ strategies constitute a NE in every subgame. It is
found through backward induction: first by finding the optimal strategy of the player who chose last
and substituting the last decision nodes by the payoffs that would have occurred if the last player had
chosen an optimal action in each subgame; then consider the penultimate decisions nodes, …
Every subgame-perfect equilibrium is also a Nash equilibrium but not every NE is subgame-perfect.
Sequential games with infinitely many stages can have multiple subgame-perfect equilibria.
RISK PREFERENCES I
The focus of risk preferences is how people deal with uncertainty in financial decision making, in labor
market decisions, in environmental decisions, …
EXPECTED UTILITY THEORY
EUT is the standard model in economics that describes how individuals deal with risky situations.
In EUT, risk preferences are given by a utility function U(x) that assigns to each outcome x a value of
“utility”. The expected utility from a lottery is derived from the sum of possible utilities multiplied by
the respective probabilities (i.e.: ½ U(1.000.000) + ½ U(32.000) or U(500.000)). The lottery with the
highest expected utility is the preferred lottery.
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- If U(x) = x, the individual is risk neutral because the person would take the risk (516.000 >
500.000).
- If U(x) = √𝑥, the individual is risk averse because he would choose the safest option (590 <
707). His utility function is concave, and it exhibits “diminishing marginal utility”.
- If U(x) = x2, the individual is risk seeking/loving and would prefer a risky lottery to any safe
lottery with the same expected monetary payoff. His utility function is convex.
The core features of the EUT are that the risk attitude only depends on the final outcome and the
probability distribution over final outcomes, expected utility is linear in probabilities and losses are
treated in the same way as gains.
ST-PETERSBURG PARADOX
A fair coin is flipped repeatedly. If heads shows up for the 1st time at the n-th toss, you win 2n €. The
expected payoff from this lottery is equal to ∞ (1+1+1+…).
• Certainty equivalent: amount of money that makes the individual indifferent between the lottery and
the safe outcome.
XXX
• Risk premium: difference between the lottery’s expected payoff and the certainty equivalent; it is the
amount of money the individual is willing to pay in order to get the lottery’s expected payoff for sure
instead of the lottery.
XXX
PROBLEMS OF EUT
• The Allais Paradox shows that people do not treat responsibilities linearly. Indeed, many individuals
stick to their behavior even if one explains their “inconsistency” to them. Consequently, expected utility
theory does not make precise predictions for all situations.
• Another problem is that for small and intermediate lotteries, EUT predicts risk neutral behavior but we
know that the typical decision maker is risk-adverse even when the lottery is small (that is, he rejects
small lotteries with substantial positive expected payoff, and thus he rejects large lottery that could be
very advantageous as well).
PROSPECT THEORY
Many alternatives risk preference model have been developed in order to avoid the problems caused by
EUT. The most convincing one is the prospect theory.
In the EUT, gains and losses are not really differentiated. However, in prospect theory, utility is defined
over gains and losses (relative to a neutral reference point) rather than over final outcomes, as in EUT.
Moreover, prospect theory assumes that losses weight heavier than gains. This causes a kink in the origin
of the utility function, which is concave in the domain of gains, and convex in the domain of losses:
𝑥𝛼 𝑖𝑓 𝑥 ≥ 0
𝑣(𝑥) = { (where 𝜆 is the coefficient of loss aversion)
−𝜆(−𝑥)𝛽 𝑖𝑓 𝑥 < 0
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Finally, utility is not linear in probabilities as in EUT but is weighted according to some decision
weight. This captures the fact that many individuals overweight small probabilities and underweight
intermediate probabilities. As an example, the Allais paradox shows that the difference between 0,99
and 1 has more behavioral impact than the difference between 0,33 and 0,34.
We consider a student who has 20.000€ to spend during his studies. After getting his 1st job, he can now
spend 60.000€ yearly. He will most likely experience the difference of 40.000€ as a gain. After a while,
he got another job that allowed him to spend 100.000€ but then he lost that job and got a 3rd one where
he now earns enough to spend 60.000€ a year. He will most likely experience the difference of 40.000€
as a loss.
- According to EUT, he should get the same utility from 60.000€ in both situations but it is
probably not the case. He is certainly happy in the 1st situation and rather unhappy in the 2nd
one.
- According to the prospect utility, he experiences the prospected utility v(40.000) in the 1 st
situation (reference point = 20.000€) and v(-40.000) in the 2nd (reference point = 100.000€).
Moreover, the student benefits less from 40.000€ less than he suffers from cutting back his
expenses by 40.000€.
RISK PREFERENCES II
We consider a risk-preference model in which the reference-point is derived endogenously from
expectations.
REFERENCE-DEPENDENT PREFERENCES
THE UTILITY FUNCTION
Let c = (c1, c2) ∈ ℝ2 be consumption and r = (r1, r2) ∈ ℝ2 a reference point. For example, c1 ∈{0,1}
denotes whether the consumer purchases a good or not, and c2 is the money left after the transaction.
Consumption may be uncertain (the consumer may not know the exact prices of the good). Let c be
drawn from the probability measure F. Also, the reference point may be stochastic. Let r be drawn from
the probability measure G. The consumer’s utility is then U(F|G) = ∬ u(c|r)dG(r)dF(c). We consider
the following specification of the utility function: u(c|r)=c1+c2+𝜇 (c1 – r1)+ 𝜇 (c2 – r2); c1+c2 is
“consumption utility” and 𝜇(c1 – r1)+ 𝜇(c2 – r2) is “gain-loss utility”. Both components are additively
separable across dimensions. Call 𝜇 a “universal gain-loss function”.
We impose the following assumption on 𝜇. They correspond to Kahneman and Tversky’s description
of the “value function” defined on c – r.
- A0 𝜇 is continuous, twice differentiable for x ≠ 0, and 𝜇(0) = 0.
- A1 𝜇 is strictly increasing.
- A2 If y > x > 0, then 𝜇(y) + 𝜇(-y) < 𝜇(x) + 𝜇(-x).
- A3 𝜇’’(x) ≤ 0 for all x > 0, and 𝜇’’(x) ≥ 0 for all x < 0.
- A4 lim 𝜇’ (-|x|) / lim 𝜇’ (|x|) = 𝜆 > 1.
𝑥→0 𝑥→0
A3 captures the fact that the marginal change in 𝜇 is larger for changes that are close to the reference
level. For some results, we have to replace this assumption by A3’ 𝜇’’(x) = 0 for all x ≠ 0.
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REFERENCE POINT AND PERSONAL EQUILIBRIUM
It is not clear where the reference point r comes from. There is very little empirical evidence on this
issue. In most models, r is just the status quo. We will assume that the reference point is given by the
consumer’s expectations. Specifically, let {DI}I∈ℝ be a continuum of choice sets and F(I) the distribution
over {DI}I∈ℝ which defines the consumer’s expectations. We introduce an equilibrium concept that
combines rational expectations and reference-dependence.
• Definition 1: A selection { 𝜎𝐼 ∈ DI} I∈ℝ is a personal equilibrium (PE) if
U(𝜎𝐼 |∫ 𝜎𝐼 dF(I)) ≥ U(𝜎′𝐼 |∫ 𝜎𝐼 dF(I)) for all I ∈ ℝ and alternative selection 𝜎′𝐼 ∈ DI.
• Definition 2: A selection { 𝜎𝐼 ∈ DI} I∈ℝ is a preferred equilibrium (PPE) if it is a PE, and
U(𝜎𝐼 |∫ 𝜎𝐼 dF(I)) ≥ U(𝜎′𝐼 |∫ 𝜎′𝐼 dF(I)) for all PE’s {𝜎′𝐼 ∈ DI} I∈ℝ.
A selection {𝜎𝐼 ∈ DI} I∈ℝ is a map that tells us what consumption bundle 𝜎𝐼 will be realized in situation
I when the choice set is DI. This selection also defines the consumer’s expectations about what she is
going to consume. The expectations in turn define the reference point. In PE, it is optimal to choose 𝜎𝐼
in situation I. Since the consumer is free to choose any (feasible) plan, he should select one that
maximizes ex-ante expected utility. This must be the case in PPE.
EXAMPLE 1: SHOPPING
Consider a consumer who has to decide whether to purchase a good or not, c1 ∈{0,1}. Denote c2 the
money left after the transaction. His endowment is given by (0,0). His utility consumption is given by
c1 + c2. We assume that 𝜇 satisfies A3’ such that 𝜇(x) = 𝜂x for all x > 0 and 𝜇(x)=𝜆𝜂x for all x < 0,
where 𝜆 > 1. The price of the good denoted by p is uncertain.
Suppose that the consumer’s expected payment is p* ≥ p (the price of the good never exceed the
consumer’s expectation) and the consumer expects to get the good with probability q. If he purchases
the good, his total utility is given by 1 – p + (1 – q) 𝜂 + 𝜂 (p* – p). If he does not purchase the good, his
total utility is given by -q 𝜂𝜆 + 𝜂p*. His net gain from purchasing the good is then 1 + 𝜂(1 – q + 𝜆q) –
(1 + 𝜂)p (1). This term is maximal if q = 1.
Suppose that the consumer’s expected payment is 0 (any positive payment that implies that gain-loss
utility is negative) and the consumer expects to get the good with probability q. If he purchases the good,
his total utility is given by 1 – p + (1 – q) 𝜂 + 𝜂𝜆p. If he does not purchase the good, his total utility is
given by -q 𝜂𝜆. His net gain from purchasing the good is then 1 + 𝜂(1 – q + 𝜆q) – (1 +𝜆 𝜂)p (2). This
term is maximal if q = 0.
1+𝜂𝜆
From (1), we get that the consumer never purchases the good if p > pmax = 1+𝜂
.
1+𝜂
From (2), we get that the consumer always purchases the good if p < pmin = 1+𝜂𝜆.
Suppose that with probability qL the price is pL < pmin and with probability qH = 1 – qL the price is pH >
pmax. What will the consumer do if he unexpectedly finds that the real price is pM ∈ (pmin, pmax)? Note
that the consumer expects to the good with probability qL and his expected payment is qL pL. If he
purchases the good at price pM, his total utility is given by 1 – pM + qH 𝜂 – qL 𝜂𝜆(pM – pL) – qH 𝜂𝜆 pM. If
does not purchase the good at price pM, his total utility is given by - qL 𝜆𝜂+ qL 𝜂 pL.
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Consider the case where pL = 0. From the last two formulas of total utility, we get that the consumer
1+𝜂 (𝜆−1)𝜂
purchases the good at price pM if and only if pM ≤ + qL . The higher is qL, the more attached
1+𝜂𝜆 1+𝜂𝜆
is the consumer to the good so that he is ready to pay higher prices.
Consider the case where pL ≥ 0 and qL = 1 (the consumer expects with certainty to purchase the good).
From the last two formulas of total utility, we get that the consumer purchases the good at price pM if
(𝜆−1)𝜂
and only if pM ≤ 1+ pL . The higher is pL, the more does the consumer expect to pay, and the less
1+𝜂𝜆
he suffers if he finds that the real price is even higher. This constitutes a violation of the law of demand.
EXAMPLE 2: LABOUR SUPPLY
In a paper, it was shown that the labor supply of NYC cabdrivers responds negatively to hourly wages.
This clearly contradicts standard models of labor supply where workers intertemporally substitute labor
and leisure. They suggest that cabdrivers have daily income targets (they do not optimize
intertemporally): after meeting the daily target, they stop working. In the following, we use our theory
of reference-dependent preferences to formalize this argument. Our theory does not assume fixed
targets, but endogenously derives targets (reference points) from expectations.
A taxi driver decides whether to go to work in the morning, and, if yes, whether to continue driving in
the afternoon. Let em ∈{0,1} be his decision in the morning, and ea ∈{0,1} his decision in the afternoon.
Both income in the morning, wm, and in the afternoon, wa, are uncertain. If the driver works in the
morning, he learns his afternoon income. Consumption utilities are wm + wa and -f(em+ ea), where f is
the per-unit cost of effort. We assume that 𝜇 satisfies A3’ such that 𝜇(x) = 𝜂x for all x > 0 and 𝜇(x)=𝜆𝜂x
for all x < 0, where 𝜆 > 1.
𝑎 𝑎
We define wages 𝑤𝑚𝑖𝑛 and 𝑤𝑚𝑎𝑥 such that the taxi driver always (never) continues working in the
𝑎 𝑎 ). Suppose that the driver’s expected (total) income is 𝑤
afternoon when wa > 𝑤𝑚𝑎𝑥 (wa < 𝑤𝑚𝑎𝑥 ̃ ≥ wm
+ wa and he expects to work in the afternoon with probability q. If he drives in the afternoon, his
additional utility is given by wa – f – 𝜂𝜆(𝑤
̃ – wm – wa) – (1 – q) 𝜂𝜆f. If he does not drive in the afternoon,
his additional utility is given by – 𝜂𝜆(𝑤̃ – wm) + 𝜂qf. His net gain from driving in the afternoon is then
wa – f – 𝜂𝜆 wa – f𝜂 ((1 – q) 𝜆 + q)(1) This term is maximal if q = 1.
Suppose that the driver’s expected (total) income is 𝑤 ̃ = 0 and he expects to work in the afternoon with
probability q. If he drives in the afternoon, his additional utility is given by wa – f – 𝜂(wm + wa) – (1 – q)
𝜂𝜆f. If he does not drive in the afternoon, his additional utility is given by 𝜂wm +q 𝜂f. His net gain from
driving in the afternoon is then wa (1 + 𝜂) – f (1 + 𝜂𝜆(1 – q) + 𝜂q)(2). This term is maximal if q = 0.
1+𝜂
From (1), we get that the driver always works in the afternoon if wa < wmin = 1+𝜂𝜆.
1+𝜂𝜆
From (2), we get that the driver never works in the afternoon if wa > wmax = f.
1+𝜂
Suppose that with probability qL the wage is 𝑤𝐿𝑎 < wmin and with probability qH = 1 – qL the wage is 𝑤𝐻𝑎
> wmax. What will the driver do if he unexpectedly finds that the real wage is 𝑤𝑅𝑎 ∈ (wmin, wmax)? In order
to capture all interesting effects, we assume that 𝑤𝐸𝑚 – 𝑤𝑅𝑎 < 𝑤𝑅𝑚 ≤ 𝑤𝐸𝑚 , where 𝑤𝐸𝑚 is the wage the driver
expected to earn in the morning, 𝑤𝑅𝑚 is the morning wage that he realized, and 𝑤𝑅𝑎 is the realized
afternoon wage. We are interested in the driver’s behavior if he works in the morning. His reference
point is as follows: the taxi driver expects to work all day with probability qH, and he expects to earn
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𝑤𝐸𝑚 + qH 𝑤𝐻𝑎 . The driver’s utility when he continues working in the afternoon is 𝑤𝑅𝑚 + 𝑤𝑅𝑎 – 2f + qL 𝜂
(𝑤𝑅𝑚 + 𝑤𝑅𝑎 – 𝑤𝐸𝑚 ) – qH 𝜂𝜆(𝑤𝐸𝑚 + 𝑤𝐻𝑎 - 𝑤𝑅𝑚 – 𝑤𝑅𝑎 ) – qL 𝜂𝜆f. His utility when he does not continue
working in the afternoon is 𝑤𝑅𝑚 – f + qL 𝜂𝜆 (𝑤𝐸𝑚 – 𝑤𝑅𝑚 ) – qH 𝜂𝜆(𝑤𝐸𝑚 + 𝑤𝐻𝑎 - 𝑤𝑅𝑚 ) – qH 𝜂f. Hence, if
1+𝜂+𝑞 𝜂(𝜆−1)
𝑤𝑅𝑚 =𝑤𝐸𝑚 , the driver continues working as long as 𝑤𝑅𝑎 ≥ 1+𝜂+𝑞 𝐿 𝜂(𝜆−1) f.
𝐻
INTERPRETATION
If qL is small, then the driver expects to work in the afternoon and expects a high income. Being short
of the income target feels like a loss. Hence, the driver is willing to work in the afternoon even at a
moderate wage 𝑤𝑅𝑎 .
If qL is large, then the driver does not expect to work in the afternoon and expects a low income. Hence,
even at a substantial wage 𝑤𝑅𝑎 , the driver will not continue working.
The model therefore makes the following prediction: if individuals expect wages to be high, then they
work more (we then have intertemporal substitution); if individuals face unexpectedly high wages, the
effect on working hours is weaker or negative.
PORTFOLIO CHOICE
We use prospect theory to explain the equity premium puzzle, which says that if people’s risk
preferences can be described by EUT, they invest too little of their wealth into stocks.
THE EQUITY PREMIUM PUZZLE
For a while, the average annual return on stocks (= risky assets) was of 7% and on bonds (= safe assets)
of less than 1%: this is inconsistent with standard economic models, the equity premium of 6% being
way too high (or meaning an incredibly high risk aversion).
The expected utility is given by 𝐸 [∑∞ 𝑡
𝑡=0 𝛽 𝑈(𝑐𝑡 )], where β ∈ [0,1] is the discount factor and U(ct) the
utility from consumption in period t.
𝐶 1−𝛼 – 1
The utility function is assumed to be U(c, α) = , where 𝛼 ∈ (0, ∞) measures the degree of risk
1− 𝛼
aversion. Reasonable estimates show that 𝛼 ≈ 1. Mehra and Prescott vary β ∈ [0,1] and 𝛼 ∈ (0, 10) in
order to obtain the highest admissible equity premium.
EXPLAINING THE EQUITY PREMIUM PUZZLE WITH MYOPIC LOSS AVERSION
Explanation for the equity premium puzzle based on prospect theory:
• Loss aversion: investors have preferences over returns (gains and losses) rather than over the
consumption profile the returns help provide.
• A short evaluation period: investors frequently evaluate their portfolios and then perceive the utility
from gains and losses. The shorter the evaluation period, the less attractive are risky assets.
Supposing a risky asset that pays an expected 7% per year with a s.d. of 20% (like stocks) and a safe
asset that pays a sure 1%. The attractiveness of the risky asset will depend on the time horizon of the
10
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