Loss Aversion and the Dynamics of Political Commitment ò · Œ Dynamic reference point formation (1/2) ªßà Dynamics of reference points not much explored. Kahneman and Tversky

Prepared with SEVISℒℐDℰS

Loss Aversion and

the Dynamics of Political Commitment

John Hassler (IIES)

Jose V. Rodriguez Mora (University of Edinburgh)

June 4, 2010

ê ò ê

http://www.econ.upf.es/~sevimora/

Summary (1/2) ãß ê

∙ Political commitment ß∙ Traditional Commitment Technologies ß∙ Our Contribution ß∙ Our explanation: Entitlement ß∙ Loss-aversion in Brief. ßà∙ Dynamic reference point formation ßà∙ Model without loss-aversion ßà∙ Taxes under commitment ßà∙ The commitment game ßà∙ Loss aversion ßà∙ Reference consumption and reference taxes ß∙ Investments ß∙ Utility of workers ß∙ Tax determination ß∙ Equilibrium: ßà∙ Backward-Looking Reference Point Dynamics ß∙ Last Period (T ) ßà

ê òê ãß

Summary (2/2) âß ê

∙ Next to last period (T − 1) ßà

∙ Previous to next to last (T − 2) ß

∙ Equilibrium characterization ß

∙ Finite horizon equilibrium: discussion ßà

∙ Infinite horizon ßà

∙ Forward-looking loss-aversion ß∙ Forward looking Reference Dynamics ß

∙ Equilibrium Definition ßà

∙ Last Period (T ) ß

∙ Next to last period (T − 1) ßà

∙ Previous to next to last (T − 2) ß

∙ Equilibrium characterization ß

∙ Discussion ß

∙ What have we learn? ßà

ê òê â ß

Political commitment ß ê

∙ At least since the work of Kydland and Prescott, it is well known

that the ability to commit is very important for political outcomes

and welfare. Optimal sequence of taxes is typically time inconsis-

tent, capital taxation example:

∙ At t fix future sequence of taxes, in particular �t+s.

∙ Planner accounts at t the distortionary effects that has on investmentbetween t and t+ s. Thus, optimally relatively low �t+s.

∙ At t + s the tax is lump sum... it is optimal to set �t+s large... unlikewhat was optimal s periods before.

∙ The (discretionary) Markov equilibrium predicts unrealistically high

taxes. In general, a big policy failure whenever there is time incon-

sistency.

∙ How do governments achieve (at least partial) commitment?

êò ê ß 160

Traditional Commitment Technologies ß ê

∙ Classical explanations;

∙ Delegation to a person with suitably chosen preferences.

∙ Repeated interaction and non-Markovian strategies.

∙ The first begs the question and the second would typically involve

complicated coordination between voters of different generations.

∙ Formally, the “threat” of a long punishment phase can sustain

“good” equilibria with e.g., low taxation on sunk capital.

∙ But are these punishment phases credible threats, particularly in

OLG settings? Renegotiation proofness takes away most (all?) of

these non-Markovian equilibria.

∙ We argue there is a need for other (complementary) explanations

relying less on intergenerational voter coordination.

ß êò ê ß 260

Our Contribution ß ê

∙ Study the dynamics of commitment:

∙ Procrastination Principle

∙ If short run costs versus long run benefits.

∙ Simply “to commit” is not a markov equilibrium.

∙ To commit takes time.

∙ No Markov eq. in pure strategies

∙ Entitlement as a Commitment Technology.

∙ introduce prospect theory in a dynamic political economy setting

∙ natural and general environment that in practice provides govern-

ments with a commitment technology

ß êò ê ß 360

Our explanation: Entitlement ß à ê

∙ Entitlement effects – people have a distaste for feeling cheated,

not getting what they feel entitled to.

∙ More specifically, individuals form reference levels for consumption.

∙ Individuals are loss averse with respect to reference levels (in the

sense of Kahneman and Tversky)

∙ losses relative to a reference point are valued strictly higher than

gains.

∙ Loss-aversion can provide a commitment mechanism, complement-

ing other ones.

ß êò ê ßà 460

Loss-aversion in Brief. (1/2) ãß à ê

∙ Following Kahneman and Twersky we assume individuals have loss-

aversion. Key features are that individuals:

∙ care more about losses relative to a certain reference point than

about gains,

Formally, there is an " > 0 such thatu(r;r,i)−(r−x;r,i)

x − u(r+x;r,i)−u(r)x ≥ " ∀x > 0.

∙ are risk-loving in losses – a 50/50 chance of loosing x or zero is

better than a certain loss of x/2.

pu (r − x; r, i) + (1− p)u (r; r, i) > u (r − px; r, i) ∀x > 0, p ∈ (0,1)

ß êò ê ßà ã å 560

Loss-aversion in Brief. (2/2) âß à ê

∙ This implies that utility has a kink at a possibly time-varying but pre-

determined reference point, and that utility is concave (convex) above

(below) the reference point.

c

Uc

r

ß êò ê ßà å â 660

Dynamic reference point formation (1/2) ãß à ê

∙ Dynamics of reference points not much explored.

∙ Kahneman and Tversky (implicitly) used the past, status quo, as

the reference point.

∙ But, reference points (entitlements) may be also partially forward-

looking and determined by expectations about the future (Koszegi

and Rabin, QJE’07).

∙ We allow for them to be either FORWARD or BACKWARD look-

ing.

ßà êò ê ßà ã å 760

Dynamic reference point formation (2/2) âß à ê

∙ BACKWARD LOOKING:

∙ Based on last period’s experiences. To establish commitment for tomorrowhas a cost today for the government.

∙ It has a short run cost to implement.

∙ The Procrastination Principle applies: it takes time to implement commit-ment.

∙ FORWARD LOOKING:

∙ An important part of politics is to affect reference levels.

∙ In our model, political candidates run only once and cannot make any formallybinding commitments. But, they are allowed to make a “promise” about thefuture tax rate, although they will not be around to implement the latter.

∙ If rationally believed, the “promise” can affect future reference levels andthereby be self-fulfilling. A seemingly empty promise with commitment value.

∙ Commitment is implemented much faster.

ßà êò ê ßà å â 860

Model without loss-aversion (1/6) ãß à ê

∙ Two-period OLG structure.

∙ In each generation, there are two types of agents

∙ workers and

∙ entrepreneurs.

∙ Time starts at t = 0 and is potentially infinite.

∙ Workers have a simple private life.

∙ Exogenous wage in their second period of life,

∙ consumption only in second period, we normalize the private

income to zero.

∙ Utility of young at t is �u(dt+1

)= �dt+1

∙ Utility of old at t is u (dt) = dt,

∙ dt: consumption of old worker in period t.


Model without loss-aversion (2/6) âãß à ê

∙ Young entrepreneurs invest.∙ Choose it with (gross) return of 1 at t+ 1∙ Investment utility cost i2t∙ Consume only in second period: ct+1.

+ They observe �t before choosing it, but only affects them insofar∙ has information on �t+1∙ (with loss aversion) affects their reference point for t+ 1

∙ Given a tax-rate �t, an entrepreneur born in period t− 1 solves

Ut = maxct+1,it

−i2t2

+ �u(ct+1

)s.t. ct+1 = it

(1− �t+1

).

∙ u(ct+1

)= ct+1.

∙ �t+1 determined in the beginning of period t+ 1, when it are sunk.

∙ Taxes are used for transfers benefiting the workers.

∙ The policymakers budget constraint: Tt+1 = it�t+1

ßà êò ê ßà å âã å 1060


∙ Two sets of decisions are taken.

∙ Private decision – investment

∙ chosen privately by young entrepreneurs after observing cur-rent tax rates.

it = arg maxit−i2t2

+ �it(1− Et�t+1

)= �

(1− Et�t+1

)∙ Collective decision – taxes.

∙ If at t policies are not previously commited, chosen in everyperiod t in order to maximize a weighted sum of the utility ofold and young living individuals.

∙ Two interpretations of the collective decision making.∙ The outcome of probabilistic voting.∙ OR planner that maximizes expected utility of living in-

dividuals without commitment.



∙ We assume that there is a political incentive to use taxes to transfer

resources to the poor workers.

∙ Extra weight > 0 to workers.

∙ Poor workers have higher marginal utility than entrepreneurs.

dt = �tit−1,

dt+1 = �t+1it,

ct = it−1 (1− �t)ct+1 = it

(1− �t+1

)



∙ Political objective function as the weighted sum of utility of living

generations of entrepreneurs and workers

Wt = W(�t, it, �t+1, it−1

)= it−1 (1− �t) + (1 + ) �tit−1 −

i2t2

+ �it(1− �t+1

)+ � (1 + ) �t+1it.

∙ Equilibrium: Markov equilibria.

∙ Limit of final horizon T when T →∞ (no trigger-strategies eq.)

∙ No young is born in final period T ,

∙ Political objective in T is simply iT−1 (1 + �T ) .

∙ Maximized with �T = 1, implying iT−1 = 0.


Model without loss-aversion (6/6) âß à ê

∙ Key for the analysis is that there is tension between:

+ ex post incentive to tax and the

+ ex ante cost of distorting investments,

... a time inconsistency in taxation.

In the absence of any commitment technology (and in partcular of loss

aversion) the only finite horizon equilibrium feature it = 0 and �t = 1 for

all t. Clearly, this is the only infinite horizon Markov equilibrium that is a

limit of a finite horizon equilibrium.


Taxes under commitment (1/2) ãß à ê

∙ Sequence of tax rates that maximize political welfare if there is full

commitment over the two period planning horizon subject to private

rationality.

∙ Full commitment: �1 and �2 can be set independently.

arg max�1,�2

W (�1, i1, �2, i0)

s.t. i1 = � (1− �2)

∙ i0 is sunk, so:

�1 = 1

�2 =

1 + 2 ≡ �c.

∙ Time inconsistent: New policymaker in t = 2 wopuld set �2 = 1

since i1 is then sunk.


Taxes under commitment (2/2) âß à ê

∙ Restricted commitment: same tax has to be set for all future periods.

∙ Commitment is costly.

∙ To put low taxes tomorrow you need low taxes today... even ifi0 is sunk

∙ Price of future commitment in terms of current payoffs.

∙ Maximizing under the restriction �1 = �2,

�1 = i0 + �2

�2 (1 + 2 ).

∙ If at t = 0 agents did anticipate this:

i0 = �

(1−

i0 + �2

�2 (1 + 2 )

)=⇒ i0 = �2 1 +

� + 2� +

�1 =(1 + �)

� (1 + ) + (1 + �)≡ �f .


The commitment game (1/6) ãß à ê

∙ Commitment game: commitment to a constant tax rate forever in-

cluding the one in the current period can be introduced at any

point in time t ≥ 1.

∙ Tempting to conjecture that �1 = �f is a Markov equilibrium in this

game.

∙ ... this is not the case

∙ t denote the commitment decision in period t,

∙ If t = 1 and t−1 = 0, commitment is introduced in period t

∙ �t+s will be equal to �t for all s ≥ 1.

∙ Requires that t+s = 1 if t = 1 for all s ≥ 1.


The commitment game (2/6) âãß à ê

Definition: A Markov equilibrium is a tax function � (it), a commitmentdecision rule t = (it) applying when t−1 = 0 and a rational investmentrule it = �

(1− Et�t+1

)such that

1. taxes and the commitment decision are set to maximize the politicalpayoff:

{� (it) , (it)} = arg max�t, r

{(1− t)W(�t, it, � (it) , it−1

)+ tW

(�t, it, �t, it−1

)} ,

subject to

2. investments are done individually rationally

it = � (1− ((1− t) � (it) + t�t))

There is no Markov equilibrium with (it) = 1 in the game with an infinitehorizon. That is, introducing the commitment technology for sure is notan equilibrium.



∙ Procrastination Principle

∙ If the next policymaker will commit to a low tax rate, the current

policymaker has no incentive to commit itself: the private agents

know it and will make large investments even if current taxes are

set high.

∙ Also, if the current policymaker knows that the next policymaker

will not commit, then there is an incentive to commit in the

current period and set �t = it+�2

�2(1+2 ), which if anticipated would

result in �t = �f .



∙ Finite horizon.

∙ Last period, no incentive for the policymaker to restrain taxation: �T = 1 if T−1 = 0.

∙ Policymaker in period T−1 knows that the next policymaker will not restrain taxa-tion and this creates an incentive be forward-looking and commit (if commitmenthas not already been introduced).

�T−1 = iT−2 + �2

�2 (1 + 2 )

∙ If agents in period T − 2 expected this, the equilibrium outcome is �T−1 = �f .

∙ Policymaker in period T − 2 now knows that commitment will be introduced inthe next period

∙ No incentive to introduce it and will instead behave myopically.

∙ ...

∙ Policy functions never converge

∙ Oscillate between being myopic and forward-looking



∙ Infinite horizon

∙ Markov equilibrium with pure strategies cannot be sustained.

∙ A mixed strategy equilibrium

∙ commitment is introduced (if it is not already introduced) with a constantprobability p

∘ and taxes are then set to �x.

∙ If commitment is not introduced, � = 1.

∙ Intuition: the more likely it is that the next policymaker will commit, the weakeris the incentive to commit today, and viceversa.

∙ For an intermediate value of p, the current policymaker is indifferent betweencommitting and randomizing with probability p.


The commitment game (6/6) âß à ê

The following is an equilibrium in the commitment game. If t−1 = 0 (no

commitment has yet been introduced)

{�(it−1

),

(it−1

)} =

⎧⎨⎩{�c

(1 +

it−1�2

),1}

with probability p ( , �)

{1,0} otherwise

where

p ( , �) =

⎧⎨⎩�(1+2 )−

√2 �(1+2 )

�−2 (1−�) if ∕= 12

�1−�

12 if = 1

2�

1−�

where we note that lim →0 p ( , �) = 1 and lim →∞ p ( , �) =√�−�

1−� , and

lim →∞ p ( , �) = lim�→1 (lim →∞ p ( , �)) = 12. Along the equilibrium path,

investments are ix ≡ p�2(1+ ) (p+�)+�(1+ ) until commitment to �x = (p+�)

(p+�)+�(1+ )is

achieved when they increase to ixp .


Loss aversion (1/7) ãß ê

∙ Given expectations of next period tax-rate �t+1, an entrepreneur born

in period t solves

Ut = maxct+1,it

−i2t2

+ Et�Ue(ct+1, rt+1, it

)s.t.ct+1 = it

(1− �t+1

)Ue is a loss-averse utility function, depending on ct+1, it and the refer-

ence point rt+1.

ßà êò ê ß ã å 2360

Loss aversion (2/7) âãß ê

Uc

r C

ßà êò ê ß å âã å 2460


Uc

h i

r C



Uc

h i

r C

∙ Discontinuity at r;

Ue (ct+1, rt+1, it) = ct+1−ℎ⋅I (ct+1 < rt+1) it,

∙ ℎ ≥ 0 parameterizes the degree

of loss-aversion. Marginal utility,

when existing, is unity.

∙ The loss associated with being

“cheated” is proportional to pre-

tax income. As investment and

pre-tax return approach zero,

loss associated with “too high”

taxes goes to zero smoothly.



∙ We can smooth it a bit.

Uc

h i

r C



∙ We can smooth it a bit.

Uc

h i

rCro


Loss aversion (7/7) âß ê

∙ ℎ > 0 implies utility is loss-averse risk-neutral. A mean-preserving

spread does not affect expected marginal utility, but all key implications

of loss aversion discussed above remain.

ßà êò ê ß å â 2960

Reference consumption and reference taxes ß ê

∙ It seems reasonable that reference consumption may depend on individ-

ual investments – if an individual invests more, she might feel entitled

to more consumption.

∙ We assume simply

rt+1 = it(1− �rt+1

),

where �rt+1 is a period t determined reference level for �t+1

∙ Implies ct+1 < rt+1 ⇔ �t+1 > �rt+1

ßà êò ê ß 3060

Investments ß ê

∙ Entrepreneurs choose investments at t to maximize utility,

∙ given expectations about taxes at t+ 1

∙ and given the reference level for taxes at t+ 1.

∙ FOC for investment is

it = �Et[1− �t+1 − ℎ ⋅ I

(�t+1 > �rt+1

)].

∙ Note that expectations of becoming “cheated” reduces the marginal

value of investments.

ß êò ê ß 3160

Utility of workers ß ê

∙ It is “good” to give consumption to workers:

Uw(G) = (1 + )G

with > 0

∙ determines the short run benefit of taxation and redistribution, asworkers have a large marginal utility.

∙ To simplify the presentation, we disregard loss-aversion for workers. Ifwe assume instead:

Uw(Gt) = (1 + )Gt − gI(Gt < Grt)

Assuming politicians also promise a level of transfer, reproduces thesame result.

∙ Intuition: there is no ex post political incentive to under-providetransfers.

ß êò ê ß 3260

Tax determination ß à ê

∙ Political objective function (alt. social welfare function)

Wt ≡ Ue(ct, rt, it−1

)+Uw (Gt)−

i2t2

+�Ue(ct+1, rt+1, it

)+�Uw

(Gt+1

),

subject to the resource constraints

Gt = �tit−1,

Gt+1 = �t+1it,

ct = it−1 (1− �t) ,ct+1 = it

(1− �t+1

),

and

it = max{�Et

[1− �t+1 − ℎ ⋅ I

(�t+1 > �rt+1

)],0}

ß êò ê ßà 3360

Equilibrium: (1/2) ãß ê

∙ Optimizing private behavior, taking into account the rational expected

public behavior.

∙ “Optimizing” public behavior, given rational private behavior.

∙ Taking into account that future reference points (and political prefer-

ences) are determined by today’s policies.

ß êò ê ß ã å 3460

Equilibrium: (2/2) âß ê

∙ Backward-looking reference formation:

�rt+1 = �t

∙ A Markov equilibrium is a collection of functions ⟨�, i⟩ , such that �t =

�(it−1, �

rt), and it = i

(�t, �

rt+1

),where � : R+ ⊗ [0,1] → [0,1] and i:

[0,1]⊗ [0,1]→ R+ satisfying simultaneously

1.- � (it−1, � rt ) = arg max�tW(�t, � rt+1, i

(�t, � rt+1

), �(i(�t, � rt+1

), � rt+1

), � rt+1; it−1, � rt

)s.t.

∙ � rt+1 = �t.

2.- i(�t, � rt+1

)= max

{0, �

(1− �t+1 − ℎ ⋅ I

(�t+1 > � rt+1

))}, where

(a).- �t+1 = �(it, � rt+1

)and

(b).- � rt+1 = �t.

ß êò ê ß å â 3560

Backward-Looking Reference Point Dynamics ß ê

∙ We proceed by backward induction.

∙ Last Period (T)

∙ Next to last period (T − 1)

∙ Previous to next to last (T − 2)

∙ Equilibrium characterization

∙ Finite horizon equilibrium: discussion

∙ Infinite horizon

ßà êò ê ß 3660

Last Period (T) (1/5) ã ñ ê

∙ In final period, the reference point is predetermined and the political

objective is

WT = cT − ℎ ⋅ I(�T > �rT

)iT−1 + iT−1�T (1 + )

= iT−1(1 + �T − ℎ

(�T > �rT

)).

∙ If �rT is low, you choose �T = 1

∙ If it is large, you choose �rT .

ßà êò ê ß ã å ñ 3760

Last Period (T) (2/5) âã ñ ê

τR τ

WT

01

ßà êò ê ß å âã å ñ 3860


τR τ

WT

01



τ *τ

WT

01


Last Period (T) (5/5) âñ ê

∙ �∗ is the value of reference point that makes the gov’t indifferent

between “cheating” and put �T = 1 and keep the taxes equal to


∙ �∗ = 1− ℎ

∙ Independent from �, depends only on:

∙ the cost of disappointing per unit of investment (h)

∙ and the benefits ( )

∙ We assume �f < �∗, you can not get your “second best”. Partial

commitment.

∙ Worthwhile to “cheat” people only if �rT is sufficiently low.

∙ Independently of period T − 1 investments.

�T = arg max�T∈[0,1]

WT =

{�rT if �rT ≥ 1− ℎ

loss of “cheating” larger than gain

1 else, gain of “cheating” larger than loss

ßà êò ê ß å â ñ 4160

Next to last period (T − 1) (1/3) ã ñ ê

∙ taxes at T :

�T =

{�T−1 if �T−1 ≥ �∗ The loss of deviating is too large.1 else, The loss of deviating is smaller than the gain.

∙ Knowing this, in period T − 1 individuals choose investment:

iT−1 =

⎧⎨⎩�(1− �T−1

)if �T−1 ≥ �∗ they know that �T = �T−1.

0 else,They know that �T = 1.Tomorrow’s temptation will be too large!

∙ In T −1, a reduction in �T−1 increases iT−1 in the range �T−1 ∈ (�∗,1]

– limited costly commitment.


Next to last period (T − 1) (2/3) âã ñ ê

∙ We can then show that;

�T−1(�T−2

)= �∗ ∀ �T−2, iT−2

∙ The tax is independent from the state variable (�rT−1 = �T−2)!!!

∙ If smaller than �∗, i = 0 as agents anticipate �T = 1. That is bad. You

do not do it.

∙ If larger than �∗, you can do better by reducing taxes (as �f < �∗ is

your preferred once-and-for-all commitment tax, if unconstrained by

loss aversion burden.


Next to last period (T − 1) (3/3) âñ ê

∙ �f < �∗, so W is decreasing ∀� ∈ [�∗,1].∙ Increase in taxes reduces investment.

∙ So, taxes �T−1 not larger than � ∗: �T−1 ≤ � ∗

∙ If �T−1 < �∗, investment iT−1 will be zero, since agents then ratio-nally expect �T = 1.∙ iT−1 (�T−1) is discontinuous at � ∗!!!!!!

∙ If �T−2 was large (�∗ < �T−2), then certainly �T−1 = �∗,∙ There is no loss aversion in reducing taxes to � ∗

∙ and the investment would fall dramatically if you reduce it further.

∙ If �f ≤ �T−2 < �∗.∙ ℎ (the utility loss due to an increase in taxes) is not large enough to prevent

an increase to � ∗.

∙ By increasing the taxes you incur in a loss,

∙ But you also increase investment (it would be zero if �T−1 < � ∗.

∙ the same if �f ≤ �T−2 < � ∗

∙ So it can not be that �T−1 < � ∗.


Previous to next to last (T − 2) ñ ê

∙ We now know that �T−1 is set independently of �T−2, i.e.,

∙ �T−1 is politically forward-looking only.

∙ Since �T−1 is set independently of �T−2, political decisions on �T−2 can

be taken without considerations about the future.

∙ The optimal choice of �T−2 is therefore identical to that of period T .

�T−2 = TT−2(�T−3

)=

{�T−3 if �T−3 ≥ �∗,

1 else,

�T−2 is politically backward-looking .

ßà êò ê ß ñ 4560

Equilibrium characterization ñ ê

∙ Continuing backward, we establish:

The only finite horizon equilibrium features

�T−s =

⎧⎨⎩�e(�T−s−1

)=

{1 if �T−s−1 < �∗

�T−s−1 if �S−s−1 ≥ �∗and s is even.

�o(�T−s−1

)= �∗ if s is odd.

ßà êò ê ß ñ 4660

Finite horizon equilibrium: discussion Þ ñ ê

∙ The equilibrium involves oscillation between forward-looking strategic behavior(the odd strategy) and complete “myopic” behavior, constrained by the previoustax rate. These oscillations are key to the equilibrium existence.

� T, myopic

� T-1, forward-looking

� T-2, myopic


� ...

� If voters and political candidates expect future voters to behave strategically,

limiting next periods taxes in order to constrain later taxes, there is no need to

be strategic already in the current period. Instead, it is better to procastinate,

behaving myopically.

� Conversely, an expectation that future voters will behave myopically, creates an

incentive to act strategically in the current period, despite its short run costs.

� Although tax policies must oscillate in equilibrium, the actual tax-rate does not.

It is constant at 1�

h

� �

�

after the �rst period.

ßà êò ê ß Þ å ñ 4760

Finite horizon equilibrium: discussion ÞÞ ñ ê


∙ T, myopic


� T-2, myopic


� ...









h

� �

�


ßà êò ê ß å ÞÞ å ñ 4760



∙ T, myopic

∙ T-1, forward-looking

� T-2, myopic


� ...









h

� �

�





∙ T, myopic


∙ T-2, myopic


� ...









h

� �

�





∙ T, myopic


∙ T-2, myopic


� ...









h

� �

�





∙ T, myopic


∙ T-2, myopic


∙ ...









h

� �

�





∙ T, myopic


∙ T-2, myopic


∙ ...

∙ If voters and political candidates expect future voters to behave strategically,limiting next periods taxes in order to constrain later taxes, there is no need tobe strategic already in the current period. Instead, it is better to procastinate,behaving myopically.





h

� �

�





∙ T, myopic


∙ T-2, myopic


∙ ...


∙ Conversely, an expectation that future voters will behave myopically, creates anincentive to act strategically in the current period, despite its short run costs.



h

� �

�



Finite horizon equilibrium: discussion Þñ ê


∙ T, myopic


∙ T-2, myopic


∙ ...


∙ Conversely, an expectation that future voters will behave myopically, creates anincentive to act strategically in the current period, despite its short run costs.

∙ Although tax policies must oscillate in equilibrium, the actual tax-rate does not.It is constant at 1− ℎ

≡ � ∗ after the first period.

ßà êò ê ß å Þ ñ 4760

Infinite horizon (1/2) ã ñ ê

∙ First, the finite horizon equilibrium does not converge to a Markov

equilibrium as the horizon is extended backwards to infinity.

∙ The logic – that expectation of future myopia gives incentives for

strategic behavior and vice versa – suggests the existence of a Markov

equilibrium in mixed strategies.

∙ The conjecture is correct.


Infinite horizon (2/2) âñ ê

∙ A Markov equilibrium exists with the following characteristics:

�t = � (�t−1) =

{�e (�t−1) with probability 1− p (�t−1)�o (�t−1) with probability p (�t−1)

,

i (�t) =

{0 if �t < � ∗

� (1− �t + p (�t−1) (�t − � ∗)) if �t ≥ � ∗

with i′ (�t) < 0 ∀�t > � ∗.

∙ Starting from any �0 ∈ [0,1] and i0 = i (�0), the equilibrium tax-rate convergeswith probability 1 to � ∗.

p (� rt ) =

⎧⎨⎩1 for � rt = 1,

p1+1

2

√((2p1)2−4p2(2(p1+ ℎ)−p2))

p2for 1 > � rt > � ∗

< for � < �∗

,

p1 = ( (1 + �) (1− �)− �� (1 + )− ℎ) andp2 = � ( (1− �)− ℎ) (1 + 2 ) .

∙ Starting from any �0 ∈ [0,1] and i0 = i (�0), the equilibrium tax-rate convergeswith probability 1 to � ∗.


Forward-looking loss-aversion ß à ê

∙ We proceed by backward induction.

∙ Forward looking Reference Dynamics

∙ Equilibrium Definition

∙ Last Period (T)

∙ Next to last period (T − 1)

∙ Previous to next to last (T − 2)

∙ Equilibrium characterization

∙ Discussion

ß êò ê ßà 5060

Forward looking Reference Dynamics ñ ê

∙ ... Framing

∙ Like Koszegi and Rabin, we assume that reference points are ra-tional expectations.

∙ We require �rt+1 to be in the set of equilibrium tax rates for t+ 1.

∙ We allow politicians to affect reference points by making “promises”about the future. But remember that the promise is empty – thepolitician does not remain in office nor runs again and he has noformal commitment power.

∙ The promise can affect the future if it is believed, in which case itbecomes the the reference point.

∙ It is believed if it is done by the winning candidate and is in the setof equilibria for next period. If the promise is not an equilibrium,�rt+1 is some element of the set of equilibrium tax rates.

∙ If the promise is not an equilibrium outcome, then it is not believed.In that case they believe taxes will be some (any) credible value.

ß êò ê ßà ñ 5160

Equilibrium Definition (1/2) ã ñ ê

A Markov equilibrium is a collection of functions ⟨�, �p, i⟩ , such that �t =

�(it−1, �

rt), �

pt+1 = �

(it−1, �

rt)

and it = i(�t, �

pt+1

),where � : R+ ⊗ [0,1] →

[0,1] and i: [0,1]⊗ [0,1]→ R+ satisfying simultaneously

1.- � (it−1, � rt ) = arg max�tW(�t, �

pt+1, i

(�t, �

pt+1

), �(i(�t, �

pt+1

), � rt+1

), � rt+1; it−1, � rt

)s.t.

∙ � rt+1 satisfying F.

2.- �p (it−1, � rt ) = arg max� pt+1W(�t, �

pt+1, i

(�t, �

pt+1

), �(it, � rt+1

), � rt+1; it−1, � rt

)s.t.

∙ � rt+1 satisfying F.

3.- i(�t, �

pt+1

)= max

{0, �

(1− �t+1 − ℎ ⋅ I

(�t+1 > � rt+1

))}, where

(a).- �t+1 = �(it, � rt+1

)and

(b).- � rt+1 satisfying F.

ß êò ê ßà ã å ñ 5260

Equilibrium Definition (2/2) âñ ê

∙ 1 and 2 imply that the policy maker explicitly takes into account that

the choice of �t and �pt+1 can affect next periods taxes,

since �t+1 = �(i(�t, �

pt+1

), �rt+1

)

∙ The private choice (3) is taken for a given �t+1 and 3(a) requires that

this value satisfies rational expectations.

ß êò ê ßà å â ñ 5360

Last Period (T) ñ ê

∙ Exactly like before:

∙ �∗ is the value of reference point that makes the gov’t indifferent

between “cheating” and put �T = 1 and keep the taxes equal to


∙ �∗ = 1− ℎ

∙ Independent from �, depends only on:

∙ the cost of disappointing per unit of investment (h)

∙ and the benefits ( )

∙ Worthwhile to “cheat” people only if �rT is sufficiently low.

∙ Independently of period T − 1 investments.

�T = arg max�T∈[0,1]

WT =

{�rT if �rT ≥ 1− ℎ

loss of “cheating” larger than gain

1 else, gain of “cheating” larger than loss


Next to last period (T − 1) (1/2) ã ñ ê

∙ The reference point at T is determined by the promises made at T −1.

Promises at or above �∗ would have been believed, forming the refer-

ence point:

�rT =

{�pT if �pT ≥ �

∗ If the promise can be believed� else. If the promise can not be believed

where � ∈ [�∗,1] is the (out of equilibrium) belief if �pT < �∗

∙ Given period T tax decisions, investments in T − 1 are

iT−1 =

{�(1− �pT

)if �pT ≥ �

∗ the promise is believed

�(1− �) else. the promise is not believed

∙ Notice! They do not depend on the taxes placed at T − 1, only on

the promises made at T − 1 on the taxes at T .

ß êò ê ßà ã å ñ 5560

Next to last period (T − 1) (2/2) âñ ê

∙ In period T − 1 political competition maximizes voter welfare.

∙ The problem of choosing �T−1 is identical to that at T ,

∙ The investment is not affected by it!

�T−1 =

{�rT−1 if �rT−1 ≥ �

∗,1 else.

∙ The “promise” at T − 1 on T that maximizes voter welfare is

�pT = max {�c, �∗}

+ Nothing below �∗ is believed

+ Going below the commitment tax rate is suboptimal.

ß êò ê ßà å â ñ 5660

Previous to next to last (T − 2) ñ ê

The problem is perfectly identical to the one at T − 1.

∙ The reference point for T − 1 depends on if the promise made at

T − 2 is believed...

∙ On whether it is larger than �∗∙ because �T−1 depends only on this.

∙ The investment level depends only on the promise for T − 1, not

on �T−2

∙ The political decision on �T−2 does not affect the investment.

∙ Thus, it depends only on whether �T−2 is larger than the refer-

ence for T − 2 (a state variable).

∙ The political decision on the promise is the max of

∙ �∗ (nothing lower is believed) or

∙ �c (it is the global max, and you want it if it is believable)


Equilibrium characterization ñ ê

∙ There is a unique equilibrium in the finite horizon case.

∙ This equilibrium is also a Markov equilibrium in the infinite horizon

∙ and features

�t = �(�pt

)=

{�pt if�pt ≥ �∗

1 else.

�pt+1 = �p

(�pt

)= max {�c, �∗}

it = i(�pt+1

)=

{�(1− �pt+1

)if �pt+1 ≥ �

∗

� (1− �) else.

for �∗ ≡ 1− ℎ and any � ∈ [�∗,1]

∙ Thus, given any starting[i0, �

p1

], then ∀t > 1 the tax is

�t = max {�c, �∗}


Discussion ñ ê

∙ Equilibrium

∙ Renegotiation proof.

∙ Does not rely on strategies with long memory.

∙ Independent of �, because it hinges on the gain today.

∙ Tomorrow’s things are determined by the promises.

∙ Promises need to be credible.

∙ Equilibrium provides with the equivalent of partial commitment.

∙ With worker loss-aversion, the equilibrium is identical.


What have we learn? Þ ò ê

1. Procrastination Principle:

∙ If there are short run cost of future commitment, It takes timeto achieve commitment in equilibrium.

� Future Extension: shocks and policy

2. Loss Aversion is a natural commitment technology.

� It allows governments to access at least partial commitment, im-

proving outcomes.

3. In the measure that reference point dynamics are forward looking

(framing), fast commitment

4. In the measure that Reference point dynamics are backward look-

ing (experience), slow commitment

ß êò ê Þ å 6060

What have we learn? ÞÞ ò ê



∙ Future Extension: shocks and policy

2. Loss Aversion is a natural commitment technology.

� It allows governments to access at least partial commitment, im-

proving outcomes.





ß êò ê å ÞÞ å 6060





2. Loss Aversion is a natural commitment technology.∙ It allows governments to access at least partial commitment, im-proving outcomes.











3. In the measure that reference point dynamics are forward looking(framing), fast commitment




What have we learn? Þ ò ê





3. In the measure that reference point dynamics are forward looking(framing), fast commitment

4. In the measure that Reference point dynamics are backward look-ing (experience), slow commitment

ß êò ê å Þ 6060

Prepared with SEVISℒℐDℰS

Loss Aversion and

the Dynamics of Political Commitment

John Hassler (IIES)

Jose V. Rodriguez Mora (University of Edinburgh)

June 4, 2010

ê ò ê

http://www.econ.upf.es/~sevimora/

Documents

Loss Aversion and the Dynamics of Political Commitment ò · Œ Dynamic reference point formation (1/2) ªßà Dynamics of reference points not much explored. Kahneman and Tversky