discount rates: the equilibrium approach · 2008-09-25 · the equilibrium price q is an...

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Beliefsheterogeneity

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

How toaggregateexpertsopinions

Impact ofstochasticshifts

Generalizations

Discount rates: the equilibrium approach

Elyès Jouini

PIMS, 2008

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The classical model

A time horizon T and a ltered probability spaceΩ,F , (Ft )t2[0,T ] ,P

Agents indexed by i = 1, ...N,

current endowment at time t denoted by ei

tVNM utility function of the form

EZ T0ui (t, ct (ω)) dt

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The classical model

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The classical model

VNM utility function of the form

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The classical model

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium

∑i=1e

det = µtet dt + σtet dWt e0 = 1

Denition (Arrow-Debreu equilibrium)

A positive price process q and optimal consumption plansy

ii=1,...,N

s.t. markets clear, i.e. ∑Ni=1 y

i = e with

y i= argmax

EhR T0 qt(y itet)dt

i0EZ T

0ui (t, ct ) dt

Characterized by

u0it, y

= λiqt

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The representative agent

Theorem (Negishi)

Let us consider u dened by

u(t, x) = max∑ xi=x

∑ λiui (t, xi ).

The equilibrium price q is an equilibrium price in the economywith 1 agent (representative agent) with an initial wealth e.

The equilibrium is characterized by

u0 (t, et ) = qt .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The riskless asset

We consider an asset with a (riskless) dynamics

dS0t = rt (t,ω)S0t dt

We have S00 = E [qtSt ]

More generally, for B 2 Fs

E [1B (qtSt qsSs )] = 0 (no arbitrage)

qS0 is a martingale and

rt = µq

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The riskless asset

rt = µq

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The riskless asset

rt = µq

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The riskless asset

rt = µq

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Short-term rate

Power utility functions

u(t, c) = expZ t

0ρsds

Short rate

rt = ρ|ztime preference rate

µ|zwealth e¤ect

121η(1+

1η)σ2| z

precautionary saving

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Short-term rate

Power utility functions

u(t, c) = expZ t

0ρsds

Short rate

rt = ρ|ztime preference rate

µ|zwealth e¤ect

121η(1+

1η)σ2| z

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Discount factor and rate

At = E [qt ] (Discount factor)

Rt = 1tlnE [qt ] (Discount rate)

If all the parameters are constant and no risk

R = ρ|ztime preference rate

µ|zwealth e¤ect

(Ramsey)

If σ 6= 0

µ|zwealth e¤ect

121η(1+

1η)σ2| z

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Discount factor and rate

At = E [qt ] (Discount factor)

Rt = 1tlnE [qt ] (Discount rate)

If all the parameters are constant and no risk

µ|zwealth e¤ect

(Ramsey)

If σ 6= 0

µ|zwealth e¤ect

121η(1+

1η)σ2| z

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Beliefs heterogeneity

Agent i maximizes EQihR T0 ui (t, ct (ω)) dt

dQ idP = M

iT and dM

it = δitM

From agent i point of view

det = µitet dt + σtet dW

Q it e0 = 1

µit = µt + δitσt

Divergence of opinion about the growth rate

ui (t, ct (ω)) = D itc1 1

η , with

D it expR t0 ρi (s,ω) ds

(heterogeneous time

preference rates)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

dQ idP = M

iT and dM

it = δitM

Q it e0 = 1

η , with

(heterogeneous time

preference rates)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

dQ idP = M

iT and dM

it = δitM

Q it e0 = 1

η , with

(heterogeneous time

preference rates)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

dQ idP = M

iT and dM

it = δitM

Q it e0 = 1

η , with

(heterogeneous time

preference rates)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Main questions

Representative agent ? (consensus belief, consensus timepreference rate)

Socially e¢ cient discount factor = average of theindividually anticipated ones?

Risk-free rates and discount rates ?

Beliefs dispersion ! additional risk or uncertainty !lower discount rates ?

DDR ? Trajectory of the decline ?

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Main questions

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Main questions

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Main questions

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Main questions

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Declining discount rate

Weitzman (1998) : « To think about the distant future interms of standard discounting is to have an uneasyintuitive feeling that something is wrong somewhere»

DDR in a deterministic setting: known changes in growthrate and/or in risk aversion

DDR with uncertainty

Uncertainty on the discount rate itself (certainty equivalentanalysis, Weitzman, 1998, 2001)Uncertain growth : Gollier (2002a and b, 2007), Weitzman(2004)

Sustainable welfare function à la Chilchinisky (1997) andLi and Löfgren (2000).

Empirical and experimental evidence: individual hyperbolicdiscounters.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Uncertainty on the discount rate itself (certainty equivalentanalysis, Weitzman, 1998, 2001)

Uncertain growth : Gollier (2002a and b, 2007), Weitzman(2004)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium

Denition

Arrow-Debreu equilibrium : a positive price process q and afamily of optimal consumption plans

ii=1,...,N

such that

markets clear, i.e.(y

i= y i

q,M i ,D i , e

∑Ni=1 y

wherey i (q,M,D, e) = argmax

EhR T0 qt(y itet)dt

i0EhR T0 MtDtu (ct ) dt

Characterized by

D itMitu0y

= λiqt

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Aggregation of individual beliefs andtime-preferences

We let N i denote the individual composite characteristic M iD i .

Theorem

We have qt = Ntu0 (et ) with N =

h∑Ni=1 γi

N itηi1/η

Furthermore, N = BDM with

dMt = δMMtdWt , δM =N

∑i=1

τi δi

dBt = ρBBtdt, ρD =N

∑i=1

τiρi

ρB =η 12

∑i=1

δi2 δ2M

Var τ (δ)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Consensus Arrow-Debreu prices and consensussocially e¢ cient discount factors

qiequilibrium price if agent i only

Corollary

We have

∑i=1

η#1/η

At E [qt ], discount factorAit E

, discount factor if agent i only

Can the socially e¢ cient discount factor At be representedas an average of the individual Ait?

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Corollary

We have

∑i=1

η#1/η

At E [qt ], discount factor

Ait Eqit

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Corollary

We have

∑i=1

η#1/η

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Corollary

We have

∑i=1

η#1/η

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Consensus socially e¢ cient discount factors

Theorem

If δi δ and ρi (s,ω) ρi (s) ,

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

equality only when divergence is deterministic (Ni/Nj isdeterministic for all i , j).

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

∑i=1

γiAitη

If η = 1, At = ∑Ni=1 γi

If η 6= 1,

At h∑Ni=1 γi

Aitηi1/η

for η < 1,

At h∑Ni=1 γi

Aitηi1/η

for η > 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Consensus socially e¢ cient discount factorsMain results

The equilibrium approach is compatible with Weitzmansassumption (arithmetic average discount factor) if

1 Logarithmic utility functions2 Each scenario/expertise corresponds to a subjectivediscount factor (di¤erent µ0i s or ρ0i s)

3 Well chosen weights

In general,

1 the right concept of average is the ηaverage2 The average is a weighted average3 A can not be reduced to this average : there is anaggregation bias (upward or downward depending on η)

Beliefs heterogeneity can be interpreted as morerisk/uncertainty or less information : same impact on thetrade-o¤ between todays consumption and futureconsumption (Gollier-Kimball 1996, Gollier, 2000)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

1 Logarithmic utility functions

2 Each scenario/expertise corresponds to a subjectivediscount factor (di¤erent µ0i s or ρ0i s)

In general,

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,1 the right concept of average is the ηaverage

2 The average is a weighted average3 A can not be reduced to this average : there is anaggregation bias (upward or downward depending on η)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,1 the right concept of average is the ηaverage2 The average is a weighted average

3 A can not be reduced to this average : there is anaggregation bias (upward or downward depending on η)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,1 the right concept of average is the ηaverage2 The average is a weighted average3 A can not be reduced to this average : there is anaggregation bias (upward or downward depending on η)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In general,1 the right concept of average is the ηaverage2 The average is a weighted average3 A can not be reduced to this average : there is anaggregation bias (upward or downward depending on η)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Consensus risk-free rates and consensus sociallye¢ cient discount ratesConsensus risk-free rates

Theorem

r f =N

∑i=1

τiρi

| z Agg time pref

+(µ+ δMσ)

η| z Agg exp. wealth

σ2| z stand prec sav

Var τ (δ)| z Agg bias

∑i=1

τir if 1

2(η 1)Var τ (δ)

= r f (stdard) +N

∑i=1

τiρi +

∑i=1

τi δi

!σ η 1

2Var τ (δ)

Average level (patience, pessimism)

CorrelationBeliefs dispersion (depends on η > 1, η < 1). For η > 1 :more risk ) more saving ) downward pressure on r f

Rt = 1t log E

R t0 r

fs dsi. Intuitively, the

properties of r f should be inherited by R.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

r f =N

∑i=1

τiρi

| z Agg time pref

+(µ+ δMσ)

∑i=1

τir if 1

2(η 1)Var τ (δ)

= r f (stdard) +N

∑i=1

τiρi +

∑i=1

τi δi

!σ η 1

2Var τ (δ)

Average level (patience, pessimism)Correlation

Beliefs dispersion (depends on η > 1, η < 1). For η > 1 :more risk ) more saving ) downward pressure on r f

Rt = 1t log E

R t0 r

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

r f =N

∑i=1

τiρi

| z Agg time pref

+(µ+ δMσ)

∑i=1

τir if 1

2(η 1)Var τ (δ)

= r f (stdard) +N

∑i=1

τiρi +

∑i=1

τi δi

!σ η 1

2Var τ (δ)

Average level (patience, pessimism)CorrelationBeliefs dispersion (depends on η > 1, η < 1). For η > 1 :more risk ) more saving ) downward pressure on r f

Rt = 1t log E

R t0 r

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Theorem

r f =N

∑i=1

τiρi

| z Agg time pref

+(µ+ δMσ)

∑i=1

τir if 1

2(η 1)Var τ (δ)

= r f (stdard) +N

∑i=1

τiρi +

∑i=1

τi δi

!σ η 1

2Var τ (δ)

Average level (patience, pessimism)CorrelationBeliefs dispersion (depends on η > 1, η < 1). For η > 1 :more risk ) more saving ) downward pressure on r f

Rt = 1t log E

R t0 r

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Long term considerations

TheoremSuppose that for all i , the individual asymptotic discount rateR i∞ lim

tT ;t ,T!∞RT ,it exists. Moreover, we suppose

γI (T ) ε > 0 for R I∞ = infR i∞; i = 1, ...N

). Then,

R∞ limtT ;t ,T!∞

RTt = infR i∞, i = 1, ...,N

The aggregation bias vanishes in the long run

The relevant asymptotic behavior is the one with thelowest discount rate

Homogeneous beliefs : lowest rate of impatience(Gollier-Zeckhauser in a deterministic setting)Homogeneous time preference rate : most pessimistic rate

γI (T ) ε > 0

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

). Then,

RTt = infR i∞, i = 1, ...,N

The aggregation bias vanishes in the long runThe relevant asymptotic behavior is the one with thelowest discount rate

γI (T ) ε > 0

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

). Then,

RTt = infR i∞, i = 1, ...,N

Homogeneous beliefs : lowest rate of impatience(Gollier-Zeckhauser in a deterministic setting)

Homogeneous time preference rate : most pessimistic rate

γI (T ) ε > 0

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

). Then,

RTt = infR i∞, i = 1, ...,N

γI (T ) ε > 0

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

). Then,

RTt = infR i∞, i = 1, ...,N

γI (T ) ε > 0

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic settingsLogarithmic case

At = ∑i γiAit (arithmetic average)

RTt = µ σ2 1tlog

γTi exp(ρi+σδi)t

#R0 = µ σ2 +∑

ρi + σδi

R∞ = µ σ2 + inf

ρi + σδi

γTi =wiρi

1 expρiT

1∑j wjρj (1 expρjT )1

R0 R∞ and RTt decreases with t

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

RTt = µ σ2 1tlog

#R0 = µ σ2 +∑

ρi + σδi

R∞ = µ σ2 + inf

ρi + σδi

γTi =wiρi

1 expρiT

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

RTt = µ σ2 1tlog

#R0 = µ σ2 +∑

ρi + σδi

R∞ = µ σ2 + inf

ρi + σδi

γTi =wiρi

1 expρiT

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

10007505002500

0.01625

0.01375

0.0125

0.01125

1 0 0 07 5 05 0 02 5 00

0 .0 2

0 .0 1 7 5

0 .0 1 5

0 .0 1 2 5

w1= w2, δ1= δ2 cov(w , δ) > 0di¤erent levels of δ Rt% with δ for small tRt& with δ, pessim. limit Rt& with δ for large tsame starting point 6= starting point

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic settingsPower utility functions, eta<1

The equilibrium discount rates dominates the averages

The ηaverage is a better approx., the distance is due tobeliefs disp. and this e¤ect may last for centuriesThe three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium discount rates dominates the averagesThe ηaverage is a better approx., the distance is due tobeliefs disp. and this e¤ect may last for centuries

The three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium discount rates dominates the averagesThe ηaverage is a better approx., the distance is due tobeliefs disp. and this e¤ect may last for centuriesThe three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic settingsPower utility functions, eta>1

The equilibrium discount rates is below the averages

The ηaverage is still a better approximationThe three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium discount rates is below the averagesThe ηaverage is still a better approximation

The three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The equilibrium discount rates is below the averagesThe ηaverage is still a better approximationThe three curves converge to the lowest discount rate

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Divergence of expertsopinions

Let us consider n experts :R ii=1,...,n as in Weitzman (2001)

N groups of agents,

wi relative size of group i ,ρi pure time preference rate of the agents in group i ,

t the time at which a cost or benet is incurred,

et lnN ((µi 12σ2i )t, σ

2i t),

log utility functions

R i ρi + µi σ2i , equilibrium discount rate if theeconomy was made of group i agents only

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

wi relative size of group i ,

ρi pure time preference rate of the agents in group i ,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

N groups of agents,

2i t),

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Rt 1tln

∑i=1

wiρi∑Nj=1 wjρj

expR i t,

∑i=1

wiρi expr i t

∑Nj=1 wjρj exp (r j t)

the consensus discount rates are averages of the individualrates (as in Weitzman 1998)

weighted averages, weights proportional to the pure timepreference rates,

bias towards the more impatient agents

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Rt 1tln

∑i=1

wiρi∑Nj=1 wjρj

expR i t,

∑i=1

wiρi expr i t

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Rt 1tln

∑i=1

wiρi∑Nj=1 wjρj

expR i t,

∑i=1

wiρi expr i t

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

In the case of homogeneous beliefs (µi = µ, σi = σ)

∑i=1

wiρi exp (ρi t)

∑Nj=1 wjρj exp

ρi + µ σ2.

the expression involves the covariance between ρi andexpρi t as in Lengwiler (2005)

it gives the expression for the consensus utility discountratedi¤erent from the one obtained by Gollier (2005) orNocetti (2008)our weights are given by wiρi expρi t instead ofλi expρi tboth approaches coincide if the Pareto weights areproportional to wiρi .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

∑i=1

wiρi exp (ρi t)

∑Nj=1 wjρj exp

ρi + µ σ2.

the expression involves the covariance between ρi andexpρi t as in Lengwiler (2005)it gives the expression for the consensus utility discountrate

di¤erent from the one obtained by Gollier (2005) orNocetti (2008)our weights are given by wiρi expρi t instead ofλi expρi tboth approaches coincide if the Pareto weights areproportional to wiρi .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

∑i=1

wiρi exp (ρi t)

∑Nj=1 wjρj exp

ρi + µ σ2.

the expression involves the covariance between ρi andexpρi t as in Lengwiler (2005)it gives the expression for the consensus utility discountratedi¤erent from the one obtained by Gollier (2005) orNocetti (2008)

our weights are given by wiρi expρi t instead ofλi expρi tboth approaches coincide if the Pareto weights areproportional to wiρi .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

∑i=1

wiρi exp (ρi t)

∑Nj=1 wjρj exp

ρi + µ σ2.

the expression involves the covariance between ρi andexpρi t as in Lengwiler (2005)it gives the expression for the consensus utility discountratedi¤erent from the one obtained by Gollier (2005) orNocetti (2008)our weights are given by wiρi expρi t instead ofλi expρi t

both approaches coincide if the Pareto weights areproportional to wiρi .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

∑i=1

wiρi exp (ρi t)

∑Nj=1 wjρj exp

ρi + µ σ2.

the expression involves the covariance between ρi andexpρi t as in Lengwiler (2005)it gives the expression for the consensus utility discountratedi¤erent from the one obtained by Gollier (2005) orNocetti (2008)our weights are given by wiρi expρi t instead ofλi expρi tboth approaches coincide if the Pareto weights areproportional to wiρi .

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

∑i=1

wiρi∑Nj=1 wjρj

∑i=1

wiρi∑Nj=1 wjρj

If ρi and bi are independent,

∑i=1

wi expr i t∑Nj=1 wj expr j t

r i = rWt

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

General properties and specic distributions

Corollary1 Rt and rt decrease with t,

2 The asymptotic equilibrium discount rates are given by thelowest individual discount rate, i.e.R∞ = r∞ = inf i r i = inf i R i .

3 ρi = ρ, and normal distribution Nm, v2

bi = µi σ2i , Rt = ρ+m v 22 t (Reinschmidt, 2002)

4 If ρi γ(α1, β1) and bi = µi σ2i γ(α2, β2)independent, then rt = α1+1

β1+t+ α2

5 As in 4. and β1 = β2 = β then R i γ(α, β) withα = α1 + α2 and rt = rWt + 1

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Corollary1 Rt and rt decrease with t,2 The asymptotic equilibrium discount rates are given by thelowest individual discount rate, i.e.R∞ = r∞ = inf i r i = inf i R i .

β1+t+ α2

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

β1+t+ α2

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

β1+t+ α2

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

β1+t+ α2

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Figure: Calibration with two independent gamma distr. on Weitzman(2001)s data. We assume that the two distributions are homotheticand calibrate in order to t the mean and the variance of theempirical distribution. Weitzman (2001)s statistical modelcorresponds to λ = 1. We maximize the log-likelihood and obtainλ = 0.4116.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

5003752501250

Figure: Marginal discount rate curve through our calibration (uppercurve) and discount rate curve of Weitzman (2001) (lower curve).The intermediate curve represents, with our calibration, theunweighted average.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The gamma distribution case

a decrease in the mean m2 or an increase in the variancev22 of the individual beliefs (bi ) decreases the marginaldiscount rate rt

same result with a decrease in the mean m1 of theindividual pure time preference rates (ρi ).

an increase in the variance v21 of the individual pure timepreference rates (ρi ) decreases the marginal discount ratert for t large enough.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

a decrease in the mean m2 or an increase in the variancev22 of the individual beliefs (bi ) decreases the marginaldiscount rate rtsame result with a decrease in the mean m1 of theindividual pure time preference rates (ρi ).

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

a decrease in the mean m2 or an increase in the variancev22 of the individual beliefs (bi ) decreases the marginaldiscount rate rtsame result with a decrease in the mean m1 of theindividual pure time preference rates (ρi ).

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The general case

1 If all the agents have the same ρi , then a FSD (resp.SSD) shift in the distribution of

R iincreases the

discount rate Rt for all horizons.

2 If all the agents have the same ρi , then a MLR shift in thedistribution of the

r iincreases the marginal discount

rate rt for all horizons.3 If all the agents have the same beliefs, then a MLR shift inthe distribution of the

R iincreases the discount rate Rt

for all horizons.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The general case

R iincreases the

discount rate Rt for all horizons.2 If all the agents have the same ρi , then a MLR shift in thedistribution of the

rate rt for all horizons.

3 If all the agents have the same beliefs, then a MLR shift inthe distribution of the

for all horizons.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

The general case

R iincreases the

discount rate Rt for all horizons.2 If all the agents have the same ρi , then a MLR shift in thedistribution of the

rate rt for all horizons.3 If all the agents have the same beliefs, then a MLR shift inthe distribution of the

for all horizons.

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Measured set of agents

Agent i has a probability measure Q it that represents thedistribution of datet aggregate consumption

Agent i has a pure time preference rate ρi , a share of totalwealth wi and a log-utility

Rt 1tlnZ wiρiR

wjρjdν(j)exp

R it t

dν(i)

where R it is the equilibrium discount rate that wouldprevail if the economy was made of agent i only

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Agent i has a probability measure Q it that represents thedistribution of datet aggregate consumptionAgent i has a pure time preference rate ρi , a share of totalwealth wi and a log-utility

Rt 1tlnZ wiρiR

wjρjdν(j)exp

R it t

dν(i)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Agent i has a probability measure Q it that represents thedistribution of datet aggregate consumptionAgent i has a pure time preference rate ρi , a share of totalwealth wi and a log-utility

Rt 1tlnZ wiρiR

wjρjdν(j)exp

R it t

dν(i)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Take-home messages

The socially discount factor is not, in general, anarithmetic average of the individually anticipated ones

ηaverageweightsbias related to beliefs and time preference rates dispersion

The arithmetic average corresponds to a utility maximizingagent that considers each individual belief as a possiblescenario while our approach corresponds to a centralplanner that maximizes the social welfare

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Take-home messages

ηaverage

weightsbias related to beliefs and time preference rates dispersion

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Take-home messages

ηaverageweights

bias related to beliefs and time preference rates dispersion

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Take-home messages

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Take-home messages

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Logarithmic case: weighted arithmetic average (à laWeitzman)Deterministic divergence (Ni/Nj deterministic): weightedηaverage (Gollier-Zeckhauser)

Aggregate pessimism and patience reduces R

Beliefs dispersion reduces R for η > 1

Long term rate: lowest discount rate

Medium term: increasing as well as decreasing yield curves

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Logarithmic case: weighted arithmetic average (à laWeitzman)

Deterministic divergence (Ni/Nj deterministic): weightedηaverage (Gollier-Zeckhauser)

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

Equilibriumapproach

Elyès Jouini

Introduction

The classicalmodel

Aggregation

Discountfactors

Discount rates

Long term

Specicsettings

Generalizations

Specic cases

discount rates: the equilibrium approach · 2008-09-25 · the equilibrium price q is an...

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