A Dynamic Model of Altruistically-Motivated Transfersmkredler/ReadGr/VeigaOnBarczykKredler14.pdfA Dynamic Model of Altruistically-Motivated Transfers Barczyk,Daniel Kredler, Matthias

A Dynamic Model of Altruistically-Motivated Transfers

Barczyk,Daniel Kredler, Matthias

Review of Economic Dynamics 2014

Presented by Ruben Veiga

October 2016

Kredler-Barczyk (RED 2014) ( Review of Economic Dynamics 2014 Presented by Ruben Veiga )Altruistically-Motivated Transfers October 2016 1 / 24

Outline

1 Introduction

2 The modelEnvironmentPiecewise-smooth profilesEquilibrium

3 Understanding players’ incentivesInstantaneous best response functionsOver consumptionPareto allocationsStrategic interactions

4 Exploiting homogeneitySpecial casesRegionsTragedy of the commons equilibriumOther resuts


Outline

1 Introduction





Introduction

Motivation: provide a tractable theory for behavior ofimperfectly-altruistic agents in a dynamic setting withoutcommitment.

Model:Simplest setting with two altruistic agentsEquilibrium concept: Markov Perfect Equilibrium.

Results/Contributions:Understanding of the incentives in a dynamic setting with savings andtransfers.Characterization of a continuum of equilibria in which:1. Poor agent get transfers out of poverty.2. Then, tragedy of the commons occurs with defined property rights.With uncertainty: Samaritan’s dilemma vs Prodigal-Son dilemma.


Outline

1 Introduction





Environment

Setting:

Continuous timeTwo agents: she (ct , gt , kt) and he (c ′t , g

′t , k′t). They consume, transfer

and save.Riskless asset k ≥ 0 with return r .

Gifts gt = (gj ,t , gm,t):

Flow transfers: gf ,t∆t.Mass transfers: gm,t ∈ [0, k].

Timing:

t: Transfers g ′t , gt are chosen simultaneously.t + ∆t: Consumption c ′t , ct are chosen simultaneously.

Realized consumption: a broke player cannot give transfers:

c∗t =

{ct if kt > 0 or g ′m,t > 0

min{ct , g ′f ,t} otherwise


Environment

Setting:






Timing:



c∗t =




Environment

Setting:






Timing:



c∗t =




Environment

Setting:






Timing:



c∗t =




Law of motions of the states (k , k ′):Flow: {

kt = rkt − c∗t − gf ,t + g ′f ,tk ′t = rk ′t − c ′∗t − g ′f ,t + gf ,t

; xt = (kt , k′t)

Mass transfer: limh→0

kt+h = kt + gm,t

Preferences {v0 =

∫∞0 e−ρt [u(c∗t ) + αu(c ′∗t )]dt

v ′0 =∫∞

0 e−ρt [u(c ′∗t ) + α′u(c∗t )]dt


Law of motions of the states (k , k ′):Flow: {

kt = rkt − c∗t − gf ,t + g ′f ,tk ′t = rk ′t − c ′∗t − g ′f ,t + gf ,t

; xt = (kt , k′t)

Mass transfer: limh→0

kt+h = kt + gm,t

Preferences {v0 =

∫∞0 e−ρt [u(c∗t ) + αu(c ′∗t )]dt

v ′0 =∫∞

0 e−ρt [u(c ′∗t ) + α′u(c∗t )]dt


Piecewise-smooth profiles ({v , c , g ; v ′, c ′, g ′})

Mass transfer regions are such that if one agent gives one, thistransfer brings us to the frontier of the region.

Rest of policy and value functions are continuously differentiable ineach region (may be discontinuous at frontiers).


Recursive formulationTransfer stage (t) (she):

v(xt) = Maxg≥0,g=0 if k=0

{v c(xt ; g , g′(xt))}

Consumption stage (t + ∆t) (she takes his strategy c ′0 as given):

v c(xt ; g , g′) = Max

c≥0{u(c∗(c , k , g ′))∆t + αu(c ′∗(c ′0(xt ; g , g

′), k ′, g))∆t

+ e−ρ∆tv(xt+∆t)}

s.t.. xt+∆t =

{kt − gm + g ′m + [rkt − c − gf + g ′f ]∆t

k ′t + gm − g ′m + [rk ′t − c ′(xt ; g , g′) + gf − g ′f ]∆t

In equilibrium c(x) = c ′0(x ; g(x), g ′(x))

Definition ((piecewise-smooth)Markov Perfect Equilibrium)

A (p-s)MPE is a piecewise-smooth profile {v , c , g ; v ′, c ′, g ′}) andconsumption-stage strategies {c0, c ′0} satisfy mutual best responding:

{v ; c , c0, g} is a best response to {c ′0, g ′}.{v ′; c ′, c ′0, g ′} is a best response to {c0, g}. Full


Recursive formulationTransfer stage (t) (she):

v(xt) = Maxg≥0,g=0 if k=0

{v c(xt ; g , g′(xt))}

Consumption stage (t + ∆t) (she takes his strategy c ′0 as given):

v c(xt ; g , g′) = Max

c≥0{u(c∗(c , k , g ′))∆t + αu(c ′∗(c ′0(xt ; g , g

′), k ′, g))∆t

+ e−ρ∆tv(xt+∆t)}

s.t.. xt+∆t =

{kt − gm + g ′m + [rkt − c − gf + g ′f ]∆t

k ′t + gm − g ′m + [rk ′t − c ′(xt ; g , g′) + gf − g ′f ]∆t

In equilibrium c(x) = c ′0(x ; g(x), g ′(x))

Definition ((piecewise-smooth)Markov Perfect Equilibrium)

A (p-s)MPE is a piecewise-smooth profile {v , c , g ; v ′, c ′, g ′}) andconsumption-stage strategies {c0, c ′0} satisfy mutual best responding:

{v ; c , c0, g} is a best response to {c ′0, g ′}.{v ′; c ′, c ′0, g ′} is a best response to {c0, g}. Full


Outline

1 Introduction





Instantaneous best response functions

No mass transfers:

ρv =αu(c ′) + (rk + g ′)vk + (rk ′ − g ′ − c ′)vk ′+

+ Maxg≥0{g [vk ′ − vk ]︸︷︷︸

µ

}+ Maxc≥0{u(c)− cvk} (1)

Continuous time simplifies the problem:

His c ′ does not affect her choice c contemporaneously. (Inmediatestrategic considerations are 2nd order).No interaction between consumption and transfer decisions problems.

Consumption FOC: uc(c) = vk

Transfer motive µ is the marginal benefit of transferring a marginalunit to him.

If µ < 0 she does not transfer.If µ = 0 she is Ok with any flow-transfer.If µ > 0 she she would like to make a mass-transfer.(does not happen)


Instantaneous best response functions

No mass transfers:

ρv =αu(c ′) + (rk + g ′)vk + (rk ′ − g ′ − c ′)vk ′+

+ Maxg≥0{g [vk ′ − vk ]︸︷︷︸

µ

}+ Maxc≥0{u(c)− cvk} (1)

Continuous time simplifies the problem:

His c ′ does not affect her choice c contemporaneously. (Inmediatestrategic considerations are 2nd order).No interaction between consumption and transfer decisions problems.

Consumption FOC: uc(c) = vkTransfer motive µ is the marginal benefit of transferring a marginalunit to him.

If µ < 0 she does not transfer.If µ = 0 she is Ok with any flow-transfer.If µ > 0 she she would like to make a mass-transfer.(does not happen)


Over consumption

How she feels about (c, c ′) in interval ∆t:

v(xt) = v(xt)−ρv∆t+Maxc{u(c)∆t + αu(c ′)∆t + vk k∆t + vk ′ k ′∆t}︸︷︷︸

H(c,c ′)

+o(∆t)

Do not internalize the effects of their actions in the other.

In this case they would be better off reducing their consumption.


Pareto allocations

Benevolent planner that weight her with η and manages the assets.

Standard savings problem:

duc(ct)

dt= (ρ− r)uc(ct)

Divides resources according to:

ct = P∗ρKt ; c ′t = (1− P∗)ρKt ; P∗ =η + (1− η)α

1 + ηα + (1− η)α′


Strategic interactions (no mass transfers)

duc(ct)

dt= (ρ− r)uc(c)︸︷︷︸

efficient

+ [vk ′ − αuc(c ′)]c ′k︸︷︷︸altruistic-strategic dist.

+ [vk ′ − uc(c)]g ′k︸︷︷︸transfer-induced incentives

(2)

(∂c′2∂k2

> 0,∂g′2∂k2

> 0)

No distortions with selfish or perfect altruism.


Strategic interactions (no mass transfers)

duc(ct)

dt= (ρ− r)uc(c)︸︷︷︸

efficient

+ [vk ′ − αuc(c ′)]c ′k︸︷︷︸altruistic-strategic dist.

+ [vk ′ − uc(c)]g ′k︸︷︷︸transfer-induced incentives

(2)

(∂c′2∂k2

> 0,∂g′2∂k2

> 0)

No distortions with selfish or perfect altruism.Kredler-Barczyk (RED 2014) ( Review of Economic Dynamics 2014 Presented by Ruben Veiga )Altruistically-Motivated Transfers October 2016 14 / 24

Outline

1 Introduction





Homogeneity

Homothetic utilities and income proportional to assets.

Change of variables

K = k + k ′ ; P =k

K

Homogeneous strategies:

c(k, k ′) = C (P)K ; gf (k, k ′) = Gf (P)K ; gm(k , k ′) = Gm(P)K ;

c0(k , k ′; g , g ′) = C 0(P; g/K , g ′/K )K

Analogous MPE definition.

Proposition (V monotone)

In any homogeneous MPE, V (P) is weakly increasing in P.

To find equilibrium we look at some special cases and characterize thedifferent possible regions (now unidimensional!).


Homogeneity


Change of variables

K = k + k ′ ; P =k

K



c0(k , k ′; g , g ′) = C 0(P; g/K , g ′/K )K






Homogeneity


Change of variables

K = k + k ′ ; P =k

K



c0(k , k ′; g , g ′) = C 0(P; g/K , g ′/K )K






Homogeneity


Change of variables

K = k + k ′ ; P =k

K



c0(k , k ′; g , g ′) = C 0(P; g/K , g ′/K )K






Homogeneity


Change of variables

K = k + k ′ ; P =k

K



c0(k , k ′; g , g ′) = C 0(P; g/K , g ′/K )K






Special casesSelf-sufficient equilibrium

No transfers: G = G ′ = 0.


CSS(P) = ρP ; C ′SS(P) = ρ(1− P) (3)

Proposition

1 SS strategies can be sustained in equilibrium only if α = α′ = 0.

2 If α = α′ = 0, then SS strategies are the only equilibrium and itsefficient.

Why not with α = α′ > 0 and P = 0.5. No subgame perfect! e.gP → 0.


Special casesSelf-sufficient equilibrium

No transfers: G = G ′ = 0.


CSS(P) = ρP ; C ′SS(P) = ρ(1− P) (3)

Proposition

1 SS strategies can be sustained in equilibrium only if α = α′ = 0.

2 If α = α′ = 0, then SS strategies are the only equilibrium and itsefficient.

Why not with α = α′ > 0 and P = 0.5. No subgame perfect! e.gP → 0.


Special casesWealth-pooling equilibrium

Remove restrictions G ≥ 0. Share resources.

Equilibrium policies (tragedy of the commons):

CWP(P) =ρ

1 + α; C ′WP(P) =

ρ

1 + α′(4)

Proposition

1 WP strategies can be sustained in MPE only if α = α′ = 1.2 If α = α′ = 1, then WP strategies are the only equilibrium. Besides:

Transfers undetermined for P ∈ (0, 1).Efficient.

Why not with α = α′ ∈ (0, 1) and P = 0.5. No subgame perfect! e.gP → 1. She becomes family-dictator.


Special casesWealth-pooling equilibrium

Remove restrictions G ≥ 0. Share resources.

Equilibrium policies (tragedy of the commons):

CWP(P) =ρ

1 + α; C ′WP(P) =

ρ

1 + α′(4)

Proposition

1 WP strategies can be sustained in MPE only if α = α′ = 1.2 If α = α′ = 1, then WP strategies are the only equilibrium. Besides:

Transfers undetermined for P ∈ (0, 1).Efficient.

Why not with α = α′ ∈ (0, 1) and P = 0.5. No subgame perfect! e.gP → 1. She becomes family-dictator.


Characterization of the regions

NT (No-transfer regions)Dominated by altruistic-strategic distortions.

SS (Self-sufficient)

FT (Flow-transfer regions)Dominated by transfer-induced incentives.

WP (Wealth-pooling)

MT (Mass-transfer regions)Left immediately. Once out, recipient stays at WP region.


Tragedy of the commons equilibrium

Iff α, α′ > 0, ∃ a continuum of homogeneous MPE s.th:1 He gives transfers in PFT ′ = [0,P1). She gives transfers inPFT = (P1, 1].

2 WP region PWP = (P1,P2). Absorbent. Transfer undetermined.

Double multiplicity (transfers and (P1,P2)).Kredler-Barczyk (RED 2014) ( Review of Economic Dynamics 2014 Presented by Ruben Veiga )Altruistically-Motivated Transfers October 2016 20 / 24

Other results

Why not and eq. in which donor delays transfers until recipient isbroke?. → Two steady states. → Both over consume.

With shock on income:

Theorem (Party Theorem)

There exist an equilibrium in which only constrained recipients receivetransfers.

When P = 0 she is broke (absorbent) and he choose preferred allocation.

When reaching P = 0 her consumption jumps down.(Samaritan’s dilemma)

Theorem (Prodigal Son dilemma: no MT when broke)

There cannot be an eq. in which a MT goes to broke player.

Unless α = α′ = 1 or father can commit.

Father cannot commit to not provide transfers after the initial MT. The sonwould consume the transfer and come back for more.


Other results

Why not and eq. in which donor delays transfers until recipient isbroke?. → Two steady states. → Both over consume.

With shock on income:

Theorem (Party Theorem)

There exist an equilibrium in which only constrained recipients receivetransfers.

When P = 0 she is broke (absorbent) and he choose preferred allocation.

When reaching P = 0 her consumption jumps down.(Samaritan’s dilemma)

Theorem (Prodigal Son dilemma: no MT when broke)

There cannot be an eq. in which a MT goes to broke player.

Unless α = α′ = 1 or father can commit.

Father cannot commit to not provide transfers after the initial MT. The sonwould consume the transfer and come back for more.


Conclusions

Usual caveat: only MPE, history dependent equilibria? commitmentmechanisms available?

The model provides a rationale for family behaving as a perfectlyaltruistic dynasty less patient than individuals.

Relation between altruism and tragedy of the commons.

Provide new insight on Prodigal Son and Samaritan dilemma beyondtwo period models.


Order 0:

Order 1 for g

Order 1 for c



Documents

A Dynamic Model of Altruistically-Motivated Transfersmkredler/ReadGr/VeigaOnBarczykKredler14.pdfA Dynamic Model of Altruistically-Motivated Transfers Barczyk,Daniel Kredler, Matthias