Fiat Money as a Public Signal, Medium of Exchange, and Punishment

Pedro Gomis-Porqueras; Ching-Jen Sun

doi:10.1515/bejte-2019-0098

Published by De Gruyter February 26, 2020

Fiat Money as a Public Signal, Medium of Exchange, and Punishment

Pedro Gomis-Porqueras and Ching-Jen Sun

From the journal The B.E. Journal of Theoretical Economics

https://doi.org/10.1515/bejte-2019-0098

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Abstract

This paper studies different welfare-enhancing roles that fiat money can have. To do so, we consider an indivisible monetary framework where agents are randomly and bilaterally matched, while the government has weak enforcement powers. Within this environment, we analyze state contingent monetary policies and characterize the resulting equilibria under different government record-keeping technologies. We show that a threat of injecting fiat money, conditional on private actions, can improve allocations and achieve efficiency. This type of state contingent policy is effective even when the government cannot observe any private trades and agents can only communicate with the government through cheap talk. In all these equilibria fiat money and self-enforcing credit are complements in the off equilibrium. Finally, this type of equilibria can also emerge even when the injection of fiat money is not a public signal.

Keywords: cheap talk; record-keeping; fiat money

JEL Classification: E00; E40; C73; D82

Acknowledgements

We would like to thank the Editor and the anonymous referees for their suggestions and feedback. We also like to thank Luis Araujo, Andrei Shevchenko, Aleks Berentsen, Sebastian Lotz, Fernando Martin, Raoul Minetti, Cyril Monnet and Chris Waller for their comments.

Appendix

A Proof of Proposition 1

First we observe that on the equilibrium path (where agents follow unconditional gift-giving), the expected payoff for each agent is

VG=σ+δ1−β(u−c).

Once the government sees a deviation, M units of money are randomly injected into the economy. Let VM0 denote the expected payoff of an agent without money and VM1 the expected payoff of an agent with money. Following Kiyotaki and Wright (1989), we have

VM0=βVM0+σMN−1[−c+β(VM1−VM0)]+δ(u−c)=σMN−1[−c+β(VM1−VM0)]+δ(u−c)1−β,and

VM1=βVM1+σN−MN−1[u+β(VM0−VM1)]+δ(u−c)=σN−MN−1[u+β(VM0−VM1)]+δ(u−c)1−β.

The expected payoff for an agent who deviates is given by

Vd=MNVM1+N−MNVM0=MNσN−MN−1[u+β(VM0−VM1)]+δ(u−c)1−β+N−MNσMN−1[−c+β(VM1−VM0)]+δ(u−c)1−β=σM(N−M)N(N−1)+δ1−β(u−c).

Hence no agent will deviate from gift-giving if −c+βVG≥βVd, which implies that the discount rate has to satisfy

β≥cc+σ(1−M(N−M)N(N−1))(u−c).

On the other hand, after a deviation, in order for agents to follow a monetary equilibrium, we require β≥βM≡cc+σN−MN−1(u−c) as derived in Kiyotaki and Wright (1989). The two inequalities hold simultaneously if

β≥cc+σmin{1−M(N−M)N(N−1),N−MN−1}(u−c)=βPO.

B Proof of Proposition 2

In order to show that the strategy profile supports gift exchange as an equilibrium, we need to establish the following two conditions:

I. No agent has an incentive to deviate on the equilibrium path

There are two conditions to be established:

(i) As agents follow unconditional gift-giving on the equilibrium path, the expected payoff for each agent is

VG=σ+δ1−β(u−c).

Pick any matched pair (i, m_t(i)) in period t, and suppose (without loss of generality) that m_t(i) likes i^'s good. If agent i follows the equilibrium path and produces for m_t(i), his expected payoff is

−c+βVG.

If i deviates from the equilibrium path and does not produce for m_t(i), both i and m_t(i) will report to the central bank. The central bank then hands out one unit of money to both i and m_t(i). Following the monetary equilibrium after deviation, i^'s expected payoff is VM=21. Accordingly, i has no incentive to deviate whenever we satisfy

−c+βVG≥βVM=21.

First we derive VM=21 explicitly. Again, following Kiyotaki and Wright (1989), VM=20 and VM=21 satisfy the following system of equations:

[1−β+2σβN−1−σ2N−1β−σβN−2N−11−β+σβN−2N−1][VM=20VM=21]=[δ(u−c)−σ2N−1cδ(u−c)+σN−2N−1u]

Accordingly, VM=21 can be solved as

VM,2=(1−β+2σβN−1)(δ(u−c)+σN−2N−1u)+σβN−2N−1(δ(u−c)−σ2N−1c)(1−β+2σβN−1)(1−β+σβN−2N−1)−σ2β22(N−2)(N−1)2=(u−c)[δ(1−β+σβNN−1)+2σ2β(N−2)(N−1)2]+(1−β)σuN−2N−1(1−β)(1−β+σβNN−1)

Hence the no-deviation condition becomes

−c+βσ1−β(u−c)≥β1−β(u−c)2σ2β(N−2)(N−1)2+(1−β)σuN−2N−11−β+σβNN−1,

which can be rearranged as:

(u−c)βσ−(1−β)cβσ−N−2N−1(u−c)2σβN−1+(1−β)u1−β+σβNN−1≥0.

This inequality holds if

(u−c)βσ−(1−β)cβσ−(u−c)2σβN−1+(1−β)u1−β+σβ≥0.

After some simplifications, we get the following quadratic inequality in β:

[c(2σ−1)+σ2(u−c)(1−2N−1)]β2+2c(1−σ)β−c≥0.

Let σ^* denote the unique σ ∈ (0, 1) such that c(2σ−1)+σ2(u−c)(1−2N−1)=0. The coefficient of β²

c(2σ−1)+σ2(u−c)(1−2N−1){>0σ>σ∗=0σ=σ∗<0σ<σ∗.

Consider N > 3 and let β1∗ denote the smallest value of β ∈ (0, 1) that satisfies the above quadratic inequality. It can be readily verified that

β1∗={−2c(1−σ)+(2c(1−σ))2+4c[c(2σ−1)+σ2(u−c)(1−2N−1)]2[c(2σ−1)+σ2(u−c)(1−2N−1)]σ≠σ∗12(1−σ)σ=σ∗.

(ii) When seeing no deviations, an agent has no incentive to unilaterally deviate from the equilibrium path and report to the central bank. This requires

VM=11≤VG.

Recall that

−c+βVM=11≥βVM=10,and VM=11=σN−1N−1[u+β(VM=10−VM=11)]+δ(u−c)1−β.

Therefore the above inequality holds as

VM=11=σN−1N−1[u+β(VM=10−VM=11)]+δ(u−c)1−β≤σ+δ1−β(u−c)=VG.

II. No agent has an incentive to deviate off the equilibrium path

There are two conditions to be established:

(i) Since the government can not observe the actions of private agents, it is important to determine whether agents have been provided proper incentives to report a deviation. To determine when that will be the case, we need to establish, for any given matched pair, it is optimal for both agents to report to the central bank after seeing a deviation. We note that after seeing a deviation from his trading partner, an agent must believe that his trading partner is the only one who deviates. The following condition makes sure that each agent in a matched pair will report to the central bank after seeing a deviation:

VM=21≥VM=10.

When there is only one unit of money in the economy, we have the following system of equations on VM=10 and VM=11:

[1−β+σβN−1−σ1N−1β−σβ1−β+σβ][VM=10VM=11]=[δ(u−c)−σ1N−1cδ(u−c)+σu]

Hence VM=10 can be solved as

VM=10=(δ(u−c)−σ1N−1c)(1−β+σβ)+(δ(u−c)+σu)σ1N−1β(1−β+σβN−1)(1−β+σβ)−σ2β2N−1.

The no-deviation inequality becomes

(u−c)[δ(1−β+σβNN−1)+2σ2β(N−2)(N−1)2]+(1−β)σuN−2N−1(1−β)(1−β+σβNN−1)≥(δ(u−c)−σ1N−1c)(1−β+σβ)+(δ(u−c)+σu)σ1N−1β(1−β+σβN−1)(1−β+σβ)−σ2β2N−1

As (1−β)(1−β+σβNN−1)=(1−β+σβN−1)(1−β+σβ)−σ2β2N−1, the inequality holds if and only if

(u−c)[δ(1−β+σβNN−1)+2σ2β(N−2)(N−1)2]+(1−β)σuN−2N−1≥(δ(u−c)−σ1N−1c)(1−β+σβ)+(δ(u−c)+σu)σ1N−1β.

After some simplifications, we get

β≤u(N−2)+cu(N−2)+c+σ(u−c)(2N−1−1)≡β2∗.

Observe that β2∗>1 when N > 3. Hence this condition holds for all β ∈ (0, 1) whenever N > 3.

(ii) If someone deviates in a bilateral meeting, both agents will report to the central bank, and two units of money will be injected into the economy. We need to make sure that agents follow a monetary equilibrium after money injection, which requires

β≥βM=2=cc+σN−2N−1(u−c).

Now, combining all conditions in I and II, we conclude that there exists an equilibrium where a threat of money injection can support gift-giving when N > 3 and

β≥βCT≡max{β1∗,βM=2}.

Note that limN→∞β1∗<1 and limN→∞βM=2<1 thus we have that limN→∞βCT<1.

C Proof of Proposition 3

(i) Recall that

βM=cc+σN−MN−1(u−c)βPO=cc+σmin{1−M(N−M)N(N−1),N−MN−1}(u−c)

Apparently βM≤βPO. For βPO=βM, we require

1−M(N−M)N(N−1)≥N−MN−1

A simplification shows that the above inequality holds if and only if M≥N.

(ii) Recall that

β1∗={−2c(1−σ)+(2c(1−σ))2+4c[c(2σ−1)+σ2(u−c)(1−2N−1)]2[c(2σ−1)+σ2(u−c)(1−2N−1)]σ≠σ∗12(1−σ)σ=σ∗,

where σ^* is the unique σ ∈ (0, 1) that solves c(2σ−1)+σ2(u−c)(1−2N−1)=0, and

βCT=max{β1∗,βM=2}.

As βM≥βM=2 for any M ≥ 2, βM≥βCT if βM≥β1∗. Given σ≥12,σ ≠ σ^* as c(2σ−1)+σ2(u−c)(1−2N−1)≥c(212−1)+(12)2(u−c)(1−2N−1)=(12)2(u−c)(1−2N−1)>0. Accordingly,

β1∗=−2c(1−σ)+(2c(1−σ))2+4c[c(2σ−1)+σ2(u−c)(1−2N−1)]2[c(2σ−1)+σ2(u−c)(1−2N−1)].

Define

a=c(2σ−1)+σ2(u−c)(1−2N−1)andb=2c(1−σ).

Then

β1∗=−b+b2+4ac2a

and βM≥β1∗ if

cc+σN−MN−1(u−c)≥−b+b2+4ac2a⇒b2+4ac≤b+2acc+σN−MN−1(u−c)

which can be further simplified to

(N−M)2N−1(u−c)+2c(N−M)−c(N−3)≤0.

Accordingly, βM≥βCT if

M≥N−(N−1)c2+u−cN−1c(N−3)−cu−c.

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Published Online: 2020-02-26

Fiat Money as a Public Signal, Medium of Exchange, and Punishment

Abstract

Acknowledgements

Appendix

A Proof of Proposition 1

B Proof of Proposition 2

C Proof of Proposition 3

References

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