貨幣與公共財

行政院國家科學委員會專題研究計畫 成果報告

貨幣與公共財

中 華 民 國 93 年 11 月 4 日

摘要 關鍵詞：貨幣；公共財；通貨膨脹稅；搜尋模型 本文討論政府使用通貨膨脹稅融通公共支出如何影響一國貨幣的流通以及福利 問題。本文考慮一個兩國的搜尋模型，政府的角色在於發行貨幣和使用通貨膨脹 政策獲取稅收，並以該收入向民間購買財貨並藉以提供公共財。在本經濟體中， 雖然政府向民間的購買提高總合需求，但也會產生排擠效果，對民間的交易產生 負面影響。在此兩國模型中，人們與本國人和外國人交易的頻率不同，而這反映 在各國規模大小和兩國經濟整合度兩項參數上。我們發現，一國採取高通貨膨脹 稅將不利於該國貨幣的流通。大國的貨幣較易成為國際通貨，而發行國際通貨的 政府可能採取較低的通貨膨脹稅，因為低膨脹稅使該國貨幣較易成為國際通貨而 享受對外國人課稅的好處，因而得以提高福利。

Abstract key words: Þat money; public goods; inßationary policy

This paper studies how government uses inßation tax to Þnance public goods affects the circulation of a national currency in a two-country search theoretic model. Agents obtain utility from the private good and the public good of his own country. Each government prints Þat money to purchase goods, taxes on money holdings, and provides public goods by purchasing private goods from its fellow citizens. While government purchases increase the demand for the private goods, it also induces a crowding-out effect by reducing the matching rates among private agents. Agents interact with home and foreign agents in different frequencies, reßecting the relative country size and the degree of international economic integration. We Þnd that, a higher inßation tax rate makes a currency less likely to circulate locally and internationally. A big country may adopt a lower inßation tax rate to induce its currency to be used as the international currency. A cost and a beneÞt accompany such a policy. The cost is that a better equilibrium associated higher inßation tax is eliminated. A beneÞt is that it also eliminates the equilibria that entail lower welfare.

1

Introduction

Historically, government policies have been designed to inßuence the value and circulation of a certain object as a medium of exchange. For example, governments may designate the legal tender status to a certain currency. In some of the medieval economies the sovereign debased coins to collect seigniorage in order to Þnance war. The widespread use of purely Þat money is a 20th-century development. One objective of government to issue Þat money is to Þnance public goods. Fiat money is an efficient means of payment for the government to purchase goods, and allows it to use inßation tax to Þnance spending. Although it is an efficient way for the government to Þnance public goods, inßation tax may deteriorate the value and realm of circulation of a currency, particularly in a world with multiple currencies. Obviously, inßation tax can not be used without limit: a government can generate seigniorage tax only if the currency issued remains in circulation. On the other hand, governments may have incentive to impose a higher inßation tax when its currency circulate abroad because the tax burden falls partially on foreigners.

This paper studies how government uses inßation tax to Þnance public goods affects the circulation of a national currency in a two-country search theoretic model. Each country consists of inÞnitely-lived private agents and a government. A representative agent obtains utility from private good, and the public good of his own country. Each government prints Þat money to purchase goods, taxes on money holdings, and provides public goods by purchasing private goods from its fellow citizens. While government purchases increase the demand for the private goods, it also induces a crowding-out effect by reducing the matching rates among private agents. Agents interact with home and foreign agents in different frequencies, reßecting the relative country size and the degree of international economic integration. Government in each country chooses a set of policies including the rate on taxing money holdings and the amount of public goods provided. Agents choose which money to hold to conduct trade. In so doing they must take into account which money yields higher frequency of trade, and the risk of conÞscation (a proxy for inßation) that each currency is subject to.

We study how inßation tax affects whether one country’s currency is more likely to circulate locally and internationally, and welfare. If the degree of economic integration is sufficiently low, national currency circulates only locally; there is no international currency and no international trade. We Þnd that if a government raises the inßation tax, its citizens are more likely to use foreign currency. Hence, under inßationary policy, staying autarchy is not the best response unless the population size is big enough to compensate the negative impacts of conÞscating currency and difficulties in trade caused by the crowding-out effect of higher inßation tax.

As the economic integration is higher, a national currency may start to circulate internationally. In particular, the country with a bigger size, representing larger economies of scale in trading, is more likely to make its currency to circulate internationally. If a country adopts a sufficiently higher inßation tax than the other country does, it may induce its citizen to abandon the use of national currency and also reduces the possibility that its currency circulate abroad. The reason is that, higher government’s purchases increase the probability for domestic sellers to sell goods, but for all agents the meeting rates with fellow and foreign citizens are reduced due to the crowding-out effects. A higher home inßation tax not only deters home agents’ incentive to accept national currency, but also reduces foreign agents’ willingness to accept it. As the degree of economic integration is higher, the ‘imported’ impact due to inßation tax and crowding-out effect would be stronger.

However, a higher inßation tax does not necessarily preclude a currency from circu-lating abroad. We Þnd from numerical experiments that, when supply of both currencies is moderate, if the smaller country adopts a higher inßation tax, the equilibrium where its currency circulates home and abroad is more likely to exist than the one in which the currency circulates only locally. Higher domestic inßation tax increases agents’ incentive to abandon national currency; however, this effect will be lower if the currency circulates abroad also, because there are more home sellers and so the trade opportunity for the fellow citizens increases. On the other hand, if the currency circulates only locally, the tax burden falls completely on local citizens and so they may stop using the national currency.

However, if currency supply is much lower, higher domestic inßation makes the national currency less likely to circulate internationally. Because of the currency shortage at home, agents of smaller country have higher incentive to accept other country’s currency.

We then consider a policy game in which the two governments choose tax rates on their respective currencies, measuring the payoff of each government by the utility of its own representative agent. Numerical examples show, among others, that the equilibrium tax rate of a currency is higher when it becomes an international currency than otherwise. Welfare may be improved for the issuing country of the international currency when its currency supply is not too low, for that circulation of a currency abroad may create currency shortage at home. We also Þnd that a government can implement a policy to interfere the existence of equilibrium so that its fellow citizens enjoy higher welfare (e.g., adopting a lower inßation tax to induce its currency circulate internationally). However, a cost and a beneÞt accompany such a policy. The cost of adopting a lower inßation tax is that a better equilibrium associated with a higher inßation tax (which generates higher seigniorage tax to provide public goods) is eliminated. A beneÞt is that it also eliminates the equilibria that entail lower welfare.

2

The Basic Model

The Environment

Time is discrete and the horizon is inÞnite. There is a [0, 1] continuum of inÞnitely-lived agents with unit mass. The agents are divided into two regions, Home and Foreign. Let n ∈ (0, 1) be the size of Home population. There are k (k ≥ 3) types of indivisible goods. Within each economy, there are equal proportions of k types of agents, who specialize in consumption, production and storage. A type i agent derives utility only from consuming type i good and can produce only good i + 1 (mod k). Agent i can only store his production good costlessly up to one unit; he can neither produce nor store other types of goods. Hence, there is no double coincidence of wants. Let u > 0 be the instantaneous utility from consuming an agent’s consumption good and δ his discount

rate.

There are two distinguishable Þat money, Home currency and Foreign currency. Each currency is indivisible. An agent can store only one unit of good or one unit of currency at a time. Let mh (mf) denote the fraction of Home agents holding the Home (Foreign) currency. The inventory distribution of Home agents can be summarized by X = (1 − mh−mf, mh, mf).Likewise, let m∗h (m∗f)denote the fraction of Foreign agents holding the Home (Foreign) currency. The inventory distribution of Foreign agents can be summarized by X∗ _{= (1− m}∗

h− m∗f, m∗h, m∗f). Let m and m∗ ∈ (0, 1) denote the supply of the Home currency per Home agent and that of Foreign currency per Foreign agent, respectively. Then,

nm = nmh+ (1− n)m∗h, (1− n)m∗ = nmf + (1− n)m∗f.

Agents are matched randomly in pairs. Let β ∈ (0, 1). Without government policy, a Home agent meets a Home agent with probability n, and meets a Foreign agent with probability β(1−n). A Foreign agent meets a Home and Foreign agent with probability βn and (1 − n), respectively. Thus, agents who live in different countries meet less frequently than a pair of agents who live in the same country. Note that the above description implies the probability of meeting a trade partner also depends on the size of country. We can interpret β as the degree of economic integration or, a measure of the trading frictions in international trade, and the country size as a measure of the trading frictions in local trade. For example, an increase in β reduces international trade frictions, because higher β makes it easier to meet with Foreign agents. Similarly, an increase in n reduces the local trade frictions in Home country because higher n makes it easier for the Home agents to meet with their fellow citizens. However, an increase in n reduces the relative size of Foreign country and so increases the local trade frictions in Foreign country.

Trade entails a one-for-one swap of inventories, and takes place if and only if both agents agree to trade. The trade partner’s type and inventory are observable, trade histories are not. Agents are unable to commit to future actions, and proposed transfers cannot be enforced. Thus, people trade when there is single coincidence of wants, and all

trades involve the use of a tangible medium of exchange.

The role of government in the provision of public goods

In each country there is a government whose role is to print Þat money, tax money holdings and provide public goods from the private goods that it purchases. An agent who holds Home (Foreign) currency is subject to a probability τh (τf) that his money would be conÞscated by the government of Home (Foreign) country. The rate τh (τf) can be interpreted as tax rate that a government imposes on money holdings in order to provide public goods. We can also interpret τh (τf) as inßationary tax.

When a government purchases goods from private agents, it prints one unit of Þat money to the producer. We assume that a government buys goods only from its fellow citizens. We assume that it is not feasible that the government pays for its purchase with private goods (i.e., no barter trade between government and private agents). Whether or not an agent accepts a particular currency is part of his strategy, and he will obtain neither extra reward nor punishment for his decision. Let nγ_{h (resp. (1 − n)γf}) denote the arrival rate to each agent by Home (resp. Foreign) government which proposes to purchase his commodity. Thus, with government policy the arrival rate to a Home agent of Home (resp. Foreign) agents is n(1 − γh) (resp. β(1 − n)(1 − γf)). A Foreign agent meets a Home (resp. Foreign) agent with probability βn(1 − γh) (resp. (1 − n)(1 − γf)). This implies that government’s involvement in the economy creates a crowding-out effect, in that government purchases reduce the meeting probability and trade among private agents.

A government transforms the private goods it purchases into public goods from which every private agent in the country enjoys the utility of φ(g) where g is the total quantity of private goods purchased by the government in a unit of time. We assume φ(0) = 0, φ0(g) _{→ ∞ as g → 0, φ}0(g) > 0 and φ00(g) < 0. Public goods are nonstorable (e.g., army service). We may assume that Home government and Foreign government have different efficiency in providing public goods. For example, assume that the quantity of public goods g is a fraction θ of total consumption goods purchased by the government,

and both countries may have different θ’s. We may also assume that Home and Foreign agents have different preferences for public goods.

Strategies and equilibrium

An agent chooses trade strategies to maximize his expected discounted utility, taking as given others’ strategies and the distribution of inventories. We restrict our attention to pure strategies which only depend on his nationality and the objects he and his trading partner have in inventory. Thus, the Home agent’s trade strategy can be described as

sab =     

1if he trades object a for b 0otherwise,

where a, b = g, h, or f, and a 6= b. Similarly, the Foreign agent’s trade strategy is given by s∗

ab = 0 or 1. We consider only time-independent strategies. Given that the physical environment is stationary and the planning horizon is inÞnite, we can therefore conÞne our attention to steady-state equilibrium.

Let Vg, Vh and Vf denote the expected discounted utility to a Home agent holding his production good, the Home currency, and Foreign currency, respectively. Let Pab (P∗

ab) denote the transition probability with which a Home (Foreign) agent switches his inventory from object a to object b. Then, the Bellman’s equations are

Vg = [(1− Pgh− Pgf)Vg+ PghVh+ PgfVf]/(1 + δ) (1) Vh = [τhVg + Phg(u + Vg) + (1− Phg − Phf − τh)Vh+ PhfVf]/(1 + δ) (2) Vf = [τfVg+ Pf g(u + Vg) + Pf hVh+ (1− Pf g − Pf h− τf)Vf]/(1 + δ). (3) Note that the Þrst terms in the RHS of equality in (2) and (3) imply that, if an agent’s currency is conÞscated by the issuing government (with probability τh and τf that his money is conÞscated by Home and Foreign government, respectively), his value becomes that of holding production good. The value functions and strategies must satisfy the following incentive compatibility constraints:

sag = 1 iff Va< u + Vg (a = hor f ) sab = 1 iff Va< Vb (a, b = h or f ).

For example, Vg > Vf is the sufficient and necessary condition for a Home agent not to trade his production good for Foreign currency.

We restrict our attention to the equilibrium where agents always accept their local currency; i.e., Home currency is accepted by the Home agents and Foreign currency is accepted by the Foreign agents. We are left four types of equilibria — no international currency, Foreign currency is the only international currency, Home currency is the only international currency and both currencies circulate in both countries. We characterize the existence conditions in terms of β and n, the extent of international and local trade frictions.

First of all, in any of these equilibria, we have Pf h = Phf = Pf h∗ = Phf∗ = 0. Given the tie-breaking rule [assume somewhere!], no two agents in the same country exchange Home currency and Foreign currency; indeed, for currency exchange to occur between two, say, Home agents, we need shf = sf h = 1, which implies Vf > Vh and Vh > Vf, a contradiction. Therefore, the only possibility for currency exchange is between agents from different countries. Due to the nature of equilibrium, this may happen only when both currencies circulate worldwide. In this case, we need to have, say, Vh > Vf and Vf∗ > Vh∗ (the opposite case has a similar consequence). If τh = τf holds, then the two currencies are perfect substitutes, and therefore, Vh = Vf and Vf∗ = Vh∗, which is a contradiction. But, if, say, τh becomes smaller (resp. greater) than τf, then Home currency is more (resp. less) attractive for both Home and Foreign agents than Foreign currency. Thus, both Home and Foreign agents have the same incentive concerning the acceptance of currency, and therefore, there is no room for currency exchange.

Before conducting equilibrium analysis, let us calculate the value functions from (1), (2) and (3):

Vg = [(δ + Pf g+ τf)PghPhg+ (δ + Phg + τh)PgfPf g] u/P, (4) Vh = [Phg(δ + Pgh)(δ + Pf g + τf) + PhgPgf(δ + Pf g) + PgfPf gτh)] u/P, (5)

Vf = [Pf g(δ + Pgf)(δ + Phg + τh) + Pf gPgh(δ + Phg) + PghPhgτf)] u/P, (6) where

P = δ [(δ + Pgh+ Phg + τh)(δ + Pf g + τf) + Pgf(δ + Phg + τh)] . Using the above value functions, we are able to state some general results. Proposition 2.1. 1. u + Vg > Vg, Vh, Vf.

2. max{Vh, Vf} > Vg.

3. Vh > (<)Vg iff Phg(δ + Pf g + Pgf + τf) > (<)PgfPf g. 4. Vf > (<)Vg iff Pf g(δ + Phg+ Pgh+ τh) > (<)PghPhg.

3

Equilibrium

3.1

Equilibrium with two local currencies: Equilibrium A

In this equilibrium a Home agent trades his production good for the Home currency, the Home currency for his consumption good, but does not accept Foreign currency (u + Vg > Vh > Vg ≥ Vf). A Foreign agent trades his production good for the Foreign currency, the Foreign currency for his consumption good, but does not accept the Home currency (u + V∗

g > Vf∗ > Vg∗ ≥ Vh∗).There is no international currency and no international trade in this equilibrium. The inventory distributions are given by X = (1 − m, m, 0) and X∗ _{= (1− m}∗_{, m}∗_{, 0). The transition probabilities in this equilibrium for a Home agent} are:

Pgh= n(1− γh)m/k + nγh, Phg = n(1− γh)(1− m)/k Pf g= β(1− n)(1 − γf)(1− m∗)/k, Pgf = Phf = Pf h= 0.

(7) Note that Pgh incorporates the opportunity to sell goods to acquire money from private agents and government with probability n(1 − γh) and nγh, respectively. If a Home agent ever holds Foreign currency, then given others’ strategies the chance that he can acquire consumption goods is from trading with Foreign sellers, of which the probability

is β(1 − n)(1 − γf)(1− m∗)/k. Similarly, the transition probabilities for a Foreign agent are: P∗ gh = (1− n)(1 − γf)m∗/k + (1− n)γf, Pf g∗ = (1− n)(1 − γf)(1− m∗)/k P∗ hg = βn(1− γh)(1− m)/k, Pgh∗ = Pf h∗ = Phf∗ = 0. (8)

Given the inventory distributions the balanced budget constraints for Home and For-eign country satisfy

nγh(1− m) = τhm,

(1_{− n)γ}f(1− m∗) = τfm∗,

(9) respectively. Thus, given the tax rate (τh, τf),the purchase of private goods from govern-ments, (γh, γf), must satisfy (9). This also implies that the supply of Home and Foreign currency per capita does not change over time.

To Þnd the existence conditions for Equilibrium A, we verify the incentive constraints u + Vg > Vh > Vg > Vf and u + Vg∗ > Vf∗ > Vg∗ > Vh∗.

From Proposition 2.1, Vg ≥ Vf and Vg∗ ≥ Vh∗ imply other inequalities. We have Vg ≥ Vf (Home agents do not accept Foreign currency) iff β ≤ βA,where

βA=

m[n(1_{− m) − mτ}h][n(1− m) − mτh+ kτh]

[(1_{− n)(1 − m}∗_{)− m}∗_{τf][(1− m)(n + δk) − mτh+ kτh]}. Likewise, Foreign agents do not accept Home currency, or V∗

g ≥ Vh∗, iff β ≤ β∗A, where β∗_A= m ∗_{[(1− n)(1 − m}∗_{)− m}∗_τ f][(1− n)(1 − m∗)− m∗τf + kτf] [n(1_{− m) − mτ}h][(1− m∗)(1− n + δk) − m∗τf + kτf)] .

The above expressions are cumbersome, and the effect of an increase in, say, τf is ambiguous. Instead of analyzing the case for which intuition does not work well, we focus on the case where agents are sufficiently patient relative to matching frequency, i.e., we study the limiting situation where δ goes to zero. Taking the limit, we obtain

lim δ→0βA= m[n(1_{− m) − mτ}h] (1− n)(1 − m∗_{)− m}∗_τf (10) lim δ→0β ∗ A= m∗_{[(1− n)(1 − m}∗_{)− m}∗_τ f] n(1_{− m) − mτ}h (11)

Given parameter values of m, m∗, k, τh, and τf, β ≤ βA, β ≤ β∗A give the existence conditions of equilibrium A on (n, β) space, shown in Figures 1-a, 1-b.1 _{To see how} a policy pair (τh, τf) affects the existence region of equilibrium, note that given other parameters, an increase in τh leads to a decrease in βA, while an increase in τf leads to an increase in βA. Likewise, an increase in τh leads to an increase in β∗A, while an increase in τf leads to a decrease in β∗A.If we interpret (τh, τf)as a proxy for the rate of inßation, then this change is intuitive, i.e., the higher (resp. lower) the rate of inßation of Home (resp. Foreign) currency is, the more likely Home agents are to use Foreign currency.

Given a policy pair (τh, τf),if the degree of economic integration is sufficiently small, national currency circulates only locally; there is no international currency and no inter-national trade. Other things being equal, this equilibrium does not survive if the country size is uneven. If n is sufficiently large, the trade with Home agents is so easy that Foreign agents would have incentives to use Home currency. For a given n, Equilibrium A does not exist when β is sufficiently high, either. The higher the degree of economic integration becomes, the easier trade with foreigners, and the higher the incentive to accept foreign currency becomes.

The change in the existence region of equilibrium compared to the case with (τh, τf) = (0, 0) depends on the relative magnitudes of τh and τf.2 If τh > τf > 0, βA curve shifts downward while β∗Arotates clockwise (see Figure 1-b). Similarly, if τf > τh > 0, β∗Acurve shifts downward while β_A rotates anti-clockwise. A higher Home inßation rate induces a higher incentive for Home agents to accept Foreign currency. The downward shift of βA implies that under inßationary policy, staying autarchy is not the best response unless the population size of the country is big enough to compensate the negative impacts of conÞscating currency (a proxy for inßation). Thus, for a given pair of (n, β), if a country adopts too high an inßation tax rate, it may destroy the equilibrium with two currency areas. To see the move of β∗_A when τh > τf > 0, notice that, though government’s

1_{The parameters are m = m}∗_{= .2, k = 10, τ}

h= τf = .1. 2_{Notice that here we are considering changes in τ}

hand τf at a time, which is different from the above comparative statics analysis.

purchases increase the probability for private agents to sell goods, it also reduces the meeting rates among private agents. This crowding-out effect works like trade frictions. As the degree of economic integration is higher, the ‘imported’ impact due to inßation tax and crowding-out effect would be stronger. Given a high β, the crowding-out effect is high so not accepting Home currency is best response for Foreign agents even when n is larger than without policy. However, when β is low, the effect of τf dominates the ‘imported’ effect by τh and so the strategy of staying autarky survives at smaller n than without policy.

3.2

Equilibrium with one local currency and one international

currency: Equilibrium F and H

We discuss the existence conditions for Equilibrium F, where Home currency is accepted only in Home country, while Foreign currency circulates in both Home and Foreign country as an international medium of exchange. Equilibrium H is the mirror image of Equilibrium F and can be characterized in a similar manner.

Equilibrium F requires u+Vg > Vh, Vf > Vg and u+Vg∗ > Vf∗ > Vg∗ ≥ Vh∗.When agents follow these strategies, mh = mand so X = (1−m−mf, m, mf)and X∗ = (1−m∗f, 0, m∗f). The steady state requires that the ratios of commodity holders to the Foreign currency holders in the two countries be equalized, i.e.,

m∗f 1_{− m}∗ f = mf 1_{− m − m}f .

From the steady state condition, mf = (1−m)m∗f.Therefore we can rewrite the inventory distributions in terms of m∗

f as X = ((1 − m)(1 − m∗f), m, (1− m)m∗f) and X∗ = (1− m∗

f, 0, m∗f). The total supply of Foreign currency must equal the total amount circulates in both countries

The transition probabilities for a Home agent are Pgh = n(1− γh)m/k + nγh Pgf = [n(1− γh)(1− m) + β(1 − n)(1 − γf)]m∗f/k Phg = n(1− γh)(1− m)(1 − m∗f)/k Pf g = [n(1− γh)(1− m) + β(1 − n)(1 − γf)](1− m∗f)/k Phf = Pf h= 0 (13)

and for a Foreign agent

Pgf∗ = [βn(1− γh)(1− m) + (1 − n)(1 − γf)]m∗f/k + (1− n)γf P∗ hg = βn(1− γh)(1− m)(1 − m∗f)/k Pf g∗ = [βn(1− γh)(1− m) + (1 − n)(1 − γf)](1− m∗f)/k P∗ gh = Pf h∗ = Phf∗ = 0 (14) where m∗

f satisÞes (12). Given the inventory distributions the balanced budget constraints for Home and Foreign country are

nγ_h(1_{− m)(1 − m}∗

f) = τhm, (1_{− n)γ}f(1− m∗f) = τfm∗,

(15)

respectively.

In the sequel, we take the limit of δ going to zero in making comparative statics and other characterizations, which implies that matching frequencies are sufficiently high relative to time preference. From Proposition 2.1, it suffices to check that Home agents accept Home currency (Vg < Vh), and that Foreign agents do not accept Home currency (V∗

g ≥ Vh∗).

First, substituting (13) into the third and forth claims of Proposition 2.1, and taking the limit of δ going to zero, we have Vh > Vg iff

f (n, β) = βF > 0 (16)

and Vg∗ ≥ Vh∗ iff

where f (n, β) and f∗(n, β)are deÞned in the Appendix. Equilibrium F exists if and only if the two incentive constraints hold, given (13), (14) and (15). We depict the equilibrium region deÞned by (16) and (17) on the space of (n, β) in Figures 2-a, 2-b.

Equations (16) and (17) are too complicated for us to sign the comparative statics in general. However, we are able to do it if we evaluate the derivatives near (τh, τf) = (0, 0). We prove that ∂βF ∂τh|τh=0,τf=0< 0, ∂βF ∂τf|τh=0,τf=0> 0, ∂β∗ F ∂τh|τh=0,τf=0> 0and ∂β∗ F ∂τf|τh=0,τf=0>

0 if k is not too small.3 Therefore, as τh (resp. τf) increases from zero, β = βF shifts downward (resp. upward), while β = β∗_F shifts upward (resp. upward). As τh increases from zero it requires a larger n (or higher β) for Home agents to accept Home currency. Similarly, Foreign agents’ not-accepting-Home-currency strategy survives even at a larger n(or higher β). An increase in τf from the policy pair (0, 0) makes Home agents less likely to abandon its national currency, while Foreign agents less likely to accept Home currency. The latter result may be somewhat surprising; however, recall that higher inßation tax also increases government’s purchases, which increases the probability for domestic sellers to sell goods. As the beneÞt compensates the cost of conÞscation of currency, an increase in τf could make the equilibrium strategy survive at a larger n.

We now consider changes in τh and τf simultaneously at one time. Note Þrst, when τh > 0and τf > 0, β∗F rotates clockwise, similar to the case in equilibrium A. However, the shift of βF depends on the relative magnitudes of τh and τf.If τhis sufficiently larger than τf, βF shifts downward and so for accepting Home currency to remain the best response it requires a higher Home country size than without policy. For other cases, βF rotates anti-clockwise, implying the ‘imported’ effect from τf has different impact at different degree of economic integration. That is, when β is high (resp. low) accepting Home currency is the best response at a smaller (resp. larger) n than without policy. When β is high, the ‘imported’ crowding-out effect is so strong to offset the trading frictions caused by inßation tax τh that Home agents would have incentive to use Home currency even at a smaller n than without policy.

3 ∂β∗F

∂τf|τh=0,τf=0> 0 iff k > k =

m∗(1−n)(1−nm)

3.3

Equilibrium with the uniÞed currency

In this equilibrium, two currencies are uniÞed and become perfect substitutes. That is, both Home and Foreign currency circulate in both countries, u + Vg > Vh, Vf > Vg, and u+Vg∗ > Vf∗, Vh∗ > Vg∗.When agents follow these strategies, X = X∗,and mh = m∗h = nm, and the steady state that the ratios of commodity holders to the Home currency holders in the two countries are equalized gives m∗h = (1− m∗)mh. The inventory distributions are given by X = (1 − mh, mh, 0) and mf = m∗f = (1− n)m∗. The transition probabilities are Pgh= nm[n(1− γh) + β(1− n)(1 − γf)]/k + nγh Pgf = [n(1− γh) + β(1− n)(1 − γf)](1− n)m∗/k Phg = Pf g = [n(1− γh) + β(1− n)(1 − γf)][1− nm − (1 − n)m∗]/k P∗ gh= nm[βn(1− γh) + (1− n)(1 − γf)]/k + (1− n)γf P∗ gf = [βn(1− γh) + (1− n)(1 − γf)](1− n)m∗/k P∗ hg = Pf g∗ = [βn(1− γh) + (1− n)(1 − γf)][1− nm − (1 − n)m∗]/k Phf = Pf h = Pf h∗ = Phf∗ = 0. (18)

We Þnd that, given any τh > 0 and τf > 0, the above conditions hold for all parame-ters, and so we conclude that this equilibrium exists for any (n, β) ∈ (0, 1)2_.

4

Policy and Welfare

We use the long-run expected utility of a representative agent as the welfare criterion. Let W (W∗) denote the welfare criterion for Home (Foreign) country, where

W = (1_{− m}h− mf)Vg+ mhVh+ mfVf + [θnγh(1− mh − mf)]1/α

W∗ = (1_{− m}∗_{h− m}∗_f)V_g∗+ m∗_hV_h∗+ m_f∗V_f∗+ [θ∗(1_{− n)γf}(1_{− m}∗_{h− m}∗_f)]1/α∗.

Let W denote total welfare, the weighted average of welfare of both countries by using the population size as the weight. Notice that the welfare criteria incorporate the expected utility of consuming public goods. The total quantity of public goods provided in Home (Foreign) country is equal to the transformation parameter θ (θ∗) multiplied by the total

private goods purchased. The Home and Foreign agents’ preference for public goods is determined by the parameter α and α∗_{,respectively. We Þrst assume θ = θ}∗ _{= 1 and α =} α∗ = 2 in the following examples.

We discuss welfare issues from the observations of numerical experiments. In the following discussions we consider both countries adopt the tax rate that each government chooses independently to maximize welfare W and W∗_{,respectively. We Þnd that, given} (m, m∗), W and W∗ are highest in equilibrium U. All agents prefer a fully integrated equilibrium because it results in highest frequency of trade. For most examples Home agents prefer equilibrium H to A to F, Foreign agents prefer equilibrium F to A to H, but the total welfare W is lowest in equilibrium A. We do, however, Þnd exceptional cases. For example, if Home country is the bigger country and issues less currency than Foreign country (m < m∗_{), the total welfare W is lowest in equilibrium H. This implies} that, though a switch from equilibrium A to an equilibrium with international currency improves trade opportunity, the welfare gain of the issuing country may not be big enough to compensate the loss of the other country. When the Home currency supply is much lower, Home agents even prefer equilibrium F to H; i.e., agents from Home country enjoy higher welfare in the equilibrium where the circulation of its currency remains locally. The currency shortage at home makes the inßow of foreign currency a bless rather than a curse.

Given the above results, one may think whether a government can implement a policy to interfere the existence of equilibria so that its fellow citizens enjoy higher welfare. From Figure 3 we observe that if τh is low enough, the only equilibria are equilibrium H and U. We also know that W is higher in equilibrium H than in equilibrium A and F when those equilibria coexist. Hence, Home government would have incentive to lower tax rate τh to eliminate other equilibria where Home welfare is lower. The magnitude of τh that can make Foreign agents start to accept Home currency depends on τf.However, one can Þnd a threshold of τh, τ ,below which Foreign agents start to accept Home currency regardless of τf. This policy ensures higher welfare to Home agents.4 Notice that, if τh > τ and τf

is not too high, there coexist equilibria A, F and H. If Home government reduces inßation tax below τ and we are in equilibrium H, welfare W is lower than that in equilibrium H with τh > τ but higher than that in equilibria A and F. Thus, the cost of adopting a lower inßation tax for a country is that a better equilibrium associated with a higher tax rate is eliminated. The beneÞt is that it also eliminates the equilibria that entail lower welfare.

5

Conclusion

Some economists have argued that inßationary bias is necessary in an environment where sovereigns have the ability to tax foreigners. Yet they are puzzled by the observation that the United States, the country that would seem best able to impose a seignorage tax on foreigners, have relatively stable monetary policy. This paper may offer an answer to the above issue. We have shown that, a big country, by adopting a lower inßation tax, can insure the existence of equilibrium where its national currency circulate abroad. This would entail higher welfare to its fellow citizen, provided it would not cause currency shortage at home, because the tax burden partially falls on foreigners. A higher inßation tax may lead to an inferior outcome where its currency circulates only locally, under which the tax burden falls completely on its citizens. Thus, the negative impact of a country’s inßationary policy on the realm of circulation of its currency provides an inßation discipline, an issue that cannot be answered in a framework without considering the endogenous emergence of an international currency.

Previous studies using models with explicit trade frictions on the issue of international currency include the following. Matsuyama, Kiyotaki and Matsui (1993), of which the model is adopted in this paper, show that as the degree of economic integration is big enough there arises international currency. Zhou (1997) considers preference shocks to induce currency exchange in a framework similar to Matsuyama et al. (1993). Wright and Trejos (2001) considers a search model with divisible goods to study the determination of

exchange rate. Soller Curtis and Waller (2003) show how currency restrictions (such as restrictions on the internal use of foreign currency) and government transactions policy affect the values of Þat currencies in a two-country model. Ravikumar and Wallace (2002) shows that a uniform currency can eliminate inferior equilibria associated with distinct currencies where output and quantities of trade are less than the optimal quantities. The major differences of this paper from previous studies is that we explicitly consider an objective for government to issue Þat money, formulate how government’s collection of seigniorage interacts with public goods provision, and study in a two-country world how the policies affect which currency is accepted as a medium of exchange and its welfare implications.

Reference

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Li, V (1995) “The Optimal Taxation of Fiat Money in Search Equilibrium,” International Economic Review 36, 927-942.

Li, Y. and R. Wright (1998) “Government Transaction Policy, the Media of Exchange and Prices,” Journal of Economic Theory, 81, 290-313.

Matsuyama, K., N. Kiyotaki and A. Matsui (1993) “Towards a Theory of International Currency,” Review of Economic Studies 60, 283-307.

Ravikumar, B. and N. Wallace (2002) “A BeneÞt of Uniform Currency,” manuscript, Pennsylvania Sate University.

Soller Curtis, E. and C.J. Waller (2000) “A Search-theoretic Model of Legal and Illegal Currency,” Journal of Monetary Economics, 45, 155-184.

Soller Curtis, E. and C.J. Waller (2003) “Currency Restrictions, Government Transaction Policies and Currency Exchange,” Economic Theory, 21, 19-42.

Wright R. and A. Trejos (2001) “International Currency,” Advances in Macroeconomics, 1 (1) Article 3.

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