Government Transaction Policy, the Media of Exchange and Prices Li and Wright (1998 JET) April 2018
Introduction Karl Menger (1892) Money has not been generated by law. In its origin it is a social, and not a state-institution. However, by state recognition and regulation, this social institution of money has been perfected and adjusted to the manifold and varying needs of an evolving commerce. Adam Smith (1776) A prince, who should enact that a certain proportion of his taxes should be paid in a paper money of a certain kind, might thereby give a certain value to this paper money; even though the term of its final discharge and redemption should depend altogether upon the will of the prince.
Introduction (con t) Lerner (1947) It is true that a simple declaration that such and such is money will not do, even if backed with the most convincing constitutional evidence of the state s absolute sovereignty. But if the state is willing to accept the proposed money in payment of taxes and other obligations to itself the trick is done.
Issues to focus on How government policies concerning acceptance of a certain money and at what price affect the determination of which objects are used as media of exchange and at what value. We introduce government into the framework in a way that preserves the spatial, temporal and informational frictions that are inherent in the search model. Government is a subset of the agents who are subject to the same constraints as private agents, but perform some exogenously specified trading policies.
Main results Ensuring the existence and uniqueness of an efficient equilibrium by accepting money at a high probability as long as the government is sufficiently big. By refusing to accept money, a big enough government can preclude an equilibrium where money has value. Even if government is not big enough to affect the number of equilibria, it will generally affect the equilibrium price of money. When there are multiple currencies, a big enough government can affect which currency is used as medium of exchange and their exchange rates.
A Model with indivisible goods The basic model is Kiyotaki abd Wright (1993 AER). Meeting technology Agents meet bilaterally according to an anonymous matching process with Poisson arrival rate α. Whenever you meet someone, there is a probability x that he consumes what you produce; and, conditional on this event, there is a probability y that he also produces what you consume. The probability of a double coincidence of wants is xy.
Value functions rv 0 = v 0 + α(1 M)xy(U C) + αmx max π(v 1 V 0 C) π rv 1 = v 1 + α(1 M)xΠ(U + V 0 V 1 ) v 1 (v 0 ) is an instantaneous utility from holding money (production opportunity). The maximization problem implies: 0 ifv 1 V 0 C < 0 π = (0, 1) ifv 1 V 0 C = 0 1 if V 1 V 0 C > 0 V 1 V 0 C takes the same sign as (1) = (1 M)(Π y) K, (2) where K = (rc + v 0 v 1 ) / (U C).
Government policy Can government policy be used to guarantee the existence of the eqm with Π = 1, and rule out the existence of the other equilibria? A fraction of the population γ is called government agents. A government trading policy is specified by T = (T gg, T mg, T gm ). Private agents continue to use the individually-maximizing trading strategies, π. Need to keep track of who holds the money. m = (m p, m g ) satisfies the steady-state condition: ΠT mg M [ΠT mg + (T gm ΠT mg ) (γ M)] m p (T gm ΠT mg ) (1 γ)m 2 p = 0.
Government policy: value functions rv 0 = v 0 + A 1 (U C) + A 2 max π(v 1 V 0 C) (3) π rv 1 = v 1 + A 3 (U + V 0 V 1 ), (4) A 1 = γ(1 m g )yt gg + (1 γ)(1 m p )y prob. of barter A 2 = γm g T mg + (1 γ)m p prob. of trading goods for money A 3 = γ(1 m g )T gm + (1 γ)(1 m p )Π prob. of trading money for goods. V 1 V 0 C in the individual maximization condition is proportional to = A 3 A 1 K. The individual maximization condition is still described by (1), but now V 1 V 0 c is proportional to = A 3 A 1 K, where = γ(1 m g )(T gm yt gg )+(1 γ)(1 m p )(Π y) K. (5)
Π = 0 equilibrium does not exist... Algorithm: Given T, solve for m and substituting into, and then verify the best-response condition. Does nonmonetary equilibrium still exist under policy T gm = 1? Result: The Π = 0 eqm does not exist under T gm = 1 iff γ > γ 1 = K + y + M(1 yt gg) 1 + y(1 T gg )
Π = 1 equilibrium exists... When will Π = 1 be an equilibrium under T gm = 1? Result: The Π = 1 equilibrium exists iff γ γ 2 = K (1 M)(1 y) (1 M)y(1 T gg ). Conclusion: The policy T gm = 1 can guarantee the existence of monetary eqm as well as rule out the existence of nonmonetary eqm iff the government is big enough, and just how big depends on k, y, M, and T gg.
Proposition 1 If and only if γ is big enough, a policy whereby government agents accept money with high probability can eliminate equilibria with Π < 1 that exist without government intervention, and can guarantee the existence of a unique equilibrium with Π = 1. This is true even if the Π = 1 equilibrium does not exist without government intervention. Such a policy is more effective if T gg is small. Also, if and only if γ is big enough, a policy whereby government agents accept money with low probability can eliminate a monetary equilibrium that exists without intervention. This policy is more effective if T gg is big.
A model with divisible goods The basic model is Trejos and Wright (1995 JPE), where a buyer makes a take-it-or-leave-it offer to the seller. Value functions: rv 0 = v 0 + (1 M)y[u(q ) q ] + M max π[v 1 V 0 Q] π rv 1 = v 1 + (1 M)Π max[u(q) + V 0 V 1, 0]. Given Q [0, ˆq], a buyer s take-it-or-leave-it offer is q = q(q) = max[d(q), 0], where D(Q) = V 1 V 0. A steady state equilibrium is a fixed point of q(q). When multiple equilibria coexist, the low price eqm Pareto dominates the high price eqm, and both dominate the nonmonetary eqm.
A model with divisible goods: government policy q s (q d ): quantity of goods that government agents supply (demand) in exchange for money. We set T = (1, 1, 1), and let q d = V 1 V 0. Consider a policy q s (0, ˆq). Value functions: rv 0 = v 0 + (1 M)y[u(q ) q ] + M max π(v 1 V 0 Q) π rv 1 = v 1 + (1 γ)(1 m p )Π max[u(q) + V 0 V 1, 0] +γ(1 m g ) max[u(q s ) + V 0 V 1, 0].
Proposition 2 If and only if γ is big enough, a policy whereby government agents supply q s in exchange for money can eliminate both a low q monetary equilibrium and the nonmonetary equilibrium, and can guarantee the existence and uniqueness of a high q monetary equilibrium. This is true even if a monetary equilibrium does not exist without intervention, although then establishing a monetary equilibrium implies the government must run a deficit. If there exist monetary equilibria without government, for big γ, setting q s = qh 0 implies q = q0 h is the unique equilibrium and entails no deficit or surplus. Also, a policy whereby government agents demand q d in exchange for money can eliminate monetary equilibria that exist without intervention if q d is low and γ big enough.