Kiyotaki and Wright (1993)

Transcription

1 Review May 2017

2 Kiyotaki and Wright (1993) Π = prob. that a random commodity trader accepts fiat money. π = best response of a representative agent. In steady state, Bellman equations are rv g = β(1 M)x 2 (u ε) +βmx max m V g, 0) π (1) rv m = β(1 M)xΠ(u ε + V g V m ) (2)

3 KW (1993): Best response conditions Strategy: accept money with prob π, where 1 if V m > V g π = [0, 1] if V m = V g 0 if V m < V g V m and V g depend on Π: individual s best response correspondence is π = π(π). An equilibrium for the model is a fixed point Π = π(π).

4 KW (1993): Multiple equilibria Claim. There are three equilibria: Π > x V m > V g π = 1: pure monetary eqm. Π = x V m = V g Π < x V m < V g π [0, 1]: mixed eqm. π = 0 : nonmonetary eqm. Multiplicity results from a coordination problem in the social choice of media of exchange. Government s policy can influence this choice (Li and Wright 1998).

5 Li and Wright (1998): 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 (3) = (1 M)(Π y) K, (4) where K = (rc + v 0 v 1 ) / (U C).

6 Li and Wright (1998): 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.

7 Li and Wright (1998): Government policy (con t) rv 0 = v 0 + A 1 (U C) + A 2 max π(v 1 V 0 C) (5) π rv 1 = v 1 + A 3 (U + V 0 V 1 ), (6) 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 (3), 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. (7)

8 Kiyotaki and Wright (1989) Definition A steady state Nash equilibrium consists of τ = (τ 1, τ 2, τ 3 ) and P satisfying Maximization: Given other s strategies and P, τ i maximizes expected utility for type i. Rational expectation: Given τ, P is the resulting steady state inventory distribution.

9 Pure strategy profile: S = (S 1, S 2, S 3 ) Let S i = 1 if i trades good i + 1 for good i + 2, and 0, otherwise. A pure strategy profile is S = (S 1, S 2, S 3 ). e.g. S 2 = Pr(agent II trades 3 for 1) 1 S 2 = Pr(agent II trades 1 for 3) i V i,i+1 V i,i+2 > 0 S i = 0

10 Steady state inventory distribution Let P 1 = P 12, P 2 = P 23, P 3 = P 31. P 1 P 2 S 1 = (1 P 1 )P 3 P 2 P 3 S 2 = (1 P 2 )P 1 solve for P = (P1, P2, P3 ) P 3 P 1 S 3 = (1 P 3 )P 2

11 Fundamental equilibrium: S = (0, 1, 0) Value functions Given others strategies and P, V 12 = 1 ( 1 + r ){ C [V 12 +P 21 (u 1 + V 12 ) + P 23 max(v 12, V 13 ) +V 12 ]} V 13 = 1 ( 1 + r ){ C [V 13 + V 13 +P 31 (u 1 + V 12 ) + P 32 max(v 12, V 13 )]}

12 Fundamental equilibrium: S = (0, 1, 0) Existence conditions: rv 12 = C [P 21u 1 + P 23 max(0, V 13 V 12 )] rv 13 = C [P 31(u 1 + V 12 V 13 ) + P 32 max(v 12 V 13, 0)] C 3 C }{{} 2 > 1 3 [P 31 P 21 ]u 1 }{{} storability marketability then V 12 > V 13 V 21 > V 23 and V 31 > V 32 for all parameter values and P i,j. Inventory distribution = (1, 1 2, 1)

13 Trejos and Wright (1995) If q = Q is taken as given, then rv s = αmx[v b V s c(q)] (8) rv b = α(1 M)x[u(Q) + V s V b ] (9) can be solved for V b = V b (Q) and V s = V s (Q) In the following, normalize αx = 1.

14 Trejos and Wright (1995): Definition of equilibrium When bargaining over q, agents take V b (Q) and V s (Q) as given. In eqm, q = Q. A steady-state equilibrium is a list (Q, V s, V b ) satisfying: (i) q = Q solves the bargaining problem q = arg max[u(q) + V s ][V b c(q)] s.t. u(q) + V s (Q) V b (Q) and V b (Q) c(q) V s (Q) }{{} participation conditions taking V b (Q) and V s (Q) as given; and (ii) V s and V b satisfy (8) and (9), taking Q as given.

15 Trejos and Wright (1995): Equilibrium The necessary and sufficient condition for q = arg max[u(q) + V s ][V b c(q)] is [V b c(q)]u (q) [u(q) + V s ]c (q) = 0. (10) We insert V b and V s, and set Q = q, (10) becomes a function q = e(q) [or, T (q) = 0]. Show T (0) = 0, T (0) > 0, T (q) < 0. By continuity, there exists a q (0, q) s.t. T (q) = 0.

16 Trejos and Wright (1995): Proposition 1 For any r > 0 and M (0, 1), there exists a nonmonetary steady state equilibrium and a unique monetary steady state equilibrium. The monetary equilibrium is unconstrained, and it satisfies u (q) > c (q).

17 Li and Wright (1998): divisible goods The basic model is Trejos and Wright (1995 JPE), except that we assume that in a meeting 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]. In any monetary em, take-it-or-leave-it offers implies Q = V 1 V 0. When multiple equilibria coexist, the low price eqm Pareto dominates the high price eqm, and both dominate the nonmonetary eqm.

18 Li and Wright (1998): government policy We set T = (1, 1, 1) and consider a policy that fixes the quantity of goods that government agents supply in exchange for money, 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].

19 Williamson and Wright (1994): Bellman s equations under private Info θ < 1 rv g = θp[θ + (1 θ)σ](u + W V g ) +(1 θ) max σ{p[θ + (1 θ)σ](u + W V g ) σ +(1 P)(W V g )} rv b = P(1 θ)σ(u + W V b )

20 International currency: Bellman s equations Π ij : probability that type i accepts money j. p ij : fraction of type i holding object j, j = 0 (goods), 1, 2 rv 10 = (p 11 + βp 21 ) max(v 11 V 10, 0) π 11 +(p 12 + βp 22 ) max(v 12 V 10, 0) π 12 rv 11 = (p 10 Π 11 + βp 20 Π 21 )(u + V 10 V 11 ) rv 12 = (p 10 Π 12 + βp 20 Π 22 )(u + V 10 V 12 ) Similar equations hold for country 2 agents.

21 Two national currencies Equilibrium (Π 12, Π 21 ) = (0, 0): no international currency. Distribution: P 12 = P 21 = 0, p 11 = 2M 1, P 22 = 2M 2. Bellman s equations: rv 10 = 2M 1 (V 11 V 10 ) rv 11 = (1 2M 1 )(u + V 10 V 11 ) rv 12 = β(1 2M 2 )(u + V 10 V 12 ) V 12 is defined as the life-time expected value that a country 1 agent had deviated from the candidate strategy; i.e. accepts and holds currency 2.

22 Two national currencies (con t) Best response conditions: { 1 if Vij > V Π ij = i0, 0 if V ij V i0. For Π 12 = 0 and Π 21 = 0 to be a best response, we need to check: V 21 V 20, and V 12 V 10, respectively. Given V 21 V 20 (i.e., Π 21 = 0), Π 12 = 0 is a best response iff β(1 + r)(1 2M 2 ) 2M 1 (1 2M 1 ) A symmetric condition holds for type 2 agents. These conditions hold if β is small (the degree of economic integration is low) but not if β is large.