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Optimal Monetary Policy in an Overlapping Generations Model with Search Theoretic Monetary Exchange

  • Ryoji Hiraguchi EMAIL logo
Veröffentlicht/Copyright: 29. April 2017
The B.E. Journal of Theoretical Economics
Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 17 Heft 2

Abstract

It is well-known that in the monetary OLG models, a deviation from the Friedman rule can improve welfare because it generates intergenerational wealth transfers; however, the rule becomes optimal if the age-specific lump-sum tax policy is available. We revisit the issue using a microfounded model of money with centralized and decentralized markets. The individuals live for two periods. The young individuals work, receive wage income and hold money and capital in the centralized market. They also trade goods in the decentralized markets either as a buyer or a seller. Only money is accepted as a means of payment in the decentralized markets. The old individuals consume all their wealth in the centralized market. The quantity in the decentralized market negatively depends on the seller’s wealth, because the marginal utility of consumption in the centralized market is diminishing, but the buyer takes it as exogenous. Therefore, the equilibrium wealth exceeds the socially optimal level under the Friedman rule. A positive nominal interest rate makes money holdings costly, reduces wealth and improves welfare, even if the government optimally uses the age-specific tax.

Keywords: search; money; overlapping generations

Appendix

In Section A, we provide proofs of propositions.

A Proofs of Propositions

A.1 Proposition 1

Equations (21), (22), (23), (24), and (25) correspond to the equilibrium conditions (15), (16), (19), (18), and (17), respectively. Therefore, the necessity is obvious. In the following, we demonstrate the sufficiency condition. Suppose we find an allocation (k∗,l∗,q∗,c∗,m∗), the government policy (τ∗,T∗), and the price of money ϕt=m∗/M¯t satisfying the five conditions. Let w∗=w(k∗/l∗) and R∗=R(k∗/l∗) denote the factor prices. In that case, eq. (24) implies the budget constraint of the young since

(52)w∗l∗=ϕtM¯t+1+k∗+T∗

In terms of the first order condition, eq. (21) immediately implies

(53)g′(l∗)w∗=R∗U′(c∗−m∗)+U′(c∗)2,

Furthermore, if we let q∗=U(c∗+m∗)−U(c∗) and λ∗={U′(c∗+m∗)u′(q∗)−U′(c∗−m∗)}/2, eq. (25) implies

(54)g′(l∗)w∗=ϕt+1ϕt[U′(c∗−m∗)+U′(c∗)2+λ∗],

Equations (53), (54), and (54) imply that (k∗,l∗,m∗) solves eq. (8). ◼

A.2 Proposition 2

The necessity condition is obvious. Here, we show the sufficiency condition. Suppose we find the allocation (k∗,l∗,c∗,q∗,m∗) satisfying eqs (21), (22), (23), and (27). In the following, we show that the allocation satisfies the five equilibrium conditions (21), (22), (23), (24), and (25). By assumption, eqs (21), (22), and (23) hold true. Therefore, we only have to show eqs (24) and (25). If we determine the money growth rate τ∗ by

R∗(1+τ∗)−1=w∗R∗2g′(l∗){U′(c∗+m∗)u′(q∗)−U′(c∗−m∗)},

Equation (25) holds true. Equation (27) then implies that the nominal interest rate R∗(1+τ∗)−1 is non-negative. If we let T∗=w∗l∗−k∗−(1+τ∗)m∗, the allocation (k∗,l∗,c∗,q∗,m∗) satisfies the budget constraint of the young, eq. (24) when the government policy is (τ∗,T∗). ◼

A.3 Proposition 3

Since k=xl and F(k,l)=Axαl, c=(Axα−δx)l. Equation (29) implies that

l1+η=A2(1−α)α2x1−2α(Axα−δx)2−y1−y.

This equality uniquely determines l={2−y1−yA2(1−α)α2(A−δx1−α)x1−α}1/(1+η), given x and y, Similarly, k=xl, c=(Axα−δx)l and m=cy are also uniquely determined. Consumption is positive if Axα>δx, and this is equivalent to x<(Aδ)11−α, and eq. (27) holds and then the nominal interest rate is positive if y≤y¯. ◼

Acknowledgment

We are grateful to two anonymous referees for their valuable comments.

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Published Online: 2017-4-29

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

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  2. Optimal Monetary Policy in an Overlapping Generations Model with Search Theoretic Monetary Exchange
  3. Getting a Job through Unemployed Friends: A Social Network Perspective
  4. Dynamic Stability of Post-Keynesian Pricing
  5. A Nonspeculation Theorem with an Application to Committee Design
  6. Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness
  7. Privatization Neutrality Theorem in Free Entry Markets
  8. Strong Forward Induction
  9. The Case of “Less is More”: Modelling Risk-Preference with Expected Downside Risk
  10. Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies
  11. Notes
  12. A Height-Based Multidimensional Extension of the Lorenz Preorder for Integer-Valued Distributions
https://doi.org/10.1515/bejte-2016-0039
Schlagwörter für diesen Artikel
search; money; overlapping generations

Artikel in diesem Heft

  1. Research Articles
  2. Optimal Monetary Policy in an Overlapping Generations Model with Search Theoretic Monetary Exchange
  3. Getting a Job through Unemployed Friends: A Social Network Perspective
  4. Dynamic Stability of Post-Keynesian Pricing
  5. A Nonspeculation Theorem with an Application to Committee Design
  6. Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness
  7. Privatization Neutrality Theorem in Free Entry Markets
  8. Strong Forward Induction
  9. The Case of “Less is More”: Modelling Risk-Preference with Expected Downside Risk
  10. Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies
  11. Notes
  12. A Height-Based Multidimensional Extension of the Lorenz Preorder for Integer-Valued Distributions
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