Ryoji Hiraguchi, Faculty of Economics, Ritsumeikan University, 1-1-1, Noji-Higashi, Kusatsu, Shiga, Japan (rhira@fc.ritsumei.ac.jp).

Article

# Optimal Monetary Policy in OLG Models with Long-Lived Agents: A Note

Article first published online: 24 JUL 2013

DOI: 10.1111/jpet.12050

© 2013 Wiley Periodicals, Inc.

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#### How to Cite

HIRAGUCHI, R. (2014), Optimal Monetary Policy in OLG Models with Long-Lived Agents: A Note. Journal of Public Economic Theory, 16: 164–172. doi: 10.1111/jpet.12050

I thank two anonymous referees for their comments.

#### Publication History

- Issue published online: 7 JAN 2014
- Article first published online: 24 JUL 2013
- Accepted manuscript online: 21 MAY 2013 10:40AM EST
- Manuscript Accepted: 29 OCT 2011
- Manuscript Received: 14 FEB 2009

- Abstract
- Article
- References
- Cited By

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

This note reexamines Crettez, Michel, and Wigniolle (2002), who studied a two-period overlapping generations model with cash-in-advance constraints and showed that a combination of saving tax and monetary policy involving positive nominal interest rates could achieve the first-best allocation. The note shows that their result does not hold if agents live for three periods. The implementation of the first best requires the Friedman rule. If agents are long-lived, saving tax cannot offset a distortion caused by the positive nominal interest rate.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

Crettez, Michel, and Wigniolle (2002, henceforth CMW) provide a two-period overlapping generations (OLG) economy in which the Friedman rule (henceforth FR) does not apply. They investigate a Diamond-type model with partial cash-in-advance (CIA) constraints and show that the optimal monetary policy is indeterminate if saving tax is available. This feature reflects a redundancy of policy instruments.

We show that one way to reintroduce the FR is to add one period in such a way that it kills the redundancy of the instrument. In the three-period OLG model, the saving tax cannot offset the distortions by the positive nominal interest rate. To achieve the first best, the government must follow the FR, and the first best can actually be implemented as a competitive equilibrium under the FR.

Numerous authors study the FR in models with CIA constraints. With respect to the infinite horizons models, Dotsey and Sarte (2000) study a model without tax in which labor supply is inelastic and show that any monetary policy attains the first best when the model is deterministic. Ireland (2003) shows the unique optimality of the FR in a similar model with elastic labor supply. Chari and Kehoe (1999) get the same result in the cash-credit model with elastic labor supply in which the government uses linear consumption tax. With respect to the two-period OLG model with inelastic labor supply, Gahvari (2007) considers the model with partial CIA constraint in which the government uses linear consumption tax and lump-sum tax. He shows that the FR is not the only optimal policy since the policy instruments are redundant. His result is close to the one in CMW. On the other hand, Bhattacharya, Haslag, and Martin (2009) investigate a random relocation model without tax in which agents are subject to the CIA constraint with some probability and show that the FR is not optimal. In these models, the government expenditure is either zero or constant, while it is proportional to the output in CMW.

In the following, Section 'Three-Period OLG Model' sets up the model. Section 'Planner's Problem' investigates the first best. Section 'Necessity of the Friedman Rule' shows the optimality of the FR. Section 'Conclusion' concludes our paper. Proofs of the propositions are in the Appendix.

### 2. Three-Period OLG Model

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

#### 2.1. Set-Up

Time is denoted by . In each period, agents with measure 1/2 are born. They live for three periods, young, middle, and old. In period 0, there are 1/2 initial old agents and 1/2 initial middle agents. The agent born in period *t* has the utility function with and . Here, and are, respectively, the consumption during young, middle, and old, and β is the discount factor. He supplies one unit of labor when he is young and middle. His budget constraints are

where () is the saving when he is young (middle), () is the money holding when he is young (middle), is the price, is the wage, () is the wage (saving) tax, and is the interest rate. By nonarbitrage conditions, the net rate of return on saving must not be smaller than the rate of return on money. In the following, we write the net wage rate as .

Following CMW, we assume that each agent holds a constant fraction of values of his second-period and third-period consumptions in money. Thus, and . In the competitive equilibrium, . In what follows, we consider the money growth rate as a policy instrument.

#### 2.2. Competitive Equilibrium

In this section, we characterize a competitive equilibrium. We first define the allocation and the policy.

DEFINITION 1. *The* *allocation* is a vector where the initial capital *k*_{0} is given. The stationary allocation is a vector .

DEFINITION 2. *The* *policy* is a vector . *The* stationary *policy* is a vector .

Next, we obtain the budget constraint of the agents. Let denote the nominal interest rate. The budget constraint of the initial old is . Thus his consumption is constant. Similarly, the intertemporal budget constraints of the initial middle is written as . Recall that , , and are given. Here, we assume that the government sets the initial money supply so that it satisfies the CIA constraints for the initial old and the initial middle. Thus and . The budget constraints of the initial old and the initial middle can be reexpressed as

- (1)

- (2)

Finally, the intertemporal budget constraint of the consumer born in period *t* (⩾0) is

- (3)

At the left-hand side of Equation (3), the saving tax appears twice, while the nominal interest rate appears only once. These two policies affect the economy differently.

Third, we obtain the first-order condition (FOC). The initial old does not make any decision. The initial middle takes the initial money supply as given.1 Thus the FOC is

- (4)

The consumer born in period 0) maximizes his utility subject to his intertemporal budget constraint. The FOCs on , , and at the competitive equilibrium are written as

- (5)

- (6)

Fourth, we describe the behavior of the firm. One competitive firm produces the final good , where is capital, is labor, and *F* is the constant returns to scale production function. Since , , and , where . Capital is fully depreciated. The government expenditure is proportional to the final output with . The resource constraint is

- (7)

Fifth, we describe the problem of the government. The government finances the expenditure by taxes, bonds, and seigniorage. Let denote the amount of the government bonds which matures in period *t*. Its budget constraint is

Finally, we determine the price sequence. The aggregate money demand must be equal to the money supply and then

Thus, the price which clears the money market is

- (8)

and the nominal interest rate, is

- (9)

The next proposition characterizes the competitive equilibrium.

PROPOSITION 1. An allocation is a competitive equilibrium allocation under the policy if and only if it satisfies the seven equations (1), (2), (3), (4), (5), (6), and (7). Here, the price and the nominal interest rate are, respectively, given by Equations (8) and (9). The factor prices are , , , and .

Proof. See the Appendix. ▪

### 3. Planner's Problem

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

Let denote the social welfare, where is a social discount factor. The planner maximizes *V* subject to the resource constraint. The first-best allocation satisfies the initial condition , following the interior conditions

- (10)

- (11)

- (12)

and the feasibility condition

- (13)

### 4. Necessity of the Friedman Rule

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

This section shows that the optimal monetary policy must follow the FR.

#### 4.1. Steady State

Here we focus on the steady state. At a stationary allocation with a stationary policy , the budget constraint of the consumer is

- (14)

where , , and . The equilibrium FOCs are

- (15)

- (16)

The resource constraint is

- (17)

The next proposition characterizes the stationary equilibrium.

Note that we have four unknown, , and *k*, and also have four equations (14), (15), (16), and (17). Thus these equations fully characterize the competitive equilibrium. The first-best stationary allocation satisfies

- (18)

- (19)

- (20)

Equations (15), (16), (18), and (19) imply that to implement the allocation as a competitive equilibrium, we must have . Hence, we must follow the FR (i.e., ). From Equations (16) and (19) we get . Equation (20) implies that the saving tax must satisfy .

We now show that can actually be implemented as a competitive equilibrium. Let , , and . Consider a stationary policy such that , , and . We can easily check that satisfies the budget constraint and the FOCs of the consumer. Thus, implements as the competitive equilibrium. Note that the policy follows the FR.

If agents live only for two periods, a deviation from the FR can be neutralized by the saving tax. However, this result does not hold in the economy with long-lived agents. To achieve the first-best allocation, we must not rely on the monetary policy away from the FR. Note that the government expenditure is assumed to be proportional to the output and then we need positive saving tax to implement the first best as a competitive equilibrium.

#### 4.2. General Case

Here, we consider a nonstationary equilibrium. We first show that if the first-best allocation is implemented as a competitive equilibrium by some policy, it *must* follow the FR. We then show that there exists a policy that actually decentralizes the first-best allocation.

Suppose that a competitive equilibrium allocation coincides with the first best under some policy . In what follows, let and . In period , the FOC between young and middle, Equation (5), is written as

- (21)

Substitution of Equation (6) into (21) yields

- (22)

Hence, for all and then we must follow the FR. If , then Equations (5) and (10) together imply that the saving tax is given by for all . Thus, the net rate of return on saving must be .

Next we show that there exists a policy which implements the first-best allocation as a competitive equilibrium allocation. From the previous arguments, the policy must follow the FR (i.e., ) and satisfy for all . Consider the policy which satisfies the following five conditions.

(1) Saving tax :

- (23)

- (24)

(2) Wage tax :

- (25)

- (26)

(3) Money growth rate :

- (27)

Here we explain how we fix the policy instruments. First, the initial saving tax (23) comes from the initial old's budget constraint (1). Second, the saving tax in period *t* (24) is necessary to decentralize the first best. Third, the initial period wage tax (25) comes from the initial middle's budget constraint (2). Fourth, the wage tax in period (26) is determined by the budget constraint of the agents born in period *t* (3). Finally, the money growth rate (27) is obtained by setting the nominal interest rate to zero in Equation (9).

The next proposition shows that the allocation satisfies the equilibrium conditions in Proposition 1.

PROPOSITION 3. Under the policy , the first-best allocation satisfies all the equilibrium conditions (i.e., Equations (1), (2), (3), (4), (5), (6), and (7)) .

Proof. See the Appendix. ▪

Thus we can conclude that the first-best allocation can actually be implemented as a competitive equilibrium and that the optimal monetary policy follows the FR.

### 5. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

We show that in a three-period OLG model with CIA constraints, the optimal policy must follow the FR. CMW show that the optimal monetary policy can deviate from the FR but they depend on the assumption that the agents live for two periods. In general, saving tax cannot offset distortions by a positive nominal interest rate.

We admit that the result in this note is weak in the sense that the result is based on the special assumptions that young and middle-aged agents offer *fully inelastic* labor supply and that middle-aged and old age consumption is subject to the *partial* CIA-constraint. As a future study, we hope to study the optimal policy in a more general economy.

### Appendix

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

Proof of Proposition 1. Necessity is obvious. Here we prove sufficiency. Suppose that the price satisfies Equation (8) and that the factor prices are and . We show that when Equations (1)–(7) hold, the allocation *A* maximizes the firm's profit and the consumer's utility, it clears money market and that it is feasible. First, the firm maximizes its profit since the factor prices are competitive. Second, every consumer maximizes his utility because the utility function is strictly concave and the allocation *A* satisfies both his budget constraint and the FOCs. Third, Equation (8) implies that the money market clears. Finally, Equation (7) implies the feasibility. ▪

Proof of Proposition 3. We show the satisfactions of the seven equations separately. Following Proposition 1, we set the price sequence such that it clears the money market. Then it satisfies for all . In this case, from Equation (26), for all . Recall that and then the net rate of return on saving is for all .

First, Equation (23) is the same as Equation (1). Second, Equation (2) is the same as Equation (25) when . Third, Equation (3) coincides with Equation (26) if . Fourth, Equation (4) holds since Equations (10), (11) and (12) imply

Fifth, Equation (12) implies

- (A1)

Since , Equation (5) holds. Sixth, Equations (10), (11), and (12) imply

Since , Equation (6) holds. Finally, Equation (7) holds because the allocation satisfies the resource constraint (13). ▪

- 1
He maximizes his utility subject to the budget constraint const.

### References

- Top of page
- Abstract
- 1. Introduction
- 2. Three-Period OLG Model
- 3. Planner's Problem
- 4. Necessity of the Friedman Rule
- 5. Conclusion
- Appendix
- References

- 2009) Optimal monetary policy and economic growth, European Economic Review 53, 210–221. , , and (
- 1999) Optimal fiscal and monetary policy, in Handbook of Macroeconomics, J. Taylor and M. Woodford, eds., pp. 1671–1745. Amsterdam: Elsevier. , and (
- 2002) Optimal monetary policy, taxes, and public debt in an intertemporal equilibrium, Journal of Public Economic Theory 4, 299–316. , , and (
- 2000) Inflation uncertainty and growth in a cash-in-advance economy, Journal of Monetary Economics 45, 631–645. and (
- 2007) The Friedman rule: Old and new, Journal of Monetary Economics 54, 581–589. (
- 2003) Implementing the Friedman rule, Review of Economic Dynamics 6, 120–134. (