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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. 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

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

2.1. Set-Up

Time is denoted by inline image. 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 inline image with inline image and inline image. Here, inline image and inline image 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

  • display math

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

Following CMW, we assume that each agent holds a constant fraction inline image of values of his second-period and third-period consumptions in money. Thus, inline image and inline image. In the competitive equilibrium, inline image. In what follows, we consider the money growth rate inline image 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  inline image where the initial capital k0 is given. The stationary allocation is a vector inline image.

DEFINITION 2. The policy is a vector inline image. The stationary policy is a vector inline image.

Next, we obtain the budget constraint of the agents. Let inline image denote the nominal interest rate. The budget constraint of the initial old is inline image. Thus his consumption is constant. Similarly, the intertemporal budget constraints of the initial middle is written as inline image. Recall that inline image, inline image, inline image and inline image 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 inline image and inline image. The budget constraints of the initial old and the initial middle can be reexpressed as

  • display math(1)
  • display math(2)

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

  • display math(3)

At the left-hand side of Equation (3), the saving tax inline image appears twice, while the nominal interest rate inline image 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 inline image as given.1 Thus the FOC is

  • display math(4)

The consumer born in period inline image0) maximizes his utility subject to his intertemporal budget constraint. The FOCs on inline image, inline image, and inline image at the competitive equilibrium are written as

  • display math(5)
  • display math(6)

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

  • display math(7)

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

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

  • display math

Thus, the price inline image which clears the money market is

  • display math(8)

and the nominal interest rate, inline image is

  • display math(9)

The next proposition characterizes the competitive equilibrium.

PROPOSITION 1. An allocation inline image is a competitive equilibrium allocation under the policy inline image 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 inline image, inline image, inline image, and inline image.

Proof. See the Appendix.     ▪

3. Planner's Problem

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

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

  • display math(10)
  • display math(11)
  • display math(12)

and the feasibility condition

  • display math(13)

4. Necessity of the Friedman Rule

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. 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 inline image with a stationary policy inline image, the budget constraint of the consumer is

  • display math(14)

where inline image, inline image, and inline image. The equilibrium FOCs are

  • display math(15)
  • display math(16)

The resource constraint is

  • display math(17)

The next proposition characterizes the stationary equilibrium.

PROPOSITION 2. An allocation inline image is a stationary competitive equilibrium allocation under the policy inline image if and only if it satisfies Equations (14), (15), (16), and (17), where inline image, inline image, and inline image.

Note that we have four unknown, inline image, 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 inline image satisfies

  • display math(18)
  • display math(19)
  • display math(20)

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

We now show that inline image can actually be implemented as a competitive equilibrium. Let inline image, inline image, and inline image. Consider a stationary policy inline image such that inline image, inline image, and inline image. We can easily check that inline image satisfies the budget constraint and the FOCs of the consumer. Thus, inline image implements inline image as the competitive equilibrium. Note that the policy inline image 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 inline image 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 inline image under some policy inline image. In what follows, let inline image and inline image. In period inline image, the FOC between young and middle, Equation (5), is written as

  • display math(21)

Substitution of Equation (6) into (21) yields

  • display math(22)

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

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

(1) Saving tax inline image:

  • display math(23)
  • display math(24)

(2) Wage tax inline image:

  • display math(25)
  • display math(26)

(3) Money growth rate inline image:

  • display math(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 inline image(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 inline image to zero in Equation (9).

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

PROPOSITION 3. Under the policy inline image, the first-best allocation inline image 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. 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 inline image and inline image. 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 inline image such that it clears the money market. Then it satisfies inline image for all inline image. In this case, from Equation (26), inline image for all inline image. Recall that inline image and then the net rate of return on saving is inline image for all inline image.

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

  • display math

Fifth, Equation (12) implies

  • display math(A1)

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

  • display math

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

  1. 1

    He maximizes his utility inline image subject to the budget constraint inline imageconst.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three-Period OLG Model
  5. 3. Planner's Problem
  6. 4. Necessity of the Friedman Rule
  7. 5. Conclusion
  8. Appendix
  9. References
  • BHATTACHARYA, J., J. HASLAG, and A. MARTIN (2009) Optimal monetary policy and economic growth, European Economic Review 53, 210221.
  • CHARI, V., and P. KEHOE (1999) Optimal fiscal and monetary policy, in Handbook of Macroeconomics, J. Taylor and M. Woodford, eds., pp. 16711745. Amsterdam: Elsevier.
  • CRETTEZ, P., B. MICHEL, and B. WIGNIOLLE (2002) Optimal monetary policy, taxes, and public debt in an intertemporal equilibrium, Journal of Public Economic Theory 4, 299316.
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  • DOTSEY, M. and P. SARTE (2000) Inflation uncertainty and growth in a cash-in-advance economy, Journal of Monetary Economics 45, 631645.
  • GAHVARI, F. (2007) The Friedman rule: Old and new, Journal of Monetary Economics 54, 581589.
  • IRELAND, P. (2003) Implementing the Friedman rule, Review of Economic Dynamics 6, 120134.