An introduction of a simple monetary policy withsavings taxation in the overlapping generations
model
Taro Ikeda

March 2018
Discussion Paper No.1810

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANAn introduction of a simple monetary policy with savings taxation in the overlapping generations model
Taro Ikeda
Graduate School of Economics, Kobe University
March 6, 2018
Abstract
In this paper, we introduce a simple monetary policy with savings taxation into Samuelson’s (1958)
overlapping generations model. In our model, we confirm that the real market interest rate increases in
response to an increase in the rate of the savings taxation as a policy lending rate.
Keywords: overlapping generations model; monetary policy; savings taxation
JEL classifications: E10; E20; E52

1. Introduction
In this paper, we introduce a simple monetary policy with the taxation of savings into Samuelson’s (1958)
overlapping generations model.1 We assume that only the central bank provides the means for individuals of
current generations to save: the central bank imposes a savings tax rate (lending rate) 𝜏 on one unit of individual
savings demand, and provides (1 − 𝜏) units of savings to the individuals of the current generation. In turn, theimprovement of the central bank’s balance sheet that results from this lending provides the same amount of realmoney supply for the young people of the next generations.2
This transfer of a real quantity of money increases the endowments of the individuals. Therefore, the realmoney transfer may improve the utility of individuals by increasing their consumption feasibility. In this sense,
our model answers the question “Why do individuals hold money?” with the response “Because it is obligatorily
given by the policy” In our model, money does not yield direct utility for individuals, but it is held by agents as anadditional endowment as a result of the monetary policy requirements.
This paper mainly focuses on the fundamental equilibrium condition defined by Samuelson (1958). The
fundamental equilibrium condition implies that savings are cleared, as was excess demand for goods, such thatthey equal zero at the equilibrium. In particular, we consider how the equilibrium market interest rate responds tothe change of the lending rate set by the savings taxation policy.3 In other words, we are interested in the mannerin which the market interest rate safeguards the equilibrium against the policy interventions.

 Research Fellow, Graduate School of Economics, Kobe University, Japan (Email: taroikd@people.kobe-u.ac.jp).
1

We restrict the discussion to the case of agents who live two periods. In addition, most of the variablesconsidered are measured in real terms.
2

Throughout the paper, we assume that 𝜏 < 1. This implies that the central bank cannot require more goods
from the individual than the quantity that he/she originally provided to the bank3

As purpose of this paper is to provide the most basic model based on Samuelson (1958), we do not discuss

welfare issues as well as production problems and others.
1This simple model suggests that the market interest rate moves in the same direction as the policy
lending rate under the fundamental equilibrium conditions.
2. Introduction of the monetary policy
Suppose that the young people of generation 𝑡 cannot make savings—that is, make consumption-loans
to themselves in old age—without transforming a portion 𝑠𝑡 (𝑡) of their 𝑡-period goods into savings throughcentral banking, where the only central bank is the sole provider of savings.
However, transforming goods into savings imposes a tax of rate 𝜏on the young: after receiving lending

revenue 𝜏𝑠𝑡 (𝑡) from the young, the central bank gives them an amount of savings equal to (1 − 𝜏)𝑠𝑡 (𝑡). Then, inturn, the young people of generation 𝑡 lend their savings (1 − 𝜏)𝑠𝑡 (𝑡) to themselves when they are old, based on
the market interest rate 𝑟(𝑡). Finally, the young people obtain their savings with an after-tax-interest rate,(1 − 𝜏)𝑟(𝑡)𝑠𝑡 (𝑡), in old age.4
The lending revenue of the central bank increases the assets side of the central bank balance sheet. Then,
we assume that the central bank issues new real money to provide for the young people of generation 𝑡 + 1. The
new real money, which is the same amount as the lending revenue, increases the liabilities side of the sheet, andbalances the account. As a result, our monetary policy satisfies:(1)

𝑚𝑡+1 (𝑡 + 1) = 𝜏𝑠𝑡 (𝑡).

By the balance sheet equation (1), an increase of the lending interest rate 𝜏 leads to an increase of the realmoney transfer to the young of the next generation, 𝑚𝑡+1 (𝑡 + 1). The real money transfer does not requiresignificant time lags because the taxation of savings in period 𝑡 is implemented at the end of the period 𝑡.
3. Individual optimization
In the purest Marshallian case, the individual of generation 𝑡 has a log-utility function expressed as𝑢𝑡 (𝑐𝑡 (𝑡), 𝑐𝑡 (𝑡 + 1)) = log 𝑐𝑡 (𝑡) + log 𝑐𝑡 (𝑡 + 1) (Samuelson, 1958). Then, the maximization problem of the agent
becomes as follows:
(2)

∑𝑡+1
𝑖=𝑡 log 𝑐𝑡 (𝑖) subject to:𝑐𝑡 (𝑡) +𝑐𝑡 (𝑡 + 1)
𝜔𝑡 (𝑡 + 1)
= [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +,(1 − 𝜏)𝑟(𝑡)(1 − 𝜏)𝑟(𝑡)where 𝑐𝑡 (𝑡) and 𝑐𝑡 (𝑡 + 1) denote the consumption amounts of generation 𝑡 for youth 𝑡 and old age 𝑡 + 1,
respectively. In addition, 𝜔𝑡 (𝑡) and 𝜔𝑡 (𝑡 + 1) denote the endowments of generation 𝑡 for youth and old age,respectively. Note the fact that, in the lifetime budget constraint, the interest rate imposed on savings 𝑠𝑡 (𝑡)corresponds to the after-tax-interest rate (1 − 𝜏)𝑟(𝑡) and the real money transfer 𝑚𝑡 (𝑡) is added to theendowment 𝜔𝑡 (𝑡) in the agent’s youth.
Then, the Euler equation of this problem becomes:𝜕𝜕𝑢𝑡 ⁄𝑢 = (1 − 𝜏)𝑟(𝑡),𝜕𝑐𝑡 (𝑡)𝜕𝑐𝑡 (𝑡 + 1) 𝑡

4

If there are no special conditions and the market interest rate 𝑟(𝑡) is constant, an increase of lending rate 𝜏 inperiod 𝑡 decreases the after-tax-interest rate (1 − 𝜏)𝑟(𝑡) in period 𝑡 + 1. Therefore, the well-known positive
relationship between the interest rate and the money supply is superficially preserved in this model.
2and, therefore, we obtain the following:(3)

𝑐𝑡 (𝑡) =1𝑐 (𝑡 + 1).
(1 − 𝜏)𝑟(𝑡) 𝑡

By equation (3) and the lifetime budget constraint, the optimal savings functions for the individual’s youth andold age, respectively, can be derived as follows:(4)1
𝜔𝑡 (𝑡 + 1)
𝑠𝑡 (𝑡) ≡ [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] − 𝑐𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] −,22(1 − 𝜏)𝑟(𝑡)(5)(1 − 𝜏)𝑟(𝑡)1[𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)].
𝑠𝑡 (𝑡 + 1) ≡ 𝜔𝑡 (𝑡 + 1) − 𝑐𝑡 (𝑡 + 1) = 𝜔𝑡 (𝑡 + 1) −2
2

4. Fundamental equilibrium conditions
This section provides the fundamental equilibrium conditions defined by Samuelson (1958).
For any 𝑡 ≥ 1, there exists 𝑁(𝑡) young of generation 𝑡 and 𝑁(𝑡 − 1) old of generation 𝑡 − 1. Then,the aggregate savings of each period yield the following fundamental equilibrium condition:(6)

0 = 𝑁(𝑡)𝑠𝑡 (𝑡) + 𝑁(𝑡 − 1)𝑠𝑡−1 (𝑡),for all 𝑡 ≥ 1.
This condition requires good market clearing, in the sense that the savings play the role of excess
demand for goods. We also assume that the population is finite and constant over time (i.e., 𝑁(𝑡) = 𝑁(𝑡 − 1) =𝑁) to avoid difficulties for the aggregations within the periods.
5. Equilibrium
The population of the economy begins in period 0. In period 0, young people of generation 0 are born,
and they are the only people who exist in this period. Then, in period 1, the young of generation 1 are born, andthey coexist with the old of generation 0 (who were young in period 0).5 Thereafter, the age distribution of theyoung and old is reproduced forever.
Here, we calculate the fundamental equilibrium condition (6) with 𝑁(𝑡) = 𝑁(𝑡 − 1) = 𝑁:
0 = 𝑠𝑡 (𝑡) + 𝑠𝑡−1 (𝑡),for each time period. Note that the real money supply 𝑚0 (0) is always zero because there were no people saving
before period 0.
Below, we present the fundamental equilibrium conditions in each period.
Period 0 (in which 𝜏𝑠−1 (−1) = 𝑚0 (0) = 0):1
𝜔0 (1)𝑠0 (0) + 𝑠−1 (0) = { [𝜔0 (0) + 𝑚0 (0)] −} + 0 = 0,22(1 − 𝜏)𝑟(0)1𝜔0 (1)𝑠0 (0) + 𝑠−1 (0) = { [𝜔0(0)] −} + 0 = 0,22(1 − 𝜏)𝑟(0)𝜔0 (0) =𝜔0 (1).
(1 − 𝜏)𝑟(0)Period 1 (in which 𝜏𝑠−1 (−1) = 𝑚0 (0) = 0 and 𝑠0 (0) = 0 by 𝜔0 (0)):
5The population of generation 0 disappears at the end of period 1.
3

(1 − 𝜏)𝑟(0)1𝜔1 (2)
1[𝜔0 (0) + 𝑚0 (0)]} = 0,
𝑠1 (1) + 𝑠0 (1) = { [𝜔1 (1) + 𝑚1 (1)] −} + { 𝜔0 (1) −2
2(1 − 𝜏)𝑟(1)22𝑠1 (1) + 𝑠0 (1) = {𝜔1 (1) −𝜔1 (2)} + {0} = 0,
(1 − 𝜏)𝑟(1)𝜔1 (2).
(1 − 𝜏)𝑟(1)𝜔1 (1) =Period 2 (preconditions are abbreviated):(1 − 𝜏)𝑟(1)1𝜔2 (3)1[𝜔1 (1) + 𝑚1 (1)]} = 0,
𝑠2 (2) + 𝑠1 (2) = { [𝜔2 (2) + 𝑚2 (2)] −} + { 𝜔1 (2) −2
2(1 − 𝜏)𝑟(2)22𝑠2 (2) + 𝑠1 (2) = {[𝜔2 (2)] −𝜔2 (3)} + {0} = 0,
(1 − 𝜏)𝑟(2)𝜔2 (3).
(1 − 𝜏)𝑟(2)𝜔2 (2) =⋮By these conditions, we can deduce the fundamental equilibrium condition for period 𝑡 as follows:(7)𝜔𝑡 (𝑡) =𝜔𝑡 (𝑡 + 1).
(1 − 𝜏)𝑟(𝑡)Therefore, the real market interest rate 𝑟(𝑡) satisfies the following relationship:(8)(1 − 𝜏)𝑟(𝑡) =𝜔𝑡 (𝑡 + 1),
𝜔𝑡 (𝑡)

under the fundamental equilibrium condition. Accordingly, the equilibrium market interest rate 𝑟(𝑡) offsets thechange of 𝜏 to retain a fraction of the individual’s lifetime endowments at each time period.6
2

1.5

𝒓 𝒕

1

𝒊(𝒕)

0.5

𝝉
0
-0.5

-0.3

-0.1

0.1

0.3

0.5

-0.5

Figure 1. Relationship between lending interest rate 𝝉 and market interest rate 𝒓(𝒕) in the fundamental
equilibrium with lifetime endowment (𝝎𝒕 (𝒕), 𝝎𝒕 (𝒕 + 𝟏)) = (𝟏, 𝟏)

6

The equilibrium market interest rate (7) clears the two sets of savings in each period (𝑠𝑡 (𝑡), 𝑠𝑡−1 (𝑡)) separately.

Therefore, considering money in this equilibrium is meaningless.
4

Figure 1 represents the response of the real gross market interest 𝑟(𝑡) to the savings tax rate 𝜏 in thefundamental equilibrium, given the lifetime endowment (𝝎𝒕 (𝒕), 𝝎𝒕 (𝒕 + 𝟏)) = (𝟏, 𝟏). In this specified economy,
the market interest rate rises in response to the increase of the policy lending rate (the rate of savings taxation).
Further, in this zero-growth equilibrium, a negative rate of savings taxation implies a negative net market interest
rate, 𝑖(𝑡) ≡ 𝑟(𝑡) − 1.
6. Constant growth within generation and equilibrium market interest rate
So far, we have investigated the proportional relationship between the market interest rate and the rateof savings taxation in the equilibrium. However, we have not yet determined the equilibrium market interest rate
itself. Therefore, let us consider the fundamental equilibrium with a constant economic growth rate within

generation, 𝑔 ≡𝜔𝑡 (𝑡+1)−𝜔𝑡 (𝑡)∈ (−1,1) (see Figure 2 for example).
𝜔𝑡 (𝑡)

𝝎𝒕 𝒕 , 𝝎𝒕 𝒕 + 𝟏
𝜔0 1

1.6
1.4
1.2

𝜔0 0

1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

8

9

10

11

𝒕
generation 0

generation 1

generation 2

generation 3

generation 4

generation 6

generation 7

generation 8

generation 9generation 10

generation 5

Figure 2. Constant growth rate within generation with 𝒈 = 𝟎. 𝟓 and 𝝎𝒕 (𝒕) = 𝟏.
From the assumption, it follows that:
(9)

1+𝑔 =

𝜔𝑡 (𝑡 + 1).
𝜔𝑡 (𝑡)

Inserting (9) into (8), the equilibrium real market interest rate satisfies the following:
(10)

𝑟=

1+𝑔
.
1−𝜏

In the fundamental equilibrium with a constant growth rate 𝑔, the market interest rate moves to keep theafter-tax-interest rate at (1 + 𝑔) in response to the change of the savings taxation rate. How? For 𝜏 < 1, themarket interest rate satisfies:
(11)

𝜕
1 2
𝑟 = (1 + 𝑔) () > 0.
𝜕𝜏1−𝜏Therefore, the equilibrium market interest rate is increasing in the rate of the savings taxation.
5

In sum, we confirm that the market interest rate increases in response to the monetary policy interest
rate increasing.
7. Conclusion
In this paper, we introduce a simple monetary policy involving savings taxation into Samuelson’s (1958)
overlapping generations model. We confirm that, in our model, the real market interest rate increases in response
to a rise of the rate of the savings taxation as a policy lending rate.
Reference
Samuelson, Paul A. 1958. “An Exact Consumption-Loan Model of Interest with or without the Social Contrivanceof Money.” Journal of Political Economy, Vol. 66, No. 6, December, pp. 467–482.

6

Appendix for “An introduction of a simple monetary policy by saving taxations in the overlapping generations

model”March 6, 2018

A1. Calculations for “Individual optimization (Section 3)”
Let us consider the following individual maximization problem:𝑡+1(1 − 𝑎)∑ log 𝑐𝑡 (𝑖) subject to𝑖=𝑡

𝑐𝑡 (𝑡) +𝑐𝑡 (𝑡 + 1)
𝜔𝑡 (𝑡 + 1)
= [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +.
(1 − 𝜏)𝑟(𝑡)(1 − 𝜏)𝑟(𝑡)The lifetime budget constraint in (1 − 𝑎) can be split into the budget constraints of young and old as follows:𝑐𝑡 (𝑡) + 𝑠𝑡 (𝑡) ≤ 𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡),𝑐𝑡 (𝑡 + 1) ≤ 𝜔𝑡 (𝑡 + 1) + (1 − 𝜏)𝑟(𝑡)𝑠𝑡 (𝑡).
At this point, note that we define the individual savings of each age as 𝑠𝑡 ≡ 𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡) − 𝑐𝑡 (𝑡), 𝑠𝑡 (𝑡 + 1) ≡𝑟(𝑡)𝑠𝑡 (𝑡).
Then, we set the Lagrangian for the maximization problem as follows:
𝑡+1ℒ = ∑ log 𝑐𝑡 (𝑖) + 𝜆 {[𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +𝑖=𝑡

𝜔𝑡 (𝑡 + 1)
𝑐𝑡 (𝑡 + 1)
− [ 𝑐𝑡 (𝑡) +]}.
(1 − 𝜏)𝑟(𝑡)(1 − 𝜏)𝑟(𝑡)Then, we obtain the first-order conditions as follows:𝜕1ℒ=− 𝜆 = 0,𝜕𝑐𝑡 (𝑡)𝑐𝑡 (𝑡)
𝜕
1
1
ℒ=
−𝜆
= 0.
(1 − 𝜏)𝑟(𝑡)𝜕𝑐𝑡 (𝑡 + 1)
𝑐𝑡 (𝑡 + 1)
Accordingly,
1
1
1
=
,
𝑐𝑡 (𝑡 + 1) 𝑐𝑡 (𝑡) (1 − 𝜏)𝑟(𝑡)𝑐𝑡 (𝑡 + 1)
1
1=⁄
= (1 − 𝜏)𝑟(𝑡).
𝑐𝑡 (𝑡)
𝑐𝑡 (𝑡) 𝑐𝑡 (𝑡 + 1)
Therefore, we obtain(2 − 𝑎)𝑐𝑡 (𝑡) =1𝑐 (𝑡 + 1).
(1 − 𝜏)𝑟(𝑡) 𝑡Inserting (2 − 𝑎) into the lifetime budget constraint, we obtain𝑐𝑡 (𝑡) + 𝑐𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +𝜔𝑡 (𝑡 + 1).
(1 − 𝜏)𝑟(𝑡)Then, we drive the optimal consumption demands for the individual when young and old as follows:1𝜔𝑡 (𝑡 + 1)
𝑐𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +,22(1 − 𝜏)𝑟(𝑡)

And,
1
1
𝑐𝑡 (𝑡 + 1) = (1 − 𝜏)𝑟(𝑡)[𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] + 𝜔𝑡 (𝑡 + 1).
2
2
In addition, we obtain the optimal saving functions for the individual when young and old as:1
𝜔𝑡 (𝑡 + 1)
𝑠𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] − 𝑐𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] − { [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] +},22(1 − 𝜏)𝑟(𝑡)1𝜔𝑡 (𝑡 + 1)
𝑠𝑡 (𝑡) = [𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] −,22(1 − 𝜏)𝑟(𝑡)And,𝑠𝑡 (𝑡 + 1) = 𝜔𝑡 (𝑡 + 1) − 𝑐𝑡 (𝑡 + 1) = 𝜔𝑡 (𝑡 + 1) − {(1 − 𝜏)𝑟(𝑡)1[𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)] + 𝜔𝑡 (𝑡 + 1)},2
2

(1 − 𝜏)𝑟(𝑡)1[𝜔𝑡 (𝑡) + 𝑚𝑡 (𝑡)].
𝑠𝑡 (𝑡 + 1) = 𝜔𝑡 (𝑡 + 1) −2
2
A2. Calculations for “Equilibrium (Section 5)”
Here, we calculate the fundamental equilibrium condition (6) with 𝑁(𝑡) = 𝑁(𝑡 − 1) = 𝑁:
0 = 𝑠𝑡 (𝑡) + 𝑠𝑡−1 (𝑡),by each period. Note that the real money supply 𝑚0 (0) is always zero because we find no people with savings
prior to period 0.
Below, we present the fundamental equilibrium conditions in each period.
Period 0 (with 𝜏𝑠−1 (−1) = 𝑚0 (0) = 0):1
𝜔0 (1)𝑠0 (0) + 𝑠−1(0) = { [𝜔0 (0) + 𝑚0 (0)] −} + 0 = 0,22(1 − 𝜏)𝑟(0)1𝜔0 (1)𝑠0 (0) + 𝑠−1 (0) = { [𝜔0 (0)] −} + 0 = 0,22(1 − 𝜏)𝑟(0)𝜔0 (0) =𝜔0 (1).
(1 − 𝜏)𝑟(0)Period 1 (with 𝜏𝑠−1 (−1) = 𝑚0 (0) = 0 and 𝑠0 (0) = 0 by 𝜔0 (0)):(1 − 𝜏)𝑟(0)1𝜔1 (2)
1[𝜔0 (0) + 𝑚0 (0)]} = 0,
𝑠1 (1) + 𝑠0 (1) = { [𝜔1 (1) + 𝑚1 (1)] −} + { 𝜔0 (1) −2
2(1 − 𝜏)𝑟(1)22(1 − 𝜏)𝑟(0)1𝜔1 (2)1
𝜔0 (1)𝑠1 (1) + 𝑠0 (1) = { [𝜔1(1)] −} + { 𝜔0 (1) −[]} = 0,(1 − 𝜏)𝑟(0)22(1 − 𝜏)𝑟(1)22𝑠1 (1) + 𝑠0 (1) = {𝜔1 (1) −𝜔1 (1) =𝜔1 (2)} + {0} = 0,
(1 − 𝜏)𝑟(1)𝜔1 (2).
(1 − 𝜏)𝑟(1)Period 2 (preconditions are abbreviated):(1 − 𝜏)𝑟(1)1𝜔2 (3)1[𝜔1 (1) + 𝑚1 (1)]} = 0,
𝑠2 (2) + 𝑠1 (2) = { [𝜔2 (2) + 𝑚2 (2)] −} + { 𝜔1 (2) −2
2(1 − 𝜏)𝑟(2)22(1 − 𝜏)𝑟(1)1𝜔2 (3)1
𝜔1 (2)𝑠2 (2) + 𝑠1 (2) = { [𝜔2(2)] −} + { 𝜔1 (2) −[]} = 0,(1 − 𝜏)𝑟(1)22(1 − 𝜏)𝑟(2)22𝑠2 (2) + 𝑠1 (2) = {[𝜔2 (2)] −𝜔2 (2) =𝜔2 (3)} + {0} = 0,
(1 − 𝜏)𝑟(2)𝜔2 (3).
(1 − 𝜏)𝑟(2)Using these conditions, we deduce the fundamental equilibrium condition for period 𝑡 as follows:(3 − 𝑎)𝜔𝑡 (𝑡) =𝜔𝑡 (𝑡 + 1).
(1 − 𝜏)𝑟(𝑡)A3. Calculations for “Constant growth within generation and equilibrium market interest rate (Section 6)”
In the main text, we obtain the equilibrium market interest rate as(4 − 𝑎)𝑟(𝜏) =1+𝑔.
1−𝜏Here, we calculate the partial derivative of (4 − 𝑎) on the domain 𝜏 < 1. Accordingly, the partial derivative of𝑟(𝜏) with respect to 𝜏 becomes(5 − 𝑎)𝜕
1 2𝑟(𝜏) = (1 + 𝑔) () > 0.
𝜕𝜏1−𝜏Therefore, we confirm that the market interest rate 𝑟 increasingly responds to the rate of saving taxation 𝜏.