1

Ref. 252

The Overlapping Generation Theory of Interest

By Hak Choi*

This paper shows that most of the existing theories cannot sufficiently explain

interest rate. It then generalizes Paul A. Samuelson’s (1958) overlapping

generation (OLG) model, and uses the results to develop an alternative theory

of interest. The generalized OLG model derives new definitions of money,

saving, investment, capital and capital formation, which clear the way to

solving for interest rate. Among others, this paper proves that any monetary

policy to influence the market interest rate must end up with the opposite rate

change, inflation or deflation, and lower income. It also explains the

international interest rate differences. The optimal OLG model also generates

economic growth. (JEL D91, E43, J26)

* Department of International Business, Chienkuo Technology University, No.1, Chiehsou N. Rd., Changhua City, (500) Taiwan. (email: hakchoi@gmail.com)

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The Overlapping Generation Theory of Interest

What is interest rate? According to Eugen von Böhm-Bawerk (1889), it is the cost of

working with the roundabout way of production. According to Irving Fisher (1907, 1930), it is

the price for being impatient. According to John Maynard Keynes (1953), it is the price for

the convenience of liquidity. According to Paul A. Samuelson (1958), it is equal to population

growth rate, which has to be either zero or negative. Knut Wicksell (1898, 1907, 1958) even

claims that there exist concurrently two different interest rates, a natural and a money rate.

Hence, there is no consensus of what interest rate should be.

The definition of interest rate is not clear, because the elements that make it up are not

clear. For example, what is saving? Is it equal only to bank deposit? Can households use their

unconsumed income to purchase securities or other assets? Most studies consider only annual

saving, and leave lifetime saving unattended. The definition of money is also not clear.

Sometimes it means cash, sometimes cash plus bank deposit. But, can peanuts also be counted

as money (Joan Robinson 1953, 86)? More fundamentally, what is capital? Is it the one

accumulated according to Robert M. Solow (1956), or some other suspected but not identified

by Robinson (1953, 1959)? It is even not clear, in what market interest rate can be found. Of

course, it can not be found in the population market. But, can it be found in the money market,

or the capital market? Hence, the small number remains a mysterious object, and sometimes

an abused tool of government intervention.

This paper considers the supply side of the interest rate problem. It uses Samuelson’s

(1958) overlapping generation (OLG) model to derive new definitions of saving and capital

supply. Then, interest rate can be determined. Apart from clarifying the above stated

3

confusing concepts, this paper also rectifies many seemingly contradictory observations. For

example, do people in developing countries really lead a more relaxed life? Is monetary

policy really effective? Why cannot international interest rate differences be evened out? Why

does high interest rate always come along with high inflation?

The rest of this paper is organized as follows. Section I reviews the existing theories of

interest. Section II generalizes the original OLG model, while Section III contains more

reflections on the generalized model. Section IV uses the generalized OLG results to develop

an alternative theory of interest. Some concluding remarks are affixed in Section V.

I. The Existing Theories of Interest

Because of space limit this section reviews only some existing theories of interest,1 they

will all be proved insufficient to explaining interest rate.

A. The Böhm-Bawerk Theory of Capital

Böhm-Bawerk’s (1889) shows that the need for fund to bridge the roundabout methods

for increased output gives rise to interest rate. For an example described by Böhm-Bawerk, a

person can obtain water by directly going to the spring. Alternatively, she can drink more

efficiently by using a pail, or even better by connecting a pipe to her house. The pail or the

pipe is the roundabout way of production. As the roundabout technology increases

productivity, the person will be ready to pay an interest for the fund required to make the

capital available. Hence, interest rate is the productivity measure of capital. Though he works

out in great details the nature of capital and the demand for it, Böhm-Bawerk does not

1 More extensive review, especially covering earlier studies, can be found in the works of Böhm-Bawerk.

4

elaborate on how its supply, the so-called subsistence fund, is formulated. This is the reason

for the amorphous concepts of saving and capital formation described above.

Böhm-Bawerk’s work was criticized by Wicksell for lack of unity (Samuelson 1967, 27),

but Wicksell’s theory is even more questionable.

B. The Wicksell Theory of Interest Rate and Price

Wicksell combines the Ricardian quantity theory of money—a larger money supply leads

to a lower interest rate—with Böhm-Bawerk’s capital theory, and argues that money interest

rate deviating from Böhm-Bawerk’s natural rate causes inflation or deflation. Wicksell (1907,

216) provides the following example to argue how price is forced to increase by a downward

deviation: “building companies … will be able to raise money, say at 4 per cent. instead of 5

per cent., and therefore, other things being the same, they can offer, and by competition will

be more or less compelled to offer for wages and materials, anything up to 25 per cent. more

than before, 4 per cent. on￡125 being the same as 5 per cent. on￡100.”

With reference to a special cost function like C=iwL, where i is interest rate, w wage rate

and L labor, such calculation can be found to rest on three assumptions: 1. a constant budget;

2. the whole benefit of the lower interest rate goes to input other than capital; 3. input demand

is price inelastic. Obviously, the foundation of Wicksell’s conclusion is flimsy. This argument

also means that higher (or lower) price is associated with lower (or higher) interest rate,

something Wicksell (1958, 87) himself disagrees. As far as production cost is concerned, a

lower interest rate means a lower supply curve. Then, output price must be lower given a

negative demand curve. Accordingly, a lower interest rate must result in deflation, not

inflation. All in all, Wicksell’s logic is full of contradictions. As he himself admits, the

coexistence of the two interest rates can never be proved. Indeed, they do not exist, for if they

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do, there must also be natural and money prices.

The picture Wicksell describes is actually the standard quantity theory of money:

increase of money supply leads to inflation, but that requires many other things, including

money velocity and interest rate, being held constant. The Ricardian quantity theory of money

looks like the Keynes theory of interest, which will be discussed shortly.

C. The Fisher Impatience Theory of Interest

In his first attempt, Fisher (1907) painstakingly introduces the concept of impatience that

yields the willingness to pay interest for borrowed money. This theory was criticized by

Böhm-Bawerk for being too simple (Samuelson 1967, 29). Subsequently, Fisher (1930)

introduces the production technology in the second attempt.

Fisher’s first attempt is like poising a single horizontal line in the capital market, whereas

in his second attempt he adds the capital marginal productivity to the graph. However, such a

picture merely describes the demand behavior of a competitive firm for some given interest

rate; it cannot determine interest rate.

Similar to Fisher’s impatience, Angus Deaton and John Muellbauer (1980, 359-65) also

derives consumer’s borrowing demand. Although households sometimes do borrow, vis-à-vis

industrialists, they are net lenders. Hence, household’s borrowing and lending behaviors

cannot determine interest rate either.

One more thing, the concept of intertemporal production possibility curve introduced by

Fisher is faulty. Unlike standard production, intertemporally inputs are not subject to the

principle of mutual exclusiveness. A car driven today, except for some depreciation, will still

be available tomorrow. Hence, there is no tradeoff of resources from one period to another.

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D. The Keynes Monetary Theory of Interest

Keynes (1953, 166-167) also adopts Fisher’s concept of time preference, but calls it

liquidity preference. Like Fisher, Keynes also equates such preference to interest rate, but the

problem here is more serious.

According to Keynes, when interest rate is low (or high), stock price is high (or low),

people will sell (or buy) the stocks and hold more (or less) liquidity, i.e., cash. This is called

the liquidity demand, which is the major component of the Keynesian demand for money. The

interaction of the demand with the discretionary money supply is supposed to determine

interest rate. For example, an increase of money supply would result in a lower interest rate,

as also argued by the Ricardian quantity theory of money. But, there are three pitfalls.

Ceteris paribus, if a decrease of interest rate can induce an increase of cash holding,

there must also be a drop of bank deposit by the same amount. Through the deposit multiplier,

there must be a larger drop in “money supply”. Hence, the Keynesian liquidity demand turns

out to be a supply function of money. This is the first pitfall.

Suppose the increase of cash holding is entirely financed by new money, the negative

effect on the “money supply” is neutralized. There is no change of “money supply”, except by

exactly the amount of the new money. There is still no demand. This is the second pitfall.

Even if selling of securities means increase of cash holding, such behavior looks more

like random walk. There is no demand schedule at all. Hence, the so-called Ricardian quantity

theory of money is also invalid. A question to be pondered upon is: Is there still a money

market? An answer will be given later in the concluding section.

E. The Samuelson Biological Theory of Interest

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Samuelson (1958, 472) argues that population must grow at the same rate as interest and

that such interest rate must be zero or negative.

The first argument is interpreted as a determination method for interest rate by simple

laymen as well as learned scholars, e.g., Olivier J. Blanchard (1985), Robert J. Barro, and

Xavier Sala-i-Matin (1995, 20), and, Gregory S. Amacher, Markku Ollikainen, and Erkki

Koskela (2002, 347). It is also supported by the golden rule theory of Edmund S. Phelps

(1961) and comes to be known as the Swan-Phelps-Robinson theorem (Samuelson 1962, 251).

However, this argument is merely an assumed identity. Using an identity as a solution is

arguing in a circle. Perhaps, it is interest rate that determines population growth. At the end of

the day, this identity does not necessarily hold. A contrast against empirical figures quickly

reveals that the required population growth rate can be very disproportionate. For example,

corresponding to a mere one percent interest rate, a one percent annual population growth is

immense for most industrialized countries. Not many countries can live with any population

growth higher than three percent in peace time (United Nations 2005). Hence, interest rate has

no direct relation with population growth and the first assertion is not endorsed.

It will be revealed in Section II (Footnote 3) below that Samuelson’s second argument is

actually the result of a mathematical mistake. Strictly speaking, interest rate can never be

determined in the population market. Regrettably, many countries base their population policy

on this theory.

Hence, all the above theories do not sufficiently explain interest rate.

8

II. The OLG Model and Its Generalization2

The preceding section shows that interest rate has not been explained, because capital

supply is not properly defined. Solow (1956) works out capital formation using an inventory

model. Strictly speaking, it is not a supply function, for it does not cover interest rate. It will

be shown that Samuelson’s OLG model provides a better definition. Because of its

extensiveness, the development is divided into two sections. This section reviews the basic

model, generalizes it to accommodate any number of periods and makes it continuous. Capital

supply and other reflections will be provided in the next section.

A. The Basic Model

In the world of Samuelson (1958), every person or generation goes through three periods

of life: a youth period of the age between 20 and 40, a middle-age period between 40 and 65,

and a retired period hence after. The person works for two periods, but has to allocate the

income for three consumptions, which are denoted as C11, C12, and C13, where the first

subscript represents the individual, while the second the period. The corresponding savings

are 1111 1 CS −= , 1212 1 CS −= , and 1313 CS −= , where the nominal disposable incomes are

assumed constant and standardized to one. The savings and consumptions are connected

together by some interest rates in the following respective relations:

(1) 0)1)(1()1( 112112213 =+++++ SiiSiS ,

2 Readers familiar with the OLG model may jump to Proposition 1 in Section III without loss of

continuity.

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and

(1′) 1132112111 1 RCRRCRC +=++ ,

where )1(1 tt iR += is the discount factor. Accordingly, the retiree must get back the

previous savings compounded with interests, or the lifetime income must cover the lifetime

consumption.

Each person hence faces the following optimization problem:

(2) Maximizing ),,( 131211 CCCU , subject to 1132112111 1 RCRRCRC +=++ .

Assuming a log-linear utility function, 131211131211 logloglog),,( CCCCCCU ++= , the

following utility-maximizing solution for the three savings is obtained (Samuelson 1958, 477,

eq.17):3

(3) 33

2 111

RS −= ,1

12 31

32

RS −= , and

22113 3

13

1RRR

S −−= .

Solution (3) also reflects the life cycle saving and consumption arrangement of the

individual (Albert Ando and Franco Modigliani 1963). However, it can be proved that

aggregation will smooth away the cyclical fluctuations.

3 Note that the re-substitution of this solution into the income constraint to solve for interest rate is a

tautology. If instead, it is substituted into a zero-interest rate income constraint, as done by Samuelson (1958,

475, eq.14), interest rate will be zero or negative. This, of course, confuses the identity of interest rate. This is the

mistake that causes Samuelson’s second argument described in Section I.

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If interest rate is constant, i=it, the third period dis-saving becomes

]1)/1)[(3/1(13 +−= RRS 3/])1()1[( 2ii +++−= . On the assumption that income of the

younger generations grows at the same rate as i, if income of the first generation is 1, income

of the second generation must be )1( i+ , while that of the third generation 2)1( i+ . Then, the

aggregate income in the third period is 2)1()1( ii +++ . Hence, the two working generations

together must always save 1/3 of the aggregate income in the third period, in order to meet the

repayment demand by the retired generation, i.e., 333 YS = , where the first subscript for the

individual is dropped. This applies to all periods hence after, i.e., 3,3 ≥= tYS tt , where Yt

can be calculated from 3for )]1()1[()1( 23 ≥++++= − tiiiY tt (Henry Aaron 1966, 373,

eq.1).

Similar results are more easily obtained in a 2-period OLG model: 5.011 =S , and

)1(5.0/5.012 iRS +−=−= . Accordingly, saving is always 1/2 of the income in both periods.

Based on these results it can be inducted that a g-period OLG model must have 1/g of the

aggregate income as saving, or:

(4) g

s 1= ,

where s is the saving rate and g the OLG order.

This induction helps get rid of cluttering with the individual savings in the first g-1

periods at the beginning of the Universe. The complexity for the solution that might occur in

higher order needs not be worried any more, and it is not necessary to retreat to lower order.

Now, any order can be accommodated.4

4 Some studies, e.g., David A. Starrett (1972) and Bommier and Lee (2003), also assume relatively high or

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According to equation (4), people with higher (or lower) OLG order require less (or

more) saving. If they intend to work until they die, i.e., ∞→g , they do not have to save at

all. If they intend to retire earlier, i.e., a smaller g, they had better save more while they are

young.

Irrespective of what the OLG order may be, the aggregate saving is always a constant

fraction of the aggregate income, independent of interest rate, and the individual saving or

consumption fluctuations are smoothed away. This result also echoes Milton Friedman’s

(1957) permanent income hypothesis: consumption is a constant fraction of income.

B. The Generalized OLG Order

The division method of people’s lifetime by Samuelson is quite rough. To be precise, the

OLG order can be obtained from the following ratio:

(5) wf

f

TTT

g−

= ,

where Tf is the number of lifetime years and Tw the number of working years. It is obvious

that ∞<≤ g1 . As an illustration, if people work for 40 years during their 60-year lifetime,

the OLG order is 3. For another example, if people work for 45 years during their 65-year

lifetime, the OLG order is 3.25. Now, even fractional order is allowed. In addition, time of

childhood can also be included, since Tf stands for the whole life and Tw does not have to be

the earlier part of the lifetime.

Equation (5) can also be adjusted to allow for altruistic motive. For example, if a

continuous OLG order, but they do not solve the simultaneous equations.

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person’s, say, John’s, original OLG order is 3, but he intends to leave a bequest equivalent to

the consumption of one period after life, he should pursue an OLG order of 2, as a result of 2

working periods and 2 retired periods. This is equivalent to extending both the life and

retirement time by one more period, and his saving rate also increases from 0.33 to 0.5.

Similarly, if another person, say, Mary, has to cover the consumption of her schooling

children or her unemployed parents, she must also maintain a higher saving rate and a lower

implicit OLG order.

Equation (5) states the exact measure of the OLG order, but it is hard to find out people’s

numbers of lifetime and working years, especially those of the living population. Without

them, the OLG order cannot be determined. Without the OLG order, the model becomes a

hollow one. Here is another method to get around this restriction. It is obvious that labor is

not exactly equal to population, P, unless everyone works for the whole life. If the OLG order

were known, the exact relation between labor and population could be calculated from

ggPL )1( −= . Hence, labor participation rate, p, can be defined as

(6) g

gPLp 1−== .

Since data of labor participation rate are more readily available, they can be used to

impute the OLG order from g=1/(1-p). For example, such rate in the USA and Japan is 52

percent, while in Mexico 40 percent (World Bank 2004); hence, their respective imputed OLG

orders are 2.1 and 1.68. Thus, data of OLG order are also readily available.

The data just described reveal that Mexicans seem to enjoy life better. The optimal OLG

model below will explain this seemingly incongruous observation.

C. The Continuous OLG Model

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The original OLG model is a discrete one, which comes along with the following odd

implication: all babies must be born on the same date, and then no more birth is allowed until,

say, 20 years later. To make the model continuous, the total population is divided by the

number of lifetime years to obtain the average number of people entering any stage of life

every year. Then, saving of the retiring generation must have grown to

(7) giii wT /])1()1()1[( 2 ++++++ L ,

where i now represents the annual interest rate. If per capita income of the younger

generations also grows annually at the same rate, the saving rate required to meet the

repayment demand is still the same as given in equation (4). Although each person lives

through g discrete periods, generations overlap continuously and annually.

THE TWO OLG ASSUMPTIONS

(i) Each person has the same numbers of lifetime and working years.

(ii) Per capita income of the younger generations grows at the same rate as interest.

Assumption (ii) replaces the Samuelson argument of population growth having the

same rate as interest.5 It also allows flexibility: if income of the younger generations has

lower (or higher) growth rate, they have to save more (or less). However, the saving must still

be a constant fraction of the aggregate income. Since the above analysis applies to both g and

g+1, the induction is established.

5 Hence, Samuelson’s theory of interest is actually an assumption.

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III. Some Reflections on the Generalized Model

In this section, the above OLG model will include time preference and will be

transformed into a new classical growth model. Then, the OLG saving will be linked to

capital and investment, and the quantity theory of money will be given an economic meaning.

Finally, the optimal OLG order will be derived.

A. The Generalized Model with Time Preference

To include time preference, the 3-period utility function is modified to

2131211131211 )1(log)1(loglog),,( ρρ ++++= CCCCCCU , where ρ is the time preference

rate. A larger ρ indicates higher preference of present to future consumption. The optimization

problem (2) with this new utility function results in the following solution for the three

consumptions:

])1(

11

11[)1( 2111 ρρ ++

+++= RC ,

(8) ])1(

11

11[])1(

)1([ 21

112 ρρρ +

++

++

+=

RRC ,

])1(

11

11[])1(

)1([ 2221

113 ρρρ +

++

++

+=

RRRC .

The last equation of this solution indicates that in the third period the required saving

must be [ ]2)1()1(1/1 ρρ ++++ of the same aggregate income, i.e., 2)1()1( ii +++ . The

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saving rate with time preference is lower. For example, if ρ=0, s=0.3333; but if ρ=0.05,

s=0.3172. Similar observation is also made by A.P. Lerner (1959, 513).

By the same induction previously established, a gth order OLG model with time

preference has a saving rate of [ ]12 )1()1()1(1/1 −+++++++= gs ρρρ L , which by

summation of the geometric series becomes

(9) 0for 1)1(

≠−+

= ρρρ

gs .

If ρ is small, this equation can be approximated by expanding the Taylor series on the

exponential component to fall back to equation (4), i.e., ggs /1)11( =−+≅ ρρ . The

negative relations between saving rate and OLG order, with and without time preference, are

depicted in Figure 1.

FIGURE 1. Saving Rate and OLG Order

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PROPOSITION 1: Based on the two OLG assumptions, the original OLG model can be

reduced to a single saving rate equation of ]1)1/[( −+= gs ρρ .

B. The New Classical Growth Model

It will be shown here that the OLG model is identical to the new classical growth model.

On the first OLG assumption each individual, say individual 1, pursues the maximum of the

following g-period utility function:

(10) 11

21,112

1111,11211 )1(log

)1(log

1loglog),,,,( −−

−− +

++

+++

+= gg

gg

gg

CCCCCCCCUρρρ

LL .

Note that most existing growth studies, e.g., Paul M. Romer (1986, 1021) and Robert E.

Lucas, Jr. (1988, 7), and Robert G. King, Charles I. Plosser and Sergio T. Rebelo (1988, 332),

formulate the utility function in an infinite time horizon, but that implies zero saving

according to Proposition 1. That should be avoided. It is also possible to replace g with Tf, and

g-1 with Tw, and the model will have the annual measures, but the retention of g helps derive

the saving rate.

The marginal rate of substitution (MRS) between any two consecutive periods, t and t+1,

is

(11) 121for )1(

1

1,1

1

1,1 ,g-,,tC

CdC

dCMRS

t

t

t

t L=+

−== ++ ρ.

The corresponding g-period lifetime income constraint is

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(12) 211

31111

1111

21112

11 )1()1(1)1()1(1 −−−−−

++

+++

++=

++

+++

++ ggg

gg

g

iy

iy

iyy

iC

iC

iCC LL ,

where 11y is the constant nominal disposable income of this individual. From this equation

the economic rate of substitution (ERS) between any two consecutive periods, t and t+1, can

be derived to be

(13) idC

dCERS

t

t −−== + 1,1

1,1 121for ,g-,,t L= .

The g-1 equalities of MRS with ERS, along with the income constraint of equation (12),

can solve for the g optimal consumptions and hence savings, as demonstrated above when

g=3. On the second OLG assumption, the solution is reduced to a single equation of saving

rate as stated in Proposition 1.

The generalized OLG model is therefore identical to the new classical growth model, but

the new growth model also produces explicit solution for consumptions and savings. Since the

saving rate is the one for the whole economy, this model evolves seamlessly from individual

utility modeling to economy-wide aggregate solution. Economic growth will be expressed in

equation (19) below.

C. Some New Understandings about Saving and Capital

According to the above development, the working generations always save a constant

fraction of their income. Where does this saving go? Neoclassical theory says that it goes to

new investment. It tells only half of the story. Here is a more complete one.

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Suppose households and industrialists are two different entities, and households are

owners of all resources and recipients of all income. Households here refer to individuals, not

families. A household and an industrialist can be the same individual; they are so called only

to distinguish their economic roles. At each period, the working households divide their

income between consumption and saving, according to the OLG arrangement.

Suppose at the beginning of some period 1 the aggregate saving, 1S , is deposited in some

banks and the industrialists come to borrow it to purchase some productive assets. In period 1,

capital and investment are identical, i.e., 11 KI = . In contrast to the neoclassical definition,

capital should include machines, buildings, software, trademarks, goodwill, human capital, etc.

Capital here is simply defined as any asset that can be resold by the end of the production

process. Contrary to the neoclassical treatment, inventory is not capital, for it is idle resource,

like idle cash. Industrialists will check the marginal productivity of each targeted asset against

the rental price of capital, before they borrow the money.

Note that the screening criterion is profit maximization, not profit itself. Hence, interest

rate is contrasted against net capital productivity, not against profit rate. An industrialist may

as well borrow to minimize loss.

However, not the whole saving becomes capital. There are various reasons for that.

Perhaps, households like to keep some idle cash for precautionary or transactional purpose.

Banks must reserve some deposit for withdrawal calls. Industrialists also use part of the

borrowing as working capital for smooth operation. Finally, households may exchange part of

their saving into foreign currency. Hence, capital cannot exceed saving. If the fraction of

saving that turns into capital is k, with 10 ≤≤ k , then 111 kSKI == , and k will be called the

capital-saving ratio.

Suppose at the end of period 1 the industrialists liquidate all the assets and pay back the

loans with interest through the banks to the lenders, among them is the retired cohort. Then at

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the beginning of period 2, a new group of working households will come up with another

saving. Another new group of industrialists will come to borrow the money to purchase assets,

including some depreciated ones this time.

Of course, capital contains some depreciated assets, but industrialists will invest in such

assets, only if their productivity is higher than the rental price. As all the old assets have been

liquidated at the end of period 1, if some of them have become less productive, they will not

appear in the investment list of period 2 any more. Hence, investment in depreciated but

productive assets is also investment, and the aggregated investment in depreciated and new

assets is capital. If the capital-saving ratio is still k, 222 kSKI == . Accordingly, saving not

only goes to new investment, it also goes to old investment.

This reasoning applies to all periods hence after, thus

(14) ttt kSKI == , for all t.

Consequently, capital formation, 1−−= ttt KKK& , can be defined as

(15) )( 1−−= ttt SSkK& .

This formation stands in sharp contrast to the neoclassical one, 1−−= ttt KSK δ& , where δ

is the depreciation rate (Solow 1956). Supposedly, capital formation comes from saving as

investment and is net of depreciation.

The major cause to the difference is the confusion of whether the borrowing needs to be

repaid or not. In the neoclassical economics, the borrowing, St-1, is not repaid; investment is

just another category of expenditure; once households turn in their fund, they have no claim to

the assets any more; the most they can get back is the interest income. In contrast, according

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to the OLG model described above, the retired cohort, along with all the other previous

working households, can claim back the whole saving with interest; even a bequest will be

reclaimed by some descendants.

As another confusion, the Solow equation does not exclude all unproductive old assets,

which are merely depreciated. A badly located restaurant attracts no customer; or a very

expensive software becomes useless in a very short time. They should both be relinquished

and eliminated from the subsequent capital calculation, even though there is little or no

depreciation. In the calculation of capital in equation (14), only the productive assets, old or

new, will survive the screening process. As stated, the Solow equation checks only the

productiveness of new investment.

Moreover, the Solow equation does not allow for idle cash, bank reserve or working

capital.

The Solow equation is at most an inventory control model, like the water level in a

reservoir. Water, which flows out to supply external use, comes from heaven; it does not have

to be repaid, become unproductive or idle. In contrast, equation (14) is a balance sheet; every

debit entry must be balanced by a credit entry, and equation (15) measures the change of two

balance sheets. In economics, capital is not just a stock concept, it is a lively stock.

One more thing, to calculate capital from the Solow equation is extremely difficult, due

to the estimation required on investment and depreciation many years back. In contrast,

according to equation (14) capital can be readily estimated from saving. Hence, the Solow

equation is very imprecise, while equation (14) is a distinctively better alternative.

So much is about the new definition of capital, but what about saving? It is modeled so

far that the only outlet for household saving is bank deposit. In reality, some of it flows to

other outlets, like securities, real estates, cars, other durable goods, or business ventures.

Purchase of securities or other financial products must also become capital. Such investment

21

is thus equivalent to bank deposit, along with its multiplying effect. When a household buys

an IPO stock, money is transferred to industrialist A, who deposits a fraction of it in Bank B,

who lends a fraction of it to industrialist C, who deposits a fraction of it in Bank D, and so on,

and so on.

When households put their money into the other outlets, assets with resale value after use,

they invest their own saving and become themselves industrialists, again with the same

multiplying effect.

Accordingly, a household’s saving is the sum of cash and all assets holdings. In that case,

it will be a multiple of annual income, contrary to the conventional understanding of it being a

fraction of annual income. The difference is caused by the time-dimension. According to

equation (7), a household’s saving is accumulated during the many working years, not just

one year. However at some present point of time, not every person has gone through the

whole working duration, Tw. Hence, the economy-wide total saving, S, must be the sum of

saving from working households with varying fraction of Tw. Though tedious, this saving can

be derived to be:

(16) af sYTS21

= ,

where Ya is the annual aggregate income.6 Note that even the time-subscript is dropped from

this equation, as all variables are contemporary.

D. The Quantity Theory of Money

6 Similar treatment can be found in Menahem E. Yaari (1965) and Thomas Wiedmer (2002). Yaari (1965,

138) has a similar saving function, while Wiedmer (2002, 503) equates capital with saving.

22

The sum of idle cash, bank reserve and working capital described above constitute the

stock of domestic currency, M. If foreign deposit is assumed zero, the quantity theory of

money can be alternatively formulated as: SkM )1( −= . Since S is a function of annual

nominal income, as just proved, the implied velocity of money is inversely proportionate to

(1-k), or directly proportionate to k. This gives the identity nature of the quantity theory an

economic meaning. Ceteris paribus, a smaller k, and thus a lower velocity, will lead to higher

interest rate, as will be proved in Section IV. Obviously, the money must be cash.

E. The Optimal OLG Model

Up till now the OLG order is treated as given. However, equation (5) shows that the

OLG order is a combination of the numbers of working and lifetime years. Although the

number of lifetime years is exogenously, heavenly fixed, people can vary their OLG order by

changing the number of working years. They will do so to pursue maximal per capita income,

Ya/P. Subsequently, some formulations for output and population are necessary.

One formulation for the annual output is the Cobb-Douglas production function:

(17) βα KALYa = ,

where A is the technology parameter, and α and β are the labor and capital coefficients

respectively. Another formulation will be considered in Section IV.

Next, according to Gary S. Becker (1960, 215), or more specifically according to Becker

and Barro (1988, 11, eq.18), fertility, N, is a positive diminishing function of annual income,

i.e., φγ aYN = , where γ>0, but 0<φ<1. Current population is previous population plus fertility

minus mortality, 11 −− −+= tttt mPNPP , where m is the mortality rate. The recursive

23

substitution of this step function, along with the second OLG assumption of constant income

growth, results in the following population function:

(18) φχ aYP = ,

where ])1/()1(1/[ φγχ im +−−= . Again, current population becomes a function of current

income.

The interaction of the saving/capital function, the labor/population function, and the

production function results in the following equilibrium annual output:

(19) [ ] βφααβ χ −−−= 11

)(]2/)[( pkTTAY wfa .

The effects of the other variables in this equation are obvious, but an interesting

observation is that a longer retirement period, wf TT − , corresponds to higher income. Similar

observation is made by Barro and Sala-i-Martin (1995, 432). The reason is that longer

retirement requires more saving, and hence more capital and higher income. This equation

also generates growth, when any one of the parameters changes correspondingly.

On differentiation of equation (19) with respect to Tw, per capita income reaches its

maximum when the following condition is fulfilled: )/( βαα += fw TT . The substitution of

this condition into equation (5) leads to the following optimal OLG order, g*:

(20) βα

+=1*g .

24

Though humble in appearance, this equation contains many interesting implications.

1. The lowest possible optimal OLG order is one, when the labor coefficient is zero. In

such case, the person spends her whole life in retirement, which of course requires sufficient

inheritance.

2. Given that the labor coefficient is not zero, if the capital coefficient is very small,

the optimal OLG order approaches infinity. The person has to work for most of his whole

lifetime. This is another extreme from the previous implication, and is also the environment

before most of the roundabout technologies have been invented. These two implications

together confirm the limits of the OLG order: ∞<≤ g1 .

3. A larger capital coefficient reduces the OLG order. The person works less, if capital

becomes more efficient.

4. A larger labor coefficient corresponds to a higher OLG order. A more capable

person tends to work longer. Along with the preceding implication, it means that people

should endeavor more to upgrade their capital productivity, in order to have higher income

and longer retirement period.

5. In the case that both the labor and capital coefficients are equally small, the OLG

order may look very small too, but it does not mean that the population is really enjoying

better life and has longer retirement period. The truth is that they have fewer job opportunities

and lower income. This is the situation of Mexico and many other developing countries.

If the income defined in equation (17) is subject to personal tax, disposable income

becomes smaller. However, since the difference is a constant fraction, the existence of this tax

will not affect the derivation and its results. Though equation (20) is derived on the

assumption of zero time preference rate, as long as the rate is small, its existence will not

affect the result, according to the Taylor series expansion. This optimal order will help

develop the alternative theory of interest in the next section.

25

IV. An Alternative Theory of Interest

Among all the existing theories of interest rate, Böhm-Bawerk’s work provides a good

foundation. Based upon that, this section adds the OLG saving just derived as capital supply

to solve for interest rate. It then explains two related paradoxes.

A. The Determination of Interest Rate

As described above, all outlets of saving except cash must turn into productive assets.

Industrialists must make sure that the net marginal productivity of each eligible asset, old or

new, is greater than or equal to interest rate, before they borrow any money. Given the

aggregate production function of equation (17), the aggregate net marginal productivity of all

eligible assets, i.e., capital, can be derived to be

(22) iKYa ≥−δβ .

Since a lower interest rate will make more assets eligible, this demand is negatively sloped, as

depicted in Figure 3. This is also known as the Keynesian marginal efficiency of capital

(Keynes 1935, Chapter 11; Dale W. Jorgenson 1967, 133).

On the other hand, the OLG model has made it clear that capital is supplied by saving

derived upon household utility consideration. Since the OLG saving is independent of interest

rate, capital supply, kS, is a vertical line. The intersection of the demand and supply curves

will result in the equilibrium interest rate.

As Ya stands for nominal income, an output price variable can be singled out and

26

attached to the numerator of equation (22), but then the same variable must also be attached to

the denominator, which also contains nominal income according to equation (16). Hence, its

effect is cancelled out. Although inflation does not affect interest rate, it reduces the assets

value of lenders, depositors, and cash holders. Although real interest rate accounts for

inflation, it does not reflect the whole loss wherefrom.

By the same token, the effect of real income on interest rate is also cancelled out. If

Böhm-Bawerk or Samuelson (1967, 28) were right on this point, richer countries would have

higher interest rate.

Since any new money must become household’s idle cash, bank reserve or working

capital, there is no increase of capital supply. Hence, monetary policy has no effect on interest

rate either. If printing money were able to reduce interest rate, such rate would be zero, or at

least very low, in countries with hyper-inflation. On the contrary, evidences in, e.g., Latin

Focus (2006) reveal that their interest rates can be astonishingly high.

Another extra source of fund is foreign reserve, earned or speculative. The movement of

such money also acts like monetary

policy with the same result: it must

become cash held by someone. Hence, it

has no effect on interest rate as just

proved.

The above listed are the

non-influencing factors. Understanding

them is sometimes more important than

understanding the influencing factors

listed below.

The interaction of equations (14)

FIGURE 2. Capital Market and Interest Rate

27

and (22), along with the saving definition of equation (16) and the optimal OLG order of

equation (20), results in the following equilibrium annual interest rate:

(23) δβα−

+=

fkTi )(2 .

According to this equation, the labor and capital coefficients have positive effects on

interest rate, while the effects of the capital-saving ratio, life expectancy and depreciation rate

are negative.

The positive effect of the capital coefficient is understandable, but that of the labor

coefficient is not immediately decipherable. A higher labor coefficient leads to more working

years according to equation (20). More working years require less saving, and hence higher

interest rate. On the other hand, a longer life expectancy requires more saving, thus lower

interest rate is resulted.

If the capital-saving ratio is smaller, interest rate will be higher. The size of this ratio

depends on market structure and economic-political stability. If the financial market is not

perfect competition, bank deposit and security investment are unsafe, while borrowing is

difficult. Then, households increase their holding of idle cash, while industrialists keep more

working capital. In face of economic or political instability, there is capital flight and banks

increase their reserve. By economic instability, inflation or currency devaluation are meant.

These all contribute to a smaller k, and a higher interest rate. As described in Section II, a

drop of the money velocity is a sign of declining k. Hence, contrary to the Keynes liquidity

preference theory, increase of cash holding actually leads to higher interest rate, ceteris

paribus.

A higher depreciation rate leads to lower net marginal productivity, and hence lower

interest rate. If a taxi is made of Samuelson’s (1958, 469) chocolate, it does not pay to invest

28

in it, or to keep it as capital. Demand for capital is low then.

Equation (23) is the basic formulation, and the following three variations can also be

contemplated. First, a parallel interest rate with a CES production function,

[ ] ξξξ αβ/1LKYa += , is:

(23-1) δβ

βαβε

−⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

−1)(2

fTki ,

where 1/(1-ξ) is the elasticity of substitution. When the elasticity is one, this equation falls

back to equation (23); but when it is greater than one, interest rate will be higher.

Second, if time preference is incorporated into equation (23), equilibrium interest rate

becomes:

(23-2) δρ

ρβ ββα

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−+

=

+

1)1(2

fkTi .

Interest rate also becomes higher, because the bracketed term of this equation is always

greater than g, according to equation (9). Obviously, to reduce this equation to ρ or to

reproduce the Fisher theory will call for a lot of restrictions.

Finally, if the output in equation (22) is subject to a sale tax, x, equation (23) becomes:

(23-3) δβα−

−+=

fkTxi )1)((2 .

Interest rate is lower on the backdrop of such tax. However, a profit tax has no impact on

29

interest rate at all. Accordingly, interest rate determination can be summarized as follows:

PROPOSITION 2: Interest rate depends on ten factors: labor productivity (+), capital

productivity (+), life expectancy (-), depreciation (-), market inefficiency (+), economic and

political instability (+), elasticity of substitution (+), time preference (+), and sale tax (-).

To have positive interest rate, the first term of equation (23), or its variations, must be

greater than the depreciation rate. Otherwise, Samuelson’s desire for zero or negative interest

rate would be fulfilled, subject to the qualification made by Fisher (1930, 192). Had the

population lived in an environment of the shipwrecked sailors, i.e., 0≅+ βα , interest rate

would be zero or negative. Hence, zero or negative interest rate is very unlikely.

According to equation (22), the income from the net marginal productivity of capital

using the borrowed money must be more than enough to pay for the interest due. Quid pro

quo is not necessary. Any worry about paying the retirees is therefore unwarranted, unless

there is mismanagement. The resultant interest rate then becomes an exogenous variable to

households and industrialists for making their appropriate decisions.

B. The First Interest Rate Paradox

As proved above, monetary policy has no effect on interest rate. But then, how can the

interest rate reduction (or hike) be explained, every time a central bank increases (or

decreases) money supply?

The answer lies in whether interest rate is honored as an equilibrium result or is used as a

policy tool. As demonstrated in this paper and as pursued by most previous studies described

above, interest rate is market equilibrium. But in practice, most central banks use it as a tool

30

to assist their operation, and it might even work. For example, when the discount rate is set

below the market level, commercial banks use any available IOU to discount for cash, and

new money is injected into the market. When it is set above the market level, they use any

available cash to redeem the IOUs or purchase bonds from the central bank, and money is

retrieved. This is simple arbitrage, no theory is required. Hence, monetary policy seems to be

effective. However, it comes along with some unpleasant and undesired consequences.

As proved above, any increase or decrease of cash does not affect the equilibrium

interest rate, hence interest rate must eventually return to that level. Here is another reason. If

unrestricted, the existence of the interest rate difference will lead to either infinite or zero

money supply. Consider another example, if the market price of gasoline is $2, but a

government sets the price to $1.99 (or $2.01), people will buy all from (or sell all to) the

government. To avoid these extremes, the government must impose quota. That applies also to

open market operation. When the quota is used up, interest rate must return to the market

level. Hence, there is no distinction between natural and money interest rate, and the Wicksell

theory is further disproved. But, the worst is still to come.

After an expansionary monetary operation, quantity of money is increased. Since the

increased money cannot become capital supply, people possess excessive holding. As money

itself does not carry utility or productivity, people exchange the excessive holding for other

things with utility or productivity. Inflation is resulted. As long as excessive holding exists,

inflation will persist. When people exchange it for foreign goods, inflation is exported. This

result follows also from the standard quantity theory of money, when the velocity is constant.

However, the latter may also be affected.

In the case of frequent exercise of the expansionary policy, people change their habit or

expectation to adapt to a larger percentage of saving as cash, i.e., a smaller k or a lower

money velocity, and interest rate will be higher. Hence, lowering interest rate must result in an

31

even higher one. This is similar to price ceiling or support. A lasting price ceiling must lead to

less production, and hence an even higher price. A lasting price support will lead to more

production, and hence an even lower price. This offers a stronger explanation of the so-called

liquidity trap: interest rate cannot be lowered despite repeated increase of money supply.

Moreover, as k becomes smaller, capital stock shrinks, output and income must also

drop.

By the same token, raising interest rate must eventually result in an even lower one and

must lead to deflation. However, output may not increase correspondingly because of trading

difficulty for lack of money. On the contrary, output must drop drastically. According to

Friedman (1968, 3), the great depression in the 1930s is caused by the “highly deflationary

policies”.

Hence, any attempt to influence the market interest rate must result in the opposite rate

change, price fluctuations, and lower income. Monetary policy should be restricted to

supplying just enough money, and let the market determine its own rate. Interest rate

intervention, like any other intervention, is in contravention of laissez faire.

C. Why Aren’t International Interest Rate Differences Evened out?

Capital is known to have high mobility; interest rate differences across border should be

swiftly evened out. However, despite rapid technology improvement in money transferring,

there is still significant difference of interest rate from one country to another. The difference

is more obvious between developed and developing countries. It would be too rough to

attribute the difference to the loosely defined “country risk”. There are more than one cause

for that.

In the first place, developing countries have lower OLG order, as evidenced by their

32

lower labor participation rate. As explained by the optimal model, the lower OLG order

means lower productivities of the inputs, and lower income. Second, these countries have

lower capital-saving ratio, due to imperfect market structure and economic-political instability.

Finally, people in these countries have shorter life expectancy. As a result of all these reasons,

interest rate in such countries is high according to (23).

Faced with the high interest rate, these countries are tempted to exercise expansionary

monetary policy by the now proven faulty Keynes theory. Consequently, even lower income,

inflation and even higher interest rates are the results, as just proved. This also explains the

positive relation between inflation and interest rate observed by Wicksell (1907, 217) and

Friedman (1968, 6-7).

Since any inflow of foreign money must also become cash, it will further aggravate the

situation. The famous Brady plan and many similar ones designed to rescue those

unsuccessful attempts by foreign monies to tap the Latin American markets, and their failures

confirm this ineradicable interest rate difference (Gregory Ruggiero, 1999). Most foreign

monies to these countries have become delinquent.

V. Conclusions

The paper points out the false reasoning of many previous theories of interest. For

example, interest rate can never be determined in the population market. Profit rate can never

be employed as an investment criterion, or linked to interest rate. Time preference is only one

of the ten factors affecting interest rate. Wicksell’s logic is incorrect: ceteris paribus, lower

interest rate must lead to lower prices. There is no such thing as natural or money prices,

likewise there is no such thing as natural or money interest rate. Liquidity preference, which

is actually financial investment, is random walk. However, this paper gives the identity nature

33

of the quantity theory of money an economic meaning: ceteris paribus, a lower money

velocity leads to higher interest rate.

This paper also clarifies many previously confusing concepts surrounding interest rate.

For example, it shows that saving is not just bank deposit; household saving is spread also in

other assets holdings, like cash, securities, real estates, cars or households’ own business

ventures. All assets holdings except cash turn into capital supply. This capital supply fills the

theory vacuum left behind by Böhm-Bawerk.

Next, it shows that capital is accumulated from, after all the old assets have been

liquidated, both old and new, but productive assets. The Solow definition, and hence Phelps’

golden rule, are both invalid. Capital, or household assets holdings net of cash, is the wealth

of a nation. As the Solow definition of capital formation is refuted, growth theory,

neoclassical or new classical, must also be revised. This paper also shows how a new classical

growth model, which is actually the generalized OLG model, should be properly designed.

Economic growth can be easily explained by the generalized model.

Then, it shows that money is composed only of cash. When some saving is transformed

into capital, through deposit or other outlets, it is not money any more. If a deposit is treated

as money, so must be a security, a car, a house, or a business. Hence, any broader definition of

money is redundant. There is a market for non-money, i.e., capital, but there is no money

market.

When these concepts and misconceptions are clarified, interest rate is found to depend

not only on some familiar factors, but also on some exotic ones. However, technology,

inflation, income, printing money, foreign money, and profit tax have no effect on it, nor does

population growth or profit rate. Hence, the high interest rate in most developing countries

cannot be made lower by any of these methods, and their people do not lead a more relaxed

life.

34

Interest rate, like any other price, is the equilibrium result of market interaction. Using

interest rate as a policy tool is an intervention to the free market, must result in the opposite

rate change, inflation or deflation, and lower income.

This paper rectifies many previously contradictory definitions and theories. This would

not have been accomplished without the light shed from the original OLG model. However,

the secret of the OLG model will remain unnoticed, if it is not generalized according to the

procedures laid down by this paper. This paper takes the old model to a new level.