Interest

Historical aspects of the interest rate

Theories of interest

BIBLIOGRAPHY

Interest is one of the forms of income from property, the other forms being dividends, rents, and profits. Interest usually originates in the payment for a loan of money over a period of time—although it can also arise from loans in kind. Interest is essentially measured by the difference between the amount that the borrower repays and the amount that he originally received from the lender (which is called the principal).

The term interest sometimes has the broader connotation of all income from property. This is the case when we speak of “the interest charge on capital,” which denotes the alternative income that can be earned on a given quantity of money capital. It is, however, with the narrower sense of the term that this article will almost exclusively be concerned.

Etymologically, interest stems from the Medieval Latin word interesse, although the meaning of interesse was sharply distinguished by the medieval canonists from what is now denoted by interest. In particular, the canonists used interesse to refer to the compensation made by a debtor to his creditor for damages caused to the creditor as a result of default or delay in the repayment of a loan; as such the compensation evolved from the quod interest of Roman law, which was the payment for damages arising from the nonfulfillment of any contractual obligation. The canonists considered interesse to be conceptually distinct from the payment for the use of a loan, which they (and Roman law) denoted by the term usura (Noonan 1957, pp. 105-106). Since canon law permitted interesse but forbade usura, the reason for the evolvement of the modern term interest is clear.

In the case of a loan repaid in one lump sum, the total amount of interest that is due depends on the principal, P, on the percentage rate of interest per unit of time, r, on the number of time units over which the loan is outstanding, h, and on the number of time units after which the interest obligation is added to the debt of the borrower, m. If this obligation is added only once, when the loan matures (that is, if h = m), the interest is said to be simple. The total amount to be repaid at maturity, S, is then

(1)S = P(l+mr).

Correspondingly, the [eta] interest payment is

(2)S-P = P(l +mr) -P = mrP.

If, in addition, the time unit with respect to which the rate of interest is measured is the same as that of the duration of the loan, then h = m = 1, and formulas (1) and (2) are further simplified.

If the interest obligation is added more than once, interest is said to be compounded. In this case h is usually a multiple of m, so that h/m represents the number of times that interest is compounded over the duration of the loan. The total amount to be repaid at the maturity of the loan in this case is

(3)S = P( 1 + mr)^λ/m

The shorter is the period after which interest is compounded (that is, the smaller m is), the larger is the amount payable, S. In the limiting case, one can conceive of compounding being carried out continuously over time, in which case the amount payable is derived from formula (3) by letting m approach zero, thus obtaining

(4)S = Pe^rh,

where e is the transcendental number 2.71828 … The variable r in this case is frequently called “the force of interest.”

In order to compare the rates of interest effectively paid on loans with different arrangements for compounding, it is necessary to express the loans in a standardized form. This form is usually that of a loan granted at a rate of interest of i per year, compounded once a year. The standardized rate of interest, i, corresponding to any single-repayment loan extending for h years, is then determined by solving for i in

(5)S/P=(l+i)^h

where S, P, and h are the actual terms of the loan. Alternatively—and frequently more conveniently —the standardized rate is the force of interest, p, as solved from

(6)S/P =p^h.

Loans need not be made in the form of money but can be made in the form of a commodity (for example, wheat, in ancient times). The rate of interest on such loans is frequently called “the ownrate of interest”—a term originated by Keynes (1936, p. 223; see also Lerner 1953, pp. 354-356). Because the anticipated price movements of various commodities differ, their own-rates need not be equal to each other or to the rate of interest on money loans. If, however, uj^re is perfect arbitrage, then in equilibrium we must have approximately r_n= r_c + s_c, where r_n is the rate of interest on money loans, rc is the own-rate of interest on the loan of any commodity, and sc is the anticipated rate of increase (net of carrying charges) in the money price of that commodity. For if (say) the right-hand side of the foregoing equation were to exceed the left-hand side, a profit could be made by borrowing money, using the proceeds to purchase wheat to be lent out at its own-rate, and subsequently selling the wheat received in repayment of the loan. Such arbitrage transactions would raise r and lower r_cand s_c until the foregoing equality was established.

If the formula is applied to commodities in general—so that sc represents the anticipated increase in the general price level—it becomes an expression of Irving Fisher’s celebrated distinction between the money rate and the real rate of interest. The real rate is the rate of interest as measured in terms of commodities and accordingly is approximated by r” - sc (Fisher 1930, chapters 2 and 19). The logic behind this relationship explains the fact that money rates of interest tend to be high in periods of inflation. It also explains the appearance in inflationary economies of so-called linked or escalated bonds—that is, bonds whose principal and/or interest payments are proportionately adjusted in accordance with some price index (Brown 1955, pp. 200-203; Finch 1956; Robson 1960).

Since they are alternatives to one another, the different forms of property income must be quantitatively related. In a world of certainty and perfect markets, the ratio of the dividends paid by a stock (including net appreciation in the value of the stock) to the original price of the stock, the ratio of the rent of an asset (net of depreciation and other operating costs) to the original price of the asset, and the ratio of the profits of a firm to the amount invested in it all have a common value, equal to the rate of interest. This equality reflects the fact that in such a world the individual is indifferent about the form in which he receives his income from property.

Conversely, the preferences that individuals in the real world have for particular forms of property income reflect the differing degrees of risk and uncertainty attached to these various forms, the individual’s own attitudes toward the risk and uncertainty, the differing degree of knowledge that individuals have of market conditions, differing anticipations of future price conditions, and the like. Of the four forms of property income, interest usually involves the least risk and the least necessity for acquiring detailed knowledge of market conditions. Assuming that the individual has an aversion both to risk and to the effort involved in acquiring information, we should on the average expect the rate of interest to be lower than the rates of return implicit in other forms of property income.

Historical aspects of the interest rate

Credit arrangements go back to prehistoric times and may even have antedated the emergence of a money economy. Similarly, there is ethnological evidence of the existence of credit in kind in primitive communities having no trace of a medium of exchange.

In prehistoric agricultural communities, loans of seed—to be repaid by a greater quantity of seed at harvest time—were undoubtedly a recognized type of arrangement. However, to the extent that repayment was not required in the case of harvest failure, such arrangements were not pure loans but had at least some aspects of what we would today call a partnership or, more generally, an equity investment. Such mixed arrangements complicate any attempt to compare modern credit arrangements with those of earlier periods. In any event, it seems possible that the natural productivity of agriculture suggested the concept of interest on loans to men in prehistoric—and then in historic—times.

Loans bearing interest are described in contracts from the Sumerian civilization (beginning about 3000 b.c.), and mentions of such loans are numerous in the descriptions of credit agreements that have come down to us from the Babylonian empire (beginning about 1900 B.C.). In the Sumerian period the customary annual rates of interest on loans of barley and silver were, respectively, 33J per cent and 20 per cent. These rates were later established as the legal maxima by the Code of Hammurabi (c. 1800 b.c.) as part of its general tendency to regulate prices, terms of contracts, and other aspects of economic life (Heichelheim [1938] 1958, vol. 1, pp. 55, 111-112, 134-136; Einzig 1949, p. 372; Homer 1963, chapters 1-2).

An even greater development of the credit system characterized the Greek city-states of the seventh century B.C. and onward. Particularly popular were “bottomry loans,” in which the money advanced by a lender was secured by the hull of a ship or by its cargo on a specific voyage. In the case of shipwreck the debt was usually canceled; if the voyage was successful the borrower paid from 22 to 100 per cent interest, depending on the length and hazardousness of the voyage. In contrast, loans secured by real estate and loans to cities frequently bore interest of 8 to 10 per cent. Similar rates prevailed during the Roman era, which was characterized by legal maximum rates, first of 8.5 per cent per year and subsequently of 12 per cent.

Most loans in ancient times were granted for what would today be considered a short period; a year or a part of a year. Since there were no business firms in the modern sense of the term, the loans were necessarily to persons or partnerships. From this, however, we should not infer that these loans were predominantly for consumption purposes. It should also be noted that the Greeks and Romans looked down upon the earning of income from interest. According to certain authorities, this contrasts with the view of the peoples of the ancient Oriental civilizations, who (it is claimed) accepted interest as a normal feature of economic life (Heichelheim [1938] 1958, vol. 1, pp. 104-105, 219).

The absolute prohibition of interest was an outstanding feature of ancient Hebrew economic legislation, as incorporated in the well-known Biblical passages “If thou lend money to any of my people that is poor by thee, thou shalt not be to him as an usurer” (Exodus 22.25) and “Thou shalt not lend upon usury to thy brother” (Deuteronomy 23.19). The rabbis of the Mishnah (200 b.c.-a.d.200) applied this proscription to commercial transactions as well. At the same time they accepted its effective evasion in certain instances by means of the legal fiction of considering the lender to be a partner entitled to profits from the enterprise financed by his funds (Mishnah: Baba Metzia v, 4).

In the Middle Ages the prohibition of interest (or, as it was then called, usury) was a central feature of canon law. But since the church did not prohibit income from property as a category, here too there rapidly developed legal fictions by which the prohibited interest from loans was converted into other—and permissible—forms of income from property. For some scholars this process represents an evasion and a vitiation of the canon law; for others it reflects the deliberate attempt of the church authorities to accommodate themselves to the business needs of the community and accordingly to place the main emphasis of the prohibition of usury on loans made for consumption purposes (Pirenne [1933] 1936, p. 146; Knight [1921-1935] 1951, p. 256; Noonan 1957, pp. 169-170, 190-195).

An important aspect of the Reformation in the sixteenth century was the movement to abolish the legal prohibition of interest. The outstanding theological protagonist of this movement was Calvin. Toward the end of Henry vIII’s reign in England, a law was enacted (1545) legalizing interest but limiting it to a legal maximum of 10 per cent. In the first half of the nineteenth century, Roman Catholic authorities also publicly decreed that the interest permitted by law could be taken by everyone (Nelson 1949, chapter 3; Noonan 1957, chapter 19).

With due allowance for the scarcity and imperfections of the data, there seems to have been a decline in interest rates in Europe from the medieval period to the Renaissance. The minimum interest rates on loans secured by land in the Spanish Netherlands from the twelfth century through the fifteenth century ranged from 8 to 10 per cent, as compared with corresponding rates of 6 to 8 per cent in the Dutch Republic during the sixteenth and seventeenth centuries. At the beginning of the eighteenth century in England, long-term government-bond yields were from 6 to 8 per cent, declining to 3 per cent in the midyears of the century and rising erratically toward the end of the century to 5 to 6 per cent. These last rates continued during the Napoleonic wars at the beginning of the nineteenth century. Subsequently, the rate fell again to somewhat above 3 per cent and declined even further toward the end of the century. Corresponding yields during the twentieth century have generally been higher and in 1955-1965 have again reached 6 per cent. Thus, there may have been a slight downward secular trend in long-term interest rates in England over the past two hundred years, although it remains to be seen if this trend has not been reversed in the period since World War ii (Homer 1963, pp. 137-138, 193, 411, 504-506).

In any event, there does seem to have been a downward trend during the past fifty to one hundred years in the share of interest in the national income of developed countries. Thus, in the United States this share fell from 5.7 per cent in 1899-1908 to 4.0 per cent in 1954-1960. More generally, the data from the United Kingdom, France, Germany, and the United States all show a marked decline over this period in the share of property income (including that of unincorporated enterprises) in total national income. Kuznets attributes this finding in part to the fact that over the same period the ratio of total wealth to national income (the capital-output ratio) has moved downward in these countries; he also conjectures that the decline was reinforced by a secular downward trend in the yield on capital.

Similarly, Kuznets shows that in 1958 the share of income from assets (excluding those of unincorporated enterprises) in total national income was slightly higher in underdeveloped countries (20.6 per cent) than in developed countries (18.4 per cent). He attributes this finding to the much higher yield on capital in the underdeveloped countries, which more than offsets their lower proportion of corporate assets. Correspondingly, the extent to which the share of property income in the underdeveloped countries exceeds that of the developed ones is much greater when account is taken of all such income—including that of unincorporated enterprises. This finding, as well as that mentioned in the preceding paragraph, would, of course, be modified if we were to take account of the capital invested in the form of education in human beings and accordingly of that part of wages and salaries which reflects the return on this capital (Kuznets 1966, tables 4.2, 4.3, and 8.1 and accompanying text). [Seecapital, human; INCOME DISTRIBUTION, article onfunctional share.]

Studies of the business cycle in the United States have shown that the rate of interest (as measured by the yield on railroad bonds) more or less conforms, with a lag, to the level of general economic activity. Similarly, it conforms, with a lag, to the movements of the price level—a fact (referred to generally by Keynes as the “Gibson paradox”) that raises certain theoretical questions discussed below, under “Interest in a money economy,” in the section “Theories of interest” (Keynes 1930, vol. 2, pp. 198 ff.; Macaulay 1938, chapter 6; Burns Mitchell 1946, p. 501; Burns 1961, p. 28; Cagan 1966).

Theories of interest

Interest in a barter economy

From the definition of the interest rate as the price paid for the use of a loan for one unit of time, it seems natural to analyze the determination of this rate in terms of the demand for and supply of loans. Although there are many theories that have not been explicitly formulated in terms of this “loanable-funds approach” to interest, they can nevertheless be examined from this viewpoint.

Investment and the demand for loans. For simplicity, let us define “households” as those economic units that engage solely in the sale of the-services of factors of production and in the consumption of goods and services, and let us define “businesses” as the locus of all production and investment activity. Then in an economy in which money exists only as an abstract unit of account (and for the present purpose such an economy is equivalent to a barter economy) any savings that a household makes will be lent out. Correspondingly, the supply of loans by households to the business sector must equal the total amount that households save. (Consumption loans will be discussed later.) Similarly, under the assumption of a perfect capital market, the net investment of the business sector can be considered as being equal to its total demand for new loans (that is, net of refundings) from the household sector, for in such a market it can make no difference either to businesses or to households whether investment is financed by means of loans, stock issues, or undistributed corporate profits.

The basic fact that lies behind businesses’ demand for loans is the productivity of capital [seecapital; investment]. By this is meant the fact that an investment project can yield over time a stream of returns that exceeds the total costs of carrying it out. In particular, if a project is planned for n years, and if S₁, S₂,..., S_n represents its expected stream of net returns or, in more technical terms, of quasi rents (that is, the expected receipts in each year from the sale of the output of the project less the corresponding current operating expenses in the form of wages, costs of raw materials, and the like—exclusive of depreciation), then the productivity of the capital invested in the project is reflected in the fact that S₁,S₂...S_n (where S_n also includes the possible scrap value) is greater than the cost of the plant and equipment, V, involved in the project.

More precisely, the marginal productivity of capital—which for our purposes can be taken as synonymous with Keynes’s marginal efficiency of capital—is defined as that rate which equalizes the present value of the stream of quasi rents with the cost of plant and equipment. It is that rate, p, which satisfies the equation

where it has been assumed that net positive receipts occur at the end of each period, whereas the total payment for plant and equipment is made at the beginning of the first period. In a world of certainty and a perfect capital market, potential investors will carry out any project whose marginal rate of return, p, exceeds or equals the interest rate, r, which is assumed to remain constant over the economic horizon and which measures the actual or imputed marginal cost of the funds invested in the project. Correspondingly, investors will not carry out any project whose rate of return is less than r. Thus, under these assumptions, the rate of interest serves as the rationing device that allocates scarce capital funds in an optimal manner among competing investments projects. From this it also follows that the lower the rate of interest, the larger the number of investment projects that it pays to carry out.

This relationship is frequently described by means of a negatively sloped curve relating increasing volumes of investment to a declining rate of interest. It should be emphasized that the concept of investment basic to this curve is that of

gross investment. The firm’s decision about a particular investment project can in no way be affected by whether this project represents “replacement” or “new capital"; nor, in a perfect capital market, can it be affected by the accounting distinction between funds made available from depreciation allowances and those made available from new loans. For our purposes, however, it will be more convenient to deal with the derivative concept “net investment,” defined as gross investment less depreciation. As a first approximation (whose accuracy increases the shorter the life of the project) the calculation of this depreciation can be assumed to be unaffected by the rate of interest. Correspondingly, the net-investment curve is obtained from the gross-investment curve by simply shifting the latter to the left by an amount equal to depreciation.

This, then, is the explanation of the negative slope of the demand curve for loans, D, in Figure 1, which under our present assumptions is identical with the net-investment curve. The vertical axis in this diagram represents the (real) rate of interest and the horizontal axis the real volume of new loans made during the specified period of time (say, a year) to which the analysis refers. By construction, any rate of interest on this curve (say, r0) corresponding to a given volume of loans (say, b0) equals the marginal productivity of capital after net investments of the designated volume have been carried out. In other words, if the stock of capital (and hence real volume of loans) in the economy at the beginning of the year in question is B0, then r0 measures the marginal productivity of a stock of capital of B0 + b0 units.

Another property of the demand curve that has been frequently assumed in the literature is that in the absence of uncertainty the demand for loans approaches infinity as the rate of interest approaches zero, for in such circumstances it would pay to carry out any investment project that would yield a perpetual stream of net income. Thus, at a zero rate of interest it would pay to level the Alps or to build a canal across the United States—if such activities would enable us to anticipate with certainty net savings in transportation costs through all future time.

It is clear from the foregoing presentation that b has the dimensions of a stock and is thus not affected by changes in the unit of time used to measure the period specified in the analysis. In other words, at the rate of interest r₀, firms will desire to contract b0 units of new loans during the year—whether the year is called a year or 12 months or 52 weeks. Alternatively, however, b can be interpreted (as it frequently is) as the average rate at which new loans are contracted during the year. Under such an interpretation, b clearly has the dimensions of a flow and is accordingly affected by the time unit used to measure the year. In this case the firms’ total stock of borrowing at the end of the year (at the rate of interest r₀) is represented by Bo where T is the length of the year as measured in the stipulated time units and where b₀T thus has the dimensions of a stock. Clearly, b₀T in the preceding expression is equal, in both dimensions and magnitude, to the b₀ of the interpretation given at the beginning of this paragraph (Patinkin [1956] 1965, pp. 515-523).

Savings and the supply of loans. The supply side of the loan market as described above represents the savings behavior of households [seeconsumption function]. In the analysis of this behavior by, for example, Marshall and Cassel, it was pointed out that an individual who saved in order to assure himself a given level of income in the future (say, after retirement) would have to accumulate a smaller capital sum the higher the rate of interest; correspondingly, such an individual would respond to a rise in interest rates by saving less. More generally—and more precisely—a rise in interest rates generates a wealth effect which, in the case of a lender, tends to offset the substitution effect. The rise makes current consumption more expensive relative to future consumption (by decreasing the present value of the cost of a unit of future consumption). Hence, it generates a substitution effect that decreases current consumption and thereby increases current saving. At the same time, it makes lenders better off by virtue of the higher interest they can earn and thus generates a wealth effect that tends to increase current consumption. If the wealth effect were to predominate (as is implicitly assumed by Marshall and Cassel in the case mentioned above), the individual’s savings curve would, in the relevant region, be negatively sloped with respect to the interest rate.

It is, however, unlikely that such a negative slope could characterize the savings curve of the economy as a whole. To every lender in a closed economy there corresponds a borrower for whom the wealth effect works in the opposite direction. Hence, in the absence of distribution effects, these wealth effects will cancel out, leaving the aggregate savings curve to reflect solely the uniformly positive influence of the substitution effects. Correspondingly, the supply curve of loans, S, in Figure 1 has been drawn with a positive slope (Bailey 1957; Bear 1961).

In the standard Fisher case of an individual with a two-period horizon, the individual is in an optimum position when he adjusts his current consumption (and hence his savings) so as to equalize his marginal rate of substitution of future (C₂) for present (C₁) consumption with 1 + r. In the case of an individual with a horizon of n periods, the planned stream of future consumption C₂, C₃,..., C_n (where C_t represents consumption at the beginning of period t) can be considered as a single, composite good. The quantity of this good, C_F, is defined as equal to the constant level of per-period consumption over the future which has the same present value as C₂, C₃..., C_n; that is,

It can then be shown that the optimum marginal rate of substitution of future (C_F) for present (C₁ consumption in this case is

For large values of n, this marginal rate can thus be approximated by r. That is, the individual in his optimum position can be induced to give up one dollar of present consumption by compensating him with a constant stream of future consumption of (approximately) r dollars per period (Leontief 1958; Liviatan 1966). In this sense, then, the rate of interest (say, r₂) on curve S corresponding to any volume of new loans made during the year (say, b₀) equals the marginal time preference of the individuals in the economy after they have carried out the savings that provide these loans.

The equilibrium rate of interest. The equilibrium rate of interest for the year in question is determined in Figure 1 (interpreted as above in terms of either a stock or a flow) by the intersection of the demand and supply curves for loans at point P, corresponding to the rate of interest of r₁. As already indicated, for the barter economy now being discussed these curves can also be interpreted as respectively representing the net investment and savings curves of neoclassical interest theory. Correspondingly, by construction of these curves the equilibrium rate measures at one and the same time the marginal productivity of capital and the marginal rate of substitution between future and present consumption. (This analysis could just as well be carried out in terms of gross investment and saving—the curves in Figure 1 could be made to represent these magnitudes by simply shifting them to the right by an amount equal to depreciation allowances; hence they would continue to intersect at the rate of interest r₁.)

The foregoing analysis can be recast in terms of the demand and supply for the total stock of loans to be outstanding at the end of the year in question. Clearly, the amount of this stock demanded at the rate of interest r₁ is B₀ + b₁ where, as before, B₀ represents the stock at the beginning of the year. Similarly, the stock supplied at the end of the year is also B₀ + b₁. Thus, this approach yields the same equilibrium rate of interest, r₁ as the preceding one and in this sense is equivalent to it.

On the other hand, the elasticity (although not the slope) of the curves in Figure 1 at each point would be less if they referred to the total stock outstanding. This elasticity is the percentage change in the total stock of loans (say) demanded following a unit percentage decrease in the rate of interest; the same absolute increase in the amount demanded is a much smaller percentage of the total stock of bonds outstanding than of the net accretion during a year. Conversely, a one per cent increase in this net accretion will require a much smaller fall in the rate of interest insisted upon by debtors than will a one per cent increase in the total stock of bonds held.

Consumption loans can be incorporated into the analysis of Figure 1 either by adding them to the demand side or by deducting them from the supply side—in which case S would represent the net supply of loans by households to the business sector. From either viewpoint, in a modern economy the relative influence of such loans on the over-all demand-and-supply situation—and hence on the equilibrium rate of interest—is a minor one. Correspondingly, it is a much better approximation of the truth to say that interest must be paid on consumption loans because the potential lender has the alternative of lending money at interest on productive loans than to say that the demand for consumption loans is the reason for the existence of interest.

It should also be emphasized that “impatience” or “time preference"—in the sense that an individual with a two-period horizon systematically prefers (say) ten units of consumption goods today and four tomorrow to the alternative combination of four units today and ten tomorrow—is not a necessary condition for the existence of interest. For the major manifestation of time preference in this sense—in an economy in which all individuals have a finite horizon and anticipate a constant stream of income payments—is that (strictly) positive savings will be forthcoming only at a (strictly) positive rate of interest. Conversely, absence of time preference manifests itself in the fact that if the rate of interest is zero, then savings are also zero (Boulding [1941], pp. 746-752 in 1948 edition; Friedman 1957, p. 12). Thus, the fact that individuals in such an economy insist on receiving interest in order to save is no evidence of a systematic preference for present goods over future ones. More generally, the fact that an individual will, at the margin, insist on receiving more than one unit of future goods to compensate him’ for forgoing one unit of present goods is not necessarily the cause of the existence of interest but its effect: the individual insists on receiving more because he has the alternative of obtaining more by lending out at interest the money that would be released from current consumption by saving.

Whether b in Figure 1 is interpreted as the net accretion to the stock of loans over the year or as the average rate of flow of new loans during the year, the equilibrium determined there is that of the short run. (Alternatively, in terms that derive from the second of the two interpretations, it is a “flow equilibrium.”) The stock of capital (and hence the stock of outstanding loans) at the beginning of the subsequent year will increase, shifting both the demand and supply curves that could be drawn in Figure 1 for that year. In particular, if technology and the quantities of other factors of production are kept constant, the law of diminishing returns will cause a leftward shift of the investment curve and hence of D. At the same time, the increased capital will generate an increased stream of income, thus increasing the amount of savings forthcoming at any rate of interest and thereby shifting S to the right. The new equilibrium situation generated by these shifts is represented by the intersection at Q of the dashed curves D’ and S’ in Figure 2. For the same reason as before, point Q also represents a position of short-run equilibrium, so the curves will shift once again.

The stationary state. A question that has frequently been raised in the literature is whether the shifts just discussed can ultimately bring the economy to a position of long-run equilibrium (or “stock equilibrium,” in the alternative terminology described above) in which no further capital accumulation will take place—and whether in this classical “stationary state” the rate of interest can be positive [seecapital],

A situation in which both these conditions are met is illustrated in Figure 2 by the intersection of D” and S” at point R on the vertical axis. At this point the marginal productivity of the existing stock of capital equals the prevailing rate of interest r_L so firms have no inducement either to augment or to reduce this stock—and with it the stock of their outstanding loans from households. (The possibility of corresponding reductions in both of these stocks at rates of interest above r_L is represented by the extension of D” to the left of the vertical axis.) Similarly, households have no inducement either to make new (net) loans to firms or to insist on the redemption of part of their outstanding loans in order to consume the proceeds.

A necessary condition for the existence of a stationary state with a positive rate of interest is that the individuals of this state (assumed to be of a given, finite life expectancy and to anticipate a constant stream of income) have a preference for present as against future consumption; for, as noted above, in the absence of such a preference these individuals would have zero savings only at a zero rate of interest—that is, their savings curve would always start at the origin of Figure 2 (cf. Pigou 1935, chapter 10).

Growth models. We must now note that—as Frank Knight has particularly emphasized ([1921-1935] 1951, pp. 262-264; 1936, pp. 614-619, 626-630; 1944)—the concept of the stationary state has no relevance to the real world, which historically has been characterized by continuous growth in total capital and (as noted above, in “Historical aspects of the interest rate”) relative stability of the rate of interest. These developments can in part be interpreted as reflecting the absence in the real world of the two basic assumptions that underlie the theory of the stationary state—namely, constancy in the quantity of other factors of production (that is, labor) and constancy of the state of technology. Indeed, a major concern of economic theory in recent years has been to analyze

the growth patterns of economies in which one or both of these assumptions do not hold.

One case of a growing economy that has been particularly studied is that of a one-good, two-factor (capital and labor), perfect-competition economy with a given constant-returns-to-scale technology in which the labor supply is expanding at a constant, exogenously determined rate of n per cent per year (Solow 1956; Swan 1956). Assume for simplicity that the individuals in this economy save a constant proportion, s, of their incomes. Represent this behavior in some initial position by the vertical line S* in Figure 3, where for convenience savings are measured as a percentage of the total existing stock of capital in that position. Similarly, let 1* represent the investment curve of the economy, also measured as a percentage of initial capital. The intersection of these curves at point A—corresponding to the rate of interest r* —then represents the equilibrium position of the economy.

It is clear, however, that this is only a short-run equilibrium, for the percentage rate of growth of capital (J/K) at point A equals m, which is less than the corresponding rate of growth of labor, n. Hence total output, total income, and (by assumption) total savings must all increase at some common rate greater than m and less than n; therefore, S/K must increase, shifting S* to the right. Similarly, the fact that the ratio of labor to capital is increasing means (assuming that we are in the region of diminishing returns) that the marginal productivity of capital is increasing accordingly, so that I* also shifts to the right. Furthermore, if we assume that the rate of interest throughout this adjustment process equals the marginal productivity of capital, then the investment curve must shift farther to the right than the savings curve, thus intersecting the latter at a higher rate of interest.

This process could continue indefinitely without the system’s ever reaching a position of long-run equilibrium. However, under certain assumptions about the nature of the production function, such a position will be reached. In this case the rightward shifts of the vertical savings curve in Figure 3 continue until the curve reaches S** at point n on the abscissa, where it is intersected at a higher rate of interest, say r**, by an investment curve (I**) that has also shifted to the right during the adjustment process. The point of intersection, B, between S** and I** represents a situation in which capital and labor—and hence total output, income, savings, and investment—are all expanding at the common rate n. Correspondingly, there will be no further shifts in the savings and investment curves of Figure 3. Point B thus represents a long-run moving equilibrium in which the labor-capital ratio and hence (by virtue of the properties of a constant-returns-to-scale production function) the rate of interest and the real wage rate remain constant. Clearly, per-capita income and consumption also remain constant in this moving equilibrium.

The same argument holds, mutatis mutandis, if the initial equilibrium position in Figure 3 corresponds to a rate of capital accumulation v greater than n. In this case output (income) will increase at a rate less than v, so that S/K shifts to the left. Similarly, the increasing ratio of capital to labor will shift the investment curve to the left, and the equilibrium rate of interest will fall. Once again, long-run moving equilibrium will be reached only if the investment curve ultimately intersects the savings curve in the vertical position S**, corresponding to a rate of growth of total output of n per cent per year.

As might be expected, a special case of this long-run equilibrium is the classical stationary state represented in Figure 2, above. This can be represented in Figure 3 by setting the rate of growth of labor, n, equal to zero and drawing S** accordingly to coincide with the vertical axis.

Returning to the nonstationary case depicted in Figure 3, let us assume that the long-run equilibrium position B is disturbed by an exogenous decrease in the savings ratio s that shifts the savings curve to S*. The initial effect of this decrease is to shift the economy to short-run equilibrium position C. Here the forces described above will again come into operation, shifting the savings and investment curves to the right and thus generating the path of short-run equilibrium positions—say,

CD—by which the economy is brought to the new long-run equilibrium position—say, D. Labor, capital, and hence total output at D are once again growing at the rate n. On the other hand, the decrease in the savings ratio has caused an increase (from r** to r***) in the long-run equilibrium rate of interest and (because of the constant-returns-to-scale assumption) a decrease in the real wage rate. Both of these changes reflect the fact that along path CD labor is expanding more rapidly than capital, so the labor-capital ratio is higher at D than at B. Similarly, along path CD output is expanding at a lower rate than labor, so per-capita output is less at D than at B.

In brief, a decrease in the savings ratio in this model will decrease the long-run equilibrium ratio of capital to labor, with corresponding consequences for factor prices. The savings ratio, however, will not affect the equilibrium growth rate of the economy, which is uniquely determined by the exogenously given rate of growth of the labor supply.

It should be emphasized that the fact that per-capita income is less at D than at B does not imply that per-capita consumption is also less at D than at B. The lower level of income might be offset by the lower savings ratio (higher consumption ratio) that by assumption prevails at D. This suggests that among long-run equilibrium positions with alternative savings ratios there exists one specific ratio that maximizes per-capita consumption. It can be shown that such a maximum is achieved when the savings ratio is such as to bring the economy to a long-run equilibrium capital-labor ratio that generates a marginal productivity of capital equal to the equilibrium rate of growth, n. The mathematical demonstration is straightforward: Per-capita consumption equals

where L is the quantity of labor and Y = F(K,L) is the production function. By the assumption of constant returns to scale, this expression can be rewritten in terms of the capital-labor ratio, K/L, as

where the second term has been multiplied and divided by K. Since only positions of long-run equilibrium are being considered, S/K can be replaced by the constant n. Maximizing the resulting expression by differentiating it with respect to K/L and setting the result equal to zero then yields

where Ft(K/L, 1) is the partial derivative of F(K/L, 1) with respect to its first argument and thus represents the marginal productivity of capital. The value of K/L that satisfies this equation is, thus, the capital-labor ratio that maximizes long-run per-capita consumption.

An economy satisfying this equality between the marginal productivity of capital and the rate of growth is said to be growing in accordance with the “golden rule of accumulation.” Under the assumption of perfect competition (in which case the marginal productivity of capital equals the rate of interest) this rule can be stated alternatively in terms of the equality between the long-run equilibrium rate of interest and the growth rate. This in turn implies that if point B in Figure 3 is on the golden-rule path, then it satisfies the relation

so that

S = I = r^**K.

That is, in a competitive economy growing in accordance with the “golden rule,” total savings or investment during any period equals total income from capital. It should be noted that this is a macroeconomic relationship and does not imply that all income from capital is saved (Phelps 1965; Marty 1964).

The foregoing graphical analysis can be generalized to the case in which the savings ratio is an increasing function, s = s(r), of the rate of interest, so that the savings function (expressed again as a percentage of the stock of capital) has the form

Such a function could be represented in Figure 3 by a positively sloped curve that would shift during the adjustment process in a manner analogous to that in which the vertical savings curve shifts.

The analysis can also be readily generalized to the case of technological change which is entirely labor augmenting (Harrod-neutral). In this case n can be considered as the rate of growth of the “effective quantity” of labor, where this rate equals the sum of the natural rate of increase of the labor force (say, n’) and the rate of increase in labor efficiency per unit of labor (say, n”). The argument proceeds exactly as before, with the sole difference that the position of long-run equilibrium (in which savings, investment, capital, and output are once again all growing at the rate n) is characterized by an increase of n” per cent per period in per-capita consumption.

Much attention has been given in the recent literature to more complicated growth models with more general kinds of technological changes or with more than one good, or both. These models are described elsewhere [seeeconomic growth, article onmathematical theory].

Interest in a money economy

An economy in which money exists as a medium of exchange is not characterized by the identity that holds in a barter economy between planned investment and the demand for loans, on the one hand, and between planned savings and the supply of loans, on the other. In a money economy, firms can plan to finance investment in plant and equipment—and households can plan to supply loans—by reducing their money balances. Similarly, the supply of loans can be affected by changes in the quantity of money in the system. Another noteworthy difference between a barter economy and a money economy is that in a money economy the wealth of households includes the real value of their money holdings (or, more generally, of their net financial assets); correspondingly, an increase in wealth in this form will increase consumption and thus (inter alia) affect the rate of interest. This is a manifestation of what has been called the real-balance effect.

The stock of money that individuals want to hold is positively dependent on their total wealth (or income) and inversely dependent on the rate of interest, which measures the individual’s opportunity cost of holding a unit of money instead of an income-yielding asset. Although the analysis of this dependence of money holdings on interest has its antecedents in the writings of the neoclassical economists, these economists did not really integrate the relationship into their expositions of the quantity theory of money, and into their analyses of the velocity of circulation in particular. This was indeed one of Keynes’s major contributions. [Seeliquidity preference; money, article onquantity theory.]

In any event, individuals will not be willing to hold the existing stock of money in the economy unless the value they attribute to the liquidity service provided by this stock at the margin equals the rate of interest that could alternatively be earned by making a loan (or, in more familiar Keynesian terms, by holding a bond). It follows that when the system as a whole is in equilibrium, the rate of interest must at one and the same time equal the “threefold margin” of liquidity services, productivity of capital, and time preference (Robertson [1924-1940], pp. 16-17 in the 1940 edition). Indeed, the money balances of a firm can be considered as part of its working capital—just like any other inventory that it holds—and must accordingly yield a corresponding marginal productivity.

The two major problems in the theory of interest that (by definition) are specific to a money economy are the effects of changes in the quantity of money and the effects of shifts in liquidity preference. These problems will now be discussed, first under the neoclassical assumptions of price flexibility and a constant full-employment level of income and then under the Keynesian assumptions of price rigidity and unemployment. For simplicity, it will be assumed that the economy’s reaction to these monetary changes is rapid enough to make it possible to study them within the framework of short-run equilibrium analysis (on the rest of this section, see Patinkin [1956] 1965).

A major theme of classical and neoclassical interest theory was that an increase in the quantity of money (generated by gold discoveries or by a government deficit) in the first instance increases the supply of loans and thus depresses the rate of interest. This is reflected by the shift from S to S’” in Figure 4. At the same time, classical and neoclassical economists argued that the increased quantity of money also raises prices and thus causes the real supply of loans to decrease once again. Ultimately—after prices have increased in the same proportion, restoring the real value of the public’s money holdings—the supply curve returns to its original position, S, so that the rate of interest is once again r1. Thus, the ultimate invariance

of the rate of interest with respect to changes in the quantity of money (that is, the “neutrality of money”) was an integral part of classical quantity-theory reasoning.

The details of the process just described were set out by classical and neoclassical economists in various ways. Among the best known expositions is that of Wicksell, who referred to the rate of interest actually prevailing as the “money” or “market” rate—as distinct from the “real” or “natural” rate, which equates planned savings with investment (rl in Figure 4). By definition, the natural rate is also the rate equating the aggregate demand for commodities with the corresponding supply. The mechanism by which the money rate becomes equated with the natural rate was called by Wicksell the “cumulative process.” The basic component of this process is the fact that the rising price level generated by the excess demand for commodities increases the demand for hand-to-hand currency, which the public satisfies by converting their bank deposits into cash. The resulting internal drain on bank reserves forces the banks to raise their lending rate until it is once again equal to the natural rate. Wicksell used the lag with which the market rate adjusts itself to the natural rate to explain the fact (noted above, at the end of the section “Historical aspects of the interest rate”) that empirically the interest rate and the price level have tended to move together (Wicksell [1898] 1936, pp. 107, 167-168; [1906] 1935, vol. 2, pp 205-207; Cagan 1965, pp. 252-259).

It should be emphasized that the invariance of the equilibrium rate of interest holds even under the Keynesian assumption that the real demand for money also depends on the rate of interest. After prices have risen in the same proportion as the quantity of money, individuals will be willing to hold this increased nominal quantity—which will then represent an unchanged real quantity—at a correspondingly unchanged rate of interest.

However, classical and neoclassical economists did recognize that under certain circumstances monetary changes could have real consequences. Thus, they sometimes argued that—as a result, say, of the lag of wages behind prices—the inflationary process generated by an increase in the money supply could change the distribution of real income in favor of profit recipients and against wage earners. The profit recipients’ presumed lower propensity to consume would then bring about a decrease in the total real consumption of the economy and hence an increase in its total real savings. Because of these “forced savings” (as they came to be called) the savings curve would not return to its original position even if prices were to increase in the same proportion as the quantity of money. In such cases these economists readily recognized that the monetary expansion was not neutral: it would increase the stock of capital in the economy and thus permanently depress the rate of interest.

Similarly, an increase in the quantity of money generated by an open-market purchase of bonds (instead of by the gold discoveries or the government deficit assumed until now) will also have a depressing effect on the equilibrium rate of interest. In the first instance, such a purchase merely replaces bonds by an equivalent amount of money in the public’s portfolio and hence leaves undisturbed total wealth, the aggregate demand for commodities, and the initial equilibrium in the commodity market. But by simultaneously increasing the demand for bonds and supply of money, it disturbs the equilibrium in both these markets and causes the rate of interest to decline. This decline reacts back on the commodity market, stimulating aggregate demand there and eventually causing prices to rise. It follows that if the reduction in the rate of interest stimulates investment relatively more than consumption, government open-market policy will have a long-run effect on the growth path of the economy.

Exactly the same reasoning can be applied to show that an increased supply of loans financed by a decrease in the desire to hold money balances (in Keynesian terms, by a decrease in liquidity preference) will also depress the equilibrium interest rate. The possibility of such an effect was not recognized by the neoclassical economists. This is another manifestation of the fact, which we have already noted, that although in their quantitytheory discussions these economists adverted to the holding of bonds and other productive assets as an alternative to holding money, they did not fully integrate this possibility into their thinking.

The initial impact of both the foregoing changes is again illustrated by means of the shift from S to S’” in Figure 4. The resulting decline in the interest rate generates an excess demand in the commodity market and hence an increase in the price level. This, in turn, reduces the real supply of loans and causes S’” to begin to shift to the left again. But in this case, unlike the case of an increase in the quantity of money generated by a government deficit, the supply curve cannot return to its original position. In that position the rate of interest would be the same as it originally was,

but prices would be higher. Since the nominal value of the public’s net financial assets is not affected by the open-market purchase or the shift in liquidity preference, this would imply the existence of a negative real-balance effect, which would preclude equilibrium in the commodity market. Thus equilibrium will be reached at a rate of interest below r₁ but above r₄.

The extent of the depressing effect on the equilibrium interest rate in both of the preceding cases clearly depends on the strength of the real-balance effect, which empirically has been shown to be fairly weak. Indeed, in an economy in which the net financial assets of the public were zero—and in which, accordingly, there was a unique rate of interest at which full-employment equilibrium in the commodity market could obtain—the equilibrium rate would be unaffected. Such an economy is one in which there is no government debt and in which the money supply is created entirely by the banking system in the process of extending loans to the private sector (the so-called “pure inside-money economy”).

Under the Keynesian assumptions of price rigidity and unemployment, the rate of interest would be affected in all of the cases discussed above. For example, if the quantity of money was increased while prices remained unchanged or rose only slightly, then the economy would be brought to a new position in which the interest rate was lower—and hence the levels of aggregate demand and employment were higher—than before the monetary expansion. It should be noted that the decline in the rate of interest in this case is of the same nature as the short-run decline specified in the neoclassical analysis described in the last section.

Once we depart from the analytical framework of an economy with a given productive capacity and turn to growth models, there are additional reasons for the nonneutrality of money. For example, it will generally be true that a growing economy whose money supply is continuously expanded at a rate that stabilizes the price level will have a different growth path than one in which the quantity of money is kept constant and the price level declines over time. In general terms, the reason for this difference is that the declining price level increases the rate of return obtained by holding money balances (by virtue of the increase in their purchasing power that it generates), and thus increases the attractiveness of holding savings in this form. Hence, even if the over-all savings ratio, s, should remain constant, the amount of savings (at any given level of output) that will take the form of physical capital goods will decrease; that is, the physical savings ratio will decrease. Thus, the effects of a declining price level can be analyzed in Figure 3 (which refers only to physical savings) in terms of a leftward shift of the (physical) savings curve from S^** to S^*; just as in the earlier analysis of such a shift, the long-run equilibrium rate of interest will then rise, and the long-run real wage rate will fall. More generally, the lower the rate of increase of the money supply, the greater the rate of decrease of prices and the smaller the long-run-equilibrium capital/labor ratio, hence the higher the corresponding marginal productivity of capital and the rate of interest. This result, however, need not obtain if the over-all savings ratio is not constant, but instead depends directly on the respective rates of return of the various assets, including the rate of return on money balances (i.e., the rate of decrease of prices) (Gurley & Shaw 1960; Enthoven 1960; Patinkin [1956] 1965, pp. 360-364; Tobin 1965; Stein 1966).

General equilibrium analysis. Although the analysis of the determination of the equilibrium interest rate was presented above in terms of the demand and supply curves for loans (the so-called loanable-funds approach), it is clear from the discussion that these curves also reflect the forces at work in the markets for money and commodities, respectively. Correspondingly—and in a completely equivalent manner—it would be possible to present a “liquidity-preference approach” that would depict the equilibrium rate of interest as occurring at the intersection of the demand and supply curves for money—and in which these curves would in turn reflect the forces at work in the markets for loans and commodities.

Thus, no substantive meaning can be attached to the choice of the particular market in which one carries out the analysis of the rate of interest. This choice does not alter the basic fact that the rate of interest influences behavior in all the markets of the economy and that accordingly its equilibrium value (like that of any other price) is determined by the necessity for all these markets to be in equilibrium. This fundamental point can best be brought home by actually carrying out the analysis within a general-equilibrium framework in which all these markets are considered simultaneously (Hicks 1939, chapter 12; Patinkin [1956] 1965, pp. 258-260, 330-334, 375-381).

A meaningful question, however, is the nature of the forces that have affected the equilibrium rate of interest over the course of time. The classical and neoclassical view has been that the major forces are changes in tastes, which affect the desire to save, and changes in factor proportions and technology, which affect the productivity of capital. On the other hand, this view attributes only minor importance to changes in the quantity of money and (by inference) to shifts in liquidity preference. It is in this sense that in the classical scheme interest is a “real” phenomenon. In contrast, the Keynesian view attaches much more importance to the monetary factors mentioned above and in this sense treats interest as a “monetary phenomenon.”

The emphasis placed on monetary factors seems to have diminished in the more recent expositions of the Keynesian theory (Robinson 1951, especially pp. 93 and 110). This might be interpreted as reflecting the fact that the Keynesian monetary theory of interest had its genesis in the great depression of the 1930s, when “excess savings” and extensive unemployment of plant and equipment discouraged thinking in terms of the scarcity and marginal productivity of capital. Conversely, the renewed emphasis on these traditional, real factors in recent years can be interpreted as a reflection of the widespread “capital shortage” of the post-World War ii boom period in developed countries—and, even more so, of the key role played by the volume of capital formation and its productivity in the development plans of underdeveloped countries.

Interest differentials. The discussion thus far has been presented in terms of a single interest rate. Actually, however, there is a whole array of interest rates in the market. The major source of differentials in interest rates is the varying degree of risk associated with different loans. Two types of risk have received particular attention in the literature: the risk of default and the risk of illi-quidity. “Illiquidity” refers to the risk that the lender may have to sell the bond under unfavorable circumstances before its maturity. This, indeed, is the basis of the Keynesian explanation of the fact that individuals hold money even though they have the alternative of holding interest-bearing bonds (Tobin 1958). In any event, the actual rate of interest on any particular bond can be viewed as being composed of the “pure” rate of interest that would be paid on a riskless loan plus an appropriate risk premium.

Insofar as the risk of default is concerned, a study of corporate bonds of different quality classes in the United States for the period 1900-1943 showed that the bonds that were rated highest by investment agencies (for example, Moody’s) yielded on the average 5.1 per cent, whereas those rated lowest (and whose default rate was eight times as great) yielded 9.5 per cent. It has been claimed that an important reason why this risk premium has been so large (4.4 per cent) is the legal and traditional limitations on large institutional investors, which have prevented them from buying the larger quantity of high-yield, low-quality bonds appropriate to a rationally diversified portfolio (Hickman 1958, pp. 10-17).

The risks of illiquidity have been analyzed particularly in connection with the tendency of the interest rate in most periods to increase with the duration of the loan. The formal analysis of this “term structure of interest rates” distinguishes between the long-term rate (say, the annual rate of interest, R, on a loan granted for n years), the forward short-term rate (the rate of interest r_t at which a one-year loan to be given in year t can now be contracted), and the spot short-term rate (the existing rate for a one-year loan). In equilibrium, (1 + R)ⁿ = (1 + r₁)(l + r₂) . . . (1 + r_n), for unless this equality obtains, individuals will find it advantageous to shift from long-term loans to an arrangement of refinancing by means of forward contracts for a series of short-term loans.

As can be inferred from the discussion above, a basic contention of liquidity-preference theory is that in order to induce lenders to make long-term loans—which in principle is equivalent to making forward contracts for short-term loans—the forward rate for year t must exceed the actually expected spot rate for that year by a “liquidity premium” that will compensate the lender for the risks involved in tying up his funds. This implies that if the short-term rate is expected to remain unchanged in the future, then the current short-term rate will be lower than the current long-term rate (Hicks 1939, chapter 11).

Although the empirical evidence is not unequivocal, it does on balance seem to support the foregoing theory. It might also be noted that the yield on short-term government securities in the United States during the past forty years has typically been lower than the yield on corresponding long-term securities. Again, the history of business cycles in the United States during the past century has shown that the short-term rate has tended to rise relative to the long-term rate near the peak of the cycle. This can be interpreted in terms of the preceding equation as reflecting the market’s opinion that the short-term rates at such a time are abnormally high and are due to fall. A corresponding interpretation holds, mutatis mutandis, for the symmetrical finding that the short-term rate falls

relative to the long-term rate near the trough of the cycle (see Kessel 1965 and references there cited).

To return to the discussion of a single rate of interest, it should be emphasized that neoclassical and Keynesian economics, although they differ in their analyses of the equilibrating mechanism, are at one in equating the rate of interest with the value of the productive services of capital as determined by the market. However, as Knight emphasized many years ago in his fundamental and incisive critique of “productivity ethics,” this does not constitute an ethical legitimation of the resulting income distribution and of its interest component in particular. Among other things, this distribution can be no more “just” than the distribution of the wealth that generates the interest income (Knight [1921-1935] 1951, pp. 41-75 and 255).

In the Marxist theory of labor value, interest has a negative ethical connotation; it is one of the forms of “surplus value” that stem from the exploitation of labor in a capitalist economy. Because of this ideological connotation, interest costs were largely ignored in the economic planning of the U.S.S.R. in the 1930s and after. In more recent years, however, widespread agreement has developed among Soviet economists and practitioners on the theoretical necessity of making some sort of interest charge on capital (although usually under a different name) in order to make an optimal choice among alternative investment projects. There also seems to be increasing evidence of the actual use of such a charge by the planning authorities (Kaplan 1952; Nove 1961, pp. 209-217, 228-231; Bergson 1964, chapter 11, especially p. 252; Barkai 1967).

Don Patinkin

BIBLIOGRAPHY

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Bergson, Abram 1964 The Economics of Soviet Planning. New Haven: Yale Univ. Press.

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Burns, Arthur F. 1961 New Facts on Business Cycles. Pages 13-44 in Geoffrey H. Moore (editor), BusinessCycle Indicators. Volume 1: Contributions to the Analysis of Current Business Conditions. Princeton Univ. Press.

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Cagan, Phillip 1966 Changes in the Cyclical Behavior of Interest Rates. Review of Economics and Statistics 48, no. 3:219-250.

Conard, Joseph W. 1959 An Introduction to the Theory of Interest. Berkeley: Univ. of California Press.

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Keynes, John Maynard (1930) 1958-1960 A Treatise on Money. 2 vols. London: Macmillan. → Volume 1: The Pure Theory of Money. Volume 2: The Applied Theory of Money.

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INTEREST

INTEREST. Usury laws, inspired both by Scripture and by a misunderstanding of monetary economics, have probably never prevented lenders from charging interest. Throughout early modern Europe, not only states, businesses, and private individuals, but even religious institutions from mosques to monasteries commonly lent and borrowed with interest. Nonetheless, usury laws have shaped the history of credit by forcing contracting parties to disguise interest payments as something else. No one, of course, is fooled by the subterfuge, but it has often determined the nature of monetary institutions and severely limited the survival of documentation through which historians might study the movement of the interest rate. By the eighteenth century, as religious objections weakened, the debate over usury laws became utilitarian rather than doctrinal. At the same time, a new debate over the determination of the interest rate became central to the economic theory of the Enlightenment, and to the rejection of earlier mercantilist policies.

USURY LEGISLATION

"Lend without expecting any return," counsels Jesus in the Sermon on the Mount (Luke 6:34–35), and though the context would suggest that one should not even expect repayment of the principal, the medieval church read his statement as a prohibition on interest. The lesson was reinforced by certain passages of the Old Testament that denounce "usury" without clearly defining the word (Exodus 22:25; Deuteronomy 23:19–20; Psalms 15:5), as well as Aristotle's doctrine that money is sterile (Politics 1:10; Ethics 5:5). Jews were often permitted to lend to Christians at interest since they fell outside the spiritual authority of the church, thus demonstrating that the original purpose of usury legislation was to protect the lender from sin, not to protect the borrower from exploitation. The reputation of Jews as moneylenders was greatly exaggerated, however, and they never played more than a minor role in credit markets before the rise of the Rothschild Bank in the nineteenth century.

A papal bull of 1425 permitted Catholics to buy and sell perpetual annuities, at least when mortgaged against real property (a distinction that was eventually ignored). The Orthodox Church also relaxed usury laws by the sixteenth century. Though the Koran also denounces usury (2:275, 3:130, 4:161, 30:39), in the early fifteenth century the Ottomans came to allow a form of perpetual annuity known as the cash waqf. Originally created to fund charitable institutions such as schools and mosques, it became a common form of private investment by the sixteenth century.

In western Europe in the sixteenth century, Protestant reformers began to chip away at the remaining religious prohibitions on interest, which they associated with Scholasticism. Luther, Calvin, and Zwingli variously argued that interest is not usurious so long as the rate charged is moderate and the contract is in accordance with the Golden Rule ("Do unto others as you would have them do unto you"). Luther, moreover, insisted that one should submit to the laws of the state and not invoke biblical usury prohibitions as an excuse for default. In Catholic Europe, the Jesuits played a similar role in promoting the toleration of interest. The effect of such teachings was not so much to extinguish as to secularize discussions of usury law, so that by the eighteenth century the debate had become almost entirely utilitarian rather than exegetical.

Some Enlightenment writers, including John Locke and Jeremy Bentham, insisted on the complete deregulation of interest. Adam Smith believed that a legal ceiling on interest rates was justified to prevent consumption loans to spendthrifts, since lenders would consider them a bad risk at the legal rate. He agreed, however, that if the ceiling were set below the market rate for commercial loans, it would be counterproductive since merchants would be forced to borrow outside the law. Without the security provided by the law courts, lenders would charge a risk premium, thus actually raising, not lowering, interest rates. Anne-Robert-Jacques Turgot made much the same point when he described an incident in Angoulême in 1769, in which a group of insolvent debtors brought financial panic, and thus extraordinarily high interest rates, on the entire city by attempting to prosecute their creditors for usury.

LONG-TERM INTEREST

Long-term bonds in early modern Europe usually took the form of perpetual annuities. The purchaser of the annuity (that is, the lender) paid a lump sum, in return for which the seller (or borrower) promised to pay a fixed coupon once a year forever. On the Continent the contract of sale had to pass before a notary (thus incurring notarial fees), as did any resale to a third party. The lender could not require the borrower to repay the principal, though the borrower could do so voluntarily at any time, and thus extinguish the loan. Throughout Europe, permissible coupon rates tended to fall from 10 percent or more in the sixteenth century to 5 percent or less in the eighteenth century. In any given period, coupon rates recorded in notarized contracts were usually simply the maximum allowed by law, and thus seemed to represent a legal fiction. That is, contracting parties presumably varied the yield rate of the bond simply by agreeing to a sales price somewhat higher or lower than that stated in the contract. For historians, the fluctuation of long-term interest rates on private bonds is thus largely unrecoverable.

European states also borrowed primarily through perpetual annuities, the most famous being the Consols with which Britain consolidated its national debt after the Glorious Revolution (1688–1689). By the eighteenth century state bonds were actively traded on national stock exchanges, and yield rates can often be inferred from the quotations printed in commercial newspapers. In Britain and the Netherlands, where representative assemblies managed the national debt to the advantage of their wealthy constituents, the risk on state bonds was essentially zero, and yield rates fell as low as 3 percent. In France, on the contrary, the monarchy issued partial defaults on its debt every few decades and could only continue to borrow by offering exceptionally high interest rates (which, of course, rendered future defaults more likely). The need for cheap credit to finance increasingly costly wars thus seems to have worked to the advantage of representative regimes, a fact that goes a long way toward explaining the widespread movement for constitutional reform in the second half of the century.

SHORT-TERM INTEREST

Starting in northern Italy at the end of the thirteenth century, merchants developed a variety of new forms of short-term credit that bore hidden interest. By far the most important were the promissory note and the bill of exchange. A promissory note is little more than an IOU by which the debtor (who in most cases is purchasing merchandise on credit rather than actually borrowing cash) promises to pay to the creditor, "or his order," a given sum on a given date. Typically written at term of two to six months, rarely more than a year, such notes make no mention of interest, but the interest is in fact included in the face value. At any given moment the market value of a promissory note is thus its face value minus the "discount," or interest over the remaining term. If, for instance, the discount rate is currently 8 percent per year (0.08), then a promissory note with a face value of 100 ducats payable in six months (0.50 year) is worth:

The discount rate thus expresses interest not as a percentage of the principal borrowed, but as a percentage of the final payment (interest plus principal). If r is the interest rate as conventionally calculated, d is the discount rate and t is the term, then:

At short term, however, the difference between r and d is negligible.

Through the bill of exchange, a merchant sells the right to collect a sum of money from his correspondent in a different city. Rather than an IOU, it is thus a sort of "he-owes-you" used to transfer funds between two geographically distant locations, either within the same country (inland bills) or in different countries (foreign bills). The value of the bill of exchange depends on the going exchange rate, expressed as a percentage premium or loss for inland bills, and as a rate of exchange between two national currencies in the case of foreign bills. As with the promissory note, the bill of exchange nowhere mentions interest, but merchants openly charged less for bills written at longer term. Even sight bills (technically payable one day after acceptance by the party on whom they were drawn) included a small amount of hidden interest, since it would take them several weeks to reach their destination through the mail. Bills payable one, two, or more months after acceptance sold at correspondingly more advantageous exchange rates. One of the curious results is that the going rate of exchange at any city A on another city B was consistently different from the rate of exchange at B on A. If a merchant purchased a bill of exchange at A on B, sent it to B, instructed his correspondent to use the funds to purchase another bill at B on A, and finally cashed the latter in A, he would end up with more than he started with, the difference corresponding to the interest on his initial outlay.

Bills of exchange and promissory notes did not require notarization. By the seventeenth century (and probably earlier) they were negotiable throughout Europe by simple endorsement. Issued by businesses large and small, they circulated widely. Unlike cash, commercial paper, with its hidden interest, constantly gained value until it came due. The portfolio of credits outstanding thus came to replace cash as the largest reserve of liquid wealth, not only for wholesale merchants but even for humble artisans and shopkeepers. The movement of the interest rate therefore directly concerned all business people.

THE MOVEMENT OF INTEREST RATES

The economic history of Europe has been written largely on the basis of grain prices. The movement of interest rates, though equally important, is less well known, largely because the habit of disguising interest makes the rates so difficult to recover. Eighteenth-century economists asserted that interest rates had fallen steadily from about 10 percent in the sixteenth century, to 6 to 8 percent in the seventeenth century, and to 5 percent or less in the eighteenth century. This long-run movement has been substantiated by the research of Sidney Homer and Richard Sylla.

The short-run movement of the discount rate at Paris and Amsterdam came to light suddenly in the eighteenth century, thanks to exchange rate quotations in The Course of the Exchange. Beginning in 1723, this British commercial newspaper printed two exchange rates at London on Amsterdam, one for sight bills and one for two-month bills. The percentage difference between the rates corresponds to the discount rate in Amsterdam, at least as it was known to London exchange agents. The newspaper similarly printed twin rates on Paris from 1740. For the period through 1789, discount rates in Paris averaged 5 percent and tended to peak in the autumn months as grain merchants borrowed heavily to finance the purchase of the harvest. Discount rates at Amsterdam averaged 4.5 percent and were not clearly tied to the agricultural cycle. Discount rates at Paris correlated poorly with those at Amsterdam, demonstrating that the two markets were not highly integrated. The most pronounced feature of each series was the sharp rise of interest rates during financial panics that tended to occur two or three times a decade.

Several studies have demonstrated that the London and Amsterdam capital markets were highly integrated with each other in the eighteenth century, and that one of the principal mechanisms of integration was interest rate arbitrage. That is, speculators frequently used the exchange market to move funds between these two cities in order to take advantage of the higher rate of return. London and Amsterdam were probably the exception, however. Interest rate arbitrage appears to have been far less significant at Paris, and the same was probably true in other financial centers.

THE EIGHTEENTH-CENTURY DEBATE

The economists of the Enlightenment shared with their mercantilist predecessors the conviction that high interest rates are a disincentive to invest, since any investment earning less than the interest rate will be unprofitable. Early modern economic policy was thus largely a set of strategies for reducing the interest rate. Enlightenment writers came to differ sharply from the mercantilists, however, in their theory of the determination of the interest rate, and thus in the specific strategies that they considered advisable.

Mercantilist writings of the seventeenth and early eighteenth centuries, including those of John Locke, are marked by a belief that the rate of interest is an inverse function of the money supply. Though often poorly articulated, this quantity theory of interest, suggestive of John Maynard Keynes's "liquidity curve," was clearly central to monetary thought and went largely unchallenged until the mid-eighteenth century. Like many late mercantilists, Montesquieu, in his Spirit of the Laws (1748), saw proof of the quantity theory of interest in the decline of interest rates from roughly 10 percent to 5 percent since the discovery of the Americas, which he thought was due to the resulting influx of silver. Thus, to encourage investment, mercantilists sought to draw bullion into the country by means of a favorable balance of trade. At times they also proposed more creative devices for increasing the money supply, such as John Law's 1705 scheme to issue a paper currency based on the value of land.

Inspired in part by the early eighteenth-century writings of Richard Cantillon and Pierre de Boisguilbert, the Enlightenment subjected the quantity theory of interest to systematic critique. Boisguilbert had pointed out that most of the money supply was quasi-money in the form of commercial paper, and that its quantity was not dependent on stocks of coin. Cantillon argued effectively that an increase in the money supply would raise prices and thus leave the real money supply unaltered, with no long-run effect on interest rates. Adam Smith, David Hume, and the French Physiocrats repeated and developed these arguments. As Hume pithily remarked in 1752, "Silver is more common than gold; and therefore you receive a greater quantity of it for the same commodities. But do you pay less interest for it?"

Enlightenment writers came thus to argue that the rate of interest is an inverse function not of the supply of money, but of the supply of productive capital. The new theory, like the old one, offered a plausible explanation of the gradual decline of interest rates since the sixteenth century. Since the supply of capital was thought also to determine the rate of profit, the hope was now that at equilibrium the interest rate would fall below the profit rate, rendering all regulation of the interest rate unnecessary. Smith asserted that in England the profit rate was currently about 10 percent, and the interest rate about 5 percent. Though he acknowledged that the relationship was not strictly linear, he believed that the interest rate would rise or fall with the profit rate in such a way as to leave investors with a reasonable net profit. Still, the Physiocrats feared that excessive government borrowing might crowd out private investment by artificially bidding up the interest rate, and consequently sought to persuade the French monarchy to reduce budget deficits.

See also Banking and Credit ; Capitalism ; Hume, David ; Law's System ; Locke, John ; Mercantilism ; Physiocrats and Physiocracy ; Smith, Adam .

BIBLIOGRAPHY

Primary Sources

Hume, David. "Of Interest." In Essays Moral, Political and Literary. Revised ed. Edited by Eugene F. Miller. Indianapolis, 1987.

Smith, Adam. An Inquiry into the Nature and Causes of the Wealth of Nations. Edited by Edwin Cannan. New York, 1937.

Turgot, Anne-Robert-Jacques. Ecrits économiques. Edited by Bernard Cazes. Paris, 1970.

Secondary Sources

De Roover, Raymond. "What Is Dry Exchange? A Contribution to the Study of English Mercantilism." Journal of Political Economy 52 (1944): 250–266.

Eagly, Robert V., and V. Kerry Smith. "Domestic and International Integration of the London Money Market, 1731–1789." Journal of Economic History 36 (1976): 198–212.

Ferguson, Niall. The Cash Nexus: Money and Power in the Modern World, 1700–2000. New York, 2001.

Heckscher, Eli F. Mercantilism. 2nd ed. 2 vols. Translated by Mendel Shapiro. New York, 1955.

Hoffman, Philip T., Gilles Postel-Vinay, and Jean-Laurent Rosenthal. Priceless Markets: The Political Economy of Credit in Paris, 1660–1870. Chicago, 2000.

Homer, Sidney, and Richard Sylla. A History of Interest Rates. 3rd ed. New Brunswick, N.J., 1991.

Mandaville, Jon E. "Usurious Piety: The Cash Waqf Controversy in the Ottoman Empire." International Journal of Middle Eastern Studies 10 (1979): 289–308.

Neal, Larry. The Rise of Financial Capitalism: International Capital Markets in the Age of Reason. Cambridge, U.K., 1990.

Rist, Charles. History of Monetary and Credit Theory from John Law to the Present Day. Translated by Jane Degras. New York, 1940.

Taeusch, Carl F. "The Concept of 'Usury': The History of an Idea." Journal of the History of Ideas 3 (1942): 291–318.

Thomas M. Luckett

Interest

When you borrow money to buy a car or a house you are not only expected to pay back that money, but to pay interest on it, too. Interest is a fee paid by a borrower to the lender for the use of money. It is calculated as a percentage of the loan amount. When you deposit money into a savings account or certificate of deposit or buy a savings bond, you are loaning money and so you are paid interest. Interest plays an important role in economics because it serves as an incentive for those with available money to lend it to those needing it. There are many different ways that this fee is expressed and calculated.

Types of Interest

The term "simple interest" refers to a percentage of the loan that must be paid back in addition to the original loan. For example, if you borrow $1,000 at 10 percent simple interest and pay it back five years later, you will pay back the $1,000 plus 10 percent or $100 additional dollars for each year, a total of $500 in interest.

Few loans, however, are actually based on simple interest. Loans are usually based on "compound interest," where the total of the outstanding original money and the accumulated interest are calculated on a regular basis to compute the interest owed. In the preceding example, if interest were to be compounded annually (i.e., once each year), the interest for the first year would still be $100. But in the second year the borrower would owe interest not just on the original $1,000, but on the additional $100 of interest that was owed to the lender for the first year. The total interest for the second year would be $110. Similarly in the third year the interest would grow to $121. By the end of 5 years the total interest, compounded annually, would be $610.51, in contrast to the $500 in simple interest. This is illustrated in part (a) of the table.

"Discount interest" is interest that is subtracted from the loan when it is first made. Following the above examples, if you borrowed $1,000 at a discount interest rate of 10 percent, you would only receive $900, but would be expected to pay back $1,000. Since in this case you have really only borrowed $900, which you end up paying $100 interest on, you are paying a higher rate of interest than with simple or compound interest. In this example, a 10 percent discount rate would be the same as an 11.11 percent compound rate.

Compound interest is not necessarily compounded on an annual basis. By compounding interest more often the lender is able to get interest on interest. This can have a significant affect on the total earnings made on a savings deposit or other loan. One thousand dollars in a savings account paying 6 percent interest compounded annually would earn $60 in interest in one year. What if that same account were compounded monthly, as shown in part (b) of the table? Six percent is not earned each month. Instead, the annual rate of 6 is divided by the number of months, 12, giving a half percent per month. So, using the table, you will note that after the first month, the account earns $5. But during second month there is now $1,005 in the account. The extra $5 also earns interest.

You can see from part (b) of the table that you earn an extra $1.68 by compounding the same savings monthly. In this case the rate is still 6 percent, but the yield is 6.168 percent. The rate is sometimes called APR and the yield APY, for annual percentage rate and annual percentage yield, respectively. Note that if a bank offers you a yield of 6 percent they are actually offering you a rate of less than 6 percent. When comparing interest rates, it is important that you do not confuse rates with yields.

So if you get a little more for computing interest monthly, what would happen if you computed it daily? The 6 percent rate would be divided by the number of days in the year (365) and the result would be the interest for one day. While this is only about $0.16, that extra $0.16 starts to earn interest too. By the end of the year, this results in $61.83 in interest being earned in the account or the yield for a 6 percent APR savings account compounded daily, day in to day out, which is 6.183 percent. Although the interest could be computed over smaller intervals, the interest calculation is a function that approaches a limit . The limit is so closely approached at the daily compounded value that there is little to be gained by compounding the interest more than daily.

Interest and Loans

Loans such as automobile, credit cards, and home mortgage loans work in a similar fashion, with the bank serving as the lender. Automobile and mortgage loans are generally fixed term , meaning that they are expected to be paid off completely at a set time in the future. Automobile loans are usually a 3-to 5-year term, while home mortgage loans can be for terms as long as 30 years. These loans allow the consumer to buy these expensive items before they have all of the money for them, and pay for them while they use them. The cost of this convenience, however, is the interest. Home mortgage loans can also be either at a fixed rate, agreed to at the start of the loan, or can be variable-rate loans, where the interest rate can change. Although variable rate loans are generally offered at a lower percentage than fixed-rate loans, the borrower must face the risk that the interest rate could be adjusted and become higher than the rate that was offered for the fixed-rate loan.

Credit card loans generally have a significantly higher interest rate and require a minimum payment each month. Borrowers frequently find that they make no progress in lowering their credit card debt because more credit is incurred as the outstanding balance is paid off.

Credit cards are an extremely convenient but expensive financial tool, and many borrowers get into serious financial trouble when they let their credit card debt grow. For example, if you have a credit bard balance of $1,000, the interest on this balance will be $15 per month if the annual interest rate of the card is 18 percent. The minimum payment that the card issuer charges will be somewhat higher than the monthly interest charge, but not by much. For this example, assume a minimum payment of $50. By paying only the minimum required payment each month, and adding interest, your balance is not significantly reduced from month to month. At this rate it could take years to pay off your balance.

When a borrower pays off a loan such as a house mortgage or a car loan, they make a payment at regular intervals, usually monthly. Some of the payment pays the interest on the loan for that month. The rest is applied to the current outstanding value of the loan and reduces it, so that the loan will be completely paid off at the agreed upon time. In the case of a 30-year home loan, a borrower will find that the largest part of the monthly payment is applied to interest payments at first and a small amount is applied to the balance of the loan, called the principal. The result is that after a few years you still owe nearly the full loan amount. Slowly the borrower makes progress against the principal, until finally after 30 years the loan is paid off.

For example, if a borrower borrows $100,000 on a 30-year mortgage at 8 percent fixed rate, after one year the borrower will have paid $8,805.17, not including taxes or insurance, but will have only reduced the principal by $835.36 (the remaining $7,969.81 goes towards interest). It is valuable to realize that if the loan has no per-payment penalty, you can significantly reduce the total cost of the loan by making additional payments in the early years, as these additional payments apply completely to the loan principal. For example, if after the first year the borrower was able to pay back an additional $904, the loan would be paid off a full year earlier, avoiding the last year of payments totaling $8,805.17. In this way a few extra payments each year or a slightly higher monthly payment can significantly reduce the term on the loan and result in significant savings.

By building a savings account that benefits from compound interest, a small investment will grow well over time. If a person who is 20 years old deposits $100 monthly into a savings account that pays 7 percent interest, they would have deposited $48,000 by age 60. But thanks to compound interest, there would be over $265,000 in the savings account.

see also Percent.

Harry J. Kuhman

Bibliography

McNaughton, Deborah J. Destroy Your Debt! Your Guide to Total Financial Freedom. Winter Park, FL: Archer-Ellison Publishing Company, 2001.

in·ter·est / ˈint(ə)rist/ • n. 1. the state of wanting to know or learn about something or someone: she looked about her with interest. ∎  (an interest in) a feeling of wanting to know or learn about (something): he developed an interest in art. ∎  the quality of exciting curiosity or holding the attention: a tale full of interest. ∎  a subject about which one is concerned or enthusiastic: my particular interest is twentieth-century poetry.2. money paid regularly at a particular rate for the use of money lent, or for delaying the repayment of a debt: the monthly rate of interest | [as adj.] interest payments. 3. the advantage or benefit of a person or group: the merger is not contrary to the public interest we are acting in the best interests of our customers. ∎ archaic the selfish pursuit of one's own welfare; self-interest.4. a stake, share, or involvement in an undertaking, esp. a financial one: holders of voting rights must disclose their interests | he must have no personal interest in the outcome of the case. ∎  a legal concern, title, or right in property: third parties having an interest in a building.5. (usu. interests) a group or organization having a specified common concern, esp. in politics or business: the regulation of national interests in India, Brazil, and Africa.• v. [tr.] excite the curiosity or attention of (someone): I thought the book might interest Eric. ∎  (interest someone in) cause someone to undertake or acquire (something): efforts were made to interest her in a purchase.PHRASES: declare an (or one's) interest make known one's financial interests in an undertaking before it is discussed.in the interests (or interest) of something for the benefit of: in the interests of security we are keeping the information confidential.of interest interesting: much of it is of interest to historians.with interest with interest charged or paid: loans that must be paid back with interest. ∎  (of an action) reciprocated with more force or vigor than the original one: he may have a reputation for getting even, with interest.

INTEREST

Interest is typically expressed as a quarterly or annual rate of percentage charged or earned on a sum of money. For example, if an individual borrows $2000 from a bank or lending institution and that institution charges an annual rate of six percent interest on the loan, the individual will pay the lender up to $60 a year for the use of the money. Paid interest is usually incorporated in monthly installments submitted by money borrowers in a scheduled repayment plan. The process is similar for interest earned. A bank may offer an annual earned interest rate of three percent to investors with a savings account. An individual with $100 in a savings account at that bank will earn $6 a year in interest.

Interest rates play an important role in lending and investment decisions. Money borrowers will look for the lowest interest rate they can find for their loan, whereas investors will look for the highest interest rate to better the return on their investments. In the economy the interest rate is often affected by factors such as a country's stage of development, productivity, and investment needs, among others. Internationally, fluctuating interest rates can cause flows of capital between financial institutions and different countries. For instance, if domestic interest rates are low in the United States and higher in Asia, investment capital will flow out of the United States and into Asia so that investors can take advantage of the higher interest rates on their investment moneys.

See also: Capitial Gain, Debt