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]eHCfG$)eKMnCCCff]lCCC&gPPPP ) ) Irving Fisher. The Rate of Interest: Its nature, determination and the relation to economic phenomena. New York: The Macmillan Company, 1907.(
When they look back on the twentieth century, historians of thought will surely rank Irving Fisher as one of the major contributors to economic theory. Samuelson (1967) described Fisher as Americas greatest analytical economist and Tobin (1987, p.369) acknowledges that Fisher is widely regarded as the greatest economist America has produced. Certainly, Fishers analytical techniques presented in The Rate of Interest (1907) and later in The Theory of Interest (1930) still permeate all corners of neoclassical economics as taught in economics departments around the world. Fisher provides the common thread that links Eugen von BhmBawerks Capital and Interest: A History and Critique of Interest Theories (1884) through Frank Knight, Paul Samuelson, Milton Friedman and Robert Solow to contemporary textbooks such as Lectures in Macroeconomics by Blanchard and Fischer (1989), Intertemporal Macroeconomics by Azariadis (1993), Foundations of International Macroeconomics by Obstfeld and Rogoff (1996) or Advanced Macroeconomics by Romer (2006).
There are also many additional links to Fisher in research on the microeconomic foundations to macroeconomics or the use of the representative agent, equilibrium real business cycle models and finance theory. A most timely illustration is the paper by Azariadis and Kaas (2007) which makes the claim that by applying the tools of dynamic general equilibrium theory to what are clearly Fishers core theoretical concepts it will soon be possible to produce a theory of everything in macroeconomics. In particular, it is claimed that the research program initiated by Robert Lucas and Edward Prescott is poised to extend well beyond the frictionless equilibrium environment of its initial versions.
A reread of Fishers Rate of Interest is therefore instructive not just for the history of economic thought but is also of direct relevance to developments in modern macroeconomics. What then was Fisher all about? In what follows I will draw on Samuelson (1967) and Dougherty (1980) to briefly summarise Fishers vision. Then I will revisit the criticisms of Fishers analysis and evaluate some properties and interpretations of his model before providing a brief overview of Fisherian analysis as it appears in modern macroeconomics. I conclude with an assessment of Fishers legacy.
Fishers theory of the rate of interest
Samuelson (1967; 2005, p. 338) identifies the defining characteristic of Fishers interest theory as the idea that the terms of trade between present and future consumption is the objective counterpart of the rate of interest. It is this concept of interest that is open to several interpretations that enable it to sidestep the technical conundrums of capital theory or to be seen as the forerunner of the neoWalrasian general equilibrium models of Hicks, Arrow and Debreu. The Walrasian foundations of Fishers work are evident in his dissertation of 1891 which was published in 1892 as Mathematical Investigations in the Theory of Value and Prices and reprinted as volume 1 of Fisher (1997). Fishers contributions to economic theory were largely neglected by the wider profession after the 1930s until they were rediscovered by Samuelson and others in the 1960s. Around the same time Fishers analysis was inevitably drawn into the capital debates. Dougherty (1980) and Cohen and Harcourt (2003) provide a useful survey of the capital theory controversies and place Fishers capital theory in context. In that respect one of the referees drew my attention to Velupillais (1975) discovery in The Rate of Interest, pp. 35253, that Fisher illustrates the concept of the reswitching of techniques, a property that, somewhat ironically, applies to his own concept of rate of return over cost as well as to BhmBawerks notion of the period of production.
In The Rate of Interest Fisher begins by outlining the flaws in simple productivity and cost theories of interest and the weakness in BhmBawerks notion of the period of production before he proceeds to outline his own theory. Over the years many have commented on the clarity of Fishers exposition and The Rate of Interest is a fine example of his work as he proceeds by a series of elementary steps that he labels the first, second and third approximations to his theory. At each step he provides a verbal, geometric and mathematical explanation with the latter two treatments confined to appendices. The mathematical treatment consists largely of counting equations and unknowns to establish that the necessary number of independent equations exist to determine the unknowns.
The first approximation treats income as a fixed endowment, the second approximation treats income as variable endowment subject to known technology and the third approximation treats income as uncertain (stochastic). The assumptions underlying the analysis are stated explicitly and it is apparent that we are dealing with a consumerproducer or selfemployed artisan who acts with perfect foresight in production and consumption and as a price taker in a perfectly competitive loans market. Many capital goods exist and represent the existing technology but they are not modelled explicitly. The same is true of money, labour and wages that are formally absent from Fishers model. This has prompted many to question how capital can be absent from what purports to be capital theory and the same question could be asked of the other missing components, particularly money. But as Dougherty (1980, p. 20 emphasis added) explains:
The answer (from a Fisherian point of view) is that the latter [capital] is a misnomer, the essence of the subject being the efficient production and pricing of a stream of consumption goods over time. The concepts of capital, investment, even income are not only superfluous but actually dangerous, at least to the extent that they distract attention from the main issues.
This explanation by Dougherty probably goes to the heart of the matter when it comes to disputes between Fisherians and their critics.
In the following outline of Fishers theory I will concentrate on the second approximation as the first and third approximations add or subtract nothing of analytical significance from the second. The second approximation is illustrated with reference to Figure 1 which is a version of Figure 29 from The Rate of Interest p. 409.
SHAPE \* MERGEFORMAT
Figure 1: Fishers integration of time preference and technology in a theory of the rate of interest.
For simplicity Fisher considers only two periods, say the present and the future, and as Samuelson (1967; 2005 p. 348) notes he never faced up to the solution of an infiniteperiod equilibrium case. That case was first tackled by Ramsey (1928) for the case of identical men. For modern treatments see Blanchard and Fischer (1989, chapter 2) or Romer (2006, chapter 2). Figure 1 represents the choices facing the representative producerconsumer with the incomes for periods 1 and 2 on each axis with incomes in all other periods assumed constant. The curve labelled U is one of a family of indifference curves and the curve ZZ represents the possible income or consumption streams given the existing capital goods and embodied technology. The capital goods are not formally modelled but they generate the two income streams when combined with labour and Samuelson (1967) describes ZZ as a production possibility or opportunity frontier. Given the real rate of interest EMBED Equation.3 determined on the perfectly competitive loan market the absolute slope of AB is equal to EMBED Equation.3 Fisher calls AB the market line.
The producer component of the representative individual maximises the present value of the technically feasible income streams by producing at point P while the consumer component maximises utility by consuming at point Q. Access to the perfect market in loans allows this individual to trade current income DP for an IOU promising to deliver DQ income for consumption next period. However, this is an analysis at the level of the individual so can hardly pass as a theory of the rate of interest as it takes the rate of interest as given. To determine the real rate of interest EMBED Equation.3 in the perfectly competitive market for IOUs Fisher therefore aggregates across all individuals and concludes that the sum of the value of the demand for IOUs equals the sum of the value of the supply of IOUs to determine a market clearing real rate of interest EMBED Equation.3 .
A modern formal restatement of Figure 1 based on the maximization of lifetime utility (modelled as present and future utility) reduces to the following expression:
EMBED Equation.3 (1)
Expression (1) is a form of the Euler equation that is found throughout modern macroeconomics with microfoundations which is it self indicative of Fishers influence. Figure 1 and expression (1) both illustrate what Dougherty (1980, p. 39) calls Fishers triple equality; that in any two time periods the marginal rate of time preference will be equal to the rate of interest and the rate of profit. For a discussion of what the rate of profit means in this context see Pasinetti (1969).
This is a rather strong interpretation that treats EMBED Equation.3 as the rate of profit. The subjective rate of time preference, say, EMBED Equation.3 , is embedded in the subjective time preference or discount factor, EMBED Equation.3 so EMBED Equation.3 . Hence the term the left hand term of expression (1) is simply the marginal rate of substitution between present and future consumption with the comparison dependent on the subjective rate of discount between the two. The middle term includes that real rate of interest as the determinant of the slope of the market line while the right hand term includes the slope of the intertemporal production function, the marginal rate of transformation between present and future consumption. Doughertys statement of the triple equality therefore implies that EMBED Equation.3 which in turn implies the special case where EMBED Equation.3 or EMBED Equation.3 .
The right hand term in expression (1), EMBED Equation.3 often leads to the interpretation of Fishers model as a onecommodity model when F(K,L) is interpreted as a form of the aggregate production function in a onecommodity world. Obstfeld and Rogoff (1996) explicitly interpret the model in this fashion and treat capital as something that can be consumed or saved. The corn economy or Knights (1944) Crusonia plant is the apt analogy. However there are well known limitations on attempts to generalise from one commodity models. Dougherty (1980, chapter 2) and Cohen and Harcourt (2003) provide a useful summary. An interesting property of this interpretation is that saving is by definition investment (what is saved is by definition invested) so there is no need for the two to be coordinated by the real rate of interest in a perfectly competitive market for loans as Fisher proposed.
Alternatively, the technology in Fishers model can be interpreted along the lines suggested by Debreu as isomorphic with the analysis of exchange. On this interpretation an endowment of capital goods exists but it is analytically equivalent to the stream of consumption goods it is technically capable of producing. Hence Doughertys explanation that it is not necessary to specify capital goods in the model. See Radners (1987) discussion of production as indirect exchange or Hirshliefers (1970, pp. 1226) description of the paradigm of production and exchange. On either interpretation the marginal rate of substitution between present and future consumption is brought into equality with the marginal rate of transformation between present and future consumption and the real rate of interest; this version of Fishers triple equality always holds in equilibrium.
Thus in Fishers model the forces of time preference and technology are said to be the ultimate determinants of the rate of interest generated by the loan market. Samulesons (1967; 2005 p. 344, emphasis added) description of Fishers interpretation of his theory has a modern ring:
But as Fisher would insist, it [the determination of the rate of interest by supply and demand] is not a superficial description of supply and demand but rather a formulation that analyses their ultimate source in taste and technology.
Lucas and Prescott would no doubt applaud.
More importantly, Samuelson (1967; 2005, p. 344) also explains how Fishers version of Figure 1 could be interpreted in the case of two types of individuals borrowers and lenders in terms of an Edgeworth Box with offer curves passing through point P and Q in Figure 1  and points out that Fishers analysis is then completely isomorphic to the microeconomic model of general equilibrium in J. R. Hicks, Value and Capital (1939). What this tells us is that Fishers vision of general equilibrium can map directly into the neoWalrasian general equilibrium systems of Hicks (1939), Debreu (1959) and ArrowHahn (1971). This is not a surprising conclusion in view of Fisher (1892) but the Walrasian vision is both its strength and its weakness. Its strength comes from the fact that the neoWalrasian structure is not open to any conundrums of capital theory (See Dougherty (1980) or Rogers (1989)) although this is disputed by Garegnani. Its weakness is the weakness of the neoWalrasian tradition, the inability to deal with money and the economy as a whole, i.e. macroeconomics. As one of the referees noted, this is ironic for Fisher who spent so much of his career writing on monetary macroeconomics. Today, of course, those weaknesses are seen by many as a virtue.
Fishers model has been subjected to various criticisms over the years. Dougherty (1980, chapter 7) provides a sympathetic overview and explains why the assumption of perfect foresight, the absence of money, capital, labour and/or wages fails to undermine support for the Fisherbased version of neoclassical economics. These factors are said to obscure the view of the essence of interest theory which is the exchange of intertemporal consumption streams. In terms of the formalism these objections would require the rejection of the ArrowDebreu model or the parables that attach to the concept of a corn economy or the Crusonia plant of Knight as a metaphor for an aggregate production function. Neoclassical economists are naturally reluctant to give these up because they are perceived to be consistent with some degree of empirical success. See the view of Solow discussed by Cohen and Harcourt (2003).
Evaluating Fishers model
Consider first the Fisher demonstration that the rate of interest is positive as a result of the bias in time preference and/or technology. This was thought to be important in explaining BhmBawerks agio or impatience theory of interest and generalising it to include a role for technology. However, in Fishers general equilibrium setting this result has no explanatory power because the model places no sign restrictions on the rate of interest. Attributing a positive rate of interest to bias in time preference and/or technology is exposed as an exercise in psychology or sociology. See Dougherty (1980, p 34) and Samuelson (1967; 2005) for examples of this reasoning.
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Figure 2. Possible general equilibria in Fishers model with zero real rate of interest despite time preference and technology bias.
To see why this is so consider Figure 2 which is the now familiar Fisher diagram with a 45o line from the origin and a 45o line representing the case where the real rate of interest or discount is zero. In the case of no time preference or technology bias the production possibility curve and the indifference curves are treated as symmetrical around the 45o line from the origin. In that case Doughertys interpretation of the Fisher triple equality holds. Consequently Figure 2 illustrates a production possibility curve that is biased in favour of future production and consumption; say it is an example of BhmBawerks roundabout method of production. Three mutually exclusive sets of intertemporal indifference curves which show negative bias in time preference are also added to Figure 2. They show a preference for future over current consumption as they are tangential to the market line above the 45o line from the origin. These three representative individuals prefer chocolates tomorrow over chocolates today; the inverse of what was assumed by BhmBawerk and Fisher. The point is that there is no reason to rule out any of the three general equilibrium solutions generated by the intertemporal indifference curves labelled I, II or III in Fishers model with heterogeneous agents.
To confirm this conclusion simply follow Samuelson (1967; 2005 p. 344) and construct an Edgeworth Box by dividing the individuals in Fishers model into two types, borrowers and lenders, (heterogeneous agents) and apply the distinction to interpret Figure 2. The representative individual (standing in for the millions of individuals who are borrowers) with indifference curve labelled III clearly has a relative preference for present consumption relative to the production point P and borrows against future production and consumption. In a general equilibrium context, Samuelsons Edgeworth Box, he must be matched by a representative lender (standing in for the millions of lenders) and the market for IOUs clears when aggregate lending equals aggregate borrowing. In fact this must happen in the Edgeworth Box even in the case of two individuals, one of each type, so long as each acts as a pricetaker. The representative individual with indifference curve labelled II produces the autarky result and representative individual with indifference curve labelled I is a representative lender to be matched with a representative borrower in general equilibrium. What this result shows is that time preference and/or technology bias do not explain why the rate of interest is positive. All the general equilibrium results in Figure 2 are compatible with a positive, negative or zero rate of interest depending on the slope of the AB market line relative to 45o. In short the model is compatible with any rate of interest. BhnBawerks argument that the rate of interest is positive because individuals have a positive time preference, that is, they prefer chocolates now to chocolates later, actually receives no support from Fisher. Fishers model with heterogeneous agents and no constraints on the utility functions places no sign constraints on the rate of interest. It also places no constraints on the number of equilibria. There will generally be a finite number of equilibria so uniqueness is a special case. Some question whether this is a theory of the rate of interest at all.
But what exactly is the rate of interest in Fishers model? It is clearly a barter intertemporal rate of exchange between consumption goods. Recall Samuelsons (1967; 2005) description of the intertemporal rate of exchange between consumption streams as the objective (real?) counterpart to the (money?) rate of interest. As Dougherty (1980, p. 38) points out the generalization of Fishers analysis from two time periods to many can also be interpreted as formally equivalent to the generalization of the traditional a temporal analysis from two commodities to many and a formal treatment can be found in Debreu (1959). But in that case why didnt Debreu use the title Theory of Interest rather than Theory of Value? The obvious answer is the correct one: the two can be one and the same.
Dougherty (1980) is well aware of this relationship and treats it as a virtue. To continue the quotation started above, Dougherty (1980, p. 20) argues:
Nor does one (even) have to introduce the concepts of rates of profit and interest, since one can conduct the analysis of value exclusively in terms of price streams.
This seems to be saying that we can have a theory of interest that does not include interest which may strike many as odd. It is nevertheless consistent with the objective determination of interest as a rate of exchange between intertemporal consumption streams and a well known property of NeoWalrasian general equilibrium theory. It is the inevitable consequence of searching for the essence of interest theory along the lines pioneered by Fisher. The relationship between the interest rate and the price streams for Doughertys (1980, p. 21) two periods appears as:
EMBED Equation.3 (2)
If the consumption good in period t is the numeraire a unit of consumption good available at the end of period t (discrete case)  then the rate of interest is simply the ratio of the quantities of consumption good consumed in each period. To confirm Doughertys explanation for the redundancy of r in Fishers model simply substitute (2) into (1) and the rate of interest vanishes. Notice also that the prices are exchange ratios defined in terms of a quantity of the numeraire commodity. In that sense Fisher is ultimately dealing with a commodity rate of interest, the quantity of chocolates today that will be traded for a unit of chocolates tomorrow, and there are as many such rates of interest as there are commodities a well known property of neoWalrasian general equilibrium models.
That brings us to the role of money and the relationship between Fishers commodity rates of interest and his real rate of interest as a nominal or money rate adjusted for the rate of inflation. Fisher discusses this issue in chapter V of The Rate of Interest under the heading of Appreciation and Interest and it is an idea for which he is justly famous. However, it is the relationship between the commodity interest rates of the moneyless general equilibrium model and the inflation adjusted real rate of chapter V that is at the heart of disputes between monetary theorists in the traditions of Real and Monetary Analysis identified by Schumpter (1954). In this respect The Rate of Interest falls into the tradition of Real Analysis money has no essential role in its determination. Throughout The Rate of Interest Fisher deals in nominal dollar values so he tacitly assumes that he is dealing with a monetary economy. But once he relies on the assumption of a perfectly competitive loan market he effectively eliminates any role for money. In effect the isomorphic relationship between Fishers and Debreus models means that the perfect market for IOUs is analytically identical to the Walrasian tatonnoment auction or its modern equivalent the time0 auction of Ljungqvist and Sargent (2004).
The absence of an essential role for money in Fishers model is intentional as he rejects money theories of the rate of interest in chapter XVI of The Rate of Interest and this is interpreted by Dougherty (1980, p. 105) as a loose form of the classical dichotomy now tacitly accepted by most neoclassical economists. In terms of such a dichotomy, money was appended to the system of equations describing the real sector of the economy, Fishers general equilibrium system, and usually in the form of the Equation of Exchange. Monetary factors were then said to cause short run disturbances but to have no influence on the longrun equilibrium determined by the real forces of productivity and thrift (time preference and technology in Fishers language). See Makinen (1977, chapter 3 for an illustration of this form of analysis. Money exists but it is neutral in the long run. This is a vision that is held by almost all neoclassical and Keynesian monetary economists. But it is a vision that is not supported by the Fisher or Debreu general equilibrium models. In these models money is redundant not merely neutral. This conclusion follows whether money is treated as exogenous or endogenous. Failure to recognise this fact has produced immeasurable confusion in monetary theory in the 20th century that continues to this day. The Fisher model cannot be interpreted as an example of a loose classical dichotomy as suggested by Dougherty and many others. The common misunderstanding of the relationship between Keynes and Fisher prevalent today is personified by Doughertys (1980, chapter 4, emphasis added) discussion of monetary influences where he states:
In the short run, monetary factors are dominant. If there is any overriding mechanism it is the liquidity preference function, and the location of this is determined by its previous behaviour. It sets the rate of interest, and this in turn determines the volume of investment, the price of investment goods relative to consumption goods, and the rate of profit. . At any rate there are no obvious incompatibilities between his [Keyness] shortrun model and the Fisherian longrun model.
Keynes has no argument with the logic of efficient allocation but he rejected entirely the application of that logic to the determination of output as a whole in a monetary economy. Rogers (1989) argues that Keyness General Theory lies in the tradition of Monetary Analysis. The relationship between Fisher and Keynes has been well documented by Dimand (1995). In the General Theory Keynes was certainly not concerned with the analysis of an economy populated by selfemployed artisans and perfectly competitive financial markets. At best, Fishers analysis in The Rate of Interest was an example of classical Real Analysis a special case of Keyness General Theory. Some modern macroeconomists nevertheless treat their modern restatement of Fishers theory as the basis for a theory of everything in macroeconomics (for macroeconomics read the business cycle as the two are synonymous in this context).
Fisher and modern macroeconomics
The Fisherian pedigree of modern nonKeynesian economics is comprehensively and conveniently illustrated in recent paper by Azariadis and Kaas (2007). The paper is written in honour of Edward Prescott and his research agenda as represented by the methodology of what began as realbusinesscycle theory. Derivation of the realbusinesscycle theory from expression (1) proceeds by introducing stochastic serially correlated impulses into the Solow residual. In that sense it can be interpreted as an application of Fishers third approximation. The tool of this research programme is dynamic general equilibrium theory (DGE) but the methodology and theory is vintage Fisher.
The defining features of the DGE tool are defined by Azariadis and Kaas (2007, p. 14) as follows:
Combining firstorder conditions for all agents with clearing of all markets, a DGE model reduces economic behaviour to a few stochastic differential or difference equations which define the economys laws of motion, much as Newtons equations characterize the motion of all macroscopic objects in a physical environment without frictions.
The methodology is said to be based on the principle of parsimony, which means in this context that the institutional detail and exogeneity assumptions should be as few and simple as the goals of the model allow. The goals of the model are to explain as many big facts about the real economy as is possible. Explain in this context means that the model must mimic the stochastic properties of the time series it is intended to explain, often with particular attention to the second moments. The theory behind Edward Prescotts contribution to DGE and modern macroeconomics is then attributed by Asariadis and Kaas (2007, p. 38) to the study of just two terms: the Solow residual and the stochastic discount factor. Theses two factors are, as you might have expected, already visible in expression (1).
The familiar stochastic discount factor is written by Asariadis and Kass (2007, p. 35) as:
EMBED Equation.3 (3)
Expression (3) is said to be the stochastic discount factor for the case of an exchange economy with identically homothetic utility functions and EMBED Equation.3 defined as the market value at t of one consumption unit available at t+s. Expression (3) is simply a reformulation of the two left hand terms in expression (1) and a combination of Fishers first and third approximations. The Solow residual is obviously related to the last element of expression (1) which can be interpreted as a measure of total factor productivity that accounts for the majority of growth in terms of a Solow growth accounting exercise, See Romer (2006, section 1.7). From this perspective it is apparent that Irving Fisher is the intellectual parent of the major nonKeynesian research programme in macroeconomics in 2007.
Here is not the place to evaluate the likely success of this research program. Nevertheless, the Fisherian heritage explains why there is so little analytical structure to modern DGE models beyond the ideas presented by Fisher one hundred years ago. Addition of concepts like money, capital, and unemployment represent unnecessary institutional detail that obscures the essence of the laws of motion of the model economy. The mapping of Fishers insights into the ArrowDebreu model has however come at some cost in the form of a gap between the theory and what we might bluntly call reality. On its own terms the DGE model has not yet achieved sufficient success at explaining the big facts to justify its use by applied macroeconomists and policy makers (assuming that is the ultimate objective). Nevertheless, Azariadis and Kaas (2007) are optimistic that the next generation of DGE models will bridge the gap and that a theory of everything in macroeconomics is at hand.
Fishers legacy
From what we have said above it is clear that Fisher casts a long shadow over modern macroeconomics. In a sense, the shadows of Keynes and Fisher cover almost all of macroeconomics in the 20th century and today it is apparent that the Fisherian vision underpins the modern classical approach to macroeconomics. As Samuelson (1967; 2005) and Tobin (1987) documented, Fisher anticipated what has become known as the neoWalrasian or ArrowDebreu model of general equilibrium. As such he is also the father of modern applied or dynamic general equilibrium theory. Despite a century of ever increasing technical sophistication it would be fair to say that virtually all of the nonKeynesian research effort represents an attempt to apply Fishers triple equality. From this perspective modern micromacroeconomics is all in Fisher (1907). Fisher (1907) provides the vision and Ljungqvist and Sargent (2004) provide the tool box.
Many Keynesians, old and new, also embrace Fisher in the form of a loose classical dichotomy when they accept his analysis for the determination of longrun equilibrium. Keynesian analysis is thereby relegated to a nonmarket clearing analysis of the business cycle. The classical tradition rejects such analysis as inconsistent with the rational behaviour of agents and market clearing and the realbusinesscycle is generated by serially correlated shocks to the Solow residual in expression (1). It has become customary to describe such models as frictionless. This disappearance of the distinction between short and long run and the adoption of frictionless equilibrium business cycle theory follows automatically from Fishers methodology.
The attraction of Fishers methodology is that his theory can be converted into a theory of the business cycle without including anything of concern to Keynesians or policy makers. And as the theory can be mapped into the ArrowDebreu world the analysis can proceed without any need to model money or capital or the need to distinguish between microeconomics and macroeconomics as obvious examples of institutional detail that have caused difficulties for theorists. The weakness is that Fishers vision can be so generalized that it is emptied of any useful content. The methodology of calibration is designed to address that weakness but as Hoover (1995) suggests the application of the methodology with representative agents is less than compelling. See also Kirman (1992) on the pitfalls facing interpretation of the representative agent.
In the recent resurgence of the classical vision classical macroeconomics has generally held fast to Fishers methodology. However, Azariadis and Kaas (2007) are concerned that the frictionless applications of Fishers triple equality in a realbusinesscycle context are unable to explain many big facts about the economy. Hence the big question to answer now is how to take the next step and improve the explanatory power of calibrated models. Azariadis and Kaas suggest that the way forward is to introduce frictions and/or place constraints on the auctioneer. But how are frictions introduced into a frictionless world?
On the face of it such an attempt seems to be inconsistent with Fishers methodology. Fisher excluded frictions from his theory because they distracted from the essence of the rate of interest. But Fisher clearly introduced frictions when applying his theory. Frictions were not part of the theory but were introduced only to explain how the theory could be reconciled with the facts. For example, failure of the money rate of interest to adjust so as to compensate for inflation was explained by Fisher in terms of the failure of individuals to act according to his assumption of perfect foresight and Makinen (1977, chapter 3) provides a detailed account of Fishers misperceptions and monetary theory of the business cycle along similar lines.
Introducing frictions into the theory is something completely different; it involves the construction of a new theory. For example, if money is to be introduced into Fishers triple equality, where would it fit? Money and its medium of exchange function can only be introduced and have a meaningful role if there is a friction that the use of money can help to overcome. Starting with a frictionless model is obviously not going to be fruitful. Some recasting of the theory will be called for and this may present opportunities for a genuine synthesis between Keynes and the modern classics.
In this respect Fishers (1933) work on debtdeflation has more in common with Keynes and the post Keynesians. This work had a profound influence on Tobin (1975) and Minsky (1982) and many aspects of the analysis seem to abandon the real general equilibrium vision underlying The Rate of Interest and the modern mainstream analyses that rest on it. It may be the case that here Fisher resolved the tension between his work on monetary macroeconomics and his underlying real general equilibrium vision and crossed over from Real to Monetary Analysis.
The issues raised by Fishers theory and methodology thus remain at the cutting edge of macroeconomics and monetary theory in 2007. Samuelsons (1967; 2005) description of Fisher as modern and Americas greatest analytic economist remains as true today as when Fisher was rediscovered in the 1960s.
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Minsky, Hyman (1982). Inflation, Recession and Economic Recovery, (Wheatsheaf: Brighton)
Obstfeld, Maurice and Rogoff, Kenneth (1996). Foundations of International Macroeconomics, (The MIT Press: Cambridge USA).
Pasinetti, Luigi. (1969). Switches in Technique and the Rate of return in Capital Theory, Economic Journal, LXXIX, pp. 50831. Reprinted in Capital Theory Volume I edited by Bliss, C, Cohen, A. J. and Harcourt. G. C. 2005. 78101.
Radner, Trout. (1987). Production as indirect exchange, in The New Palgrave: General Equilibrium, ed. By John Eatwell, Murray Milgate and Peter Newman, (Macmillan: London).
Ramsey, Frank. P. (1928). A Mathematical Theory of Saving, Economic Journal, 38, 543559.
Rogers, Colin. (1989). Money, Interest and Capital: A study in the foundations of monetary theory, (Cambridge University Press: Cambridge).
Rogers, Colin. . (2006a). Doing without money: a critical assessment of Woodfords analysis, Cambridge Journal of Economics, 30, 293306.
Rogers, Colin. (2006b). The principle of effective demand: the key to understanding the General Theory, paper presented at the IEA Conference: Keyness General Theory after 70 years, held at Santa Colomba and Siena, July 3 6, 2006.
Romer, David. (2006) Advanced Macroeconomics, third edition, (McGraw Hill: New York).
Samuelson, Paul. (1967) Irving Fisher and the Theory of Capital, in Ten Economic Studies in the Tradition of Irving Fisher, chapter 2, (John Wiley: New York). Excerpts published in Capital Theory Volume I, edited by Bliss, C. Cohen, A. J. and Harcourt, G. C. (2005), pp. 337349.
Schumpeter, Joseph A. (1954). History of Economic Analysis, (Oxford University Press: New York).
Sharp, Clifford. (1981). The Economics of Time, (Martin Robertson: Oxford).
Solow, Robert. M. (1969). On the rate of return: Reply to Pasinetti, Economic Journal LXXX, 42328. Reprinted in Capital Theory Volume I, edited by Bliss, C. Cohen, A. J. and Harcourt, G. C. 2005, pp. 102107.
Tobin, James (1975). Keynesian Models of Recession and Depression, American Economic Review, 65 (May), 195202.
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( I am grateful to two referees and to Geoff Harcourt for comments that greatly improved an earlier draft.
The Rate of Interest , with a new editorial introduction and postscript, appears as volume 3 of The Works of Irving Fisher, (1997) edited by Willian J Barber, assisted by Robert W. Dimand and Kevin Foster, with consulting editor James Tobin.
Dimand (1995, p. 261) asks, Why didnt we all become Fisherians?. Clearly many modern macreconomists have.
The evolution of Fishers capital theory can be traced through Knight (1944) and Dewey (1965) through Hirshliefer (1970) to Solow and modern macroeconomics texts. Knight introduced the notion of a Crusonia plant to avoid the problems with the measurement of aggregate capital. Hirshliefer (1970, p. 159) defines the Crusonia plant as a vast, edible fungus, that grows continuously and automatically at an overall constant rate except for the bits that are consumed. Hishliefer (1970) extends Fishers model to deal with firms but it is not clear why this extension is necessary in view of Fishers methodology.
The exchange between Pasinetti (1969) and Solow (1969) and reproduced in Bliss, Cohen and Harcourt (2005) illustrates several of the points made below.
For a modern application of this analysis to a representative country see Obstfeld and Rogoff (1996 chapter 1, Figure 1.3). Again a referee drew my attention to an interesting paper by Thomas Humphrey (1988) that traces the history of the application of Fishers diagram to two countries instead of two periods.
There is no reason why we couldnt use periods 1 and 21 as suggested by Sharp (1981, p. 183). Fishers first approximation takes a fixed point on or inside the ZZ locus so may not coincide with point P as the choice of technique that maximises the present value of future consumption.
Samuelson (1967) draws all his diagrams with consumption rather than income on each axis. Recall Doughertys explanation that nothing of analytical significance is involved by this switch in the context of the model.
I am ignoring here the well know fact that aggregation across individuals in these models (just two will produce this result) generally cannot sustain the monotonic form of the individual net excess demand functions so multiple equilibria are the norm in aggregate versions of the model. See Dougherty (1980, pp. 3233) or Azariadis (1993, chapter 11).
The example is taken from Obstfeld and Rogoff (1996) who offer some justification for the form of the utility function they employ. See Ljungqvist and Sargent (2004) for a formal treatment.
As Samuelson (1967; 2005 Bliss et. al.) notes, in The Rate of Interest Fisher described his theory as an impatience theory but came to realise the role of technology when he revised and republished his work in 1930 as The Theory of Interest: As Determined by Impatience to Spend Income and the Opportunity to Invest It. Figure 1 thus represents Fishers integration of the concept of time preference as embodied in the intertemporal indifference curves with the technology as embodied in the income or production possibility frontier, ZZ. In this fashion it could be said that the forces of productivity and thrift interact to determine the rate of interest in a perfectly competitive market for loans. Clearly, for Fisher (1930, p. 282) both time preference and technology play a role: To adopt a simile of Alfred Marshall, both blades of a pair of scissors are needed to make the scissors work  quoted by Dougherty (1980, p. 34.).
Dougherty (1980, p. 34) argues incorrectly that positive technology and/or time preference bias ensures that the rate of interest is positive. His argument is based on a partial equilibrium interpretation as a single representative agent of Fishers diagram (his two Figures 3.9a and 3.9b) and overlooks the general equilibrium solution obtained by completing the Edgeworth Box. In any event it is not just time preference that determines the slope of the indifference curve.
This is another way of explaining why Woodfords (2003) search for interest rate rules in a frictionless model is a willothewisp. See Rogers (2006a).
See Hahn (1982, chapter 1).
Azariadis (1993) describes Fishers IOUs as inside money but the model is best described as an illustration of McCallums (1985) accounting system of exchange. McCallum correctly labels such economies as nonmonetary as the medium of exchange function of money is missing. The analysis of the IOU market under risk is the stepping off point for modern finance theory. See Arrow (1963) and Dewey (1965).
See for example Bibows (2001, pp. 608609) explanation of how the elimination of time in neoWalrasian models eliminates any attempt to make sense of loanable funds theory and price flexibility or Rogerss (2006a) critique of Woodford (2003).
See Rogers (2006b) for a recent statement of Keyness monetary theory in the tradition of Monetary Analysis as opposed to Keynesian theory based on the loose classical dichotomy in the tradition of Real Analysis.
See Hoover (1995) and Romer (2006, section 4.9) for a discussion of the methodology of calibration and numerical simulation.
The incoherent relationship between the frictionless and frictions versions of Woodfords (2003) models illustrates this point. Woodford introduces money via a cash constraint which converts money from something that overcomes frictions in the real world into something that causes friction in his model.
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