# Equivocation in mathematical economics.

Professions develop their own social conventions. One in economics is that arguments should be expressed in mathematical form where possible. This may be an attempt to make an often controversial field appear more "scientific." Physics appears to be the model for a science. The typical format for a theoretical article, or the theoretical section of an empirical article, starts by representing each variable by a symbol. The facts behind the argument are then expressed through equations that use these symbols. The theory is then developed by combining and manipulating these equations. It will be argued here that this social convention has made it much easier for the logical error of equivocation to be committed. Economists traditionally take great pride in the high degree of mathematical sophistication economics displays. They believe this gives the field a high degree of logical rigor. Here, it will be argued that the widespread use of symbolic notation makes the field peculiarly vulnerable to what logicians call the "fallacy of ambiguity (equivocation)." Engel (1, p. 90) offers the following discussion of equivocation, "Equivocation is from the Latin for, 'equal' (equi) 'voice' (vox). A word is used univocally if it has the same meaning throughout a given context, equivocally if one or more other meanings are equally possible." He offers the following humorous example:

Only man is rational.

No woman is a man.

Therefore no woman is rational.

In the above example, the two propositions containing "man" are put adjacent to one another. Thus, the error is very easy to detect. The word man is used with two different meanings. Normally the human mind carries along with a word a set of associations, perhaps a formal definition, perhaps even a picture. When the same word is used in adjacent sentences, these associations conflict and the error is discovered.

Here, it will be argued that the traditional use of symbols and algebra in economics makes similar errors of equivocation harder to detect. The reason is that in examining a mathematical argument, the symbols are divorced from any meaning, and we examine only the formal correctness of the mathematics. This is both the strength of mathematics and its weakness. The strength is that the compactness of the notation makes it easy to follow complicated arguments. The weakness is that equivocation errors are easy to make.

In the leading economic journals the usual mode of argumentation is through symbol manipulation. The assignment of symbols is typically rather casual. Such phrases as "M is the quantity of money", "K is capital", "L is labor" are common. Almost never does the theoretical section state the units in which the various quantities are measured. The rules of mathematics are then used to combine these equations and the new results announced. If the article is to be empirical, data measuring the variables of interest is inserted near the end of the process and results announced.

The result is a very compact mode of expression, with the theoretical section of a paper consisting of a short section establishing the notation and then a few lines of equations to present a model. Since mathematics is rigorous, the result is supposedly free from logical error. It will be argued that this mode of discussion can easily lead into logical error, especially that of equivocation. Here equivocation takes the form of using a single mathematical symbol for two different concepts in different parts of the argument. An example will deal with Keynesian macroeconomic theory where money (M) is often undefined. Many trained in mathematics believe that if A = B (Equation 1), and B = C (Equation 2) it follows immediately that A = C (Equation 3). Wrong! It follows only if B stands for the same thing in both equations. If B stands for different things in the two equations, one cannot argue that A = C. If the symbols have been initially stated to stand for words that are undefined or which have multiple meanings, the supposed proof that A = C can be no more convincing than the same argument expressed in words.

It may be thought that the logical error of one symbol standing for two different concepts in the same argument is too elementary to be common, and is certainly not to be found in the leading journals. Let us look at a case from monetary economics, Keynesian monetary theory, where a theory in mathematical form has become widely accepted even though the key terms appear to be used with different meanings in different parts of the argument. The discussion can be brief since the substantive points have already appeared in print elsewhere. Money

The term money is used with many definitions. Statistics are normally presented for several definitions often identified as |M.sub.1~, |M.sub.2~, etc. However, many arguments that take a mathematical form do not bother to state which concept of money the M in their arguments stands for, running an appreciable risk of committing the logical fallacy of equivocation. It is easy to use one definition in discussing the demand for money and another one when discussing the supply of money. It seems obvious that for equilibrium the supply and demand must be equal. However, if the definitions of money differ in the supply and demand equations, the error of equivocation has been committed. Discussions of money demand usually outline several reasons for holding money including to make transactions and to serve as a store of value. Not all forms of money are equally suited for these purposes. Medium of exchange is normally only coins, currency, and funds in checking deposits. Other types of deposits are suitable for use as a store of value or as a way of holding funds when other securities are considered too risky.

The importance of different motives for holding money, and the strength of the different factors that affect these motives will depend on which types of deposits are included within the category of money. If money is restricted to coins, currency, and bank accounts, the factors affecting the transactions demand will be very important. If money includes types of accounts that are held as a long term store of value in competition with stocks and bonds, other factors will be important, including the relative rates of return on such deposits relative to the returns on other stores of value.

Likewise, in a supply function for money the letter M must stand for some quantitative concept. Often the supply function is very simple. A fixed money supply is asserted, often expressed as M = |M.sub.0~ where |M.sub.0~ is this fixed supply.

The idea of a constant quantity of money is plausible. The plausibility arises because the word money calls to mind an image of gold coins, or dollar bills, and it is very easy to imagine the quantity of such physical objects remaining unchanged as they are passed from hand to hand. If the author really means by money such physical means of exchange, he probably needs relatively little justification for writing M = Mo.

However, today checking accounts are used for medium of exchange. These are merely bookkeeping entries. Most deposits used as stores of value are computer bookkeeping entries. It is not obvious why the quantity of deposits in a nation's computers should remain constant as the public's demand to hold assets in different forms changes, or as funds flow from one account to another. Failure to be explicit about how a key term is defined has made it possible to slip a key assumption into a model unexamined.

The Simple Keynesian Liquidity Trap

Let us consider the simplest Keynesian monetary model, one where the economy is in a liquidity trap. Wonnacott (10, Chapter 5) provides a typical exposition. Most of the discussion is devoted to developing a speculative demand for money function. Money is asserted to be held as an alternative to bonds as a store of value. Investors are concerned about the risk involved in holding bonds. When interest rates are low relative to historical values, investors believe that rates cannot fall much further, but that they could rise. Thus, at low interest rates investors choose to hold money as an alternative to bonds. At higher interest rates, they are willing to hold bonds in increasing amounts. This leads to a speculative demand for money function in which as interest rates decline from a high to a low level, the demand for speculative holdings of money increases. At low interest rates, the curve flatten outs and the demand for money grows to an infinite amount.

Keynes also provided for a transaction demand for money which is a function of income. The higher the income the more money is required for transactions. A demand curve is obtained by adding these two sources of demand. A vertical supply curve is obtained by assuming there is a fixed stock of money, M. The interest rate is set where the supply of money is equal to the demand for money. Once the interest rate is determined, the level of investment is read from a marginal efficiency of capital schedule. This in turn determines the level of income.

The question is not asked of exactly what M stood for, or whether there is any M for which the various equations would all be reasonable. If M is a narrow |M.sub.1~ definition (such as coins, currency, and checking accounts), a transactions demand for money roughly proportional to income is plausible. Given the high reserve required against checking accounts and the likelihood that the ratio of coins and currency to checking deposits would remain constant (with each medium of exchange used for certain types of transactions) as income changed, a constant quantity of such money is plausible. However, it is not plausible that there would be an appreciable speculative demand for such money. As a store of value, non-interest paying checking deposits (as well as coins and currency) were dominated by interest paying assets which did not fluctuate in value when long run interest rates fluctuated, such as Treasury bills and savings deposits.

An appreciable store of value function for money could be obtained if money was defined to include savings deposits suitable for holding funds for a period of time. However, if this is done it is no longer feasible to casually assume a fixed quantity of money. The reserves held against saving deposits are much smaller than those held against checking deposits. A decision to save involves transferring funds from checking accounts or currency to saving deposits. This creates excess reserves. Profit maximizing banks lend out the excess reserves, causing the quantity of money to increase. Thus, with a broad money definition the apparently innocuous assumption of a fixed quantity of money breaks down. With the usual "mathematical" analytical procedure of asking if each individual equation is plausible and then whether the mathematics is correct. there appears no problem with the argument. However, there may still be the error of equivocation.

The IS/LM Analysis

Let us look at the standard IS/LM apparatus model used in many textbooks. The Wonnacott (10) version will be used because it provides a concrete case for discussion. It also happens to be the one I taught the IS/LM apparatus from. The LM curve is derived from two relations. There is a transactions demand in which the demand for money for transactions is a function of income, represented by Wonnacott as |L.sub.t~=kY where |L.sub.t~ is the transactions demand for money, Y is income, and k is a constant of proportionality. There is also a speculative demand for money in which |L.sub.s~=f(i) where |L.sub.s~ is the speculative demand for money and i is the interest rate. The demand for money is considered to be the sum of the transactions demand and the speculative demand. Thus, a third equation is that |L.sub.s~ + |L.sub.t~ = M where M is the fixed quantity of money. Unfortunately, money is not defined. Again the key question is whether there is a single definition for money which is consistent with a large speculative demand for money and with the quantity of money being constant. Keynes' Paradox of Thrift

Let us look at the well known paradox of thrift described in most textbooks. Textbooks describe how if consumers decide to increase their savings, income will be lowered. Some of the money received as income by individuals is saved rather than being spent. This lowers the income of other individuals and causes an income contraction which continues until incomes have fallen to where savings just equals investment.

Using the Wonnacott diagrams this is represented in a shift of the precautionary/speculative demand for money upwards. One effect is to raise interest rates and hence to reduce investment. The other effect is to increase the precautionary/speculative holdings of money. With the fixed total quantity of money this implies an equal reduction in the quantity of money available for transactions, and hence a decline in income. So innocuous is the assumption of a fixed quantity of money that it does not appear to be questioned in the large U.S. Keynesian literature.

Yet on closer examination one should question it. If money is limited to currency, coins, and checking deposits, it is unlikely that an increase in thrift would raise the holdings of money. The funds would be promptly invested in ways promising a higher rate of return. If funds saved were put into other types of deposits, the institution receiving the funds would be expected to lend them out. If they were used to purchase existing assets, stocks, bonds, etc. they would bid security prices up. This would cause someone else to increase spending. Either he could now raise investment funds at a lower cost, or with higher asset prices he would be wealthier and would save less.

If money is defined to include time deposits, money could serve as a store of value. Saved funds would be put into time deposits. With a constant supply of broad money, saving would imply reducing funds available for transactions, and hence, lower income.

However, the assumption that funds could be transferred from checking deposits to time deposits while leaving the total constant isn't correct given U.S. reserve requirements. Reserve requirements on time deposits are zero now (and were low when Keynesian economics was most popular). The transfer of funds from transactions oriented deposits to time deposits would create excess reserves. This would increase bank lending. The resulting expansion of the money supply would offset most, if not all, of the excess lending. Thus, the Keynesian paradox of thrift lacks a logical basis with either definition of money. This problem appears not to have been noticed because every equation in the system was plausible with M standing for some definition of money, and the mathematics used to combine the equations was correct. Yet the logical error of equivocation has been committed. Its discovery has been made more difficult by the use of symbols, especially when there is no explicit statement of what the symbols stand for.

Incidentally, the above error of equivocation in this form does not occur in Keynes' original work (6). In a footnote in the General Theory (3, p. 167) he defines money to include all bank deposits (as did other British economists of the time). British bankers (then as now) did not distinguish between the reserve backing for different types of deposits and worked to the same customary reserve against all types of deposits. This made it plausible to argue that the quantity of money remained fixed even if the recipients of income chose to hold funds received as checks for future use. If the funds were moved to a time deposit, the total quantity of money was not affected.

Incidentally, Keynes own work is not free from the problem of equivocation, although it is not related to the use of mathematics. His definition included all bank deposits, thus including interest paying time deposits. Yet in some of his writing (4, p. 215-216) he referred to money as non-interest paying. This usage would only make sense if he was mentally visualizing money as being coin, currency, and checking accounts.

However there is a charitable interpretation of what Keynes was doing. In the thirties the interest rate on British deposit accounts (time deposits) was very low (about 1/4%), a figure that was much closer to the zero rate on currency than the prevailing rates on long term bonds (say 2 1/2%). Thus it was a valid question as to why deposit accounts were held when the bond rates were a multiple of the rates paid on them. Still, Keynes created considerable confusion by lumping these low interest rate deposits in with non-interest paying checking deposits, calling the combination "money," and then referring to it as non-interest paying.

As another example let us examine the monetary implications of a surge in investment such as Keynes believed drove the business cycle. This was directly or indirectly financed out of money held for the speculative motive. If the investment was financed by funds already held as money, the speculative holdings were directly reduced. If the financing was by selling bonds the effect was to raise the interest rate, lowering bond prices. With lower bond prices investors thought prices were more likely to rise than fail, and drew on their speculative balances to buy the bonds. With a fixed quantity of money, the reduction in speculative holdings had to be offset by an increase in money held for the transactions motive. This implied an increase in income. The increase in income resulted from the investment spending itself and the operation of the multiplier. Again the weak point in the argument is the assumption of a fixed quantity of money.

With U.S. institutions, this conversion of funds held for the speculative motive to funds held for the transactions motive would be accomplished by transferring funds from time deposits to checking deposits (and to a certain extent to currency and coin). This would increase required reserves and cause the total money supply to decrease, causing only a small fraction of the invested funds to end up as transactions money. Again, with British institutions, a reduction of money held for the speculative motive would be offset by an increase in money held for the transactions motive and a corresponding increase in income. The tradition of not defining M in Keynesian economics was established early. Both Hicks (2) and Modigliani (9) developed similar systems of equations (which were essentially the IS/LM systems of the textbooks) containing a fixed quantity of M without stating how M was defined, or why its quantity was taken to be fixed. Fortunately, even if no definition of money works perfectly for the United States, the IS/LM apparatus can be salvaged by letting M stand for the monetary base (5).

Money is a widely used term with several meanings. Economists are very vulnerable to the error of equivocation when plausible sounding statements are converted to equations containing the letter M, and then these equations are combined using the laws of mathematics. The obvious implication is that the word money, or the symbol M, should not be used without a definition.

Capital is another commonly used term with several meanings, as is efficiency decline. Since, the ambiguity in these terms, and which definitions are appropriate for which problems has been discussed elsewhere (7, 8), it is not necessary to repeat the argument here.

Finally, equal in the mathematical sense is frequently used to mean "approximately equal to." Most mathematical results are only true if "=" really is "equal." For instance, the argument that A = E if A = B, B = C, C = D, and D = E becomes incorrect if the equal sign is replaced by (meaning approximately equal). If approximately equal is taken to be within 10%, A is only guaranteed to be within |(1.1).sup.4~ - 1 or 46.41% of E. Thus, it can not be deduced that A |is congruent to~ E from A |is congruent to~ B, B |is congruent to~ C, C |is congruent to~ D, and D |is congruent to~ E with |is congruent to~ interpreted as meaning within 10%. Many involved arguments become unconvincing when all the "='s" which really mean "approximated by" are replaced by the phrase "approximated by". The very compactness of mathematical notation conceals how many approximations are involved in such arguments. Implications

Great care should be exercised before believing economic arguments expressed in mathematical form. There is a temptation to find an argument persuasive if each of the verbal statements from which it is derived (assumptions) appears correct, and if no mathematical errors are committed in combining the statements. This is logically incorrect since the possibility of the logical error of equivocation has not been excluded. Each of the assumptions may be correct, but if the words and related symbols refer to different concepts, the conclusions that result from using the laws of mathematics to combine the different equations may be incorrect.

Thus, should an author wish to construct a logically correct mathematical argument he must show that the symbols consistently stand for the same concept. In addition, the symbols must correspond to the concepts implied by the words used initially in expressing the assumptions. This requires either using only words that have but a single meaning (difficult to do in economics where many words have multiple meanings), or explicitly defining terms. Explicitly defining terms appears the safest course. Two terms that should always be defined are capital and money. Likewise, one of the steps referees and editors should use in evaluating manuscripts is to ask if terms are defined and consistently used. Of course, there are other benefits to defining terms when they are ambiguous. Even if a word with multiple meanings is used consistently throughout an argument, a reader may not immediately know which of several meanings the word (and the associated symbol) has. An explicit definition can save the reader (and referees) considerable time and prevent confusion.

The cost of providing an explicit definition is low, often only a sentence or two. If the reader would have guessed without being told which definition is being used, all that is lost from making it explicit is a little time. This is minor in contrast to the confusion that occurs when the reader assumes one definition when another is meant.

In many cases, for terms such as capital, a definition would require stating what units the concept is measured in. In other words, it is necessary to state what would constitute one unit of the quantity, and then to state how the ratio of the quantities of other items would be related to the item under consideration.

Of course, it is not to be suggested that merely expressing arguments in verbal form prevents equivocation. (Remember Keynes' use of the term money). It is still easy to use one word with several meanings in the course of an argument. However, the error appears a little less likely with words than with symbols. The reason is that the mind carries along with a word a definition or definitions, and often a somewhat concrete image. If the word is used in different meanings in the same argument there is a moderate chance that the associated definitions and concepts will come to mind, revealing the shift in meanings. Symbols are less likely to be associated with definitions or images and are more likely to be manipulated as symbols using the laws of mathematics. When combining two equations the author (and reader) is unlikely to ask if the symbols stand for the same concept in each equation.

Conclusions

Correct use of mathematics as an aid to reasoning requires avoiding the error of equivocation or the use of the same word (or symbol) with two or more meanings in the same argument. Because mathematical symbols do not carry with them associated definitions, the error of equivocation is easy to make when using mathematics in economics. In microeconomics, ambiguity in the word capital often leads to error. In macroeconomics, money is a term which often leads to error. Such errors can, and should be avoided by explicitly defining terms and stating which units are being used.

References

Engel, Morris, With Good Reason, New York: St. Martin's Press, 1982.

Hicks, John R., "Mr. Keynes and the 'Classics'; A Suggested Interpretation," Econometrica, 5, April 1937, No. 2, pp. 147-1160.

Keynes, John Maynard, The General Theory of Employment, Interest, and Money, New York: Harcourt, Brace. and Co., 1936.

-----, "The General Theory of Employment," Quarterly Journal of Economics, 51 (February) 1937, pp. 209-23.

Miller, Edward, "An Extension of the Economics of Keynes to the United States," Southern Economic Journal, 50 (January 1984) No. 3, pp. 781-801.

-----, "Keynesian Economics as a Translation Error: An Essay on Keynes' Financial Theory," History of Political Economy, 17 (Summer 1985) No. 2, pp. 265-286.

-----, "Robinson's Classic Question Revisited: How to Measure Capital?"--in The Joan Robinson Legacy, edited by Ingrid Rima, Armonk, New York: M. E. Sharpe, 1991, pp. 136-151.

-----, "Is Efficiency Decline Rent Decline or Capacity Decline?" Southern Economic Journal, 58, No. 3 January 1992, pp. 635-643.

Modigliani, Franco, "Liquidity Preference and the Theory of Interest and Money," Econometrica, 12, January 1944, No. 1, pp. 45-88.

Wonnacott, Paul, Macroeconomics, Homewood, Illinois: Irwin-Dorsey, 1978.

Edward M. Miller Professor of Economics and Finance, University of New Orleans.