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Extending the equation of exchange: dual equations, for output and input.



As a framework to organize thinking about inflation, deflation, and recession or depression, an equation of exchange is often useful: either Irving Fisher's MV * PT where T is an index of all the economy's transactions during a time period (final real output, inter-firm, resource input, and even existing assets), or the more modern MV * PY where Y is an index of final real output only. (1) But neither equation directly and explicitly considers resource input prices and quantities. Doing so can illuminate a variety of economic events and issues.

Because the all-transactions equation of exchange integrates less easily with modern macroeconomic models that explain Gross or Net National Product, our point of departure is the final output equation of exchange, MV * PY. We treat it as an identity that summarizes a larger macro-economic model, not as a non-tautological "quantity theory of money" that holds only in monetary equilibrium. (2)



An equation analogous to the equation of exchange for output can be defined for resource inputs, giving dual equations of exchange:

MV * PY and MV * WX,

where W represents an index of resource input prices, and X an index of real resource inputs used. (The equation MV * PT has no such unexploited analogue, because it already encompasses all the economy's transactions.) Substituting and rearranging shows that nominal output equals nominal income, (3) and the "productivity ratio" of output to input quantities equals the "input reward ratio" of input to output prices:

PY * WX and Y/X * W/P.

Assume also a production function, relating the economy's output capacity to the resource inputs available:

Y = f(X).

The U.S. government routinely estimates the value of M,P,Y, and hence of V * PY/M, and private research has extended these estimates back to the 1800s, but there are no annual estimates of W and X for the entire economy, although at least in principle they could be estimated. Meanwhile, albeit empty empirically, (4) they are useful constructs.

Dual equations of exchange make it possible to discuss directly not only production but also incomes--and income issues can fascinate students for whom the equivalence of real output and income may be a dimly understood abstraction.

Illustrating Dual Equations of Exchange

Dual equations of exchange formalize the "circular flow of economic activity": while resource inputs flow from the household to the producing sector, and real output flows from the producing to the household sector, money in exchange flows in the opposite direction. This circular flow depicts a world in which households ultimately own everything and therefore buy all final output while supplying all resource inputs and receiving all income: on the diagram's top half, MV represents the flow of money out of the household sector and PY represents the flow of money into the producing business and government sectors on the diagram's bottom half, MV (again) represents the flow of money out of the producing business and government sector and WX represents the flow of money incomes into the household sector. Clearly, MV * PY and also MV * WX, because payments must equal receipts, for output and for resource inputs. Also, PY * WX and Y/X * W/P, because households own everything and thus receive all funds that flow in to the producing sector.

The Ownership Perspective: Justification and


For students, the diagram's ownership perspective needs some justification and elaboration: Workers of course belong to households all private businesses are owned by some person or group all governments are controlled (the essence of ownership) by some person or group (be they voters, politicians, bureaucrats, or despots), who by definition are part of the household sector. Thus the economy's production not only of consumer commodities and goverment output but also of new capital goods that will be used by either business or government--the entire Gross National Product--flows to households. In exchange, household payments to businesses include not only consumption spending, but also saving spent on new financial investment in business and government (such as the purchase of newly issued stocks and private and government bonds), and taxes (net of transfers) and various fees and other payments to government. (5) Finally, since the household sector ultimately owns all assets used by business and government, it supplies all resource inputs to the business and government producing sector. In exchange, households receive the entire gross national income: wages and salaries, royalties, rents, interest payments, dividends, and all other profits (including retained earnings, which business managers retain only with the at least tacit permission of business owners).



Considering output quantities and prices and input quantities and prices permits investigating many interesting questions. For example:

Inflation and Money and Velocity Growth

Suppose a "short-run" situation: the economy's resource inputs X are fixed, and its real output Y is also fixed. Any increase in M or V or both will raise not only output prices P but also input prices W:

[Mathematical Expression Omitted!

Many students are surprised that when output prices rise, at least some households receive higher incomes from higher resource input prices, and that as a group, while households pay more for the output they buy, they will receive equally more for the resource inputs they sell. Thus because the percentage increase in P and in W will be equal, the inflation will neither harm nor benefit the "average" household: if all households owned the same mix of resources, inflation would have no effct on their relative welfare.

Of course it is important to emphasize that, in reality, households don't all own the same mix of resources, so inflation does alter relative welfare, sometimes severely.

Students have little trouble comprehending a money supply increase, but a velocity rise is more difficult, until MV is seen as representing spending, so that any increase in nominal aggregate demand not generated by a larger money stock is symbolized by higher velocity. Examples help. Velocity rises as private spending increases in response to fears of prospective wartime shortages, as in Lebanon and elsewhere in the Middle East when military tensions rise, or in the U.S. in 1950 when President Truman announced U.S. involvement in the Korean War. Velocity rises with an effectively expansionary fiscal policy. Velocity falls if higher interest rates decrease aggregate demand (in addition to shifting its composition from long-lived to short-lived assets and to perishable goods and to services). All these affect not only output prices, but also resource input prices.

In response to expected inflation, aggregate demand for goods and services, and hence velocity, may either rise or fall. Inflation encourages people to "stock up" on long-lived assets, acquisition of which is a form of saving and investment, but they can't "stock up" on perishables: extra haircuts this week can't be used next year. If people fear that a long-lived inflation will reduce future productivity and real income growth, making them unable to afford future perishables (for example, college tuition for today's pre-schooler), their fearful expectations may persuade them to save more, increasing the demand for real or financial assets (possibly including money (6)). If the resulting attempt to acquire financial assets is sufficiently large, velocity declines.

A reasonable assumption might be that in an inflation's initial stages, velocity rises as households and firms increase their demand for all godds and services. But then if the inflation becomes long-lasting, velocity fall if desired saving in financial assets increases enough.

Price Responses to Quantity Variables

Despite familiarity with the assertion that inflation results from "too much money chasing too few goods," students (and others) often overlook the goods part of this statement. Dual equations of exchange show that output and input prices depend not only on the money supply and its velocity, but also on real output and resource inputs, and on productivity Y/X. If real output Y changes only because resource inputs X change, Y/X remaining constant, P and W will move together, proportionately: for example, with constant MV,

[Mathematical Expression Omitted!

But productivity changes alter the relationship between P and W: for example, again with constant MV,

[Mathematical Expression Omitted!

Analogous but opposite results occur if real output Y falls, resource inputs X fall, or productivity Y/X falls.

We are used to real output Y rising over time, as resource inputs X increase with saving and investment in real capital, population growth by immigration and by birth, demographic changes that increase the working age proportion of the population, natural resource discoveries, territorial acquisitions, and so forth. And productivity Y/X also commonly rises with the growth of scientific and technical knowledge. But material progress is not inevitable: real output Y can fall, as resource inputs X decrease or productivity Y/X declines. Demographic changes can lower the working-age proportion of the population (as in the U.S. over the next century). Was and natural disasters reduce the stocks of real capital and labor (of innumerable examples, the Black Death of the Middle Ages is particularly dramatic). Productivity (not merely its growth rate) declines because of disastrous economic and social policies (third world countries provide many examples, but Spain under Ferdinand and Isabella with their Inquisition against Moors and Jews, plus their 10 percent alcabala turnover tax, also may qualify). Note that productivity may decline not only "exogenously," but also as a result of long run inflation. [7! (Spain again illustrates, particularly after it began importing and monetizing enormous quantities of gold and especially silver from New World mines.)

Expectational Inflation

Suppose households, business firms, and goverment agencies all expect future inflation. As they build their expectations into long term contracts, future output prices P and future resource input prices W will in fact rise accordingly--a self-fulfilling prophecy. The resulting inflation is not really demand-pull, because excess commodities demand is not the proximate cause. Neither is it administered inflation, (8) because it does not result from monopoly power: the households and firms that set future prices in accordance with their expectations may be operating in highly competitive markets. We call this scenario "expectational inflation."

Suppose that future output prices are set no lower than the expected future output price inflation rate. Then if the money supply together with its velocity MV rises at the expected inflation rate plus the real output growth rate, inflationary expectations will be exactly "ratified": output prices will rise at the expected inflation rate, and resource input prices will rise at the MV growth rate less the resource input growth rate,

[Mathematical Expression Omitted!

If MV rises more rapidly than this ratification rate, demand pull inflation will raise output prices and input prices higher than forecast, perhaps raising inflationary expectations for the next period,

[Mathematical Expression Omitted!

But if MV rises less rapidly than this ratification rate, then given the expectational inflation hypothesis that opened this paragraph, real output produced and real resource inputs used must fall,

[Mathematical Expression Omitted!

This last scenario caused the relatively severe recession of 1981-1982. Because of the long and continually worsening inflation during the 1970s, at the beginning of the 1980s most people expected substantial future inflation, and their plans and agreements assumed rapidly rising output prices and resource input prices. Households paid continually higher prices set by firms firms paid continually higher wages demanded by households households and firms, expecting nominal asset prices to continue rising rapidly, borrowed money at historically very high nominal interest rates. Then when the Federal Reserve decreased the money supply growth rate, and velocity also declined, (9) the growth rate of MV dropped sharply below the increases in output prices P and resource input prices W. The result was the worst recession (10) in Y and X since World War II.

Distributing the Fruits of Progress

If additional real output Y is produced from additional resource inputs X, while productivity Y/X * W/P remains constant, output prices and resource input prices move proportionately lower than they would be otherwise. But if additional output results not from additional resource inputs but from improved productivity, the ratios Y/X * W/P rise, and input prices rise relative to output prices--or output prices fall relative to input prices. Depending on what MV is doing, the benefits of higher productivity will be distributed as either higher input prices W (as since the late 1930s), or lower output prices P (as in the post Civil War 1800s), a contrast that throws new light on U.S. history for many students.

In the United States in the second half of the 1800s, Mv grew more slowly (M2 rose while V fell) than did real output Y, and even more slowly than did real resource inputs X (using some very rough "back of the envelope" estimates for the growth of X). Consequently, input prices W remained roughly constant or fell slightly, while productivity improvement drove output prices down. Over a 28 year period from 1869 to the turn of the century, the compounded annual rates of change were approximately (11)

[Mathematical Expression Omitted!

The contrast with the U.S. during its most recent 28 year period is striking. From 1961 through 1989, MV grew much more rapidly (M2 rose greatly while V rose slightly) than did real output Y, and even more rapidly than did real resource inputs X (again, the X estimates are very rough). Input prices W and output prices P both rose dramatically, but productivity improvement caused input prices to rise even more rapidly than did output prices. The compounded annual rates of change were approximately (12)

[Mathematical Expression Omitted!

Incidentally, these very crude growth estimates suggest that in both periods, early and recent, roughly 70 percent of real output growth was due to additional resource inputs X while 30 percent was due to increased productivity Y/X. (The data actually suggest a slight rise in the role of productivity growth relative to resource growth, but the estimates are not nearly accurate enough to warrant drawing any such conclusion.)


Dual equations of exchange can contribute much to student understanding of fundamentally important macroeconomic concepts, particularly in principles classes. (They may be useful for review in more advanced courses as well). While simple enough to not be frightening, they are complex enough to serve as a vehicle for serious ideas.

Suppose the principles of macroeconomics course begins with the usual microeconomic introduction (resource scarcity, production possibilities, supply and demand), and a brief introduction to macroeconomic measurement (national income accounting for gross and net national product and income price indices, and employment and plant capacity use statistics). After this opening material, the dual equations of exchange can preview in general terms most of the basic ideas that the course will later re-examine with more formal and complex models. Its presentation of the consequences of monetary change provides an excellent motivation for money and banking and monetary policy, and the effects of government fiscal actions on velocity can "foresahdow" later discussions of fiscal policy.

Repetition may be the "soul of learning," but mere repetition usually is boring. Using the dual equations of exchange to introduce macroeconomic concepts, before using more detailed and formal models to develop more complete analyses, is one way to get the repetition without the boredom.

(1.) In either equation, M is the money supply stock, V is an appropriate definition of money's "velocity" or turnover rate, and P is an appropriate defined price level. For further discussion and background, see note 9 in the accompanying essay, or bordo [1987!, Humphrey [1974 and 1984, reprinted in 1986!, and Blaug [1985, pp. 632, 636, and chapters 1, 5, 15!.

The money supply usually is assumed to be dependent on the actions of government itself or a governmentally controlled central bank, although a privately supplied money is at least conceptually possible see Selgin [1988!.

(2.) As an identity, the equation of exchange can be interpreted as summarizing a larger macroeconomic (static or dynamic) model containing money. Solving the entire structural model including the equation of exchange gives a "reduced form" equation for V = V (*). Then with a given money supply, velocity increases if the nominal value of final output increases for any reason--in terms of a more advanced and detailed formal model, either because of a shift of or a movement along either the aggregate demand or the aggregate supply functions.

For example, Milton Friedman's "Theoretical Framework for Monetary Analysis" [1970!, revised and reprinted in Gordon [1974!, added the MV = PY equation of exchange to Hick's ISLM model [1937!, which has its own (structural) money demand function and its own equilibrium condition for money. But Friedman continued to argue, along with other monetarists, that the income velocity of money is sufficiently stable that the equation of exchange can be a non-tautological "quantity theory of money" model of the macroeconomy.

For an elaborated discussion, see notes 12 and 13 in the accompanying essay.

(3.) Of course one could reason in the opposite direction: if MV * PY, then because nominal national output equals national income, PY * WX, it follows that MV * WX. If PY and WX are defined as nominal gross national product and income, they include " capital consumption allowances" (mostly depreciation). Subtracting these allowances would define PY and WX as nominal net national product and nominal "net" national income, respectively, making velocity somewhat smaller. ("Net" national income is larger than the Department of Commerce's National Income at factor cost, because indirect business taxes less subsidies to government enterprises, business transfer payments, and the statistical discrepancy have not been deducted.)

(4.) To be a proxy for W, the producer price index includes too many intermediate commodities. To get a proxy for X, raw materials and labor force and plant capacity statistics--availability and use--would need to be combined, and data on government's productive assets included.

The government does estimate labor and productivity, which is the flow of real output Y per labor hour input. But these data are not the same as Y/X because X includes not only labor but also all non-labor resource inputs. Recent government estimates of real capital productivity, and multifactor (labor plus capital) productivity, apply only to the private sector.

(5.) Many circular flow discussions use the concepts of expenditure demand "leakages" (household saving, tax payments, and net imports from foreign countries) and "injections" (business purchases of new capital goods, government spending on transfers and on goods and services), where the leakeges and injections are disconnected. More realistically, flows through financial markets connect such "leakages" and "injections" together.

(6.) Although macroeconomic discussions often assume that a shift in the demand for goods and services requires an opposite shift in the demand for money, this equivalence need not hold if other assets (tangible real or personal property, financial equities, or debt) are considered. The effect of expected inflation on the demand for money and on velocity depends on how risk averse households and firms (both lenders or borrowers) are, and on which of a variety of inflationary risks they fear most. See Folsom (forthcoming).

(7.) For an excellent listing of inflation's real costs, see Hughes [1982!. Also see Feldstein [1979!.

(8.) Administered inflation is discussed in the accompanying essay, although for reasons of space, not here.

(9.) The decline is blamed variously on mysterious unknowns, on financial deregulation that permitted banks to pay interest on household checking deposits, on attempts to increase saving in financial assets for reasons explained above at note 6, and on rational adaptation to the oil shocks of the 1970s. For this last see Gonzalez and Folsom [1989!.

(10.) Unless this "worst recession" is put into perspective, students may roughly equate it to the Great Depression. But in 1981-1982 real output fell less than 3.4% from its 1981 quarterly peak to its 1982 trough, while unemployment rose from about 7.4% to 10.7% of the labor force. The corresponding 1929-1933 annual numbers were a 31% decline in real output, while unemployment rose from 3.2% to 24.9% of the labor force.

(11.) These percentage changes compare annual averages for 1869-1873 and 1897-1901 (single year data not being available)--28 years (1871-1899) which exclude the Civil War's immediate aftermath and the inflation that began in 1902, and which are described by continuous data series.

Over this period, the money supply (the broader M2 definition because M1 data are not available prior to 1915) rose 305% velocity fell 39% output prices fell 36% real output rose 289%. Resource inputs X rose 165%, which is a guess given by a weighted average of growth in "gainful workers" 10 years of age and older (up 120%, 2.85% compounded) and real capital growth (reproducible wealth, up 329%, 5.3% compounded), where the weights are 78.6% for labor and 21.4% for capital, using estimates of the percent distribution of national income by type of income for 1900 (the earliest year available). (Income of unincorporated enterprises all is allocated arbitrarily to labor.) Given these other changes, resource input prices W are computed to have fallen 5% over the period. For further discussion, see the accompanying essay at note 28.

Source: U.S. Department of Commerce [1975!, series A7 D26, 705, 735, 736 E40, 52, 135 F1, 3, 5, 71, 98, 125, 186-191, 447 X415, particularly the italicized series. Detailed calculations are available in Folsom [1990!.

(12.) These percentage changes compare annual data for 1961 and 1989. The M2 money supply (chosen for comparability with the 1871-1899 analysis) rose 863% velocity rose 2% output prices rose 305% real output rose 141%. Resource inputs X rose 84%, which is a guess given by a weighted average of growth in the labor force (up 74%, 2.0% compounded) and real capital growth (nonhousehold fixed reproducible tangible wealth net of retirements and depreciation, plus business inventories, up 131% 3.0% compounded), where the weights are 81.9% for labor and 18.1% for capital, using estimates of the percent distribution of national income by type of income for 1989. (Proprietor's income all is allocated arbitrarily to labor). Given these other changes, resource input prices W are computed to have risen 533% over the period.

Sources: Economic Report of the President, 1990 (tables C-1, C-2, C-3, C-19 C-32) U.S. Department of Commerce, Business Statistics: 1961-1988 (p. 70) and the Survey of Current Business, October 1989 (page 33 table 4), July 1990 (pp. 40-89, NIPA tables 1.1, 1.2, 1.14, 7.4 p. S-15), and August 1990. Detailed calculations are available in Folsom [1990!, which also gives alternative (private sector only) estimates based on Bureau of Labor Statistics multifactor productivity data.


Blaug, Mark. Economic Theory in Retrospect. 4th ed. London: Cambridge University Press, 1985.

Bordo, Michael D. "The Equation of Exchange," in The New Palgrave: A Dictionary of Economics, vol. 2, London: MacMillan Press, 1987, 175-177.

Feldstein, Martin S. "The Welfare Cost of Permanent Inflation and Optimal Short-Run Economic Policy." Journal of Political Economy, vol. 87, August 1979, 749-768.

Folsom, Roger Nils. "Portfolio Adjustment and the Demand for Money in the Face of Expected Inflation: An Uncertainties Perspective. "Forthcoming in the Journal of Post Keynesian Economics.

_____. "U.S. Economic Growth, 1871-1899 and 1961-1989: A Dual Equations of Exchange Comparison." Unpublished, October 1990.

Friedman, Milton. "A Theoretical Framework for Monetary Analysis." Journal of Political Economy, vol. 78, March/April 1970, 193-238. Revised and reprinted in Gordon [1974!.

Gonzalez, Rodolfo Alejo, and Roger Nils Folsom. "Responding to the Oil Shocks: The U.S. Economy Since 1973." The Freeman, vol. 39, February 1989, 62-65.

Gordon, Robert J., ed. Milton Friedman's Monetary Framework: A. Debate with His Critics. Chicago: University of Chicago Press, 1974.

Hicks, J.R. "Mr. Keynes and the 'Classics': A Suggested Interpretation." Econometrica, vol. 5, 1937, 147-59. Reprinted in William Fellner and Bernard F. Haley, editors for the American Economic Association, Readings in the Theory of Income Distribution. Philadelphia: The Blakiston Company, 1946, 461-476. Although a classic, Hick's analysis is summarized only partially in most intermediate macroeconomics textbooks.

Hughes, Dean W. "The Costs of Inflation: An Analytical Overview." Federal Reserve Bank of Kansas City Economic Review. November 1982.

Humphrey, Thomas M., "The Quantity Theory of Money: Its Historical Evolution and Role in Policy Debates," and "Algebraic Quantity Equations before Fisher and Pigou." Federal Reserve Bank of Richmond Economic Review, May/June 1974 and September/October 1984, respectively. Reprinted in his Essays on Inflation. Fifth edition. Richmond: Federal Reserve Bank, 1986, 1-18, 278-287.

Selgin, G. A. The Theory of Free Banking: Money Supply Under Competitive Note Issue. Totowa, New Jersey: Rowman and Littlefield, 1988.

U.S. Department of Commerce, Bureau of the Census. Historical Statistics of the United States, Colonial Times to 1970. Washington, D.C.: U.S. Government Printing Office, 1975.

U.S. Department of Commerce, Bureau of Economic Analysis. Business Statistics, 1961-1988. Washington, D.C.: U.S. Government Printing Office, December 1989.

_____. Survey of Current Business. October 1989 July and August 1990.

U.S. President. Economic Report of the President. February 1990.
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Title Annotation:economic model during a time period
Author:Folsom, Roger Nils
Publication:Economic Inquiry
Date:Jan 1, 1991
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