The Theory of Income Determination

Samuelson's first Econometric Society meeting as a professor, rather than as a graduate student, was in December 1940, in New Orleans, where he took part in a session on Keynesian economics.

Samuelson responded by discussing three meanings of Say's Law, the term used by Keynes for the idea that there can be no shortage of aggregate demand. The first, “most zealously held,” was purely metaphysical and irre­futable: “supply is demand since goods exchange against goods.”¹ This was empirically meaningless. The second was that purchasing power is indestruc­tible: that what is not spent on consumption is automatically invested. This was empirically false, because effective purchasing power is constantly chang­ing. The final meaning, which related directly to Lange's paper, was that involuntary unemployment was impossible if prices were sufficiently flexible.

Samuelson’s most substantial argument against Lange was that while there might be a sufficiently low price level that would generate full employment, a falling price level might not eliminate unemployment. He was making a dynamic argument, present in the General Theory, but widely neglected in subsequent debates, because it was hard to make with a simple mathematical model such as Lange was using.

The third speaker was Hansen, though it is not clear whether Hansen addressed this question, for Econometrica did not publish an abstract of his paper, merely a note that its substance would be included in his forthcoming book, Fiscal Policy and Business Cycles (1941a). This was a project in which Samuelson had become involved. The project had begun with Hansen writ­ing a manuscript, “Fiscal Policy in Relation to the Business Cycle,” which was discussed by leading economists at a conference organized by the Social Science Research Council in June 1939. During the following year, Hansen discussed the manuscript with many colleagues and students at Harvard, including Samuelson.² The published version opened with a statement of the relationship between depression and war, a situation then engulfing Europe and increasingly dominating American economic policymaking.

The war now afflicting, directly and indirectly, the entire world, can­not be explained by overly simplified dogmas running in terms of competitive capitalism and imperialistic rivalries. But it has, nonethe­less, an economic basis—the inability of the great industrial nations to provide full employment at rising standards of real income. The disas­trous economic breakdown of the thirties let loose forces which have set the world in flames. The ultimate causes of the failure to achieve a world order in the political sphere must be sought against the facts of economic frustration.³

Economic policy could not be separated from international relations, an argu­ment Samuelson would take up a year later, when he taught students at the Fletcher School about the relationship of economic activity to war.^a

Perhaps the key point in Hansen’s book was the emergence of a new aim for fiscal policy—ensuring full employment through high levels of govern­ment expenditure, financed either from progressive taxation or by a rise in the public debt.

a. See chapter 16 this volume. income—that there was a consumption function—he made the case for high investment. He illustrated this with diagrams showing the relation­ship between consumption and income, including one similar to the diagram Samuelson had used in one of his articles on the business cycle.^5b Unlike Samuelson’s diagram, Hansen's consumption function was a straight line, but more significantly, there were numbers on the axes: a theoretical idea had been quantified.

Hansen had been able to do this using data collected by the National Resources Committee on the proportion of income spent on consump­tion by households with different income levels. There was, however, a problem that could not be solved using such data. Hansen calculated that raising national income from $50 billion to $80 billion would double the proportion saved, from 6.9 percent to 14.9 percent. However, data col­lected by Simon Kuznets showed that the proportion of income saved did not rise in the long term: the consumption function he had calculated held only when income changed within a short period of time. A different type of empirical analysis was needed. This was provided in an appendix, under Samuelson’s name, titled “A Statistical Analysis of the Consumption Function.” In this appendix, discussed from a different perspective in chapter 17 this volume, Samuelson used multiple regression analysis to estimate alternative forms of the consumption function from aggregate data.^c It is remarkable that Hansen’s chapter does not cite Samuelson’s results, and Samuelson does not discuss the chapter to which his work was an appendix. However, the publication of two sets of estimates, derived using different methods, testifies to the importance Hansen attached to the consumption function, a concept that was central to both wartime and postwar planning.

A central element in Hansen’s argument, as in virtually all analyses of income determination at this time, was the multiplier. The concept was dis­cussed in detail in the General Theory and previous literature, but Samuelson believed that the theory was still not fully understood. So in the summer of 1941, he wrote a long paper, eventually published as “Fiscal Policy and Income Determination.”⁶ The paper’s short second sentence sketched a view of progress in economics that he could well have picked up from Schumpeter, then working furiously on his history of economic analysis—that economic analysis advances discontinuously, taking large steps forward, and then need­ing time for gains to be consolidated.⁷ Samuelson claimed that the theory

b. See figure 13.1 earlier in this volume.

c. These estimates are also discussed in chapter 17 this volume. of the multiplier had immediate appeal because it “neatly expressed latent vague and intuitive notions of ‘purchasing power,' ” but because it was over­simplified, it received much criticism.⁸ Though referring only to the multi­plier, rather than to the Keynesian system as a whole, the sentence in which he made this remark encapsulated his view of what his graduate student Lawrence Klein was soon to label “the Keynesian revolution.” The theory took off because it formalized ideas that were already in circulation, and it was opposed because it was oversimplified in ways that made it incorrect. The implication was that when the theory was elaborated—a task to which Samuelson's paper was to contribute—there could be a consensus. Samuelson thus set himself two tasks: to clear the ground by “isolating some current misapprehensions” and to take account of complications not discussed in oversimplified versions of the theory.

The first, short step was to isolate the multiplier from policy recommen­dations. Despite its use by Hansen and other advocates of fiscal stabilization policy, the multiplier was not the “rationalization of a free spending policy” that many economists thought it to be.⁹ The doctrine had no implications with respect to public spending, for if a rise in government spending induced a fall in private investment, the multiplicand might be negative, meaning that there would be no case for raising government spending.

Samuelson's central message concerning the multiplier was that it was crucial to match the multiplicand—the expenditure being multiplied—by the appropriate multiplier. This was illustrated by the problem of govern­ment spending, where the multiplicand might be taken either as govern­ment spending on goods and services or as spending net of taxes (the deficit). If the former were used, then the multiplier had to be adjusted to allow for the additional taxes that would be paid as income rose.^d The appropri­ate multiplier for government spending treated both saving and taxation as leakages, giving a much lower multiplier and a lower rise in taxation. It was impossible, Samuelson argued, for induced rises in taxation to cover the costs of the initial rise in government spending. As Samuelson put it, “Not even so powerful an agency as the Treasury can lift itself by its own bootstraps.”¹¹

The crucial point was that the marginal propensity to tax (the amount of tax raised by each dollar of extra income) was less than ι.

This being the case, if a government spends, without at the same time making autonomous changes in tax rates, it cannot raise the national income in a stable system without at the same time raising deficits by some amount. The induced increase in taxes resulting throughout all time from given expenditure must fall short of that expenditure. Of course, the larger the propensity to tax, the less the Treasury will lose, but there must always be some finite loss.¹²

Samuelson immediately went on to point out that it might be possible to revise the tax system so as to reduce saving, making it feasible to maintain full employment even with a balanced budget.^e

Having argued that raising government spending would increase the defi­cit if there were no rise in tax rates, Samuelson turned to the financial impli­cations of deficits. His main aim was to explain how it had been possible to have a growing government debt at the same time as continuing low interest rates. There was simply no evidence that the Treasury or the central bank had “rigged” the market to achieve this; the purchases and sales of securi­ties that would have been necessary to keep interest rates low had not taken place. The explanation he gave was that if the rate of increase in the debt was constant, as would be the case with a constant deficit, the interest rate would remain constant. It took an increase in the rate of growth of the deficit to push up interest rates. A higher deficit might raise interest rates, but those rates would not rise indefinitely. Once again, this was an argument about dynamics, but no doubt frustrating many mathematically competent readers of Econometrica, he gave no mathematical model to substantiate his claims, choosing instead to discuss U.S. policy on gold.

d. Suppose the marginal propensity to consume was ¾, so that the “basic” multiplier was

1/ (1 — 34) = 4, and that the marginal rate of tax was 40 percent. It was fallacious to argue that a $2.5 billion rise in government spending would raise output by $10 billion, and that this would raise $4 billion in taxes, resulting in a reduced deficit.

e. He did not specify what changes in the tax system might achieve this.

Samuelson’s next main point was to counter the argument, made by Hansen, that public works spending (on roads, hospitals, and other public projects) was more effective in raising income than was spending on relief and Social Security.¹³ One argument was concerned with the immediate effect of any government spending. Samuelson pointed out that this could work either way. If the money allocated to public relief and Social Security would be spent more quickly than money allocated to public works, then relief expenditure would be more effective—the opposite of what Hansen was claiming. Another argument made by Hansen was that concentrating public spending on large projects would make it more effective. Samuelson’s response was that being more visible did not mean that its effects would be greater, for numerous small projects could, taken together, have just as large an effect.

Samuelson also criticized what he chose to call the “velocity approach,” more often called the quantity theory of money. The twentieth-century ver­sion of the quantity theory, expounded by Irving Fisher, centered on the equation MV = PT, where M is the stock of money, V the velocity of cir­culation, P the price level, and T the volume of transactions. If V and T are constant, then a change in M must cause P to change in the same proportion. “Income velocity,” the term used by Samuelson, is the velocity obtained by defining T to include the transactions that enter national income, so that PT is national income.^f He began with a mathematical point:

It is unfortunate that ancient astronomers selected the period of revo­lution of the earth around the sun as the conventional unit of time reckoning, because with present financial habits this yields a figure for the income velocity of money of two or three, not dissimilar to the figure usually derived for the multiplier.¹⁴

However, though this gave Samuelson an opportunity to write with irony, this was irrelevant to his main point, which was that exponents of the veloc­ity approach were making the same mistake as had many Keynesians. They were not distinguishing sufficiently enough between propositions that were true by definition and what were refutable hypotheses. The velocity approach, Samuelson argued, was based on the assumption that the velocity of circula­tion was stable. At full employment, this implied that changes in the money supply would induce proportionate changes in the price level—a theory

f. In the 1950s, Milton Friedman revived the quantity theory by arguing for a more flexible interpretation of this equation, in which V did not need to be constant.

Samuelson considered very important, even if requiring some modifications— but when there was unemployment, monetary changes would result, at least in part, in changes in output.^g He argued that attempts to reconcile income velocity with the multiplier were unpersuasive, for they rested on manipulat­ing identities and failed to explain anything. Even the argument that veloc­ity played a role in the process of adjustment to a change in investment was not right, because normal velocity figures presumed stable payment habits— precisely what would not be found in a period when the economy was out of equilibrium. “At best,” he wrote, “the normal speed of turnover of money is one minor limiting factor; at worst, it is irrelevant and misleading.”¹⁵

Samuelson wrote with great confidence, boldly criticizing a theory with a long history; even his characterization of the quantity theory as the velocity approach served to trivialize it. He stressed that he had “avoided glossing over fundamental differences of opinion and logic,” and he implied that he had understated the objections to the velocity approach when he wrote that he had made no effort “to indicate the substantial unanimity now achieved by informed writers on many issues.”¹⁶ Yet behind this clearly implied claim to authority, the issues about which he was writing were still not sorted out.

The theory of the multiplier and income determination had still not stabi­lized, with the result that his paper was disjointed, driven by the positions he was criticizing: it reads as a series of loosely connected points. There is a great contrast here with his papers on consumer theory and international econom­ics, for both of which there was an established set of assumptions that defined a theory that he could systematize by applying more rigorous mathemati­cal analysis than was being employed by his contemporaries. When it came to the multiplier, he wrote as though he were doing the same thing: using formal mathematical analysis to cut through the confusion found in previ­ous work. However, his position was different because, despite the existence of mathematical models that encompassed Keynesian and classical systems, he had not found a single model from which he could derive the results he wanted to derive. This meant that although his thinking about dynamics was informed by his thinking as a mathematician, he did not present a dynamic model from which his conclusions could follow.

g. Using the notation defined here, if there is full employment, T, which measures real activity, cannot change, and so a rise in MV must cause P to rise; in contrast, if there is unemployment, monetary changes may cause T to rise or fall, implying that P need not change. The claim that V and T are constant, and that changes in P are therefore proportional to M, is a testable hypothesis; the claim that MV = PT, with nothing said about V and T, is true by definition.

The editors of the American Economic Review gave Samuelson an opportu­nity to link such ideas to the dispute over Keynes and classical economics, to which he had contributed in his AEA debate with Lange, when they invited him to review Employment and Equilibrium, the latest book by the Cambridge economist A. C. Pigou (1941). Samuelson contended that, in debating what classical economics was, Keynes's interpreters were “very much in the posi­tion of the man who, having lost his donkey, had no recourse but to ask himself what he would do if he were a jackass, and then do the same thing.”¹⁷ As the economist whom Keynes had chosen to exemplify classical theory, Pigou could tell Keynesians how a jackass thought.^18,11 Samuelson praised the book highly as “one of the most important books of recent years.”¹⁹ He thought the book's methodology was “almost ideal,” and it revealed com­mon ground between classical and Keynesian economics. The concluding words of the review were that Pigou's book “reveals with remarkable force the extent to which the Keynesians all along have been speaking classical prose, at the same time that ‘classicists' have thought in Keynesian poetry.” The review shows, more clearly than does his previous work with Hansen on the cycle, or his discussion of fiscal policy, how Samuelson was thinking about the Keynesian system in 1941.

The concepts of saving and investment were being widely discussed and were causing great confusion. Samuelson conceded that Pigou was correct in accepting the Keynesian definitions, according to which saving and invest­ment were defined as being equal.²⁰ However, there might still be disequilib­rium, in that the amount that households wished to save out of their income might not be the same as that which entrepreneurs wished to invest at the same level of income. Because it was necessary to deal with magnitudes that were not necessarily observed, he suggested that, rather than using the termi­nology advocated by the Swedish economists—of saying that ex ante saving and investment might differ even though ex post they had to be equal—the terminology of virtual and observable would seem more suitable.

The reason Samuelson thought the book almost ideal from a method­ological point of view was that it derived comparative statics results, and it focused attention on dynamic processes—methods that had been central to his dissertation, defended earlier in the year. Disequilibrium, or the “inap­propriateness” of the desired saving and investment, was what brought about change, something that was recognized in the “fruitful detailed

h. He used the jackass analogy when discussing his own education at Chicago (see chapter 5 this volume). An analogy he first used to establish Pigou's authority was later used to establish his own.

time-sequence analysis” of “Robertson, Kahn, J. M. Clark, Lundberg,” and, critically, “Keynes at an earlier stage.”²¹ Samuelson argued that there was no inconsistency in analyzing a short-run equilibrium, even though it was changing in the long run. To support this he alluded to the paradoxes of the ancient Greek philosopher Zeno that he had used previously in criti­cizing Keynes: just as an arrow moves even though at any time it is at a particular place in the air, so saving and investment could be equal and yet changing.²²

In early 1942, Samuelson followed up this review by writing a note in which he claimed that he and another reviewer, Nicholas Kaldor, had been right to argue that investment would depend on the level of employment.²³ He adopted the strategy he commonly used in such cases of starting with a simple model and generalizing it. The simplest model adopted an aggregate production function in which output was a function of labor and capital.^[38] The rate of interest was assumed to equal the marginal product of capital, which depended on employment and the capital stock. Investment (the growth rate of the capital stock) could be anything—the investment function was hori­zontal. He then followed Pigou in postulating different production functions for the production of consumption and investment goods. Pigou might be right to argue that as production of investment goods rose, the marginal pro­ductivity of capital in producing investment goods would fall, giving a nega­tive relationship between investment and the rate of interest. The problem with the argument was that it failed to take account of relationships between the two sectors.

Samuelson’s first conclusion was that Pigou's assumptions defined a system that was much more complicated than Pigou realized, and that these addi­tional complications added nothing substantial to his argument. However, there was a more interesting message. Pigou was presenting a classical system in which full employment could be achieved. Samuelson argued that high investment would lead to high employment and a high marginal product of capital. What he called “perfectionist diddling of the market interest rate” might ensure that precisely the right level of investment was undertaken to achieve full employment. However, even if monetary policy could achieve this perfect result, it was achieved by flouting “the tyranny of the strict accel­eration principle.” Samuelson was implicitly using a non-Keynesian concept, taken from traditional American business cycle theory, to criticize Pigou’s classical theory.

As in his earlier writings, Samuelson was critical of Keynes. Not only had Keynes abandoned the dynamic analysis of his earlier work, which was important for deriving operational theorems, but he had also accepted too uncritically the law of diminishing marginal productivity—the idea that when employment increased, the wage employers would be prepared to pay would go down. However, he also found fault with Pigou. Perhaps the most important problem was that he was skeptical of Pigou's assumption that the central bank could determine the level of money income: cheap money might fail to raise demand, either because it might be impossible to lower interest rates or because investment did not respond to changes in the rate of inter­est. The message of his later note was that American business cycle theory, represented by Hansen, had focused on important relationships that neither Keynes nor Pigou understood.