Those to whom the doctrine of multiplier seems an alien morsel in the corpus of economic doctrine should remember that it is merely a disguised form of the ordinary supply schedule of free capital, but with the level of income treated as a variable.
— (Harrod Reference Harrod1937, pp. 77–8)I. INTRODUCTION
The correspondence between John Maynard Keynes and Roy Harrod regarding the classical theory of the rate of interest demonstrates their difficulty in coming to a common understanding on the interdependence of markets. In the original version of Chapter 13 of the General Theory, Keynes writes that the Marshallian theory of the rate of interest “makes no sense.”Footnote 1 For Keynes, the theory of the multiplier, which explains the level of equilibrium in the goods market, led to the questioning of the coherence of the classical theory of the market for financial assets. To which Harrod immediately responded:
The view that I object to lies in the argument that because saving must always and necessarily equal net investment (which I accept) there is “no sense” in the view that interest is a price which equates the demand for saving in the shape of investment to the supply which results from the community’s propensity to save (Harrod, 1 August 1935, in Keynes CW Vol. XIII, p. 530).
At stake in the controversy was the coherence of the classical theory of interest rate, without supposing full employment. According to Harrod, the multiplier theory allows one to enrich the classical theory of interest rate, but does not contradict it. Harrod understood that if the multiplier is at work, then we are short of one equation to determine the interest rate. He agreed that Marshallian theory explains the level of interest rate only if the level of income is assumed to be given;Footnote 2 therefore, the level of interest rate is indeterminate if the level of income is variable. He agreed with the positive contribution of Keynes, that it is necessary to introduce the liquidity preference for the determination of the levels of both the income and the interest rate. As far as Harrod was concerned, the classical theory was certainly incomplete, but it was not incoherent. Keynes’s inference is the opposite. For Keynes, the classical theory makes no sense if it does not succeed in explaining the level of interest in a world without full employment. He refused to interpret his own theory as complementary to this theory.
This paper shows that Keynes’s criticism of classical theory of interest rate is based on his multiplier theory and is separated from his own theory of interest rate based on liquidity preference analysis.Footnote 3, Footnote 4 Our study attempts to demonstrate the relevance of Keyne’s criticism and throws new light on its significance.Footnote 5
O’Donnell (Reference O’Donnell1999) and Besomi (Reference Besomi2000) have convincingly argued that it was Harrod who originally suggested to Keynes to draw a diagram with a family of savings curves in order to think about the Marshallian theory of interest rate in a variable income regimeFootnote 6—a diagram which prefigures the “only diagram in the General Theory” where Keynes demonstrates that the equilibrium of financial market is indeterminate once the level of income varies with the level of investment. But neither O’Donnell nor Besomi has analyzed the reason why Keynes was able to utilize Harrod’s suggestion against Harrod’s own goal. The present study reveals that Harrod had not grasped the ensemble of characteristics of interdependence between markets that are at work in the General Theory.
In sections 2 and 3, we draw upon the aggregation rule of budget constraints to throw new light on the Harrod-Keynes controversy by comparing the classical and Keynesian analyses of interdependency between the markets for goods and for financial assets. In sections 4 and 5, we then show that the criticism of the classical theory of the rate of interest formulated by Keynes is valid once the level of income is not given; and that it does not assume adherence to the theory of liquidity preference. Furthermore, in section 6, our analysis emphasizes that this critique presupposes equilibrium in all markets with the exception of the labor market. In section 7, we show that Harrod did not completely grasp its significance, which explains both a lacuna in Harrod’s modelling of the General Theory (1937), and the fact that he did not see that the introduction of the market for cash balances was insufficient to determine the rate of interest. In section 8, we introduce Hicks (Reference Hicks1937) to clarify the conditions that must hold for Keynesian criticism of the classical theory of interest rate to be valid. In section 9, we infer that if there is relevance in Keynes’ criticism of the classical theory, then a similar criticism can be directed at Keynes’s own theory. At the first instance this seems paradoxical. Our inquiry reveals that the indeterminacy of the equilibrium in a classical model where the multiplier is at work is a special case of possible indeterminacy of equilibrium in the Keynesian model. This gives a new perspective to the controversy, and the mutual misunderstanding between Harrod and Keynes. Section 10 contains some brief concluding remarks.
II. THE MARSHALLIAN THEORY OF THE RATE OF INTEREST, THE MARKET FOR GOODS, AND THE FINANCIAL MARKET
Although presented as partial equilibrium analysis, the Marshallian theory of the rate of interest involves two markets. One of these is of course the market for financial assets, but the theory also involves, symmetrically, the market for goods. Indeed, if the rate of interest, which equilibrates savings and investment, secures the mutual adjustment of the demand for financial assets (savings) and the supply of financial assets (investment), then it also governs the distribution of production—the supply of goods—between consumption goods and investment goods in satisfying the demand for these two respective types of goods. This little-noticed characteristic of the Marshallian analysis of the rate of interest can be highlighted by introducing Say’s Law.
The equations of equilibrium in the markets for goods and financial assets are
where c is real consumption; 
the fixed level of income; R the long-term rate of interest; i real investment; T d, 
and T s are respectively the demand for, the initial stock of, and the supply of financial assets.
We have a system of two equations for one variable, the rate of interest R, for which there is a solution since the two equations are linearly dependent. Equation (2) is equal to Equation (1) multiplied by the interest rate R.
Indeed, the two equations for equilibrium in the markets for goods and financial assets are linked by Say’s Law I below and it is obtained by aggregating the budget constraints of firms and households. We speak of “Say’s Law I” because in the following pages we will refer to various forms of Say’s Law, which relate to different hypotheses: fixed or variable income; lack or presence of the labor market:
![$$\eqalign{ & {\rm Firms} : i(R) = {1 \over R}T^s (R) \cr & {\rm Households} : c(\bar y,\;R) + {1 \over R}T^d (\bar y,\;R) = \bar y + {1 \over R}\bar T \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & {\rm Say’s \,\,law \,\,I} : \left[ {c(\bar y, R) + i(R) - \bar y} \right] + {1 \over R}\left[ {\left({T^d (\bar y, R) - \bar T} \right) - T^s (R)} \right] = 0 \cr}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160203113856959-0731:S1053837210000180_equ1.gif?pub-status=live)
where I assume that we have infinitly bonds assets
The rate of interest, i.e. the relative price of a current good and a future good, is therefore a simultaneously adjusting variable between the markets for financial assets and for (current) goods. Let us assume an initial equilibrium E 1 (see Figure 1 where the 45° slope line on the goods market represents the production possibility curve).
Figure 1. The classical interdependancy between financial and goods markets
When the marginal efficiency of capital rises, the demand for investment goods and the supply of financial assets increase (1). The rise of the rate of interest (2) stimulates savings; the demand for financial assets rises (moving along the curve T d) and the demand for consumption goods falls (3). At the new equilibrium E 2, the rate of interest has increased, the supply of and the demand for financial assets and the supply of investment goods have risen [(1) and (4)], while the demand for and supply of consumption goods have fallen.
III. THE MULTIPLIER VERSUS MARSHALLIAN THEORY OF INTEREST RATE
The theory of the multiplier developed in Chapter 6 of the General Theory overturns this analysis. If the marginal efficiency of capital is growing the resulting increase in demand for investment goods prompts a rise in the production of investment goods, and therefore, of distributed income, which in turn prompts an increase in the demand for consumption goods. Here income is a variable: 
The increase in income ceases once savings, representing a leakage from the mechanism whereby demand tracks income, are equal to investment. Here the variation of income ensures the adjustment of the supply of goods to their demand, and also the demand for financial assets to their supply. If one proposes, as Keynes does initially,Footnote 7 that savings, and therefore the demand for financial assets, depend only on income, then the contrast with Marshallian analysis is very striking. In this case, the equations for equilibrium in goods and financial markets are as follows:
Taking into account the linear dependency of the two equations, every level of income which secures equilibrium in the goods market likewise assures equilibrium in the financial market. Every level of the rate of interest is linked to a level of investment which generates a level of income such that the savings consequent upon this income are necessarily equal to investment. This result leads Keynes to maintain that the classical theory of the rate of interest “makes no sense”:
I still maintain that there is “no sense” in the view that interest is a price which equates saving and investment; or at any rate if one could invent a sense for it, it would be quite remote from anything intended by the classical theorists. Perhaps the clue is to be found where you allege that I am doing great violence to the accepted and familiar when I maintain that “two independent demand and supply functions won’t jointly determine price and quantities,” for my whole point is that the functions in question are not independent (Keynes to Harrod, 9 August 1935, in Keynes CW Vol. XIII, p. 538).
The Marshallian analysis of every market determines two unknowns (price and quantity transacted) via two independent functions of supply and demand. In the financial market the two functions of investment (the supply of financial assets) and saving (the demand for financial assets)—both of them functions of the rate of interest—determine the rate of interest and the quantity of financial assets issued (investment) and subscribed (savings). If, according to Keynes, this theory makes no sense, it is because “the functions in question are not independent” (ibid.) for, according to the analysis of the multiplier, the amount saved depends on the amount of income, which itself depends on the sum invested. Hence Keynes was of the view that it was necessary to abandon the classical theory of the rate of interest, and seek another one.Footnote 8
IV. THE MARSHALLIAN MODEL AND THE INDETERMINACY OF EQUILIBRIUM
Let us summarize. Marshallian theory is a partial equilibrium theory. One market is modelled and the assumption of ceteris paribus deals with all other markets and the variables of adjustment on those markets. In the particular market that is analyzed there are two variables: price and the quantities exchanged. But, by allowing for the interdependence of markets, the price in the financial market—here the inverse of the interest rate—also becomes the adjusting variable for a second market, that for goods. In addition, Keynes introduces income as another adjusting variable in the second market.
Furthermore, if income is an adjusting variable in the goods market, taking into account the linear dependence of the two equations,Footnote 9 then it also serves as the adjusting variable in the financial market; consequently, for Keynes, the rate of interest is not capable of playing this role. Hence the critique of the Marshallian construction. The correspondence with Harrod, to whom Keynes gave the entire first draft of his book for comment, shows how much Harrod wished to persuade Keynes to abandon his vehement criticism of classical theory. He thought that on this point Keynes was wrong and its publication would be detrimental for the dissemination of the positive theoretical content of the General Theory (Besomi Reference Besomi2000). But Harrod’s arguments had the opposite effect: they assured Keynes in his conviction that his critique was well-founded.
Discussion quickly moved on to the possibility of constructing a savings curve within a Marshallian perspective with varying levels of income as a function of the rate of interest, which could be independent of the investment curve that is itself a function of the interest rate. The intersection of the two curves would permit the equilibrium rate of interest to be determined. Having first noted that the construction of the savings curve presented no problems if income were fixed, which suited Keynes, Harrod suggested a way in which it might be constructed, assuming varying income levels. In his view, this would render the Marshallian and Keynesian analysis compatibleFootnote 10:
But generally when you draw a supply curve α = f(β), it is assumed that you are treating α as a function of a single variable, price, and other things including income were equal. That is the classical supply curve. To relate the classical supply curve to yours, you would have to draw a family of classical supply curves corresponding to different levels of income and to show that the value of each corresponding to a given rate of interest was identical with that of the demand curve, owing to the operation of the multiplier affecting the level of income. The value (i) of demand [investmentFootnote 11] for each different value of R (interest) makes (via the multiplier) income and that when you draw the classical supply curve for that level of income (i.e. schedule of saving propensity at various interest rates at that level of income) the value(s) of this supply curve [savingsFootnote 12] for the value of R used in computing that level of income is identical with the value (i) of the demand curve [investmentFootnote 13]. I have tried to put this in other words again in note at end (Harrod to Keynes, 30 August 1935, in Keynes CW Vol. XIII, p. 555).
By proposing to trace a family of supply curves which intersected with the demand curve, Harrod was seeking a three-dimensional Marshallian schemaFootnote 14 capable of determining three variables: the interest rate, income, and the amount of savings (and investment). Seeking to establish complementarities between the analyses of Alfred Marshall and of Keynes, Harrod here described savings as a simultaneous function of the rate of interest (as in Marshall) and of income (as in Keynes). In the note added to the end of the letter, which he refers to above, he reversed the causality between the variables and set up the same scenario by making the interest rate an unknown which is a function of savings and of incomeFootnote 15:
Let R 1, R 2 etc. be rates of interest and y 1, y 2 etc. incomes corresponding to them (y 1 being derived from R 1 via marginal efficiency of cap. and the multiplier). For each value of y draw a classical supply curve, of which each curve shows amount of saving corresponding to various values of R at a given level of y. Then according to you it will be found that the value of R at which the [supplyFootnote 16] curve appropriate to income y r intersects the demand curve is in fact R r , where R r represents any given rate of interest whatever. The so-called supply curve in the passage from your letter which I have quoted Footnote 17 is the locus of points on the classical supply curves for that value of R corresponding to the level of income on the assumption of which each was drawn (Harrod to Keynes, 30 August 1935, in Keynes CW Vol. XIII, P. 556–7).
Keynes wrote in the margin of this letter: “the demand curve doesn’t intersect at all. It lies along it throughout its length.”Footnote 18 Keynes is right here, since each of the points of the classical (savings) supply curve that Harrod constructs is also on the demand curve (investment); the two curves are so coincident that they do not intersect at one point,Footnote 19 but intersect at every point. Equilibrium is indeterminate. This was what Keynes had already argued in his letter of 27 August, and which Harrod had acknowledged in his answer:
Without bringing in liquidity preference the position of equilibrium is entirely indeterminate.Footnote 20 I agree. And when I am seduced by the view that you give liquidity preference too much work, the view that I would then fall back on is that equilib. is indeterminate! (Harrod to Keynes, 30 August 1935, in Keynes CW Vol. XIII, p. 554).
This resulting indeterminacy of the equilibrium becomes evident when one writes Say’s Law I bis, obtained by aggregating the budget constraints under the assumption of variable incomeFootnote 21:
![$$\eqalign{ & {\rm Firms} : i(R) = {1 \over R}T^s (R) \cr & {\rm Households} : c(y, R) + {1 \over R}T^d (y, R) = y + {1 \over R}\bar T \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & {\rm Say’s \,\,law\,\, I\,\, bis} : \left[ {c(y, R) + i(R) - y} \right] + {1 \over R}\left[ {\left({T^d (y, R) - \bar T} \right) - T^s (R)} \right] = 0 \cr}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160203113856959-0731:S1053837210000180_equ2.gif?pub-status=live)
If there are two unknowns (y and R), there is only one independent equation:
![$$\left[ {\left({T^d (y, R) - \bar T} \right) - T^s (R)} \right] = R.\left[ {y - \left({c(y, R) + i(R)} \right)} \right]$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160203113856959-0731:S1053837210000180_equ3.gif?pub-status=live){kind=small}
The supply of and demand for financial assets match in the same way that the demand for and supply of goods do. The equilibrium conditions are identical in the two markets. One equation is not sufficient to determine two unknowns: y and R. Equilibrium here is indeterminate. Referring to the macroeconomic schema used by Hansen (Reference Hansen1953), one would say that in the plane (y, R)Footnote 22 the curves i–s and T–T, which describe respectively the equilibrium conditions in the goods and financial markets, are superimposed: if there is equality in the demand for and supply of goods, there is equality in the supply of and demand for financial assets. Since there is an infinite number of combinations of y and R which secure equilibrium in the goods market, they simultaneously realize equilibrium in the financial market.Footnote 23 Harrod’s suggestion was aimed at the construction of a curve for the supply of savings which turned out to be entangled with the investment curve, leading to two “Marshallian” curves having the same properties. The superposition of the two “Marshallian” curves noticed by Keynes highlights the indeterminacy of equilibrium.Footnote 24
V. INDETERMINACY OF EQUILIBRIUM AND THEORICAL INCOHERENCE
The thoughts of Keynes and Harrod converge with regard to the indeterminacy of equilibrium, but diverge on its interpretation. According to Keynes, it indicates that the classical theory is incoherent, whereas for Harrod it is incomplete but not incoherent. Recall that the two authors have removed the classical tacit assumption of given income.Footnote 25
For Keynes, as soon as income is variable, the classical theory “makes no sense” because it cannot explain the equilibrium level of interest rate. This is the subject of the sole diagram of the General Theory.Footnote 26 There are two curves:
• the supply curve of financial assets which is a decreasing function of the rate of interest, and
• the demand curve for financial assets which is an increasing function of the rate of interest,
where the latter depends on the level of income. Assume the existence of an initial equilibrium and suppose a change in the marginal efficiency of capital. The supply curve shifts, prompting a shift of the demand curve by an unknown amount.Footnote 27 As Keynes explained:
Now if the investment demand-schedule [i.e. financial assets’ supply scheduleFootnote 28] shifts, income will, in general, shift also. But the above diagram does not contain data to tell us what its new value will be; and, therefore, not knowing which is the appropriate Y-curve [i.e. financial assets’ demand scheduleFootnote 29], we do not know at what point the new investment demand-schedule [i.e. financial assets’ supply scheduleFootnote 30] will cut it
(Keynes Reference Keynes and Moggridge1936, p. 181).We do not have the two independent functions necessary to determine the two unknowns: savings (and investment) on the one hand, the interest rate on the other. The equilibrium is indeterminate since, as we have seen, at each point of the supply curve for financial assets one is also at a point on the demand curve for financial assets [Y curve] which is related, via income, to this level of investment. Every point on the supply curve for financial assets [the investment demand schedule] is an equilibrium. In the very first version of Chapter 14, Keynes wrote:
… whereas it is perfectly easy to name a price at which the supply and the demand for a commodity would be unequal, it is impossible to name a rate of interest at which the amount of saving and the amount of investment could be unequal (Keynes 1935 in CW Vol. XIV, p. 476).
In the letter to Harrod on 27 August, he had already come to the conclusion that “savings and investment are the same things” (Keynes 1935 in CW Vol. XIII, p. 560). Recall again that the level of income is variable. As we have seen in the previous section, the investment curve and the Marshallian saving curve suggested by Harrod are superimposed.
However, for Harrod, classical theory, as far as it presupposed full employment—therefore given incomeFootnote 31—remains coherent. In his eyes, Keynes’ rejection of the classical theory of employment should not involve a rejection of classical theory in its entirety. The introduction of the multiplier into the analysis of the interdependence of markets does not lead to the rejection of classical market theory—it rather widens its horizon. Harrod repeated this argument in his 1937 article. Before dealing with this, it will be useful to introduce some more precise details concerning the interdependence of the markets. This controversy has to do with the violation of Say’s Law in the case of the missing labor market.
VI. SAY’S LAW AND THE LABOR MARKET
Harrod does not discuss Keynes’s thesis by adding the labor market to markets for goods and financial assets with a view to the analysis of market interdependence. But if one does add the labor market, Say’s Law states that the markets for goods and financial assets can be in equilibrium only if the labor market is in equilibrium. The outcome of this is that there is only one possible equilibrium situation for goods and financial markets—not an infinity of them. In this case, the interest rate and income are perfectly determined by classical theory. Indeed, the aggregation of budget constraints, taking account of the supply of and demand for labor, gives Say’s Law II:
![$$\eqalign{ & {\rm Firms} : i(R) + wN^d (w) = y + {1 \over R}T^s (R) \cr & {\rm Households} : c\left(\,\,{f(N^s ), R} \right) + {1 \over R}T^d \left(\,\,{f(N^s ), R} \right) = wN^s (w) + {1 \over R}\bar T \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & {\rm Say’s \,\,law \,\,II} : \left[ {c\left({f(N^s ), R} \right) + i(R) - y} \right] + {1 \over R}\left[ {\left({T^d \left(\,\,{f(N^s ), R} \right) - \bar T} \right) - T^s (R)} \right] \cr\hskip++12pt+ w\left[ {N^d (w) - N^s (w)} \right] = 0 \cr}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20160709215039-36662-mediumThumb-S1053837210000180_equ4.jpg?pub-status=live)
where w designates the real wage rate, N s the supply of labor, N d the demand for labor; and y = f (N d).
We have three interdependent conditions for equilibrium:
(i) c(f(Ns), R) + i(R) = y
(ii)
(iii) Nd (w) = Ns (w)
Here, however, the model is recursive via the equation for labor market equilibrium. Indeed, equation (3) determines the equilibrium rate of real wages w*, hence the level of employment N*, hence the level of income y* = f(N*). The equation for the goods market, or that for the financial market, now contains only one unknown: the rate of interest. The equilibrium rate R* is therefore determined.Footnote 32, Footnote 33
Keynes’ contradictory result is owed to the absence of the labor market from his analysis of simultaneous equilibrium in different markets; an absence which is not questioned by Harrod who shares the idea that household demand is constrained by their effective supply of labor 
and not by their notional supply N s. We have :
where u designates unemployment. The effective supply of labor 
is equal to, because it is determined by, the effective demand for labor 
from firms; firms that are constrained both by the level of real wages (first postulate) and by the outlets that they have for their products ![$[f(\hat N^d) = c + i]$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160203113856959-0731:S1053837210000180_eqi8.gif?pub-status=live){kind=small}
.
The reader will verify that the equilibrium condition for the labor market disappears by substituting the effective demand for and supply of labor (
and 
) for the notional demand and supply (Nd and Ns) in the budget constraintsFootnote 34 above; he will obtain Say’s Law I bis (see section 4, p. 12, above).
Thus, introducing the effective supply of labor (rejecting the second postulate) renders the equilibrium indeterminate and this underpins Keynes’s criticism of the classical theory of the interest rate. Such criticism appears coherent and does not depend, contrary to Milgate’s contention (1977), on the theory of liquidity preference. The argument of Chapter 14 does not presuppose the conclusions of Chapter 15, but those of chapters 2 and 6Footnote 35. From this point of view, Keynes’s criticism of classical theory, which relies on the multiplier theory and contests what Carabelli (Reference Carabelli, Bateman and Davis1991, Reference Carabelli and Vecchi2004, and Reference Carabelli and Marcuzzo2006) calls the classical “tacit assumption of independence,” appears to be an external criticism, not an internal one. We have yet to see how Chapter 15 solves the problem of the indeterminacy of equilibrium highlighted in chapter 14.
VII. HARROD’S INTERPRETATION
As in his 1935 correspondence with Keynes and Robertson,Footnote 36 in his 1937 Econometrica article “Keynes and Traditional Theory,” or in his 1951 biography of Keynes (1951: 453), Harrod consistently argued that it was not really justifiable to refer to the classical theory of interest rate as incoherent, and thought that Keynes was wrong to reject it. Harrod (Reference Harrod1937) sought to present Keynes’ theory in such a way that it could be seen as complementary to what he called traditional theory. This is presented in two equations:
• The first is the equation for the demand for capital ρ = f 1(i) . It is deduced from the decreasing relationship between the productivity of capital and the level i of investment.
• The second is the equation for the supply of capital: s = f 2(R). It indicates that the amount of savings is an increasing function of the rate of interest.
A priori, there seem to be four unknowns: ρ, i, s, and R. However:
• Harrod on the one hand also writes the equation of equality between the interest rate R and the marginal productivity of capital ρ, such that R = ρ. This allows us to treat, in the equation for the demand for capital, ρ as the rate of interest.Footnote 37 To make the remainder of the article easier to follow we retain the roman letter R; the equation for the demand for capital ρ = f 1(i) becomes R = f 1(i).
• On the other hand, Harrod continually substitutes i for s, and s for i;Footnote 38 one therefore has the implicit equation s = i which eliminates the unknown i. The equation for the demand for capital R = f 1(i) becomes R = f 1(s).
In the end, the Traditional Theory contains two equations [R = f 1(s) and s = f 2(R)] for two unknowns: s and R.
Thus [in the Traditional TheoryFootnote 39] there are two unknowns, the rate of interest and the volume of saving, and sufficient equations to determine them. …
This treatment of interest and saving is analogous to that of the price of a particular article and the amount of it produced. The treatment depends on the short-cut assumption of ceteris paribus. … Among the “other things” which are supposed to be “equal” is the level of income in the community under discussion
(Harrod Reference Harrod1937, p. 240).With the two equations (1) and (2) for two unknowns, classical theory determines in a coherent manner (according to Harrod) the amount of savings (and of investment) as well as the rate of interest. But it assumes a given level of income—an assumption that, according to Harrod, Keynes removed.
I suggest that the most important single point in Keynes’s analysis is the view that it is illegitimate to assume that the level of income in the community is independent of the amount of investment decided upon. No results achieved by the short-cut of such an assumption can be of any value
(Harrod Reference Harrod1937, p. 240).According to Harrod, if one argues in terms of variable income, the two equations (1) and (2) coherently determine the amount of savings, but neither the savings rate s/y, nor, a fortiori, the level of income y. To determine this third unknown, in Harrod’s view, we have to augment Traditional Theory by modifying equation (2). Whence comes the introduction of income y in f 2a (see the table above) which is transformed into f 2b:
In this form the second equation shows itself as the doctrine of the multiplier. The multiplier is the reciprocal of the fraction expressing the proportion of any given income, which, at a given rate of interest, people consume
(Harrod Reference Harrod1937, p. 241).Having done this, a third unknown is introduced and a third equation appears to be necessary:
Meanwhile, since there are three unknowns and but two equations in the saving/interest complex, another equation is needed
(Harrod Reference Harrod1937, p. 241).Harrod writes the function f 3 for liquidity preference [equation (3)], encouraged by Keynes’s theory, where the interest rate is an increasing function of y and a decreasing function of m, which gives three equations for three unknowns: s, R, and y. In Harrod’s view, Keynes completes classical theory. Neither the one nor the other is incoherent. Both contain the equations necessary to determine the variables of the theory. There are two equations for two variables (s and R) in the Marshallian model; Keynesian theory includes a supplementary equation (liquidity preference) since there is one more variable (y). It is not because the Marshallian model does not determine the three variables of the Keynesian model that it is incoherent.
But Harrod is wrong to think that the three equations of the Keynesian model are sufficient. There is a hidden variable. Harrod did not include a fourth variable, the price level P, which is however necessarily introduced at the same time as money in equation f 3. His exposition is confused on this point. He suggests that the variable P is integrated in y, which means that y designates nominal, and not real, income:
Ought not the price level to come in also? That may be taken to be subsumed under y, the level of income, in a manner that I shall presently explain
(Harrod Reference Harrod1937, p. 242).This implies that the quantity of money m corresponds to the nominal quantity of money M, and not to the real quantity, M/P. However, instead of clarifying how the price level comes into equation (3), the text that follows explains that the quantity of money available for speculative motives falls as income rises, as a result of which the interest rate increases as income rises. Then he concludes:
We now have three equations to determine the value of three unknowns, level of income, volume of saving (= volume of investment) and rate of interest (marginal productivity of capital)
(Harrod Reference Harrod1937, p. 243).This is evidently mistaken, since the variable y in equations (1) and (2) is different from the variable y of equation (3). In the first two equations we have real income; in the third equation we have nominal income. In fact the model contains four variables, and not three: real investment, interest rate, real income, and price level. We are short of one equation. Harrod’s three-equation system becomes coherent once again if we suppose that y corresponds to real income in each of the three equations, and that m designates not the nominal quantity of money M, but the real quantity of money, which would be m = M/P. This explicitly raises the question of the determination of the price level P. Harrod sees the difficulty, which he touches on a few pages later:
But if the m in R = f 3(y, m) is indeterminate, there are too many unknowns in the interest/savings set of equations. Thus it is necessary to the validity of Keynes’s solution of the problem of investment and interest that the amount of money available for liquid reserves should be determinate, and that involves that the price level should be determined otherwise than by the monetary equation. And so in Keynes’s system it is
(Harrod Reference Harrod1937, pp. 248-9). But Harrod is mistaken about Keynes’s solution to this problem. He never once mentions the level of exogenous monetary wages 
. Instead, he takes us through a vague general account of the determination of the price level P:
The matter may be put thus: The saving/interest equations suffice to determine the level of activity, subject to the proviso that the quantity of money which appears in the liquidity preference equation is a known quantity; and this will be known if the price level and therefore the amount absorbed in active trade is known. The equations in the general field suffice to determine the price level, subject to the proviso that the level of activity is known. Thus there is after all mutual dependency. The level of activity will be such that so much money is absorbed in active trade that the amount left over enables interest to stand at a rate consistent with that level of activity
(Harrod Reference Harrod1937, pp. 248–9).And he concludes:
The mutual interdependency of the whole system remains, but the short-cuts indispensable to thinking about particular problems, as Keynes has carved them, remain also
(Harrod Reference Harrod1937, pp. 248–9).Harrod relies on the interdependence of markets to determine the price level, but does not get around to writing the missing equation necessary for his system to describe this interdependence in a satisfactory way. His modelling of the General Theory is therefore defective. While he appreciated that Keynes had raised the question of the interdependence of goods and financial markets, he did not grasp all the consequences of the fact that Keynes’ criticism of the classical theory of the interest rate was based on the absence of the labor market in the definition of general equilibrium. He did not see that if the equilibrium is undetermined in Marshallian analysis as soon as the multiplier theory is introduced, the same type of indeterminacy was implied by the modelling of Keynesian analysis that he proposed. The attempt by Harrod to demonstrate that Keynes had generalized Marshallian theory therefore fails. But this attempt did influence Hicks’ article (1937).Footnote 40
VIII. HICKS’S SOLUTION
Hicks maintains that the Classical Theory is a particular case of a General Theory which likewise had another particular case—which he called “Mr. Keynes’s Special Theory”—where the rate of interest depends exclusively on liquidity preference as a motivation for speculation. The idea of a break between classical analysis and Keynes is once more disputed. While pursuing the same object as Harrod, Hicks abandons the assumption of fixed income in presenting the Classical Theory. And if he succeeds where Harrod fails, it is because he introduces into this classical theory a crucial supplementary assumption, also presented in Keynes’ book, regarding the level of exogenous monetary wages 
. This assumption permits, by resorting to the first postulate, the introduction of the following equation determining the price level:
This additional equation (d) is inserted to equations (a) and (c) in the table below which represent equilibrium conditions in the markets for goods and for money (cash balances).
where 
designates the rate of exogenous monetary wages; P the general price level; 
the quantity of nominal money (= the money supply); and L the nominal demand for money.
The absence of equation (b) representing the financial market is without a priori importance inasmuch as it is linearly dependent on equations (a) and (c).Footnote 41 In the table above one should note that the two special cases (Classical Theory and Keynes’s Special Theory) and the general case (the General Theory) differ in equation (a) and (c) for the market equilibrium in goods and money, but not in equations (d). We restrict our analysis to the Classical Theory.
It seems that in the Classical Theory as presented by Hicks:
• The equilibrium condition for the goods market is written in nominal terms. This presence of price mathematically links equation (a) to equations (c) and (d). Subject to this link is not falling afoul of the criticism by Lange and Patinkin of the classical dichotomy.Footnote 42
• The labor market is absent.Footnote 43 Monetary wages are not endogenous.
• Taking account of the dichotomy, the equation for the financial market (not written) maps on to the equation for the goods market.
Mathematically, we have three equations for three unknowns. Contrary to Harrod’s analysis, the indeterminacy of equilibrium is here resolved. With respect to the monetary equation (c), k being given, the quantity 
of exogenous money determines the nominal income P.y [i.e. the position of the hyperbolic equilateral of the equation 
in the plane (P, y)]. One can then indifferently:
• substitute the value of y for P obtained in the equation (c) for y in equation (d), which gives the price level P. Equation (c) then gives the level of income y, and equation (a) then gives the level of interest rate R; or
• substitute the value of P for y obtained in equation (c) for P in equation (d), which gives the level of income y. Equation (c) then gives the price level P, and equation (a) then gives the interest rate R Footnote 44.
The graphic resolution is shown in Figure 2.Footnote 45
Figure 2. Hicks’s solution
Without equation (d), it is clear that equilibrium is indeterminate in the Classical Theory presented by Hicks:
• Equation (a) stripped of the variable P is incapable of determining two variables (y and R).
• The system of the two equations (a) and (c), from which one cannot exclude the variable P, is not sufficient to determine three variables (y, R, and P).
Keynes’ criticism of classical theory is therefore well-founded. However, it seems that the very same criticism can be directed at “Mr. Keynes’s Special Theory” and at the “General Theory”: the equations (a) and (c) are not sufficient to determine three variables: y, R, and P. Equation (d) is needed. In all three cases the solution which resolves the indeterminacy is identical: an assumption is made concerning the level of exogenous money wages. If it is thought that the solution applies to “Keynes’s Special Theory,” it must also apply to the “Classical Theory.” Here of course it is a question of “classical theory” in the very particular interpretation of Keynes, Harrod, and Hicks, i.e. a theory in which there would be market equilibrium in all markets with the exception of the labor market!
IX. MARKET FOR CASH BALANCES, LABOR MARKET, AND WALRAS’S LAWFootnote 46
What the preceding highlights is that the introduction of the market for cash balances is not enough to resolve the indeterminacy of the interest rate, and hence income, in the classical theory as understood by Keynes (or by Harrod, or by Hicks). This is because an additional variable, the price level, is introduced together with the market for cash balances. There is still one equation missing. It is a characteristic feature of Keynesian macroeconomics: the aggregation of budget constraints generates as many equilibrium conditions for markets as there are variables. But while these equilibrium conditions are linearly dependent, there are n – 1 independent equations with n variables.Footnote 47 In this respect, the indeterminacy of equilibrium in the classical model governed by Say’s Law I bis (above, p. 12) is a particular case of the indeterminacy of equilibrium in the Keynesian model governed by Walras’s Law I (below). The latter is obtained by aggregating the budget constraints of the Keynesian model, i.e. by taking account of the effective supply of and demand for labor (and not the notional supply and demand).
![$$\eqalign{ & {\rm Firms} : i(R) + w\hat N^d (y,w) \,=\, y + {1 \over P}{1 \over R}T^s (R) \cr & {\rm Households} : c\left({y, R} \right) + {1 \over P}L(y, R) + {1 \over P}{1 \over R}T^d \left({y, R} \right) \,=\, w\left({N^s (w) - u} \right) + {1 \over P}\bar M + {1 \over P}{1 \over R}\bar T \cr & {\rm where} \,\,w\, = \,\,{W \over P} , u\, = \,N^s (w) - \hat N^d (y,w)\, {\rm and} \,\,y \,=\, f\left({\hat N^d } \right) \,=\, \,f\left({N^s (w) - u} \right) \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & {\rm Walras’s \,\,law \,\,I} : \left[ {c\left({y, R} \right) + i(R) - y} \right] + {1 \over P}{1 \over R}\left[ {\left({T^d \left({y, R} \right) - \bar T} \right) - T^s (R)} \right] \cr\hskip12pt+ {1 \over P}\left[ {L(y, R) - \bar M} \right] = 0 \cr}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary-alt:20160709090502-91133-mediumThumb-S1053837210000180_equ7.jpg?pub-status=live)
We have three linearly dependent equations (is – TT – LM) for three unknowns: y, R, and P. The equilibrium is undetermined. To resolve this indeterminacy, we need to add one more equation to the market equations. Hence the equation (d).
We can note that the labor market equation disappears with the aggregation of budget constraints, since the demand of households and firms is constrained by their effective, not their notional, supply. The reader can easily verify this by substituting notional for effective labor demand and supply in the constraints of households and firms, obtaining Walras’s Law II as below:
![$$\eqalign{ & {\rm Walras’s \,\,Law \,\,II} : \cr & \left[ {c\left({y, R} \right) + i(R) - y} \right] + {1 \over P}{1 \over R}\left[ {\left({T^d \left({y, R} \right) - \bar T} \right) - T^s (R)} \right] + {1 \over P}\left[ {L(y, R) - \bar M} \right] \cr\hskip12pt+ {1 \over P}W\left[ {N^d (w) - N^s (w)} \right] \,=\, 0 \cr}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160203113856959-0731:S1053837210000180_equ8.gif?pub-status=live)
The reader will verify that the model is recursive with respect to the labor market and that the indeterminacy of equilibrium has been resolved without a supplementary equation!
X. IN CONCLUSION
The classical theory criticized by Keynes in Chapter 14 of his General Theory is not the classical theory of Chapters 1 and 2, “… which dominates the economic thought, both practical and theoretical, of the governing and academic classes of this generation, as it has for a hundred years past” (Keynes Reference Keynes and Moggridge1936, p. 3). In Chapter 14 Keynes’s criticism bears on only one aspect of the dichotomy inherent in classical analysis—the determination of the interest rate by the real forces of investment and saving. But the theory that he presents and criticizes has a strong scent of Keynesianism:
• In the first place, the model is not recursive with respect to the labor market, whereas the classical theory presented in Chapters 1 and 2 of the General Theory is. In the classical theory criticized in Chapter 14, the goods and financial markets can be in equilibrium while the labor market is not.
• In the second place, the condition for equilibrium in the financial market is linked to the condition for equilibrium in the goods market which, itself, is given by the theory of the multiplier outlined in Chapter 6.
The classical theory criticized by Keynes in Chapter 14 can be summarized in a system of two equations with two unknowns:
These two equations are linearly dependent: they are linked by Say’s Law I bis (above, p. 12) obtained by aggregating the “Keynesian” budget constraints of the real choices of firms and households.Footnote 48 Two linearly dependent equations cannot determine two variables, so it seems that equilibrium is undetermined; or more exactly, there are infinitely many equilibria. Although Keynes did not express it in this way, he saw this but did not succeed in convincing Harrod that if one accepted (Chapter 6) that the goods market determines income as a function of the interest rate, then traditional theory, according to which the supply (saving) and demand (investment) of capital determines the interest rate, made no sense. Harrod saw that Keynes made use of the interdependence of the markets, but he interpreted this as an extension of Marshallian analysis and did not grasp all of the consequences. The gaps in his modelling of Keynesian theory serve to illustrate this. Harrod did not see the consequences for a general equilibrium model of the modifications introduced by Keynesian theory to the establishment of budget constraints; he counted the equations but did not take sufficient account of the Say’s and Walras’s Laws. He did not grasp all of the properties of Keynesian equilibrium, i.e. a general equilibrium of markets excluding the labor market. Ultimately, he did not perceive the importance of the assumption that the rate of money wages was exogenous for removing the indeterminacy of equilibrium. Basically, Harrod did not understand that the indeterminacy of equilibrium in the “classical theory” criticized by Keynes in Chapter 14 of the General Theory was not due to the absence of the market for cash balances, but to the absence of the labor market.
Hicks modelled the classical theory as a special case of a general theory where the labor market was missing. If, in this classical case, the dichotomy remains, the equilibrium condition of the goods market is nonetheless expressed in monetary terms: one has P.i(R) = S(P.y, R) and not i(R) = s(y, R); that is to say, an equation for three unknowns (P, y, R). Hicks introduces the price level into the function g 1, which becomes the function h 1 (see below); he adds the equation h 3 for market for cash balances equilibrium and the equation h 4 which equates the rate of exogenous money wages with the marginal productivity of labor expressed in money. On the other hand, he does not write the equation h 2(P, y, R) = 0 for financial market equilibrium, which, in the classical case, is linearly dependent on the equation h 1. A system of three independent equations for three unknowns is obtained:
Hicks therefore presented a classical theory in which, leaving aside the Lange-Patinkin criticism of the classical dichotomy, equilibrium is not indeterminate.
It is plain that in the absence of equation h 4, the system of equations h 1 and h 3 has no unique solution; something that also applies to the “General Theory” presented by Hicks as well as for the other special case, “Mr. Keynes’s special case.” This is because of the absence of the equilibrium condition for the labor market, or of the assumption by which firms and households are constrained by their effective supply, and not by their notional supply. Hence the need to introduce the rate of exogenous monetary wages to achieve a solution.
The criticism made by Keynes of the “classical theory of the rate of interest” appears thus analytically well-founded, but directed to a special case of his own theory. This is where the equilibrium equations for the goods and financial markets are expressed in real terms,Footnote 49 where they are independent of the equilibrium conditions of the market for cash balances, and where the assumption of a given level of exogenous money wage rate is therefore without effect.
