Those who are strongly wedded to what I shall call “the classical theory”, will fluctuate, I expect, between a belief that I am quite wrong and a belief that I am saying nothing new. It is for others to determine if either of these or the third alternative is right. (Îšeynes, General Theory, p. v)
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It is usually considered as one of the most important achievements of the Keynesian theory that it explains the consistency of economic equilibrium with the presence of involuntary unemployment. It is, however, not sufficiently recognized that, except in a limiting case to be considered later, this result is due entirely to the assumption of “rigid wages” and not to the Keynesian liquidity preference. (Modigliani, 1944, p. 65)
Many economists, soon after the publication of the General Theory (1936), set out to formulate and, at the same time, to clarify the difficult and often confusing content of the book. Among the first models that were specified was that of John Hicks (1937 and 1983), which was to constitute the backbone of what today came to be known as macroeconomics.  In his article Hicks sought to express the central propositions of the General Theory in terms of equations and graphs in the effort to illuminate the relation between the theory of effective demand and liquidity preference. Furthermore, Hicks clarified these relations with the aid of two curves the SI and the LL, which later became known as the IS-LM curves. Hicks’s model became particularly popular in the US through the work of Paul Samuelson (1948) initially and subsequently through Alvin Hansen (1953). These two economists contributed more than anybody else to the popularisation of the Keynesian analysis and way of thought. The IS-LM conceptual apparatus has displayed remarkable longevity and resilience to various critiques and since the late fifties or early sixties continues to be part of the formal education of economists. At the same time, the IS-LM model plays a significant role by virtue of the fact that macroeconomic analyses, regardless of the approach, are cast to a great extent, in terms of the IS-LM representation of the economy. This is not to imply that the IS-LM model is without its problems; on the contrary, many economists expressed scepticism on the validity of the IS-LM as a representation of the General Theory and the way in which the economy works. 
In what follows we present and evaluatse the Hicksian IS-LM model and continue with Keynes’s reaction to the Hicksian restatement of the General Theory. Next, we introduce Modigliani’s version of the Keynesian model and the chapter ends with some concluding remarks.
11.2 Hicks’s Analysis of IS-LM
Hicks’s analysis focuses on the relation between savings and investment and seeks to establish the simultaneous determination of income and the rate of interest in both the real and monetary economy. According to Keynes’s analysis in General Theory income constitutes the principal variable in his analysis; nevertheless one would continue to be in the spirit of Keynes by considering the important role of the rate of interest. Thus, Hicks argued that investment (Î™ ) is a function of the rate of interest (i ) and also income (Y). Formally, we write the following function:
Similarly, for the saving function (S ), we have:
The equilibrium condition is:
From the above equality, we derive the following particular functional formalisation, which is called IS and it is defined as the locus of points that determine a relation between the rate of interest and the level of income, when investment and savings are equal to each other. The IS curved is formed in the way we show in Figure 1, where we have the saving and the investment functions for each income level.
Figure 1. Equilibrium in the Goods Market and the IS Curve
Let us suppose that we are in an initial equilibrium point such as A and let us further suppose that income increases from Y1 to Î¥2. It follows that the savings and investment schedules-both have positive their first derivatives with respect to income-shift to the right and their intersection at point Î’ determines the new equilibrium point. It is important to stress that the savings function is much more sensitive to variations in income, and therefore it shifts to the right by more than the investment function.  The two equilibrium points (i1, Y1) and (i2, Y2) are portrayed in Figure 1(b). In a similar fashion, we generate a series of such points, which when connected form the IS curve. 
Hicks furthermore incorporates in his analysis the money market, where the supply of money (M) is exogenously determined, i.e., Îœ=Îœ0 /P, where Îœ0 is the exogenously given nominal money supply and P is the price level. The demand for money depends on income and the rate of interest, i.e., L=L(i, Y). By invoking the balancing condition M=L, we arrive at
M0=L(i, Y )
Figure 2 illustrates the equilibrium position in the money market, where the supply of money, for reasons of simplicity and clarity of presentation, is depicted with a vertical line indicating its exogenous character.  The demand for money, as we know, is inversely related to the rate of interest, a relation whose details have been analysed in the previous chapter. When income increases it follows that much more liquidity is required for the needs of transactions and therefore the interest rate will increase for any given level of money supply. In terms of a graph we have:
Figure 2. Equilibrium in the Money Market and the LM Curve
We observe that with the supply of money given the demand for money for transaction purposes is directly related to income. The crucial question here is that while we refer to the money market the discussion is in terms of the bond market. In particular, we know that the excess demand for any good leads to an increase in its price until excess demand becomes zero and thus we get the equilibrium point. Since in the case of money market the equilibrium interest rate is derived in the market for bonds (see ch. 9), then how can the same interest rate equilibrate the money market? In Keynes’s analysis it seems that there is an implicit portfolio stock exchange constraint, which can be written as follows:
(L – M ) + (Bd – Bs) = 0
Where Î’ symbolises the bond market, while the superscripts d and s symbolise the demand for and the supply of bonds, respectively. Consequently, we have the total demand for wealth (L + B d) equal to its supply (M + Bs). If we, further, suppose Walras’s Law, then the above equality necessarily holds and if the rate of interest brings equilibrium in the market for bonds then on the basis of Walras’s Law we conclude that equilibrium will be also established in the money market, that is L = M. As a consequence, we can follow Keynes, who argued that interest rates are determined in the money market. Because of the Walras Law, equilibrium in the bond market and equilibrium in the money market is one and the same. If, for example, i > i*, then Bd > Bs and because of the stock constraint we get L < M, that is there is an excess supply of money in the economy.
Returning to the above equilibrium relations, we end up with a system of four equations and four unknowns: Y, i, I, S. The equations IS and LM represent the reduced form of the above system of simultaneous equations, whose solution gives the equilibrium income together with the equilibrium interest rate. In the same figure, we present the interest rate that corresponds to the liquidity trap (iLT), where the demand for money is infinitely elastic. Consequently, the LM curve is essentially the solid line.
Figure 3. Equilibrium in the Market for Goods and Money
The intersection of the two curves at point B determines the equilibrium pair of interest rate and income. Any point above the IS curve indicates excess supply of goods and every point below the IS curve indicates excess demand for goods. As for the LM curve, every point to the right indicates excess demand for money and every point left to the LM curve indicates excess supply of money. The intersection of the two curves defines four quadrants, which are portrayed in Figure 3 above and in each quadrant we indicate the excess demand or supply in the goods and money markets. The mechanism that establishes equilibrium in the economy works as follows: let us suppose that for some reason the economy is out of equilibrium at a point on the quadrant II. In such a case, savings exceed investment and thus income has a tendency to decrease, while the demand for money is greater than the supply and the interest rate tends to increase. The changes are expected to lead the economy towards equilibrium at point B. In an analogous way, we can describe the mechanism that restores equilibrium at points in the other quadrants and this is left as an exercise.
11.3 Hicks and Keynes
Hicks’s article was published in 1937, eight months after the publication of the General Theory. Keynes already knew the content of the article since he was among the first that the article was given to for comments before its publication to the Economic Journal. Keynes never disapproved directly and explicitly the presentation in terms of the IS-LM apparatus. Don Patinkin (1922-1995) in a series of articles argues the fact that Keynes never said anything negative for the formalisation of his theory by Hicks and that this ipso facto implies an adoption of this presentation on his part (Patinkin, 1990). If Keynes disagreed then he would have every reason to emphatically express his disagreement. After all Hicks’s presentation in a sense was provocative, since Keynes’s General Theory in it was viewed as a special case of the neoclassical true general theory.
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Post Keynesian economists claim that the fact that Keynes did not exercise a negative critique can be attributed to his idiosyncrasy that would not pay attention to anyone’s writings which might concern his General Theory. On the other hand, Keynes did not have any reason, to express, at least in the beginning, his strong disagreement to Hicks’s presentation. It is possible that he did not think that Hicks’s article would meet the success that it finally met (Put footnote 6 here).  It is certain that he disagreed with Hicks’s view as this can be judged by a careful reading of his correspondence with Hicks and from the article that he wrote in the Quarterly Journal of Economics (1937), where he summarised his views. Specifically, he placed special emphasis, once again, on the fact that economies are characterised by uncertainty.
Hicks’s approach, is characteristically different from that of Keynes’s. We know from Pasinetti (1973) that Keynes followed a sequential analysis starting from the marginal efficiency of capital, and then to the interest rate, to investment and through the investment multiplier to the equilibrium level of income. By contrast, in Hicks, all of the above take place simultaneously, as we show in Figure 3. Furthermore, Hicks in his formulation of the demand for money refers to a single interest rate. In the General Theory, however, we know that Keynes refers to two interest rates, the current and the expected in the long run. Consequently, Keynes’s analysis is in sharp contrast to Hicks’s and on top of all we have the issue of uncertainty that permeates the General Theory and is completely absent in Hicks’s presentation.
Another important difference is that Hicks does not refer to the problem of unemployment equilibrium which is so central in Keynes-and really differentiates him from the classics-. Instead, Hicks locates the difference between Keynes and the classics to the interest rate and the issue of whether it increases with investment or not (Barens and Caspari, 1999, p. 219). According to Hicks, in periods of stagnation the interest rate is particularly low and under these circumstances speculators are not willing to hold non-liquid assets; consequently, their demand for money is so high that it absorbs whatever quantity of money is available. Thus, every increase in the supply of money is counterbalanced by a corresponding increase in the demand for money and the rate of interest remains constant. Monetary policy therefore is completely ineffective and it cannot restore the economy to full employment equilibrium. Hicks notes,
there are conditions in which the interest-mechanism will not work. The special form in which this appears in the General Theory is the doctrine of a floor to the rate of interest – [the liquidity trap] as Sir Dennis Robertson has called it. (Hicks, 1957, p.287)
If we suppose that the economy is in the liquidity trap  , then a monetary policy, regardless of how active it might be, cannot shift the economy beyond the initial equilibrium point. In terms of Figure 4, if the economy is in equilibrium at point A, an expansionary monetary policy will shift the LM curve, for example to the position LMÎ„, with no consequence what so ever for the initial equilibrium position.
Figure 4. Equilibrium in the Markets for Goods and Money
Consequently, Hicks in his model claims that the General Theory is not so general as Mr. Keynes thought, but rather a special case of the neoclassical theory, where the liquidity trap has a prominent position. The truth, however, is that the idea of the liquidity trap is very hard to pin point in the writings of Keynes; of course, there are some sporadic hints in the General Theory, as for example is the following:
There is the possibility, for reasons discussed above, that, after the interest rate has fallen to a certain level, liquidity-preference may become virtually absolute in the sense that almost everyone prefers cash to holding a debt which yields so low a rate of interest. In this event the monetary authority would have lost effective control over the rate of interest. But whilst this limiting case might become practically important in future, I know of no example of it hitherto. (Keynes, 1936, p. 207) 
However, Keynes does not discuss this case in any detail so as to claim that this is the hallmark of his theory. What is certain, however, is that the liquidity trap is more Hicks’s and subsequently Hansen’s (1953, pp. 122-3) idea rather than Keynes’s.  Consequently, the view that the liquidity trap is the essence of Keynes’s theory is due to the influence that the Hicksian model exerted on macroeconomics and much less to Keynes and his writings.
Suppose, now, that for some reason investment increases, and then the increase in the rate of interest follows suit, a result which is consistent with neoclassical theory and with Hicks’s argument. It is true, that in Keynes the arrow of causality is different from that in neoclassical economics. However, it continues to be true that, under normal conditions, the interest rate increases when investment increases except for the case of the liquidity trap, where only income changes in every change in investment. The trouble, however, with Hicks’s view is that for Keynes the rate of interest is determined by monetary forces, while in the IS-LM framework the interest rate is determined by real forces. This is an issue that Keynes pointed out in his letter to Hicks. For example we read:
From my point of view it is important to insist that my remark is to the effect that an increase in the inducement to invest need not raise the rate of interest. I should agree that, unless the monetary policy is appropriate, it is quite likely to. In this respect I consider that the difference between myself and the classicals lies in the fact that they regard the rate of interest as a no-monetary phenomenon, so that an increase in the inducement to invest would raise the rate of interest irrespective of monetary policy. (Keynes, 1973, p.80)
A final point relates to the inclusion of current income in the investment function. Keynes objected to this idea for the reason that income was already included in the definition of the marginal efficiency of capital through the prospective yields. The following quotation from his letter to Hicks, shows that Keynes was not only acquainted with the IS-LM apparatus but also as a modern econometrician argued against the inclusion in the same specification of both income and interest rate. Specifically, Keynes notes:
At one time I tried the equations, as you have done, with I in all of them. The objection to this is that it overemphasizes current income. In the case of the inducement to invest, expected income for the period of investment is the relevant variable. This I have attempted to take into account of in the definition of the marginal efficiency of capital. As soon as the prospective yields have been determined, account has been taken of income, actual and expected. But, whilst it may be true that entrepreneurs are over-influenced by present income, far too much stress is laid on the psychological influence, if present income is brought into such prominence. It is of course, all matter of degree. (Keynes, 1973, pp. 80-81).
Barens and Caspar (1999) in their discussion of Hicks and Keynes note that while Hicks accepted all of Keynes’s points he nevertheless insisted in his own formulation for merely pedagogical reasons.
10.4 Modigliani’s Synthesis
Hicks’s model does not refer explicitly to the labour market; it is simply confined to demonstrating that there is equilibrium in only two markets that is the market for goods and the market for money. In his model, Hicks explicitly argues that the money wage as well as the general price level are exogenously given. Franco Modigliani (1944) extended Hicks’s model by including the labour market and the production function. Modigliani argued that the assumption of equilibrium with unemployment cannot be supported on the basis of the liquidity preference theory except for the particular case of the liquidity trap. In general, however, the Keynesian hypothesis can be supported on the assumption of the rigidity in the money wage. For Modigliani, the equilibrium in terms of the IS-LM model implies a pair of interest rate and money income that clears simultaneously the money and good markets. Consequently, we must take into account that the money income (Î¥ ) is equal to the price level (P ) times the level of the real income (X ). As a result, we may write:
The level of real income (or output) is a function of the level of employment of labour (Î). Consequently, we have:
The level of employment in turn is determined at the point, where the marginal product of labour is equal to wage. Consequently, we have:
Up until now we have a system of 7 equations (the three equations above together with the system of 4 simultaneous equations of the IS-LM) with 8 unknowns, that is I, S, i, Y, X, W, P. More specifically, we have the 4 equations of Hicks’s model:
And the three new equations suggested by Modigliani:
w=P F -1(Î)
The system is overdetermined by one equation, the missing equation is the supply of labour. Modigliani in his article invokes Keynes’s assumption of the given money wage. More specifically, the money wage is given if, and only if, the economy is at a level of output less than full employment. We know that in the neoclassical analysis the supply of labour is a function of the real wage N=F(w/P) so the money wage can be written as w=F-1(N)P Formally, Modigliani stated his condition in the labour market in the following way:
w=awo +bPF -1(Î)
Where, a=1, b=0 if Î < Îf
a=0, b=1 if Î = Îf
The last equation indicates that if the current employment in the economy is smaller than full employment (Îf ), then Keynes’s view for the rigidity of money wage holds indeed, that is we have (a=1 and b=0). Money wage is viewed as a “datum a result of history or of economic policy or of both” (Modigliani, 1944, p. 47). If, however, the economy is at full employment, then the money wage becomes flexible (a=0 and b=1) and the last equation becomes an ordinary supply of labour function. Consequently, the money wage will be determined from the supply of labour at the point of full employment.
In Modigliani’s presentation we find that the central assumption is the rigidity of the money wage, an assumption which, as with the liquidity trap does not really find any justification in the General Theory, where the nominal wage is being used simply to determine the price level. By contrast, in Modigliani’s presentation the nominal wage has another important role to play. This is revealed if we express Modigliani’s system of simultaneous equations in terms of wage units or alternatively in terms of labour commanded.  Thus, we have:
investment is given in terms of labour commanded
savings is given in terms of labour commanded
equilibrium in the goods market
equilibrium in the money market
income given in terms of labour commanded
the production function, which is by definition in real terms
the real wage is equal to the marginal product of labour
the supply of labour
Hence, we have a system of 8 equations and 8 unknowns (I/w, S/w, i, Y/w, Î, N, w, P ). If, for a moment, we disregard the fourth equation and focus our attention on the remaining 7 equations, we observe that these can determine all the variables but one, that is the money wage. The result is that the supply of money determines the money wage; since this is the only variable that remains to close the system. Such a determination is due to the quantity theory of money. Consequently, Modigliani’s system of equations is dichotomised into the real economy-which includes all the equations except the fourth one-and the money economy, that is the equation of equilibrium in the money market. The real economy gives solutions in real terms (7 equations with 7 unknowns, that is I/w, S/w, i, Y/w, X, N, w/P) while the money supply:
determines the nominal wage, since the other variables are determined in the real economy. Consequently, the money supply determines the nominal wage and through the real wage it also determines the general price level. Thus, monetary policy may affect real magnitudes in the Keynesian model, contrary to Hicks’s reasoning according to which the money supply does affect the real economy.
Modigliani’s analysis leads to the conclusion that flexibility in prices and money wages establishes full employment in the economy. The mechanism that restores full employment works as follows: the existence of unemployment drives down nominal wages and therefore incomes fall. The demand for money for transaction purposes, being directly related to income, falls as well, and with a given supply of money the rate of interest falls as well. From thereon investment increases and the economy moves toward the full employment level of output. Modigliani managed to formalise Keynes’s argument about the results of the flexibility in money wages. It is important to stress that the pivotal variable in this formalisation of the theory of employment is the idea of inflexibility of money wage. A corollary of this theory is that the role of money is not neutral. For example, the increase in the supply of money affects the price level and reduces the interest rate and thus output and employment are increased. If the nominal wage were perfectly flexible, then money’s role would be neutral since it does not influence the interest rate i, or the liquidity preference L and output remains the same. Consequently, under conditions of a fully flexible nominal wage the increase in the supply of money leads only to an increase in the general price level. Consequently, Îœodigliani concludes that Keynes’s theory works only in case of inflexibility of the nominal wage. If, however, the money wage is flexible then we derive the usual neoclassical results, where the real economy determines the level of output and employment and the money economy determines the nominal variables of the economy. This does not imply a rejection of Keynes’s theory; on the contrary, economists accept the idea of inflexibility of the money wage as a stylised fact of modern economies and thus, Keynesian policy is viewed as both theoretically valid and necessary. The problem, however, relates to the theoretical consistency of the Keynesian system that once again became a special case of the general neoclassical model according to which the economy exhibits a sufficient flexibility in prices of commodities and the factors of production.
10.5 Summary and Conclusions
In an overall evaluation of the two models we see that they both represent aspects or partial arguments of the General Theory. Nevertheless their major problem in terms of the General Theory is the simultaneity issue and also the treatment of uncertainty. In Hicks’s article we find an explanation of unemployment and recession as a result of the liquidity trap, which differentiates Keynes’s theory from the (neo)classical one. In Modigliani, by contrast, the recession is the result of the inflexibility of money wage and not of the lack of effective demand. Commenting on this kind of revision of the General Theory Paul Samuelson in the third edition of his popular text Economics, notes:
In recent years 90 per cent of American economists have stopped being ‘Keynesian economists’ or ‘anti-Keynesian economists’. Instead they have worked towards a synthesis of whatever is valuable in older economics and in modern theories of income determination. The result might be called neoclassical economics and is accepted in its broad outlines by all but about 5 per cent of extreme left-wing and right-wing writers.” (Samuelson, 1955, p. 212).
These efforts to cast Keynesian theory in terms of IS-LM, Samuelson called neoclassical synthesis, since it puts together the neoclassical analysis of investment and savings and the market for labour with the analysis of Keynes about the interaction between the money market and the real level of economic activity. The neoclassical synthesis became the dominant presentation of the General Theory. According to this view, when there is unemployment, then we have Modigliani’s supply of labour function with a=1 and b=0, and as a result of the exercise of appropriate monetary and fiscal policy the economy approaches the level of full employment. When the economy approaches the level of output that corresponds to full employment, then once again the neoclassical theory becomes relevant.
If our central controls succeed in establishing an aggregate volume of output corresponding to full employment as nearly as is practicable, the classical theory comes into its own again from this point onwards. (Keynes, 1936, p. 378)
In general, economists of the neoclassical synthesis argue that although the economy returns to full employment through the price mechanism, nevertheless this is a long run process. Consequently, for immediate results active fiscal and monetary policies are necessary.
Modigliani’s ideas, which became the foundation of the neoclassical synthesis, and which essentially constitute a Marshallian partial equilibrium approach, became the object of criticism from Walrasian authors. They posited the following question: how is it possible to have equilibrium in all the markets but one? The protagonists of this critique of the neoclassical synthesis are Alex Lejonhufvund and Robert Clower, whose contributions we discuss in the next chapter.
Other criticisms included the phenomena of unemployment and later of the stagflation in the late sixties or seventies. Some economists, the monetarists for example, tried to fix the weaknesses of the model and others such as the New Classical economists claimed that the premises on which the IS-LM framework is based are dubious, while New Keynesian economists in the 1980s revived the old Keynesian models by injecting realism and by basing them on microeconomic foundations which simply were not used in the initial models. Whatever happens to the current macroeconomic debates and the various criticisms launched against the IS-LM models, one thing is certain, that these will continue to be part of the formal education of future generations of economists.
Questions for Discussion and Thought
Write down the Hicksian system of equations.
Draw a graph with the IS-LM system of equations and assuming a disequilibrium situation describe the dynamics of attaining equilibrium.
To what extend does Hicks’s model represent Keynes’s General Theory?
What was Keynes’s reaction to Hicks’s IS-LM representation of the General Theory?
Discuss Modigliani’s Neoclassical Synthesis. To what extent does his model differ from Keynes’s?
What are the major similarities and differences between Hicks’s and Modigliani’s models?
Critically evaluate the following statement: “I will assume all markets with the sole exception of the labour market are in equilibrium.”
Notes on Further Reading
Hicks’s (1936) article is easy to follow, but the exposition of ideas (not “visions”) is really dry. The reader discerns an effort on the part of Hicks to express Keynes’s ideas in terms of equations and graphs without, however, the proper textual documentation. As we have mentioned, at the time that Hicks presented his article in the econometric society meeting at Oxford, two other related papers were presented in the same meeting by Meade and Harrod. Darity and Young (1995) present the details of these three articles a
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