Franco Modigliani's dissertation at the New School for Social Research, written under Jacob Marschak, proposed to complete the exercise John Hicks (1937) began by including, in addition to the ISLM equations, the missing labor market and production function equations. His results were subsequently published in a famous 1944 Econometrica article, "Liquidity Preference and the Theory of Interest and Money". Modigliani (1944) proceeded to write up a macromodel as a system of simultaneous equations similar to the following:
(1) Y = C(Y) + I(r) C_{Y} > 0, I_{r} < 0
(2) M/p = L(r, Y) L_{Y} > 0, L_{r} < 0
(3) Y = F(N) F_{N} > 0 F_{NN} < 0
(4) N = f(w/p) f｢ < 0
(5) N = g(w/p) g｢ > 0
Presumably, the first equation is the IS relation, the second the LM relationship and the third is the production function. The fourth equation is, of course, the labor demand function. All these equations (1)(4) are effectively within Keynesian ISLM system, however, the absence of crucial equations, such as a bond market or Walras's Law for stocks, makes this not only a rather poor caricature of Keynes's system but indeed, it is logically flawed. Nonetheless, Modigliani proposes that it is equation (5), the labor supply function as a function of the real wage, that is the exception. The way he has written it implies that, together with (4), he has imposed labor marketclearing, i.e. N^{d} = N^{s}_{ }= N. This, Modigliani notes, is a "Classical" assumption he has imposed within the ISLM system to see whether "Classical" results can be derived from an ostensibly "Keynesianlike" set of equations. What he wants to prove, as he himself notes, is that:
"the liquidity preference theory is not necessary to explain underemployment equilibrium; it is sufficient only in a limiting case, the "Keynesian case". In the general case it is neither necessary nor sufficient; it can explain this phenomenon only with the additional assumption of rigid money wages." (F. Modigliani, 1944: p.756)
Thus, let us turn to what Modigliani calls the "general" case. We have five equations and five unknowns (Y, r, N, w, p) with M as exogenous (K, not written and normalized out, is also exogenous). We can get rid of (3) and express all the terms as:
F(N)  C(Y)  I(r) = 0
M/p  L(r, F(N)) = 0
N  f(w/p) = 0
N  g(w/p) = 0
or totally differentiating each equation with respect to M:
(1C_{Y})F_{N}(dN/dM)  I_{r}(dr/dM) = 0
1/p  (M/p^{2})(dp/dM)  L_{r}(dr/dM)  L_{Y}F_{N}(dN/dM) = 0
dN/dM  (f｢ /p)(dw/dM) + f｢ (w/p^{2})(dp/dM) = 0
dN/dM  (g｢ /p)(dw/dM) + g｢ (w/p^{2})(dp/dM) = 0
or, in matrix form:
(1C_{Y})F_{N} 
 I_{r} 
0 
0 
dN/dM 
0 

L_{Y}F_{N} 
L_{r} 
M/p^{2} 
0 
dr/dM 
= 
1/p 

1 
0 
(w/p^{2})f｢ 
 f｢/p 
dp/dM 
0 

1 
0 
(w/p^{2})g｢ 
g｢/p 
dW/dM 
0 
Now, after some calculation, we will find that the determinant of the matrix of coefficients is:
A = I_{r}(M/p^{3})[f｢  g｢ ]
Now, the rest of the exercise revolves around finding what the terms dp/dM, dr/dM and dN/dM are. As we know, from Cramer's rule:
dp/dM = A_{3}/A
where A is the determinant of the original matrix and A_{3} is the determinant of a transformed matrix, i.e. the matrix A with the solution column replacing the third column in the matrix, i.e.
(1C_{Y})F_{N} 
 I_{r} 
0 
0 

A_{3} 
= 
L_{Y}F_{N} 
L_{r} 
1/p 
0 
1 
0 
0 
 f｢ /p 

1 
0 
0 
g｢ /p 
which is not too difficult to solve. Expanding by third column:
(1C_{Y})F_{N} 
 I_{r} 
0 

A_{3} 
= 
(+1/p) 
1 
0 
f｢ /p 
1 
0 
 g｢ /p 
then expanding by the second column:
A_{3} = 1/p [I_{r}(g｢ /p + f｢ /p)]
or simply:
A_{3} = I_{r}(f｢ g｢ )/p^{2}
Thus dp/dM = A_{3}/A = I_{r}(f｢ g｢ )/p^{2}A where A as we know is I_{r}(M/p^{3})[f｢  g｢ ], thus:
dp/dM = I_{r}(f｢ g｢ )/p^{2}[I_{r}(M/p^{3})[f｢  g｢ ]]
= p/M
or simply:
(dp/dM)(M/p) = 1
so the elasticity of price with respect to the money supply is one, i.e. an increase in money leads to a proportional increase in the price level  the very claim of the old Neoclassical Quantity Theory of Money! To verify whether this is indeed the case, i.e. whether money supply is neutral, we ought to conduct the same exercise for dr/dM and dN/dM. Doing so via Cramer's rule (we shall not go through the math here again), we should realize that:
dr/dM = 0
dN/dM = 0
i.e. an increase in money supply M leads to no change in interest rate and no change in the level of employment  and, consequently, as Y = F(N), no change in the level of output. Thus, in this Modigliani model, money is completely neutral as no real variables (interest, employment and output) are affected and the price level changes oneforone with the money supply. Hence, Modigliani concludes, Neoclassical results are restored within an ostensibly Keynesianlike ISLM model.
Of course, this should be no surprise  Keynes (1936) himself had argued that the Neoclassical theory should come into its own at full employment (i.e. labor market clearing), which has been assumed in the inital set of equations. However, Modigliani (1944) then asks how it might be possible to restore "Keynesian" results. In his view, "in the Keynesian system...the supply of labor is assumed to be perfectly elastic at the historically ruling wage rate, say w_{0}." (Modigliani, 1944: p.47). Consequently, the labor supply equation is no longer a function of the real wage, thus we can throw that equation out (and with it the full employment assumption) and replace the money wage rate w with w_{0} so that the labor demand function becomes the labor employment function, but with money wage at w_{0}, i.e. N = f(w_{0}/p) alone.
What Modigliani is effectively proposing is that if money wages are assumed rigid, then this model ought to yield nonneutrality of money. Thus, in his view, the Keynesian system only works if there are sticky money wages. To see how he obtains this, let money wages by sticky at w = w_{0}. Consequently let us rewrite the system as:
(1) Y = C(Y) + I(r) C_{Y} > 0, I_{r} < 0
(2) M/p = L(r, Y) L_{Y} > 0, L_{r} < 0
(3) Y = F(N) F_{N} > 0 F_{NN} < 0
(4) N = f(w_{0}/p) f｢ < 0
where, note, the labor supply curve (and thus the full employment assumption) has been omitted and money wages, w, are assumed rigid at w = w_{0}. So we have four equations and four unknowns (Y, r, N, p) with M, K and w_{0} taken as exogenous. Expressing everything in implicit form and totally differentiating with respect to M, we obtain:
(1C_{Y})F_{N}(dN/dM)  I_{r}(dr/dM) = 0
1/p  (M/p^{2})(dp/dM)  L_{r}(dr/dM)  L_{Y}F_{N}(dN/dM) = 0
dN/dM + f｢ (w_{0}/p^{2})(dp/dM) = 0
or, in matrix form:
(1C_{Y})F_{N} 
 I_{r} 
0 
dN/dM 
0 

L_{Y}F_{N} 
L_{r} 
M/p^{2} 
dr/dM 
= 
1/p 

1 
0 
(w_{0}/p^{2})f｢ 
dp/dM 
0 
The determinant of the matrix of coefficients is (expanding by the last row):
A = I_{r}(M/p^{2})  (w_{0}/p^{2})f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}]
Assuming (1C_{Y}) > 0, then the first term on the left is negative (I_{r}(M/p^{2}) < 0) and the entire term (w_{0}/p^{2})f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}] is positive. Thus, A < 0.
Let us now find dp/dM via Cramer's rule again, so dp/dM = A_{3}/A and A_{3} is the determinant of the matrix A with the solution column replacing the third column:
(1C_{Y})F_{N} 
 I_{r} 
0 

A_{3} 
= 
L_{Y}F_{N} 
L_{r} 
1/p 
1 
0 
0 
so expanding by the third column:
A_{3} = I_{r}/p
which we know is negative as I_{r} < 0. Thus:
dp/dM = A_{3}/A = I_{r}/p[I_{r}(M/p^{2})  (w_{0}/p^{2})f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}] > 0
This is positive because the numerator A_{3} and the denominator A, as we saw, are negative. Now, the money elasticity of price is (dp/dM)(M/p), thus multiplying by (M/p) and rearranging:
(dp/dM)(M/p) = 1  MI_{r}/w_{0}f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}] > 0
or:
1 (dp/dM)(M/p) = MI_{r}/w_{0}f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}] > 0
because the numerator is negative, MI_{r} < 0, and the denominator is negative, w_{0}f｢ [(1C_{Y})F_{N}L_{r} + I_{r}L_{Y}F_{N}] < 0. Thus, this implies :
1 > (dp/dM)(M/p) > 0
i.e. a rise in the nominal money supply is followed by a lessthanproportional rise in the price level  and thus there is a change in the real money supply. Thus, one precept of the Quantity Theory of Money is invalidated. To verify the nonneutrality of money, it is worthwhile verifying the value of dr/dM. Doing so via Cramer's rule we obtain, in the end:
dr/dM = (1C_{Y})F_{N}[(w_{0}/p^{3})f｢ ]/A
We know that the denominator is negative A < 0 and it is easy to notice that the numerator is positive as (1C_{Y})F_{N}[(w_{0}/p^{3})f｢ ] > 0, thus dr/dM < 0. In short, an increase in the nominal supply of money will reduce the rate of interest. What about employment and output? Well, applying Cramer's Rule again, we see that:
dN/dM = (1/p)I_{r}(w_{0}/p^{2})f｢ /A
where the numerator is negative (as I_{r}, f｢ < 0) and we know the denominator is negative A < 0, thus dN/dM > 0. Thus, we find that real employment N increases if the money supply rises. By the chain rule, we know dY/dM = (dY/dN)ｷ(dN/dM) = F_{N}(dN/dM), so output also rises with a rise in the money supply. Thus, in this case, money is not neutral as real terms r, N and Y will be affected by the supply of money. It seems, then, that "Keynesian" results are restored.
In sum, Modigliani proposed that with sticky wages (w = w_{0}), money is nonneutral: an increase in the nominal money supply M raises the price level lessthanproportionally, decreases the interest rate and raises employment and output. If money wages are fully flexible, as in the earlier case, then money is neutral  it affects neither interest nor employment nor output and increases the price level proportionally. Thus, Modigliani (1944) concludes, Keynes's theory only works if there is sticky or rigid money wages.
Modigliani's 1944 paper caused barely a ripple of outrage and was quietly incorporated into the orthodoxy. Only a few protests as to the logical coherence of this model  such as those by Frank H. Hahn (1955) and Don Patinkin (1948, 1956)  were registered but subsequently ignored. Most other economists, proceeded quite as if Modigliani had "accurately" represented the General Theory in his system of simultaneous equations.
The Keynesian system, the conventional wisdom thus went, was only applicable under sticky wages. Of course, this did not deny the efficacy of Keynesian policy: sticky money wages were seen as a "real problem" and thus the Keynesian policy conclusions were still seen as valid and relevant. But the Keynesian system was no longer theoretically valid  it became merely a "special case" of the more general "Neoclassical" one. If wages were flexible, the Neoclassical conclusion of full employment and neutrality of money would hold true. Many economists suggested that wage flexibility emerged in the "longrun"  a slippery and still largely undefined concept  and thus, things tended to full employment and were barred from reaching it only via an "imperfection" in the market system, such as that introduced by sticky money wages. Until Axel Leijonhufvud (1968) launched his allout attack on this "NeoclassicalKeynesian" Synthesis, most economists had really no idea that Keynes's General Theory was in fact substantially different from this caricature.
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