The Dynamic Interactions Between the Phillips Relation and the IS-LM Model

The Dynamic Interactions Between the Relation and the IS-LM Model*

IAN McDONALD

University of Melbourne, Parkville, Victoria 3052

In this paper the IS-LM model is extended dynamically by the addition of a price-adjustment relation (the Phillips relation) and a quantity-adjustment relation. Three specifications of the Phillips relation are considered. I t is shown that the nature of the price-adjustment 'process has important implications for stability; in fact the price-adjustment processes which yield clockwise (counter-clockwise) Phillips loops may also be unstable (are not unstable). On the other hand, the speed of the price-adjustment process is not relevant for stability. The speed of quantity adjustment is only important in 'the expectations case and there the slower the speed o f adjustment the more likely is instability. No clear conclusions emerge, in general, concerning the slopes of the IS-LM relations and stability. In conclusion the results are related to the debate on the micro-foundations of Keynesian economics.

I In the standard income expenditure model,

i.e. the Hicksian IS-LM analysis, quantities are assumed to adjust freely whilst the price level is fixed. In the last 20 years the determinants of price changes have been studied in the Phillips curve literature (recently surveyed in Laidler and Parkin 119751). Some papers have been concerned with the dynamic implications of combining the Phillips curve with the income-expenditure model. Laidler ( 1973), Tobin (1975) Taylor (1977), Pyle and Turn- ovsky (1976), Turnovsky (1978), (1977) and Yarrow (1977) have used the expectations Phillips curve with varying degrees of sophistication of the demand for money specification.1 However, while there have .been these attempts to improve the price flexi-

* I would like to thank Stephen Turnovsky, David Vines and a referee for helpful comments.

1 The papers by Turnovsky allow for the accumulation of wealth and the government budget constraint-issues which are not explored here.

Phillips

bility assumptions of the standard theory, only in Tobin's paper has an attempt been made in the direction of improving the quantity flexibility assumption. And yet this is clearly an important extension. Friedman (1970) has argued persuasively that macro-models should incorporate specifications of how both output and prices adjust to disequilibrium.

This paper examines the stability conditions of a model in which the adjustment of both output and price is determined endogenously. The study is a comparative one. Three specifications of the Phillips curve are used: the Lipsey (1960) specification, which stresses the dispersion of demand over markets; the Kuska (1966) specification, which stresses the slug- gish response of the price level in both upward and downward directions; and the expectations- augmented Phillips curve in which expectations are determined by the adaptive mechanism first put forward by Cagan (1956). In each case an output adjustment process is specified in which output adjusts to the difference between planned effective demand and output.

3 69

370 THE ECONOMIC RECORD DEC.

In the original formulation of Lipsey and

Kuska the variables were wage inflation (-)

and unemployment (u) . The Phillips relation is specified here as a relation between output

inflation (-) and the level of output ( Y ) in

order that the link with the IS-LM model be established. However, when identifying a ‘Phillips loop’ we define its direction of rotation as the direction which would be

observed in (-, - Y) space and not. (-, Y)

space since the former corresponds to the

(-, u ) space in which Phillips loops are

usually discussed. The local stability of full employment equi-

librium is of fundamental importance in macro- economics. Tobin has argued that the proposi- tion Keynes attacked was not the existence of full employment equilibrium but the ability of a private market economy to steer itself to the full employment equilibrium. (See Tobin [1975], pp. 195-6, and also Leijonhufvud [ 19681 for a similar view.) Under this interpretation it makes little difference whether the Pigou, or real-balance, effect guarantees the existence of equilibrium if that equilibrium is unstable. Furthermore, if the full employment equilibrium is unstable a justification for macroeconomic stabilization policies exists. It should be noted that it is not the case that an unstable model i s obviously unrealistic since actual economies do not experience unemployment rates of zero or 100 per cent or inflation rates of plus or minus infinity. Fluctua- tions in a model which is locally unstable around the equilibrium position may be bound- ed by ceiling and floors. Also, the fluctuations will be as rapid or violent as the speeds of adjustment of output or prices allow. Slow speeds of adjustment can generate slow movements in income and the price level even if the economy is in disequilibrium.

Since a collection of Phillips curve formulations is analyzed it is helpful to anticipate the major conclusion. In the models studied

W

W

p‘

P

P P

P P

W

W ,

the Phillips curve formulations which have been put forward to explain counter-clockwise loops are also stable. Those formulations which have been put forward to explain clockwise loops yield the possibility of instability. In the conclusion this result is com- pared with the ideas of Leijonhufvud (1968) and others, on the relationship between stability and output/price flexibility.

I1 It will be supposed that the adjustment of

aggregate output (Y) is a function of the difference between the planned level of expenditure (E) and the actual level of output.

(1) dY . - = Y = A ( E - Y), A, > 02. dt

Planned expenditure is the sum of planned consumption (C) and planned investment ( I ) expenditure. These aggregates are related to the level of output, the nominal rate of interest ( i ) and the expected rate of inflation (n) by the following specifications. (We abstract from the accumulation of capital and wealth.) C = a (Y, i, n) (2)

The rate of interest is determined by money market equilibrium which is assumed to hold at all points in time. The money supply ( M s ) is the product of the money base ( M ) and the money-base multiplier, q ( i ) ; the latter is positively related to the nominal rate of interest, i.e. q1 > 0. The money demand function contains output, the nominal rate of interest and the expected rate of inflation as its argument. Money market equilibrium implies

al > 0, a, < 0, a3 > 0 Z = b (n - i ) bl > 0. (3)

Ms M.q( i )

P P - - - ___ = L (Y , i, nf L, > 0,

L, < 0 L, < 0. It is convenient to rewrite this relation in the following form

i = g (Y, P , rr) (4) where -1

L

2 Subscripts denote derivatives.

1978 PHILLIPS RELATION AND IS-LM MODEL 371

L3 -1 , g , = - e < o L

and 41 L2

4 L Combining (1) to (4) we get the output

o z - - -

adjustment relation

y = A { a ( Y , g ( Y , P , rr), rr) + b(n - g ( Y , P, n)) - Y}. (5)

This output adjustment relation will be used throughout. However, in order to provide a contrast with the expectations case and in order to follow the original specifications the expected rate of inflation will be considered exogenous in the Lipsey and Kuska cases. If the expected rate of inflation. is exogenous the dynamic movements implied by the output adjustment equation can be determined by the construction of a locus of points for which Y = 0. The slope of this locus is downward sloping, i.e.

( 1 - 4 (; ;) P.L, p _ - _ --

L . In the area to the right of this locus the rate of interest is high enough to cause planned expenditure to fall short of the level of output. Output will fall. The converse is true in the area to the left of Y = 0 locus. The locus is shown in Figures (1)-(3).

- - _-- d p I . dY 1 Y = 0 (b,-a,)

I I I The original Phillips relationship (Phillips

[19SS]) has undergone many extensions and revisions since it first appeared. Labelling any aggregate relation between inflation and aggregate excess demand as a Phillips relation, three particular specifications, each of which can describe the Phillips loops, have received attention in the literature. They are:

P -= B ( Y , Y ) P B ( Y * , 0 ) = 0 (6a)

Bj > 0, B, 3 0

P -= B ( Y , Y ) B, > 0, B, > 0 P B(Y, 0) = 0. (6c)

The Lipsey (1960) formulation is an exampIe of the first, the Kuska (1966) formulation is an example of the second and the expectations theory is an example of the thud. No doubt alternative micro-foundations can be posited to explain each of the formulations-we, however, will follow our identification. The inter- action of each with the IS-LM model will be considered in turn.

The Lipsey Formulation Lipsey (1960) observed the existence of

both clockwise and counter-clockwise loops around the Phillips curve. He explained these loops by appealing to an aggregation argument. If the Phillips curve in each of many micro- labour markets is convex to the origin then counter-clockwise loops will be present in the aggregate data if the dispersion of demand over markets increases as aggregate demand increases and decreases as aggregate demand decreases. Clockwise loops on the other hand can be explained if increases in aggregate demand are accompanied by decreases in the dispersion of demand over micro-labour markets. Using output as a measure of aggregate demand and assuming that prices are determined by a constant markup over wages over the cycle,3 then the Lipsey Phillips curve can be written as 6(a) where if B2 > 0 (B3 < 0) the loops are counter-clockwise (clockwise).

By substituting the 9 relation into the Lipsey Phillips relation we obtain a second differential equation (of course this equation is not independent of the first-however the dynamic behaviour implied by these two dependent equations can be studied using the standard methods). Note that the expected rate of inflation variable is suppressed in the output adjustment equation.

3 If the markup is positively related to aggregate demand the degree of clockwiseness of the price inflation loops would be somewhat less (or the degree of counter-clockwiseness would be somewhat greater) than the degree of cloclcwiseness (or of counter-clockwiseness) of the wage inflation loops.


P - = B (Y, A { d Y , s(Y, PI) P

4- b ( - g ( Y , P)) - Y}). (7)

A Iocus of points in ( P , Y) space for which k = 0 can be constructed. The slope of this locus is easily shown to be related to the slope of the Y = 0 locus in the following fashion.

"j = - Bl p (??) dY P = 0 BZA, ( b l - ~ a ) 4

dP I +GI L o . (8)

As noted above, clockwise loops imply B2 < 0 and thus, as inspection of (8) indicates, a downward sloping P = 0 locus which is steeper than the Y = 0 locus. In Figure 1 the two loci are drawn for this case with the arrows showing the dynamic movements of P and I in each of the areas A X ) . The curves intersect at Y = Y*-this 'is the equilibrium position. The area of positive inflation is above and to the right of the P = 0 locus.

If there are no Phillips loops i.e., B, = 0, the P = 0 locus will be a vertical line. If the loops are counter clockwise then B, > 0. In- creasing B,, i.e., considering counter-clockwise cycles of greater magnitude, rotates the = 0 locus in a clockwise direction. In Figure 2

P

ce el

I I I Y* Output

Price level

I P = O . .

Y = O

1 I I

output Y *

FIGURE 2

the case where B, > 0 but small is shown. The P = 0 locus is positively sloped. In Figure 3 the P = 0 locus associated with a larger value of B, is shown,. The locus is nega- tively sloped and cuts the Y = 0 from below. ( In the limit as B , + 00 the slope of P = 0 approaches the slope of Y = 0.)

It is easy to see that the phase diagram of the last case (Figure 3) yields a stable equilibrium position and paths to this equilibrium which are direct, i.e., non-cyclical. The other two cases cannot be analyzed so easily so we use the results of the algebraic analysis presented in the Appendix.

C I

I

P.0

. i = 0

I I I

output Y*

FIGURE 3 FIGURE 1


It is shown in the Appendix that the model

> c (9)

will be stable if

Clearly if B2 > 0 this condition is satisfied- counter-clockwise loops will yield a stable equilibrium. If B2 < 0 then the interpretation of (9 ) is rather more complicated. We may say that, ceteris paribus, stability is more likely:

The smaller the absolute value of B2, i.e. the smaller the size of the clockwise Phillips loops; the smaller is al , the marginal propensity to consume; the smaller is the absolute value of u2, the interest responsiveness of planned consumption demand; the larger is L, , i.e. the steeper the transactions demand for money schedule; the larger the absolute value of L,, i.e. the less steep the liquidity preference schedule; 1 the larger is ql , i.e. the more responsiveness is the money supply to changes in the rate of interest; and the smaller is bl, i.e. the steeper the marginal efficiency of capital schedule.

Rather surprisingly, stability does not de- pend on the speed of adjustment of output (A , ) or the steepness of the Phillips curve ( B l ) . This is commented on below.

The Kusku Formulation It is often asserted that wages and prices

are inflexible downward, or at least are difficult to reduce. Kuska (1966) points out that wage reductions may yield costs to the firm because of the consequent loss of workers’ morale. Kuska goes on to argue that downward wage rigidity may also imply upward wage rigidity, at least in the short run. When deciding on a wage increase the firm will have to consider the consequences of the possibility that in the future a decline in demand may leave’it with a wage which is too high. If the same considera- tions apply to the firm’s output pricing policy4

Marshall’s analysis of the role of the norm(a1) price and of the fear of entrepreneurs of ‘spoiling the market’ can be captured by this formu1,ation of the Phillips relation. Marshall (1920) pp. 374-7.

then the Phillips relation may be written

P - = j ( Y - Y*) - k ( P -P , , ) , j , k > 0. P

Po, the ‘norm’ price is not the same as the expected price of the next period. It is a longer- run concept-the average price over the trade cycle. It is a price around which actual prices are expected to fluctuate over some period of time. Kuska describes the ‘norm’ wage as the average wage which is expected to rule over the trade cycle. Since the monetary base and the fiscal balance are held constant over the cycle in the current model, an analogous interpretation of the ‘norm’ price is legitimate in the current context.6

The loops generated under this hypothesis are counter-clockwise; at the peak of the output cycle

. . - dt

and so a counter-clockwise cycle in P. The slope of the P = 0 locus is

dPI - - i > O . E ( P = O k

The Y = 0 locus remains the same as for the Lipsey case. Thus Figure 2 can also serve as the phase diagram for the Kuska case. I t is shown in the appendix that the Kuska model is always stable.

The Expectations-Augmented Phillips Curve For- mulation

Recent literature (e.g. Phelps et al. [1970]), has emphasized the impact of endogenously-determined inflationary expectations on the rate of inflation. The usual formulation (e.g. Laidler [ 19731, Grossman [1974]) argues that a discrepancy between actual and expected inflation will lead to a change in the latter, that is

where n represents the speed of adjustment of inflationary expectations. If the actual inflation is the sum of an excess-demand function and expected inflation,

5 In an economy with a positive rate of increase of the general price level over the cycle the ‘norm’ price would have to be appropriately modified.

3 74 THE ECONOMIC RECORD DEC.

t = j ( Y - Y*> +n, j > 0 (15) P where j = the slope of the short-run Phillips curve, then the Phillips relation is a 2nd-order differential equation

d g ) = n j w - Y*> + j Y. . , - dt

This is a particular version of equation (6[c]). The Phillips trade off is between aggregate demand and the acceleration of inflation rather than just inflation itself. Clearly the loops will be clockwise as pointed out by, for example, Brechling (1968), Grossman (1974). At the peak of the output cycle

Y = O a n d Y > Y*impliesd

The differential equation system comprising the output-adjustment relation. and equations (14) and (15) is analyzed in the Appendix. It is shown that the following condition is necessary and sufficient for stability. 1 1 ’ ->

Ceteris paribus, stability is more likely: (a) the larger is A,, i.e. the greater the

speed of adjustment of output; (b) the smaller is al , i.e. the smaller the

marginal propensity to consume; (c) the larger is (--a2), i.e. the greater the

interest sensitivity of planned consumption expenditure;

(d) the smaller is a3, i.e. the smaller the sensitivity of planned consumption expenditure to changes in the expected rate of inflation;

(e) the larger is L,, i.e. the steeper the transactions demand for money schedule;

( f ) the smaller is ( - L 2 ) , i.e. the smaller the interest sensitivity of money demand;

(g) the smaller is ql, i.e. the smaller the responsiveness of the money supply to changes in the interest rate;

(h) the smaller is ( - L 3 ) , i.e. the smaller the responsiveness of money demand to changes in the expected rate of inflation; and

(i) the smaller is n, i.e. the smaller the speed of adjustment of inflationary expectations.

The size of (+), the responsiveness of investment to the real rate of interest, is ambiguous. However a large b, (Le. a shallow marginal efficiency of capital schedule) will tend to be stabilizing if -a2 S u3, that is, if the sensitivity of consumption demand with respect to the expected rate of inflation is greater than the interest sensitivity of consumption demand.

As in the Lipsey clockwise case, the steepness of the short-run Phillips curve or the responsiveness of the price level to the level of aggregate demand is irrelevant for stability.

More simple forms of this stability condition can be related to other papers. If the supply of money and the demand for money are assumed independent of the rate of interest, i.e. q1 = L, = 0, the stability condition reduces to

1 L3

-’-t n

which is the Cagan stability condition (see Cagan, [1956], Taylor [1977], Yarrow [19771). Thus the general stability condition derived here is more restrictive than the Cagan case. In Yarrow’s model it is assumed that 6 , = -a2 = u3 and that A, = 00 and q, = 0. The stability condition becomes

1 L, L3 ->---- n L L

which is equivalent to Yarrow’s condition (Yarrow [1977], equation [ 2 6 ] ) . The insights of these more simple models are reinforced in this model-that is, the smaller is the responsiveness of money demand to interest rates and to the expected rate of inflation, and the smaller the speed of adjustment of expectations, the more likely is stability.6 On the other hand

e B u t note that in the Lipsey clockwise loop case the larger the responsiveness of money demand to interest rates the more likely is stability. See also Section IV below,


it should be noted that as the degree of realism is increased the likelihood of instability is also increased. I n particular the incorporation of a less than instantaneous output adjustment term into the expectations case makes the likelihood of the economy approaching full employment equilibrium under its own steam less likely.

IV The models studied in this paper suggest a

link between the nature of the Phillips loops and the stability of the economy which has passed unnoticed in the previous literature. The functional forms of the Phillips curve which have been put forward to explain counter-clockwise Phillips loops yield a stable equilibrium when integrated with the IS-LM model whatever the magnitude of the functional relations (given that the usually accepted sign conditions hold) underlying the IS-LM analysis and whatever the speed of adjustment of output to aggregate demand. On the other hand the functional forms of the Phillips curves which have been put forward to explain clockwise Phillips loops yield the possibility of instability. For these latter cases the speed of price response to the level of aggregate demand does nor enter the stability condition. That is to say, the slope of the short-run Phillips curve is irrelevant to the question of stability. What is relevant is the size of the price change response to the change in aggregate demand in the Lipsey clockwise case, and the speed of the adjustment of the expected rate of inflation to the actual rate in the expectations-augmented Phillips curve case.

All models were specified to include the possibility that output does not adjust instan- taneously to the level of aggregate demand. But this adjustment coefficient on!y entered the stability condition for the expectations-augmented Phillips curve case. For t!iis case it is slow adjustment of output which can yield instability. For the Lipsey cloclcwlse loop case the stability condition is indepxident of the output-adjustment coefficient.

There is a striking difference between the stability implications of the magnitudes of the IS-LM relations in the Lipsey clockwise case and,the expectations case, which deserves men- tion. In the Lipsey clockwise case the smaller

the interest responsiveness of consumption and investment expenditures (i.e. the steeper the IS curve) and the larger the interest responsiveness of the demand for money and the supply of money (i.e. the flatter the LM curve) the more likely is stability. In the expectations case the reverse is true. The greater the interest elasticity of consumption demand and the smaller the interest responsiveness of the demand for and the supply of money, the more likely is stability. Thus in general no clear conclusions for stability emerge from assuming either the mone- tarist or the Keynesian empirical specifications of the IS and LM curves. In the Lipsey case Keynesian empirical specifications are the more stable, whereas in the expectations case mone- tarist empirical specifications are more stable.

Leijonhufvud (1968) has argued that the crucial element in Keynes’ analysis that yielded the conclusion that market forces may not generate a stable, full employment solution is Keynes’ reversal of the Marshallian adjustment speeds. That is to say, Keynes assumed that quantities adjusted at a faster rate than prices. Tobin (1975) has argued along similar grounds, emphasizing the contrasting dynamic implications of Marshallian and Walrasian adjustment processes, In Figure 4 the two types of adjustment process are illustrated. An increase in demand has shifted the demand curve to D2. Under the Walrasian adjustment process output initially rises from Q,, to Q,. Subse- quently price will rise and the adjustment path

Price

S

I I I I 1 ! I I I I I I

FIGURE 4


will be along the demand curve leading to the equilibrium position , (P2 , Q2). The Walrasian adjustment path is ABC.? The Marshallian adjustment path is different. Initially price adjusts to P, following the rise in demand. Then the adjustment path is along the demand curve to the equilibrium position (P.,, e l ) . Thus the Marshallian path is ADC (cf Leijonhufvud 119681, p. 51).

There are several points which should be noted about the adjustment paths. The ‘pure’ Marshallian path (path A D C ) requires that price adjust infinitely fast in response to disequilibrium. The ‘pure’ Walrasian path (path ABC) requires that quantity adjust infinitely fast. It is this difference between the two processes which Leijonhufvud focuses on (e.g.: ‘In the Keynesian macrosystem the Marshallian ranking of price-and quantity-adjustment speeds is reversed: In the shortest period flow quantities are freely variable, but one or more prices are given, and the admissable range of variation for the rest of prices is thereby limited. The “revolutionary” element in the General Theory can perhaps not be stated in simpler terms’. Leijonhufvud [1968], p. 5 2 , original italics.) There are, however, other differences between the two adjustment processes which are also important. Under the Marshallian process price responds to demand and output responds to price. Under the Walrasian process output responds to demand and price responds to output. (Cf Tobin 119751, p. 196.) It should be clear that the dynamic specification adopted throughout this paper is Walrasian in form-the output-adjustment equation specifies output adjustments as a function of excess aggregate demand whilst the functional forms of the Phillips curve that have been used specify price changes as a function

Assuming all firms follow the pure Walrasian adjustment process ( A B C ) then the aggregate adjustment process can be captured in the Lipsey model by setting A , = m and B3 = 0 or in the Kuska model by setting A , =, co and

‘This article follows Tobin (1975) in calling such an adjustment path Walrasian. The question whether an adjustment process of this type should be ascribed to Walras is not examined here.

s The aggregate price-adjustment relations each assume the existence of one level of aggregate output for which the price level is constant. For

of output.

j = 0.8 In both cases the condition for stability becomes

The usual sign restrictions will ensure that this condition is met. Thus we cannot conclude that Walrasian adjustment processes are sufficient to generate instability. Under this dynamic specification of the Walrasian adjustment process market forces will ensure that the economy will approach full employment.

Marshall modified the rather simple price- adjustment process outlined above by intro- ducing the concept of the normal price (Marshall E19201, Book V, Chapter V). Kuska has applied this concept to the Phillips curve literature. In this paper the juxtaposition of the concept of normal price with a framework of adjustment of price and quantity which is essentially Walrasian in nature is not one that obviously fits very closely into any particular doctrinal category. In a pure sense such a mixture is neither Marshallian nor Walrasian. However, from a practical or theoretical point of view the resulting adjustment process is not unappealing. In a Walrasian scheme where output adjustment is instantaneous price changes occur as firms move along the demand curve facing them. That is to say, price and

output greater (smaller) than this level, prices continually rise (fall). This requires that the aggregate supply curve is vertical. Now even if the aggregate supply curve is vertical and if only shifts in the firm’s demand curve induced by aggregate demand shifts are considered, it is an open question whether the individual firm should be postulated to have a vertical supply curve. In the ex poste sense the supply curve would be vertical in the face of shifts in aggregate demand (as opposed to changes in the composition of demand away from or towards the firm). How- ever, one should not necessarily assume that the firm can distinguish between aggregate demand induced shifts and shifts induced by changes in the composition of aggregate demand. In Figure 4 the upward sloping curve reflects the supposition that the firm cannot make such a distinction. The supply curve chown is the firm’s ex ante supply curve. Note that a change in the general price level or a change in the expected price level will shift this supply curve.


output adjustments occur simultaneously. Now, following a rise in demand, if the entrepren- eur’s conception of normal price exceeds current price then the movement along the demand curve is faster than if the entrepren- eur’s conception of normal price falls short of current price. I n terms of the static demand and supply curves of Figure 4 the addition of the concept of normal price would yield no perceivable difference to the adjustment path. However the resulting aggregative model is stable.

In both the Lipsey clockwise loop case and the expectations-augmented Phillips curve case there is a possibility of instability. If the speed of adjustment of output is assumed infinite then firms are following Walrasian adjustment paths of the type illustrated in Figure 4. However, the crucial difference between these two cases and those above is that a’change in output has an accelerating effect on price. In the Lipsey clockwise case the initial effect of an increase in output on price is ambiguous-the higher level of output exerts upward pressure whilst the change (increase) in output exerts downward pressure. Subsequently, if the size of the increases in output falls the increase in the price level accelerates. In the expectations case an initial increase in output does increase the price level. The price level increase will then raise the expected rate of inflation and this will accelerate the increase in the price level. The acceleration effect captures the idea that prices are ‘initially sticky’ when output changes. It is price stickiness or price rigidity in this xeme that can yield the possibility of instability. On the other hand the speed of adjustment of the price level to the level of output (the slope of the short-run Phillips curve) has no effect on the question of stability. The speed of output adjustment also has no effect on stability in the Lipsey clockwise lcop case, and in the expectations case the faster the speed of output adjustment the more like‘y is stability. This suggests that the relative spc-ds of adjustment of prices and quantities are not important for stability questions.

Clearly the specification of a type of dynamic adjustment using static concepts yields various dynamic interpretations. There is some ambigu- ity involved in going from one to another. No doubt dynamic specifications different from those adopted here can be put forward and can

be claimed to reflect the Walrasian adjustment process. In any event it can be concluded that the dynamic implications of the Walrasian process are not at all clear. All the adjustment processes analyzed in this paper are basically Walrasian. The closeness with which they ap- proximate the pure or extreme Walrasian path, i.e. path ABC in Figure 4, has no bearing on stability. It would seem then that the use of the static framework of demand and supply theory to classify adjustment processes is not very useful for drawing implications about macroeconomic stability. On the other hand, for the cases studied in this paper, the nature of the Phillips loops generated by the alternative price-adjustment specifications is a good guide to the possibility of stability. Whether this classification would stand up to further analysis is, of course, an open question. For example the effects of capital and wealth accumulation have not been studied; the incorporation of these might upset this result.

The analysis of this paper suggests the in- formation about the Phillips loops yields implications about the stability of the economy. Many researchers claim that the nature of the loops are indentifiable from empirical data. For example, it has been asserted that for wage inflation data the Phillips loops have changed from a counter-clockwise to a clockwise pattern in the present century for the UK (Lipsey [1960]) and for the US (Grossman [1974]). If the same change has simultaneously occurred for the price inflation data then the analysis of this paper suggests that these economies have become more liable to fluctuations in the level of activity. However, Matthews (196s) has argued that the cyclical fluctuations of output in the UK have not increased over the same time period. Grossman (1974) has also asserted that cycles have not increased in amplitude. Since the model presented here has assumed no changes in government policy the conclusion is suggested that government fiscal and monetary policy in recent times should be ascribed a more significant counter-cyclical role than simple tests indicate. The post Second World War economy may be more prone to fluctuations and, in the abscnce of government policy, may have yielded a greater cyclical movement than the economy of the earlier period.


APPENDIX

A The Lipsey Model

9 = A~a(Y ,g (Y ,P) )+b~-g (Y ,P) ) - YI P - = B(Y,Y) P Linearizing around the equilibrium point ( Y = Y*, P = P*)

Then the characteristic equation is 0 = D2+D(Alx- A lgZ(u2-b l )P*B2~-Al~z (a2-b l~

P*BI. For stability the coefficients must have the same sign. The crucial necessary and sufficient condition is

This expression reduces to the condition in the text. B The Kuska Model

Y = A{a( Y,g( Y,P))+ b( -g( Yip))- Y ) P - = j(Y- Y*)-k(P-P,). P Linearizing around the equilibrium point (Y = Y*, P = P*) the svstem becomes

1 -ua,+g,(b,-ua3+g,(b,-uadP*Bz >*o.

The Eharacteristic equation is 0 = D' + D[A,x+ P*k] +Alp* [xk +g2 j(bl :.ax)]. All the coefficients of this equation are positive and so the system is stable. C The Expectations-Augmented Phillips Curve

r = A(a(Y,s(Y,P,rr),x)t..b(n-g(Y,P,K))- Y l

P - =j(Y-Y*)+n P

. P z = n(B - n)

Linearizing

The coefficients of the characteristic equation

0 = aQD3+alD2+alD+a5 are a, = 1 a1 = A,{ 1 - 0 1 +g,(b, - az)} ax = Alj(P*g,(bl-n*)-n(uxg,fa,+bi-big3} a, = A&P*g,(bi- a&. Necessary and sufficient conditions for stability are (Samuelson [1947] p. 433) a,, a,, a3, a,@,- aoa, > 0.

Thus the crucial condition is A,%l - a , + g l ( b l - a z ~ ~ ~ P * g x ( b ~ - - a , ) - ~ ~ ~ ~ g ~ + ~ 3 + bl- b,g3)} - AljnP*g,(6, - a2) > 0. This reduces to the condition in the text.


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