OUTPUT/INFLATION DYNAMICS
Derivations and Analysis:
Dynamic Aggregate Demand (DAD) curve and the
Dynamic Short-run Aggregate Supply (DSAS) curve
Jeremiah Allen
©1982; 2004
[NOTE: I derive the DSAS curve from the Phillips curve and Okun’s law. BJM develop the
Phillips curve from theory. I like the first approach – my approach – for two reasons: 1) This is
how macroeconomic theory actually developed historically; and 2) It has theory develop to
explain the observed world, rather than having theory develop out of thin air. Note also that
BJM don’t actually derive a DSAS curve as such; they have both the Phillips curve and
Okun’s Law and use both in their solutions in Chapter 12. The combination of the Phillips
curve and Okun’s law is the DSAS curve, but BJM don’t take the final step. So they don’t
have graphs like we will have, with inflation, , on the vertical axis, and output, Y, on the
horizontal.]
[NOTE ALSO: BJM use different symbols than I do. I will stick to mine and not try to show
where the two differ. You will use my symbols in all problems, SSExs and exams. Read BJM
for history and policy. Their model is the same as ours except for symbols.]
II.
DSAS
The Model
The derivation of the Dynamic Short-run Aggregate Supply curve (DSAS) begins with
an observed empirical regularity, the Phillips curve. In 1958 A. W. Phillips observed a
remarkably stable relation between the unemployment rate, ut, and the rate of change of
money-wages, (Wt – Wt-1)/Wt-1 , in the UK for nearly 100 years: 1861 –1957. There were three
interesting features of the Phillips relation: 1) It appeared to be stable over 100 years. 2) It was
non-linear. 3) Each business cycle followed counter-clockwise loops. Richard Lipsey, a few
years later, provided a simple Demand and Supply model, with downward wage rigidities, as the
theoretical basis for the Phillips curve. In an additional lovely piece of theory, Lipsey also
provided an explanation for the loops. (This explanation of the loops forms the basis for some
of the labour market dynamics we’ll see later.)
For the theoretical analysis, I ignore the loops and treat the Phillips curve as linear. There
are two DSAS curves: the Flexible-Price DSAS(FP) and the New-Keynesian DSAS(NK). As
with the MAS curves, with FP, Aggregate Prices, P, can fall; that is, inflation, , can be
negative. So the DSAS(FP) is a straight line. With NK, Aggregate Prices can not fall; that is,
inflation is constrained:   0. So the DSAS(NK) is horizontal at  = 0. NK is more realistic
since Aggregate Prices have not fallen measurably in an industrial market economy since 1933,
and it is closer to the curve of the original Phillips curve.
The equation of a linear approximation to the Phillips curve is:
(Wt – Wt-1)/Wt-1 = – r(ut).
(II. i)
Transform the Phillips curve with the observation that money-wage increases are the sum of
productivity increases (prod) and inflation (): (Wt – Wt-1)/Wt-1 = (prod) + t . Substituting:
t + (prod) = – r(ut).
Subtract (prod) from the left-hand side. This shifts the horizontal axis up from the line [(Wt –
Wt-1) / Wt-1 = 0] to the line [ = 0]. Now denote the point where the transformed curve crosses
the new abscissa as un. un is “full” employment. In Friedman’s language, “the natural rate of
(II. ii)
DAD and DSAS
page 9
unemployment”, or, a more neutral term, the “NAIRU” (Non-Accelerating Inflation Rate of
Unemployment.) It is the level of unemployment associated by Okun’s Law with potential
output, Yn.
The linear approximation to the transformed Phillips curve is:

t = –r(ut)
Now subtract un from ut : this has no algebraic effect except to shift the parameter value:
(FP): t = –r’(ut – un)
(NK): t = –r’(ut – un) if ut < un);t = 0 if ut ≥ un .
(II. iii)
These are shown on Figure II-1.
Figure II-1
t
transformed
Phillips
curve
linear approximation (II. iii)
NK
un
ut
FP
Finally, substitute Okun’s Law for the right hand side. Okun’s Law is:
ut = un –s[(Yt – Yn)/Yn] , or (ut – un) = –s[(Yt – Yn)/Yn]
(II. iv)
where s is “Okun’s constant”. Okun’s constant was once considered to be around 1/3, which is
the value used by BJM, but indications are that it is now somewhat higher. Substitute Okun’s
Law, equation II. iv, into equation II. iii. This gives a linear DSAS:
DSAS (FP):t =  [(Yt – Yn)/Yn],
DSAS (NK):t =  [(Yt – Yn)/Yn], if Yt > Yn);t = 0 if Yt ≤ Yn
where  = (r’s). The two DSAS curves are shown on Figure II-2.
Figure II-2
(II. v)
DAD and DSAS
page 10
t
MAS
DSAS(FP&NK)
DSAS(NK)
0
Yt
Yn
DSAS(FP)
Once again, and for exactly the same reasons as given in Part I-B above, in the analyses
that follow, I will describe only the NK model. Remember from those reasons given above that
to do so is, implicitly, also to do the analysis for the FP model.
III.
EA-DSAS
A. The Model
The seeming stability of the Phillips curve, and thus of the DSAS curve, was challenged
initially by Milton Friedman, in the late 1960s. The “challenge” was formalized by Edmund
Phelps a few years later. The basis of the challenge is that the DSAS curve represents
unacceptably stupid behavior, if inflation persists.
The underlying model of the Phillips curve, as developed by Lipsey, is simple Demand
and Supply in labour markets. Over the period Phillips observed his curve, inflation never
persisted. There were always cycles; expansions always ended quickly. Money-wages
increased during expansions, when excess demand of labour was high (unemployment was
low). But as the economy shifted from expansion to contraction, the rate of money-wage
inflation quickly dropped as excess supply of labour (unemployment) increased.
Friedman argued – and virtually all macro-economists now agree – that if governments tried
to hold the rate of unemployment at less than un for some length of time by keeping inflation
constant at a positive level, then that level of inflation would become expected. This would
cause the Phillips curve, and thus the DSAS curve, to shift up. The shift is caused by people
– both buyers and sellers in the labour market – coming to expect the level of inflation that the
government is maintaining. During the period Phillips observed, this had never happened
outside of the two world wars, and the wars were considered unusual – behavior was different.
But once governments began to think they could permanently trade-off a constant level of
inflation for a level of unemployment below un, it could happen, and probably did.
This is nicely described in the Bruce reading, but I’ll describe it briefly here too. The
argument is simple: that the Phillips curve, and thus the DSAS curve, will shift up or down
such that they intersect the MAS at the expected rate of inflation,e. This means that the two
curves – Phillips and DSAS – are no longer stable. So they can shift. We model these shifts
as taking place in discrete time, just like our shifts of the DAD. So from now on, we have to give
the DSAS with a time subscript, t: DSASt. [At this stage, having derived the DSAS curve from
the Phillips curve, I will do the analysis using only DSASt.]
DAD and DSAS
page 11
Consider Figure III-1 below. Suppose the government were to attempt to hold the economy
at Point A, with inflation of x% and output of YX ↔ uX < un. The government would do this if it
believed that DSAS0 was permanent and stable, and if it wanted to keep the unemployment rate
at uX, which is determined by Okun’s Law from level of output, YX . But after a time period,
people begin to expect inflation of x% (e1 = x%). The DSAS curve shifts up – from DSAS0 to
DSAS1 – to cross Yn at  = x% .
Figure III-1
t
MAS
NK
DSAS1
1
DSAS0
x%
0
0
YX
FP
Yt
Yn
This insight led to the Expectations-Augmented DSAS curve or EA-DSAS. The EADSAS is the short-run aggregate supply curve we will use from here on. The equation of the
EA-DSAS is:
EA-DSAS:
t =  [(Yt – Yn)/Yn] + et.
(III. i)
--------------------------------------------------------------------------------------------------------This now forces attention to how expectations are formed. Notice that we now have two
equations, EA-DSAS and DAD, but we now have three unknowns: Yt , t , and et . To able to
solve these we need a third equation. The equation I use here is the one developed by
Friedman and Phelps, and is called “adaptive expectations”. The basic idea is that people
develop expectations on the basis of experience, and that it takes some time for experience to
change expectations. The equation of adaptive expectations is:
AE:
et = et-1 + g(t-1 – et-1) ; g  1
(III. ii)
Equation (III. ii ) gives a value of et which is determined exclusively by lagged values of itself
and . All of the latter are pre-determined. Note that if g = 1, et = t-1 ; any period’s inflation
immediately becomes its next period’s expected inflation.
[NOTE: Expectations formation is still controversial. Variations include the strangely-labeled
Rational Expectations models. Note also, that while it hasn’t been explicitly formulated, or put
into a model, it’s clear from observations – especially in the major recessions of the early 1980s
and early 1990s – that inflation falls, and quickly, when unemployment becomes large. Thus, it
would be both “adaptive” and “rational” to have lower inflationary expectations when
unemployment is substantially higher than un. In a more fully developed model, then, (ut – un)
DAD and DSAS
page 12
would be an argument in the equation of expectations. We won’t do that; this works well and is
simpler to work with.]
We have three equations, DAD, EA-DSAS and AE, and three unknowns: Yt , t , and et .
The system has a solution. Reduced-form equations, in either Yt or t , can be found from these
three equations. [The equation for et is already a reduced-form equation.]
__________________________________________________
EXERCISE: Find the reduced-form first-order difference equations for Yt and t . What will be
the dynamic behavior of each? Your answer to that last question for Yt will turn out to be wrong.
Can you guess why?
---------------------------------------------------------------------------------One solution, the one that will be the starting point for two-thirds of the analysis below, is shown
on Figure III-2:
Figure III-2
t
DAD0
MAS
EA-DSAS0
0
Yn
Yt
To work out solutions for this system, I suggest the following algorithm:
1.
You begin with all variables in Period 0: Y0 , 0, ande0.
2.
Find e1 . This is entirely determined by lagged values, of  and itself; that is,
values of  and e in Period 0.
EITHER
3.
Fiscal or exogenous-monetary policy occurs. Put the values of a1 or m1 into DAD1
and solve.
4.
Substitute the EA-DSAS1: 1 =  [(Y1 – Yn) / Yn] + e1 , for 1 in DAD1. You now
have an equation in just Y1 and numbers. Solve this for Y1 .
5.
Substitute this value of Y1 into the EA-DSAS1, and solve for 1 .
This is recursive, so just keep repeating, substituting t=2 for t=1; then t=3 for t=2; then etc.
OR
3. 4.and 5: If there has been a supply shock, 1 > 0 exogenously. Substitute this value of
1 into DAD and solve for Y1. This works ONLY for Period 1. For all periods after,
follow steps 2 thorough 5 above.
This is recursive, so just keep repeating, substituting t=2 for t=1; then t=3 for t=2; then etc.
DAD and DSAS
page 13
B. Analysis
The analysis is all in the Y,  space. Our model is now fully defined in that space. The
curves we use are the DAD, the EA-DSAS(NK), the MAS(NK), and the AE equation. The basic
model is shown in Figure III-2 above, beginning with a long-run equilibrium of full employment,
Y0 = Yn, and no inflation, 0 = 0. This is the starting point for the first two parts of the analysis
below. Those two parts are actions which cause inflation.
Once again, I will describe only the NK model; once again, what happens in the FP model is
implicit in that description. And once again, because I don’t want to give you the answers to
Problem #4, I will describe only Period 1 and half of Period 2. I will show only DAD0 and DAD1 ,
and EA-DSAS0, EA-DSAS1 which is the same as EA-DSAS0 , and EA-DSAS2 . (Since we’ve
been through the fact that the DSAS curve must be Expectations-Augmented, to simplify I’ll
just refer to it as DSAS from now on.)
I.A on Problem #4: Expansionary policies
This is illustrated in Figure III-3. The economy begins with full employment, Y0 = Yn, and no
inflation, 0 = 0. In Period 1, the economy is hit with expansionary policy, either fiscal or
exogenous-monetary, and DAD shifts to the right. This gives a Short-run value of Y, Y1 , that is
greater than Yn. It must be Short-run since it is greater than Yn – that is, it is not on the MAS
curve. It also gives a value of , 1 , that is greater than 0: 1 > 0. Because 1 > 0, if g = 1 the
DSAS curve shifts up by the amount, 1 , for Period 2. Since e2 = 1, DSAS2 crosses MAS at
1 .
What happens next depends on the monetary response and the values of the parameters, ,
g and . With NO accommodation, and with the special parameter values of Problem #4, where
 = 1, g = 1, and  = Yn, the DAD curve won’t move. DADt = DAD1 for all t = 2,n. As t increases
from 2, each DSASt will be a bit higher than the last, and eventually DSASt will cross DAD
where DAD1 crosses MAS. The system will converge to Yn and MRE . Thus, as you can see,
another condition for an inflationary equilibrium is that the Short-run Aggregate Supply curve
shift as expectations develop.
You’ll work out what happens with accommodation as you do Problem #4, but you can
already see that it is nowhere nearly this simple.
Figure III-3
t
DAD0
DAD1
MAS
DSAS2
2
MRE
DSAS0,1
1
1
0
0
Yn Y1
Yt
DAD and DSAS
page 14
I.B on Problem #4: Supply Shock
This is illustrated on Figure III-4. The economy begins with full employment, Y0 = Yn, and no
inflation, 0 = 0. In Period 1, the economy is hit with a supply shock. Prices rise such that
inflation in Period 1 = 1 . The DSAS curve shifts up, to the left, because 1 is now greater
than zero. DSAS1 crosses DAD at 1 where Y = Y1. (This happens because during this time
period, shift from t = 0 to t = 1, inflation is exogenous.) This gives a short-run value, Y1, that is
less than Yn. What happens after this depends on the monetary response and the values of the
parameters, , g, and .
II. on Problem #4: Disinflation
This is illustrated on Figure III-5 for Case 1: “cold turkey”. The economy begins with full
employment, Y0 = Yn, and 10% inflation, 0 = .10. In Period 1, the government instructs the
monetary authorities to set m1 = 0. The DAD curve shifts down, to the left. Inflation is reduced
in Period 1, so DSAS2 is below DSAS0,1 . On Figure III-5, the special case where g = 1 is
illustrated, so DSAS2 crosses MAS at 1 . Output is also reduced in Period 1: Y1 < Yn. What
happens after this depends on the values of the parameters, , g, and . The values given in
Problem #4 will have the system move to zero inflation, MRE = 0, and unemployment, YMRE < Yn
in the NK model; full employment YMRE = Yn in the FP model. Aggregate prices, P, have to fall,
that is inflation has to be negative ( < 0), for full employment to be restored.
Figure III-4
MAS
t
DAD0,1
DSAS1
DSAS0
1
1
0
0
Yt
Y1 Yn
Figure III-5
t
DAD0
DAD1
MAS
0
DSAS0,1
DSAS2
0 = .10
1
1
2
0
Y1
Yn
Yt