OUTPUT/INFLATION DYNAMICS
Derivations and Analysis:
Dynamic Aggregate Demand (DAD) curve and the
Dynamic Short-run Aggregate Supply (DSAS) curve
Jeremiah Allen
©1982; 2004
II.
SAS
The Model
The derivation of the Dynamic Short-run Aggregate Supply curve, DSAS, begins with
the derivation of the static Short-run Aggregate Supply curve, SAS. The derivation of the
SAS begins with an observed empirical regularity, the Phillips curve. [Note: BJM begin here
too. They add mark-up pricing to get to their Aggregate Supply curve in Chapter 9. This isn’t
necessary; we need only the empirical fact that Prices don’t fall.] In 1958 A. W. Phillips
observed a remarkably stable relation between the unemployment rate, u, and the rate of
change of money-wages, W, 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 a lovely piece of theory, Lipsey also provided an
explanation for the loops. [This explanation of the loops forms the basis for the labour market
dynamics we’ll see later.]
For the theoretical analysis at this stage, I ignore the loops and treat the Phillips curve as
linear. There are two SAS curves: the Flexible-Price SAS(FP) and the New-Keynesian
SAS(NK). As with the MAS curve, with FP aggregate prices, P, can fall; that is, inflation, ,
can be negative. So the SAS(FP) is a straight line. With NK aggregate prices can not fall; that
is, inflation is constrained:   0. So the SAS(NK) is horizontal at  = 0. NK is more realistic
since Prices have not fallen measurably in an industrial market economy since 1938, and it is
closer to the curve of the original Phillips curve.
For the derivation of the SAS curve, first transform the Phillips curve with the observation
that money-wage increases are the sum of productivity increases (prod) and inflation (): W =
(prod) + . This shifts the horizontal axis up from the line [W = 0] to the line [ = 0], and
changes the vertical axis from W to . Now write the equation of a linear approximation to the
transformed Phillips curve:
W = – r(u), where u is the unemployment rate and r is a parameter
(II. i)
Add the transformation: t + (prod) = – r(u); and subtract (prod) from the left-hand side:
t = – r(u).
(II. ii)
Now denote the point where the transformed linear approximation crosses the new abscissa –
the u axis here – as un. un is “full employment”. In Friedman’s language it is “the natural rate of
unemployment”, or, in a more neutral term, it is the “NAIRU” (Non-Accelerating Inflation Rate of
Unemployment.) This is the level of unemployment associated by Okun’s Law with potential
output, Yn. Add this transformation:
The linear approximation to the twice transformed Phillips curve is:
t = –ř(u – un).
which is shown on Figure II-1.
(II. iii)
DAD and DSAS
page 2
Figure II-1
transformed
Phillips
curve

linear approximation (II. iii)
u
Finally, substitute Okun’s Law for the right hand side. Okun’s Law is:
u= un –v[(Y – Yn)/Yn]  (u – un) = –v[(Y – Yn)/Yn]
(II. iv)
where v is “Okun’s constant”. Okun’s constant was once considered to be around 1/3, but
indications are that it is now somewhat higher. Substitute Okun’s Law, equation II. iv, into
equation II. iii. This gives a linear SAS:
SAS: = f [(Y – Yn)/Yn],
(II. v)
where f = (ř v). The two SAS curves are shown on Figure II-2.
Figure II-2

MAS
SAS(FP&NK)
SAS(NK)
0
Y
Yn
SAS(FP)
DAD and DSAS
III.
page 3
DSAS
A. The Model
The seeming stability of the Phillips curve, and thus of the SAS 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 SAS represents unacceptably
stupid behavior, if inflation persists. Their extension of the Phillips curve analysis makes it
dynamic, and that analysis is where we begin the derivation of the Dynamic Short-run
Aggregate Supply curve, the DSAS curve
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 supply of labour (unemployment) was low or
negative. But as the economy shifted from expansion to contraction, the rate of money-wage
inflation quickly dropped as excess supply (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 SAS 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. 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 SAS curve, will shift up or down as
expectations rise or fall. The DSAS curve builds those shifts into its equation. The equation
has the DSAS curves intersect the MAS curve at the expected rate of inflation,e. We model
these shifts as taking place in discrete time, just like our shifts of the DAD. So we start by giving
time subscript, t, to the variables in SAS, which becomes
SASt :t = f [(Yt – Yn)/Yn] .
(II. v)
[At this stage, having derived the SAS curve from the Phillips curve, I will derive DSASt
from the SASt, rather than beginning with the Phillips curve again.]
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 > Yn ↔ uX < un The government would do this if
it thought SAS0 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 period of inflation
equal to x%, people begin to expect inflation of x% (e = x%). The SASt curve shifts up – from
SAS0 to SAS1 – to cross Yn at  = x% .
This insight led to the DSAS curve, which is derived by having the SAS curve augmented
with expectations. The DSASt is the short-run aggregate supply curve we will use. The
equation of the DSASt is:
DSASt:
t = f [(Yt – Yn)/Yn] + et .
(III. i)
Note that this is not yet quite dynamic; there are no explicit lags in the DSASt equation. The lag
structure has to enter through expectations – that is, by making et a function of lagged
DAD and DSAS
page 4
variables. (There is a lag in DSASt; Yt is a function of Yt-1, which is a function of t-1 , but it’s
implicit not explicit.)
Figure III-1

MAS
NK
SAS1
1
SAS0
x%
0
0
YX
FP
Y
Yn
This now forces attention to how expectations are formed. Notice that we now have two
equations, DSASt and DADt, but we now have three unknowns: Yt , t , and et . To able to
solve these we need a third equation. The equation I will 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:
AEt:
et = et-1 + g(t-1 – et-1);
0<g≤1
(III. ii)
Equation (III. ii ) gives a value of et which is determined exclusively by lagged values of itself
and  ; these are all pre-determined. This also converts DSASt to be explicitly dynamic; et is a
function of t-1 , giving an explicit lag structure. Note that if g = 1, et = t-1 ; any period’s
inflation immediately becomes the next period’s expected inflation. But if g < 1 and t-1 is held
constant, expectations converge to t-1.
[NOTE: Expectations formation is still controversial. Variations include the strangely-named
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)
would be an argument in the equation of expectations. We won’t do that; adaptive expectations
works well and is much simpler to work with.]
We have now three equations, DADt, DSASt and AEt, 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.]
DAD and DSAS
page 5
__________________________________________________
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

DAD0
MAS
DSAS0
0
Yn
Y
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. These values are always
given.
2.
A good short cut is to substitute the DSASt: t = f [(Y1 – Yn) / Yn ] + et-1 , for t in
DADt and solve for a reduced form equation in Yt. You will have an equation in just
Yt , Yt-1 , et , and numbers. [This equation is given on Problem 4; it is: Yt = [Yt-1 +
at + mt + f + et]/[1 + (f /Yn)] The divisor is all numbers, so find this equation
as numbers and variables. You will use this a lot, so keep it handy; it will make all
solutions easier. Call it the working equation.]
3.
Find e1 . This is entirely determined by lagged values, that is, by values of e0 and
0 .
4.
Substitute the values of Y0 and e1 into the working equation; that equation is now
all numbers and Y1 ; solve for Y1.
5.
Substitute this value of Y1 into DSAS1; that equation is now all numbers and 1 ;
solve for 1 .
This is recursive, so just keep repeating, substituting t=2 for t=1; then t=3 for t=2; then etc.
EXCEPT:
New Keynesian: 4. and 5: IFt < 0, set t = 0, and substitute t = 0 into DADt and solve for Yt.
OR
DAD and DSAS
page 6
IF there has been a supply shock, then 1 > 0 > 0 exogenously. Substitute the exogenous
value of 1 into DAD1 and solve for Y1. This works ONLY for Period 1. For all periods after,
follow steps 3 thorough 5.
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 DADt, the DSASt, the MAS(NK), and the AEt 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 are actions which cause inflation.
The NK model and the FP model are the same for Periods 1 and 2, so the descriptions below
apply to both. Because I don’t want to give you the answers to Problem #4 yet, I will describe
only Period 1 and Period 2. I will show only DAD0 , DAD1 and DAD2 , and DSAS0 , DSAS1
which is the same as DSAS0 , and DSAS2 . I will use only g = 1 in the descriptions below.
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 DAD1 shifts up, to the right, of DAD0. This gives a Short-run value,
Y1 , that is greater than Yn : 1050. 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 1 , that is greater than 0: 1 = .05. 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 = .05.
What happens next depends on the monetary response and the values of the parameters, f
and . With NO accommodation, and with the special parameter values of Problem #4, where f
= 1 and  = Yn, the DAD curve shifts back. With these parameter values, it shifts halfway back.
So the DAD and DSAS in Period 2 intersect at Y2 = Yn and 2 = .05.
If there is lagged accommodation, in Period 2 the DAD curve stays at DAD1. The DSAS
curve has shifted up, as above. The intersection is halfway up the DAD curve from the
equilibrium of Period 1. This is shown as the asterisked values of “2”, “2*” on Figure III-3 below.
This continues for all time periods, the DSAS curve in each period shifting so the equilibrium
moves a distance halfway up the DAD from the old equilibrium to the Medium Run Equilibrium,
MRE, which is where DAD1 (= DADt for all t) crosses MAS.
Figure III-3
DAD2

DAD1
2*
DAD0
2
MRE
MAS
DSAS2
DSAS0,1
2*
1,2
1
0
0
Yn Y1
Y0,2
Y
DAD and DSAS
page 7
Y2*
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 exogenously
such that inflation in Period 1 is 1 =.10. The SAS curve shifts up, to be flat at 1 . This gives a
short-run value of Y1 that is less than Yn. What happens after this depends on the monetary
response and the values of the parameters, f, g, and .
For g =1 and NO accommodation, the DAD curve shifts down to the left, crossing the Output
axis at 900. DSAS shifts up to cross MAS at .10. The two intersect at y = 900 and  = 0. With
accommodation, DAD2 is the same as DAD1 and DSAS2 is the same as with no
accommodation. The case with accommodation is shown as the asterisked values of “2”, “2*”
on Figure III-4 below.
Figure III-4
MAS
DAD2
DAD0,1

SAS1
2*
DSAS2
DSAS0
1
1
0
0
2
Y1,2 Yn
Y
MAS(NK)
2*
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 mt = 0 for all t. DSAS1 is the same as DSAS0 because it intersects
MAS at e1 0 . The DAD curve shifts down, to the left as m1 is set to zero. Output is reduced
in Period 1: Y1 = 950 < Yn. Inflation is reduced in Period 1 because of the leftward shift of DAD1
: 1 = .05.
In Period 2, DSAS2 is below DSAS0,1 . On Figure III-5, the case where g = 1 is illustrated, so
DSAS2 crosses MAS at 1 = .05. The values given in Problem #4 will have the system move to
zero inflation, 2 = 0, and unemployment, Y2 = 950 < Yn.
DAD and DSAS
page 8
In future periods, t = 3,n : in the NK model, Yt = 950, and t = 0, for all t; in the FP model,
Aggregate prices, P, will fall so that inflation will be negative ( < 0), for several time periods.
With FP, eventually full employment and zero inflation is reached.
Figure III-5
DAD2

DAD0
DAD1
MAS
0
DSAS0,1
DSAS2
0 = .10
1
1
2
0
Y1,2
Yn
Y