ECONOMICS 3012
Notes III: Symbols and basic Macroeconomic Algebra
Prices and Output
Part B: Short-Run Aggregate Supply, the SAS curve and Price-Adjustment Dynamics:
Dynamics and symbols:
Since the analysis of the Short-Run is dynamic, I’ll begin with a description of how time works in
dynamic analysis. For this unit, all subscripts on Variables denote the time period.
Time is broken into discrete time periods, which are denoted t. All Variables of interest in dynamic
analysis are subscripted with a t, ie with a time period. Each variable takes on a single value in each
time period, but, unless in equilibrium (which means we’ve moved to the Medium-Run), each
subscripted endogenous variable takes on different values in each different time period.
The symbol, t, is used two ways. In the first, a t can be number, an integer: 0, 1 ,2, 3, …, n. Used
this way, 0 is always followed by 1; 1 is always followed by 2; … n-1 is always followed by n; etc. But
using them like this is often awkward. The second way the symbol, t, is used is more general. In this
way, we have t-1, t, t+1, etc. t-1 is always followed by t which is always followed by t+1, but t can have
any integer value.
The figure below illustrates the two uses of the symbol, t. First note that everything begins at t = 0,
or “period 0”, no matter where in real time we are beginning. Now, to make sense of the time period
subscripts, consider a variable, Y, in period 2: this is labeled Y2. It has a unique value in period 2. If we
are in period 2 then t=2. Yt-1 is the unique value of the variable, Y, in period 1; Yt-2 is the unique value
of the variable, Y, in period 0; Yt=1 is the unique value of the variable, Y, in period 3; and Yt+2 is the
unique value of the variable, Y, in period 4. Similarly, in period 13, t-1 is period 12, and t+1 is period
14. In general, in period n, t-1 is period n-1, and t+1 is period n+1.
time
0
1
2
3
12
13
14
n-1
n
n+1
A digression on the “natural rate” of unemployment, and “natural” levels of employment and Output
Before I begin the analysis, I will derive the “natural” rate of Output, Yn. In class I showed Yn as the
limit to Y from the Production Possibilities Curve (or Frontier). It is. But the question that we need to
answer in macro-economics is: How do we know where Yn actually is? Or to rephrase the question:
How do we know when an increase in Aggregate Demand will lead to a rise in Prices, P, rather than
only an increase in Output, Y? The answer, which is what Chapter 9 of BJM is all about, is that this
point is determined in the labour market. In the “real world”, it is a rate of unemployment which is
operationally defined by its relation to inflation. If an increase in Aggregate Demand causes the
unemployment rate to fall below the “natural” rate, Prices, P, rise as Output increases.
Theoretically, the “natural” rate of unemployment is the point on the wage-setting relation where
Wage-setting is equal to Price-setting. Begin with Wage-setting, which I rewrite as linear: W = Pe(w0 –
w1u). Since Prices are constant at the “natural rate” by our definition, Pe = P and we can rewrite
Wage-setting as: W = P(w0 – w1u) 
(B-1) W/P = w0 – w1u.
From Price-setting: P = (1+)W  P/W = (1+) 
(B-2) W/P = 1/(1+).
Econ 3012: Notes on Symbols and Algebra
page 2
The “natural rate” of unemployment is where (B-1) = (B-2): w0 – w1un = 1/(1+), so
“natural rate” of unemployment
un = [w0 –1/[(1+)]/w1
[And I will echo BJM here in pointing out the absurdity of the name. It should be called the “structural
rate” of unemployment because it is determined by the structure of the economy. For example, that
rate is higher in Canada than in the US because Canada has a smaller population scattered around a
larger country. Milton Friedman called it “natural” because he was making a polemical point, and the
word “natural” has a greater impact on non-economists than does the word “structural”.]
From this and definitions we get the “natural” level of employment. U is the number of units of
unemployed labour, N is the number of employed “units” of labour, and L be the number of “units” of
labour participating in the labour force. u  U/L = (L – N)/L = 1 – (N/L). BJM and I, here, assume that L
is constant, so un =1 – (Nn/L), so (Nn/L) =1 – un and:
“natural” level of employment
Nn = L(1 – un).
From this and the production function, Y = N, we get:
“natural” level of Output
Yn = Nn
Short-Run Aggregate Supply: the SAS curve:
To the MAS curve we now add the Short-Run aggregate supply curve, the SAS curve. [Note: in
the text this is just labeled the AS curve. The text does MAS curve implicitly, leaving only one AS
curve. But I think it is much easier to understand this if we draw formal curves for both time periods:
Short-Run and Medium-Run.] The SAS curve is derived from the Wage-setting relation, the Pricesetting relation, and the Production Function. These are described in detail in BJM, Chapter 9, so I
need not repeat that. The curve itself is derived in Chapter 10.1. As is their custom, BJM derive the
SAS curve using functional notation, because they have it as a curve, rather than a line. I give a linear
equation, which is an approximation to the true shape, which is a curve.
The Short-Run analysis to the right of Yn is inherently dynamic. So the variables on the SAS curve
must have time subscripts.
SAS curve
This can be rewritten as
Pt = Pe + f[(Yt – Yn)/Yn]
Pt = Pe + f [(Yt /Yn) – 1],
where f is, as per our convention, a parameter. The parameter f determines by the percentage
increase in Prices, P, given a percentage increase in Y above Yn .
We now have another equation with two simple parameters, e and  two exogenous variables, Pe
and Yn ; and two endogenous variables, Yt and Pt , for each time period. Pe is the expected Price level
– that is, the level of Prices, P, that workers in period t-1 expect for period t. I will follow the book and
set Pe = Pt-1, except for a few unusual circumstances.
Linear AD curve
But first, to make life easy for you, I give you a linear approximation to the AD curve. The non-linear
one works ok for some purposes, like Problem Set 5, Part II and Problem Set 6. But for finding shortrun equilibria, a linear approximation is necessary.
We have the non-linear AD curve from Notes II, Part D:
Y = A +  (M/P).
To make this linear I take the inverse relation between Y and P and make it a negative relation
Y =A + M – P.
Finally, because the short-run analysis is dynamic, I add time subscripts to variables:
Econ 3012: Notes on Symbols and Algebra
Linear approximation AD curve
page 3
Yt =At + Mt – Pt
This is an equation with three complex parameters, , , and ; two endogenous variables, each
representing policy, At and Mt ; and two exogenous variables, Yt and Pt , for each time period. These
are the same two endogenous variables as in the SAS curve, so we can have a solution. This solution
is the Short-Run equilibrium. Note that there is one solution for each time period. Because both
“curves” are linear, the solution is relatively easy.
Solution:
I think the best way to do this – it is the more intuitive of the two possibilities because the norm is to
have a new AD curve as the initial change – is to substitute Pt from the SAS curve into the linear,
dynamic-like, AD curve:
Yt =At + Mt – {Pe + f [(Yt /Yn) – 1]}.
I won’t do the algebra here, and you are not responsible for it, but this simplifies to:
Yt = [Yn/(Yn +f)] At + Mt + f]– [Yn/(Yn +f)] [Pe].
To make things easier, set [Yn/(Yn +f)] = .  is a complex parameter. The solution can now be written:
Short-Run equilibrium solution
Yt = At + Mt + f]– Pe.
Price-Adjustment Dynamics: Expansionary Policy:
While I’ve referred to the solution as a “Short-Run equilibrium” it does NOT have the properties of a
normal equilibrium. In most of your economics courses, “equilibrium” is where the system is headed, it
is a “stopping point”. A normal equilibrium is the point where two (or more) curves meet. Once that
point is reached, the economic forces work to keep the system at that point. The “Short-Run” equilibria
here are not stopping points. The fact of moving to a new Price level, P, shifts up the SAS curve,
which will cause a new Short-Run equilibrium in the next period. The Short-Run equilibria here, then,
are not stopping points but points on a path to a stopping point – the Medium-Run equilibrium. Finding
the stopping-point is relatively easy. This is what is described in Part A above, and is what you do in
Problem Set 6.
Finding the Short-Run equilibria – there are, theoretically, an infinite number of these (but relax, I
won’t ask you to find all of them) – is a dynamic process. That is, it is a process that occurs in formal
time periods. We call it “Price-Adjustment”.
The way Price-Adjustment works is that you begin with a period, the initial period, t = 0. In the initial
period there exists an initial equilibrium. The initial equilibrium is a Medium-Run equilibrium.
In period 1 either fiscal or monetary policy changes, which means either A1 > A0 or M1 >M0. Either
will cause the economy to move to a new Medium-Run equilibrium. Either will also cause a Short-Run
equilibrium with Yt > Yn . This will be followed by a series of Short-Run equilibria as the system moves
toward the new Medium-Run equilibrium. Below I describe how to do these, the Price-Adjustments,
algebraically. Theoretically they go on for an infinite number of periods. On Problems I will have you
do only two or three periods. Below I describe how to do the first three periods. The analysis for period
t = 2 is just the same as for period t = 1, except that all “0”s become “1”s, and all “1”s become “2”s. The
analysis for periods t = 3,n are just repeating what you do for the period t = 2, but changing the t
subscript numbers appropriately.
---------------------------------First, to make life easy, find  from the information given.
Period 1:
With  known write out the Short-Run solution with the correct values of A1 and M1 . Use that ShortRun solution to find Y1 . Remember that Pe is just Pt-1 , so use the value given for P0 for Pe in the Short-
Econ 3012: Notes on Symbols and Algebra
page 4
Run solution. Substitute this value of Y1 into the SAS curve and find P1. If you have been given an LM
curve, substitute the values of M1 and P1 into it to find i1 .
Period 2:
Just repeat. Begin by writing out the Short-Run solution with new values: A2 = A1 and M2 = M1 unless
you are given different information, and Pe is now P1 . Use that Short-Run solution to find Y2 .
Substitute this value of Y2 into the SAS curve and find P2. Remember that you have a new SAS curve
because Pe is now P1 . If you have been given an LM curve, substitute the values of M2 and P2 into it to
find i2 .
Period 3:
Just repeat, again. Begin by writing out the Short-Run solution with new values: A3 = A2 and M3 = M2
unless you are given different information, and Pe is now P2 . Use that Short-Run solution to find Y3 .
Substitute this value of Y3 into the SAS curve and find P3. Remember that you have a new SAS curve
because Pe is now P2 . If you have been given an LM curve, substitute the values of M3 and P3 into it to
find i3 .
It’s obvious from the process in time described above what to do with Periods 4,n.
-----------------------------------------------------------------------The way Price Adjustment works is roughly shown on Figure III-B-1 below.
Figure III-B-1
P
MAS
AD0
P2
AD'
2
SAS2
Pmre
SAS0,1
P1
1
P0
0
Yn Y1
Y2
Y
Initial equilibrium, Period 0, is Y = Yn, and P = P0 . A change occurs; either expansionary fiscal or
expansionary monetary policy. This shifts AD from AD0 to AD'. The new Medium-Run equilibrium is P
= Pmre, and Y = Y*. Price adjustment is about how we get there.
In Period 1, move up SAS1 , to where it intersects AD' at Point 1. At Point 1, Y = Y1 and P = P1 .
These are found, as described above.
In Period 2, the SAS has shifted up to SAS2 because the previous highest Price, Pt-1 , is now P1. Move
up SAS2 to where it intersects AD' at Point 2. At Point 2, Y = Y2 and P = P2 . These are once again
found as described above.
Repeat for Period 3. Keep repeating for Periods 4, 5, …, n.
---------------------------------------------------------------
Econ 3012: Notes on Symbols and Algebra
page 5
This system of dynamic equations continues at t = infinity. As time approaches infinity, Y
approaches Yn and P approaches Pmre . This is convergence. Figure III-B-2 shows more convergence,
by enlarging the part of the graph where the action is taking place.
Figure III-B-2
MAS
P
P4
AD'
Pmre
P3
P2
P1
1
AD0
P0
Yn
Y2
Y1
Y
Y4
Y3
Values of Y approach Yn, and values of P approach Pmre , but in ever-decreasing amounts. That is,
Y4 is closer to Y3 than Y3 is to Y2 , and Y3 is closer to Y2 than Y2 is to Y1 . The higher the value of the tsubscript, the closer the Yt is to Yn. And P4 is closer to P3 than P3 is to P2 , and P3 is closer to P2 than
P2 is to P1 . The higher the value of the t-subscript, the closer the Pt is to Pmre . This is convergence,
and here we know exactly to what values the variables are converging.
Price-Adjustment Dynamics: Price Shock
BJM also show you price-adjustment dynamics in the case of a Price Shock. In my opinion this is
unnecessary. The movement to the Medium-Run Equilibrium is smooth and doesn’t offer any
interesting insights, unlike the situation with expansionary policy. In fact, the process shown in BJM is
incoherent if one tries to work it out with algebra. Which means, in my opinion, that it doesn’t show the
real world at all. In Problems you will do Price Shocks only in the Medium-Run.
A Brief Look at Instability: Accommodating Monetary Policy:
The two important questions in dynamics are first: “Does the system converge? – that is, “Is the
system stable?” Here the answer is “Yes”. The second question is: “To what values does it converge?”
Here that answer is the Medium-Run equilibrium.
We now look at one simplified version of a situation that is unstable. That in when the government,
through its agent the Central Bank, “accommodates” the rising Prices that follow expansionary policy
that takes the economy, in the Short-Run, to levels of Output higher than the “natural” level, Yn.
“Accommodating” monetary policy means that the government, or its agent the Central Bank, increases
M as P rises in such a way as to hold M/P constant. There could be two reasons for this: 1) the
government is convinced that it can, and wants to, maintain the level of Output, Y1 . More likely, the
government is convinced that it can, and wants to, maintain the level of unemployment associated with
level of Output, Y1. Remember that as we move up the AD curve, from point 1 to the Medium Run
Equilibrium, the LM curve is steadily shifting up to the left as P increases. This causes the interest
rate, i, to rise. The government might not want the interest rate to rise.
Econ 3012: Notes on Symbols and Algebra
page 6
Either of these motives can cause the government to “accommodate” the rising Prices. To repeat, it
does so by having M increase as P increases, holding M/P constant. This will cause the LM curve to
remain in a fixed position. But, because M is increasing, the AD curve will shift up. I will show the
results for three periods on Figure III-B-3 below. If the government were to hold Mt/Pt constant, the
model would have no solution and the situation is essentially incoherent. So here, as I describe this,
and as you do the algebra, what the government does is set the percentage increase in M in period t
equal to the percentage increase in P in period t-1. The algebra of this is that the government causes
Mt = Mt-1{1+[(Pt – Pt-1)/Pt-1]} . [NOTE: In Problem 7, since we start with P0 = 1.0, this can be simplified
to setting Mt = Mt-1Pt-1 .] This causes the AD curve to shift up as the SAS curve shifts up, so that the
economy does not move along the AD curve. The situation is unstable because it doesn’t converge.
The economy just keeps going up the dashed line,
, which is vertical above a level of
output, Yq , where Yq > Yn . Prices rise forever.
Figure III-B-3
MAS
P
P3
AD3
AD2
SAS3
3
AD0 AD1
SAS2
2
P2
SAS0,1
P1
1
P0
0
Yn
Y1 =
Y2 =
Y3 =
Yq
Y
As before, we start with the initial equilibrium with AD0 , Y =Yn , and prices fixed at P0 .
Expansionary fiscal or monetary policy, probably triggered by the belief that unemployment could be
reduced to lower than un , shifts the economy to AD1 . The economy moves up along SAS1 to a ShortRun equilibrium at point 1, with Y = Y1 and P = P1
Left alone, the price-adjustment dynamics would have the SAS curve continually shift and the
economy would move up along AD1 to the new Medium-Run equilibrium as described above. But that
movement does two things: 1) it causes Y to steadily fall, ultimately back down to Yn , which means
unemployment rises, ultimately back up to un ; and 2) the interest rate steadily increases, causing a
crowding-out effect. The government doesn’t like either 1) or 2) or both, so rather than leaving the
economy alone, it accommodates.
As the economy begins to move up along AD1 , accommodation sets M2 such that there is a new AD
curve, AD2 . Meanwhile, the increase in Price from P0 to P1 has caused the SAS curve to shift up to
SAS2 . There is a new Short-Run equilibrium at point 2 , with Y = Y2 = Y1 , and P = P2 such that (P2 –
P1) = (P1 – P2) .
Again, left alone, the price-adjustment dynamics would have the SAS curve continually shift and the
economy would move up along AD2 to the new Medium-Run equilibrium as described above. But,
again, the government doesn’t like either 1) or 2) or both, so rather than leaving the economy alone, it
Econ 3012: Notes on Symbols and Algebra
page 7
accommodates. As the economy begins to move up along AD2 , accommodation sets M3 such that
there is a new AD curve, AD3 . Meanwhile, the increase in Price from P1 to P2 has caused the SAS
curve to shift up to SAS3 . There is a new Short-Run equilibrium at point 3, with Y = Y3 = Y2 = Y1 , and
P = P3 such that (P3 – P2) = (P2 – P1) = (P1 – P0) . Prices never converge, they just keep rising.
And so on forever.
Except not, because as people begin to expect this constant inflation, the SAS curve begins to shift
up faster. And if the government continues to accommodate, the AD curve shifts up faster. Prices
accelerate. Then, people begin to expect accelerating Prices and … . The final result is constantly
accelerating inflation. This is known as “hyper-inflation”. This is modeled in Economics 4012.