1. The Mathematics Behind Monetary Policy
and the Phillips Curve
Christopher Braden Vernet
Fall 2015
1

2. 1 Introduction
In economics there is this idea called creative destruction. A new technol-
ogy challenges an entire industry by creating a better and cheaper product.
This forces competition to either adapt the new technology or perish. Sim-
ilarly economics goes through a similar cycle. A certain economic theory or
idea will reign supreme until a massive economic downturn happens then a
new theory will rise from the ashes of the old. For example the Great De-
pression saw Adam Smith’s classicalism change to Keynesianism which was
replaced with Milton Friedman’s Monetarism, which was ditched during the
Great Recession in 2008 with the implementation of Quantitative Easing.
One of the ideas that has adapted and survived was known as the Phillips
curve. The basic idea is that in the short run inflation and unemployment
are negatively related. Higher inflation leads to lower unemployment and
vice versa. Tamara Todorova explored the mathematics behind the Phillips
Curve in her paper The Economic Dynamics of Inflation and Unemployment.
In her paper she discusses the stability of critical points in the Phillips Curve
model depending on different linear economic theories and models. In this
paper I want to expand that analysis with an emphasis on different mone-
tary policies. Instead of a constant monetary policy, the central bank adapts
a monetary policy based on unemployment and inflation levels, which can
become a nonlinear relationship.
2 Methods
For each of these models we will start with the mathematics to hypothesize
about the long term behavior of the solution. Then we will populate these
models with real and artificial variables. Artificial variables are used to
illustrate the behavior and don’t have any basis in the real world. On the
other hand real variables will be based on the behavior of the US dollar
between January 1990 and December 2007. This period was chosen because
it was a relatively calm time for US Economics. It was some time after the
early 1980s energy crisis and ends right before the Great Recession. The
data was obtained from the St. Louis Federal Reserve and the US Bureau of
Labor Statistics. All the data can be accessed at the url [9]. All models will
be graphed using Matlab’s ODE45 or ODE15s if the behavior of the solutions
are intractable.
2

3. 3 Initial Model
Our initial model of the Phillips curve is based off the Todorova’s Extended
model [Todorova, 135]. It assumes that expected inflation is equal to the
actual inflation. This may not necessarily be true in the short run and will
be discussed later. Her model is based off four factors: The unemployment
rate (U), the natural rate of unemployment (UN ), the inflation rate (q) and
the change in the money supply (m).
Unemployment is a function of aggregate demand. Aggregate demand is
the sum of a nation’s consumption, investment, government spending, and
net exports. When aggregate demand goes up, more goods and services are
demanded from the economy, meaning employers need to hire more staff
decreasing unemployment. Therefore, aggregate demand is a function of the
nominal money supply (M) divided by the price level (p). The nominal
money supply is how many dollars exist in the economy. On the other hand
the price level is how much stuff that money can buy. This level is calculated
via goods baskets. Every month the Bureau of Labor Statistics sends agents
around the country collecting prices on different goods. The average of these
prices is called the Consumer Price Index. When aggregate demand goes
rises, demand for labor also rises which decreases unemployment. However
because of diminishing returns, this relationship is logarithmic as opposed to
linear.
U = γ − β ln
M
p
β, γ > 0 [Todorova, 135] (1)
Take the derivative by time and we get
dU
dt
= −β(
d ln(M)
dt
−
d ln(p)
dt
) = −β(
M
M
−
p
p
) = −β(m − q) (2)
where U is the rate of unemployment, m = M
M
is the percent change in the
money supply and q = p
p
is the change in the price level divided by the price
level which is the definition of inflation.
In modern economies m (the percentage change in the money supply) is
based on the policy of a country’s central bank. If the central bank increases
the money supply then m is positive, on the other hand if the central bank
decreases the money supply then m is negative. There are many schemes
that a central bank can use to determine monetary policy and no two coun-
tries have the exact same monetary scheme. In this paper we are going to
3

4. examine three general schemes: constant, free floating, and inflation target-
ing. Free floating schemes are where the central bank is given free reign on
monetary policy. However they will almost always expand the monetary base
when unemployment is high, and reduce it when inflation is high. Inflation
targeting is when the central bank tries to bring inflation into a certain range
[Truman 5]. Constant monetary policy is when the central bank will expand
the monetary base at a constant rate without regard to unemployment or
the inflation rate.
For now we will assume that the central bank will have a constant mone-
tary policy. The change in unemployment is related to the difference between
the inflation rate and the monetary policy. If inflation is lower than the mon-
etary expansion, this will encourage spending, investment and exports which
will bring unemployment down. In essence the central bank is “tricking”
the economy into lower unemployment. In effect the central bank convinces
people that they have more purchasing power then they actually do. This
“overheats” the economy. On the other hand when inflation outstrips the
monetary policy, the average person loses purchasing power so consumption
decreases which increases unemployment.
In the long run the unemployment rate will reach an equilibrium which
Milton Friedman called the natural rate of unemployment (UN ) [Friedman,8].
It is “the level of unemployment that occurs in long term overall equilibrium
if market imperfections exist on the labour and commodity markets [Bofinger
101].” This rate can be different between countries and can even change over
time. However changes in the natural unemployment rate are usually due
to government policies or shifting demographics [Blanchard 176]. In other
words, the natural rate of unemployment cannot be changed by monetary
policy. However, in the short run it is possible for the unemployment rate to
deviate from the natural rate of unemployment by manipulating the interest
rate. From this we get the equation
dq
dt
= −α(U − UN ) α, UN > 0 (3)
When the unemployment level is lower than the natural rate of unemploy-
ment then people are spending and investing more and thus demand for all
goods and services go up. This causes prices to rise which increases inflation.
If unemployment is higher than the natural rate then people are spending
and investing less so inflation goes down. Therefore we now have a system
of linear equations that can be solved.
4

5. From (2) and (3) we get the following system of equations.
dq
dt
= −α(U − UN )
dU
dt
= −β(m − q)
(4)
The main assumptions in Todorova’s paper is that all the parameters are con-
stant. The central bank has a constant monetary policy, the natural rate of
unemployment doesn’t change, and the relation between inflation and unem-
ployment doesn’t change. Immediately we can see that the nonhomogenous
linear system has a steady state solution (m, UN ). The linear part of (7) is
represented by the matrix
0 −α
β 0
We find that the eigenvectors for this matrix ±
√
−βα
2
= ±i
√
βα
2
. Because β and
α are positive, the real part of the eigenvalues are zero. The solution to this
system form a family of periodic orbits with the critical point as their center.
3.1 Sample Model
To illustrate the system’s stability, we can populate the model with vari-
ables. Using the variables UN = 5, m = 2, β = .08 and α = .23. UN is based
off the US natural unemployment rate which is considered to be between 5%
and 6%. m is based off the Federal Reserve’s target inflation of 2% [2]. α and
β are based off a linear regressions taken from annual data taken between
1990 and 2007. [Cite data dropbox]. Starting with the initial condition of
2.1% inflation and an unemployment rate of ranging from 5% to 5.4%. and
letting t go from 0 to 40 we get Figure 1.
According to this model the economy is periodic. This makes sense math-
ematically because the eigenvalues of this matrix are purely imaginary. Also
the solution moves in a counterclockwise direction. This means that when
unemployment is low, inflation is increasing, because the economy is heating
up. When unemployment gets too high, then the economy starts to cool
down and inflation decreases. This simulates the business cycle, where un-
employment increases for a while, then decreases. Furthermore, Hoesk and
Zhan hypothesize that the economy would move in this direction in their
unemployment-inflation cycle [Hosek and Zahn 248]. However in real life the
economy does not fluctuate this regularly.
5

6. Figure 1: Red dot is the critical point. Black points are the initial conditions.
Notice that the solutions move in a clockwise direction.
4 Variable Monetary Policy
Holding m to be constant is a legitimate hypothesis. “The Federal Open
Market Committee judges that inflation at the rate of 2 percent ...is most
consistent over the longer run with the Federal Reserve’s mandate for price
stability and maximum employment.”[2] So taking a constant monetary pol-
icy would be a reasonable assumption. And during normal economic times
this tends to be true. This hypothesis breaks down during difficult economic
times. Looking at Figure 2 we can see that before 2008 that the growth of
the money supply was constant and stable. During these periods the un-
employment rate and inflation rates were relatively normal. However during
the Great Recession that started in August 2008 we can notice that the
monetary policy expanded drastically a few times. During this period the
Fed increased the money supply to bring down unemployment in a program
called Quantitative Easing.
The Fed should behave according to a few rules The monetary policy is
a function of unemployment and inflation: m = f(q, U). We assume that m
has the following properties
1. m( ¯m, UN ) = ¯m
2. mU ≥ 0
6

7. Figure 2: US Adjusted Monetary Base Feb 1984-Sept 2015[3]
3. mq ≤ 0
It is assumed that the Fed has a target monetary policy that it won’t deviate
from when the economy is doing well. This goal is denoted as ¯m and defined
as m( ¯m, UN ) = ¯m. When the economy is stable then the central bank does
not need to deviate from their stated goals. From our quote above we can see
that the value of ¯m is 2. If unemployment gets too high, then the Fed will
increase the monetary base in order to stimulate the economy. Therefore
mU ≥ 0. Lastly, if inflation gets too high then the Fed will decrease the
money supply in order to bring prices back down. Therefore mq ≤ 0.
Under these assumptions our linear system (4) becomes
dq
dt
= −α(U − UN )
dU
dt
= −β(m(q, U) − q)
(5)
Notice that ( ¯m, UN ) is still an equilibrium solution by property 1 of m. Using
these assumptions we can update our linearization matrix to include variable
monetary policy.
0 −α
−βmq( ¯m, UN ) + β −βmU ( ¯m, UN )
7

8. If we assume that the monetary policy is constant then mU = mq = 0. This
would give our original linearization matrix. Furthermore we can update our
eigenvalues.
−βmU ± (βmU )2 − 4αβ(1 − mq)
2
(6)
We know that α, β and 1 − mq are positive (property 3) therefore the eigen-
valuses real parts are negative whenever mU (UN , ¯m) < 0. This means that
solutions initially near the critical point will will always converge to the
critical point. How the solution converges depends on the relation between
(a)= (βmU )2
− 4αβ(1 − mq) and zero. If (a) is greater than zero then the
solution converges along one of the eigenvectors of the matrix. On the other
hand if (a) is less than zero then the solution spirals towards the critical
point.
4.1 m(U, q) = ¯m + ec1(U−UN )
− ec2(q− ¯m)
This is a simplistic model that a central bank could use to determine its
monetary policy. When the unemployment level is above the natural rate
the ec1(U−UN )
gets large and will start to dominate. This is equivalent to the
central bank expanding the money supply to decrease unemployment. On
the other hand when inflation exceeds the central bank’s target, −ec2(q− ¯m)
will dominate. The central bank is constricting the money supply to reduce
inflation. c1 and c2 are positive real numbers that represent the central bank’s
approach to monetary policy. A relatively high c1 represents an activist
central bank who are willing to reduce unemployment at the expense of
higher inflation. A relatively high c2 represents a stability-oriented central
bank that tries to maintain a low inflation rate[Bofinger 113]. In order to
maintain the three properties, c1 and c2 are both positive. At the critical
point (UN , ¯m), m(UN , ¯m) = ¯m+1−1 = ¯m, mU = c1 > 0 and mq = −c2 < 0.
Thus m fits our profile.
This equation has a few properties that make it ideal to study the three
rules effects on the system. First it is easy to calculate and adjust mU and
mq. Second, higher deviations lead to disproportionately larger reactions.
For example with
m = ¯m + c1(U − UN ) − c2(q − ¯m)
the increase in money supply is the same if unemployment goes from 5%
to 6% as it would if it went from 6% to 7%. This isn’t realistic. A 6%
8

9. unemployment rate is relatively benign while a 7% unemployment rate is
considered problematic. The following equation would also have increasing
responses.
m = ¯m + c1(U − UN )k1
− c2(q − ¯m)k2
where k1 and k2 are odd integers (even integers violate assumptions 2 and
3). The problem with this is that this complicates the modeling. Instead of
having to determine two variables, we would have to look over four. There
is nothing wrong with this equation, however the original m is simpler.
Replacing m into the original equations we get the system
dq
dt
= −α(U − UN )
dU
dt
= −β( ¯m + ec1(U−UN )
− ec2(q− ¯m)
− q)
(7)
giving us the coefficient matrix
0 −α
c2β + β −c1β
From this we can get the eigenvalues of
−c1β±
√
(c1β)2−4αβ(c2+1)
2
. If (c1)2
β >
4α(c2 + 1) then the solution will converge directly along some vector to the
fixed point. On the other hand if (c1)2
β < 4α(c2 + 1) then the solution will
spiral in towards the fixed point. The central bank’s approach to monetary
policy will affect the movement of the economy. The long term solution
remains constant.
4.2 An Illustration on Different policies
We will be reusing the numbers from section 3 in our updated model. This
gives us the following system.
U = −.08(2 + ec1(U−5)
− ec2(q−2)
− q)
q = −.28(U − 5)
(8)
From our analysis above, if c1 <
c2
2−14
14
then we have a stable node, however
if c1 >
c2
2−14
14
then we have a stable focus (See Figure 3).
Let’s assume that the central bank holds deviations from equilibrium
equally. In other words, a 6% unemployment is just as bad as a 3% inflation.
9

10. Figure 3: Bifurcation diagram for c1 and c2 between 0 and 10
If for c1 = c2 then for any real number h, m(UN + h, ¯m + h) = ¯m. Starting
simply c1 = c2 = 1, we get negative complex eigenvalues (−.08±
√
.992i
2
) so the
critical point is a stable focus. This is true for all c’s less than 14.93. What
Figure 4: c1 = c2 = 1 Figure 5: c1 = c2 = 14.94
happens if the central bank handles deviations more harshly? As you can
see in Figure 5, the solution converges directly when c1 = c2 = 14.94. Next,
we want to study other values that would produce a stable node. By taking
c1 = 7 and c2 = 1 our chart in Figure 6 that the critical point should be as
10

11. stable node. All the initial conditions arrived along the same vector.
Figure 6: c1 = 7 c2 = 1
4.2.1 Inflation Targeting
So far the examples have supposed that the central bank will try to
influence the economic output. This is not always the case. In Inflation
targeting the central bank attempts to achieve price stability by attempting
to bring inflation to a certain level [Truman 6]. In other words the central
bank ignores the unemployment rate and achieves long term stability by
keeping the inflation rate low. This policy means that that c1 = 0, m =
3 − ec2(q−2)
and our model becomes.
dq
dt
= −α(U − 5)
dU
dt
= −β(3 − ec2(q−2)
− q)
(9)
Because the eigenvalues are now purely imaginary, we need to see use other
methods to determine the long term behavior. However se see that (9) is a
Hamiltonian system with Hamilton
V =
(U − 5)2
β
+
1
α
q
2
ec2(s−2)
+ s − 3 ds (10)
11

12. ec2(s−2)
+ s − 3 is positive when s > 2 and negative when s < 2. Therefore
V (q, U) → ∞ as ||(q, U)|| → ∞.
V =
(U − 5)2
2β
+
1
α
q
2
ec2(s−2)
+ s − 3ds
˙V =
2U (U − 5)
β
+
2(ec2(q−2)
+ q − 3)q
α
˙V =
2(U − 5)(−β)(−ec2(q−2)
− q + 3)
β
+
2(ec2(q−2)
+ q − 3)(−α)(U − 5)
α
˙V = 2(ec2(q−2)
+ q − 3)(U − 5)) − 2(ec2(q−2)
+ q − 3)(U − 5)) = 0
(11)
It can be shown that this equation forms a closed shape that is symmetric
around U=5. If we are given δ > 0 determined by V (q0, U0), we can get
δ =
(U − 5)2
2β
+
2
c
ec(q−2)
+ q2
− 6q + 8 − 2
c
α
(U − 5)2
= 2β(δ −
2
c
ec(q−2)
+ q2
− 6q + 8 − 2
c
α
)
(12)
Because δ is constant and the integral
q
2
ec(s−2)
+s−3ds → +∞ as |q| → ∞
we can see that V is a closed path that is symmetric around U=5. That
means that the solution cannot diverge off to infinity nor can it converge to
the critical point. Thus the solution is periodic with a stable closed orbit
around the critical point.
By setting c2 to 0, 1, 2, 4 we can see how the central bank’s policy will
affect the economy. In figure 7 the blue path was the shows our original
model when m = ¯m = 2. By increasing c we can show that the range
of inflation’s will decrease while the range of unemployments will remain
relatively constant. From this we can see that by increasing c1 the solution
will converge faster than if we increase c2.
5 Expected inflation
One of the first major hurdles of the Phillips Curve was Stagflation in the
1970s. In an attempt to bring unemployment down, the FED increased the
money supply. The problem was people started to expect higher inflation,
so the effect of expanded monetary policy was reduced. In other words, to
12

13. Figure 7: Blue-c2 = 0, Black-c2 = 1, Red-c2 = 2, Cyan-c2 = 4
reduce the unemployment by the same amount, the FED needed to increase
inflation to higher levels. This phenomenon is called expected inflation(π).
We are going to first see how the model works with a constant monetary
policy, then we will create a experiment with the pseudo-monetary policy
discussed in section 4.
Our expectation inflation equation is based on the idea of adapted ex-
pectations. People will update their expectations when new information is
presented. In this context people have their idea of what inflation is going
to be based on past experiences. Thus the expected inflation will look like
π = j(q − π) j ∈ (0, 1] [Todorova, 134] (13)
If j=1 then people will base their expectations on the last period’s infla-
tion. The higher j is, the faster people will change their expectations. Lower
j’s lead to more stable expectations The expected inflation will have an effect
on both inflation and the unemployment rate. Up until now we have been
assuming that q = π. If the central bank is honest and consistent then as
time goes on, then q should approach π. Todorova simplifies the inflation to
a linear function of unemployment and expected inflation.
q = c − αU + hπ h ∈ (0, 1] [Todorova, 134]
π = j(c − αU + (h − 1)π) j ∈ (0, 1]
(14)
13

14. Figure 8: Expected inflation over time. Notice it starts to converge to somewhere
between 1% and 2%
What this means is that if we can show that U and π converge, then q
will also converge. In Todorova’s original model, U’ was a function of the
monetary policy and inflation. “When inflation is high for too long, this may
discourage people from saving, consequently reduce aggregate investment
and increase the the rate of unemployment [Todorova, 134].” However, this
sounds more like expected inflation than actual inflation. When we were
assuming expected inflation equaled actual inflation this was fine. However
now we need to update our unemployment function.
U = −β(m − π) (15)
The monetary policy should still be the same. In real life it is easier to
calculate actual inflation than expected inflation. The government has many
incentives to maintain data on inflation. Therefore there tends to be more
up to date data on actual inflation than expected inflation. Thus it is easier
to obtain inflation data than expected inflation data. However, because
we are looking in terms of U and π, we need to update our function from
m = f(q, U) to m = g(U, π) we get m = f(U, c − αU + hπ) where f holds the
same properties as m. This gives us the equations
mU = fU − αfq ≥ 0
mπ = h · fq ≤ 0
(16)
14

15. Therefore even though m now must be in terms of π and U, it still holds its
original traits. We can therefore check the stability of our system.
−j(1 − h) −jα
−βhfq + β −βmU
We get the eigenvalues of
−(j(1−h)+βmU )±
√
(j(1−h)+β ˙mU )2−4[β ˙mU j(1−h)−jβ2(hfq−1)]
2
.
Because all the variables except fq are positive and h < 1, we can conclude
that U, π, and, by extension, q, converge. The method of convergence de-
pends on the sign of (j(1 − h) + βmU )2
− 4(jβ2
(hfq − 1)). If this is greater
than or equal to zero then the series converges linearly, however if it is less
then zero the solution spirals in towards the center.
5.1 Examples
Up until this point we have defined the natural unemployment rate as
the rate of unemployment when the economy is steady. However there are
many ways to determine the natural unemployment: “the natural rate of
unemployment is the unemployment such that the actual inflation rate is
equal to the expected inflation rate.” [Blanchard 170] Using data from 1990
to 2007 I estimated h to be equal to .84. If we want to keep the natural rate
of unemployment we simply need to find c such that
q = c − (.23 · 5) + .84q
This is satisfied with c=1.47. Lastly, we will take a j=.5. j in this case
is an artificial variable because the actual variable was very small. In our
scenario, the expected inflation will change based on 1/2 of the difference
between expected and actual inflation. we get the following system.
q = 1.47 − .23U + .84qπ
U = −.08(2 + ec1(U−5)
− ec2(−.53−.23U+.84π)
− q)
π = .5(1.47 − .23U − .16π)
(17)
Thus our eigenvalues are equal to
−.08 − .08mU ± (.08 + .08mU )2 − 4(.0064mU − .0032(−.84c2 − 1))
2
15

16. Convergence will depend on wether 4(.0064mU −.0032(.84fq −1)) is greater or
less than (.08 + .08mU )2
.Starting with a constant monetary policy we know
that mU = 0 and fq = 0 so .0064 − 4 · .0032 < 0 so the series will spiral
towards the critical point (Figure 9).
Figure 9: Convergence when m = 2
Bringing in the previous function for variable monetary policy we can
see that fu = c1 and fq = c2 converges directly to the fixed point. We
can see that the traits of the free floating bankng policies the path’s con-
vergence didn’t change. However, the inflation targeting model (Figure 13),
actually converged. By factoring in the expected inflation we have actually
increased the convergence of the model. Before a constant monetary policy
was periodic. Furthermore inflation targeting is convergent not periodic.
6 Conclusion
When accounting for expected inflation, the monetary policy will converge
to the critical point. Furthermore more aggressive policies will lead to faster
convergence. However there are a few assumptions that must be taken into
account. First this is still a simple model, there are many things that it
doesn’t take into account, such as energy prices, foreign prices, interest rates
16

17. Figure 10: c1 = c2 = 1 Figure 11: c1 = c2 = 14.94
Figure 12: c1 = 7, c2 = 1 Figure 13: c1 = 0, c2 = 4
economic bubbles and busts etc. All of these can affect the inflation and
unemployment rate. Nowhere in this model does it show that a country’s
main export suddenly dropped in value, causing a spike in unemployment
and inflation. Second, this assumes that the central bank knows and targets
the natural rate of unemployment which might not always be the case. Also
the natural rate of unemployment might change over the long run. Lastly,
central banks tend not to use a formula to determine economic policy. If that
was the case, then most central banks would be run by a computer.
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17

18. [2] Board of Govoners of the Federal Reserve. Federal Reserve, Web. 30 Nov.
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[9] https://drive.google.com/folderview?id=
0ByXZyPgM4REnUWUweGhCRUs4MFU&usp=sharing
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