Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
Page 1 of 7
Suggested Solutions for Problem Set #5
1. (2 points total)
The Phillips Curve equation can be described by the equation π = πe - (u - u*) + ss.
A. (½ point) Suppose the expected inflation rate is 2 percent,  = 0.6, the natural rate of unemployment is 5 percent, and
there are no supply shocks. When the unemployment rate is 5 percent, what is the inflation rate? When the unemployment
rate is 4 percent, what is the inflation rate?
When u = 5%, π = 0.02 – 0.6(0.05-0.05)+0 = 0.02 = 2%
When u = 4%, π = 0.02 – 0.6(0.04-0.05)+0 = 0.026 = 2.6%
B.
(½ point) Explain why the inflation rate increases when the unemployment rate decreases.
When output increases and thus labor demand increases and thus (assuming no change in labor force) the unemployment
rate decreases, firms will need to offer higher wages in order to entice workers to work for them. These higher wages that
result from the lower unemployment rate are then passed on to consumers in the form of higher prices in all types of
goods and services. By definition, an aggregate increase in price levels means an increase in the inflation rate
We are using a linear Phillips Curve to make the math easier. Most estimates of the Phillips curve are nonlinear, with a
steeper PC at very low unemployment and a relatively flat PC at very high unemployment. To see why it is usually nonlinear (but not for us, just to keep the math easier), think about how much wages change when unemployment is either
extremely high or extremely low. When unemployment is extremely high, firms can hire workers without offering higher
wages – indeed, if unemployment is very, very high, firms can probably hire workers at wages that are lower than what the
firm is currently paying. Workers may feel lucky to simply have a job, and are unlikely to be picky about the wages that are
offered. Conversely, when the availability of workers is relatively low because unemployment is low, firms that want to
hire additional workers will need to offer much higher wages in order to lure newly hired workers away from their existing
jobs. Again, the changes in wage costs are passed on to customers in the form of higher prices.
C. (½ point) Draw a Phillips Curve and label it PC1. Label the point on PC1 when u=u* and there are no supply shocks.
Suppose the expected inflation rate increases. Draw the new Phillips Curve on the same set of axes and label it PC2.
An increase in the expected inflation rate shifts the Phillips Curve up.
D.
(½ point) Draw a Phillips Curve and label it PC1. Label the point on PC1 when u=u* and there are no supply shocks.
Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
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Suppose wages and prices become more flexible. Draw the new Phillips Curve on the same set of axes and label it PC2.
An increase in price & wage flexibility increases the size of  and makes the Phillips Curve steeper. But there is no change
in u* and no change in the expected inflation rate, so the Phillips Curve “pivots” around that common point where u=u*
and  = e.
2. (2 points total)
The Taylor rule can be described by the equation r = r0 + r( - t). Suppose the Fed’s “normal” baseline value of the real
interest rate is 3 percent, r = ½, and the Fed’s target inflation rate is 2 percent.
A. If the actual inflation rate is 2 percent, what real interest rate will the Fed set? If instead the actual inflation rate is 5
percent, what real interest rate will the Fed set?
Our Taylor rule is: r = r0 + rπ(π – πt)
Substituting the values from the prompt: r = 0.03 + 0.5(π – 0.02)
If π = 2%, then r = 0.03 + 0.5(0.02 – 0.02) = 0.03 + 0.5(0) = 3%
If π = 5%, then r = 0.03 + 0.5(0.05 – 0.02) = 0.03 + 0.5(0.03) = 4.5%
B.
Explain why the Fed would increase the real interest rate if the inflation rate rose.
Quite simply, the Fed does not like inflation and it understands the connection between inflation and unemployment (that
you explained in #1B). The Fed knows that the way to combat inflation is to raise the unemployment rate, reducing
pressure on wages and prices. By raising the real interest rate, the Fed can lower output (via the IS curve) and thus raise
unemployment (via Okun’s law).
C. Draw a graph of the Taylor rule and label it TR1. Suppose the composition of the FOMC changes and becomes more
“dovish.” Draw the new Taylor rule and label it TRdove.
D. Suppose instead the composition of the FOMC changes and becomes more “hawkish.” On the same set of axes you used
in part C, draw the new Taylor rule and label it TRhawk.
The terms “dovish” and “hawkish” refer to how committed the central bank is to fighting inflation. As you argued in (B),
combating inflation requires incurring the economic costs of lower output and higher unemployment. In our Taylor rule,
Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
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the Fed’s attitude towards inflation is captured in the parameter r π.
TRhawk
r
• A “dovish” Fed views the economic costs of fighting inflation as
large and will not change the real interest rate much in response to
changes in inflation. That corresponds to a small rπ and a relatively
flat Taylor rule.
• A “hawkish” Fed is committed to fighting inflation. It will change
the interest rate a lot in order to ensure inflation remains near its
target level. That corresponds to a large rπ and a relatively
steep Taylor rule.
TR1
TRdove
r0
Note that regardless of the dovishness or hawkishness of the Fed,
there is one common point to the Taylor Rule graph: when inflation
equals the Fed’s target rate (when 𝜋 = 𝜋 𝑡 ), then the Fed’s target
for the interest rate will equal the “normal” baseline value of the
real interest rate (𝑟 = 𝑟0 𝑤ℎ𝑒𝑛 𝜋 = 𝜋 𝑡 ).
πt
3.
(1 point)
Read and think about this May 2009 article by Glenn Rudebusch of the SF Fed: http://www.frbsf.org/economicresearch/publications/economic-letter/2009/may/fed-monetary-policy-crisis/
From that site, download the excel spreadsheet linked to just above the first sentence of the article. Look at the estimated
coefficients on inflation and on the output gap.
Write and attach a typed double-spaced one-page essay in which you discuss, based on Rudebusch’s article, whether the Fed
followed the Taylor Rule before the crisis began, whether the Fed was relatively dovish or hawkish over the period Rudebusch
studied, and what the post-2008 data tell you about the implementation of monetary policy.
I’m not planning to write an essay here. That was your task!
The Taylor Rule equation that Rudebusch estimated was
FFR = 2.07 + 1.28 x Inflation - 1.95 x (Unem. - CBO natur.)
He has simplified the equation relative to our expression of it by not including a separate term for the target inflation rate.
If we assume, as is reasonable, that the target inflation rate over this period was 2 percent, then Rudebusch’s equation is
equivalent to
FFR = 4.63 + 1.28 x (Inflation – 2% target inflation) - 1.95 x (Unem. - CBO natur.)
This Taylor rule tells us that the Fed reacted more strongly to a 1 percentage point deviation of unemployment from u*
(CBO natur.) than it did to a 1 percentage point deviation of inflation from their target: 1.95 > 1.28. That tells us the Fed
was relatively dovish over the estimation period.
Post December 2008 the predicted Taylor rule was negative. The Fed however respected the zero lower bound (ZLB) and
never set their target below 0. That is, interest rates were not as low post-2008 as economic conditions suggested they
would be if the Fed had continued to follow the same Taylor rule and broken the ZLB.
Denmark, Sweden, and Switzerland all set negative nominal target rates, showing that the ZLB is not sacrosanct. This
2015 VOX article might be of interest: http://voxeu.org/article/monetary-policy-zero-lower-bound or this 2015 IMF working
paper, which suggests a different approach https://www.imf.org/external/pubs/ft/wp/2015/wp15224.pdf. There is lots of
discussion of the IMF paper on the web;search “breaking through the zero lower bound” and follow the links.
π
Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
4.
Spring 2017
Professor Olney
Page 4 of 7
(1.5 points total)
Taylor rule,
𝑟 = 𝑟0 + 𝑟𝜋 (𝜋 − 𝜋 𝑡 )
IS equation,
𝑌=
Okun’s Law,
A.
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
−(
)𝑟
1 − 𝑀𝑃𝐸
1 − 𝑀𝑃𝐸
𝑌 − 𝑌∗
𝑌∗ − 𝑌
𝑢 = 𝑢∗ − 0.4 (
)
=
𝑢
∗
+0.4
(
)
𝑌∗
𝑌∗
(1 point) Derive the equation for the monetary policy reaction function: u = u0 + φ(π - πt)
Deriving the monetary policy reaction function (MPRF) is straightforward, but quite messy algebraically.
Step (1): Plug the Taylor rule into the IS curve…
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
[𝑟 + 𝑟𝜋 (𝜋 − 𝜋 𝑡 )]
𝑌=
−
1 − 𝑀𝑃𝐸 1 − 𝑀𝑃𝐸 0
𝑌=[
(𝐼𝑟 + 𝑋𝜖 𝜖𝑟 )𝑟𝜋
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
−
𝑟0 ] −
(𝜋 − 𝜋 𝑡 )
1 − 𝑀𝑃𝐸
1 − 𝑀𝑃𝐸
1 − 𝑀𝑃𝐸
Step (2): Plug the equation for Y from step (1) into Okun’s law…
𝑌∗
0.4
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
𝑢 = 𝑢 ∗ +0.4 ( ∗ ) − ∗ [(
−
𝑟0 ) −
𝑟 (𝜋 − 𝜋 𝑡 )]
𝑌
𝑌
1 − 𝑀𝑃𝐸 1 − 𝑀𝑃𝐸
1 − 𝑀𝑃𝐸 𝜋
0.4
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
0.4 𝐼𝑟 + 𝑋𝜖 𝜖𝑟
𝑢 = [𝑢 ∗ +0.4 − ∗ (
−
𝑟 )] + ∗ (
) 𝑟 (𝜋 − 𝜋 𝑡 )
𝑌 1 − 𝑀𝑃𝐸 1 − 𝑀𝑃𝐸 0
𝑌 1 − 𝑀𝑃𝐸 𝜋
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑢 = 𝑢0 + 𝜙(𝜋 − 𝜋 𝑡 ), 𝑤ℎ𝑒𝑟𝑒
𝑢0 = [𝑢 ∗ +0.4 −
0.4
𝐴0
𝐼𝑟 + 𝑋𝜖 𝜖𝑟
0.4 𝐼𝑟 + 𝑋𝜖 𝜖𝑟
(
−
𝑟0 )] 𝑎𝑛𝑑 𝜙 = ∗ (
)𝑟
∗
𝑌 1 − 𝑀𝑃𝐸 1 − 𝑀𝑃𝐸
𝑌 1 − 𝑀𝑃𝐸 𝜋
Let’s take a moment to make sense of what we just did. First, what is u0? The unemployment rate will be equal to u0 when
the actual inflation rate π is equal to the central bank’s target inflation rate πt. When the actual inflation rate π is equal to
the central bank’s target inflation rate πt, we know from the Taylor rule that the Fed sets r equal to r 0. In other words, u0 is
the level of the unemployment rate consistent with the Fed setting the interest rate at r 0.
What about φ? φ is the sensitivity of unemployment to inflation and is a combination of four forces:
1) How much the Fed changes the real interest rate in response to changes in inflation, r π.
2) How much planned expenditure changes when the interest rate changes, I r + Xr
3) How much output changes when there is an initial change in planned expenditure. That is, the multiplier, 1 / (1 – MPE)
4) The Okun’s law coefficient (0.4) and potential output (Y*), which determine how much of an effect the change in output
has on the unemployment rate.
B. (½ point) Using words, not the equations, explain how and why the slope of the MPRF changes when there is a decrease
in tax rates. In your explanation, be sure it’s clear what “the slope of the MPRF” means in terms of the relationships between
unemployment and inflation in the economy.
When tax rates are decreased, the spending multiplier is larger. That is, for any given change in autonomous spending,
there will be a larger change in equilibrium output when tax rates are lowered. This is because people are able to retain a
Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
Page 5 of 7
larger share of any increase in income, thus allowing increased consumption spending, thus increasing the injections into
the economy in each round of the multiplier process. A larger change in output means there will be a larger change in
unemployment because unemployment changes in proportion to the change in output.
Therefore when there is an increase in inflation and the central bank responds by increasing interest rates, the decrease in
output and resulting increase in unemployment will be larger at a lower tax rate. This makes the MPRF flatter.
5.
(2.5 points total)
Suppose inflationary expectations are static (never change). Suppose the economy can be described by the following:
Fed’s target inflation rate = 2%
A0 = 4,900
supply shocks = 0
expected inflation rate = 2%
Ir = 18,000
=2
rπ = ½
Xεεr = 12,000
r0 = 3%
u* = 4%
MPE = 0.6
Y* = 10,000
A.
(1 point) What are the equilibrium values of the inflation rate and the unemployment rate? What real interest rate
does the Fed set? What is the equilibrium value of real output?
First, let’s translate these numbers into equations:
Phillips curve:
π = πe – β(u – u*) + ss
π = 0.02 – 2(u – 0.04) + 0
π = 0.10 – 2u
Taylor rule:
r = r0 + rπ(π – πt)
r = 0.03 + ½ (π – 0.02)
r = 0.02 + ½ π
IS Equation:
A0
I  Xr
 r
r
1  MPE 1  MPE
4,900 18,000  12,000
4,900 30,000
Y

r

r
1  0.6
1  0.6
0.4
0.4
Y  12,250  75,000r
Y
Okun’s Law:
 Y * Y 

u  u   0.4

 Y

Y
 10,000  Y 
u  0.04  0.4
  0.04  0.4(1)  0.4
10,000
 10,000 
 Y 
u  0.44  0.4

 10,000 
Now let’s combine our simplified expressions of the Taylor rule, IS equation, and Okun’s law to get our MPRF…
Step (1): Plug Taylor rule into IS equation
Y  12,250  75,000(0.02  1 / 2 )
Y  12,250  1,500  37,500  10,750  37,500
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Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
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Step (2): Plug this expression for Y into Okun’s law and we have the MPRF:
 10,750  37,500r 
u  0.44  0.4

10,000


u  0.44  0.4(0.85  3.75 )  0.44  0.43  1.5
u  0.01  1.5
So now we have two equations in two unknowns (u & π):
MPRF:
u = 0.01 + 1.5π
PC:
π = 0.10 – 2u
Plugging the PC into the MPRF gives us equilibrium unemployment:
u = 0.01 + 1.5(0.10 – 2u) = 0.01 + 0.15 – 3u
4u = 0.16
ueq = 0.04
or
4.0%
And then you can plug u = 4.0% into the PC equation to derive the inflation rate ( = 2.0%)
Or, you can start by plugging the MPRF into the PC to get equilibrium inflation:
π = 0.10 – 2 (0.01 +1.5π) = 0.10 – 0.02 - 3π
4π = 0.08
πeq = 0.02
or
2.0%
And then you can plug π = 2.0 % into the MPRF equation to derive the unemployment rate of 4.0%.
To find the real interest rate, we plug the equilibrium inflation rate into the Taylor rule:
r = 0.03 + 0.5(π – 0.02)
r = 0.03 + 0.5(0.02 – 0.02)
req = 0.03
or
3.0%
To find the equilibrium value of real output, we plug the equilibrium interest rate into the IS equation:
Y = 12,250 – 75,000r = 12,250 – 75,000(0.03)
Y = 12,250 – 2,250
Yeq = 10,000
B.
(1 point) Suppose government spending rises by 200. If the Fed didn’t change interest rates, what would be the new
equilibrium value of real output? The Fed, though, will take action. What are the new equilibrium values of the inflation rate
and the unemployment rate? What real interest rate does the Fed set?
With no change in interest rates, an increase in government spending of 200 increases A 0 by 200. Going back to the IS
curve and setting the interest rate equal to 3 percent, we have
𝑌=
(4900+200)−30,000(0.03)
1−0.6
=
4,200
0.4
= 10,500
But the Fed will raise interest rates to prevent the resulting increase in inflation.
1
The new MPRF starts with the same Taylor rule: 𝑟 = 0.02 + 𝜋
5,100−30,000𝑟
2
1
And a new IS equation: 𝑌 =
= 12,750 − 75,000𝑟 = 12,750 − 75,000 (0.02 + 𝜋) = 11,250 − 37,500𝜋
1−0.6
2
And then plug this IS equation into Okun’s law
𝑢 = 0.44 − 0.4 (
11,250−37,500𝜋
10,000
) = 0.44 − 0.4(1.125) + 0.4(3.75𝜋) = −0.01 + 1.5𝜋
Now solve for the joint solution by combining the MPRF with the PC:
𝜋 = 0.10 − 2(−0.01 + 1.5𝜋) = 0.12 − 3𝜋
4𝜋 = 0.12
Department of Economics
University of California, Berkeley
Economics 100B Problem Set #5 Solutions
Spring 2017
Professor Olney
Page 7 of 7
𝜋 = 0.03
And therefore
u = -0.01 + 1.5(0.03) = -0.01 + 0.045 = 0.035 = 3.5%
The interest rate the Fed will set is r = 0.02 + ½(0.03) = 0.02 + 0.015 = 0.035 = 3.5%
C.
(½ point) Why does the Fed change interest rates in part B?
To fight inflation, the Fed raises interest rates from 3 to 3.5 percent. They do so knowing that a higher interest rate will
loser Investment and Gross Exports and, through the multiplier, affect Consumption and Imports, lowering GDP (here
from the 10,500 it would have reached if they had done nothing to 10,125). The increase in unemployment (here, from the
2 percent it would have fallen to if the Fed had done nothing to 3.5 percent) will slow the growth of wages and prices,
resulting in an inflation rate of 3 percent (rather than the 6 percent it would have risen to if the Fed had done nothing).
Summarizing:
Original conditions: u = 4%, 𝜋=2%, r = 3%, Y = 10,000
After change in G, if Fed does nothing: u = 2%, 𝜋=6 %, r = 3%, Y = 10,500
After change in G, with Fed’s reaction: u = 3.5%, 𝜋 = 3%, r = 3.5%, Y = 10,125
6.
(1 point total)
Suppose inflationary expectations are static (never change). For each part, draw an initial MPRF and Phillips Curve, with initial
equilibrium at u=u*=uo and π = πe = πt . Then, for each part, show the effect of the event described.
A.
(½ point) The government increases its spending.
π
An increase in government spending (G) shifts the IS curve to the right, causing
the MPRF to shift to the left. The economy jumps to point B (lower
unemployment and higher inflation). In response to the higher inflation, the
Fed will raise interest rates and the economy converges to its new equilibrium
at point C. The net effect will be a lower unemployment rate and a higher rate
of inflation.
MPRF2
MPRF1
B
C
πe=πt
A
PC1
u
u*=u0
B.
(½ point) There is a negative supply shock – a war disrupts oil production and raises oil prices.
π
The increase in oil prices results in an increase in the inflation rate. This shifts
the Phillips curve up, increasing the inflation rate at every level of
unemployment. The economy jumps to point B (higher inflation, no change in
unemployment). In response to higher inflation, the Fed raises interest rates
and the economy converges to its new equilibrium at point C. The net effect
will be a higher unemployment rate and a higher inflation rate (DOUBLE
OUCH!).
MPRF1
B
C
πe=πt
A
PC2
PC1
u*=u0
u