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Detailed Philadelphia (Prism) Forecast Overview
June 2011
Michael Dotsey
Keith Sill
Forecast Summary
The FRB Philadelphia DSGE model denoted Prism, projects that real GDP growth will
rebound strongly over the forecast horizon, with output growth approaching 7 percent by mid
2011. Even with such strong output growth, inflation is well contained, reaching 2.2 percent by
the end of the forecast horizon. Because inflation is the dominant determinant of the federal
funds rate in the model, the forecast is for a gradual increase in the funds rate. Policy begins to
tighten by the third quarter of 2011, and the funds rate reaches 3.2 percent by the fourth quarter
of 2013. Currently, many of the model’s state variables are well below their steady-state values
(see figure 8). In particular, consumption, investment, and the capital stock are low relative to
steady state, and absent any shocks, the model would predict a swift recovery. Further, the wage
rate, and hence marginal cost, is below steady state, implying that, absent any shocks, wages and
marginal cost are expected to increase, leading to an increase in inflation. Also, these state
variables have been below steady state since the end of the recession. The relatively slow
recovery to date and the low inflation that has recently characterized U.S. economic activity
require the presence of shocks to offset the strength of the model’s internal propagation channels.
Overview of the Great Recession and Early Part of the Recovery
Before proceeding to the forecasts, it may be useful to describe how the model accounts
for the Great Recession and what features drove the early stages of the recovery. Because the
model does not contain a financial sector, the past recession is a challenge for the model to
explain. The recession was marked by a severe impairment in financial intermediation that
affected many countries. As a result, risk spreads increased and the cost of funds for both firms
and consumers rose. In addition, many financial markets seized up. The deterioration in the
economy’s ability to allocate resources led to a rapid decline in both investment and
consumption, which was accompanied by a large drop in employment. To capture these
phenomena, the model must identify various shocks that allow it to match the data. Without a
financial sector that endogenously feeds in to the other behavioral relationships, the model is
forced to find ways of reducing consumption, investment, and hours worked.
To match the weakness in desired consumption, the model identifies a shock to the
parameter that governs the way individuals intertemporally allocate their consumption. Thus, the
model identifies an increase in consumers’ rate of time preference, or a decrease in their discount
factor, causing current consumption growth to be weak relative to future consumption growth.
To account for the weakness in investment, the model identifies a negative shock to the
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efficiency of investment. In particular, a unit of investment produces less capital than it normally
would, making investing less desirable. We interpret this shock as one that reflects the efficiency
with which investment funding is allocated across firms. Inefficient allocations of capital lead to
a less productive economy and less desired investment. Also, the extreme fall in employment
cannot be accounted for solely by the discount factor shock and the negative shock to the
efficiency of investment, although both shocks work in the right direction. The model also
requires that individuals desire more leisure. This is a somewhat unattractive feature of the
model, and it is fair to say that additional research needs to be done regarding the way labor
markets are modeled. Finally, a negative shock to productivity is also required in order for the
model to generate sufficient economic weakness. A recent history of the model’s key identified
shocks is presented in figure 6.
The shock to the marginal efficiency of investment and the discount factor shock are the
shocks most closely aligned with the financial crisis. The discount factor shock directly affects
asset prices, and we believe that in our one-sector model, the estimated efficiency of investment
shock is in part the result of an inefficient allocation of investment in the economy. At the
aggregate level, the impediments caused by adverse financial factors probably resulted in an
inefficient allocation of capital, implying that investment was less productive. Also, lack of
consumer credit and relatively unfavorable financing terms were factors in weak consumption
growth, and the negative shock to the discount factor reduces current consumption growth. It
would certainly be desirable to have a structural model of the financial system, but absent that, it
is at least reassuring that some of the key shocks the model identifies as important for causing
the recession are ones that have the tightest association with financial factors.
Inflation did not fall precipitously during the recession period. The primary factor that
contributed to declining inflation was the discount factor shock. The negative discount factor
shock also results in declining wages and marginal cost, which tend to reduce inflation pressures.
Further, negative shocks to firm markups contributed to the fall in inflation in late 2008 and early
2009. Somewhat offsetting the effects of these shocks was the labor supply shock, which raised
wages, marginal cost, and hence inflation as well. Because inflation is the most important
variable in the model’s estimated policy reaction function, these same shocks are the primary
drivers of the funds rate path.
The early stages of the recovery were driven by strong investment and consumption
growth. Replicating this behavior required a waning of the shocks that accounted for the
recession. Thus, the early stages of the recovery were driven by a decline in the magnitude of
both the discount factor shock and the marginal efficiency of investment shock coupled with
strong productivity growth. Productivity grows strongly because in the data output is increasing
while employment remains weak, requiring significant growth in total factor productivity. The
weak employment growth once again is produced through an increased desire for leisure on the
part of individuals in the model. The decline in inflation that accompanies the economy’s
transition into recovery is accounted for by the markup shock turning negative, implying that
firms in the model are facing a reduction in pricing power. Further, the continued presence of
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negative shocks to the discount factor, which helps hold back consumption growth, also helps to
restrain inflation through the usual demand channels. Countering somewhat the negative effects
that these two shocks have on inflation is the positive shock to leisure, which is responsible for
raising marginal cost and inflation, but the magnitude of this shock is falling as employment
picks up. The lowering of inflation is what keeps the funds rate near zero even as the economy
recovers.
The Current Forecast and Shock Identification
The current forecast is shown in figures 7a-7c, which also displays the shock
decomposition. The shock decomposition is a fairly complicated object and will be discussed in
detail in the next section. The key identified shocks are shown in figure 6. In the current forecast,
output is projected to grow robustly, well above the model’s long-run average of 2.7 percent. By
the second half of 2011, output is predicted to grow at about 7 percent. Part of this strength is
due to the model’s state variables having been well below their estimated steady state for a
considerable time, which implies strong growth as the model economy returns to steady state. A
negative government spending shock was a big factor holding down 2011Q1 growth, and since
this shock has low persistence for real output growth, it implies a strong bounce-back in 2011Q2.
The model predicts consumption growth (nondurables+services) will run at close to 4 percent in
2011Q2 driven primarily by discount factor shocks that make current consumption relatively
attractive. As well, investment continues its strong rebound from its 2010Q4 decline. Over the
medium term, the model predicts very strong investment growth that peaks in early 2012.
Consumption growth peaks in 2011Q2 and then declines slowly toward steady state over the
forecast horizon. Essentially, the extremely strong Prism growth forecast is driven by moderately
strong consumption growth and very strong investment growth.
Turning to inflation, the recent ramping up of employment growth led to an estimated
shift away from leisure and toward hours worked on the part of individuals. A positive labor
supply shock tends to push down inflation in the model because it lowers marginal cost. As the
labor market continues to strengthen, it implies less of a cumulative upward pressure on
inflation. However, the labor shock effect is more than offset by the persistent negative effects
that past negative discount factor shocks have on projected inflation. Negative shocks to price
markups also help explain the weakness of inflation over the second half of 2010, but their
effects are not very persistent. On net, inflation is predicted to pick up modestly, rising to 2.0
percent at the beginning of 2012. The slow growth in inflation implies that the funds rate will be
raised only gradually over the forecast horizon.
The Current Forecast in More Detail
To understand the forecast in more detail requires a brief look into the model and the
impulse response functions associated with the key shocks (a detailed description of the model
and its equations are contained in supporting appendices). One also needs to understand why
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these particular shocks are being identified as the primary drivers in the current forecast, as well
as what shocks were important in the past. The need to know past shocks is driven by the fact
that the model’s response to most shocks is persistent. In this document, we display only the
most important shocks. For a description of how every shock affects economic activity in the
model, please refer to the supporting appendix.
The model is a fairly standard New Keynesian model with sticky prices and sticky wages.
Firms and workers are able to reset their prices and wages at random intervals, and those that do
not reset can index their current price (wage) to steady-state inflation. The important implication
of these nominal rigidities is that prices will be a markup over marginal cost, and wages will be a
markup over the marginal disutility of working. The model contains an AR(1) shock to the price
markup, and this shock affects inflation directly. The shock also acts as an inefficient wedge in
the model and therefore affects economic activity — an increase in either markup lowers output.
Production in the model is fairly standard — intermediate goods firms use capital and
labor to produce intermediate goods that are then aggregated via a Dixit-Stiglitz aggregator into a
final good that in turn is used for investment, consumption, and government spending.
Multifactor productivity, or technology shocks, is nonstationary and its growth rate is modeled as
an AR(1). The shock affects output directly and inflation indirectly through its influence on
markups. Thus, increases in productivity increase output growth and reduce inflation.
Investment is subject to adjustment costs that are also influenced by a disturbance, μt.
Specifically,
⎛
⎛ I ( j) ⎞ ⎞
Kˆ t ( j ) = (1 − δ ) Kˆ t −1 ( j ) + μt ⎜⎜1 − S ⎜ t
⎟ ⎟⎟ I t ( j ),
⎝ I t −1 ( j ) ⎠ ⎠
⎝
where μt is a shock to investment efficiency, which may be caused by financial frictions.
Basically, a positive shock implies that fewer resources must be devoted to investment in order
to build new capital. This shock is also specified as an AR(1).
Individuals get utility from consumption, which enters via internal habit formation, and
get disutility from working. Individuals also receive utility from money balances, but this part of
the model is not essential and can be ignored. Preferences are subject to two important shocks:
one to time preference, bt and the other to the labor supply, φt. Thus, momentary utility is given
by
1−ν m
⎡
φt
χt ⎛ M t ( j) ⎞ ⎤
1+ν l
bt ⎢ln(Ct ( j ) − hCt −1 ( j )) −
Lt ( j ) +
⎜
⎟ ⎥.
1 +ν l
1 −ν m ⎝ Z t Pt ⎠ ⎥
⎢⎣
⎦
The preference shocks are composed of a shock to the preference for leisure and a shock
to the discount factor. Both shocks are AR(1) as well.
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Inflation’s relation to output can be seen through the model’s derived Phillips curve. It is
not a structural equation of the model, but it is useful to compare this equation across models to
get an idea of the inflation dynamics in the model. Note that there is no lagged inflation term in
the equation, since we do not index current prices to past inflation. The estimates indicate that
the Phillips curve is quite flat for this model over the sample period 1984Q1 to 2010Q3, and
marginal cost increases are not aggressively passed through to prices.
π t = 0.997 Etπ t +1 + 0.011mct + υt
Also of importance, especially when analyzing the effects that various shocks have on the
funds rate, is the estimated policy rule. As one can see, the movement of inflation relative to
target is the primary driver of monetary policy.
Rt = .81Rt −1 + (1 − .81)(2.25π t + 0.06 y%t )
In order to understand the contributions of the various shocks to output growth, inflation,
and nominal interest rates, it is helpful to look at the impulse responses of the variables in the
measurement equation (see equation (22) in the technical appendix). The impulse responses to
the currently most relevant shocks are shown in figures 1 through 4. We then attempt to link
these responses and the recent behavior of these variables (figure 5) to form an idea of why the
model is identifying certain shocks as important. While far from perfect, this exercise sheds
some light on why the model is identifying a particular state of the economy at a particular time.
The estimated shocks are shows in figure 6. Further, the varying persistence of each shock’s
effect on output growth, inflation, and the interest rate helps to understand the shock
decompositions that are given in figures 7a, 7b, and 7c.
Toward the end of 2009, output grew fairly strongly, but employment did not. To capture
this behavior, the model placed substantial weight on positive technology shocks. However,
weak growth in 2010Q2 was accounted for by increased disutility from work together with
negative contributions from the discount factor shock, government spending, and monetary
policy and an easing of the TFP shock. The model attributes the stronger growth in the third and
fourth quarters in part to an easing of the financial (discount factor) shock. To help the model fit
the weak growth in hours worked also required positive shocks to leisure over the last few
quarters. Further, the recent strong growth in investment relative to consumption caused the
model to identify negative shocks to the rate of time preference (financial shocks) in the first
three quarters of 2010, but the sign of the shock is reversed by the Q4 strength in consumption.
Although investment grew strongly in 2010, the growth slowdown toward the end of the year is
in part accounted for by negative shocks to the marginal efficiency of investment, which helped
to counter the positive effects of negative shocks to the discount rate. In addition, although the
negative discount rate shock and the positive leisure shock move inflation in opposite directions,
near term, the net effect is negative. By 2011Q1, the mei shock is positive as investment growth
showed a rebound and the financial shock has been running positive over the last couple quarters
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as well. To help account for the falling rate of inflation, the model attaches some importance to
negative markup shocks. Finally, to dampen somewhat the positive effect of the (negative)
financial shock on investment, the model attaches some weight to a negative efficiency of
investment shock, which also has a negative effect on inflation.
The forecast is then a product of past shocks working through the model and the fact that
the model’s state variables are below their steady-state values. At first glance, it is hard to see
why output is predicted to grow so robustly. The reason is that the response of output growth to
both the discount factor shock and the marginal efficiency of investment shock involves some
overshooting. Thus, much of the strong output growth in the projection is due to negative shocks
to the discount factor and investment efficiency in 2009 and 2010. Thus, although the initial
response to these two shocks is quite negative, about four or five quarters in the future the
economy undergoes significant bounce-back in response to these two shocks (see figures 2 and
4).
Regarding inflation, the impulse response functions for the TFP and MEI shocks indicate
that these shocks have a small quantitative effect on inflation for a typical-sized shock. Positive
productivity shocks lower marginal cost somewhat and negative MEI shocks also lower inflation
through their negative effect on demand. But these effects are small. Positive labor supply shocks
have a more significant effect. They push up marginal cost and inflation because higher wages
are needed to attract labor, while the negative discount rate shocks and the negative markup
shock act to offset these pressures. On net, the model, therefore, forecasts only a slight increase
in inflation.
The shock decomposition for the funds rate is similar to that of the discount rate in that
the discount factor shock is a key driver of the forecast – pulling down the funds rate below its
steady-state value for the next 3 years. Note, though, the effect of the efficiency of investment
shock. That shock generates overshooting in both output growth and inflation, although the
overshooting occurs after about 5 quarters in the case of inflation and about 17 quarters in the
case of the funds rate. Consequently, the sequence of large negative MEI shocks over 2007-2009
push up inflation over the forecast horizon but are still pulling down the funds rate. The markup
shock impulses indicate a bit of near-term overshooting on inflation and a steady decline toward
steady state for the funds rate. The negative markup shocks at the end of 2010 lead to a slightly
positive impact on inflation in 2011 and a steady negative effect on the funds rate.
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Philadelphia DSGE Forecast Model Impulse Responses
(one-standard-deviation shocks)
Figure 1: Response to technology shock
Inflation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
Percent
Percent Annualized
-0.08
-0.12
5
9
13 17 21
Aggregate Hours Worked
0
-0.05
-0.1
-0.15
9
-0.04
1
Federal Funds Rate
5
0
21
0.05
1
0.04
13 17 21
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0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1
5
9
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RTDRC DSGE Model
Figure 1 (continued): Response to technology shock
Investment Growth
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
1
0.8
0.6
0.4
0.2
0
-0.2
1
5
9
13
17
21
2
1.5
1
0.5
0
-0.5
1
5
Percent Annualized
Real Wage Growth
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
5
9
13 17 21
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RTDRC DSGE Model
Figure 2: Response to investment efficiency shock
Inflation
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
1
5
9
13
17
21
1
Federal Funds Rate
0.20
0.8
0.15
0.6
0.10
0.05
0.00
0
-0.2
9
13 17 21
0.2
-0.10
5
9
0.4
-0.05
1
5
Aggregate Hours Worked
Percent
Percent Annualized
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
13 17 21
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RTDRC DSGE Model
Figure 2 continued: Response to mei shock
Investment Growth
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
1
5
9
13
17
21
10
8
6
4
2
0
-2
-4
1
5
Real Wage Growth
Percent Annualized
0.4
0.3
0.2
0.1
0
-0.1
-0.2
1
5
9
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RTDRC DSGE Model
Figure 3: Response to leisure shock
Inflation
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
5
9
13
17
21
0.4
0.3
0.2
0.1
0
1
Federal Funds Rate
5
9
13 17 21
Aggregate Hours Worked
0.5
0
0.4
-0.2
Percent
Percent Annualized
0.5
0.3
0.2
0.1
-0.4
-0.6
-0.8
0
-1
1
5
9
13 17 21
1
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RTDRC DSGE Model
Figure 3 continued: Response to leisure shock
Investment Growth
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0.2
0
-0.2
-0.4
-0.6
-0.8
1
5
9
13
17
1
0.5
0
-0.5
-1
-1.5
-2
1
21
Percent Annualized
Real Wage Growth
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
1
5
9
13 17 21
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RTDRC DSGE Model
Figure 4: Response to discount factor shock
Inflation
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
1
5
9
13
17
0.3
0.2
0.1
Federal Funds Rate
Percent
Percent Annualized
0.4
0.3
0.2
0.1
0
5
9
5
9
13 17 21
Aggregate Hours Worked
0.5
1
0
1
21
13 17 21
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0.5
0.4
0.3
0.2
0.1
0
-0.1
1
5
9
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RTDRC DSGE Model
Figure 4 continued: Response to financial shock
Investment Growth
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
2
1.5
1
0.5
0
-0.5
1
5
9
13
17
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
1
21
Real Wage Growth
Percent Annualized
0.6
0.5
0.4
0.3
0.2
0.1
0
1
5
9
13 17 21
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Figure 5a
Series used to estimate model
gdp growth
consumption growth
2
2
0
0
-2
-4
1980
1990
2000
2010
2020
-2
1980
1990
2000
2010
2020
aggregate hours
investment growth
5.5
10
5.4
0
5.3
-10
1980
1990
2000
2010
2020
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1980
1990
2000
2010
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Figure 5b
real wage growth
core inflation
4
1.5
2
1
0
0.5
-2
1980
1990
2000
2010
2020
fed funds rate
15
10
5
0
1980
1990
2000
2010
2020
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1980
1990
2000
2010
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Figure 6
Estimated Shock History (in standard deviations)
labor shock
discount factor shock
3
2
2
0
1
-2
0
-1
2006
2008
2010
2012
-4
2006
TFP shock
2008
2010
2012
mei shock
4
1
0
2
-1
0
-2
2006
-2
2008
2010
2012
-3
2006
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Figure 7a
Real GDP Growth
8.0
6.0
4.0
y=2.7
percent
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
2007
2011Q1
2008
tech
2009
labor
2010
mei
fin
2011
gov
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2012
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2013
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Figure 7b
Core PCE
3.5
3.0
2.5
2.0
p=2.0
percent
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
2007
2011Q1
2008
tech
2009
labor
2010
mei
fin
2011
gov
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2012
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Figure 7c
Fed Funds Rate
6
5
r=4.8
4
percent
3
2
1
0
-1
-2
-3
-4
2007
2011Q1
2008
tech
2009
labor
2010
mei
fin
2011
gov
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2012
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Figure 8
Endogenous State Variables
c
i
10
50
0
0
-10
1980
1990
2000
2010
2020
-50
1980
1990
kbar
2
0
0
1990
2000
2010
2020
-2
1980
1990
w
2
0
0
1990
2000
2020
2000
2010
2020
2010
2020
z
10
-10
1980
2010
R
50
-50
1980
2000
2010
2020
-2
1980
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Figure 8
Endogenous State Variables
φ (leisure pref shock)
μ (capital adjust shock)
50
5
0
0
-50
1980
1990
2000
2010
2020
-5
1980
b (intertemporal pref shock)
5
0
0
1990
2000
2010
2020
-5
1980
f
λ (markup shock)
0.5
0
-0.5
1980
1990
2000
2010
2000
2010
2020
g (govt spending shock)
5
-5
1980
1990
2020
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Technical Appendix: PRISM Model
Documentation
Keith Sill
June 6, 2011
1
Model Structure
The FRBPHIL DSGE forecasting model (PRISM) is developed and maintained
by the Real Time Data Research Center (RTDRC) and by the Research Department of the Federal Reserve Bank of Philadelphia. The model is medium-scale
and features nominal and real frictions that include wage and price stickiness,
habit formation, and capital adjustment costs. This section of the model documentation describes the DSGE model, which is essentially the model in Del
Negro, Schorfheide, Smets, and Wouters (2007).
1.1
Final Goods Producers
There is a final good Yt that is produced as a composite of a continuum of
intermediate goods Yt (i)using the technology:
1
Z
Yt =
Yt (i)
1
1+λf,t
1+λf,t
(1)
0
with λf,t ∈ (0, ∞) following the exogenous process:
ln λf,t = (1 − ρλf ) ln λf + ρλf ln λf,t−1 + σλf ?λ,t
(2)
The variable λf,t is the desired markup over marginal cost that intermediate
goods producers would like to charge. From the first-order conditions for profit
maximization and the zero-profit condition (final goods producers are perfectly
competitive firms) the demand for intermediate goods is given by:
Yt (i) =
Pt (i)
Pt
f,t
− 1+λ
λ
f,t
Yt
(3)
with the composite good price given by:
Z
Pt =
1
Pt (i)
− λ1
0
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1.2
Intermediate Goods Producers
There is a continuum of intermediate goods indexed by i. They are produced
using the technology:
1−α
Yt (i) = max{Zt1−α Kt (i)α Lt (i)
− Zt Φ, 0}
(5)
where Zt is exogenous technological progress that is assumed non-stationary. We
define zt = ln(Zt /Zt−1 )and assume that it follows the process:
(zt − γ) = ρz (zt−1 − γ) + εz,t .
Prices are assumed to be sticky and adjust following Calvo (1983). Each firm
can readjust prices optimally with probability 1 − ζp in each period. Firms that
are unable to reoptimize their prices Pt (i)adjust prices mechanically according
to:
ι
1−ι
Pt (i) = (πt−1 ) p (π∗ ) p
(6)
where πt = Pt /Pt−1 and π∗ is the steady state inflation rate of the final good.
Those firms that re-optimize price choose a price level P˜t (i)that maximizes the
expected present discounted value profits in all states of nature in which the
firm maintains that price in the future:
maxP˜t (i) Ξpr P˜t (i) − M Ct Yt (i)+
∞
(7)
P
1−ι
ιp
π∗ p − M Ct+s )Yt+s
Et
ζps β s Ξpt+s P˜t (i)(Πsl=1 πt+l−1
s=1
subject to
f,t
− 1+λ
λf,t
1−ιp
ιp
s
˜
Pt (i) Πl=1 πt+l−1 π∗
Yt+s (i) =
Yt+s
Pt+s
where πt ≡ Pt /Pt−1 , β s Ξpt+s is the household’s discount factor and M Ct is the
firm’s marginal cost. Markets are assumed to be complete so all households face
the same discount factor. All firms that can re-adjust price face an identical
problem. We will consider only a symmetric equilibrium in which all adjusting
firms choose the same price – which means that we can drop the iindex. It then
follows that the aggregate price level can be expressed as:
− λ1 −λf
− λ1
ιp
1−ιp
f
˜
f
Pt = (1 − ζp ) P
+ ζp πt−1 π∗ Pt−1
.
In the estimation, we shut down inflation indexation by setting ιp = 0.
1.3
Households
The objective function for household j is given by:
"
1−νm #
∞
X
χt+s
Mt+s (j)
ϕt+s
1+νl
Et
bt+s ln(Ct+s (j) − hCt+s−1 (j)) −
Lt+s (j)
+
1 + νl
1 − νm Zt+s Pt+s
s=0
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where Ct (i)is consumption, Lt (i)is labor supply, and Mt (j)is money holdings.
Household preferences are subject to three shocks: an intertemporal shifter bt , a
labor supply shock ϕt , and a money demand shock χt . All preference shocks are
assumed to follow an AR(1) process in logs. The household budget constraint,
written in nominal terms, is given by:
Pt+s Ct+s (j) + Pt+s It+s (j) + Bt+s (j) ≤ Rt+s Bt+s−1 (j) + Mt+s−1 (j)+
k
ˆ t+s−1 (j) − Pt+s a(ut+s (j))K
ˆ t+s−1 (j)
Πt+s + Wt+s (j)Lt+s (j) + Rt+s
ut+s (j)K
ˆ t (j)is capital holdings, ut (j)is the rate of capital utiwhere It (j)is investment, K
lization, and Bt (j)is holdings of government bonds. The gross nominal interest
rate paid on government bonds is Rt and Πt is the per-capita profit the household
gets from owning firms. Household labor is paid wage Wt (j)and households rent
ˆ t−1 (j). In return, they
an “effective” amount of capital to firms Kt (j) = ut (j)K
k
ˆ
receive Rt ut (j)Kt−1 (j). Households pay a consumption cost associated with
ˆ t−1 (j). Capital accumulation is governed
capital utilization given by a(ut (j))K
by:
It (j)
ˆ
ˆ
It (j)
Kt (j) = (1 − δ)Kt−1 (j) + µt 1 − S
It−1 (j)
where δis the rate of depreciation, S(·)is the cost of adjusting investment (S 0 >
0, S 00 > 0), and µt is a stochastic shock to the price of investment relative to
consumption, assumed to follow an AR(1) process in logs.
1.4
The Labor Market
The labor market has labor packers that buy labor from households, combine
it, and resell it to the intermediate goods producing firms. Labor used by the
intermediate goods producers is a composite:
1
Z
Lt =
1
1+λw,t
Lt (j) 1+λw,t
0
The labor packers maximize profits in a perfectly competitive environment,
which leads to the labor demand:
Lt (j) =
Wt (j)
Wt
w,t
− 1+λ
λ
w,t
.
Combining labor demand with the zero-profit condition leads to the aggregate
wage expression:
Z 1
λw,t
1
λw,t
Wt =
Wt (j)
di
.
0
In the estimation, we fix λw,t = λw ∈ (0, ∞). Households have market power,
but wage adjustment is subject to a rigidity as in Calvo (1983). Each period, a
fraction 1 − ζw of households re-optimize their wage. For those that are unable
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to re-optimize, Wt (j)adjusts as a geometric average of the steady state rate
increase in wages and last period’s productivity times last period’s inflation.
For those households that can re-optimize, the problem is to choose a wage
˜ t (j)that maximizes utility in all states of nature in which the household wage
W
is to be held at its chosen value:
∞
X
ϕt+s
s
1+νl
maxW
(ζw β) bt+s −
Lt+s (j)
+ ...
˜ t (j) Et
1 + νl
s=0
subject to
∗
˜ t (j)
Wt+s (j) = Πsl=1 (π∗ )1−ιw (πt+l−1 ezt+l−1 )ιw W
for s = 1, . . . , ∞ as well as to the household budget constraint and the labor
demand condition. In the estimation, we shut down nominal wage indexation
by setting ιw = 0.
1.5
Government Policies
The government consists of a fiscal authority and a monetary authority. The
monetary authority sets the nominal interest rate according to the feedback
rule:
ρ " ψR ψY #1−ρR
Rt
Rt−1 R
πt
Yt
=
R,t .
R
R
π∗
Y
The fiscal authority balances its budget by issuing short-term bonds. Government spending is exogenous and given by:
Gt = (1 − 1/gt )Yt
where the government spending shock gt is assumed to follow an AR(1) process.
1.6
Exogenous Processes
There are seven exogenous shocks in the model. The follow the processes:
• Technology process. Let zt = ln(Zt /Zt−1 )
(zt − γ) = ρz (zt−1 − γ) + σz z,t
• Preference for leisure:
ln φt = (1 − ρφ ) ln φ + ρφ ln φt−1 + σφ φ,t
• Money demand (this shock is shut down in the estimation and the model
is not estimated using a monetary aggregate):
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ln χt = (1 − ρχ ) ln χ + ρχ ln χt−1 + σχ χ,t
• Price-markup shock:
ln λt = (1 − ρλ ) ln λ + ρλ ln λt−1 + σλ λ,t
• Capital adjustment cost (marginal efficiency of investment):
ln µt = (1 − ρµ ) ln µ + ρµ ln µt−1 + σµ µ,t
• Intertemporal preference shifter:
ln bt = ρb ln bt−1 + σb b,t
• Government spending shock:
ln gt = (1 − ρg ) ln g + ρg ln gt−1 + σg g,t
• Monetary policy shock:
R,t
1.7
Log-Linearized Model
Variables are detrended where appropriate and expressed as deviations from
steady state.
• Detrending:
¯ t /Zt ,
yt = Yt /Zt , ct = Ct /Zt , it = It /Zt , kt = Kt /Zt , k¯t = K
k
k
˜
rt = Rt /Pt , wt = Wt /(Pt Zt ), w
˜t = Wt /Wt , ξt = Ξt Zt ,
ξtk = Ξkt Zt , zt = log(Zt /Zt−1 )
• Marginal cost:
mct = (1 − α)wt + αrtk .
(8)
• Phillips curve:
πt = β Et [πt+1 ] +
1
(1 − ζp β)(1 − ζp )
mct + λf,t ,
ζp
ζp
with normalization:
˜ f,t
λf,t = [(1 − ζp β)(1 − ζp )λf /(1 + λf )]λ
˜ f,t .
and λf is the steady state of λ
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• Capital-labor ratio:
kt − Lt = wt − rtk
(10)
• Marginal utility of consumption:
(eγ − hβ)(eγ − h)ξt = −(e2γ + βh2 )ct + βheγ Et [ct+1 + zt+1 ]+
heγ (ct−1 − zt ) + eγ (eγ − h)˜bt − βh(eγ − h)Et [˜bt+1 ]
(11)
with the normalization:
¯bt = eγ (eγ − h)/(e2γ + βh2 )bt .
• Consumption euler equation:
ξt = Et [ξt+1 ] + Rt − Et [πt+1 ] − Et [zt+1 ].
(12)
• Capital accumulation:
kt = ut − zt + k¯t−1
k¯t = (2 − eγ − δ)[k¯t−1 − at ] + (eγ + δ − 1)[it + (1 + β)S 00 e2γ µt ]
(13)
• Investment:
1
β
1
it =
[it−1 − zt ] +
Et [it+1 + zt+1 ] +
(ξ k − ξt ) + µt (14)
1+β
1+β
(1 + β)S 00 e2γ t
where ξtk is the value of installed capital, evolving according to:
k
k
ξtk − ξt = βe−γ (1 − δ)Et [ξt+1
− ξt+1 ] + Et [(1 − (1 − δ)βe−γ )rt+1
− (Rt − πt+1 )]
• Capital utilization:
ut =
r∗k k
r .
a00 t
(15)
• Optimal real wage:
w
˜t = ζw βEt [w
˜t+1 + ∆wt+1 + πt+1 + zt+1 ]+
1−ζw β
1
˜
(ν
l Lt − wt − ξt + bt + 1−ζw β ϕt )
1+νl (1+λw )/λw
(16)
• Real wage:
wt = wt−1 − πt − zt +
1 − ζw
w
˜t .
ζw
(17)
• Production function:
yt = (1 − α)Lt + αkt
(18)
c∗
i∗
rk
ct + (it + γ ∗
ut )] + gt
y∗
y∗
e −1+δ
(19)
Rt = ρR Rt−1 + (1 − ρR )(ψ1 πt + ψ2 yt ) + σR R,t .
(20)
• Resource constraint:
yt = (1 + g∗ )[
• Monetary policy rule:
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2
Empirical Application
We use post-1983 U.S. data to estimate the DSGE model. We begin with
a description of our data set and the prior distribution for the DSGE model
parameters.
2.1
Data and Priors
Seven series are included in the vector of core variables yt that is used for the
estimation of the DSGE model: the growth rates of output, consumption, investment, and nominal wages, as well as the levels of hours worked, inflation,
and the nominal interest rate. These series are obtained from Haver Analytics (Haver mnemonics are in italics). Real output is computed by dividing the
nominal series (GDP ) by population 16 years and older (LN16N ) as well as
the chained-price GDP deflator (JGDP ). Consumption is defined as nominal
personal consumption expenditures (C) less consumption of durables (CD). We
divide by LN16N and deflate using JGDP. Investment is defined as CD plus
nominal gross private domestic investment (I). It is similarly converted to real
per-capita terms. We compute quarter-to-quarter growth rates as log difference
of real per capita variables and multiply the growth rates by 100 to convert
them into percentages.
Our measure of hours worked is computed by taking non-farm business sector
hours of all persons (LXNFH ), dividing it by LN16N, and then scaling to get
mean quarterly average hours to about 257. We then take the log of the series
multiplied by 100 so that all figures can be interpreted as percentage deviations
from the mean. Nominal wages are computed by dividing total compensation
of employees (YCOMP ) by the product of LN16N and our measure of average
hours. Inflation rates are defined as log differences of the core PCE deflator
index (JCXFE ) and converted into percentages. The nominal interest rate
corresponds to the average effective federal funds rate (FFED) over the quarter
and is annualized.
Our choice of prior distribution for the DSGE model parameters follows
DSSW and the specification of what is called a “standard” prior in Del Negro
and Schorfheide (2008). The prior is summarized in the first four columns
of Table 1. To make this paper self-contained we briefly review some of the
details of the prior elicitation. Priors for parameters that affect the steady
state relationships, e.g., the capital share α in the Cobb-Douglas production
function or the capital depreciation rate are chosen to be commensurable with
pre-sample (1955 to 1983) averages in U.S. data. Priors for the parameters of
the exogenous shock processes are chosen such that the implied variance and
persistence of the endogenous model variables is broadly consistent with the
corresponding pre-sample moments. Our prior for the Calvo parameters that
control the degree of nominal rigidity are fairly agnostic and span values that
imply fairly flexible as well as fairly rigid prices and wages. Our prior for the
central bank’s responses to inflation and output movements is roughly centered
at Taylor’s (1993) values. The prior for the interest rate smoothing parameter
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ρR is almost uniform on the unit interval.
The 90% interval for the prior distribution on υl implies that the Frisch labor
supply elasticity lies between 0.3 and 1.3, reflecting the micro-level estimates
at the lower end, and the estimates of Kimball and Shapiro (2003) and Chang
and Kim (2006) at the upper end. The density for the adjustment cost parameter S 00 spans values that Christiano, Eichenbaum, and Evans (2005) find when
matching DSGE and vector autoregression (VAR) impulse response functions.
The density for the habit persistence parameter his centered at 0.7, which is the
value used by Boldrin, Christiano, and Fisher (2001). These authors find that
h = 0.7enhances the ability of a standard DSGE model to account for key asset
market statistics. The density for a00 implies that in response to a 1% increase
in the return to capital, utilization rates rise by 0.1 to 0.3%.
2.2
State Space Representation
The state space representation for the model estimation is given by:
St = T St−1 + R et
with measurement equation:
∆ ln(yt )
∆ ln(ct )
∆ ln(It )
ln(Ht )
∆ ln(Wt )
πt
Rt
(21)
= D + Z ∗ St
Note that we do not allow for measurement error in the estimation.
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3
Parameter Estimates
Table 1: Prior and Posterior of DSGE Model Parameters (Part 1)
Name
Density
h
a00
νl
ζw
400 ∗ (1/β − 1)
B
G
G
B
G
α
ζp
S 00
λf
B
B
G
G
400π ∗
ψ1
ψ2
ρR
N
G
G
B
Prior
Para (1) Para (2)
Household
0.70
0.05
0.20
0.10
2.00
0.75
0.60
0.20
2.00
1.00
Firms
0.33
0.10
0.60
0.20
4.00
1.50
0.15
0.10
Monetary Policy
3.00
1.50
1.50
0.40
0.20
0.10
0.50
0.20
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Mean
Posterior
90% Intv.
0.76
0.26
1.91
0.74
1.124
[
[
[
[
[
0.71
0.10
1.07
0.58
0.37
,
,
,
,
,
0.81
0.43
2.69
0.87
1.86
]
]
]
]
]
0.16
0.90
5.30
0.16
[
[
[
[
0.13
0.89
3.22
0.01
,
,
,
,
0.19
0.92
7.25
0.31
]
]
]
]
3.31
2.25
0.06
0.81
[
[
[
[
2.60
1.90
0.04
0.77
,
,
,
,
4.17
2.64
0.08
0.86
]
]
]
]
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Table 1: Prior and Posterior of DSGE Model Parameters (Part 2)
Name
Density
400γ
g∗
ρa
ρµ
ρλ f
ρg
ρb
ρφ
σa
σµ
σλf
σg
σb
σφ
σR
G
G
B
B
B
B
B
B
IG
IG
IG
IG
IG
IG
IG
Prior
Para (1)
2.00
0.30
0.20
0.80
0.60
0.80
0.60
0.60
0.75
0.75
0.75
0.75
0.75
4.00
0.20
Para (2)
Shocks
1.00
0.10
0.10
0.05
0.20
0.05
0.20
0.20
2.00
2.00
2.00
2.00
2.00
2.00
2.00
Mean
1.66
0.28
0.25
0.85
0.16
0.96
0.91
0.71
0.63
0.39
0.17
0.35
0.50
9.08
0.14
Posterior
90% Intv.
[ 1.17 , 2.13 ]
[ 0.13 , 0.41 ]
[ 0.14 , 0.36 ]
[ 0.80 , 0.90 ]
[ 0.07 , 0.26 ]
[ 0.95 , 0.98 ]
[ 0.87 , 0.95 ]
[ 0.56 , 0.91 ]
[ 0.56 , 0.71 ]
[ 0.32 , 0.45 ]
[ 0.15 , 0.20 ]
[ 0.31 , 0.39 ]
[ 0.36 , 0.62 ]
[ 3.44 , 14.16 ]
[ 0.12 , 0.16 ]
Notes: Para (1) and Para (2) list the means and the standard deviations for
the Beta (B), Gamma (G), and Normal (N ) distributions; the upper and lower
bound of the support for the Uniform (U) distribution; s and ν for the Inverse
2
2
Gamma (IG) distribution, where pIG (σ|ν, s) ∝ σ −(ν+1) e−νs /2σ . The joint
prior distribution is obtained as a product of the marginal distributions tabulated in the table and truncating this product at the boundary of the determinacy region. Posterior summary statistics are computed based on the output of
the posterior sampler. The following parameters are fixed: δ = 0.025, λw = 0.3.
Estimation sample: 1984:I to 2010:I.
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RTDRC DSGE Model
4.1 Impulse Responses
Impulse responses are to a 1-standard deviation shock.
Figure 1: Response to technology shock
Inflation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
Percent
Percent Annualized
-0.05
-0.1
-0.15
FRBPHL Research Dept
-0.08
-0.12
5
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Aggregate Hours Worked
0
9
-0.04
1
0.05
5
0
21
Federal Funds Rate
1
0.04
13 17 21
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
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RTDRC DSGE Model
Figure 1: Response to technology shock
Investment Growth
1
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0.8
0.6
0.4
0.2
0
-0.2
1
5
9
13
17
21
2
1.5
1
0.5
0
-0.5
1
5
Percent Annualized
Real Wage Growth
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
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RTDRC DSGE Model
Figure 2: Response to mei shock
Inflation
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
21
1
Federal Funds Rate
0.20
0.8
0.15
0.6
0.10
0.05
0.00
-0.10
-0.2
FRBPHL Research Dept
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0.2
0
5
9
0.4
-0.05
1
5
Aggregate Hours Worked
Percent
Percent Annualized
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
13 17 21
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RTDRC DSGE Model
Figure 2: Response to mei shock
Investment Growth
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
1
5
9
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10
8
6
4
2
0
-2
-4
1
5
Real Wage Growth
Percent Annualized
0.4
0.3
0.2
0.1
0
-0.1
-0.2
1
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RTDRC DSGE Model
Figure 3: Response to leisure shock
Inflation
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
21
0.4
0.3
0.2
0.1
0
1
Federal Funds Rate
0.5
0
0.4
-0.2
0.3
-0.4
0.2
-0.8
0
-1
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-0.6
0.1
1
5
Aggregate Hours Worked
Percent
Percent Annualized
0.5
13 17 21
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RTDRC DSGE Model
Figure 3: Response to leisure shock
Investment Growth
0.2
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0
-0.2
-0.4
-0.6
-0.8
1
5
9
13
17
21
1
0.5
0
-0.5
-1
-1.5
-2
1
5
Percent Annualized
Real Wage Growth
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
1
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RTDRC DSGE Model
Figure 4: Response to financial shock
Inflation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
21
0.3
Percent
Percent Annualized
0.4
0.2
0.1
0
FRBPHL Research Dept
9
0.1
0
5
9
13 17 21
Aggregate Hours Worked
0.5
5
0.2
1
Federal Funds Rate
1
0.3
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0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
1
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RTDRC DSGE Model
Figure 4: Response to financial shock
Investment Growth
2
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
1.5
1
0.5
0
-0.5
1
5
9
13
17
21
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
1
Real Wage Growth
Percent Annualized
0.6
0.5
0.4
0.3
0.2
0.1
0
1
FRBPHL Research Dept
5
9
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5
9
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RTDRC DSGE Model
Figure 5: Response to markup shock
0.10
Inflation
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
0.00
-0.10
-0.20
-0.30
-0.40
1
5
9
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0.8
0.6
0.4
0.2
0.0
-0.2 1
0.40
0.05
0.30
0.00
0.20
0.10
-0.10
-0.10
-0.15
5
FRBPHL Research Dept
9
9
13 17 21
-0.05
0.00
1
5
Aggregate Hours Worked
Percent
Percent Annualized
Federal Funds Rate
1.0
13 17 21
1
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RTDRC DSGE Model
Figure 5: Response to markup shock
Investment Growth
0.10
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0.05
0.00
-0.05
-0.10
-0.15
-0.20
1
5
9
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0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
1
5
Real Wage Growth
Percent Annualized
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
1
FRBPHL Research Dept
5
9
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9
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RTDRC DSGE Model
Figure 6: Response to monetary policy shock
Inflation
0.4
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
0.0
-0.4
-0.8
-1.2
1
5
9
13
17
0.00
-0.02
-0.04
-0.06
-0.08
21
1
Federal Funds Rate
5
9
13 17 21
Aggregate Hours Worked
1.20
0.1
0
0.80
Percent
Percent Annualized
0.02
0.40
0.00
-0.1
-0.2
-0.3
-0.4
-0.40
-0.5
1
5
FRBPHL Research Dept
9
13 17 21
1
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RTDRC DSGE Model
Figure 6: Response to monetary policy shock
Investment Growth
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
1
5
9
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1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
1
5
Real Wage Growth
Percent Annualized
0.10
0.00
-0.10
-0.20
-0.30
1
FRBPHL Research Dept
5
9
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RTDRC DSGE Model
Figure 7: Response to government shock
Inflation
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Output Growth
1
5
9
13
17
0.012
0.008
0.004
0.000
21
1
Federal Funds Rate
0.05
0.40
0.04
0.30
0.03
0.02
0.10
0.00
0.00
5
FRBPHL Research Dept
9
9
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0.20
0.01
1
5
Aggregate Hours Worked
Percent
Percent Annualized
0.016
13 17 21
1
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RTDRC DSGE Model
Figure 7: Response to government shock
Investment Growth
0.05
Percent Q-to-Q Annualized
Percent Q-to-Q Annualized
Consumption Growth
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
1
5
9
0.10
0.00
-0.10
-0.20
-0.30
13 17 21
1
Real Wage Growth
Percent Annualized
0.04
0.03
0.02
0.01
0.00
-0.01
1
FRBPHL Research Dept
5
9
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5
9
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@FedFRASER