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Internet Appendix
Downward Nominal Rigidities and Bond Premia*
Francois Gourio†
Phuong Ngo‡
March 24, 2024
* The views expressed here are those of the authors and do not necessarily represent those of the Federal
Reserve Bank of Chicago or the Federal Reserve System.
† Federal Reserve Bank of Chicago; email: francois.gourio@chi.frb.org.
‡ Cleveland State University; email p.ngo@csuohio.edu.
1
Contents
1
2
3
4
Quantitative model: additional results
1.1 State-dependence . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Impulse response function . . . . . . . . . . . . . . .
1.1.2 Macro-finance moments as a function of inflation .
1.2 Asymmetry in price setting is the key parameter . . . . . .
1.3 Asymmetry in the model . . . . . . . . . . . . . . . . . . . .
1.4 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Comparative statics: risk aversion and of shock persistence
Model extension: demand shocks
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Basic moments . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Impulse responses . . . . . . . . . . . . . . . . . . . . . . .
2.5 Macro-finance moments as a function of average inflation .
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Model extension: asymmetric wage rigidities
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Macro-finance moments with both wage and price asymmetries
3.6 Deriving the wage Phillips Curve . . . . . . . . . . . . . . . . . .
Additional Empirical Results
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38
2
1
Quantitative model: additional results
In this section we present some additional results from the quantitative model. First,
we study the model state-dependence, i.e. how its implications vary as a function of
inflation, when inflation is driven by realizations of z as opposed to different levels of
R∗ . Second, we present results for the symmetric model, which demonstrates that our
results are driven by the asymmetry. Third, we illustrate the implications of the model
for asymmetry. Fourth, we illustrate the implications of a different monetary policy rule.
And finally, we present comparative statics across risk aversion and the persistence of the
technology shock.
1.1
State-dependence
1.1.1
Impulse response function
Figure 1 presents generalized impulse responses (GIRFs) to a one-standard deviation
shock to productivity, conditional on inflation being initially low (2%, red line) vs. high
(4%, black dashed line).1 Here, the conditioning is effectively a conditioning on past productivity shocks. The patterns are quantitatively similar to that of figure (6) in the main
text: when inflation is high, output and inflation are more responsive to a productivity
shock.
1.1.2
Macro-finance moments as a function of inflation
Figures 2 and 3 depict the macro-finance moments as a function of inflation (when inflation is driven by z), and compares them to the ones of the main text (where differences
of inflation come from R∗ , and correspond to different “equilibria” as opposed to “histories”). The statistics are calculated in the same way as the data: we simulate the model
and construct rolling windows (of length 72 periods), over each of which we calculate
the macro-finance moments and the mean of inflation. (We then average across many
simulations.) The main message from the two figures is that the two approaches generate
very similar implications. The one moment where a difference emerges is the volatility
of inflation and interest rates. This is because these variables are highly persistent, and
hence, for these, the short sample of the rolling windows lead to an estimate of standard
1 The
GIRF of a variable x for a shock e at horizon k defined as GIRF ( x, s, k, e) = Et ( x (t + k)|s(t) =
s + e) − Et ( x (t + k)|s(t) = s) where s is the state. It reflects that in a nonlinear model, the response in
general depends on the size (and sign) of the shock e as well as the initial condition s - unlike in a linear
model. Moreover, the change in conditional expectation is not the same as the realization of a single path.
3
TFP shocks
1.15
GDP
0.8
1.1
0.7
1.05
0.6
%
%
1
0.95
0.5
0.9
0.4
0.85
0.8
0.3
0
5
10
15
20
25
30
0
Inflation
-0.1
5
10
15
20
25
30
Policy rate
0
-0.1
-0.2
%
%
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
TFP: low inflation
TFP: high inflation
-0.6
0
5
10
15
20
25
30
0
Quarter
5
10
15
20
25
Quarter
Figure 1: Generalized impulse response functions in the baseline model.
4
30
3
3
2.5
2
2
1.5
1
1
0
0
2
4
6
8
3
0
2
4
6
8
0
2
4
6
8
0
-0.2
2
-0.4
-0.6
1
-0.8
0
-1
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 2: Macro-finance moments in the benchmark model when inflation is driven
by changes in inflation target (as in the main text) or by productivity shocks, 1/2. Red
triangles correspond to the main text calculation (inflation-target driven inflation), blue
line to the history-driven inflation.
deviation that is much smaller than the population (or ergodic) standard deviation.2 Figure 4 depicts the term premium as a function of inflation for the two approaches. Here
too they are similar. (The figure also includes results from sections 2 and 3 to be discussed
below.)
1.2
Asymmetry in price setting is the key parameter
This section presents further evidence that the asymmetry in price setting is the driving
force of our result. We show this by solving the model for ψ = 0, keeping all other parameters unchanged. As shown in tables 3 and 5 in the main text, the symmetric model
2 Hence, if we were to adopt this approach as target, we might have to recalibrate the volatility of shocks.
5
0
50
40
-1
30
-2
20
-3
10
-4
0
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
5
10
4
8
3
2
6
1
4
0
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 3: Macro-finance moments in the benchmark model when inflation is driven
by changes in inflation target (as in the main text) or by productivity shocks, 2/2. Red
triangles correspond to the main text calculation (inflation-target driven inflation), blue
line to the history-driven inflation.
6
3
2.5
%
2
1.5
1
0.5
0
0
1
2
3
4
5
rolling mean
6
7
8
9
(%)
Figure 4: Mean term premium as a function of average inflation. The red triangles are
the baseline model (main text) with inflation driven by R∗ . The blue line is the baseline
model where inflation is driven by z. The black line is the symmetric model. The purple
line is the model with demand shocks. The green line is the model with wage and price
asymmetries.
7
has high, but nearly constant bond premia, and hence does not generate the predictability evidence (Fama-Bliss regressions). To demonstrate where this comes from, figure 5
presents the impulse response function: the response of the economy to a productivity
shock is nearly constant - that is, unlike the benchmark model, there is no state dependence. Finally, figures 6 and 7 reproduce our key experiment of changing the average
level of inflation through the inflation target, and show that the symmetric model generates almost no change in macro-finance moments as the average level of inflation changes.
Figure 4 shows this is true for the term premium as well. Overall, while this symmetric
model has attractive implications along many dimensions (e.g., a high average risk premium), it fails to reproduce the secular and cyclical changes in bond premia that are the
focus of this paper, because the market price of risk is nearly constant (and the bond risk
is nearly constant).
1.3
Asymmetry in the model
A direct way to illustrate the asymmetry implicit in the model, on the other hand, is to
look at the histograms of model variables (figure 8), which show a large positive skewness
for inflation and interest rates, and a large negative for output. In contrast, the histograms
of the symmetric model are approximately normally distributed.
As explained in the text, for small shocks the model is symmetric. Figure 9 compares the impulse response function to large positive and negative shocks, namely threestandard deviation shocks. For such large shocks, the model does exhibit some asymmetry. The asymmetry, in fact, results from the nonlinearity - the effect of a 3-sd positive
shock is less than the sum of 3 one-sd shocks, as the elasticity of the economy to shocks
falls as z rises. Yet, even with such large (and unlikely) shocks, the amount of asymmetry
is not tremendous. Hence, we do not think that looking at asymmetric responses is the
right approach to test this model.
1.4
Monetary Policy
Figure 10 illustrates the effect of changing the monetary policy rule on the distribution
of output, inflation, and interest rates. The most striking feature is that the distribution
of inflation loses most of its skewness, as does that of the interest rate to a lesser extent.
Conversely, output loses some of its negative skewness. Not visible in this figure is the
slight difference in the average level of output that is created by this policy change.
8
TFP
1.1
1.05
0.75
1
0.7
%
%
GDP
0.8
0.95
0.65
0.9
0.6
0.85
0.8
0.55
0
5
10
15
20
25
0
30
5
Inflation
-0.34
10
15
20
25
30
Short-term interst rate
-0.4
-0.36
-0.45
%
%
-0.38
-0.4
-0.42
-0.5
-0.44
TFP: low inflation target: E( )=2%
TFP: high inflation target: E( )=4%
-0.46
-0.55
0
5
10
15
20
25
30
0
Quarter
5
10
15
20
25
30
Quarter
Figure 5: Impulse response function to a productivity shock. Model with symmetric
price adjustment costs.
9
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0
2
4
6
8
3
0
2.5
-0.2
2
-0.4
1.5
-0.6
1
0
2
4
6
8
0
2
4
6
8
-0.8
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 6: Macro-finance moments in the symmetric model (black) and in the benchmark model (red triangles), 1/2.
10
0
50
-1
40
-2
30
-3
20
10
-4
0
2
4
6
8
40
20
30
15
20
10
10
5
0
0
2
4
6
8
0
2
4
6
8
0
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 7: Macro-finance moments in the symmetric model (black) and in the benchmark model (red triangles), 2/2.
11
0.12
A1. Policy rate - Symmetric cost
1.5
A2. Policy rate - Asymmetric cost
0.1
0.08
1
0.06
0.04
0.5
0.02
0
-10
0.15
0
-5
0
5
10
15
5
B1. Inflation - Symmetric cost
10
15
B2. Inflation - Asymmetric cost
0.5
0.4
0.1
0.3
0.2
0.05
0.1
0
0
-10
0.08
-5
0
5
10
15
0
C1. Output gap - Symmetric cost
0.15
5
10
C2. Output gap - Asymmetric cost
0.06
0.1
0.04
0.05
0.02
0
-20
-10
0
10
0
-20
20
-15
-10
-5
0
5
Figure 8: Histogram of macroeconomic variables in the symmetric model (left panel) and
in the baseline model (right panel).
12
A. TFP shock
-2.4
B. GDP
-1
-1.2
-2.8
-1.4
%
%
-2.6
-3
-1.6
-3.2
-1.8
-2
-3.4
0
20
0
30
C. Inflation
0.9
0.8
0.7
0.7
0.6
0.6
10
20
30
D. Policy rate
0.8
%
%
10
TFP: positive
TFP: negative
0.5
0.5
0.4
0.4
0.3
0.2
0.3
0
10
20
30
0
Quarter
10
20
30
Quarter
Figure 9: Impulse response functions to positive and negative shocks with asymmetric
price adjustment costs. Impulse response to a three-standard deviation of productivity
innovations shock when the shock is positive (red solid line) vs. negative (black dashed
line). The responses to positive shocks are displayed with the reverse sign. The economy
is initially at the deterministic steady state.
13
A1. Policy rate -
=5
A2. Policy rate - Benchmark (
1
= 2)
1.5
0.8
1
0.6
0.4
0.5
0.2
0
0
4
5
6
7
B1. Inflation -
8
5
=5
0.8
10
15
B2. Inflation - Benchmark
0.5
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
1
2
3
4
C1. Output gap -
5
0
=5
0.1
0.15
5
10
C2. Output gap - Benchmark
0.08
0.1
0.06
0.04
0.05
0.02
0
-20
-10
0
0
-20
10
-15
-10
-5
0
5
Figure 10: Histogram of macroeconomic variables in the model with ϕπ = 5 (left panel)
and in the baseline model (right panel).
14
∆ log Y
π
y $ (1)
y$(40)
y (1)
y(40)
Stock return
tp(40)
tp$(40)
Skewness(π)
Prob(π < 1%)
Data
Mean
Sd
– 3.03
3.14
1.97
5.63
3.20
7.36
2.97
–
–
–
–
8.12 16.32
–
–
–
–
1.55
–
1.71
–
Model
Mean
Sd
0.00 2.37
3.14 1.97
5.63 1.97
7.36 2.12
2.37 0.34
2.42 0.19
8.98 16.08
0.02 0.21
1.67 0.68
1.55
–
1.78
–
α=0
Mean
Sd
0.00 2.74
3.81 2.72
7.38 3.18
7.34 2.41
3.51 0.52
3.54 0.34
3.87 38.78
0.01 0.01
0.06 0.03
0.84
–
11.0
–
ρz = .92
Mean
Sd
0.00 3.23
3.31 1.98
6.54 3.11
7.37 0.93
3.11 1.31
3.48 0.37
3.96 20.67
0.39 0.08
0.97 0.17
1.77
–
0.00
–
Table 1: Comparative statics. As in table (3), columns 2-3 report the mean and standard deviation from U.S. data over the sample 1979q4-2008q4, and columns 4-5 report
the mean and standard deviation for the benchmark model. Columns 6-7 and 8-9 report
the same statistics respectively if α = 0 and if ρz = 0.92, respectively (while keeping all
other parameters at the benchmark values).
1.5
Comparative statics: risk aversion and of shock persistence
Table 1 shows the model moments when we set risk aversion to be low (α = 0), or when
we set a lower persistence of the technology impulse z (ρz = .92), together with the data
and benchmark model. Unsurprisingly, the low risk aversion model has a flat yield curve,
with very low and nearly constant risk premia, and the average stock return is close to the
average (log) yield on a risk-free bond. More interestingly, low risk aversion also increases
the mean and volatility of inflation. This is because with an unchanged monetary policy
rule, keeping the same R∗ (Taylor rule intercept) when the “neutral risk-free rate” has
gone up leads to more inflation. The lesson here is that monetary policy needs to offset
variation in the risk-free rate that is due to risk aversion.
The second comparative static we want to highlight is the lower persistence of technology shock. In this case, the real term premium increases significantly, since there is
now more predictable variation of growth. On the other hand, the total term premium
falls, as these shocks are less “scary” for investors. This suggests to us that a model with
more shocks may be do better at matching the data - some low persistence shocks help in
some dimensions, while higher persistence helps in others.
15
2
Model extension: demand shocks
In this section, we extend the model to allow for “demand shocks” (on top of the productivity shocks, or “supply shocks”, which we keep throughout). There are several motivations for considering this extension. First, demand shocks are typically found to be
important in accounting for macroeconomic fluctuations. Second, demand shocks generate a positive comovement between output and inflation, and hence can potentially contribute to term premia, in particular in explaining a potential sign switch where bonds
become “hedges” and term premia negative (as they might have become in the 2000s).
This section is a simple exercise to assess how demand shocks could affect our results.
The answer is nuanced. The main results appear robust - see for instance, figure 4 which
shows that as average inflation rises, the model with demand shocks also generates a
substantial increase in the term premium. As we explain below, this comes because while
the demand shock generates a positive covariance of inflation and output, the magnitude
of this covariance does not depend greatly on the level of inflation. As a result, the overall
covariance of inflation and output (reflecting both demand and supply shocks) is still
decreasing in the level of inflation, as we showed in the main text, since the effects of
supply shocks remain the same as in the baseline model. That said, some other results
change, at least in our current simple calibration. For instance, the volatility of output
may fall, rather than rise, with average inflation.
Since our main point does not require demand shocks, we chose to abstract from demand shocks in the main text. (In particular, while a model with demand shocks does
better at matching some statistics such as the stock-bond covariance, it also raises some
novel issues - for instance, matching the average slope of the yield curve is more difficult
- the yield curve remained on average steep throughout the 2000s, even as the stock-bond
covariance became negative.) We also want to note that there is room for improvement in
this section, so these results could possibly change if we refine the model or the calibration.
2.1
Model
To incorporate demand shocks, we add a disturbance directly in the Euler equation. This
can be motivated (as in Fisher (2014)) as a time-varying convenience yield for bonds.
Under this interpretation, the only modification to the model is to the Euler equation,
which becomes:
16
i
h
$
1
R
M
Et ξ −
t
t
t+1 = 1,
(1)
where ξ t is the liquidity process, reflecting “convenience” demand for bonds. We assume
that this process follows a log-normal AR(1):
log ξ t = ρξ log ξ t−1 + εξ,t ,
(2)
with εξ,t i.i.d N (0, σ2ξ ).
2.2
Parameters
To set the parameters, we follow a calibration approach similar to that of the benchmark
model in the main text. We set a number of parameters a priori as before, and set the persistence of demand shocks to 0.9, a common value in the literature. We then calibrate the
same parameters we did in the main text (i.e., α,β,σz ,ψ,Π,R$ ) as well as the new parameter σξ to match the same statistics we did in the main text (i.e., the mean and volatility of
inflation, the mean of short-term and long-term interest rates, the skewness of inflation,
the probability that inflation is less then 1%) as well as an additional moment, the correlation of output growth with the change of inflation. We choose this moment because it
captures the reduced-form Phillips correlation that is informative about the relative importance of supply vs. demand shocks: inflation increases (resp. decreases) with output
if there are demand (resp. supply) shock. We match these moments nearly perfectly, as
shown in table 3. (The probability of inflation less than 1% is somewhat too high at 5% vs.
1.7% target.) The parameters used are shown in table 2. Matching the data now requires
much higher risk aversion, because demand shocks generate a negative slope.
2.3
Basic moments
Table 3 presents the basic moments. We see that the model generates still significant, but
smaller, volatility of the term premium, and of long-term interest rates. The model still
undershoots on GDP volatility.
2.4
Impulse responses
The impulse responses of selected macroeconomic variables to both productivity and demand shocks are presented in figure 11. As in the text, we vary the inflation target so the
average inflation is high (4%) or low (2%). The patterns for the productivity shock are
17
Parameter
Description
Value
A. Taken from the literature
σ
υ
χ
ε
ϕπ
ϕy
ρξ
ρz
ϕ
Preferences: inverse IES
Preferences: labor supply
Preferences: labor supply
Preferences: substitution across goods
Monetary policy rule: weight on inflation
Monetary policy rule: weight on output
Shock: persistence of demand shock
Shock: persistence of TFP
Adj. cost of prices: size parameter
2
1.5
40.66
7.66
2
0.125
0.9
0.99
78
B. Calibrated to match key moments
R∗
σz
σξ
ψ
Π
β
α
Monetary policy rule: intercept
Shock: std. dev. of TFP innovation
Shock: std. dev. of demand innovation
Adj. cost of prices: asymmetry parameter
Adj. cost of prices: location parameter
Preferences: subjective discount factor
Preferences: Epstein-Zin curvature (note: CRRA=169)
1.0093
0.643
0.294
396
1.012
0.9906
-241
Table 2: Model parameters in the model with demand shocks. The time period is one
quarter.
E(π )
σ(π )
Skewness(π )
Prob(π < 1%)
E ( y $ (1) )
E(y$(40) )
ρ(∆π, ∆ log C )
Data
3.14
1.97
1.55
1.71
5.63
7.36
0.1
Model
3.14
1.97
1.55
5.03
5.63
7.36
0.11
Table 3: The table shows the data moments used for the calibration of the model with
demand shocks, and the corresponding model moments. Data moments are calculated
over the sample 1979q4-2008q4.
18
∆ log(Y )
π
y $ (1)
y$(40)
y (1)
y(40)
tp(40)
tp$(40)
Data
Mean
Sd
– 3.03
3.14 1.97
5.63 3.20
7.36 2.97
–
–
–
–
–
–
–
–
Model
Mean
Sd
0.00 2.37
3.14 1.97
5.63 1.97
7.36 2.12
2.37 0.34
2.42 0.19
0.02 0.21
1.67 0.68
Table 4: Data and model moments in the model with demand shocks. Columns 2 and
3 report the mean and standard deviation from U.S. data over the sample 1979q4-2008q4.
Columns 4 and 5 report the mean and standard deviation for the model with demand
shocks. All statistics are reported in annualized terms.
similar to that in the main text. A demand shock generates lower inflation, lower GDP,
and lower interest rates. When inflation is high, prices are less sticky, and consequently
demand shocks have less impact on GDP and more on inflation, which is intuitive. Interestingly, the overall covariance of output and inflation remains basically unchanged.
Hence, the overall effect of demand shocks on our key result is limited - the results subsist
as long as there are productivity shocks.
2.5
Macro-finance moments as a function of average inflation
Finally, in figures 12 and 13 we conduct the main experiment of the paper: showing how
macro-finance moments change with average inflation. For clarity, we compare in these
figures the benchmark model (with productivity shocks only) to the model with demand
shocks (which, to be clear, includes both productivity and demand shocks). As noted
above, the results are mixed. The models does still very well on the volatility of inflation,
and is qualitatively right on many of the other moments. Some moments are quantitatively off. The most glaring discrepancy is that the volatility of output now falls, rather
than rise, with inflation. This is because while the effect of productivity shocks increase
with inflation, the effects of demand shocks decrease with inflation.
19
A2. Short-term interst rate
A1. Short-term interst rate
0
-0.1
%
%
-0.5
-0.15
-1
Demand: low inflation target: E( )=2%
Demand: high inflation target: E( )=4%
TFP: low inflation target: E( )=2%
-0.2TFP: high inflation target: E( )=4%
-1.5
0
10
20
30
0
B1. GDP
0
10
20
30
B2. GDP
0.4
%
%
-0.2
0.3
-0.4
-0.6
0.2
0
10
20
30
0
C1. Inflation
0
%
%
20
30
C2. Inflation
-0.1
-0.2
10
-0.15
-0.4
-0.2
-0.6
0
10
20
0
30
20
30
D2. TFP shock
D1. Demand
0.3
10
0.6
%
%
0.2
0.1
0.55
0.5
0
0
10
20
0
30
Quarter
10
20
30
Quarter
Figure 11: Impulse response functions with asymmetric price adjustment costs in the
model with demand shocks, when the inflation target is low (2%) and high (4%).
20
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
3
0
2.5
-0.2
2
-0.4
1.5
-0.6
1
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 12: Macro-finance moments in model with demand shocks (purple line) and in
the benchmark model (red triangles), 1/2.
21
0
40
-1
30
-2
20
-3
10
0
-4
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
5
10
9
4
8
3
7
2
6
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 13: Macro-finance moments in model with demand shocks (purple line) and in
the benchmark model (red triangles), 2/2.
22
3
Model extension: asymmetric wage rigidities
In this section, we present an extension of our model that incorporates both downward
nominal price and wage rigidities. In the paper, we focus on the case of price asymmetries, but in reality, both wages and prices are asymmetric. (Indeed, at the micro level, the
evidence for asymmetries is stronger for wages than prices.) As the figures below illustrate, introducing wage stickiness and wage asymmetries in the model allows to generate
many of the similar facts, while reducing the degree of asymmetry assumed for prices. (In
that sense, this shows that the asymmetry we assume for prices in the baseline model may
reflect asymmetry in wages.) (Here too, the results, while accurate and very supportive,
could be further improved.)
3.1
Model
We introduce wage stickiness as in Kim and Ruge-Murcia (2011), who build on Kim and
Ruge-Murcia (2009) and Erceg et al. (2000). Some sections that are essentially identical to
the baseline model are denoted with an asterisk in the subsection title. We kept them here
to make this section independent of the main text.
Composite labor
Firms use composite labor to produce intermediate differentiated goods. Composite labor
is created by aggregating a variety of differentiated labor indexed by h ∈ [0, 1] using a CES
technology
ϵ ϵw−1
Z 1
w
h ϵwϵw−1
dh
,
(3)
Nt =
Nt
0
where εw determines the elasticity of substitution among differentiated types of labor. The
profit maximization problem is given by
max Wt Nt −
Z 1
0
Wth Nth dh,
where Wth and Nth are the wage and quantity of differentiated labor of type h.
Profit maximization and the zero-profit condition give the demand for labor of type h
Nth =
Wth
Wt
23
!−ϵw
Nt ,
(4)
and the aggregate wage level
Wt =
1
Z
0
Wth
1− ϵ w
1−1ϵ
dh
w
.
(5)
Final consumption goods*
To produce consumption goods, households buy and aggregate a variety of differentiated
intermediate goods indexed by i ∈ [0, 1] using a CES technology
Yt =
1
Z
0
Yt (i )
ϵ −1
ϵ
ϵ−ϵ 1
di
,
where ε determines the elasticity of substitution among intermediate goods. The profit
maximization problem is given by
max Pt Yt −
Z 1
0
Pt (i ) Yt (i ) di,
where Pt (i ) and Yt (i ) are the price and quantity of intermediate good i.
Profit maximization and the zero-profit condition give the demand for differentiated
intermediate good i
Pt (i ) −ϵ
Yt ,
(6)
Yt (i ) =
Pt
and the aggregate price level
Pt =
1
Z
0
Pt (i )
1− ϵ
1−1 ϵ
di
.
(7)
Household h’s problem
There is a unit mass of households. Each household indexed by h ∈ [0, 1] provides type-h
labor and is competitively monopolistic in the labor market. It is costly to adjust wages.
Without loss of generality, we assume that households pay wage adjustment costs which
have a general form
!
h
W
t
Φth = Φ
Wth Nth ,
Wth−1
where Φ′ (·) > 0 and Φ′′ (·) > 0.
In this paper, we follow Kim and Ruge-Murcia (2009) and use the linex function to
model wage adjustment costs. Specifically,
24
Φth
=Φ
Wth
Wth−1
!
exp −ψ
= ϕ
Wth
Wth−1
−Π
+ψ
Wth
Wth−1
ψ2
− Π − 1
,
(8)
where ϕw is the level parameter and ψw is the asymmetry parameter. If ψw > 0, the
wage adjustment cost is asymmetric. In particular, the cost to lower a wage is higher
than to increase it by the same amount. When ψw approaches 0, this function becomes a
symmetric quadratic function
Φ (x) =
Household h choose Cth , Nth , Wth , Bth
Vth
= (1 −
2
ϕw
x−Π .
2
∞
t =1
to maximize the inter-temporal utility
β) u(Cth , Nth ) +
βEt
(Vth+1 )1−α
1
1− α
with the flow utility
1− γ
u (Ct , Nt ) =
1+ η
χNt
Ct
−
,
1−γ
1+η
subject to the labor demand (4) and the budget constraint as described below.
If the parameters we use lead to a negative flow utility u(Ct , Nt ), we define utility as:
Vth
= (1 −
β) u(Cth , Nth ) −
βEt
−Vth+1
1−α 1−1 α
.
The budget constraint is:
1 h
h h
h
h
h
h
Pt Cth + R−
B
=
W
N
1
−
Φ
t
t t
t + Bt−1 + Dt + Tt .
t
Given W0 and B0 .
(9)
(10)
A symmetric solution to this optimization problem, i.e. Wth = Wt and Nth = Nt , implies a
New Keynesian Phillips curve for wages and the Euler equation (see derivation in section
3.6):
25
η +1
0 =
(1 − ε w ) (1 − Φ ( Π w
t ))
"
+ Et
Nt − Φ
′
w
(Πw
t ) Π Nt
+ εw χ
Nt
−γ
wt Ct
(11)
#
w 2
Φ′ Πw
Π
(
)
t +1
Mt,t+1
Nt+1 ,
Π t +1
Et Mt,t+1
Rt
Π t +1
= 1,
(12)
where wt = Wt /Pt is the real wage; Πt = Pt /Pt−1 is gross inflation; Πw
t = Wt /Wt−1 is
gross wage inflation. Wage inflation and and the stochastic discount factor are given by
wt
Πt ,
w t −1
Πw
t =
(13)
−α
Mt,t+1 = β
Ct+1
Ct
−γ
h
Vt+1
Et
Vt1+−1α
i 1
1− α
,
(14)
Note that when ϕ = 0 and εw → ∞, equation (11) becomes a standard marginal rate
of substitution between labor and consumption
η
χNt
−γ
Ct
= wt .
Intermediate goods producer i′ s problem*
There is a unit mass of intermediate goods producers that are monopolistic competitors.
Suppose that each intermediate good i ∈ [0, 1] is produced by one producer using the
technology
(15)
Yti = Zt Nti ,
where α ≥ 0; Nti is composite labor input used by firm i; and
ln ( Zt ) = ρ Z ln ( Zt−1 ) + ε Z,t ,
2
ε Z,t ∼ i.i.d N 0, σ Z .
(16)
Following Rotemberg (1982), we assume that each intermediate goods firm i faces
costs of adjusting prices in terms of final goods. The adjustment cost function is in a
26
general form
Pti
Pti−1
Γt = Γ
!
Yt ,
where Γ′ (·) > 0 and Γ′′ (·) > 0.
We also use the linex function to model price adjustment costs. Specifically,
exp −ψ p x − Π + ψ p x − Π − 1
,
Γ (x) = ϕ p
ψ2p
(17)
where ϕ p , ψ p are parameters that determines the level and the asymmetry of price adjustment costs. If ψ p > 0, the price adjustment cost is asymmetric. Particularly, the cost to
lower a price is higher than to increase it by the same amount. The linex function nests
the symmetric quadratic cost when ψ p approaches 0, i.e. it becomes a quadratic function
Γ (x) =
ϕp
x−Π
2
2
,
which is popularly used in the ZLB literature.
The problem of firm i is given by
∞
max
{Yti+ j ,Nt+ j ,Pti+ j }∞
j =0
Et
∑
j =0
(
"
Mt,t+ j
Pti+ j
Pt+ j
!
Yti+ j − wt Nti
−Γ
Pti+ j
Pti+ j−1
!
#)
Yt+ j
(18)
subject to its demand (6) and production function (15). In a symmetric equilibrium where
all firms choose the same price and produce the same quantity (i.e., Pti = Pt and Yti = Yt ).
The optimal pricing rule then implies the New Keynesian Phillips curve,
wt
′
1 − ε + ε − Πt Γ (Πt ) Yt + Et Mt,t+1 Πt+1 Γ′ (Πt+1 ) Yt+1 = 0.
Zt
(19)
Monetary policy*
The central bank conducts monetary policy by setting the interest rate using a simple
Taylor rule:
GDPt ϕy Πt ϕπ
∗
Rt = R
(20)
GDP∗
Π∗
where GDPt ≡ Ct denotes the gross domestic product (GDP); GDP∗ and Π∗ denote the
target GDP and inflation, respectively; R∗ denotes the intercept of the Taylor rule.
27
Equilibrium systems
With the Rotemberg price setting, the aggregate output satisfies
Yt = Zt Nt ,
(21)
As in the benchmark model, we assume that price and wage adjustment costs are
rebated to households. Hence, the aggregate resource constraint is given by
Ct = Yt
(22)
The equilibrium system for the model consists of a system of six nonlinear difference
equations (11), (12), (13), (19), (20), (21), (22) for six variables wt , Ct , Rt , Πt , Πw
t , Nt , and
Yt .
3.2
Calibration
We have three new parameters compared to the baseline model, that govern the size,
location and asymmetry of wage adjustment costs. For parsimony, we assume that the
location is the same as for price, and we set the size of the adjustment cost to a value
consistent with the literature (i.e., wages are stickier than prices). This leaves us with one
additional parameter ψw , and we use as additional moment to calibrate it the skewness
of wage growth, which is also significant in our sample.3
Table 5 presents the parameters, and the data targets and model moments used for
calibration are in table 6. We can see from this table that the model matches the data fairly
well.
3.3
Moments
Table 7 reports the basic moments. As in the baseline model, there is a large term premium, but the volatility is now lower.
3.4
Impulse responses
The impulse responses of selected macroeconomic variables to supply shocks in the model
with both price and wage asymmetry are presented in figure 14, where we vary the infla3 Because this model is significantly more difficult to solve numerically, some of the results here are based
on parameter values that differ slightly from the baseline (because we have not yet updated this part), but
the differences should not be material.
28
Parameter
Description
Value
A. Taken from the literature
σ
υ
χ
ε
ϕπ
ϕy
ρz
ϕp
ϕw
Preferences: inverse IES
Preferences: labor supply
Preferences: labor supply
Preferences: substitution across goods
Monetary policy rule: weight on inflation
Monetary policy rule: weight on output
Shock: persistence of TFP
Adj. cost of prices: size parameter
Adj. cost of wages: size parameter
2
1.5
40.66
7.66
1.75
0.065
0.97
70
200
B. Calibrated to match key moments
R∗
σz
ψp
ψp
Π
β
α
Monetary policy rule: intercept
Shock: std. dev. of TFP innovation
Adj. cost of prices: asymmetry parameter
Adj. cost of wages: asymmetry parameter
Adj. cost of prices: location parameter
Preferences: subjective discount factor
Preferences: Epstein-Zin curvature (note: CRRA=87)
1.019
0.643
280
350
1.0099
0.991
-120
Table 5: Model parameters in the model with both wage and price asymmetries. The time
period is one quarter.
σ(∆ log Y )
E(π )
σ(π )
Skewness(π p )
Skewness(π w )
Prob(π < 1%)
E ( y $ (1) )
E(y$(40) )
Data
3.03
3.14
1.97
1.55
1.91
1.74
5.63
7.36
Model
3.03
2.97
1.72
1.56
1.92
6.67
5.68
7.25
Table 6: Data and model-based moments in the model with both price and wage asymmetry. Data over the sample 1979q4-2008q4.
29
∆ log(Y )
π
y $ (1)
y$(40)
y (1)
y(40)
tp(40)
tp$(40)
Data
Mean
Sd
– 3.03
3.14 1.97
5.63 3.20
7.36 2.97
–
–
–
–
–
–
–
–
Full sample
Mean
Sd
0.00 3.15
2.97 1.73
5.68 2.46
7.25 1.05
2.56 1.11
3.16 0.29
0.59 0.11
1.52 0.23
Table 7: Data and model moments in the model with both price and wage asymmetry.
Columns 2 and 3 give the mean and standard deviation from U.S. Data over the sample
1979q4-2019q4. Columns 4 and 5 give the mean and standard deviation using simulated
data from the model.
tion target so the average inflation is high (4%) or low (2%). As in the benchmark model,
the conditional covariance driven by supply shocks is dampened (or less negative) when
inflation is low, leading to smaller bond premium. However, the magnitude of the change
appears to be less than in the baseline model, which explains why there is less volatility
of the term premium.
3.5
Macro-finance moments with both wage and price asymmetries
Finally, figures 15 and 16 show our key experiment. Here too, varying inflation leads
macro-finance moments to vary in a substantial way.4 Overall, the model with price and
wage asymmetries seems to do quite well, surpassing in many dimensions the baseline
model, despite much lower asymmetries.
3.6
Deriving the wage Phillips Curve
V h Bth−1 , Wth−1 , Zt =
(Cth )
1− γ
( Nth )
1+ η
(1 − β )
1− γ − χ 1+ η
h
1−α i 1−1 α
h
h
h
h
h
h
h
+ β Et V Bt , Wt , Zt+1
{Ct ,Nt ,Wt ,Bt }
Max
|{z}
4 Due
(23)
to numerical difficulties, we have not yet been able to vary the level of inflation as much as in the
other calculations.
30
A. TFP
1
B. GDP
0.8
0.6
%
%
0.8
0.4
0.6
0.2
0.4
0
0
10
20
0
30
C. Inflation
0
10
20
30
D. Policy rate
0
-0.2
-0.2
%
%
-0.4
-0.4
-0.6
-0.8
-0.6
At high inflation
At low inflation
-1
-0.8
0
10
20
30
0
Quarter
10
20
30
Quarter
Figure 14: Impulse response functions in the model with both price and wage asymmetry
at low and high inflation target.
31
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0
2
4
6
8
3
0
2
4
6
8
0
2
4
6
8
0
2.5
-1
2
-2
1.5
1
-3
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 15: Macro-finance moments in model with wage asymmetric rigidity (green line)
and in the benchmark model (red triangles), 1/2.
32
0
60
-2
40
-4
20
-6
-8
0
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
8
10
9
6
8
4
7
6
2
0
2
4
rolling mean
6
8
(%)
rolling mean
(%)
Figure 16: Macro-finance moments in model with wage asymmetric rigidity (green line)
and in the benchmark model (red triangles), 2/2.
33
subject to
Pt Cth
+
1 h
R−
t Bt
=
(1 + τ w ) Wth Nth
Nth =
1 − Φth
Wth
Wt
+ Bth−1 + Dth + Tth + ACth
(24)
!−ϵw
Nt
(25)
where
Wth
Wth−1
Φth = Φ
!
exp −ψw
= ϕw
Wth
Wth−1
−Π
+ ψw
Wth
Wth−1
ψ2w
− Π − 1
.
(26)
Let λth1 and λth2 be the Lagrangian multipliers for the budget constraint and the labor
demand at time t, respectively. The first-order conditions include
−γ
(Ct ) : (1 − β) Cth
− Pt λth1 = 0,
(27)
η
( Nt ) : − (1 − β) χ Nth + (1 + τ w ) Wth 1 − Φth λth1 − λth2 = 0,
(28)
−α
Et V h Bth , Wth , Zt+1
h
i 1
1
−
α
1
−
α
Et V h Bth , Wth , Zt+1
(Wt ) : 0 = β
(1 + τ w ) Nth 1 − Φth − (1 + τ w ) Wth Nth Φth
′
Bth , Wth , Zt+1
1
Wth−1
!
λth1 − εw
(29)
Nth h2
λt .
Wth
−α
Et V h Bth , Wth , Zt+1
h
i 1
1
−
α
1
−
α
Et V h Bth , Wth , Zt+1
( Bt ) : β
Vwh
34
1 h1
VBh Bth , Wth , Zt+1 − R−
t λt = 0. (30)
The envelope theorem implies:
∂V h Bth−1 , Wth−1 , Zt
≡ Vwh Bth−1 , Wth−1 , Zt
∂Wth−1
=
∂V h Bth , Wth−1 , Zt
(31)
(1 + τ w ) Wth Nth Φth
Wth
′
2
Wth−1
!
λth1
≡ VBh Bth−1 , Wth−1 , Zt = λth1
∂Bth−1
From equations (27) and (28)
λth1
λth2
−γ
h
C
1
−
β
(
)
t +1
; λth1+1 =
;
Pt+1
η
h −γ
h
h
h (1 − β ) Ct
= − (1 − β) χ Nt
+ (1 + τ w ) Wt 1 − Φt
Pt
(1 − β) Cth
=
Pt
−γ
Equation (29) can be simplified to
!
h −γ
1
−
β
C
(
)
1
t
0 =
(1 + τ w ) Nth 1 − Φth − (1 + τ w ) Wth Nth Φth
Pt
Wth−1
!
η
(1 − β ) C h − γ
Nth
t
−εw h − (1 − β) χ Nth + (1 + τ w ) Wth 1 − Φth
P
Wt
t
−α
h h h
E
V
B
,
W
,
Z
t
t
+
1
t
t
×
+β
1
h
i
1
−
α
1
−
α
Et V h Bth , Wth , Zt+1
!2
−γ
h
′ W h
1
−
β
C
(
)
t +1
t +1
(1 + τ w ) Nth+1 Φth+1
,
Pt+1
Wth
35
′
(32)
(33)
(1 + τ w ) Nth
0 =
+εw χ
Nth
η +1
Wth
1 − Φth
− (1 + τ w ) Nth
Φth
′ W h
t
h
Wt−1
!
Pt
h
h
1
−
Φ
−
ε
1
+
τ
N
(
)
w
w
t
t
h −γ
Ct
−α
Et V h Bth , Wth , Zt+1
×
1
h
i
1
−
α
1
−
α
Et V h Bth , Wth , Zt+1
!2
−γ
h
′ W h
C
Pt
t +1
t+1
(1 + τ w ) Nth+1 Φth+1
,
−
γ
h
Pt+1
Wt
Cth
+β
0 = (1 + τ w ) (1 − εw ) Nth 1 − Φth − (1 + τ w ) Nth
′ W h
h η +1
N
t
t
Φth
+ εw χ
−γ (34)
Wth−1
wth Cth
−α
−γ
h h h
Cth+1
E
V
B
,
W
,
Z
t
t
+
1
t
t
+β
1
h
i
−
γ
h
1− α
1− α
Ct
Et V h Bth , Wth , Zt+1
!2
′ W h
t +1
(1 + τ w ) Nth+1 Φth+1
Pt ,
h
Pt+1
Wt
×
(35)
Equation (30) becomes
−α
Et V h Bth , Wth , Zt+1
λh1 − R−1 λth1
t +1
t
1
h
i
1−α 1−α
Et V h Bth , Wth , Zt+1
−α
−γ
h h h
(1 − β) Cth+1
E
V
B
,
W
,
Z
t
t
+
1
t
t
0 = β
i 1
h
Pt+1
1
−
α
1
−
α
Et V h Bth , Wth , Zt+1
h −γ
−1 (1 − β ) Ct
− (1 + i t )
Pt
0 = β
36
(36)
(37)
In a symmetric equilibrium the optimal wage setting becomes the wage Phillips curve:
η +1
0 =
Nt
χ
+ εw
1 + τ w wt Ct−γ
!#
2
( Φ t +1 ) ′ Π w
t +1
,
Nt+1
(1 + τ w )
Πt
(1 − εw ) (1 − Φt ) Nt − Φ′t Πw
t Nt
"
+ Et Mt,t+1
(38)
and the optimality condition for bonds satisfies:
Et Mt,t+1
Rt
Π t +1
= 1,
(39)
where wt = Wt /Pt is the real wage, Πw
t = Wt /Wt−1 is the gross wage inflation, Πt =
Pt /Pt−1 is the gross (price )inflation rate, and the stochastic discount factor is given by
−α
Mt,t+1
= β
(V (Wt , Zt+1 ))
h
Et (V (Wt , Zt+1 ))
37
1− α
i 1
1− α
Ct+1
Ct
−γ
.
(40)
4
Additional Empirical Results
Tables 8, 9, and 10 present some robustness analysis for the results of section 6 of the
paper. Each table corresponds to a different horizon h equal to 4, 8 or 12 quarters. Specifically, we estimate equation (28) from the main text, but with the following modifications:
first, removing the quadratic time trend; second, changing the measure of inflation Zt for
the interaction variable. (We always use the same price measure as outcome variable.) In
our baseline specification, we used the 2-year PCE inflation. Instead, we show what happens if we use either the 1-year or 3-year PCE inflation, or the 2-year core PCE inflation.
Regarding the quadratic time trend, as we explained in the main text, its main effect
is to reduce the magnitude of standard errors by removing some low-frequency variation
in the GDP and inflation series. Looking across the two outcomes (GDP and price level)
and three tables (different horizons h), its effect on the point estimates is usually limited.
Regarding the alternative inflation measures, they matter relatively little for the key coefficient of interest γh (with perhaps one or two exceptions out of 18 cases (across the
three tables) where the coefficient becomes smaller). Overall, the results are stable with
respect to these changes. As in the main text, the results are only statistically significant at
h equal to 4 or 8 quarters, but the signs and magnitudes remain economically meaningful
in almost all cases even for h=12 quarters.
38
Baseline
No time trend
2y Core
3y
1y
Real GDP
βz,h
s.e.
t-stat
0.25**
(0.12)
2.13
0.30*
(0.16)
1.92
0.40***
(0.12)
3.26
0.29**
(0.12)
2.37
0.26**
(0.12)
2.28
γh
s.e.
t-stat
0.13***
(0.03)
4.75
0.14***
(0.04)
3.79
0.10**
(0.04)
2.26
0.13**
(0.05)
2.41
0.08***
(0.03)
2.95
Obs.
256
256
232
252
260
Core PCE price index
βz,h
s.e.
t-stat
-0.06
(0.06)
-0.92
-0.04
(0.06)
-0.65
-0.11
(0.09)
-1.27
-0.09
(0.07)
-1.30
-0.01
(0.05)
-0.22
γh
s.e.
t-stat
-0.09***
(0.03)
-3.40
-0.06*
(0.03)
-1.78
-0.09**
(0.03)
-2.55
-0.08**
(0.03)
-2.47
-0.07***
(0.02)
-3.29
Obs.
236
236
232
236
236
Table 8: The table reports the estimates of βz,h and γh from equation (28) in the main
text, for horizon h = 4 quarters, for y = log GDP (top panel) or the log core PCE index
(bottom panel), for different specifications: the baseline, reported in the main text (column 1); the same, but omitting the quadratic time trend (column 2); and using either the
2-year core inflation, the 3-year inflation, or the 1-year inflation as measure of initial inflation π t , rather than the 2-year inflation as in the baseline (columns 3-5). The sample is
1953q1:2019q4. Standard errors are Newey-West with 8 lags.
39
Baseline
No time trend
2y Core
3y
1y
Real GDP
βz,h
s.e.
t-stat
0.47***
(0.16)
3.03
0.56**
(0.25)
2.28
0.46***
(0.14)
3.27
0.50***
(0.14)
3.55
0.49***
(0.17)
2.93
γh
s.e.
t-stat
0.08
(0.06)
1.36
0.10
(0.07)
1.45
0.11
(0.07)
1.49
0.07
(0.08)
0.96
0.01
(0.06)
0.17
Obs.
252
252
228
248
256
Core PCE price index
βz,h
s.e.
t-stat
-0.01
(0.09)
-0.09
0.04
(0.10)
0.39
-0.08
(0.12)
-0.67
-0.05
(0.10)
-0.47
0.05
(0.09)
0.52
γh
s.e.
t-stat
-0.12**
(0.05)
-2.58
-0.05
(0.06)
-0.80
-0.13**
(0.07)
-2.02
-0.10*
(0.06)
-1.80
-0.09**
(0.04)
-2.02
Obs.
232
232
228
232
232
Table 9: The table reports the estimates of βz,h and γh from equation (28) in the main
text, for horizon h = 8 quarters, for y = log GDP (top panel) or the log core PCE index
(bottom panel), for different specifications: the baseline, reported in the main text (column 1); the same, but omitting the quadratic time trend (column 2); and using either the
2-year core inflation, the 3-year inflation, or the 1-year inflation as measure of initial inflation π t , rather than the 2-year inflation as in the baseline (columns 3-5). The sample is
1953q1:2019q4. Standard errors are Newey-West with 8 lags.
40
Baseline
No time trend
2y Core
3y
1y
Real GDP
βz,h
s.e.
t-stat
0.17
(0.17)
0.98
0.29
(0.29)
1.01
0.30
(0.19)
1.59
0.18
(0.17)
1.08
0.17
(0.18)
0.96
γh
s.e.
t-stat
0.11
(0.08)
1.44
0.14
(0.09)
1.52
0.08
(0.10)
0.80
0.11
(0.09)
1.19
0.04
(0.07)
0.49
Obs.
248
248
224
244
252
Core PCE price index
βz,h
s.e.
t-stat
0.11
(0.16)
0.67
0.20
(0.15)
1.32
0.04
(0.17)
0.24
0.08
(0.15)
0.53
0.14
(0.16)
0.88
γh
s.e.
t-stat
-0.10
(0.07)
-1.55
0.01
(0.10)
0.06
-0.10
(0.09)
-1.15
-0.06
(0.08)
-0.83
-0.08
(0.06)
-1.38
Obs.
228
228
224
228
228
Table 10: The table reports the estimates of βz,h and γh from equation (28) in the main
text, for horizon h = 12 quarters, for y = log GDP (top panel) or the log core PCE index (bottom panel), for different specifications: the baseline, reported in the main text
(column 1); the same, but omitting the quadratic time trend (column 2); and using either
the 2-year core inflation, the 3-year inflation, or the 1-year inflation as measure of initial
inflation π t , rather than the 2-year inflation as in the baseline (columns 3-5). The sample
is 1953q1:2019q4. Standard errors are Newey-West with 8 lags.
41
References
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42