Appendix B Numerical solution of the OLG model
B.4 Updating the forecasting equations
Using the simulated time-series (after discarding the first 100 observations) we estimate the following OLS regressions, for every pair of productivity shock (Ut), depreciation shock (ηt), tax rate (τK,t) and bond supply (Bt) realizations,
ln(kt+1) = q10+ q11ln(kt) (B.10) and
ln(Pt+1B ) = q20+ q21ln(kt) + q22ln(PtB) (B.11) This gives us 32 equations and 32 sets of coefficients that forecast state-contingent capital (kt+1) and bond prices (Pt+1B ). We iterate the code until we have converged on the coefficients and on the R-squared of these regressions. For the first set of equations (B.10) we obtain R-squared values around 99.99%. For the second set of equations (B.11), the R-squared values are in the 90%− 95%
range when we only use ln(kt) as a regressor, but increase to more than 99% when we add ln(PtB).
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Table 1: Panel A reports values from the baseline model and aggregate U.S. data. Government debt is the U.S. federal debt held by the public between 1952 and 2002. Panel B reports the standard deviation of consumption (Campbell, 1999 annual data) and output (National Income and Product Accounts data). Panel B also reports the standard deviation of stockholders’ and non-stockholders’
consumption growth rates from the baseline model and from the data (Consumer Expenditure Survey, numbers are from Malloy, Moskowitz and Vissing-Jørgensen (2005) and annualized).
Panel A: Share of Output (percent)
Model Data
Consumption 58.7 59.4
Investment 20.7 20.2
Government 20.5 20.4
Government Debt 35.2 35.8
Panel B: Standard deviation of growth rates (percent)
Model Data
Aggregate Output 3.9 4.3
Aggregate Consumption 3.0 3.2
Stockholders Consumption 4.4 6.9
Non-Stockholders Consumption 3.2 2.7
Table 2: Table reports tax revenue (in percent of output) by source from the baseline model and aggregate U.S. data from the BEA National accounts.
Model Data (1929-2006) Data (2006)
Capital 5.45 5.26 5.78
Labor 8.00 6.80 8.71
Consumption 7.64 8.53 7.77
Table 3: Unconditional asset pricing moments from the baseline model.
Variable Moment Model Data
Mean 1.81 1.23
rf Std. Dev. 1.21 3.83
Mean 7.04 7.77
rm Std. Dev. 19.74 20.11
rm− rf Mean 5.23 6.54
Table 4: Wealth Distribution. The table reports the percentage of each group’s total wealth held within a given percentile range. Data source: 2001 Survey of Consumer Finances. Wealth is the net worth of households as defined in the text and stockholders are defined as households who own stocks directly or through mutual funds either in taxable accounts or in tax-deferred pension plans.
Non-stockholders Stockholders All
Percentile Model Data Model Data Model Data
10th 0.00 0.009 0.50 -0.021 0.00 -0.080
20th 0.00 0.009 2.50 0.41 0.00 -0.007
50th 0.39 2.42 20.73 5.60 0.12 2.90
50th-80th 6.05 18.03 36.81 16.62 30.93 14.55
80th-100th 93.56 79.55 42.46 77.78 68.96 82.54
90th-95th 9.69 14.56 10.70 11.83 17.45 12.20
95th-99th 42.59 22.26 10.69 25.94 18.35 25.26
99th-100th 34.11 26.42 3.59 26.86 6.57 32.14
Table 5: Variables responses to fiscal expansion over different horizons. The responses are given as average in percent per year using the discount rate of zero (no discounting).
Time 10 years 20 years 50 years
dC -0.06 -0.10 -0.06
dY 0.04 -0.02 -0.02
dK 0.12 -0.06 -0.05
dCns 0.04 -0.02 -0.02
dCs -0.14 -0.15 -0.08
Table 6: Variables responses to fiscal expansion over different horizons for different discount rates.
The responses are given as average in percent per year using various discount rates.
Time 10 years 20 years 50 years
r 0% 2% rk 0% 2% rk 0% 2% rk
dC -0.06 -0.05 -0.03 -0.10 -0.07 -0.05 -0.06 -0.04 -0.02 dY 0.04 0.05 0.05 -0.02 -0.01 0.00 -0.02 -0.01 0.00 dK 0.12 0.13 0.14 -0.06 -0.03 0.01 -0.05 -0.03 0.00 dCns 0.04 0.05 0.05 -0.02 -0.01 0.00 -0.02 -0.01 0.00 dCs -0.14 -0.05 -0.03 -0.15 -0.12 -0.09 -0.08 -0.06 -0.04
Table 7: Cross-sectional impact for different wealth cohorts to baseline intervention scenario over different horizons.
q1 (low) q2 q3 q4 (high)
20 years
∆C -0.02% -0.03% -0.04% -0.17%
(eq. weighted %) 7.0% 11.1% 16.9% 65.0%
50 years
∆C -0.01% -0.02% -0.03% -0.10%
(eq. weighted %) 8.8% 11.7% 16.5% 62.9%
Table 8: Cross-sectional impact for different age cohorts to baseline intervention scenario over different horizons.
Panel A
20-35 36-50 51-65 66-80 20 years
∆C -0.02% -0.04% -0.08% -0.15%
(eq. weighted %) 6.5% 14.2% 29.0% 50.3%
50 years
∆C -0.01% -0.02% -0.04% -0.08%
(eq. weighted %) 8.9% 13.6% 25.6% 51.9%
Panel B
30 45 60 75
∆C (15 years) -0.02% -0.07% -0.12% -0.17%
(eq. weighted %) 4.5% 18.5% 32.1% 44.9%
Table 9: Variables responses to fiscal expansion over different horizons for longer intervention. The responses are given as average in percent per year using the discount rate of zero (no discounting).
Intervention 10 Years Baseline (5 years)
Time 10 years 20 years 50 years 10 years 20 years 50 years
dC -0.05 -0.19 -0.13 -0.06 -0.10 -0.06
dY 0.13 -0.04 -0.04 0.04 -0.02 -0.02
dK 0.34 -0.13 -0.13 0.12 -0.06 -0.05
dCns 0.13 -0.04 -0.04 0.04 -0.02 -0.02
dCs -0.19 -0.31 -0.19 -0.14 -0.15 -0.08
Table 10: Variables responses to fiscal expansion over different horizons for lower foreign demand for government bonds. The responses are given as average in percent per year using the discount rate of zero (no discounting).
Foreign Demand 10% Baseline (25%)
Time 10 years 20 years 50 years 10 years 20 years 50 years
dC -0.11 -0.12 -0.06 -0.06 -0.10 -0.06
dY -0.04 -0.06 -0.03 0.04 -0.02 -0.02
dK -0.11 -0.17 -0.08 0.12 -0.06 -0.05
dCns -0.04 -0.06 -0.03 0.04 -0.02 -0.02
dCs -0.17 -0.16 -0.09 -0.14 -0.15 -0.08
Table 11: Variables responses to fiscal expansion over different horizons for larger increase in gov-ernment debt. The responses are given as average in percent per year using the discount rate of zero (no discounting).
Gov. Debt Increase 35% Baseline (25%)
Time 10 years 20 years 50 years 10 years 20 years 50 years
dC -0.06 -0.12 -0.07 -0.06 -0.10 -0.06
dY 0.08 -0.02 -0.02 0.04 -0.02 -0.02
dK 0.21 -0.06 -0.05 0.12 -0.06 -0.05
dCns 0.09 -0.02 -0.02 0.04 -0.02 -0.02
dCs -0.17 -0.19 -0.11 -0.14 -0.15 -0.08
20 30 40 50 60 70 80 90 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Age
Consumption Gini coefficients over life cycle
Nonstockholders Stockholders All
Figure 1: Consumption inequality over the life-cycle. The figure shows the cross-sectional gini coefficients for each age group.
20 30 40 50 60 70 80 90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Age
Wealth Gini coefficients over life cycle
Nonstockholders Stockholders All
Figure 2: Wealth inequality over the life-cycle. The figure shows the cross-sectional gini coefficients for each age group.
0 10 20 30 40 50
Figure 3: Impulse responses to fiscal intervention. Baseline scenario.
0 5 10 15 20 25 30 35 40 45 50
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2
Agg. Cons.
Years
Response (% dev. from s.s.)
0 5 10 15 20 25 30 35 40 45 50
−1
−0.5 0 0.5 1
Capital (Shr. of Init. Cap.)
Years
Response (% dev. from s.s.)
0 5 10 15 20 25 30 35 40 45 50
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
Output
Years
Response (% dev. from s.s.)
Figure 4: Impulse responses to fiscal intervention. Solid line: longer intervention, dotted line:
shorter intervention