A.4 Narrative Monetary Policy Shocks
2.5 Calibration
This technique reduces the number of endogenous states tremendously. As shocks to idiosyn-cratic productivity or the age distribution are neglected in this baseline version, we assume their marginal distributions to be time-constant as well. Hence, instead of perturbating the full distri-bution withnS–g r idF =2560points, we only have to perturbate around the marginal distribution of assets withnRS–g r id =nTA=80points. In comparison to Reiter (2009), the full distribution function maintains its shape.
Reduction of the control space relies on (inverse) discrete cosine transformations (DCT) of stationary equilibrium value function and consumption policies. Simply put, the idea behind DCT transformation is to identify pieces of information in e.g. the consumption policy that can be effectively “discarded” without seriously compromising its informative content (akin to image compression as e.g.jpeg). This approach reduces the size of the control-space substan-tially.16
2.5 Calibration
We calibrate the economic and demographic structure of the model to the US economy. Aggre-gate data used for calibration spans the post-Volcker disinflation time until 2008. One model period equals a quarter of a year, given that many macroeconomic variable that are relevant for stabilization policy run at this frequency. The following tables summarize the choice of param-eters and their corresponding target in the data.
Table 2.1.Calibrated Household Parameters
Parameter Value Description Source
Households
β 0.993 Discount factor K/Y = 285%
γ 0.75 Inv. Frisch elasticity Chetty et al. (2011)
σ 4 Coeff. Relative Risk Aversion Kaplan, Moll, and Violante (2018) Idiosyncratic Income
ρH 0.979 Persistence of Income Process Standard Value
σH 0.08 STD of Income Process Standard Value
16However, as the number of coefficients retained from the compression step is implicitly chosen by the researcher, robustness of the results should be checked.
2.5.1 Household Preferences
An overview over the parameters relevant for the household problem is provided in Table 2.1.
Households’ felicity over the composite goodxis of constant-relative risk aversion (CRRA) form.
Their degree of risk aversion is set toσ=4, as in Kaplan and Violante (2014). The inverse Frisch elasticity of labor supply isγ=0.75, and builds on the analysis in Chetty et al. (2011) that finds values that typically range between0.5and1. For a discount factor ofβ=0.993that matches the annual real interest of the regarded time period, the annual capital output ratio produced by the model is too low. As Table 2.2 shows, the sum of both liquid assets, capital and bonds, relative to output only amounts to 1.98 instead of 2.86 as in the data. The quarterly standard deviation of shocks to idiosyncratic labor productivity is set to 0.08 and the degree of quarterly persistence to 0.979 which follows Storesletten, Telmer, and Yaron (2004). The probability to leave the high-income entrepreneur state follows Guvenen, Kaplan, and Song (2014) and is set to 6.25%. This corresponds to their annual probability of 25% to drop out of the top 1%
income group. The share of household’s in the high-income state is set to 0.5% of all households.
However, as Table 2.2 shows, the model does not perform well in matching US wealth inequality as the implied-Gini coefficient of the model is only 0.58 instead of 0.78.17
Table 2.2.Model-Implied Moments and Data
Targets Model Data Source Parameter
Asset-to-Output Ratio (K+B)/Y 1.98 2.86 NIPA Discount factor Mean government expenditures (G/Y) 0.19 0.2 NIPA Discount Factor Mean government revenue (T/Y) 0.21 0.2 NIPA Tax Rate
Gini total wealth 0.59 0.78 SCF Fraction Entrepreneurs
2.5.2 Aging
The number of households in each age group is not necessarily equal. To capture differences between young and old societies, the mass of older relative to younger households has to vary.
When younger persons age faster or older persons age slower the average lifetime increases.
Hence, the transition probabilityξj =PJ(j+1|j)from one to another age-group will vary in size.
17The too-low wealth inequality is driven by the assumption that only old, dying households enter the high income state and then consume all assets. Hence, there is no high-income inequality in earlier stages that accumulates over the life-cycle. This has been done for numerical reasons such that the dynamic response of consumption to a monetary policy shock remains plausible.
2.5 Calibration | 53
Table 2.3.Parameters for Economies with Different Age Composition
Parameter Value Description Source
Young Economy
Xy 70.4 Life expectancy, years UN Data: 1965-1970
ξ1,2, ξ2,3 1.25% Age prob. states (1,2) to (2,3) own calculations
ξ3,4 4.25% Age prob. state 3 to 4 own calculations
ϕy 15% Old-age dep. ratio UN Data: 15.7% (1965)
Middle-aged Economy
Xm 78.9 Life expectancy, years UN Data: 2010-2015
ξ1,2, ξ2,3 1.25% Age prob. states (1,2) to (2,3) own calculations
ξ3,4 2.5% Age prob. state 3 to 4 own calculations
ϕm 25% Old-age dep. ratio UN Data: 22.1% (2015)
Old-aged Economy
Xo 86.5 Life expectancy, years UN Data: 2065-2070
ξ1,2, ξ2,3 1.25% Age prob. states (1,2) to (2,3) own calculations
ξ3,4 0.96% Age prob. state 3 to 4 own calculations
ϕo 51% Old-age dep. ratio UN Data: 49.3% (2065)
The number of age states is set toJ=4to keep the model specification parsimonious. The transition probability matrix will thus correspond to (2.2). Agents are born at age 20 and start to work immediately. In the young society, average life expectancy is around 70 years, while in the grey society people get 86 years old on average. I assume that the increase in life expectancy only matters for the old. Using equation (2.3), the probabilities{ξ1,2,ξ2,3}as in equation (2.2) are thus set to1.25%, which corresponds to 20 years of young and middle-aged life, respectively.
On the contrary, agents spend between 10 and 26 years in their last phase of life. Thus,ξ3,4 should be set to values between4.25%and0.96%.18This corresponds to an old age-dependency ratio between 0.15 and 0.7 which is in line with projected US population statistics. The mass of dying households, i.e. agents that are in their final life phase varies between 0.5% and 0.1%
of total population.
18Age transition probabilities correspond to the quarterly frequency of the model, as in equation (2.1).
2.5.3 Firm Sector
Table 2.4.Calibrated Firm Parameters
Parameter Value Description Source
Intermediate Goods
δ 0.0135 Depreciation Rate NIPA: Fixed Assets and
Consumer Durables
α 0.7018 Labor Share Standard Value
ρA 0.9 Persistence of TFP Shock Standard Value
σA 0.026 STD of TFP Shock STD(Y) = 1
θ 11.4 Capital Adj. Cost Investment-Volatiliy
Ratio = 4.5
Vendors
κ 0.085 Degree of Nominal Rigidity Avg. price duration 4 quarters
µ 0.95 Mark-up on Marginal Cost Standard Value
The labor share adjusted for profit income is set to1 –α=0.702, and hence, the capital share equalsα=0.298, which are standard, long-run averages for the U.S. economy. The rate of de-preciation is set toδ=0.0135, which corresponds to an annual consumption of fixed capital of 5.4%, which has been the long-run average over the regarded time period in the data. Resellers sell intermediate goods as final goods at a markup of5%percent, which implies marginal cost ofµ= η–1η =0.95in steady state. The degree of price stickiness,κ=0.085is set to target an average stickiness of four quarters. Both parameters, markup and price stickiness assume val-ues which are standard in the literature on New Keynesian Models (compare e.g.Christiano, Eichenbaum, and Evans (1999), Galí (2008)). We choose capital adjustment cost ofφ=11.4. This comes close to an investment volatility of 4.5 in response to TFP shocks, as found in U.S.
data. The TFP shock is highly persistent. The degree of persistence is set toρA=0.9and its standard deviationσA=0.026.
2.5.4 Government Sector
Treasury
The treasury issues debt according to a rule that is similar as in Woodford (1995) or Bi, Leeper, and Campbell (2013). The tax rate and government expenditures are jointly calibrated to match