C.2 Additional figures
3.3 Quantitative analysis
3.3.1 Functional forms and parameters
Since the model needs to be solved numerically, functional forms and parameters have to be specified.18
The model is calibrated by choosing suited parameter values from related studies and by targeting selected statistics of the income, wealth, and durable distribution observed for the United States, similar to Diaz and Luengo-Prado (2010), based on data from the Survey of Consumer Finances 2016 for the year 1998. The parameter values are summarized in Table (3.1).
Parameter Value Target
αK 1.04 Real interest rate4%
β 0.90 Debt to durables
γ 0.80 Empirical LTV ratio
σc 2.00 Standard value σd 2.00 Standard value
πR,S×100 0.05 Gini coefficient income πS,R×100 0.70 Gini coefficient wealth
¯
d 0.12 Relative durables distribution φd 0.10 qd/c¯
Table 3.1: Model parameters in chapter 3
The empirical counterpart of durable consumption in this model is not defined only as residential housing but adds vehicles as well, given that these two categories account for the majority of collateral
used for household credit. For the household utility function, a additive separable specification is assumed as in Diaz and Luengo-Prado (2010) or Guerrieri and Iacoviello (2017) given by
u(ct, dt) = c1−σc t 1−σc +φdd 1−σd t 1−σd (3.6)
where σc >0 and σd >0 are the inverse intertemporal elasticities of substitution and φd >0 captures
preferences for durables. Forσc andσd, a standard value of two is chosen. The discount factorβ is 0.9
to closely match the average ratio of secured debt to durable goods in the first quartile of the wealth distribution.19 The production function is assumed to be additive separable in capital and labor and is given by
F(Kt−1, Lt) =αKKt−1+αLLt (3.7)
where αK > 0 and αL > 0 denote the marginal product of capital and labor. Under this production
function the real interest rate and the wage rate are time-invariant and given by rt = αK −1 and wt=αL.20 The parameterαK is set to1.04to get a real interest rate of 4%whileαL is set to one such
that the individual labor supply state ei,t is identical to individual labor incomewtei,t. The fraction of
seizable collateral γ is set at 0.8, implying an empirically plausible loan-to-value ratio of80% (see Diaz and Luengo-Prado (2010)).
The support for the efficiency units of labor EL and the associated transition probabilities π are chosen to match the Gini coefficients for income wtei,t (0.43) and (net-)wealth b+qd (0.8). As is well
known in the literature (see e.g. Di Nardi et al. (2015)), without additional assumptions, a standard Bewley-Aiyagari-Huggett-type incomplete-markets model fails to match important features of the wealth distribution, the concentration of wealth at the top in particular. To address this shortcoming, Diaz and Luengo-Prado (2010) is followed and assumed that individual efficiency of labor supply follows a log-normal AR(1) process,
lnei,t=ρlnei,t−1+σεi,t,
with autocorrelationρ= 0.9895and standard deviationσ= 0.1257and, additionally, that there is also a small probabilityπR,S of transitioning to a relatively high efficiency state that results in a "superstar"
income, which is left with probability πS,R. Note that, due to a time-invariant wage rate equal for all
households, the relation of individual labor supply to individual labor income is time-invariant and equal for all households. In the following, w.l.o.g., the exogenous individual states are given by individual
19The average ratio of secured debt, given by home-secured debt and vehicle installment loans, to durable goods, given
by housing and vehicles, in the first quartile of the wealth distribution for the US household sector in 1998 is about 0.74 (see Survery of Consumer Finances).
relative durables distribution
[0%,25%) [25%,50%) [50%,75%) [75%,100%]wealth quartiles
0 0.2 0.4 0.6 0.8 1d
i/
d
i Model SCF 1998Figure 3.1: Relative durable holdings for different wealth quartiles (data vs model)
labor income yi,t with yi,t:=ei,twt, and not by individual labor supply ei,t.21 While the AR(1) process
provides a good fit for most of the population, it cannot suitably account for the top 1% of the labor income distribution. While the "regular" income states y1 to y6 are obtained by discretizing the AR(1) process via the method proposed by Tauchen and Hussey (1991), the superstar income value y7 is set to match the empirical ratio y7/y6 = 6and the transition probabilities are πR,S = 0.0005and πS,R =
0.0075. Combining these values with the transition probabilities for the regular income states, obtained by discretizing the AR(1) process, yields the transition probabilitiesπ(yi,t+1|yi,t). The aggregate supply
of the durable good d¯= 0.12 is chosen to provide a reasonable fit for the durables distribution in the benchmark version, as given by Figure 3.1. Figure 3.2 shows the distribution of net-wealth for the model and the data. Lastly, the preference parameter for durables is set toφd = 0.1to get qd/c¯ ≈1.6 which is roughly in line with Diaz and Luengo-Prado (2010).22