Data And Observation Equations

t = {1, 2, . . . , n^q} 7! {1964Q1, . . . , 2007Q4} and T = {1, 2, . . . , n^T} 7! {1964, . . . , 2007}

denote the quarterly and annual time spans of the sample, respectively. Ten (oq) quarterly series are used. Per capita real growth of output, consumption, and investment, along with labor hours, the Federal Funds Rate, the GDP deflator, and the growth rate of the compensation index are obtained from the sources described in Smets and Wouters (2007)⁹.

Two quarterly measures of real per capita household debt, loaded with measurement error, aim to discipline the evolution of debt. Home mortgages (HM_t) and consumer credit debt (CCt) are obtained from Fred. The second series is loaded with a factor b. The observation equations are reported below, where dlnXt⌘ 100(lnX^t lnXt 1).

2

Debt detrending is data driven and captures the fact that the sample average growth rates of home mortgage and consumer credit debt (1% and 0.7%, respectively) are well above the growth rate of output (measured at 0.4% within the sample)¹⁰.

9The observation equations for that block of observables are similar to those in Smets and Wouters (2007), with the inclusion of measurement error, ✏^j_t⇠ N(0, µ²j), for each generic j series being the di↵erence.

10Imposing the balanced growth path trend on debt series would yield persistent and volatile measurement

Moreover, I introduce observables for the annual top 10% income and wealth shares obtained from the World Inequality Database and the work of Piketty and Saez (2003) and Saez and Zucman (2016). Table (1.1) gives inklings of the correlation of income and wealth inequality with macroeconomic and debt series. Income inequality is strongly pro-cyclical, whereas wealth inequality exhibits a small degree of counter-cyclicality. Moreover, the correlation of both income and wealth inequality with output are stronger in the post-84 period than in the pre-84 period. Furthermore, income inequality is positively correlated with the multiple measures of household debt. Although wealth inequality does not commove with debt across the entire sample, it is actually positively correlated with debt in the years before 1984 but becomes negatively correlated after 1984.

Table 1.1: Descriptive Statistics, Correlations

T IS T W S Y ⇡ HM CC

1954-2009 T IS 1.64 0.30 0.49 -0.16 0.34 0.32 T W S 0.30 1.63 -0.07 -0.33 0.02 -0.07 1954-1983 T IS 1.50 0.32 0.43 -0.19 0.37 0.43

T W S 0.32 1.90 -0.01 -0.27 0.21 0.08 1984-2009 T IS 1.69 0.15 0.68 0.37 0.23 0.19 T W S 0.15 1.14 -0.22 -0.21 -0.56 -0.47

Notes: Growth rates, annual frequency. Mnemonics: T IS: Top 10% Income Share, T W S: Top 10%

Wealth Share, Y : Real GDP per capita, ⇡: inflation, HM : Real Home Mortgage Debt per capita, CC:

Real Consumer Credit Debt per capita. Data sources: FRED Economic Data, World Inequality Database.

The model-implied pre-tax top income share that is consistent with the observed series (T IST) encompasses wages, profits (entrepreneurial income), and interest (capital income) over aggregate pre-tax income, and is pinned down by:

T IS_T = X3

j=0

n^⌧Y_{t j}^⌧ / X3

j=0

n^⌧Y_{t j}^⌧ + n^µY_{t j}^µ

⇡ tis + X3

j=0

(⌫j · ctist j) + (1· ^tis· t) + ✏^tisT (1.19)

errors in (1.18); see earlier versions of this paper.

where ctist= ctis^ea_t + ctis^b_t+ ctis^pr_t . The approximation (1.19) is obtained in equilibrium, after converting the ratio in terms of stationary variables, and taking a Taylor expansion (Appx.

A.4). The steady state top income share is given by tis = (1 s) + tis_b: 1 s is the top 10% wage share, and tisb ⌘ n^⌧b^⌧(R e ⇧)/(e ⇧w^rLR) stands for bond income flowing to the top. Steady state profits are zero. ctist stands for the cyclical component of the top income share; its weight is given by ⌫j ⌘ e^{(3 j)} /[e³ + e² + e¹ + 1]. Measurement error,

✏^tis_T ⇠ N(0, µ²tis), is included. tis· t approximates higher-order terms in the expansion, and aims at capturing the non-stationary evolution of T IST after the mid-80s; consequently, the indicator function, 1, is non-zero only during that period. {ctis^ea_t , ctis^b_t, ctis^pr_t } drive the swings of the top income share; they stand for earnings, bond income, and profits, respectively.

They are given by:

Table 1.2: Cyclical Swings, Top Income Share

earnings channel tisc^ea_t = s¯· bst (wb_t^r+ bLt) tisb

(1.20) bond income channel tisc^b_t = (bb^⌧_{t 1} b⇡t) [n^⌧b^⌧/(e ⇧w^rL)]

(bb^⌧_t brt bt^b) [n^⌧b^⌧/(w^rLR)]

(1.21) profits channel tisc^pr_t =bvt[(y/w^rL) (n^⌧!^⌧ tis)] (1.22)

According to (1.20–1.22), the top income share falls below its steady state if there is an increase in the wage share of the bottom (bs^t), the average wage, employment, or inflation.

In contrast, increases in the bottom borrowing and in profits (bv^t) boost capital income flowing to the top and, in turn, the top income share.

The model-implied top wealth share that is consistent with the observed series (T W ST) is given by the sum of shares and assets over the value of all shares (n^⌧⌦^⌧_tQt+ n^µ⌦^µ_tQt= Qt) since family debt positions net out to zero:

T W ST = X3

j=0

n^⌧

"

Qt j⌦^⌧_{t j}+ B_{t j}^⌧ /Pt j

e ^{bt j}Rt j

# /

X3 j=0

Qt j

⇡ tws + X3

j=0

(⌫j · dtws^tws_{t j}) + (tws· t) + ✏^twsT (1.23)

where dtws^tws_t = dtws^!_t + dtws^b_t+ dtws^q_t. The approximation (1.23) is obtained after converting the ratio in terms of stationary variables, and taking a Taylor expansion. (Appx. A.4). The steady state top wealth share is given by tws = n^⌧!^⌧+twsb: n^⌧!^⌧ is the top profit share, and twsb ⌘ n^⌧b^⌧/(Rq) is the top’s outstanding assets to the value of shares in terms of the final good. dtws^tws_{t j} stands for the cyclical component of the top wealth share. Measurement error,

✏^tws_T ⇠ N(0, µ²tws), is included. _tws · t approximates higher-order terms in the expansion, and aims at capturing the fact that T W ST exhibits a small positive sample growth rate.

The terms {dtws^!_t, dtws^b_t, dtws^q_t} drive the fluctuations of the top wealth share; they stand for shares, bonds, and asset price gains/losses, respectively. They are given by:

Table 1.3: Cyclical Swings, Top Wealth Share

real assets (shares) channel twsd^!_t =!b_t^⌧ (n^⌧!^⌧)

(1.24) bonds channel twsd^b_t = (bb^⌧_t br^t bt^b) twsb (1.25)

asset price gains/losses channel twsd^q_t = qbt twsb (1.26)

According to (1.24–1.26), the top wealth share overshoots its steady state when the profit shares or the intra-household assets of the top increase. It undershoots it, however, when the interest rate, or the risk premium shock, or the asset price (bqt) increase since all those changes decrease the contribution of outstanding bonds to the top wealth share.

Few assumptions underlie equations (1.18, 1.19, 1.23). (1.18) implies that the observed debt pertains to the bottom 90% of the income distribution¹¹. This mapping is supported by data from the Survey of Consumer Finance [Ravenna and Vincent, 2014], while including multiple debt indicators with measurement error helps extract their part that is relevant for the model. In addition, measurement error in both top shares (1.19 and 1.23) mitigates potential inconsistencies from assuming that the top of the income and wealth distributions coincide. Consistently with the model’s foundations, there is no feedback from inequality to the growth rate along the balanced growth path that is determined by technology.