Corollary 3 Marginal Rates of Substitution of the Tax Rates for the Log- Utility Case) Let the utility function of the representative household be logarith-
2.1 High Net Worth Households
We first specify our model for the high net worth households which are consumption smoothing households. We here, as Brunnermeier and Sannikov (2014), specify our model for the purpose of presentation, we take the instantaneous payout,c1;t, and leveraging,˛1;t;as decision variables.11 The model is stated in Eqs. (1)–(3), where preferences for the consumption smoothing households are given by (1), the
8For details on roughly zero or negative saving rates of low income households, see Dynan et al.
(2004).
9For further details of the Stein model and the derivation of this, see Gross and Semmler (2017).
10Stein (2012) adds something important to this type of model by providing a static solution for the model in terms of a mean variance approach that allows to measure then excess leveraging, see Schleer et al. (2014).
11Note that Brunnermeier and Sannikov (2014) usext as a notation for their decision variable leveraging and assume thatxt> 1:
dynamics of the capital stock,ktfor the appropriate group is left in the background,12 but we have in (2), net worth,x1;t, and in (3), the stochastic shock process,zt.
Note that we have written here the stochastic process in discrete time, since we solve the model in discrete time anyway. In our solution method we adopt a discrete–
time framework and decision horizon ofNperiods, executed for each time period, tD1 : : :Ttime periods, andhtime steps and our model is given by
VDmaxfc1;t;˛1;tg
XN kD0
.1C/tU.c1;tx1;t/ (1)
such that
x1;tDx1;t1.1Ch/.˛1;t.y1C1lnztCr1/
C.1˛1;t/.i2lnzt/'.x1;t/c1;t/ (2)
ztDexp.lnzt1Ct/: (3)
In the above model (1)–(3),c1and˛1are our two decision variables,13with the payoutc1 DC1=x1, and˛1 D1Cf1;withf1Dd1=x1the leverage ratio, measured as liability over net worth,d, is debt,y1;capital gains, driven by a stochastic shock, 1lnzt: Furthermore, r1; is the return on capital,i; an interest rate paid on debt, also driven by a stochastic shock,2lnzt,14'.x1;t/is a convex adjustment cost,;a persistence parameter, with D 0:9;andt is an i.i.d. random variable with zero mean and a variance, D 0:05. Moreover, note that in the current context of high net worth households, that dominantly borrow for asset purchases, we can have
˛1 D 1Cf1;thusf1 > 0:If˛1 > 1there is no short sales constraint of risk free assets and thus there can be active borrowing for investment.15
12In the solution procedure used here, we neglect the dynamics of the capital stock for this type of households as for the households borrowing for consumption. In Brunnermeier and Sannikov (2014), it represents the aggregate capital (withg the growth rate and ı the resource use for managing the assets) of financial specialists and households. A larger fraction of it will be held by financial specialists, since they can borrow. Those details can be neglected here. Aggregate capital is more specifically considered below, and see also Brunnermeier and Sannikov (2014).
13Note that in our model version here, we neglect the dynamics of (2). In Brunnermeier and Sannikov (2014), it represents the aggregate capital of financial specialists and households (with gthe growth rate of capital, another decision variable, andıthe resource used for managing the assets). A larger fraction of the assets will be held by financial specialists, since they have a lower discount rate, and with˛t> 0, they can borrow. Those details can be neglected here. The dynamics of aggregate capital is more specifically considered in Brunnermeier and Sannikov (2014).
14Stein (2012) posits that the interest rate shocks are highly negatively correlated with capital gains’
shocks, we have thus a negative sign in (3) for the effect of the shocks on interest rates. We here also assume that the interest rate shocks have smaller variance than the capital gains’ shocks.
15Note that wage income in such a portfolio model could be treated as in Chiarella et al. (2016).
Fig. 1 Borrowing by investors: Path of net worth,dark line, and stochastic process,red line
We can solve the model variant (1)–(3) through a stochastic version of NMPC, see Grune et al. (2015). Figure 1, the black line presents the path of net worth, the red line the stochastic process, and leveraging ˛1;t D 1 Cf1;t, is generated separately, which is always above one. We want to note that we solve here only for optimal leveraging and optimal consumption. Since˛1;t is a choice variable, both Brunnermeier and Sannikov (2014) and Stein (2012) assume that debt is redeemed in each period and, without cost, can be obtained on the market without frictions.
The leveraging of this group is not without perils. There could be a macro feedback loop, namely an externality, i.e., endogenous volatility, that is triggered below some steady state, see Brunnermeier and Sannikov (2014), which creates an unstable steady state and unstable downward spiral. In Brunnermeier and Sannikov (2014), such feedback loops can arise from large shocks below some steady state, triggering fire sales of assets, fall of asset prices and fall in net worth, generating a downward spiral.16Although model (1)–(3) does not yet directly model instability as a result of an over leveraging of the high net worth households, Fig.1depicts the volatility only in the stochastic process, the issue not pursued here further.17
Through our numerical computations we can also observe the impact of lever- aging on the accumulation of capital, not depicted here, see BS (2014, sect I). As
16In Stein (2012) the vulnerabilities and possibly adverse feedback loops are triggered by over leveraging, capital losses and rising borrowing cost.
17Excess leveraging in Stein (2012) is driven by actual leverage over and above the optimal leverage, caused by a sequence of persistent shock to increase capital gains and lowering interest rates, both giving rise to excess leveraging, For further amplifying feedback loops, as Mittnik, see Mittnik and Semmler (2016). Stein’s model can distinguish between optimal debt, actual debt and excess debt. For empirical measures of those, see Schleer et al. (2014).
Brunnermeier and Sannikov (2014) properly state, through leveraging, the capital share of the active investors in total capital—the share of financial experts in their terms—rises with leveraging, even at the stochastic steady state. This is also what creates in Brunnermeier and Sannikov (2014) the source of the wealth distribution and the endogenous risk.
The increasing wealth distribution is also observed in the empirical analysis in Sect.3.2by several notable trends. First, the share of high yielding assets owned by high income earners is substantially larger than that of lower income households.
For example, stocks and bonds become an increasing component of the net worth of such households while lower income earners may rely more upon pension funds or simply checking accounts, as discussed next. Further discussion of such observations in SCF data is reserved for Sect.3of the paper.