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-
2.2 Low Net Worth Households
Next we discuss the second part of the model, which defines the less wealthy households that may have some small amount of assets but mainly borrow for consumption. Thus this second group of households have a lower amount of wealth, lower returns on capital, lower capital gains, lower labor income, and obtain income from a risk free rate. These households are not consumption smoothers in the optimal sense, but borrow to finance consumption.18 In this case, consumption is not necessarily optimal and may in fact result in larger than optimal consumption, given the household constraints.
VDmaxfc2;t;˛2;tg
XN kD0
.1C/tU.c2;tx2;t/ (4)
such that
x2;tDx2;t1.1Ch/.˛2;t.y2C2lnztCr2/
C.1˛2;t/.i2lnzt/'.x2;t/ Qc2/ (5) ztDexp.lnzt1Ct/: (6) Here we may normally have˛2 D .1Cf2/;with˛2 < 1; which means there is income from the risk free rate as households may deposit savings in a bank with risk free returns. For these types of households we also could have for certain time
18Note that recently many financial instruments, such as sub-prime loans, collateralized debt obligations (CDOs), mortgage-backed securities (MBOs) and credit cards, have been developed to induce the low net worth households to borrow and to consume.
Fig. 2 Borrowing by consumers: Path of net worth,dark line, and stochastic process,red line
periodscQ2 D const > coptt .19 We could interpret this as a constant propensity to consume as in some text books of Keynesian economics. Here it is, however, the propensity to consume out of (net) wealth whereby the propensity to consume is taken as constant.20
Note that in the context of this section we use NMPC for solving the stochastic difference equations (5) and (6).21In other words, we don’t use the optimal control problem, this is only simultaneously computed, since we want to know that the conditioncQ2 > coptt holds. Hence, we solve in the system (4)–(6) only Eqs. (5), including (6), a stochastic difference equation. Thus we employ NMPC only as a stochastic difference equation solver. So Eq. (5), where a simulation is run, is solved with non-consumption smoothing behavior or our second type of agents with Q
c2 >coptt , which is shown in Fig.2.
Yet, in order to see whether the conditionQcDconst>coptt is fulfilled, whereby coptt is the solution of Eqs. (4)–(6), we need also to look at the solution of Eqs. (4)–
(6) using NMPC. Then we can check if actually, for some time period, consumption spending for this type of households maybe undertaken with borrowing that does not necessarily fulfill a longer-term sustainability condition of the household debt.
19This is for example what some literature has stated on the consumption boom of low income households before 2007/8, see Cynamon and Fazzari (2016).
20For this group of households Eq. (4) could be disregarded, since those households are not consumption smoothers. Formally then an additional dynamics, the evolution of debt as in Mittnik and Semmler (2016), may need to be introduced to account for the interest payment on debt. Here too wage income in such a portfolio model could be treated as in Chiarella et al. (2016).
21NMPC can also be used to solve stochastic difference equations when instead for the optimal solutioncDcopta constant term is used in Eqs. (5).
Here we can also use the fact that the interest rate paid on bank accounts is stochastic,i2lnzt. The same holds for capital gainsy2 C2lnzt. Running now the stochastic model (5)–(6), using NMPC, and checking this with the solutions for (4)–(6), we obtain the results as shown in Fig.2. There we can observe how a lower return (from capital income), lower saving rates—actually mostly negative saving rates—lead to a declining net worth and thus dissipating net wealth.
As can be observed from Fig.2the borrowing by low net worth households, to fulfill their consumption needs, produces a declining path of net worth, the dark line. The stochastic process is again the red line in Fig.2. The declining share of net wealth of low income, low net worth, households is also found in the empirical data demonstrated in Sect.3.2.
Moreover, the empirical observations found in SCF data in Sect.3.2 also demonstrates that normal income becomes a decreasing percentage of net worth for high income earners. However, this income constitutes a much greater percentage of net worth for low income households that show a declining share of total wealth in the U.S. Section3also demonstrates empirical findings that this trend has become more extreme in recent years.
Although model (5)–(6) does not yet directly model instability, the leveraging and over leveraging of this group of households is also not without macroeconomic perils. This is nicely demonstrated in Cynamon and Fazzari (2016) where a fall of asset prices and fall in net worth, in particular after 2007/8 amplified the downward spiral. Figure2, upper line, depicts the volatility only in the stochastic process of Eq. (6), the issue not pursued here further.
These observable effects of a persistently downward trend, lower line, may lead to the question of why the households don’t rationally expect this trend of dwindling net worth and debt possibly surpassing assets, producing the threat of bankruptcy and default risk. In a model of infinite horizon behavior such a situation would not occur. Through the optimal consumption and the transversality condition this would be prevented.
Since we are working here with a model of finite time horizon, though there is an expectation on some shorter time horizon by households and the lenders are loan pushers for consumption and possibly mortgages for low net worth households, we implicitly assume that there is some near-sightedness allowing for the emergence of over leveraging and the vulnerability of those households.22 In fact it is this assumption that makes our model more realistic when compared to infinite time horizon models.
22For empirical trends like this in the US since the great recession, see Cynamon and Fazzari (2016).