-40 -20 0 20 40 60
jun/91 jun/92 jun/93 jun/94 jun/95 jun/96 jun/97 jun/98 jun/99 jun/00 jun/01 jun/02 jun/03
Brazil
Figure 18: Brazilian RRA at the per capita consumption level. Hodrick-Prescott filter used to smooth consumption (C → ς).
Average RRA
jun/91 jun/92 jun/93 jun/94 jun/95 jun/96 jun/97 jun/98 jun/99 jun/00 jun/01 jun/02 jun/03
Brazil
Figure 19: RRA integrated over the Brazilian consumption distribution, su-posed to be exponential with mean equal to the per capita consumption level.
Hodrick-Prescott filter used to smooth consumption (C → ς).
level X = aς most of the time, thus guaranteeing the general equilibrium.
The mix between risk-seeking and risk-averse agents is such that, as a whole, the economy can still find its equilibrium.
As U (C) is the utility of annual consumption, its convexity below the habit level only means that agents operating in that region prefer to throw a coin and, next year, consume c + ε, if it turns out head, or c − ε, if it turns out tail (ε small compared to c), instead of a guaranteed consumption of c.
It doesn’t mean that they would behave like this when confronted with any lottery offering immediate payoff of money or other benefit. They are trying to catch up with the long run standard of living of the representative agent, not necessarily seeking immediate rewards, although it is a known fact that the lower classes tend to buy (actuarially disadvantageous) lottery tickets more frequently than the upper classes....
Due to the convexity of the lower part of the utility function, consumers who operate in that part and are not close enough to the per capita consump-tion level maximize their utility not by satisfying the first order condiconsump-tion (eq.
14), but by restricting their consumption at t=0 to the subsistence minimum and investing all the rest, that is, for them the subsistence level is binding.
As a consequence, they are willing to accept any amount of credit they are offered.
Figures 26 and 27 show the spikes in 1974 and 1980 corresponding to the two oil shocks (and a smaller one corresponding to the firs Gulf War);
notice how they are present in the models that use power (CRRA) utility and Epstein-Zin too. They alone are responsible for half of E(Q) in the period considered (1951-2001), which is the hallmark of the socalled "black swan" phenomenon, as popularized by Nassim Taleb (see, for instance, Taleb (2008a) and Taleb (2008b)) and put in doubt some of the basic assumptions of the class of models considered in the present article (and most of the literature on the equity premium puzzles).
Figures 4 and 5 suggest that, as people are born and progressively have their income raised, they behave like waves that hit a beach (the central bump), thus prone to breaking and generating turbulence and, as a conse-quence, financial bubbles; notice that this wouldn’t happen with the usual power utility (CRRA), whose RRA function is a constant horizontal line.
Thus, pressure from the risk-seeking new generation driving the government to support bad credit loans via Fannie Mae and Freddy Mac (the prosaic case of the Californian strawberries picker who made US$14K per year and was offered a US$700K mortgage became a saloon anedocte) may be the ultimate responsible for the crash; further research should confirm or reject the hypothesis that all major financial crashes are caused essencialy by this shock of generations phenomenon.
The RRA function of my utility corresponds to what is observed in animal psychology: coming from higher to lower consumption values, the represen-tative agent is first taken by panic (increased RRA), then despair (negative RRA) and, thus, disposition to risk everything, and finally desolation (neg-ative RRA, but small in module).
The implied fact, by the final model (external habit), that ∼70% of the population has negative RRA, may be an explanation for the high debt levels that this extract is willing to take, which, by its turn, could explain phenom-ena like the so-called "credit feast" in Brazil and the "real state credit crunch"
in US.
The model economy of this paper works at the edge of chaos (in a loose sense), since the operating point of the representative agent oscillates around Xi’s threshold for the existence of equilibrium, with the two points being statistically indistinguishable (the sample average of their difference is within one standard deviation from zero). The median income households operate in the chaotic regime, the representative agent barely escaping this by operating slightly above the third quintile, except around the second oil shock (Iranian revolution and Iran-Iraq war). These features are in accordance with a power law behaviour for the tail of the probability distribution of the pricing kernel and the existence of very influencial spikes present in its time series.
The fact that an S-shaped utility function can, in a way, solve the equity premium puzzle pushes the research direction towards models of the economy
akin to neural networks, in which neurons are fired whenever a threshold of excitation is crossed. If this is the right direction, only future works can say.
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8 Figures
Figure 20: Consumption smoothing.
Figure 21: Fitting to Fama and French’s 25 size and book-to-market portfolio.
Figure 22: RRA of the representative agent.
Figure 23: Expected value of the RRA over the consumption distribution.
Figure 24: Consumption of the representative agent over the habit level X = aς.
Figure 25: Consumption of the representative agent over Xi’s threshold Ctg. When C > Ctg, the general equilibrium is guaranteed to exist.
Q
Figure 27: Q=M/β (pricing kernel over the subjective discount factor).
Figure 28: Parametric space. Parameter a of our utility runs horizontally from 0 to 1 towards the right. Parameter b runs vertically from 0 to 10 towards the bottom. The stain is the region where FF% < 70%.
Figure 29: Parametric space. The vertical axis shows the average of E[RRA]
in the period 1946-2004. The roughly horizontal one, shows parameter b running from 0 to 10 (1 to 49 in the figure). The depth axis shows parameter a running from 0 to 1 (série1 to série 41 in the figure).
Figure 30: Parametric space. Parameter a of our utility runs horizontally from 0.7 to 1 towards the right. Parameter b runs vertically from 0 to 2 towards the bottom. The stain is the region where FF% < 30%.
Figure 31: The precedent mapping using the vertical axis to represent FF%.
The best fit to Fama-French portfolio is in the central depression (lowest FF%). Parameter b runs from 0 to 2 in the roughly horizontal axis. Para-meter a runs from 0.7 to 1 in the depth axis (0.7 to S43).
Figure 32: Parametric space. The vertical axis shows the average of E[RRA]
in the period 1946-2004. The roughly horizontal one, shows parameter b running from 0 to 2 . The depth axis shows parameter a running from 0.7 to 1 (0.7 to S49 in the figure).