6 Models with permanent and temporary shocks
Condition 7 Suppose individual incomes follow a process with permanent and tem- tem-porary shocks
8.5 Proof of welfare change measure
Proof: The Euler equation is
π0,texp −Ahcht
So total utility is
Uh(x) =
So consider equilibria before and after (*) innovation. We want to choose θ such that Uh(C + θ) = Uh(C∗) . I.e.
Taking logs, we get
ch0 + θ − ch∗0 = 1
Since both ch0 and ch∗0 are both equilibrium allocation, if we take averages, they must be equal, which gives us
θ = 1
Proposition 3: Consider two economies: (i) An economy as described above; (ii) an economy that is the same except that the assets are long-lived and have dividend streams (dj,t)t=1,...,Tj=1,...,J where the dj,t’s follows ARMA processes with innovations xj,t. Then the equilibrium consumption allocations are the same in the two economies.
Proof: The basic idea is that given the long-lived assets, one can synthesize the short-lived assets and vice versa. The proof is by recursion.
In the equilibrium of economy (ii) at period T − 1, the conditional payoff on asset j is dj,T which equals ET −1(dj,T) + exj,t. So, an individual can synthesize the one-period asset with payoff xj,t by buying asset j and shorting the riskless asset by ET −1(dj,T). The price for the synthesized asset must be the same in equilibrium (ii) as in equilibrium (i) by proposition 2. By arbitrage, the price of the long-lived asset must be
Pj,T −1= π0,T −1 E
T −1(dj,T) + πj,T −1.
Clearly, conditional on the wealth distribution, the two equilibria must be the same.
In period T − 2, the conditional payoff of the long-lived security in period T − 1 is dj,T −1+ Pj,T −1. By the fact that dj,T −1 is an ARMA process, Pj,T −1 is linear in xj,T −1
so the payoff on the long lived asset is affine in xj,T −1. Thus, one can synthesize a one-period asset paying off xj,T −1. Alternatively, with just the one period asset, one can synthesize the payoff on the long-lived asset. So again, the two equilibria must be the same.
References
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Study RRAa
1 3 10
Carroll and Samwick(1996)b 0.56672 1.7412 8.4603 Hall and Mishkin(1982)c 0.5648 1.7759 11.9715 Heaton and Lucas(1996)d 0.49595 1.4121 3.1785
MaCurdy(1982)e 0.9107 2.9495 f
Table 2: Welfare losses due to incomplete markets as a percentage of consumption.
aWe consider different risk aversion and assume the following: δ = .96, T = 1000, kht = 0, E yht
= 20, 000
bCarroll and Samwick’s estimated an equation of the form described in Condition 7 in logs. We take a log-linear approximation and use their estimates. They estimate the variance of permanent shocks to be 0.0217 and the variance of transitory shocks to be 0.0440.
cAs reported in Caballero(1990), Hall and Mishkin estimate the income process (again in logs) to be
yth=yht−1+ηht − 0.360ηt−1h − 0.080ηht−2− 0.060ηht−3 whereσ2= 0.041.
dHeaton and Lucas estimate a first-order AR process in logs. The AR coefficient is 0.529 and the standard deviation of the error is 0.251
eMaCurdy estimates the earnings process in logs to be:
yth=yt−1h +ηth− 0.411ηt−1h − 0.106ηht−2 whereσ2= 0.054.
fThis specification implied negative interest rates; the comparison with other series is no longer informative.
200 400 600 800 1000 1200 1400 0.8
0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
Number of periods
Delta
σ/I=.1,RRA=3
0.02 0.02 0.02
0.04
0.04 0.04
0.08
0.08 0.08 0.08
0.16
0.16 0.16 0.16
0.32
200 400 600 800 1000 1200 1400
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
Number of periods
Delta
σ/I=.3,RRA=3 0.2
0.2 0.2
0.4
0.4 0.4 0.4
0.8
0.8 0.8 0.8
1.6
1.6 1.6 1.6 1.6
3.2
Figure 1: IID shocks: Welfare gain from completing markets as a percentage of consumption. Plot shows iso-welfare curves. When shocks are IID over time, the welfare losses from incomplete markets are minimal. In the second panel, the standard deviation of the shock is 30 percent of income, yet even with a low discount rate (.92) and a short time horizon (20 periods), the welfare loss is still less than one percent of
50 100 150 200 250 300 350
Figure 2: IID shocks: Welfare gain from completing markets as a percentage of consumption. Plot shows iso-welfare curves.
With higher risk aversion (two right-hand plots), the effects of incomplete markets are somewhat more significant. For
short-37
10 20 30 40 50 60
Figure 3: IID shocks: Standard deviation of consumption divided by standard deviation of income. Plot shows iso-variance curves. Value is variance N periods before the end of time. In general, as the number of period gets larger and δ gets higher, the variance gets smaller. In the lower right panel (the high variance, high risk-aversion case), as N gets larger, the variance
38
5 10 15 20 25 30
Figure 4: IID shocks: Difference in interest rates from complete markets. Plot shows iso-difference curves. The presence of
39
0
20
40 0 1000 2000 3000 4000 5000 6000
0 1000 2000 3000 4000 5000 6000
std. dev. of income Number of periods
std. dev. of consumption
Figure 5: IID shocks: Variance of consumption for differing variances of income for RRA=10, δ = .96. This plot shows near the end of time, the variance of consumption may become a decreasing function of the variance of income. Precautionary saving drives this seeming paradox. In the last period of the model, no self-insurance is possible; agents demand large amounts of precautionary savings pushing the interest rate so far down that consumption smoothing actually becomes easier in earlier
40
50 100 150 200 250 300 350
Figure 6: The probability of negative consumption. Lines are iso-probability contours. Even with low risk aversion and a high
41
0
0.2
0.4
0.6
0.8 1
0.8 0.85
0.9 0.95
1 0 2 4 6 8 10
ρ σ/I=.1,RRA=3
δ
% of consumption
0
0.2
0.4
0.6
0.8 1
0.8 0.85
0.9 0.95
1 0 20 40 60 80
ρ σ/I=.3,RRA=3
δ
% of consumption
Figure 7: Welfare gain from completing markets as a percentage of consumption, for shocks of differing persistence. ρ is the coefficient on a first-order AR-process δ is the discount factor. As persistence of shocks increases, the benefits of completion grow non-linearly. At ρ = .5, the benefits are minimal, but at ρ = .8 they have become
42
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8: Welfare gain from completing markets as a percentage of consumption, for shocks of differing persistence. Plots show iso-welfare curves. ρ is the coefficient on a first-order AR-process. δ is the discount factor.(wfig5.eps)
0 0.2 0.4 0.6 0.8 1
Figure 9: Welfare gain from completing markets as a percentage of consumption, for shocks of differing persistence. Plots show
44
0 500 1000 1500 2000 0
500 1000 1500 2000 2500
std. dev. of permanent shock
std. dev. of consumption
RRA=3 σ=.1
0 500 1000 1500 2000
0 500 1000 1500 2000
std. dev. of permanent shock
std. dev. of consumption
RRA=10 σ=.1
0 500 1000 1500 2000
0 500 1000 1500 2000 2500
std. dev. of permanent shock
std. dev. of consumption
RRA=3 σ=.3
0 500 1000 1500 2000
0 500 1000 1500 2000
std. dev. of permanent shock
std. dev. of consumption
RRA=10 σ=.3
Figure 10: Variance of consumption as a function of variance of permanent shock. Different lines represent different rates of time-preference (fromδ = .8 at the top-left
45