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The Fragile Beneﬁts of Endowment Destruction

John Y. Campbell

Harvard University and National Bureau of Economic Research

John H. Cochrane

University of Chicago, Hoover Institution, and National Bureau of Economic Research

The beneﬁts of endowment destruction documented by Ljungqvist and

Uhlig ð2015Þ, and the related possibility that consumption can lower hab-

its, are fragile. Both issues result from a particular way of discretely ap-

proximating the underlying continuous-time model or of adapting it to

jumps. Other ways of calculating the discrete-time approximation or ex-

tending the model to jumps easily overturn the results, while making no

difference to the model’s description of asset prices and quantities. This

analysis gives an example of how to extend models so that the jump gives

the same result as a jump limit of continuous-sample-path movements.

I. Beneﬁcial Endowment Destruction?

If a consumer has preferences with habits, lowering consumption today

can lower future habits and potentially raise utility overall. Is habit per-

sistence this strong in the Campbell and Cochrane ð1999, 2000Þ model?

Suppose that a Campbell-Cochrane consumer at time t 5 0 is at the

steady-state log surplus consumption ratio s

5 s. Suppose that log en-

dowment grows steadily at the rate g for periods 0, 1, 2, that is, y

52g,

5 0, y

5 g. Suppose that the government destroys some of the time

We thank Darrell Dufﬁe, Lars Ljungqvist, Stavros Panageas, Monika Piazzesi, and Philip

Protter for helpful comments and discussion.

Electronically published September 9, 2015

[ Journal of Political Economy, 2015, vol. 123, no. 5]

1214

This content downloaded from 165.193.178.102 on July 25, 2016 22:05:49 PM

All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

from log consumption each day for 30 days. We then restore consump-

tion the same way, producing a V-shaped daily consumption pattern that

bottoms out at the same w value on day 30. We simulate the model for-

ward as before.

In this daily simulation, the Ljungqvist and Uhlig pattern disappears.

Output destruction is always harmful, and transfers are always welcome.

Still smaller time intervals lead to visually indistinguishable results.

To produce ﬁgure 1, we simulate the monthly or daily version of ð1Þ,

using j

5 0:015D, where D 5 1/12 or D 5 1/360 is the simulation in-

terval. We then recursively calculate the surplus consumption ratio,

t1D

5 ð1 2 f

Þs 1 f

1 lðs

Þðc

t1D

2 c

2 g DÞ; ð2Þ

where

lðsÞ ;



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 2 2ðs 2 s Þ

2 1 s ≤ s

max

0 s ≥ s

max

ð3Þ

5 log ðS

Þ 5 log



2 X



;

5 logðC

Þ is log consumption, X

is habit, and s 5 log



S 5 logð0:057Þ,

f 5 0:87, g 5 0.0189, and s

max

5 s 1 1=2ð1 2



Þ are parameters. We

then evaluate the utility function

U 5

1 2 g

t50;D;2D;:::

ðC

2 X

12g

1 2 g

ð12gÞðc

;

with d 5 0:89, g 5 2:00 by averaging over simulations. This is the

Campbell-Cochrane ð1999Þ model and parameters.

Why are the results of the daily simulation so different from those of

the monthly simulation? Examine ð2Þ closely. During the daily simula-

tion D 5 1/360, the surplus consumption ratio responds to each little bit

of consumption each day, and a new lðs

Þ is recomputed each day. Dur-

ing the monthly simulation the same lðs

Þ applies to all the daily changes

from t 5 0tot 5 1/12, and the same lðs

1=12

Þ applies to all the daily changes

from t 5 1/12 to t 5 2/12. A daily simulation that uses the beginning-of-

the-month value of lðs

Þ rather than the continuously evolving one would

generate Ljungqvist and Uhlig’s result.

1216 journal of political economy

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All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).

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