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Comment on “Growth cycles and market crashes”
ARTICLE IN PRESS
Journal of Economic Theory 111 (2003) 147–148
Comment on ‘‘Growth cycles and market crashes’’ Adrian Peralta-Alva Department of Economics, University of Minnesota, 1035 Heller Hall 271, 19th Avenue South, Minneapolis, MN 55414, USA Received 20 March 2002; final version received 24 April 2002
Boldrin and Levine [1] develop a model in which the unexpected obsolescence of productive assets, in a world where new technologies are costly to implement, leads to a drop in the market value of existing capital. Their quantitative analysis shows that in a realistic range of parameter values, it is possible to generate falls in the stock market on the order of 10–20%. These numbers are, however, not correct. Once a mistake in the computations of the paper is rectified, the model is capable of generating falls in the stock market on the order of 25–50% for reasonable parameterizations. The basic mistake is in the third equation of p. 32. This reads: d VM ¼ y1 ð1Þ ð1yÞ=ð1þyÞ y yþ1 ½1 ðd g Þ which is meant to be the equilibrium value of the capital stock before the obsolescence shock arrives. That it is erroneous can be seen by substituting the yþ1
equilibrium value of gr ¼ g d into the value function VM ; which is correctly reported in the paper as the second equation on p. 32: d : VM ¼ y1 y 1=ð1þyÞ yþ1 Þ ð1 ðdðgrÞ Þ This substitution yields the right expression for VM as a function of the consumption growth rate, g; and the discount factor d: d VM ¼ : ð2Þ y½1 dgy yþ1 The numerical experiments summarized on the tables of p. 33 are also wrong, as they were performed using Eq. (1). We replicated those simulations using the correct E-mail address: [email protected]. 0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-0531(03)00222-9
ARTICLE IN PRESS 148
A. Peralta-Alva / Journal of Economic Theory 111 (2003) 147–148
expression for VM ; Eq. (2), and found that the variations in the market value of the stock of capital caused by the bad shock, at different levels of risk aversion, are much larger than those reported in the paper. For the first experiment, suppose that gb ¼ 1 so that there is no growth following a negative shock. The discount factor is taken to be d ¼ 0:95; so that the interest rate in the bad state is 5%. The equilibrium growth rate of consumption is also 5%. Hence, the first table on p. 33 should be
y 1 V0 =VM Interest at M Interest at 0
0:1 8.40% 9.10% 5.00%
0:2 15.21% 8.60% 5.00%
0:4 24.53% 7.70% 5.00%
0:8 24.59% 5.90% 5.00%
The published version of the table implies a market drop of roughly 10% in response to bad news, versus the 24% reported here. The second table, on the same page, is based on a more drastic fall in the rate of consumption from 10% to 0%. Its correct version is
y 1 V0 =VM Interest at M Interest at 0
0:1 16.53% 12.80% 5.00%
0:2 30.52% 12.00% 5.00%
0:4 55.27% 10.30% 5.00%
0:8 — 6.80% 5.00%
Here the market drops are of the order of 30–40%. Furthermore, for values of y smaller than 0:55; the percentage fall is larger than 90%. The published version of the table reported drops on the order of 20%. Finally, in the third table the case when growth goes from 5% to 0% is reanalyzed, with d ¼ 0:98: The rectified results are
y 1 V0 =VM Interest at M Interest at 0
0:1 21.85% 6.20% 2.00%
0:2 40.77% 5.80% 2.00%
0:4 86.77% 4.80% 2.00%
0:8 — 4.40% 2.00%
Here values of y smaller than 0:41 yield drops in market value larger than 95%. The published version of the table reported drops on the order of 30%.
References [1] M.C. Boldrin, D.K. Levine, Growth cycles and market crashes, J. Econom. Theory 96 (2001) 13–39.
Journal of Economic Theory 111 (2003) 147–148
Comment on ‘‘Growth cycles and market crashes’’ Adrian Peralta-Alva Department of Economics, University of Minnesota, 1035 Heller Hall 271, 19th Avenue South, Minneapolis, MN 55414, USA Received 20 March 2002; final version received 24 April 2002
Boldrin and Levine [1] develop a model in which the unexpected obsolescence of productive assets, in a world where new technologies are costly to implement, leads to a drop in the market value of existing capital. Their quantitative analysis shows that in a realistic range of parameter values, it is possible to generate falls in the stock market on the order of 10–20%. These numbers are, however, not correct. Once a mistake in the computations of the paper is rectified, the model is capable of generating falls in the stock market on the order of 25–50% for reasonable parameterizations. The basic mistake is in the third equation of p. 32. This reads: d VM ¼ y1 ð1Þ ð1yÞ=ð1þyÞ y yþ1 ½1 ðd g Þ which is meant to be the equilibrium value of the capital stock before the obsolescence shock arrives. That it is erroneous can be seen by substituting the yþ1
equilibrium value of gr ¼ g d into the value function VM ; which is correctly reported in the paper as the second equation on p. 32: d : VM ¼ y1 y 1=ð1þyÞ yþ1 Þ ð1 ðdðgrÞ Þ This substitution yields the right expression for VM as a function of the consumption growth rate, g; and the discount factor d: d VM ¼ : ð2Þ y½1 dgy yþ1 The numerical experiments summarized on the tables of p. 33 are also wrong, as they were performed using Eq. (1). We replicated those simulations using the correct E-mail address: [email protected]. 0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-0531(03)00222-9
ARTICLE IN PRESS 148
A. Peralta-Alva / Journal of Economic Theory 111 (2003) 147–148
expression for VM ; Eq. (2), and found that the variations in the market value of the stock of capital caused by the bad shock, at different levels of risk aversion, are much larger than those reported in the paper. For the first experiment, suppose that gb ¼ 1 so that there is no growth following a negative shock. The discount factor is taken to be d ¼ 0:95; so that the interest rate in the bad state is 5%. The equilibrium growth rate of consumption is also 5%. Hence, the first table on p. 33 should be
y 1 V0 =VM Interest at M Interest at 0
0:1 8.40% 9.10% 5.00%
0:2 15.21% 8.60% 5.00%
0:4 24.53% 7.70% 5.00%
0:8 24.59% 5.90% 5.00%
The published version of the table implies a market drop of roughly 10% in response to bad news, versus the 24% reported here. The second table, on the same page, is based on a more drastic fall in the rate of consumption from 10% to 0%. Its correct version is
y 1 V0 =VM Interest at M Interest at 0
0:1 16.53% 12.80% 5.00%
0:2 30.52% 12.00% 5.00%
0:4 55.27% 10.30% 5.00%
0:8 — 6.80% 5.00%
Here the market drops are of the order of 30–40%. Furthermore, for values of y smaller than 0:55; the percentage fall is larger than 90%. The published version of the table reported drops on the order of 20%. Finally, in the third table the case when growth goes from 5% to 0% is reanalyzed, with d ¼ 0:98: The rectified results are
y 1 V0 =VM Interest at M Interest at 0
0:1 21.85% 6.20% 2.00%
0:2 40.77% 5.80% 2.00%
0:4 86.77% 4.80% 2.00%
0:8 — 4.40% 2.00%
Here values of y smaller than 0:41 yield drops in market value larger than 95%. The published version of the table reported drops on the order of 30%.
References [1] M.C. Boldrin, D.K. Levine, Growth cycles and market crashes, J. Econom. Theory 96 (2001) 13–39.