TY - JOUR
AU - Brock,William A.
AU - Rothschild,Michael
AU - Stiglitz,Joseph E.
TI - Stochastic Capital Theory I. Comparative Statics
JF - National Bureau of Economic Research Technical Working Paper Series
VL - No. 23
PY - 1982
Y2 - May 1982
DO - 10.3386/t0023
UR - http://www.nber.org/papers/t0023
L1 - http://www.nber.org/papers/t0023.pdf
N1 - Author contact info:
William Brock
Department of Economics
University of Wisconsin
1180 Observatory Drive
Madison, WI 537061393
E-Mail: wbrock@ssc.wisc.edu
Michael Rothschild
531 14th Street
Santa Monica, CA 90402
Tel: 310-394-6010
Fax: 310-593-4401
E-Mail: mrothsch@princeton.edu
Joseph E. Stiglitz
Uris Hall, Columbia University
3022 Broadway, Room 212
New York, NY 10027
Tel: 212/854-0671
Fax: 212/662-8474
E-Mail: jes322@columbia.edu
AB - Introductory lectures on capital theory often begin by analyzing the following problem: I have a tree which will be worth X(t) if cut down at time t. If the discount rate is r, when should the tree be cut down? What is the present value of such a tree? The answers to these questions are straightforward. Since at time t a tree which I plan to cut down at time T is worth e[to the power of rt]e[to the power of ?rT]X(T), I should choose the cutting date T* to maximize e[to the power of -rT]X(T); at t < T* a tree is worth e[to the power of rt]e[to the power of -rT*]X(T*). In this paper we analyze how the answers to these questions of timing and evaluation change when the tree's growth is stochastic rather than deterministic. Suppose a tree will be worth X(t,w) if cut down at time t when X(t,w) is a stochastic process. When should it be cut down? What is its present value? We study these questions for trees which grow according to both discrete and continuous stochastic processes. The approach to continuous time stochastic processes contrasts with much of the finance literature in two respects. First, we obtain sharp aomparative statics results without restricting ourselves to particu,ar stochastic specifications. Second, while the option pricing literature seems to imply that increases in variance always increase value, we show that an increase in the variance of a Tree's growth has ambiguous effects on its value.
ER -