Stochastic Capital Theory I. Comparative Statics

56 Pages Posted: 24 Jul 2007 Last revised: 27 Jul 2010

See all articles by William A. Brock

William A. Brock

University of Wisconsin, Madison - Department of Economics; University of Missouri at Columbia - Department of Economics

Michael Rothschild

Princeton University; National Bureau of Economic Research (NBER)

Joseph E. Stiglitz

Columbia Business School - Finance and Economics; National Bureau of Economic Research (NBER)

Date Written: May 1982

Abstract

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.

Suggested Citation

Brock, William A. and Rothschild, Michael and Stiglitz, Joseph E., Stochastic Capital Theory I. Comparative Statics (May 1982). NBER Working Paper No. t0023. Available at SSRN: https://ssrn.com/abstract=579731

William A. Brock (Contact Author)

University of Wisconsin, Madison - Department of Economics ( email )

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University of Missouri at Columbia - Department of Economics ( email )

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Michael Rothschild

Princeton University ( email )

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National Bureau of Economic Research (NBER) ( email )

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Joseph E. Stiglitz

Columbia Business School - Finance and Economics ( email )

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(212) 662-8474 (Fax)

HOME PAGE: http://www.josephstiglitz.com

National Bureau of Economic Research (NBER) ( email )

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