Long-run investment under uncertain demand
Introduction
In this paper, we compare the long-run average rate of capital accumulation determined under the assumption of a monopolistic firm facing an isoelastic demand function with multiplicative shocks with the rate calculated under the assumption of a linear demand function with additive shocks.
In the literature on irreversible investment under uncertainty, the majority of scholars use an isoelastic profit function1 where shocks enter multiplicatively, i.e. a log linear profit function.2 The use of a log linear profit function has the advantage of leading to closed form solutions for both the short-run optimal investment rule and the long-run average rate of capital accumulation (see e.g. Hartman and Hendrickson, 2002).3
To the best of our knowledge, only a few authors4 use a linear demand function with shocks entering additively. In this case, in contrast with the log linear one, the firm may face scenarios where, due to capacity in excess, the profit flow is negative. The firm may, of course, adjust operations by choosing not to utilize some capacity. Several situations may arise: the firm may, for instance, temporarily suspend production and, if profitable, restart it later, or flexibly adjust the capacity so that losses are avoided. In both cases, however, once the corresponding operative options have been accounted for, a closed form solution for the short-run optimal investment policy is not available. In addition, when studying the long-run average rate of capital accumulation, an analytical solution does not exist.
In this paper, our contribution to the analysis of the linear case is threefold. First, we derive and present an analytical solution of the short-run investment rule for the case where the firm can i) temporarily and costlessly suspend operation whenever the profit flow is negative and ii) costlessly restart later if positive. Second, we show how to use such solution in order to derive an approximation of the corresponding i) steady-state distribution of the optimal stock of capital and ii) the long-run average rate of capital accumulation. Third, we compare the long-run rates of capital accumulation determined under both assumptions, i.e. isoelastic vs. linear demand. We do it within a plausible range of parameter values used in the literature, and notice that there may be significant differences.
Our analysis allows us to conclude that under a linear demand function, excess capacity matters, whatever the policy of the firm, i.e. suspension of operations or production reduction. Therefore, the choice of a log linear profit function has a non-trivial impact on the magnitude of the long run rate of capital accumulation.
The structure of this paper is as follows. In Section 2, we introduce a general model of irreversible investment under uncertainty. Section 3 presents the long-run average rate of capital accumulation for the case of an isoelastic demand function with multiplicative shocks and for the case of a linear demand with additive shocks. In Section 4 we compare the rates obtained under both assumptions. Section 5 concludes.
Section snippets
The basic model
Let's start by modeling the general investment problem. Consider a risk-neutral monopolistic firm that costlessly produces a flow of non-storable goods (or services). Production, Qt, is based on a linear technology using only capital Kt as input factor, i.e. Qt = Kt.5 Assume that the firm faces a demand function of the following form6
Long-run rate of capital accumulation
In this section, we determine the optimal investment policy, Y∗(Kt),13 set by a firm facing two different types of demand shocks, namely isoelastic demand with multiplicative shock and linear demand with additive shocks. We will then study the effects of increased uncertainty on the optimal investment policy and the long-run average growth rate of capital accumulation.
Multiplicative vs. additive shocks: a numerical comparison
In order to better emphasize the difference between ω and λ, let's consider the case where the linearization point is such that the two demand functions have the same elasticity.20 That is:
Denote by Ω the ratio between the long-run average rates of capital accumulation under both demand specifications.
Conclusion
In this paper we study the long-run average growth rate of capital for the case of a linear demand function with a random shock entering additively. This case has not received much attention in the literature of irreversible investment under uncertainty where the use of isoelastic demand functions, i.e. with random shock entering multiplicatively, has generally been privileged. Using an isoelastic demand (and the corresponding log linear profit function) provides a clear advantage when it comes
References (28)
- et al.
An exact solution for the investment and value of a firm facing uncertainty, adjustment costs, and irreversibility
J. Econ. Dyn. Control.
(1997) Irreversible investment under uncertainty in oligopoly
J. Econ. Dyn. Control.
(1998)Irreversible investment
Res. Econ.
(1998)Investment and capacity choice under uncertain demand
Eur. J. Oper. Res.
(1999)A simplified treatment of the theory of optimal regulation of Brownian motion
J. Econ. Dyn. Control.
(1991)Irreversible investment with uncertainty and scale economies
J. Econ. Dyn. Control.
(1995)- et al.
Optimal partially reversible investment with entry decision and general production function
Stoch. Process. Appl.
(2005) - et al.
Optimal partially reversible investment
J. Econ. Dyn. Control.
(2002) - et al.
Profit sharing and investment by regulated utilities: a welfare analysis
Rev. Financ. Econ.
(2008) Real options, product market competition and asset returns
J. Financ.
(2009)
A class utility functions yielding linear demand functions
Am. Econ.
Open-loop equilibria and perfect competition in option exercise games
Rev. Financ. Stud.
Firing costs and labour demand: how bad is eurosclerosis?
Rev. Econ. Stud.
Irreversibility and aggregate investment
Rev. Econ. Stud.
Cited by (5)
Price limits and corporate investment: The consumers' perspective
2015, Economic ModellingThe effects of uncertain forest conservation benefits on long-run deforestation in the Brazilian Amazon
2018, Environment and Development EconomicsAppraisements of material handling system in context of fiscal and environment extent: A comparative grey statistical analysis
2017, International Journal of Logistics ManagementTax Competition, Investment Irreversibility and the Provision of Public Goods
2015, German Economic Review