Elsevier

Economic Modelling

Volume 41, August 2014, Pages 80-89
Economic Modelling

Long-run investment under uncertain demand

https://doi.org/10.1016/j.econmod.2014.04.023Get rights and content

Highlights

  • We consider a firm facing a linear demand function with additive shocks.

  • We define an analytical approximation of the long-run average rate of capital accumulation.

  • We compare the long-run rates of capital accumulation.

  • We notice significant differences with respect to log linear profit function.

Abstract

In the literature investigating the impact of uncertainty on short-run and long-run investment, most authors have used a log linear profit function. This functional form has been generally considered a reasonable approximation for a more general one and has the advantage of providing closed form solutions for both short-run investment rule and long-run rate of capital accumulation. In this paper, we consider the profit function for the case of a monopolistic firm facing a linear demand function with additive shocks. Under this assumption, analytical solutions, for both short-run investment rule and long-run rate of capital accumulation, are not available. We then 1) propose an analytical approximation of the short-run investment rule and 2) show how such approximation can be used in order to derive the corresponding i) steady-state distribution of the optimal stock of capital and ii) the long-run average rate of capital accumulation. Finally, we compare the long-run rates of capital accumulation calculated under both profit function specifications. We find that, within a plausible range of parameter values, the two rates are significantly different. Hence, we conclude that the choice of a log linear functional form has a non-trivial impact on the magnitude of the long run rate of capital accumulation.

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 K˜Y˜ is such that the two demand functions have the same elasticity.20 That is:Y˜/K˜η=1+η/γ.

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)

  • G. Alperovich et al.

    A class utility functions yielding linear demand functions

    Am. Econ.

    (1996)
  • K. Back et al.

    Open-loop equilibria and perfect competition in option exercise games

    Rev. Financ. Stud.

    (2009)
  • S. Bentolila et al.

    Firing costs and labour demand: how bad is eurosclerosis?

    Rev. Econ. Stud.

    (1990)
  • G. Bertola et al.

    Irreversibility and aggregate investment

    Rev. Econ. Stud.

    (1994)
  • Cited by (5)

    View full text