Nonlinearities and Inactivity in Aggregate Investment: Some Theoretical Analysis and Time-Series Evidence

The theoretical analysis of investment under uncertainty has been revolutionized over the last decade by the importation of ideas from finance. If investment is irreversible, there is a return to waiting. So although circumstances may suggest that it is profitable to invest, there may also be an incentive to postpone the decision until better opportunities arise. Identifying and valuing the option to invest has become the standard way to solve the firm's irreversible-investment problem. Empirical studies of investment that incorporate the insights of the real-options approach are now beginning to appear. These show that investment can have a nonlinear relationship to q and may show insensitivity for some threshold level to the shadow value of investment (Barnett and Sakellaris 1998). Abel and Eberly (1997) and Böhm and Funke (1999) have also shown how the real-options approach to investment can be combined with the traditional q approach. In this case the relationship between q and the rate of investment is discontinuous. Over a range of inaction there will be no investment, although q is in excess of one.This paper builds a theoretical model that explains the determinants of this investment discontinuity. In contrast to much of the literature, we use a mean-reverting stochastic process, of which the geometric Brownian motion process is a special case. Under the assumption of a production function with constant returns to scale and a specific functional form for the investment adjustment function, it is possible to derive a tractable analytical form for the shadow value of the investment project. We then analyze the comparative properties of the value of q under different assumptions about the stochastic process governing output. The advantage of using a mean-reverting process is that it better captures the undoubted persistence in the shocks that face firms, especially at the macroeconomic level.We then consider what the implications would be for the aggregate relationship between investment, q, and the business cycle. We first carry out Monte Carlo simulations of a discrete version of the theoretical model. We find that for many parameter values, aggregating suppresses any nonlinearities in the micro adjustment processes. Moreover, where we do detect nonlinearity at the aggregate level, it varies with the type of stochastic process. It is greatest when this is a random walk-corresponding to the Brownian motion in continuous time-and least when the stochastic process follows an i.i.d. process. Mean reversion lies in between. We turn finally to an empirical examination using aggregate data and explore how sensitive investment is to q in different regimes. To do this, we apply a generalization of the Granger-Lee method (Arden et al. 2000) that uses a linear spline function to approximate different regions for investment.