Optimal investment policy with fixed adjustment costs and complete irreversibility
Introduction
Investment is lumpy at the plant level with long periods of inactivity punctuated by infrequent and large adjustments. And, smaller plants have lumpier investment patterns: both the probability of a spike and the share of total investment represented by spikes are decreasing in establishment size.1
We consider a model of lumpy investment that has received a lot of attention in both applied and theoretical works.2 Existing theoretical models cannot account for size-dependent lumpiness. Fixed investment costs are proportional to the existing capital stock in Caballero and Leahy (1996), proportional to current profitability in Abel and Eberly (1998) and proportional to current output in Caballero and Engel (1999). Such fixed costs do not become irrelevant as the firm grows but imply that the firm size does not matter for investment dynamics.
We extend this theoretical literature by considering a fixed adjustment cost not proportional to the existing capital.3 We provide a characterization of the optimal policy and the value function under general assumptions on uncertainty and technology. The main difficulty created by the fixed cost is that the value function is no longer concave and hence standard arguments (see Stokey et al., 1989) need to be extended. Our contribution is threefold. First, we show that the concept of -concavity introduced by Scarf (1960) can be applied to an investment model to derive the optimal policy which is of the generalized form.4 Other existing proofs of the optimality of policies in investment models use a continuous-time setting.5 Yet, discrete time allows us to characterize the optimal policy in a rigorous and parsimonious way whereas continuous-time models can be impeded by measure-theoretic issues. In addition, discrete-time models are familiar to a broader audience since they are the basis of most modern macroeconomics and empirical work. Second, we show that the closeness of the space of -concave functions follows almost directly from the property that -concavity is preserved after maximization of a -concave function minus a fixed cost . Last, a substantive contribution is to show that under plausibly calibrated values the model delivers more lumpiness for smaller plants, a robust feature of plant-level data, while in existing theoretical models lumpiness is not systematically related to the plant size.
An important restriction we impose throughout the paper is complete irreversibility. Relaxing this assumption creates technical difficulties: the optimal policy is no longer guaranteed to be of the form and the value function is not -concave.
The rest of the paper is organized as follows. Section 2 presents the model. Section 3 derives the optimal decision rule and its properties. Section 4 presents the implications of the model for the lumpiness of investment by firm size. Section 5 concludes.
Section snippets
Assumptions
Time is discrete and indexed by . At each period, the plant’s manager decides to invest or not over an infinite time horizon. She is risk-neutral and discounts future profits at a constant rate . Her decision depends on the level of capital inherited from the previous period and the plant’s profitability . The level of capital at the start of the next period, , is: where is a positive depreciation rate. The one-period profit function is:
Optimal decision rule
The discontinuity of the one-period profit function (due to the fixed adjustment cost) implies that the value function is not concave. Nevertheless the concept of -concavity introduced by Scarf (1960) can be used to characterize the optimal decision rule.
Definition A real-valued function is called -concave for , if for any and any .
-concave functions have many useful properties.7 In particular, it is weaker than concavity.
Lumpy investment and firm size
Smaller plants have lumpier investment patterns. To illustrate that it is a feature of our model, we simulate the model using functional forms and parameter values from Cooper and Haltiwanger (2006). Cash flows are specified as . Profitability is a finite-state Markov chain that is the sum of two processes and , and a permanent component : with and . and . as in Midrigan and Xu (2014)
Conclusion
This paper characterizes the optimal policy in a model of investment with a fixed cost independent of existing capital. The true fixed-cost model implies that smaller firms have a lumpier investment pattern.
There are several possible extensions to this paper. For instance, -concavity does not allow us to characterize the optimal policy when firms are allowed to sell capital. Also, modeling capital heterogeneity is a relatively unexplored line of research (Bachmann and Ma, 2013 is an exception
Acknowledgments
The author would like to thank an anonymous referee, Jean-Marc Robin, Cuong Le Van, Francois Gourio, Simon Gilchrist, Russell Cooper, Daniel Hamermesh, Ben Skrainka, Jake Zhao and William Brock for helpful comments.
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