Adjustment costs and threshold effects in factor demand relationships

ABSTRACT

It has been recently argued that producers may not respond to every input price change in the way that a linear factor demand model would predict. This lumpy response is due to adjustment costs that are inherent in the act of adjusting the mix of inputs applied in the underlying production technologies. This study aims to provide a solid conceptual framework for these nonlinearities in factor demand relationships. Industry-specific implications of convex and non-convex adjustment costs for the linearity of the factor demand relationships as well as price and substitution elasticities are explored. A two-regime threshold system of factor demand equations is estimated for several manufacturing industries in the United States. Empirical results suggest significant threshold effects in the factor demand relationships in most nondurable goods sectors. The size and the nature of thresholds depend upon industry characteristics, including input composition and (non)convexity of underlying adjustment costs. Complete matrices of price and substitution elasticities for each industry are derived using estimates of threshold factor demand systems. Discussion of two contrasting cases in greater detail sheds light on how the effect of price shocks on factor demand relationships varies across industries with different adjustment cost structures.

KEYWORDS:

• Adjustment costs
• factor demand systems
• threshold nonlinearities
• price and substitution elasticities

JEL CLASSIFICATION:

• D24
• E23

Acknowledgement

This article is derived in part from a note published in Applied Economics Letters on February 12, 2015, available online: http://wwww.tandfonline.com/10.1080/13504851.2014.993126.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Hamermesh (1989) and Hamermesh and Pfann (1996) discuss the role of convex adjustment costs in partial adjustment models of employment demand. Chirinko (1993) and Caballero (1999) survey their use in empirical investment models. Hall (2004) estimates an industry-level model of production with quadratic adjustment costs assumed for both capital and labour. Abel (1990) provides a synthesis of the neoclassical investment model with convex adjustment costs.

² Caballero and Engel (1999) examine generalized (S, s) investment policies rationalized by stochastic fixed adjustment costs. Khan and Thomas (2003) consider non-convex adjustment costs in business cycle models. King and Thomas (2006) show that non-convex adjustment costs stagger the adjustments undertaken by firms in response to shocks. Abel and Eberly (1994, 1996), Caballero, Engel and Haltiwanger (1995) and Cooper, Haltiwanger and Power (1999) argue that non-convex adjustment costs play a central role in the investment process.

³ The cost function is not estimated directly in this article in order to conserve degrees of freedom.

⁴ Diewert and Wales (1987) note that $\mathrm{\Gamma }$ itself need not be negative semidefinite to satisfy concavity since $-[(S)-s{s}^{\prime }]$ will be negative semidefinite as long as the shares are positive.

⁵ Unlike the AES, the MES is not a symmetric measure of input substitution. Inputs that are classified as AES complements may be classified as MES substitutes, although inputs classified as AES substitutes will continue to be classified as MES substitutes. A discussion on the conceptual desirability of the MES as compared to the AES and other properties of MES can be found in Chambers (1988).

⁶ It should be noted that although there are theoretical models that explicitly calculate such adjustment costs based on an assumed functional form, it is not within the scope of this article to do so. Instead, the evidence of threshold sensitivity in parameter estimates is conceptually attributed to the adjustment costs.

⁷ Provided that ITSUR is chosen as the estimation method, estimates are invariant to which equation is deleted (Berndt and Savin 1975).

⁸ Details on using Monte Carlo integration for imposing inequality restrictions can be found in Geweke (1986) and Chalfant, Gray and White (1991).

⁹ In general, economic theory should provide guidance for choosing among alternative threshold variables. There might be other empirically plausible threshold variables to represent input price changes. Instead of the maximum of the absolute price changes, one could, for example, use an average price change measure to define the threshold variable. I estimate the same empirical models using a threshold variable that is defined as the absolute value of changes in the Theil–Thörnqvist price index. The summary results are reported in Supplemental online material.

¹⁰ Minimizing $log\left|\mathrm{\Sigma }(c)\right|$ is equivalent to maximizing the log-likelihood function.

¹¹ The application was limited to 10 nondurables sectors due to space limitations.

¹² Another set of KLEMS data is developed by Jorgenson (1995, Chapter 1). Since the BLS data cover more years and the government data are often more thoroughly documented, the BLS data set is used in this analysis.

¹³ The scale of prices makes no difference in a translog model; in particular, the results are transparent whether they are in current or constant dollar form.

¹⁴ Directly collected data on purchased business services are relatively unreliable, and for that reason, they have been ignored in similar studies in the past (see e.g. Feng and Serletis 2008). However, there is evidence of an increased use of purchased business services by industries over the post-war period. There are two issues that require attention. First, an input with a growing cost share should not be completely ignored in productivity measurement in order to avoid underestimating aggregate inputs. Second, one must recognize the possibility of substitution between capital, labour and services purchased from outside. Examples are the substitution of leased equipment for owned capital and substitution of purchased accounting services for services performed by payroll employees.

¹⁵ The BLS has recently updated its productivity database with new data containing KLEMS from 1987 to 2006. However, the new data are based on a new industrial classification system, the North American Industrial Classification System (NAICS). BLS has not converted the SIC data between 1949 and 1986 to be consistent with the NAICS. Such conversion may not accurately reflect the manufacturing industries in terms of an NAICS basis and may not capture fundamental changes in the manufacturing industries as the data collected under the SIC and NAICS systems are based on somewhat different concepts.

¹⁶ Yang (1997) uses an alternative measure of capital costs to solve the issue of constant returns to scale: Capital expenditures were calculated by adding the corporate and non-corporate capital consumption allowances, and net interest expenditures reported in the unpublished data contained in The Department of Commerce’s Gross Product Originating (GPO) database. This measure of capital expenditures was found to be negative for some industries and therefore has not been used in this study.

¹⁷ The elasticity estimates for the remaining nondurable industries are reported in Supplemental online material.

¹⁸ Caner and Hansen (2001) argue that, in general, the threshold variable should be predetermined and ergodic with a continuous distribution.

¹⁹ The choice of these two industries is based upon their distinctive nature. Results for all other industries are presented in the Supplemental online material.