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Lubik, Thomas A.. "How Large Are Returns to Scale in the U.S.? A View across the Boundary." Economic Quarterly. Federal Reserve Bank of Richmond. 2016. HighBeam Research. 19 Oct. 2018 <https://www.highbeam.com>.
Lubik, Thomas A.. "How Large Are Returns to Scale in the U.S.? A View across the Boundary." Economic Quarterly. 2016. HighBeam Research. (October 19, 2018). https://www.highbeam.com/doc/1G1-497908856.html
Lubik, Thomas A.. "How Large Are Returns to Scale in the U.S.? A View across the Boundary." Economic Quarterly. Federal Reserve Bank of Richmond. 2016. Retrieved October 19, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-497908856.html
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In this article, I investigate the size of the returns to scale in aggregate U.S. production. I do so by estimating the aggregate returns to scale within a theory-consistent general equilibrium framework using Bayesian methods. This approach distinguishes this article from much of the empirical literature in this area, which is largely based on production-function regressions and limited-information methods. The production structure within a general equilibrium setting, on the other hand, is subject to cross-equation restrictions that can aid and sharpen inference. My investigation proceeds against the background that increasing returns are at the core of business cycle theories that rely on equilibrium indeterminacy and sunspot shocks as the sources of economic fluctuations (e.g., Benhabib and Farmer 1994; Guo and Lansing 1998; Weder 2000).
Specifically, the theoretical literature has shown that multiple equilibria can arise when the degree of returns to scale is large enough. At the same time, the consensus of a large empirical literature is that aggregate production exhibits constant returns. However, equilibrium indeterminacy is a characteristic of a system of equations and can therefore not be assessed adequately with production function regressions. Instead, empirical researchers should apply full-information, likelihood-based methods to conduct inference along these lines. as not allowing for indeterminacy leaves the empirical model misspecified. I therefore estimate the returns to scale in a theory-consistent manner using econometric methods that allow for indeterminate equilibria. I apply the methodology developed by Lubik and Schorfheide (2004) to bridge the boundary between determinacy and indeterminacy and estimate a theoretical model over the entire parameter space, including those parameter combinations that imply indeterminacy. This view across the boundary allows me to detect the possibility that data were generated under indeterminacy and provides the correct framework for estimating the returns to scale.
I proceed in three steps. First, I estimate a standard stochastic growth model with increasing returns to scale in production. In this benchmark specification, I estimate the model only on that region of the parameter space that implies a unique, determinate equilibrium to get an assessment of what a standard approach without taking into account indeterminacy would result in. The estimated model is based on the seminal paper of Benhabib and Farmer (1994). The mechanism that leads to increasing returns is externalities in the production process: individual firms have production functions with constant returns, but these are subject to movements in an endogenous productivity component that depends on the production decisions by all other firms in the economy. The key assumption is that individual firms take this productivity component as given and thereby do not take into account that increases in individual factor inputs also raise this productivity component. In the aggregate, the feedback effect from this mechanism can lead to increasing returns in the economy-wide production function. Benhabib and Farmer (1994) show analytically that if the strength of this feedback effect, tied to an externality parameter, is large enough, the resulting equilibria can be indeterminate in the sense that there are multiple adjustment paths to the steady state.
In this benchmark model with externalities, I find estimates that are tightly concentrated around the case of constant returns. Moreover, I also find that aggregate labor supply is fairly inelastic. This finding presents a problem for the existence of indeterminate equilibria due to increasing returns. It can be shown algebraically that the threshold required for an indeterminate equilibrium to arise depends on how elastic the labor supply is. Even with only mildly increasing returns, crossing the boundary into indeterminacy requires a perfectly elastic labor supply, both of which factors I can rule out from my estimation. Based on this baseline model with externalities, it would therefore seem unlikely that equilibrium indeterminacy would arise since the parameter estimates are far away from their threshold values.
In the second step, I therefore estimate a modified version of the benchmark model that allows for variable capacity utilization based on the influential paper by Wen (1998). He shows that the indeterminacy threshold is considerably closer to the constant-returns case when production is subject to variable capacity utilization, that is, when firms can vary the intensity with which the capital stock is used. Given typical parameter values from the literature, the required degree of increasing returns for an indeterminate equilibrium is within the range of plausible empirical estimates. When I estimate the model with variable capacity utilization, I find mildly increasing returns, but the statistical confidence region includes the constant-returns case. As in the benchmark model, I find an inelastic labor supply. In Wen's model, the threshold value of the returns-to-scale parameter is a function of the labor supply elasticity. The threshold attains a minimum for a perfectly elastic labor supply but rises sharply when labor becomes less elastic. Even with mildly increasing returns, these results indicate that indeterminacy will likely not arise in the framework with variable capacity utilization on account of the labor supply parameter.
A caveat to this conclusion is that the results are obtained by restricting the estimation to the determinate region of the parameter space. If the data are generated under parameters that imply indeterminacy, the thus-estimated model would be misspecified and the estimates biased. This potential misspecification would manifest itself as a piling up of parameter estimates near or at the boundary between determinacy and indeterminacy (Canova 2009; Morris 2016) or it might not be detected at all if there is a local mode of the likelihood function in the determinacy region.
In a third step, I therefore apply the methodology developed by Lubik and Schorfheide (2004) that takes the possibility of indeterminacy into account and allows a researcher to look across the boundary. (1) Reestimating the two models over the entire parameter space leave the original results virtually unchanged. Using measures of fit, I find that it is highly unlikely that U.S. data are generated from an indeterminate equilibrium and are driven by nonfundamental or sunspot shocks. The combination of at best mildly increasing returns and inelastic labor supply rule out indeterminacy even after correcting for potential biases in the estimation algorithm. (2)
The article is structured as follows. In the next section, I specify the benchmark model, namely a standard stochastic growth model with externalities in production, and I discuss how this can imply increasing returns to scale and equilibrium indeterminacy. Section 2 describes my empirical approach and discusses the data used in the estimation. In the third section, I present and discuss results from the estimation of the benchmark model, while I extend the standard model in Section 4 to allow for variable capacity utilization. I address the issue of an indeterminate equilibrium as the source of business cycle fluctuations within this context in Section 5. The final section concludes and discusses limitations and extensions of the work contained in this article.
1. A FIRST PASS: THE STANDARD RBC MODEL WITH EXTERNALITIES
The benchmark model for studying returns to scale is the standard stochastic growth model with an externality in production. I use this model as a data-generating process from which I derive benchmark estimates for the returns to scale from aggregate data. Moreover, this model has been used by Benhabib and Farmer (1994) and Farmer and Guo (1994) to study the implications of indeterminacy and sunspotdriven business cycles. It will therefore also serve as a useful benchmark for capturing the degrees to scale when the data are allowed to cross the boundary between determinacy and indeterminacy.
In the model economy, a representative agent is assumed to maximize the intertemporal utility function:
[E.sub.0] [[infinity].summation over (t=0)] [[beta].sup.t] [log [c.sub.t] - [[chi].sub.t] [[n.sup.1+[gamma].sub.t]/1 + [gamma]]], (1)
subject to sequences of the budget constraint:
[c.sub.t] + [k.sub.t+1] = [A.sub.t] [[bar.e].sub.t] [k.sup.[alpha].sub.t] [n.sup.1-[alpha].sub.t] + (1 - [delta])[k.sub.t], (2)
by choosing sequences of consumption [{[c.sub.t]}.sup.[infinity].sub.t=0], labor input [{[n.sub.t]}.sup.[infinity].sub.t=0], and the capital stock [{[k.sub.t+1]}.sup.[infinity].sub.t=0]. The structural parameters satisfy the restrictions: 0 < [beta] < 1, [gamma] [greater than or equal to] 0, 0 < [alpha] < 1, 0 < [delta] < 1, whereby [beta] is the discount factor, 7 the inverse of the Frisch labor supply elasticity, [alpha] the capital share, and [delta] the depreciation rate.
The externality in the production process, [[bar.e].sub.t], is taken parametrically by the agent. Conceptually, this means that when computing first-order conditions for the agent's problem, [[bar.e].sub.t] is taken as fixed. It is only when equilibrium conditions are imposed ex post that the functional dependence of [[bar.e].sub.t] on other endogenous variables is realized. (3) I assume that [[bar.e].sub.t] depends on the average capital stock [[bar.k].sub.t] and labor input [[bar.n].sub.t]:
[[bar.e].sub.t] = [[[[bar.k].sup.[alpha]. …
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