Aggregate Returns to Scale: Why Measurement Is Imprecise

The extent to which there are aggregate returns to scale at the level of aggregate production has important implications both for the types of shocks generating business cycles and for optimal policy. However, prior attempts to measure the extent of these returns using instrumental variable techniques have yielded quite imprecise estimates. In this article, we show that the production shocks implied by a range of returns to scale that encompasses both large increasing returns and large decreasing returns are almost identical. This makes clear that there is a fundamental reason for the imprecision of prior estimates and casts doubt on our ability to generate more precise estimates. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The value of aggregate returns to scale—the percentage change in output from a given percentage change in factor inputs—has important implications for the sources of shocks that lead to business cycle fluctuations. With constant or decreasing returns to scale, business cycle models driven largely by technology shocks are consistent with a number of business cycle facts, in particular, with procyclical labor productivity. In contrast, with constant or decreasing returns to scale, business cycle models driven primarily by monetary shocks are inconsistent with procyclical productivity. With constant or decreasing returns to scale, the marginal product of labor is diminishing; therefore, an increase in labor input brought about by a monetary shock alone drives down productivity. With increasing returns, however, monetary shocks can generate procyclical productivity in otherwise standard models. Moreover, if the value of returns to scale is sufficiently large, equilibria may not be unique, and as Benhabib and Farmer (1994) show, self-fulfilling beliefs, or animal spirits, alone can generate fluctuations that are difficult to distinguish from fluctuations in the standard real business cycle model driven by technology shocks. In fact, business cycle fluctuations in economies with a sufficiently large returns to scale value can be due to virtually any shock that moves factor inputs. Thus, our ability to evaluate the importance of the sources of business cycle fluctuations depends on the value of aggregate returns to scale. While the value of returns to scale is important for evaluating the sources of business cycle shocks, measuring returns is difficult. First, there is an identification problem: model economies with any value of returns to scale are observationally equivalent if the unobserved stochastic process generating the shocks is unrestricted. Second, although researchers have come up with various ways of confronting the identification problem to measure returns to scale, the resulting estimates often cover a wide range of values, including significant decreasing and large increasing returns to scale. Moreover, the estimates often have large standard errors and corresponding wide confidence intervals. Consequently, firm conclusions about the value of returns to scale are hard to draw; thus, the importance of various sources of business cycle shocks are hard to evaluate. In this article, we analyze the measurement of aggregate returns to scale. We first show why there is an identification problem and discuss what assumptions are required to solve that problem. We then conduct a simple analysis that sheds light on how precisely returns to scale can be measured. In this analysis, we compare the technology shocks inferred from aggregate production functions that are identical except for the value of returns to scale. If we can measure returns to scale precisely, then the technology shocks should be sensitive to changes in the value of returns to scale. With precise measurement, the technology shock inferred from the assumption of decreasing returns should be different from the technology shock inferred from the assumption of increasing returns. Alternatively, with imprecise measurement, the shocks should be insensitive to changes in the value of returns to scale. We conduct our analysis for values ranging from significant decreasing returns up to substantial increasing returns. Our main finding is that the technology shocks inferred from this range of values are nearly identical. For this range of values of returns to scale, the correlation of the shocks is close to one and the shocks have the same serial correlation properties and similar variances. Unfortunately, these results have negative implications for how precisely we can measure aggregate returns using standard measures of inputs and output. The similarity of the series suggests that the likelihood functions researchers use to estimate returns to scale are insensitive to variation in this parameter and, consequently, that measurement of returns to scale will not be precise. To conduct our analysis, we construct a model economy similar to models used in the literature, derive an observational equivalence result, and show how researchers have restricted the stochastic process for the shocks to identify returns to scale. We then show how restricting the shock process also implies, in principle, sharp restrictions on the covariance properties of the technology shocks. We go on to show that the covariance properties of the innovations to the technology shock are nearly the same for a wide range of returns to scale values. The Model In this section, we construct our basic model economy and characterize a set of functions that constitute an equilibrium. Our basic model is similar to the one used by Benhabib and Farmer (1994). Our model has a measure 1 number of identical households. The households’ preferences are given by (1) E{ ∞ t=0 βu(ct,1−lt,dt)} where β is the discount factor, u is the utility function, ct is consumption of the single physical good produced in the economy, 1 − lt is leisure (nonmarket time), and dt is a preference (home production) shock. The household’s budget constraint is given by (2) yt + (1−δ)kt = ct + kt+1 + bt+1 − rtbt + τt where bt and rt denote the household’s borrowing level and the gross interest rate, respectively, and τt denotes a lump-sum tax. Per capita production of the household is given by (3) yt = λtF(kt,lt)Y φ t where kt and lt denote the household’s levels of capital and labor and Yt is aggregate per capita output. 2 Following other work in the literature, we assume that the production function F( ) is a linear homogeneous CobbDouglas function. The term λt is the aggregate technology shock. The parameter φ determines the value of the externality and, consequently, aggregate returns to scale. The economy is defined as neoclassical if φ = 0. For generality, we allow for different sources of shocks that might lead to business cycle fluctuations— technology shocks, government spending shocks, preference shocks, and shocks to extraneous factors (sunspots). We define εt as a 4 × 1 vector of independent and identically distributed random variables, with ε ≡ {εs} t s=0. We let the period t realization of the technology shock, the government spending shock, the preference shock, and, where relevant, the sunspot variable be given by λt(ε ), gt(ε ), dt(ε ), and vt(ε ). The government’s budget constraint is