Aggregating Capacity-Limiting Separable Inputs

Abstract

1. Introduction In many empirical studies on productivity, researchers frequently need to use data on capital goods in their analyses. Unfortunately, capital goods are extremely heterogeneous and data on them are difficult to capture in a single measure. There are many different types and vintages of capital goods in the same firm. It is very difficult to gather all information on all capital goods. In practice, capital goods are aggregated into one input, named 'capital'. Recent studies that aggregate capital goods are Fare, Grosskopf, and Tyteca (1996), who used the installed generating capacity as capital; Acemoglu and Zilibotti (2001), who applied the perpetual inventory method to recover capital from investment; Mathijs and Swinnen (2001), who adopted the deflated book value of fixed assets as capital; and Taymaz and Saatci (1997), who used the depreciation allowance as capital. Though each method has its merits, there may be information loss during the process of aggregating capital goods. Five possible types of errors during the aggregation process are identified here. They are: (i) Error of separability: There is a close relation between input aggregation and separability. When different capital goods are separable from other inputs, it is possible to find an aggregate of capital goods without information loss. In this case, there is no error of separability. If capital goods are not separable from other inputs, however, and we aggregate them, then actually we are approximating the production function by a separable function. Mak (1988) noted that the success of approximating a nonseparable function depends on whether that function is close enough to a separable function. Thus, when the technology is not separable, the error of separability occurs. When the production function is not close to a separable function, the error of separability is even larger. (ii) Error of formula: Suppose the production function is approximated by a separable function. Then there is a way to aggregate capital goods. If the formula we use to aggregate capital goods cannot give us an aggregation consistent with that separable function, then we face the error of formula. (iii) Error of measuring opportunity costs: Some measures of capital require information about prices of capital goods. In economic theories, all prices of inputs are the opportunity costs of using those respective inputs. If the prices used do not reflect opportunity costs, then this error occurs. (iv) Error of the deflator: When we need to add up capital goods bought in different periods by using price information, we have to deflate the values. If an inappropriate deflator is chosen, then there is an error of the deflator. (v) Error of accounting practices: Information reported by firms follows bookkeeping rules. The value of fixed assets, for example, may not be the same as the concept of capital in economics. Thus, investment and fixed assets may be biased. This problem is larger when firms in different industries or countries are compared. Whether or not there is the error of separability depends on the structure of the technology. It is beyond the control of researchers. Though we must cope with the error of separability, what we can do is to avoid the other four errors. In this paper, I link the literature on separability with that on maximum capacity in favor of using the output capacity of plants to aggregate heterogeneous capital goods. We find that when some inputs are separable in the production function, their maximum capacity can be used as an aggregate measure for them. This aggregate measure is economically meaningful and empirically tractable. This paper is organized as follows: Section 2 discusses the properties of capacity-limiting inputs. Section 3 relates capacity-limiting inputs to separability. Section 4 comments on some applications of aggregating captial goods. Section 5 concludes the paper. …