The Effects of Operating and Capital Subsidies on Total Factor Productivity: A Decomposition Approach.

• Author: Obeng, K.

K. Obeng [*]

R. Sakano [+]

Past studies relied on ad hoc associations to establish relationships between productivity on one hand and operating and capital subsidies on the other. This article deviates from these studies. It builds on recent research based on private cost to derive a total factor productivity formula that includes subsidy effects. It specifies an empirical model to estimate the required parameters to apply the formula. The application to urban transit systems shows that the effects of these subsidies on productivity through technical change reinforce the decline in productivity.

1. Introduction

   Sakano, Obeng, and Azam (1997) applied the stochastic frontier method to public transit data and decomposed allocative inefficiency between internal factors and subsidies. They found that the internal factors contributed more toward allocative inefficiencies, and subsidies accounted for less than 25% of the technical inefficiencies in transit firms. These authors also found operating subsidies to be the major sources of the inefficiencies and that capital subsidies mostly acted as a counterbalancing factor in reducing the inefficiencies. These findings suggest that both subsidies play opposing roles in affecting performance and inefficiency and that while capital subsidies increase partial productivity measures, operating subsidies may reduce them. These effects of subsidies on partial productivity will also affect total factor productivity (TFP). Therefore, there is a link between operating and capital subsidies and TFP, and we must find a way to decompose the changes in TFP such that we can identify the ef fects of subsidies. This decomposition is the motivation for this article; we develop a decomposition method for TFP that permits the contributions of subsidies to be isolated and apply the method to public transit data. The article, therefore, makes contributions in both methodology and application.

   The basis of the decomposition is an extension of the methods discussed in Obeng (1994), Obeng and Azam (1997), and Sakano, Obeng, and Azam (1997) to derive a TFP measure that depends on subsidies. [1] These methods hold that firms maximize output subject to a net cost constraint, where net cost is total cost less operating and capital subsidies. The dual of this constrained output maximization is that firms minimize their private cost subject to a production function constraint. This objective is different from those frequently adopted in the public transit and public economics literature. For example, for public transit systems, Berechman (1993) suggests cost minimization as the objective, while some firms, such as London Transport, use output maximization subject to a deficit constraint as the objective (Button 1993). On the other hand, Niskanen (1968) advocates budget maximization as the objective of bureaucrats and argues that bureaucrats derive utility from hiring inputs since that increases their budg ets. Although these objectives are valid representations of transit systems' objectives, the effect of subsidies on productivity is best represented in a net cost minimization model, where net cost is total actual cost less subsidies. As we show in this article, this objective allows us to distinguish between implied cost and actual cost and enables us to establish a relationship between the two. Furthermore, from the relationships we derive a decomposition method for total factor productivity that accounts directly for subsidies.

   Because the distinction between implied and actual costs relates to shadow prices, we can employ the nonparametric approach of Fare, Grosskopf, and Nelson (1990) to calculate the cost distortions from the subsidies. This assumes that at least one input market is efficient, and that the distortions do not vary by the sizes of the subsidies. Furthermore, it implies that the entire distortion in cost is due to subsidies, which may not be true. For example, expense preference behavior on the part of management in the presence of subsidies will further distort input prices (Obeng 2000). Therefore, the nonparametric approach may not be appropriate for determining allocative distortions from these subsidies. But, Berg, Forsund, and Jansen (1992), Hjalmarsson and Veiderpass (1992), and Thirtle, Piesse, and Turk (1996) show that the nonparametric approach, particularly Malmquist indices, can be used to decompose total factor productivity among frontier shifts (technical change) and shifts in a firm's position relativ e to the frontier, that is, a catch up or inefficiency effect. On the other hand, our parametric approach provides a TFP formula that separates the effects of subsidies from other factors.

   The paper is organized as follows: The next section deals with theoretical considerations and is followed by the empirical model specification, data, and results. The last two sections give the results of the TFP decomposition and conclusion, in that order.

2. Theoretical Considerations

   In two recent publications, Obeng (1994) and Obeng and Azam (1997) derived cost functions for a transit firm that minimized net cost subject to an output constraint. Net cost in these studies was total cost less subsidies from all sources. This firm received both operating and capital subsidies. [2] Capital subsidies, [A.sub.K](K), depended on fleet size (K), and operating subsidies, [A.sub.0](L, G), depended on labor (L) and fuel (G) used as a proxy for material and energy. From this cost function, these authors showed that the private price or the implied price of each input to a firm is

   [[W.sup.*].sub.i] = [W.sub.i](1 - [[theta].sub.i][F.sub.i]) = [W.sub.i] - ([[theta].sub.i] . A/[X.sub.i]), (1)

   where [W.sub.i] is actual input price, [[W.sup.*].sub.i] is private or implied input price, [[theta].sub.i] is the estimated elasticity of the subsidy with respect to input i, and [F.sub.i] is the share of the subsidy in the cost of an input (i.e., A/[[X.sub.i][W.sub.i]]). Equation 1 shows that the private price (or implied price) of each input to a firm is less than its actual input price after the government offers subsidies. Because of this, firms behave as if they were faced with [[W.sup.*].sub.i] and, therefore, demand more inputs than they would at the price of [W.sub.i]. This behavior is illustrated by Figure 1. At the implied price of [[W.sup.*].sub.i] the quantity of an input actually demanded is [X.sub.i], and the implied input cost is [[W.sup.*].sub.i][X.sub.i]. If firms were to perceive their costs correctly at [W.sub.i], they would demand the quantity [[X.sup.*].sub.i]. Thus, [X.sub.i] min us [[X.sup.*].sub.i] is the excess quantity demanded because of the subsidies. Because [W.sub.i] is the actual price paid for the input, the actual resource cost is [W.sub.i][X.sub.i], and [[theta].sub.i][W.sub.i][F.sub.i] is how much a subsidy reduces the cost of each input.

   Multiplying both sides of Equation 1 by the quantity, [X.sub.i], of input i and taking the sum over all inputs gives the relationship between total private or implied cost, [C.sup.*] = [[sigma].sup.i] [[W.sup.*].sub.i]X, and total actual cost (C = [[sigma].sup.i][W.sub.i][X.sub.i] as

   [C.sup.*] = C(1 - [[[sigma].sup.i].sub.1] [S.sub.i][[theta].sub.i][F.sub.i]) = C - [[[sigma].sup.i].sub.1] [[theta].sub.i]A([X.sub.1], [X.sub.2], ..., [X.sub.i]) (2)

   where

   [[[sigma].sup.i].sub.1] [[theta].sub.i] = 1.

   Here, [S.sub.i], is the share of an input in total actual cost (i.e., [W.sub.i][X.sub.i]/C) compared with the share [[S.sup.*].sub.i] (i.e., [[W.sup.*].sub.i][X.sub.i]/[sigma] [[W.sup.*].sub.i]X), of the same input in total private cost. Equation 2 also can be derived directly from Figure 1. Furthermore, although we do not do so in this article, Equation 2 is the exact equation form that will be obtained if we were to assume a Cobb-Douglas production function as the underlying technology in the net cost minimization problem. [3] The term in the first set of parentheses (1 - [[sigma].sup.i] [S.sub.i][[theta].sub.i][F.sub.i]) is the share of private cost in actual cost. Thus C [[sigma].sup.i][[theta].sub.i][S.sub.i][F.sub.i] represents the amount that subsidies reduce total implied cost.

   Equations 1 and 2 assume operating subsidies are allocated to labor and fuel (which is a proxy for materials and fuel), and capital subsidies to capital only; in practice this may not happen. For example, capital subsidies may be used to purchase services from private firms and to pay for supervisors on capital projects. The federal government recognizes this problem and allows capital subsidies to be used for labor assigned to capital projects. For this reason, we modify Equations 1 and 2 to make capital subsidies a function of fleet size and labor. This modification introduces additional terms into the equations. Also, Equation 2 assumes the subsidies are linear functions of inputs. However, the federal subsidy formula indicates that the subsidies are functions of other variables, such as population and population density, which are independent of the inputs. Therefore, we include the term [epsilon] as a random error to account for omitted variables in the subsidy functions. Graphically, this input-neutral portion of the subsidies rotates the input demand curve outward and induces more use of inputs in Figure 1. Thus,

   [[W.sup.*].sub.i] = [W.sub.i](1 - [[theta].sub.oi][F.sub.oi] - [[theta].sub.ki][F.sub.ki])

   so that

   [C.sup.*] = C([[[sigma].sup.i].sub.1] [S.sub.i](1 - [[theta].sub.oi][F.sub.oi] - [[theta].sub.ki][F.sub.ki])) . [e.sup.[epsilon]] (3)

   where

   [[[sigma].sup.i].sub.1] [[theta].sub.oi] = 1. [[[sigma].sup.i].sub.1] [[theta].sub.ki] = 1.

   The additional subscripts are for operating and capital subsidies, respectively. These changes in the implied input price and implied cost do not affect the results in Figure 1, except that the implied prices are lowered further, and increase the quantity demanded of an input more than in the previous case.

   ...