بهره وری اقتصادی هنگامی که قیمت ها ثابت نیست: رفع ابهام کمیت و بهره وری قیمت

This paper proposes an approach to compute cost efficiency in contexts where units can adjust input quantities and to some degree prices so that through their joint determination they can minimise the aggregate cost of the outputs they secure. The model developed is based on the data envelopment analysis (DEA) framework and can accommodate situations where the degree of influence over prices ranges from minimal to considerable. When units cannot influence prices at all the model proposed reduces to the standard cost efficiency DEA model for the case where prices are taken as exogenous. In addition to the cost efficiency model, we introduce an additive decomposition of potential cost savings into a quantity and a price component, based on Bennet indicators.

Traditional models for computing cost and revenue efficiency date back to Farrell [9] and will be called Farrell cost or revenue efficiency models. Since the appearance of Data Envelopment Analysis (DEA) in 1978 (see [5]) cost efficiency and revenue efficiency have been computed through linear programming models, when an option for the use of non-parametric models is taken. The alternative is to compute cost or revenue efficiency based on the definition of parametric revenue functions or cost functions like the Cobb Douglas function or the translog function (see e.g. [16]). In both cases, the underlying economic model of the firm typically assumes that it operates in a competitive market, where prices of inputs and outputs are exogenously given, often taken at the level of actual prices observed at the operating decision making unit (DMU). As a result, cost efficiency and revenue efficiency reflect cost savings or revenue gains that can accrue from changes in input or output quantities given their (externally fixed) prices. Units where input or output prices are exogenously fixed are said to be price takers. In this paper we depart from the notion of production units being price takers and consider the case where the units to be assessed for cost or revenue efficiency either can influence to some extent the input or output prices they secure and/or can secure prices other than those they actually did even if they do not influence their levels directly themselves. This in turn means that such production units can improve their cost or revenue efficiency by inter alia securing better prices for their inputs and/or outputs. We focus on cost efficiency in this paper, but the extension of the proposed approach to revenue efficiency is straightforward. Modelling economic efficiency in situations where units are not price takers has been addressed before, but mainly in a context of endogenous prices (e.g. the level of outputs of a DMU is assumed to impact the price per unit of output, so that maximum output levels may not necessarily be compatible with maximum revenue). For example, Cherchye et al. [7] considered a situation of endogenous and uncertain prices, while Johnson and Ruggiero [17] considered only the situation of endogenous prices. On the other hand, Kuosmanen and Post [18] analysed just the situation of price uncertainty. Price endogeneity as modelled hitherto has relied on some explicit functional form e.g. between output quantities and corresponding output prices. We depart in this paper from this modelling paradigm for two reasons. Firstly, to deal with situations where markets are not competitive and demand functions are not known, and secondly even if the markets are perfectly competitive the units concerned may not be able to access the fully competitive prices which in any case may not necessarily be explicitly known. We would argue that prices manifested at unit level apart from endogeneity or exogeneity as the case may be, also incorporate an element of managerial efficiency and local factors, which can make the task of predicting prices attainable at unit level very difficult. That is, whether prices are endogenous or exogenous the ability of a unit to secure optimal prices reflects, to an extent, also the unit's ability to assess the pricing context in which it is operating and take appropriate action to optimize the prices it secures. In this context we take the prices manifested across the DMUs as the best available evidence of prices that can be secured, and we replicate in the context of input prices what is done for input-output levels in a non parametric context of efficiency measurement. In traditional DEA models no assumption is made of the functional relationship between input and output levels. The production possibility set is built with reference to observed input-output level correspondences using certain assumptions (e.g. [26, p. 255]). In the same manner in this paper we use observed input prices to derive prices that are in principle attainable by DMUs. The larger the number of DMUs within a ‘pricing environment’ the better the prices attainable by DMUs will be revealed. To see how the prices that have been observed can incorporate some component of inefficiency, even in competitive markets, consider the case of a number of hypothetical hospitals in a given large city each one looking to hire a doctor of the same skills at the same point in time. It is by no means certain what would be the theoretically minimum salary at which such a doctor can be hired. The salary each hospital will end up paying to the doctor it recruited will to some extent depend on the negotiating skills of the candidate and of the hospital staff involved, on whether the post was over or under specified by each hospital relative to the skills actually needed, and on externalities such as location of the post relative to a candidate's residence, perceived culture of the employer etc. Similarly, in a banking context, where input prices can be interest paid on deposits, the ability of management to devise financial products in terms of the interplay of interest rate and withdrawal or other restrictions will affect the rate at which a bank can secure funds, whether or not there is a free competitive banking market. Thus different banks drawing funds from the same pool of savings can secure funds at different rates, in effect meaning securing inputs at different prices. In both these examples the actual recruitment salaries paid for doctors or the interest rates paid to depositors by banks offer us the best available empirical evidence of the ‘input’ prices that might have been attainable. Thus in this paper we address situations where units are not strictly price takers. Rather, prices paid reflect inter alia the degree of managerial effectiveness in securing optimal prices for their inputs. We argue that when the performance of such units is assessed, managerial ability to arrive simultaneously at an optimal price and quantity mix so as to minimise aggregate input costs should be captured. We develop a cost model for the simultaneous optimisation of input quantities and prices. Such a model requires that data on prices and quantities are available, and assumes that DMUs have some degree of influence both over prices and over quantities of inputs. The paper is structured as follows. In the next section we review literature that is more directly related with the work developed in this paper, and will further address the motivations of the paper. In Section 3 we propose a new model for computing cost efficiency and show how to decompose this measure in Section 4. In Section 5 an illustrative example is used to highlight the differences between ours and existing approaches for computing price efficiency. Section 6 concludes the paper.

This paper has addressed the case where organisational units use one or more inputs to produce one or more outputs and we wish to identify the minimum cost at which each unit could have secured its outputs. We assume observed input prices exist but we also assume that units are not input ‘price takers’. That is a unit might have been able through its own actions to secure its inputs at prices other than those that in the event materialised. This could for example have been through a combination of skilful negotiation with providers, timing purchases, optimising lot sizing and so on. We further assume that we have no information to model in a deterministic fashion potential input prices that might have been available to a unit. In this context the only information we have from which to deduce the input prices that might have been available are the observed prices at the units being compared. For the foregoing context we provide a general purpose DEA model which for the first time in this literature minimises simultaneously the quantities of inputs used and the prices paid in order to estimate a minimum aggregate cost of securing the outputs of the unit. Taking its lead from traditional DEA models, in which feasible in principle input-output correspondences are derived as convex combinations of observed such correspondences, our model extends this idea to prices. Specifically we assume that convex combinations of observed prices for each input are feasible in principle for that input. This creates a space of feasible prices from which the model draws input prices so that choosing at the same time a most appropriate mix of input levels for those prices, the unit could minimise the aggregate cost at which it secures its inputs. We go on to suggest that in a typical real life situation there would generally exist additional restrictions on prices. For example it may not be acceptable or feasible at a given unit for input prices to vary by more than a certain factor up or down from those observed. Other information such as quantity price discounts or trade offs between prices if applicable can also be modelled. The model developed upon solution yields optimal quantities, prices, output levels and the minimum aggregate cost of securing the outputs of a unit. For the case where potential cost savings are identified for a unit we have provided a formula for their decomposition. The decomposition identifies two additive components, one reflecting the scope for savings through implementing input quantity changes and the other through implementing price changes. We argue that when units are not strictly price takers the traditional concept of allocative efficiency loses its meaning as it relates to keeping prices fixed and identifying an optimal input mix for those prices. The decomposition suggested in this paper is the more suitable when both input quantities and their prices can be optimised simultaneously to arrive at minimum aggregate costs.