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**The Concept of Efficiency and Frontier Models**

**The Concept of Efficiency and Frontier Models**

In microeconomic theory a production function is defined in terms of the maximum output that can be produced from a specified set of inputs, given the existing technology available to the firms involved (Battese, 1992). The maximum possible output becomes relevant in order to answer certain economic questions such as the measurement of efficiency of firms, hence the introduction of frontier production functions which estimates the maximum output as function of inputs. Similarly, a cost frontier function would give the minimum cost as a function of output quantity and input prices. The papers by Debreu (1951) and Koopmans (1951) mark the origin of discussion on the measurement of productivity and efficiency in the economic literature. The work 45 of Debreu and Koopmans was first extended by Farrell (1957) in order to perform the measurement of productivity and efficiency.

The productivity of an economic agent can be measured simply as a scalar ratio of outputs to inputs that the agent uses in its production process. Productivity could be measured either as partial productivity such as yield per hectare (land productivity) or output per person (labour productivity) or more appropriately as total factor productivity (TFP) which is defined as ratio of aggregate outputs to aggregate inputs. An economic agent’s productivity may vary based on differences in production technology, in the efficiency of the production process, in the environment in which production occurs, and finally in the quality of inputs used by the agent (Haghiri, 2003). On the other hand, efficiency is measured by comparing observed and optimal values of the agent’s outputs and inputs. Prior to Farrell’s work, efforts were made to measure efficiency by interpreting the average productivity of inputs, then to construction of efficiency indexes. However, these methods were found unsatisfactory by economists and agricultural economists as the methods suffered from one shortcoming to another. The use of the traditional least squares methods for estimating the production function has been critiqued as this is not consistent with the definition of the production function. The estimated functions could at best be described as average or response functions because such regression estimates the mean output (rather than the maximal output) given quantities of inputs (Schmidt, 1986). This led to the development of a better-founded theoretical method for measuring efficiency, i.e. the frontier method. Frontiers models are described as bounding functions (Coelli, 1995b).

**Non-Parametric Frontier Approach**

A non-parametric approach neither specifies a functional form for the production technology nor makes an assumption about the distribution of the error terms. In other words it is robust with respect to the particular functional form and to the distribution assumptions. The non-parametric approach is mainly deterministic in nature. In a deterministic production frontier model, output is assumed to be bounded from above by a deterministic (non-stochastic) frontier. However, the possible influence of measurement errors and other statistical noise upon the shape and positioning of the estimated frontier is not accounted for. The original work of Farrell (1957) serves an important starting point for discussion of non-parametric frontiers. Farrell illustrated the measurement of efficiency using an input-oriented approach. His argument is embodied in figure 3.1. This illustration was done by considering a firm using two inputs x1 and x2 to produce output y, such that the production frontier is ),( 21 = xxfy Assuming constant returns to scale , then one 47 can write = 21 yxyxf )/,/(1 , that is the frontier technology can be characterized by a unit isoquant and this is denoted SS ′ in figure 3.1.

Knowledge of the unit isoquant of a fully efficient firm permits the measurement of technical efficiency. For a given firm using *),( 21∗ xx defined by point A )/*,/( 1 2 ∗ yxyx to produce a unit of output y * , the ratio OQ/OA measures technical efficiency and it defines the ability of a firm to maximize output from a given set of inputs. The ratio measures the proportion of ),( 21 xx needed to produce *y . Technical efficiency takes a value between zero and one and therefore provides an indication of technical inefficiency. Thus, the technical inefficiency of the firm, 1-OQ/OA, measures the proportion by which *),( 21∗ xx could be reduced (holding the input ratio 21 / xx constant) without reducing output. A firm that is fully technically efficient would lie on the efficient isoquant (example, point Q) and it takes a value of 1.

**Deterministic Non-Statistical Frontiers**

**Deterministic Non-Statistical Frontiers**

Few people adhered to the non-parametric approach by Farrell (1957). Almost as an after thought, Farrell (1957) proposed a second approach. In this approach, Farrell proposed computing a parametric convex hull of the observed input-output ratios. He recommended the Cobb-Douglas production function for this purpose given the limited selection of functional form then. He acknowledged the undesirability of imposing a specific (and restrictive) functional form on the frontier but also noted the advantage of being able to express the frontier in a simple mathematical form. This suggestion was however not followed up by Farrell. Aigner and Chu (1968) were the first to follow Farrell’s suggestion. In order to express the frontier in a mathematical form, they specified a Cobb-Douglas production frontier, and required all observations to be on or beneath the frontier.

Their model may be written as: 54 ln yi = i i α);(ln − uxf ; 0 u ≥ (3.7) where i y is the output of the ith sample firm; i x is the inputs of the ith firm, i u is a one-sided non-negative random variable associated with firm-specific factors that contribute to the ith firm inability to attain maximum efficiency of production. The one sided error term, i u forces )( ≤ xfy . The elements of the parameter vector, α , may be estimated either by linear programming (minimizing the sum of the absolute values of the residuals subject to the constraint that each residual is non-positive) or by quadratic programming (minimizing the sum of squared residuals, subject to the same constraint). Although Aigner and Chu (1968) did not do so, the technical efficiency of each observation can be computed directly from the vector of residuals, since u represents technical efficiency.

**Stochastic Frontiers**

They emerged as an improvement over average functions and deterministic frontiers. In the deterministic frontiers, all variations in the firm performance are attributed solely to variation in firm efficiencies relative to the common family of frontiers, be it production, cost or profit frontiers. Thus, the idea of a deterministic frontier shared by all firms ignores the very real possibility that a firm’s performance may be affected by 57 factors that are entirely outside its control such as bad weather, input supply breakdowns etc as well as factors under its control (inefficiency). To lump these effects of exogenous shocks, both fortunate and unfortunate, together with the effects of measurement error and inefficiency into a single one-sided error term, and to label the mixture inefficiency is questionable and is a major weakness of deterministic frontiers.

**Panel Data**

**Panel Data**

Cross sectional data provides a snapshot of producers and their efficiency. Panel data provides more reliable evidence on their performance, because they enable one to track the performance of each producer through sequence of time periods. In the Panel data model, a time varying or time invariant inefficient effect may be specified. Also, the model may assume either a fixed or random effect. A significant advantage of panels is that given consistently large time periods, they permit consistent estimation of the efficiency of individual producers, whereas the Jondrow et al. (1982) technique does not generate consistent estimators in a cross-sectional context (Kumbhakar and Lovell, 2000). Another advantage of the panel data is that the distributional assumptions about the efficiency term upon which stochastic frontier rely is no longer necessary. Also the assumption of independence between the inefficiency term and input levels is unnecessary with panel data. Again, panel data increases degrees of freedom for estimation of parameters and it permits the simultaneous estimation of technical change and technical inefficiency changes over time. However, the dearth of panel data on farmers especially in developing country agriculture has constrained the use of panel data methodologies.

**TABLE OF CONTENTS :**

- DEDICATION
- DECLARATION
- ACKNOWLEDGEMENT
- ABSTRACT
- TABLE OF CONTENTS
- LIST OF TABLES
- LIST OF FIGURES
- CHAPTER
- INTRODUCTION
- 1.1 Background to the Study
- 1.2 Problem Statement
- 1.3 Objectives of the Study
- 1.4 Hypotheses
- 1.5 Justification for the Study
- 1.6 Organization of the Thesis

**CHAPTER A REVIEW OF AGRICULTURAL POLICIES AND PROGRAMMES IN NIGERIA**- 2.1 Introduction
- 2.2 Agricultural Policy and Programmes in Nigeria
- 2.2.1 The Pre-1970 Era
- 2.2.2 The 1970-1985 Era
- 2.2.3 The 1986-1999 Era
- 2.2.4 The Post 1999 Era
- 2.3 The Performance of Nigerian Agriculture
- 2.4 Summary and Conclusions

**CHAPTER LITERATURE REVIEW ON EFFICIENCY, MEASUREMENT AND EMPIRICAL APPLICATIONS**- 3.1 Introduction
- 3.2 The Concept of Efficiency and Frontier Models
- 3.3 Non-Parametric Frontier Approach
- 3.4 Parametric Frontier Approach
- 3.4.1 Deterministic Non-Statistical Frontiers
- 3.4.2 Deterministic Statistical Frontiers
- 3.4.3 Stochastic Frontiers
- 3.4.3.1 Panel Data
- 3.4.3.2 Duality Considerations and Cost System Approaches
- 3.4.3.3 Production Frontier and Efficiency Decomposition
- 3.4.3.4 Distance Functions and Efficiency Decomposition
- 3.5 Empirical Studies on Efficiency Measurement
- 3.5.1 Empirical Comparative Studies in Agriculture
- 3.5.2 Empirical Comparative Studies in other Sectors involving Distance Functions
- 3.5.3 Recent Empirical Efficiency Studies in Nigerian Agriculture

**CHAPTER ANALYTICAL FRAMEWORK AND EMPIRICAL SPECIFICATIONS**- 4.1 Introduction
- 4.2 Analytical Framework
- 4.2.1 The Production Frontier and Efficiency Decomposition
- 4.2.2 Distance Function Approach to Efficiency Decomposition
- 4.2.2.1 The Parametric Stochastic Input Distance Function
- 4.2.1.2 The Non-Parametric Input Distance Function
- 4.3 Empirical Models
- 4.3.1 Parametric Stochastic Input Distance Function (SIDF)
- 4.3.2 Non-parametric Input Distance Function
- 4.3.3 Parametric Stochastic Frontier Production Function (SFPF)
- 4.3.4 Technology and Policy Impact on Efficiency

**CHAPTER STUDY AREA, SURVEY DESIGN AND SOCIO-ECONOMIC CHARACTERISTICS OF THE SAMPLE HOUSEHOLDS**- 5.1 Introduction
- 5.2 The Study Area
- 5.3 Survey Design and Sampling Procedure
- 5.4 Data Collection
- 5.5 Variable Description
- 5.6 Household and Farm Characteristics of Study Sample

**CHAPTER COMPARISON OF RESULTS FROM ALTERNATIVE APPROACHES**- 6.1 Introduction
- 6.2 Parameter Estimates and Efficiency Scores from the SIDF Model
- 6.3 Parameter Estimates and Efficiency Scores from the SFPF Model
- 6.4 Efficiency Scores from the Non-parametric Input Distance Models
- 6.5 A Visual Comparison of Efficiency Estimates from Different Frontier Models
- 6.6 Sensitivity of Efficiency Scores to Estimation Approaches: Formal Tests
- 6.7 Input Usage Ratios
- 6.8 Technological Innovation and Efficiency: Comparison of Alternative Models
- 6.9 Conclusions

**CHAPTER AN INTEGRATED INPUT DISTANCE MODEL FOR EFFICIENCY AND POLICY ANALYSIS**- 7.1 Introduction
- 7.2 The Integrated Model
- 7.3. Results and Discussion
- 7.3.1 Final Efficiency Scores and Distribution from the Integrated Model
- 7.3.2 Impact of Technological Innovation on Efficiency Estimates from the Integrated Model
- 7.4 Conclusions
- SUMMARY, CONCLUSIONS AND POLICY IMPLICATIONS
- 8.1 Summary and Conclusion
- 8.2 Policy Implications
- 8.3 Limitations of the Study and Areas for Future Research
- REFERENCES
- APPENDIX 1: QUESTIONNAIRE

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Sensitivity and Integration of Efficiency Estimates from Input Distance Functions and Stochastic Production Frontiers: Application to Maize Production in Benue State Nigeria