หลักสูตรอบรม การวัดประสิทธิภาพและผลิตภาพของการผลิตสินค้าเกษตร ด้วยแบบจำลอง DEA ผศ. ดร. ศุภวัจน์ รุ่งสุริยะวิบูลย์ คณะเศรษฐศาสตร์ มหาวิทยาลัยเชียงใหม่

Lecture 4: ขอบเขตเนื้อหา Metafrontier การวิเคราะห์ Metafrontier ด้วยแบบจำลอง DEA การวิเคราะห์ Metafrontier ด้วยแบบจำลอง SFA

Metafrontier When all producers in different groups of a given industry have a potential access to the same technology but each producer may choose to operate on a different part of their technologies depending on circumstances such as the natural endowments, relative prices of inputs and the economic environment, then the assessment of producer’s efficiency and productivity can be measured using a metafrontier concept. Hayami and Ruttan (1970) initially proposed a metaproduction function which is defined as the envelope of commonly conceived neoclassical production functions. Thus, it is a common underlying production function that is used to represent the input- output relationship of a given industry.

Metafrontier Consider there are two different groups of technologies, namely A and B. Let points A 1, A 2, A 3 and A 4 indicate the input-output bundles of four producers in group A. These points are used to construct a frontier for production technology in group A or T A. Similarly, points B 1, B 2, B 3 and B 4 show the input-output bundles of four producers in group B. These points are used to construct a frontier for production technology in group or T B. If each group of producers has potential access to the same technology, the grand frontier which envelops the two group-specific frontiers can be represented by line A o A 1 A 2 B 2 B 3 B o. This line is referred as a metafrontier function or T *.

Metafrontier y x A1A1 T*T* AoAo B o B3B3 B2B2 B1B1 B4B4 A3A3 A2A2 A4A4 A1A1 TATA TB TB

Metafrontier The metafrontier function can be measuring using both nonparametric and parametric approaches. The metafrontier function using DEA constructs piece-wise linear convex production technology by enveloping all observed data from each group-specific technology. It does not require specified functional form for each group-specific technology. The metafrontier function using SFA constructs a smooth production technology by tangenting a specified functional form of production functions from each group- specific technology. The metafrontier using SFA is a smooth function and not a segmented envelope of each group-specific technology.

Metafrontier y x A3A3 A1A1 A2A2 A4A4 B2B2 B3B3 B4B4 T*T* TATA TBTB AoAo B1B1 y x A1A1 T*T* TB TB TATA AoAo B o B3B3 B2B2 A3A3 A2A2 B1B1 A4A4 B4B4 A1A1 x DEA SFA

Decomposition of TE under metatechnology B 1 ** B4B4 A 3 *** B oo A 4 * AoAo A oo T * T A T B y x A3A3 A2A2 B1B1 B2B2 B3B3 BoBo A3*A3* B 4 ** B4*B4* A 4 ** A4A4 A 3 ** A1A1

Decomposition of TE under metatechnology The metatechnology (T * ) which is constructed from the two production technologies, T A and T B, is represented by line A o A 1 A 2 B 2 B 3 B oo. The boundary of the metaechnology represents a metafrontier. Consider the production technology T A where point A 1, A 2 and A 4 lie on the frontier but point A 3 lies below the frontier. TE o A of the point A 1, A 2 and A 4 = 1 TE o A of the point A 3 = A 3 * A 3 / A 3 * A 3 ***. When the metafrontier (T * ) is considered, TE o * of the point A 1, A 2 = 1 TE o * of the point A 3 = A 3 * A 3 / A 3 * A 3 ** TE o * of the point A 4 = A 4 * A 4 / A 4 * A 4 **.

Decomposition of TE under metatechnology Similarly, consider the production technology T B where point B 1, B 2 and B 3 lie on the frontier but point B 3 lies below the frontier. TE o B of the point B 1, B 2 and B 3 = 1 TE o B of the point B 4 = B 4 * B 4 / B 4 * B 4 ** When the metafrontier (T * ) is considered, TE o * of the point B 2, B 3 and B 4 is still the same as TE o B TE o * of the point B 1 = B o B 1 / B o B 1 ** When the TE o is measured relative to the group-specific technology and metatechnology, it can occur a gap between the two technologies used as a reference. This gap is called a technology gap which is defined as the ratio of the distance function using an observed data based on the metotechnology T * to the group-specific technology T k.

Metafrontier Using the output orientation, the technology gap ratio can be defined as or it can be written as The metafrontier (T * ) can be decomposed into the product of the TE measured with respect to the k-th group technology (T k ) and the technology gap ratio.

Metafrontier For example, consider point A 3 in the figure, TE with respect to T A can be measured by the ratio of the distances between A 3 * A 3 to A 3 * A 3 ***. The TE o A = 3.1/5.6 = implying that all outputs could be possibly produced by 45% more from the given inputs by using T A as a reference. The TE with respect to T * can be measured by the ratio of the distances between A 3 * A 3 to A 3 * A 3 **. The TE o * = 3.1/6.8 = implying that all outputs could be possibly produced by 54% more from the given inputs by using T * as a reference. Therefore, TGR o k = 0.456/0.554 = implying that the possible output for the T A is 82.3 percent of that represented by the metafrontier (T * ).

DEA Approach to Metafrontier First, calculate the group-specific technology If the group k consists of data on I k producers, the linear programming (LP) problem that is solved for the i-th producer in the k-th group at the t-th time period is given as follows. st where θ k ≥1. The inverse of θ k is used to define an output-oriented TE scores of the i-th producer in the k-th group at the t-th time period or TE k o,it (x, y). 0 ≤ TE k o,it (x, y) ≤ 1.

DEA Approach to Metafrontier Second, calculate the metafrontier technology The metafrontier is constructed based on the pooled data of all producers in all groups. The LP problem that is solved for the i-th producer at time period t is given as follows. st where θ * ≥1. The inverse of θ * is used to define an output-oriented TE scores of the i-th producer at the t-th time period using the metafrontier as a reference or TE * o,it (x, y). 0 ≤ TE * o,it (x, y) ≤ 1 and TE * o,it (x, y) ≤ TE k o,it (x, y).

SFA Approach to Metafrontier First, the stochastic production frontier for each group is estimated and compared with that for all producers. Then, a statistical test is performed to examine whether all producers in different groups have potential access to the same technology. If the group k consists of data on producers, the stochastic production frontier model for the i-th producer at time period t based on the group-specific data and the pooled data is given as follows. where c = k refers to the stochastic group-specific production frontier model when the data for the i-th producer in the k-th group at the t-th time period are used, and c = p refers to the stochastic pooled production frontier model when the data for all producers in all groups for all time periods are used.

SFA Approach to Metafrontier Following Battese and Coelli (1992), the stochastic group- specific and pooled production frontier models, taking the log-quadratic translog functional form under a non-neutral TC assumption can be written as follows. An estimate of output-orientated TE for the i-th producer at the t-th time period is given by

Tests of Hypotheses If the stochastic frontiers across groups do not differ, then the stochastic pooled frontier function can be used as a grand technology for each group. A test based on the hypothesis that all producers in different groups have potential access to the same technology can be conducted using the likelihood ratio (LR) test. H o : All producers in different groups have potential access to the same technology H a : All producers in different groups do not have potential access to the same technology The test statistic is calculated as where L(H 0 ) and L(H a ) are the value of the likelihood function under the null and alternative hypothesis.

SFA Approach to Metafrontier The second step will involve estimating the metafrontier function. The parameter estimates of the metafrontier function are estimated by solving the following LP problem. such that where ß k s are the estimated coefficients obtained from the stochastic group-specific frontiers ß * s are parameters of the metafrontier function to be estimated.

SFA Approach to Metafrontier Once the ß * parameters of the metafrontier function are estimated, the decomposition of TE under the metafrontier can be calculated. The technology gap for the i-th producer in the k-th group at the t-th time period can be obtained by Then, a measure of the output-oriented TE relative to the metafrontier, TE * o (x, y). can be obtained using equation

Exercise Panel data of conventional and organic farms –28 Farms: 14 farms are conventional farms and 14 farms are organic farms –15 periods from 1991 to 2005 –1 output: The gross output value of farming aggregates physical output from seven grain crops and twelve economic crops. –6 inputs: capital, labor, chemical fertilizer, pesticide, plastic film and irrigation

Estimated Parameters Parameters a Stochastic Frontier Metafrontier b OrganicConventionalAll ß0ß (0.0465) ( ) ( ) ( ) ß1ß (0.0317) ( ) ( ) ( ) ß2ß (0.0614) ( ) ( ) ( ) ß3ß (0.0727) ( ) ( ) ( ) ß4ß (0.0519) ( ) ( ) ( ) ß5ß (0.0392) ( ) ( ) ( ) ß6ß (0.1193) ( ) ( ) ( ) ßtßt (0.0059) ( ) ( ) ( ) ß (0.0355) ( ) ( ) ( ) ß (0.0510) ( ) ( ) ( ) ß (0.0520) ( ) ( ) ( ) ß (0.0442) ( ) ( ) ( ) ß (0.0359) ( ) ( ) ( ) ß (0.0775) ( ) ( ) ( ) ß (0.1112) ( ) ( ) ( ) ß (0.1230) ( ) ( ) ( ) ß (0.0713) ( ) ( ) ( ) ß (0.0570) ( ) ( ) ( ) ß (0.1500) ( ) ( ) ( ) ß (0.2384) ( ) ( ) ( )

Estimated Parameters (continued) Parameters a Stochastic Frontier Metafrontier b OrganicConventionalAll ß (0.0637) ( ) ( ) ( ) ß (0.0959) ( ) ( ) ( ) ß (0.5484) ( ) ( ) ( ) ß 1t (0.0048) ( ) ( ) ( ) ß 2t (0.0085) ( ) ( ) ( ) ß 3t (0.0103) ( ) ( ) ( ) ß 4t (0.0058) ( ) ( ) ( ) ß 5t (0.0047) ( ) ( ) ( ) ß 6t (0.0122) ( ) ( ) ( ) ß tt (0.0011) ( ) ( ) ( ) σ2σ (0.0019) ( ) ( ) γ (0.0633) ( ) ( ) η (0.0120) ( ) ( ) Log-L

Average TE and TGR FarmTE k TGRTE * FarmTE k TGRTE * Organic Conventional Beijing Shanxi Tianjin Inner-Mongolia Hebei Anhui Liaoning Jiangxi Jilin Henan Helongjiang Hunan Shanghai Guangxi Jiangsu Sichuan Zhejiang Guizhou Fujian Yunnan Shandong Shaanxi Hubei Gansu Guangdong Qinghai Xijiang Ningxia Average Average