Return to Scale Chapter 5.1-5.4 Saara Tuurala 26.9.2007
Index Introduction Baic Formulation of Retur to Scale Geometric Portrayal in DEA BCC Return to Scale CCR Return to Scale Summary Home Aignment
Introduction 1/2 Return to Scale (RTS) Mittakaavatuoto, der Skalenertrag [1] It i a technical property of a production function y f ( x1, x2,..., x m ) Decribe what happen when the cale of production increae [2]
Introduction 2/2 The main quetion can be formulated a [3]: When we double all input, doe output: double (contant return to cale, CRTS) more than double (increaing return to cale, IRTS) le than double (decreaing return to cale, DRTS)?
Baic Formulation of RTS 1/3 y production function y/x average productivity (a.p.) dy/dx marginal productivity (m.p.) Aumption: technical efficency i alway achieved
Baic Formulation of RTS 2/3 Maximum value of a.p. with repect to x d( y / x) xdy / dx y dx e( x) x y dy dx x 2 1 e i called elaticity and it meaure the relative change in output compared to the relative chance in input (IRTS when e(x) > 1, DRTS when e(x) < 1 and CRTS when e(x) 1 0 (1) (2)
Baic Formulation of RTS 3/3 In the cae of multiple input with a calar output y we get elaticity of cale ε: y ε f ( θx θx θx ) 1, 2,..., θ dy y dθ ( θ ) (4) θ i a calar to repreent increae in cale when θ > 1 In the cae of multiple input and output we need a maximum output intead of the maximal value of y. Thi can be achieved with the help of Pareto- Koopman definition of efficiency (in Chapter 3) m (3)
Geometric Portrayal in DEA 1/3 Piecewie linear production function the failure of m.p. at x o Maximum output can be achieved with the help of upporting hyperplane
Geometric Portrayal in DEA 2/3 Supporting hyperplane H 0 in a (m + ) dimenional input-output pace paing through the point repreented by the vector (x o, y o ): H 0 : u ( y y ) o v( x x o ) 0 (5) u R, v R m u 0 uy o vx o (6)
Geometric Portrayal in DEA 3/3 Hyperplane divide the pace into two halfpace and i upporting when one of the halfpace contain the production poibility et Theorem 5.1 If a DMU (x o, y o ) i BCC-efficient, the normalized coefficient vector (v, u, u 0 ) of the upporting hyperplane to P at (x o, y o ) give an optimal olution of the BCC model and vice vera.
BCC Return to Scale 1/3 Theorem 5.2 Auming that (x o, y o ) i on the efficient frontier the following condition identify the ituation for RTS at thi point: (i) IRTS if and only if u 0* < 0 for all opt. ol. (ii) DRTS if and only if u 0* > 0 for all opt. ol. (iii) CRTS if and only if u 0* 0 in any opt. ol.
BCC Return to Scale 2/3 It can be complex to find all optimal olution. Thi can be avoided by eliminating the efficiency aumption (we can tudy point like D) Aume that a olution i available from the BCC model. Thi give the information needed to proect (x o, y o ) into a point on the efficient frontier: xˆ yˆ io ro θ x B y ro io + + r i, i 1,..., m, r 1,..., (7) (8)
BCC Return to Scale 3/3 Suppoe that u o* < 0 (from the 1t tage of BCC model). Then the dual problem of the BCC model can be replaced with (if : u o* > 0, then min) max t. u i 1 0 u m i 1 i 1 0 m m i v i i v xˆ x io i v xˆ io 0, v + + 1, i r 1 r 1 u u r 1 0, u r r r u y yˆ r r ro yˆ 0 u ro 0 u 0 u 0, 0, 0 1 1,..., n 0 (9) (10) (11) (12)
CCR Return to Scale 1/3 Theorem 5.3 Let (x o, y o ) be a point on the efficient frontier. Employing a CCR model in envelopment form to obtain an optimal olution (λ 1 *,, λ n *), RTS at thi point can be determined from the following condition: (i) IRTS if Σ (λ *) < 1 in any alternate optimum (ii) DRTS if Σ (λ *) > 1 in any alternate optimum (iii) CRTS if Σ (λ *) 1 in any alternate optimum
CCR Return to Scale 2/3 Thi doe not allow u to tudy the point that lie inide the production poibility et. To tudy thee point, let u eliminate the aumption that (x o, y o ) i on the efficient frontier We utilize the information from the firt tage olution of the dual CCR model
CCR Return to Scale 3/3 (17), ˆ 0 (16) ˆ 1 (15) ˆ ˆ (14) ˆ ˆ. (13) ˆ ˆ ˆ min 1 1 1 1 1 1 y y x x t n n o n o o n m i r r i + + + + λ λ λ λ θ ε λ Aume λ * > 1 (from CCR model). Then the CCR problem can be formulated into (if λ * < 1, then max):
Summary Return to cale refer to a technical property of production. It examine change in output ubequent to a proportional change in all input [2]. In the cae of imple output the RTS i achieved by calculating the elaticity In the cae of multiple output there are theorem to tudy the RTS at the point on efficient frontier (CCR and BCC model) Eliminating the aumption that technical efficiency i alway achieved give u tool to tudy the point that lie inide the production poibility et
Reference [1] NetMot [2] Wikipedia, 25.9.2007 [3] http://cepa.newchool.edu/het/eay/product/return.htm, 25.9.2007 All other information i from: Introduction to Data Envelopment Analyi, Cooper William W, 2nd Edition
Home Aignment 1/2
Home Aignment 2/2 Study the point E: With CCR θ o* 27/32 for either λ B* 9/4 or λ C* 9/8 while all other variable 0. Which condition in Theorem 5.3 i applicable at the point E? (Ue equation (13)-(17) to formulate an optimization problem (3 p.) and olve it (2 p.)) With BCC θ o* 7/8 for λ C* λ D* 1/4 while u o* u * v * 1/4. Which condition in Theorem 5.2 i applicable at the point E? (Ue equation (7)-(12) to formulate an optimization problem (3 p.) and olve it (2 p.))