INVESTMENT AND DYNAMIC DEA

Transcription

1 INVESTMENT AND DYNAMIC DEA PIERRE OUELLETTE Département des sciences économiques, Université du Québec à Montréal LI YAN * Département des sciences administratives, Université du Québec en Outaouais RePAd Working Paper No April 2005 * Corresponding autor: Prof. Li Yan, Département des sciences administratives, Université du Québec en Outaouais, Pavillon Lucien Brault, 101 rue Jean Bosco, Gatineau QC, Canada J8Y 3G5. li.yan@uqo.ca. Telepone: ext Fax: We tank Pierre Fortin, Robert Gagné and Pierre Lasserre for teir comments on a preliminary version of tis paper. Te usual disclaimer applies. Tis paper can be downloaded from RePAd.org: ttp://ideas.repec.org/p/pqs/wpaper/ tml

2 Abstract A dynamic version of Data Envelopment Analysis (DEA) is developed in te present paper. Our model introduces investment in traditional DEA and imposes intertemporal cost minimization. Adding an intertemporal adjustment constraint into te cost minimization problem, we derive te relation between te DEA variables of te cost function and tose of te primary production frontiers coefficients. Te augmented DEA model can be solved using standard linear programming. Tis dynamic framework enables computing te production frontiers, measuring te productive efficiencies and evaluating te potential economies all in te presence of adjustment costs. JEL Classification: D24, L23 Keywords: Adjustment cost; Data envelopment analysis; Efficiency; Multiple outputs/inputs; Quasi-fixed inputs

3 1 1. Introduction Te data envelopment analysis (DEA), pioneered by Carnes, Cooper, and Rodes (1978), is an alternative to te traditional parametric approac of quantitative analysis. It offers a robust tool in studying of production frontiers and evaluating te performance of decision-making units (DMU). DEA is an optimization metod based on matematical programming generalizing te Farrell (1957) single-input/single-output tecnical efficiency measure to te multiple-input/multiple-output case by constructing a relative efficiency score as te ratio of a single virtual output to a single virtual input. DEA as rapidly extended to returns to scale, dummy or categorical variables, discretionary and non-discretionary variables, incorporating value judgments, longitudinal analysis, weigt restrictions, stocastic DEA, non-parametric Malmquist indexes, tecnical cange in DEA and many oter features. But until now, most existing DEA models rarely take into account intertemporal adjustment costs. Consequently, te estimated frontiers may be biased, even strongly biased in presence of ig adjustment costs. To correctly calculate te production frontier, te adjustment restrictions and costs from period to period must be taken into account. Nemoto and Goto (2003) proposed a DEA model wit adjustment costs incorporated. Te autors implicitly impose te following assumption on te adjustment: wen investments occur, te fictive best DMU s adjustment cost is often strictly iger tan tat of te considered DMU. Because of tis, te efficiency of te considered DMU could be overestimated. Wen disinvestments occur, te fictive dominant DMU as an adjustment cost lower tan tat of te considered DMU, wic is insufficient to be best DMU and te efficiency score so obtained could be anyting. Briefly, biased results could rise from te autors irrelevant assumption. Te present paper focuses on introducing a new version of dynamic DEA model wit weaker

4 2 and more realistic assumptions. First, we define a temporary production function. By temporary, we mean te production frontier existing at a given time. Te term frontier refers to an optimal capacity of transformation of te inputs into outputs. Several definitions are possible. We can define te frontier by te maximal quantity tat a production unit can produce from te given quantities of inputs. We can also define te same frontier by te minimal quantity of inputs necessary to produce a given vector of outputs. Tere are still oter alternatives. For example, we may be satisfied to optimize on a subset of inputs. Banker and Morey (1986) used tis strategy in teir efficiency analysis for exogenously fixed inputs and outputs. Wen completely or partially fixed or out of te manager s control, an input is said to be quasi-fixed or nondiscretionary. Te quasi-fixed inputs, suc as te size of buildings or certain large and bulky pieces of equipment are fixed in te sort run, but tey can vary in te long run following depreciation and/or new acquisitions. Tis aspect of te quasi-fixed inputs sould be modeled. Of course, investment is an input (possibly a vector), wic must be taken into account in te definition of tecnology. Te impact of investment on te production level is known as adjustment costs. Te temporary (or one-period) production function depends on bot investment and variable inputs (under te immediate control of te manager) and quasi-fixed inputs (te capital). Secondly, we define a variable cost function. Hence, we ave two representations of tecnology: te temporary production frontier and te variable cost frontier. Interesting links can be establised between te two. Except for te introduction of investment and adjustment costs, our DEA model is simply a modification of te model of Banker and Morey (1986). However, tis modification is not minor because it includes te analysis of te impact of te current decisions on te firm s future prospects. To invest implies tat future outputs will

5 3 increase and future costs and profits will be affected. Unless we were ready to deal wit a sortsigted DMU, te future impacts ougt to be built-in in te current decision-making. Te coice of intertemporal decision rule is itself an interesting subject of researc especially in te public sector. Adding an intertemporal adjustment constraint into te firm s discounted cumulative cost minimization problem, we derive te relation between te DEA variables of te cost function and tose of te temporary production function. Finally, we also sow ow to recover te returns to scale and te implicit prices of bot te capital and te investment. More importantly, we tackle te question of te measurement of te efficiency; bot a novelty and a difficult task compared to te static case. 2. Temporary Production Frontier, Adjustment Cost and Cost of Tecnical Inefficiencies Let F(y, x, k, i) = f(y)-g(x, k, i) = 0 be te production frontier. In order to determine it, te following minimization problem must be solved for eac DMU ( = 1,, H), Min θ s.t. f(y ) g(θ x, k, i ) θ > 0, (P) were y (y 1,, y M )' is te M-vector of outputs; x (x 1,, x N )' is te N-vector of variable inputs; k (k 1,, k L )' is te L-vector of quasi-fixed inputs; i (i 1,, i L )' is te corresponding investment vector. We assume tat adjustment costs are present, so g/ i < 0 for eac i 0. θ, a scalar, is a measure of efficiency, and = 1,, H is a DMU index. Te constraint inequality f(y ) g(θ x, k, i ) describes a free disposal ull (FDH) production possibility set. Problem (P) identifies weter te level of variable inputs can be reduced given te available 3 For clarity, we omit te time indexes for te matrices and rigt-and side elements of te following linear program.

6 4 level of quasi-fixed factors and investment witin te same production possibility space. Te production frontier involved in (P) can be approximated by te combination of te yperfacets linking te outer DMUs. For simplicity, we assume symmetry between te investment cost and disinvestments cost so tat te investment can be measured wit its absolute value. Te linear approximation of (P) can be computed using H linear programs as follows: 3 TE = Min θ (TE-primal) θ,λ s.t. Y 'λ y (α y ) I 'λ i (α i ) X'λ θ x (α x ) K 'λ k (α k ) 1'λ = 1 (α c ) λ 0, were TE stands for tecnical efficiency, te α s are corresponding associated variables and y 11 y 12 y 1M y 1 ' y 21 y 22 y 2M y 2 ' Y= = y H1 y H2 y HM y H ' is te output matrix of te H DMUs of dimension H M ; similarly, X is te variable input matrix of dimension H N x 11 x 12 x 1N x 1 ' x 21 x 22 x 2N x 2 ' X= = ; x H1 x H2 x HN x H ' K is te quasi-fixed inputs matrix of dimension H L k 11 k 12 k 1L k 1 ' k 21 k 22 k 2L k 2 ' K= = ; k H1 k H2 k HL k H ' I is te H L matrix of te absolute value of investment-disinvestment i 11 i 12 i 1L i 1 ' i 21 i 22 i 2L i 2 ' I = = ; i H1 i H2 i HL i H ' and 1 is a vector of ones wit appropriate dimension. If not mentioned, te dimension of vector

7 5 1 is equal to te number of DMU. Te α's are te Lagrange multipliers associated wit te corresponding constraints. All te DMUs on te final frontiers will be caracterized by teir optimal value of θ =1. Te first four groups of restrictions in TE-primal constitute te DEA form of te FDH production possibility set. Te fift one is a convexity restriction. Te Lagrange function of te problem TE-primal can be written as L TE = θ +α y ' (y -Y 'λ)+α i ' ( i - I 'λ)-α x ' (θ x -X 'λ)-α k ' (k -K 'λ)+α c (1' λ -1) = (α y 'y +α i ' i -α k 'k -α c )+θ (1-α x 'x )-[α y 'Y '+α i ' I '-α x 'X '-α k 'K '-α c 1']λ. At tis stage, it is wortwile to igligt our assumption about te symmetry between investment and disinvestment costs. Witout tis ypotesis, te constraint in (P) takes its original form of two inequalities, I + 'λ i + and I - 'λ i -, were i + R L + is te investment vector, i - R L + is te absolute value of te disinvestment vector of te DMU, R L + is te subset of te vectors wit non-negative components in te L-dimension Euclidian space, and I + and I - are te matrices of, respectively, investment and disinvestment of all DMU s. So, i =i + - i - and I=I + -I -, i =i ++i - and I =I + +I -. Witout te symmetry assumption, te Lagrange function of (P) is L TE =θ +α y ' (y -Y 'λ)+α + i '(i + -I + 'λ)+α - i ' (i - -I - 'λ)-α x ' (θ x -X 'λ)-α k ' (k -K 'λ)+α c (1'λ-1). Te symmetry between te investment cost and disinvestment cost means tat g/ i + = g/ i - for te marginal adjustment costs measured in terms of outputs. Formally, g(i 0 )/ i = - g(-i 0 )/ i for any i 0. Tis corresponds to te equality α i + = α i - wenever we use TE-primal as te linear approximate of (P). Let α i = α i + = α i -. Ten L TE =θ +α y ' (y -Y 'λ)+α i ' (i + +i --I + 'λ -I - 'λ)-α x ' (θ x -X 'λ)-α k ' (k -K 'λ)+α c (1'λ-1) =θ +α y ' (y -Y 'λ)+α i ' ( i - I 'λ)-α x ' (θ x -X 'λ)-α k ' (k -K 'λ)+α c (1'λ-1) =L TE. Te assumption of symmetry between te investment and disinvestment costs simplifies te writings of our linear approximation model, but it is not essential. One may re-decompose i and

8 6 I into investments and disinvestments wenever needed. Te dual problem of (TE-primal) is Max α (α y 'y + α i ' i -α k 'k - α c ) s.t. α x 'x = 1 (θ ) Yα y + I α i Xα x Kα k α c 1 0 (λ) α y, α i, α x, α k 0 α c free. (TE-dual) In standard notations, te tecnology sould be written as F(y, x, k, i) = 0, especially in te absence of symmetry of adjustment costs. However, for te linear approximation of te tecnology to be comparable wit our DEA model wit symmetric adjustment costs, we use F(y, x, k, i ) = 0 rater tan F(y, x, k, i) = 0. At a frontier point O = (y 0, x 0, i 0, k 0 ), te first-order Taylor expansion is F(y, x, k, i ) F(y 0, x 0, k 0, i 0 )+F y (y 0, x 0, k 0, i 0 ) (y - y 0 ) +F x (y 0, x 0, k 0, i 0 ) (x - x 0 )+F k (y 0, x 0, k 0, i 0 ) (k -k 0 ) + F i (y 0, x 0, k 0, i 0 ) ( i - i 0 ), were te partial derivatives F y, F x, F k, and F i are row vectors; (y, x, k, i ) is an arbitrary point in an appropriate neigborood of O. Of course, te smootness of F(y, x, k, i ) is required trougout te paper, as it is usually assumed in te literature. On te oter and, te Lagrange function of te problem (P) is, for DMU, Note tat L TE L, i.e. L = θ + µf(y,θ x, k, i ) θ +µ[f(y 0, x 0, i 0, k 0 )+F y (y -y 0 )+F x (θ x -x 0 )+F k (k -k 0 )+F i ( i - i 0 )]. θ + α c (1'λ -1)+α y ' (y -Y 'λ)-α x ' (θ x -X'λ)-α k ' (k - K 'λ)+α i ' ( i - I 'λ) θ +µf(y 0, x 0, i 0, k 0 )+µf y (y -y 0 ) +µf x (θ x -x 0 )+µf k (k -k 0 )+µf i ( i - i 0 ). (1) To ensure a good approximation in (1), we need to carefully coose a frontier point O for te given DMU. For example, O may be taken as (y,ψ x, i, k ) wit 0< ψ θ, for any given DMU. Let us igligt tis point wit te elp of Figure 1.

9 7 Te tangency yperplan at O and te small facet are approximately parallel. Te gradients of te facet at te fictive point DMU λ and tat of te teoretical frontier surface at O sould be considered equal. If DMU λ is an extreme point, te α s are not unique. However, various values of α s do not greatly differ wen te number of te observed DMU s are sufficiently great, because we ave assumed te smootness of F. Furtermore, under te assumption of second order differentiability of te function F, one can evaluate te partial derivatives of F at O by tose correspondingly at DMU λ. Te error incurred by tis approximation is at most (θ -ψ) x, were is te Euclidian norm. Hence, (1) yields: α y =µf 0 y', α i =µf 0 i ', α x = -µf 0 (θ x)', α k = -µf 0 k', if we identify = wit. On te oter and, te first-order conditions of (P) give L / θ = 0, i.e., 1 + µf (θ x) x = 0 or µ = -1/(F (θ x) x ). Terefore, te links between te tecnology and te dual variables of tecnical efficiency problem are α y = -F y ' /(F (θ x) x ),α i = -F i ' /(F (θ x) x ), α x = F (θ x) '/(F (θ x) x ), α k = F k '/(F (θ x) x ). Furter, from F (θx) ' = α x (F (θ x) x ), it follows tat F x 1 N =(1 N 'α x )(F x x ), wic is valid at any frontier point 4. We can now rewrite tese relations between te partial derivatives and te α s. 4 1 N is a N-vector of ones; N being te number of variable inputs.

10 8 F y /(F x 1 N ) = - α y '/(1 N 'α x ), F i /(F x 1 N ) = - α i '/(1 N 'α x ), F x /(F x 1 N ) = α x '/(1 N 'α x ), (2) F k /(F x 1 N ) = α k '/(1 N 'α x ). Te measure of tecnical efficiency can be expressed in terms of variable input levels. Te cost of tecnical inefficiency is measured by (1-θ )C obs = (1-θ ) w ' x, were C obs is te observed variable cost of DMU and w n is te market price of te nt variable input. Te tecnically efficient cost can be calculated as TE C obs=θ C obs. 3. Te Variable Cost Function in te Presence of Adjustment Cost and te Cost of Allocative Inefficiencies Te cost of tecnical inefficiency results from a coice of inputs tat is below te production frontier. Oter costs migt result because all coices on te production frontier are not equivalent. Given te price vector w of variable input factors x, some points on te frontier will yield a minimal production variable cost. To determine tese points, te following variable cost minimization problem must be solved: C min = Min x { w x f(y ) g(x, k, i )} (C) It can be approximated wit te following linear program: C AE C (w, k, y, i ) (AE-primal) =Min µ w 'x E xe, s.t. Y 'µ y (β y ) I 'µ i (β i ) X 'µ x (β x ) K 'µ k (β k ) x E -X'µ = 0 (β E ) 1'µ = 1 (β c ) µ 0

11 9 were x E is te solution to te variable cost minimization problem. Te cost of allocation inefficiencies is te difference between te efficient cost calculated in te previous section and te minimal cost, tat is (θ C obs - C min). In te traditional way, te allocative inefficiency (AE ) expressed as a percentage of te efficient cost (C obs) is AE =C min /(θ C obs). Added togeter, te allocation and tecnical inefficiencies yield te total cost surplus. Similarly, te product of te allocation and tecnical inefficiencies yields te global inefficiency as a percentage of te observed cost: OE =TE AE =θ AE =C min/c. Te Lagrange function of (AE-primal) is L AE = w 'x E +β y ' (y -Y'µ)+β i ' ( i - I 'µ)-β x ' (x -X'µ)-β k ' (k -K µ) -β E ' (x E -X'µ)+β c (1'µ -1) = [β y 'y + β i ' i -β x 'x -β k 'k -β c ] -[β y 'Y'+β i ' I '-(β x '+β E ' )X'-β k 'K'- β c 1']µ + (w '-β E ')x E. Te dual problem of (AE-primal) is written as Max β [β y 'y +β i ' i -β x 'x -β k 'k -β c ] s.t. w = β E (x E ) Yβ y + I β i -X(β x +β E ) -Kβ k - β c 1 0 (µ) β y, β i, β x, β k 0 β c free. Te Envelope Teorem sows tat at te optimum, (AE-dual) C AE/ w = L AE / w = x E ' C AE/ y = L AE / y = β y ' C AE/ i = L AE / i = β i ' (3) C AE/ k = L AE / k = - β k ' At te optimal level of te problem (C), C min = w 'x + φ F(y, x, k, i ), were φ is a Lagrange multiplier. By Envelope Teorem, C min/ w = x'. From first-order conditions, w ' = -φ F/ x. If te sum of te variable input prices is normalized to 1, 4 ten 1= w '1 N = -φf x 1 N, tus φ = -1/F x 1 N. Again by using Envelope Teorem on optimized C min, we ave tat 4 Naturally, it would be possible to impose oter types of normalization. For example, we could impose tat te nt price be equal to 1. Tis implies an asymmetry in te treatment of inputs. For tat reason, we preferred te above normalization. Neverteless, it sould be clear tat te results must be modified accordingly to te normalization rule.

12 10 C min/ y = φ F/ y = -F y / F x 1 N; C min/ i = φ F/ i = -F i /F x 1 N; (4) C min/ k = φ F/ k = -F k / F x 1 N. Under approximation sense, C AE = C min. We can ave, from (6) and (7), tat F y / F x 1 N = - β y ' F i / F x 1 N = - β i ' (5) F x / F x 1 N = w ' F k / F x 1 N = β k ' Tis is an alternative way to express te first-order derivatives of te production function. From (2) and (5), an interesting relationsip between te dual variables of tecnical efficiency problem and allocation efficiency problem can be drawn under te normalization, β y = α y /1 N 'α x, β i = α i /1 N 'α x, β E = w = α x /1 N 'α x, β k = α k /1'α x. (6) 4. Intertemporal Decisions Te investment decisions are te result of an optimization over several periods. For any specific firm (for simplicity, index is omitted in tis section), T Min { i τ } { τ= t ρ τ [C τ (w τ, k τ, i τ, q τ )+ q τ 'i τ ] } s.t. k τ+1 = (E- )k τ + i τ, τ = t, t +1,, T (7) were ρ is te discount factor, E is te (L L)-identity matrix, is te (L L)- rate-of-depreciation matrix (diagonal) wic is supposed constant and defined by ll = δ l and lv = 0, for v l, and q τ is te investment-price vector. Serving as te control instrument, investment i τ may be optimally decided to be positive or negative or null. Unlike te previous sections of tis capter, it is better to let te investments take teir algebraic values in te above intertemporal decision model, instead of absolute values. Te Hamilton function of (7) can be written as H=ρ τ [C τ (w τ, k τ, i τ, q τ )+q τ i τ ] - ϕ τ+1 ' (i τ - k τ ),

13 11 were te Hamilton multiplier vector (of dimension L), ϕ τ+1, is already actualized. At optimum, te following necessary conditions will be verified. Te terminal conditions are ϕ T+1 = 0 (L 1). ρ τ [ C τ / i τ'+q τ ] = ϕ τ+1, for τ = t, t +1,, T (8) ϕ τ+2 -ϕ τ+1 =ρ τ+1 C τ+1 / k τ+1 '+ ϕ τ+2, for τ = t, t +1,, T-1. (9) k τ+1 - k τ = i + τ - k τ, for τ = t, t +1,, T. Consider now te case of τ = t. From (9), or, wit te delay-operator defined by Lz t = z t-1, (E- )ϕ t+2 - ϕ t+1 = ρ t+1 C t+1 / k t+1 ', [(E- )L -1 - E]ϕ t+1 = ρ t+1 C t+1 / k t+1 '. Under common assumption, te norm of (E- )L -1 <1. Tis allows us to ave te following expansion according to te basic spectrum teorem of linear operators, ϕ t+1 = - s=0 (E- ) s L -s (ρ t+1 C t+1 / k t+1 '). (10) (8) and (10) lead to C t / i t ' = -(1/ρ t ) [ s=0 (E- ) s (ρ t+s+1 C t+s+1 / k t+s+1 ')] q t. Denote 1/(1+r) s+1 = ρ t+s+1 /ρ t. We get C t / i t '= -1/(1 + r){ s=0 [(E- )/(1 + r)] s ( C t+s+1 / k t+s+1 ')} q t. (11) (11) plays te key role in te links of decisions from period to period. It sows ow te investment prices, te capital sadow prices and te adjustment sadow prices are organically related. Unfortunately it seems tat, in general, even if te future sadow prices were available, te sum in (11) migt be too difficult to be calculated witout any furter assumption. We ten make te following simplification assumption. Assumption 1: C t+s+1 / k t+s+1 C t / k t = -b kt ', for all s.

14 12 Under tis assumption, (11) can be rewritten as C t / i t ' = 1/(1 + r){ s=0 [(E - )/(1 + r)] s b kt } q t For te lt component of C t / i t, we get = 1/(1+r){E-[(E- )/(1 +r)]} -1 b kt q t = (re + ) -1 b kt q t C τ / i lτ ' = b kl,t /(r + δ l ) q lt. (12) Tis is just wat we ave sougt: te intertemporal adjustment restriction describing te relationsip between te capital s sadow prices and its marginal adjustment costs. Until now, i lτ takes its algebraic value in te intertemporal decision model. To incorporate tis dynamic relationsip into DEA models were we use te absolute value of te investments, we must find te partial derivates of te variable cost function w. r. t. i t. C t / i lt as te same sign wit i lt for every l =1, 2,..., L. Actually according to (12), te firm invests in lt quasi-fixed input (i lt >0) wen b kl,t /(r +δ l )-q lt >0; it disinvests in lt quasi-fixed input (i lt <0) wen b kl,t /(r +δ l )-q lt <0 and it does not cange lt quasi-fixed input (i lt =0) wen b kl,t /(r +δ l ) - q lt =0. Ten it is always true tat for eac l = 1, 2,, L, tat is, C t / i lt ' = b kl,t /(r +δ l )-q lt, b k1, t /(r +δ 1 )-q 1t b k2, t /(r +δ 2 )-q 2t C t / i t '=.... (13) b kl, t /(r +δ L )-q Lt 5. Dynamic DEA Model Now it is possible to incorporate te intertemporal adjustment restrictions we ave obtained in (13) on our DEA models (TE-primal or dual) and (AE-primal or dual). Witout tese intertemporal restrictions, te decision maker can independently make te production plan for

15 13 eac period. Te only factor tat influences te current investment quantity is te current purcase price of te quasi-fixed inputs. But wit te restrictions (16), e sould take te impacts of future capitals sadow prices into account, wic affect te coice of current investment via te marginal adjustment costs Tecnical Efficiency Under Assumption 1: C t+s+1 / k t+s+1 ' -b kt = β kt. In terms of DEA variables, β k(t+s) β kt, for all s 0. (14) Anoter indirect impact of tis assumption comes from (13) and (14) and can be written as follows. For eac l, l =1, 2,, L, and s, s =0,1, 2, β il t+s = C t+s / i lt+s '= (r+δ l ) -1 b klt q lt. (15) Assumption 1 also as impacts on α s. Tis may be seen from (6) and (15). In fact, we can easily sow, using te normalization 1 N 'w = 1, tat And for eac l, l=1, 2,, L, α k(t+s) /1 N 'α x(t+s) = β k(t+s) = β kt = α kt /1 N 'α xt α k(t+s) = (1 N 'α x(t+s) / 1 N 'α xt ) α kt, for all s 0. (16) α ilt /1 N 'α xt =β ilt = (r+δ l ) -1 b klt q lt α ilt = 1 N 'α xt(r+δ l ) -1 b klt -1 N 'α xtq lt = (r+δ l ) -1 α klt -1 N 'α xtq lt. (17) It is convenient to write (17) in vector form in order to rewrite our DEA models. We take off te absolute by using te sign function. Tus, α ilt =[(r+δ l ) -1 α klt -1 N 'α xtq lt ] sign[(r+δ l ) -1 α klt -1 N 'α xtq lt ]. were α it =Z t [(re+ ) -1 α kt -1 N 'α xtq t ], (18) sign(i 1t ) Z t =. sign(i Lt ) since sign((r+δ l ) -1 α klt -1 N 'α xtq lt ) = sign(i lt ) for every l =1, 2,..., L. We ave i t = Z t i t. From (17)

16 14 and (18), te lt component of α i (t+s) is written as α il (t+s) = (r+δ l ) -1 (1 N 'α x (t+s) / 1 N 'α xt )α klt -1 N 'α x q (t+s), for all s 0. Te TE-dual problem in section I can be written in matrix form as s.t. y TE =Max[α y ' α i ' α x ' α k '] i (TE-dual) α 0 -k α y 0' 0' x ' 0' α i 1 (θ ) Y I -X -K α x. (λ ) α k 0 Substituting te above restriction in tis problem we get, for te period t and for eac DMU, s.t. y t TE = Max [α yt ' α xt ' α kt '] -q t'1 N Z t' i t (TE-dual restricted ) [(r t E+ ) -1 Z t' i t -k 0' x t' 0' α yt 1 (θ t) α xt. Y t -( I t Z tq t1 N '+X t ) [ I t Z t (r t E+ ) -1 -K t ] α kt 0 (λ t) We can now rewrite te primal problem as follows: s.t. θ t TE = Min [1 0'] (TE-primal restricted ) θ t, λ t λ t 0 Y t ' y t x t -( I t Z tq t1 N '+X t ) ' θ t -1 N q t'z t' i t 0 [ I t Z t(r t E+ ) -1 -K t ] ' λ t [(r t E + ) -1 Z t' i t -k t] 0 1' Allocative Efficiency Similar to wat we ave done wit TE model, we can now incorporate te intertemporal constraints into te AE problem. Te direct way to do tis is also to start wit rewriting of AE-dual problem, as it is te dual variables tat carry te explicit intertemporal information. Note tat from (6) and (18), β it =Z t [(r t E+ ) -1 β kt - q t]. Put tis relation into te AE-dual model:

17 15 y t -x t C AE = max[β yt ' β xt ' β kt ' β Et ' β 0t ] [(r t E+ ) -1 Z t i t -k t ] - q t'z t i t. (AE-dual restricted ) β 0 1 s.t. β yt β xt w t (x E t ) 0' 0' 0' E 0' β kt. Y t -X t [ I t Z t(r t E+ ) -1 -K t ] -X t 1 β Et I t Z tq t (λ t) β 0t Hence te primal problem becomes x Et C AE = min [w t' ( I t Z tq t) ' ] -q t'z t i t. (AE-primal restricted ) x E, λ λ t s.t. 0 Y t ' y t 0 -X t ' x E t -x t 0 [ I t Z t (r t E+ ) -1 -K t ] ' [(r t E+ ) -1 Z t i t -k t]. E -X t ' λ t 0 0 1' 1 0 E 0 6. Results 6.1. Final Version of te Dynamic DEA Model For te sake of clarity, we add from now on te DMU s full index and t for all te DEA variables, as well as for matrix Z and vector q. Now let us present te final form of te tecnical and allocative models in bot primal and dual settings. In tese models, te intertemporal constraints are introduced. Te tecnical efficiency primal problem is TE = Min θ, λ θt s.t. Y t 'λ t y t (α yt) x tθ t-( I t Z t q t1 N '+X t ) 'λ t -1 N q t'z t' i t (α xt) (19) G t ' λ t g t (α kt) (20) 1'λ t 1 λ t 0

18 16 were Z t' i t = i t and I t Z t is te -oriented investment matrix, tat is, te sign of i lt, i.e., sign(i lt) is imposed upon te lt quasi-fixed input of every DMU, l=1, 2,, L; G t=[ I t Z t(r t E+ ) -1 -K t ] and g t=[(r t E+ ) -1 Z t i t -k t]. Te TE-dual model is TE = max{α yt'y t - α xt'1 N q t'z t' i t + α kt'g t} s.t. x t'α xt = 1 (θ t) Y t α yt ( I t Z t q t1 N ' +X t )α xt +G tα kt 0. (λ t) Te AE-primal problem can now be rewritten as C AE +q t'z t i t =min[w t'x E t + q t'z t I t ' λ t ]. (AE-primal restricted ) x E,λ s.t. Y t ' λ t y t -X t ' λ t -x t G t' λ t g t X t ' λ t = x E t 1' λ t 1 λ t 0 It is remarkable tat te objective function as been modified. Te problem is now to minimize te total cost for eac period, including te investment expenditures. Its dual model is C AE + q t ' i t = max{β yt'y t - β xt'x t + β kt'g t + β 0t } s.t. Y t β yt - X t β xt + G t β kt + 1β 0 X t w t + I t Z t q t (λ t) All te investments take teir absolute values Some Furter Expositions for te Dynamic Models Now bot of te tecnical efficiency and te allocative efficiency models take account of cross period constraints. How do tese constraints make a firm s intertemporal beaviors different from tose in static cases? Let s focus firstly on (19). It can be arranged as

19 17 x tθ t -X t 'λ t 1 N q t'z t( I t ' λ t - i t ). (21) Te rigt-and side of te inequality (21) sould be positive for eac of its elements according to te original constraints of te problem (TE-primal). Te investment implies an adjustment cost tat can be defined in terms of quantities of variable factors becoming non productive. For an efficient DMU tat does not invest, it must be true tat x t = X t 'λ t and I t ' λ t = i t = 0. For an efficient firm tat does invest, te quantities of variable inputs sould be greater tan tose tat don t in order to compensate te adjustment cost. In tis case, we ave, from (19), tat x t = X t 'λ t + 1 N q t' ( I t ' λ t - i t) > X t 'λ t and te term 1 N q t' ( I t ' λ t - i t) corresponds to te adjustment cost. Wenever a firm s performance is not efficient, te term x t sould be corrected by θ t x t. Tis explains wy (19) plays te role of variable inputs constraints in te dynamic DEA model for tecnical efficiency. Anoter remarkable difference between our dynamic models and static ones is (20). Te original form of tis constraint is [ I t Z t (r t E+ ) -1 -K t ]'λ t [(r t E+ ) -1 Z t i t - k t ]. To simplify our interpretations, let s also consider te case of positive investments and tus Z=E. Take out a single inequality from it in order to igligt its economic sense, for example, te constraint for te lt quasi-fixed input, {[I t (r t E+ ) -1 -K t ]' λ t } ll [(r t +δ l ) -1 i lt -k lt], (22) Tis is equivalent to [k lt -(r t +δ l ) -1 i lt] {[ K t -I t (r t E+ ) -1 ]' λ t } ll. Multiplying bot sides wit (1-δ l ), we can write te left and side in form of te following series. (1-δ l )k lt -(1-δ l )(r+δ l ) -1 i lt =(1-δ l )k lt τ= 1 [(1-δ l )/(1+r t )] τ i lt (23) =k l(t+1) i lt τ =1 [(1-δ l )/(1+r t )] τ i lt (24)

20 18 =k l(t+1) =0 τ [(1-δ l )/(1+r t )] τ i lt. Te passageway from (23) to (24) is due to te capital movement law. Ten, we may put (22) into te value form by multiplying te investment price q lt. q ltk l(t+1) =1 τ [(1-δ l )/(1+r t )] τ (q lti lt) q lt(1-δ l ){[ K t -I t (r t E+ ) -1 ]'λ t } ll. (25) At te beginning of te initial time τ=0 wic corresponding to t, te DMU invests i lt. Te discounted residual value of i lt for te future is equal to te sum of te series. Te difference at te left and side of (25) represents te productive part of te lt capital stock during te period t, and terefore a cost of te current production. Being one of te constraints of te cost-minimization problem, (25) means tat for DMU, te productive part of te capital stock is, at least, as great as tat of te best DMUs, in oter words, te DMU s production plan is feasible. Finally, te objective of te allocative efficiency problem is also augmented in te presence of quasi-fixed factors. Now it is to seek te minimum of total cost wile te intertemporal adjustment expenditures are considered. Te unique difference between te first two static models and te dynamic ones we ave studied in tis capter is tat, in te dynamic model, te multipliers β and α are tied by te intertemporal restrictions. 7. A Comparison wit Nemoto and Goto (1999 and 2003) Model Recall te dynamic DEA model tat Nemoto and Goto (1999 and 2003) proposed. Our model bears some resemblances to teirs, but we argue tat our model is more general.

21 19 First, instead of working on te variable of investment i t, Nemoto and Goto (2003) imposes tat, among oters tat are te same as ours, for eac quasi-fixed input, (for simplicity, we assume in tis section tat tere is only one quasi-fixed input): K t 'λ t k t (26) K t+1 'λ t k t+1, (27) were K t is te column vector of te considered quasi-fixed input of te different DMU s and k t is te considered quasi-fixed input of te considered DMU. Let δ be te depreciation rate for tis quasi-fixed factor. It follows from (26) and (27) tat K t+1 'λ t -(1-δ)K t 'λ t k t+1 -(1-δ)k t. Tis means tat restrictions (26) and (27) imposed in Nemoto and Goto (2003) imply our constraints wen te investments are non-negative, tat is, K t 'λ t k t (28) I t 'λ t i t. (29) were I t is te column vector of te investment in te considered quasi-fixed input of te different DMU s and i t is te investment in te quasi-fixed input of te considered DMU. Note tat (28) and (29) do not imply (27). In fact, (28) implies tat tere exists a non negative γ t suc tat (1-δ)K t ' λ t +γ t = (1-δ)k t. Tis equation and (29) yield K t+1 ' λ t +γ t = [(1-δ)K t + I t ]'λ t +γ t (1-δ)k t +i t = k t+1, but not K t+1 'λ t k t+1. In oter words, our restrictions on te capital movement, (28) and (29), are weaker tan tose of Nemoto-Goto s and tus te feasible set of Nemoto-Goto s model is a subset of ours. Consequently, te cost function determined in te present paper is smaller tan tat determined in te way sown in Nemoto and Goto (2003). In general, te production tecnology determined in tis paper is more productive tan tat drawn from Nemoto and Goto (2003) and ence closer to te teoretical one wen te investment is non-negative. Tus, one can conclude tat te model of Nemoto and Goto (2003) leads to an overestimated efficiency

22 20 score in case of non-negative investment. Wen investment i t =[k t+1 -(1-δ)k t ] is negative for eac DMU, te reasonable constraints on te investment sould be -I t 'λ t -i t since te adjustment cost increases wit te disinvestment quantity and te fictive DMU suffers from an adjustment cost at least as tat of te considered DMU. Tis inequality is just our restriction. It implies, togeter wit (26), K t+1 'λ t k t+1, and not (27) (Nemoto-Goto restriction). Consequently, te feasible set of te model of Nemoto and Goto (2003) is incorrect and so for te efficiency score. A second contribution is wort noting. Te model of Nemoto and Goto (2003) includes, for eac DMU, one constraint for eac input and eac output, and tat for every period. For example, in case of one variable input/one quasi-fixed input/one output, if tere are T periods, te model includes 4*T constraints, tat is T constraints for te variable input, 2*T constraints for te quasi-fixed inputs, and T constraints for te output (plus one additional constraint if variable returns to scale are imposed). Obviously, tis may represent a numerical burden. Furtermore, tere is no reason a priori to suppose tat te time orizon for te firm is equal to te number of period of observations. Also, te interpretation of te results becomes quite troublesome. Tis model, if we are ready to accept it as suc, is meaningful only for te first period. Period 2 to T restrictions are only expected (even if we assume perfect foresigt). Tis model as a meaning only if we are ready to assume tat wat as been planned in period 1 for periods 2 to T is realized as suc in reality. In oter words, te acquisition of information after period 1 will not cange te decision for periods 2 to T taken in period 1. Every decision is taken in period 1. Certainly, tis is a very restrictive beavioral assumption. In our model, te decisions for period t are taken in period t, and new information is explicitly introduced.

23 21 8. Generalization of Efficiency Measurements Recall te traditional measures of efficiencies TE, AE and OE previously defined. Te measure of tecnical efficiency of DMU in a given year is actually te ratio of te cost of a tecnically efficient variable input bundle over observed variable cost. TE t = θ t. Te measure of allocative efficiency of DMU is AE t=c min-t /(TE t C obs-t). And, te measure of global efficiency is defined as te product OE t = TE t AE t = C min-t /C obs-t. Te performance of a given DMU is efficient if and only if OE = 1. We ave seen tat te dynamic and te static allocative models were different: te investment expenditures are taken into te objective function and tus te problem is actually to minimize te total costs of current production, not only te variable costs. Tis is one of te significant impacts of te intertemporal restrictions. In fact, it is impossible to consider te variable cost independently on and separately from te investment costs. To capture te nature of dynamic context, we sould appropriately redefine tose efficiency measures. At tis point, te objective function of te dynamic allocative efficiency model AE-primal is euristic. From it, we define now (for DMU at time t): AE d t = Optimal total cost Tecnically efficient total cost w t'x E t + q t'z t I t 'λ t =. TE t C obs-t+ q t'z t i t Te relevant tecnical efficiency measure is defined by Tecnically efficient total cost TE d t = Observed total cost TE t C obs-t + q t'z t i t =. C obs-t+ q t'z t i t

24 22 Te product TE d t AE d t defines te global efficiency measure: OE d t = TE d t AE d t Optimal total cost = Observed total cost w t'x E t + q t'z t I t 'λ t =, C obs-t+ q t'z t i t were C obs-t+ q t'z t i t is te observed total cost. Te tree new efficiency measures degenerate to static ones if no investment occurs. Moreover, it is not difficult to see, from te definitions, te following interesting relations between te static measures and dynamic measures. TE d t -1 = (TE t -1) [C obs-t /(C obs-t+ q t' i t )], = (TE t -1) (Sare of te observed variable cost in te observed total cost), AE d t -1 = (AE t -1) [(TE t C obs-t) / (TE t C obs-t+ q t' i t )] + q t' (Z t I t ' λ t - i t ) / (TE t C obs-t+ q t' i t ), = [(AE t -1) (Sare of te tecnically efficient variable cost in te tecnically efficient total cost)] + (Sare of inefficient investment cost in te tecnically efficient total cost), OE d t -1= (OE t -1) [C obs-t / (C obs-t+q t'i t )] + q t'(z t I t 'λ t - i t ) / (C obs-t+q t'i t ), = [(OE t -1) (Sare of te observed variable cost in te observed total cost)] + (Sare of inefficient investment cost in te observed total cost). It must be noted tat if massive disinvestments occur, te great value of sold or consumed equipment, tat enters te cost function in negative values, becomes a part of te DMU s revenue and reduces significantly te value of total cost. As a consequence, te new measures of efficiency TE d t, AE d t and OE d t may be greater tan one. Moreover, it is possible tat tese new efficiency measures take negative values. Tis penomenon is impossible in static settings. Te negative scores and tose greater tan one contradict somewat te usual concept: an efficiency score sould preferably be between zero and one. Anyway, it is matematically easy to define an efficiency score in order to conform better to te usual custom. Before to do tis, we must answer te following questions.

25 23 Wit dynamic settings, wat is te criterion for a given DMU to be efficient, tat is, to be on te frontier? Is it still te case tat a DMU is efficient if and only if OE d =1? An efficient DMU must ave a score equal to one, neiter less nor greater tan one. Tis is because any score OE d different from one means some kind of deviation from optimal performance. As te efficiency score may deviate one from te left as well as from te rigt, our assumption about te symmetry between investment cost and disinvestment cost implies tat te scores 1-ε and 1+ε means te same inefficiency level, for any ε > 0. Based on tis observation, we introduce te following index of global efficiency: 1 GE t = 1+ 1-OE d t. Obviously, te global efficiency score GE is always a positive number between zero and one; a greater GE corresponds a more effective DMU; and a DMU is efficient if and only if its GE is equal to one. Te relation between GE and OE d is illustrated in te following figure. If tere is neiter investment nor disinvestment, OE d does not differ from OE. Tis corresponds to te part of te curve GE above te interval [0, 1]. Te particular case of OE = 0 corresponds to GE =1/2, instead of zero. GE offers only an alternative coice of te efficiency measurement to conform te custom if tis is necessary. GE 1 1/2 0 1 OE d Figure 2 Relationsip between GE and OE d

26 24 9. Tecnology Measurements 9.1. Implicit Prices Te implicit prices of quasi-fixed inputs are just te negative value of C t/ k t, i.e., β kt = - C t/ k t', were β kt is evaluated at te optimal level and can be obtained from te solution of te dual problem of allocative efficiency. As for te implicit prices of investments, one needs to get β kt, at first. Ten, te implicit prices can be calculated by using te relation β it = (r t E+ ) -1 β kt -q t, since C AE/ i ' = β it. Precisely, β it,l = β kt,l/(r t +δ l )- q t, l = 1, 2,..., L. 9.2 Returns to Scale Te solution of te dual problem of tecnical efficiency gives us α yt and α xt, wic are te transposed vectors of te partial derivatives of te production frontier F x and F y. Of course, one can easily compute returns to scale, sadow prices and potential economies all by using te values of te dual variable of te tecnical efficiency model. Anyway, one can also coose anoter alternative metod if needed. In fact, by using te relations between α 's and β 's obtained in (9), we can obtain te elasticity of returns to scale wit β 's. Indeed, (6) reflects te duality between te production function and cost function. Substituting te relations (6), we can recover te returns to scale by using any of te following: RTS = - [F x (x E t ) + F k k t + F i i t ] (F y y t) = -{[α xt'/(1 N 'α xt)] x E t +[α k'/(1 N 'α xt)]k t -[α it'/(1 N 'α xt)]i t} {-[α yt'/(1 N 'α xt)] y t} = [β E ' x E t +β k ' k t -β i ' i t] [β y 'y t] = [w ' x E t +β k ' k t -β i ' i t] [β y 'y t].

27 Conclusions We ave introduced a dynamic version of DEA. Te augmented model allows us to measure tecnical and allocative efficiencies as in te standard model wile te inter-temporal restrictions are considered. It as been sown tat te necessary data set is made up of contemporary prices and quantities of inputs (including investment) and outputs so tat te calculation is fairly easy to perform. Of course, te linear program as exactly te same form tan te standard model, so tat no particular programming is required. Under dynamic circumstance, we define new measures of efficiency tat take account of investment expenditures. Finally, we develop explicit formula of implicit prices of quasi-fixed inputs and investments, and returns to scale. References Banker, R. D., and R. C. Morey (1986). Efficiency Analysis for Exogenously Fixed Inputs and Outputs. Operations Researc, 34: Carnes, A., W.W. Cooper and E. Rodes. (1978). Measuring Efficiency of Decision-Making Units. European Journal of Operational Researc, 2(6), Farrell, M.J. (1957). Te Measurement of Productive Efficiency. Journal of te Royal Statistical Society, Series A General, 120(3), Nemoto, J. and M. Goto. (1999). Dynamic Data Envelopment Analysis: Modeling Intertemporal Beavior of a Firm in te Presence of Productive Inefficiencies. Economics Letters, 64, Nemoto, J. and M. Goto. (2003). Measurement of Dynamic Efficiency in Production: An Application of Data Envelopment Analysis to Japanese Electric Utilities. Journal of Productivity Analysis, 19,