Mat-2.4142 Seminar on Optimization Data Envelopment Analysis Economies of Scope 21.11.2007
Economies of Scope Introduced 1982 by Panzar and Willing Support decisions like: Should a firm... Produce a variety of products Concentrate on some, spin of the others
DEA and economies of scope DEA to determine if diversification is superior to specialization Evaluate merger and acquisition possibilities Comparisons of business models
Definition: Economies of Scope Economies of Scope between to products (y 1,y 2 ) are present, if the cost of producing both products by one firm is less than producing them separately in specialized firms C(y 1,y 2 ) < C1(y 1,0) + C2(0,y 2 )
Degree of economies of scope (DES) DES j > 0 implies that firm j exhibits economies of scope DES j < 0 implies diseconomies of scope DES j = 0 means that the costs are additive in nature
Diversified and specialized firms
Checking for Economies of Scope diversified firms: production possibility set Group with all diversified firms (Group D) Outputs (y 1j, y 2j ) Input c j (j = 1,..., n) (costs)
Checking for Economies of Scope specialized firms: production possibility set There a also p specialized firms Group S 1 produces only Product 1 (z 1j ) using input (cost) v 1j for j = 1,..., p Group S 2 produces only Product 2 (z 2j ) using v 2j (costs) as input for j = 1,..., q
Checking for Economies of Scope diversified firms: eliminate cost-inefficiency For each DMU in D following LP can be solved to eliminate cost-inefficiency Let the optimal solution be θ. Then (c 0,y 10,y 20 ) produces at least (y 10,y 20 ) at minimum cost c 0=C(y 10,y 20 )
Checking for Economies of Scope creating virtual firms (1/2)
Checking for Economies of Scope creating virtual firms (2/2) Create virtual diversified firms by Combine firms out of each group V = {(v j,z 1j,z 2j ) j = 1,..., r(= p q)} (v 1k,z 1k ) S 1 and (v 2h,z 2h ) S 2 Virtual firm cost: v=v 1k +v 2h products: z 1k and z 2h Production possibility set
Checking for Economies of Scope virtual firms (1/2) c, is obtained from [cost] 0 Let the optimal θ be θ. Using this optimal θ, we define C 1 (y 10,0)+C 2 (0,y 20 ) =c 0 θ.
Checking for Economies of Scope virtual firms (2/2) Four possible cases θ < 1 (c 0,y 10,y 20 ) is enveloped by P v -> inefficient with respect to group V θ = 1 (c 0,y 10,y 20 ) is on the radially efficient frontier of P v θ > 1 (c 0,y 10,y 20 ) is not enveloped by P v -> super efficient with respect to group V The LP is not feasible (eλ = 1 cannot be satisfied) In this case assumption θ = 1 and C 1 (y 10,0)+C 2 (0,y 20 )=c 10=C(y 10,y 20 )
Checking for local economies of scope Using C(y 10,y 20 ) and C 1 (y 10,0)+C 2 (0,y 20 ) we can check the local economies of scope at DMU 0 C(y 10,y 20 ) < C 1 (y 10,0)+C 2 (0,y 20 ) Economies of scope exists C(y 10,y 20 ) = C 1 (y 10,0)+C 2 (0,y 20 ) Indifference C(y 10,y 20 ) > C 1 (y 10,0)+C 2 (0,y 20 ) Diseconomies of scope exists Degree of Economies of Scope is calculated: DES j = θ - 1
Virtual firms creation hint Notice: It is not necessary to create all combinations of specialized firms in S 1 and S 2. The virtual production possibility set P v can be formed form the combination of only the BCC-efficient DMUs in S 1 and S 2. Theorem 14.1
Example (1/4) specialized firms Specialised firms group 1 Company Name Cost Generation AEP Generating 253 832 Southern Electric Generating 179 682 South CaroHna Generating 104 443 North Atlantic Energy 103 273 Ocean State Power 74 120 Ocean State Power II 69 124 Holyoke Water Power 35 98 Safe Harbor Water Power 30 97 Specialised firms group 2 Company Name Cost Customers served PECO Energy 951 1529 Potomac Electric Power 549 718 Baltimore Gas & Electric 456 1162 Boston Edison 412 687 Duquesne Light 203 586 Central Maine Power 159 560 Commonwealth Electric 95 355 Orange & Rockland Utilities 90 211
Example (2/4) specialized firms (BCC-Efficient) Specialised firms group 1 Company Name Cost Generation AEP Generating 253 832 Southern Electric Generating 179 682 South CaroHna Generating 104 443 North Atlantic Energy 103 273 Ocean State Power 74 120 Ocean State Power II 69 124 Holyoke Water Power 35 98 Safe Harbor Water Power 30 97 Specialised firms group 2 Company Name Cost Customers served PECO Energy 951 1529 Potomac Electric Power 549 718 Baltimore Gas & Electric 456 1162 Boston Edison 412 687 Duquesne Light 203 586 Central Maine Power 159 560 Commonwealth Electric 95 355 Orange & Rockland Utilities 90 211
Example (3/4) virtual firms Virtual firms group V Firm Cost Generation Customers V1 1204 832 1529 V2 709 832 1162 V3 412 832 560 V4 348 832 355 V5 344 832 211 V6 1130 682 1529 V7 635 682 1162 V8 338 682 560 V9 274 682 355 V10 270 682 211 V11 1055 443 1529 V12 560 443 1162 V13 263 443 560 V14 199 443 355 V15 195 443 211 V16 980 97 1529 V17 485 97 1162 V18 189 97 560 V19 125 97 355 V20 120 97 211
Example (4/4) results Diversified firms group D Company Name Cost Generation Customers served Allete Inc. 277 771 132 Atlantic City Electric 246 250 512 Central Vermont Public Service 79 38 146 El Paso Electric 314 779 313 Kentucky Power 183 575 173 New York State Electric & Gas 335 26 811 Northwestern Energy 104 114 300 Ohio Edison 574 820 1,01 Efficiency Score and degree of economies of scope Company θ DES Economies of Scope Allete 1,1340 134 Yes Atlantic City Electric 0,8960-104 No Central Vermont Public Service 1,5140 514 Yes El Paso Electric 1,0220 22 Yes Kentucky Power 1,2900 290 Yes New York State Electric & Gas 0,9320-68 No Northwestern Energy 1,2230 223 Yes Ohio Edison 1,0940 94 Yes
Checking a virtual merger (1/3) Similar (reverse) procedure (economies of scope) Consider to merge Product 1 focused firm (v 1k,z 1k ) with a Product 2 specified firm (v 2h,z 2h ) Apply input orientated BCC model to remove cost inefficiency. Values (v * 1k,z 1k ) P 1 and (v* 2h,z 2h ) P 2 Form a virtual firm input (cost) (v= v * 1k +v* 2h ) and two outputs (z 1 = z 1k, z 2 = z 2h )
Checking a virtual merger (2/3) In case of infeasibility, we set C 1 (z 1,0)+C 2 (0,z 2 )=C(z 1,z 2 )
Checking a virtual merger (3/3) Using C(z 1,z 2 ),C 1 (z 1,0) and C 2 (0,z 2 ) we can check the attractiveness of the merger: C(z 1,z 2 ) > C 1 (z 1,0)+C 2 (0,z 2 ) Merger is unfavorable C(z 1,z 2 ) = C 1 (z 1,0)+C 2 (0,z 2 ) Indifference C(z 1,z 2 ) < C 1 (z 1,0)+C 2 (0,z 2 ) Merger is favorable
Home Assignment Generate virtual firms (5p) Specialised firms group 1 Specialised firms group 2 Company Name Cost Output1 Company Name Cost Output2 Special Company 1 81 321 Focused Company 1 92 214 Special Company 2 107 311 Focused Company 2 444 1111 Special Company 3 98 412 Focused Company 3 949 1524 Special Company 4 185 687 Focused Company 4 160 560 Special Company 5 321 871 Focused Company 5 201 586 Focused Company 6 547 716 Evaluate merger of South Carolina Generating with Commonwealth Electric (5p)