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Estimating economies of scope using the profit function: A dual approach for the normalized quadratic profit function
Economics Letters 100 (2008) 418–421
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Estimating economies of scope using the profit function: A dual approach for the normalized quadratic profit function Zhifeng Gao a, Allen M. Featherstone b,⁎ a b
University of Florida, United States Kansas State University, United States
A R T I C L E
I N F O
Article history: Received 11 July 2007 Received in revised form 11 February 2008 Accepted 6 March 2008 Available online 13 March 2008
A B S T R A C T Theoretical relationships between parameters of the normalized quadratic profit and cost functions are derived allowing for economies of scope to be calculated using profit function estimates. An empirical example confirms that the cost function is recovered using the estimated profit function. © 2008 Elsevier B.V. All rights reserved.
Keywords: Duality Economies of scope Normalized quadratic cost and profit function JEL classification: D01 D2
1. Introduction Economies of scope (EOS) measure the percentage of cost savings of producing several products in a single firm compared to producing the same products separately. The sources of economies of scope lie in the complimentary property among inputs. Since Baumol et al. (1982), economies of scope have become an important concept for measuring cost savings for multiproduct firms. The common approach involves estimating a cost function, and comparing the cost of producing multiproducts jointly with the cost of producing all the products individually. The normalized quadratic functional form is often used in the study of economies of scope (Featherstone and Moss, 1994; FernandezCornejo et al., 1992; Jin et al., 2005; and Cohn et al., 1989). One of the disadvantages of the parametric approach is that the data used to estimate cost functions are not always on the efficient frontier. Because scope economies are defined only on the efficiency frontier, testing economies of scope using data off the frontier could confound scope economies with X-efficiencies (Berger et al., 1993b). In addition, imposing curvature in a profit function is easier than in a cost function. Normally, concavity in outputs and convexity in inputs are imposed for
two sub-matrices of the Hessian matrix, and off diagonal sub-matrices are not considered. Using the profit function makes it easier to impose curvature on the off diagonal sub-matrices (Marsh and Featherstone, 2004). Berger et al. (1993a) also argue that measuring scope economies from a cost function doesn't consider whether the output bundle is optimal. Therefore, they suggest that more research should concentrate on estimating economies of scope from the profit function, which includes both the revenue and cost sides of production. Berger et al. (1993a) provided a new concept of optimal scope economies, which determines “whether a firm facing a given set of prices and other exogenous factors should optimally produce the entire array of products or specialize in some of them”. Using an unrestricted profit function, the optimal quantities of outputs can be derived using Hotelling's lemma. If the optimal quantities of outputs are determined to be positive at given exogenous prices, optimal scope economies exist at that point. Following Berger et al.'s suggestion, we provide a way to estimate economies of scope using the profit function. Different from Berger et al.'s (1993a) approach, we use the classic concept of scope economies that was first provided by Baumol et al. (1982). 2. Duality and recovering cost function from unrestricted profit function
⁎ Corresponding author. Department of Agricultural Economics, Kansas State University, 313 Waters Hall, Manhattan, Kansas, 66506-4011, United States. Tel.: +1 785 532 4441; fax: +1 785 532 6925. E-mail address: [email protected] (A.M. Featherstone). 0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2008.03.013
To determine economies of scope (EOS), the cost of producing multiproducts jointly and the sum of the cost to produce these products individually are compared. Economics of scope measure the
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
savings that occur if the products are produced jointly rather than separately. Specifically, EOS is: X C ðYi Þ C ðY Þ EOS ¼
i
;
C ðY Þ
where C(Yi) is the cost of producing only Yi in a separate firm, and C(Y) is the cost of producing all outputs by a single multiproduct firm. If EOS is positive, economies of scope exist and firms are more cost efficient by diversifying production. Duality theory indicates that a profit maximizing firm also minimizes cost, and the unrestricted profit function contains the same economic information as the indirect cost function (Mas-Colell et al., 1995). Theoretically, it's possible to link the parameters of the profit function to the parameters in the cost function. Lau (1976) provides Hessian identities where under perfect competition, a restricted profit (cost) function or production function can be recovered from an unrestricted profit function and vice versa. Lusk et al. (2002) examined the relationship between the parameters of production function, unrestricted profit function and restricted profit function empirically. The Hessian identities provide a relationship to determine the quadratic effects but do not provide a mechanism for determining the intercept and the linear terms that are needed to estimate economies of scope from the profit function. We use the normalized quadratic functional form to determine that relationship. Starting with a cost function, we use the maximization process to calculate the unrestricted profit function. If the parameters of profit function can be expressed using the parameters of the cost function, an inverse relationship can be obtained, which expresses the parameters of the cost function using the parameters from the profit function.
2
a11 6 a21 6 6: AA ¼ 6 6: 4 : an1
a12 a22 : : : an2
419
3 a1m a2m 7 7 : 7 7: : 7 5 : anm
N N N N : : : N N
The Hessian matrix of the normalized quadratic cost function for input prices and output quantities are BB and CC respectively. The curvature and symmetry conditions together imply that BB and CC are negative semi-definite symmetric matrices and positive semi-definite symmetric matrices, respectively. Assume both input and output markets are perfectly competitive, the unrestricted profit function can be obtained as a result of following maximization problem: P ¼ max PTY C ðW; Y Þ:
ð2Þ
where P is a vector of exogenous output prices, P = [p1 p2 …… pm]. The first order conditions allow us to determine the optimal output AP AY ¼ W;Y Þ P V AC ðAY ¼ 0 by solving a set of equations. For the normalized quadratic cost function (1), the first order conditions are: P V¼ A Vþ CCTY þ AA VTW;
ð3Þ
and the optimal output quantities are determined by solving for Y are: Y4 ¼ CC1 TðP V A V AA VTW Þ:
ð4Þ
Plugging Y ⁎ into the original cost function (1) to solve for the cost at the optimal output quantities: C ðW; YTÞ ¼ b0 þ B TW þ A T CC1 TðP V AV AAVTW Þ þ 0:5T W VT BB T W 1Tn
nT1
1Tm
1Tn
mT1
nTn
nT1
V þ 0:54 CC1 TðP V AV AAVTW Þ T CC T CC1 TðP V AV AAVTW Þ
3. Theoretical relationship between cost and unrestricted profit functions
mTm
1Tm
mT1
ð5Þ
þ W VT AA T CC1 TðP V AV AAVTW Þ :
Suppose that we have a normalized quadratic indirect cost function C(W,Y) that is continuous in (W,Y) and differentiable in W and Y, linear homogenous and concave in W, and convex in Y. The normalized cost function with n + 1 inputs and m outputs is expressed as: C ðW; Y Þ ¼ b0 þ B T W þ A T Y þ0:5T W VT BB T W 1Tn
nT1
1Tm
mT1
1Tn
nTn
nT1
ð1Þ
þ 0:5T Y V T CC T Y þ W VT AA 4 Y ; 1Tm
mTm
mT1
1Tn
n4m
mT1
where C(W,Y) is the normalized cost, W is a vector of input normalized prices and Y is the measure of output. The cost and input prices are normalized by the n + 1 input price which imposes the homogeneity condition. Formally,
mT1
Expanding via multiplication results in: C ðW; Y4Þ ¼ b0 þ BTW þ ATCC1 TP V ATCC1 TAV A4CC1 TAAVTW h i V þ0:5TW VTBBTW þ 0:5T ðP V AV AAVTW Þ VT CC1 TðP V AV AAVTW Þ
ð6Þ
þ W VTAATCC1 TP V W VTAATCC1 TAV W VTAATCC1 TAA VTW :
Because CC and BB are symmetric matrices, (CC− 1)′ is equal to CC− 1 and (BB)′ is equal to BB. Further expanding Eq. (6), the cost function is: ð7Þ
þBTW ATCC1 TAA VTW þ 0:5TW VTAATCC1 TA V
W ¼ ½w1 w2 N N wn
þ 0:5TATCC1 TAA VTW W VTAATCC1 TA Vþ ATCC1 TP V
A ¼ ½a1 a2 N N am
0:5TATCC1 TP V 0:5TPTCC1 TA Vþ 0:5TW VTBBTW
Y ¼ ½y1 y2 N N yn b11 b12 6 b21 b22 6: : 6 BB ¼ 6 : 6: 4 : : bn1 bn2 2 c11 c12 6 c21 c22 6 : 6: CC ¼ 6 6: : 4 : : cm1 cn2
nTm
C ðW; YTÞ ¼ b0 ATCC1 TA Vþ 0:5TATCC1 TA V
B ¼ ½b1 b2 N N bn
2
1Tn
N N N N : : : N N N N N N : : : N N
3 b1n b2n 7 : 7 7 where bij ¼ bji to satisfy symmetry : 7 7 5 : bnn 3 c1m c2m 7 7 : 7 7wherecij ¼ cji to satisfy symmetry; and : 7 5 : cmm
þ 0:5TW VTAATCC1 TAAVTW W VTAATCC1 TAAVTW þ 0:5TPTCC1 TP V 0:5TW VTAATCC1 TP V 0:5TPTCC 1 TAAVTW þ W VTAATCC1 TP V: Each term in Eq. (7) is a scalar, thus we can simplify the above equation to: C ðW; YTÞ ¼ b0 0:5TATCC1 TA Vþ B ATCC1 TAA V TW 1 þ 0:5TW VT BB AATCC TAA V TW þ 0:5TPTCC1 TP V:
ð8Þ
420
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
Plugging in the optimal output Y⁎ (Eq. (4)) and the above cost function into the direct profit function (Eq. (1)) results in:
Table 1 Estimated profit function and the comparison of the true and recovered cost functions
P ¼ PTCC1 TðP V A V AA VTW Þ ½b0 0:5TATCC1 TA Vþ B ATCC1 TAA V TW þ0:5TW VT BB AATCC1 TAA V TW þ 0:5TPTCC1 TP V:
Estimated parameters in profit function
ð9Þ
By simplifying Eq. (9), we obtain P ¼ b0 þ 0:5TATCC1 TA V ATCC1 TP V þ ATCC1 TAA V B TW þ 0:5TPTCC1 TP V þ 0:5TW VT AATCC1 TAA V BB TW W VTCC1 TAAVTP V:
ð10Þ
Eq. (10) is the unrestricted profit function expressed by the parameters of a normalized cost function. The final step is to reverse the process to recover the cost function from the unrestricted profit function. The unrestricted normalized quadratic profit function is: P ¼ pb0 þ PA T P V þ PB T W þ0:5T P T PCC T P V 1Tm
mT1
1Tn
nT1
1Tm
mTm
mT1
ð11Þ
þ0:5T W VT PBB T W þ W VT PAA T P V : 1Tn
nTn
nT1
1Tn
nTm
mT1
By solving for the corresponding cost function and simplifying, we determine the following relationships: pb0 ¼ b0 þ 0:5TATCC1 TA V; PA ¼ ATCC1 ;
ð12Þ
PB ¼ ATCC1 TAA V B; PCC ¼ CC1 ; PB ¼ ATCC1 TAAV B and PBB ¼ AATCC1 TAAV BB: With these relationships between the parameters from the cost and unrestricted profit functions, the parameters for the cost function can be recovered from the profit function. 4. An empirical example: A case of three inputs and two outputs Suppose that the normalized quadratic cost/profit function has three inputs (w1 w2 w3) and two outputs (y1 y2), and the input prices w1, w2 and cost C(W,Y) are normalized on the third input price w3. The cost function is w1 y1 C ðW; Y Þ ¼ b0 þ ½ b1 b2 T þ ½ a1 a2 T w2 y2 w1 b11 b12 þ 0:5T½ w1 w2 T T b12 b22 w2 c11 c12 y1 þ 0:5T½ y1 y2 T T c12 c22 y2 a11 a12 y1 þ ½ w1 w2 T T : a21 a22 y2
ð13Þ
Economies of scope can be expressed as EOS ¼
b0 þ b1 Tw1 þ b2 Tw2 þ 0:5Tb11 Tw21 þ b12 Tw1 Tw2 þ 0:5Tb22 Tw22 c12 Ty1 Ty2 : C ðW; Y Þ
ð14Þ The unrestricted profit function corresponding for the cost function is w1 p1 þ ½ pa1 pa2 T P ¼ pb0 þ ½ pb1 pb2 T w2 p2 w1 pb11 pb12 þ 0:5T½ w1 w2 T T pb12 pb22 w2 pc11 pc12 p1 þ 0:5T½ y1 y2 T T pc12 pc22 p2 pa11 pa12 p1 þ ½ w1 w2 T T : pa21 pa22 p2
pb0 pb1 pb2 pa1 pa2 pb11 pb12 pb22 pc11 pc12 pc22 pa11 pa12 pa21 pa22
−29.340 −9.912 −34.901 −0.147 −0.616 0.101 0.080 0.749 0.250 −0.001 0.309 −0.045 −0.031 −0.032 −0.040
Recovered parameters in cost function
True parameters in cost function
b0 b1 b2 a1 a2 b11 b12 b22 c11 c12 c22 a11 a12 a21 a22
b0 b1 b2 a1 a2 b11 b12 b22 c11 c12 c22 a11 a12 a21 a22
30.000 10.000 35.000 0.600 2.000 −0.090 −0.075 −0.740 4.000 0.010 3.241 0.180 0.100 0.130 0.130
30.000 10.000 35.000 0.600 2.000 −0.090 −0.075 −0.740 4.000 0.010 3.240 0.180 0.100 0.130 0.130
Following the theoretical results from Eq. (12), the parameters in the cost function (13) can be calculated from the unrestricted profit function (15). Also, note that the optimal output y1 and y2 in Eq. (14) are calAP culated using Hotelling's lemma, Ap ¼ yi . Thus, economies of scope at i the optimal quantity can be calculated using the recovered cost function parameters and the optimal output supplies. To empirically demonstrate the results, 500 data points for the three input prices and two output prices were generated using a Monte-Carlo procedure. The parameters of the normalized cost function (true cost function) were specified so that the cost function satisfied homogeneity, symmetry, and curvature conditions. Homogeneity was satisfied by normalizing all prices and cost by the third input price, symmetry was imposed by letting the Hessian matrices of input prices and output quantities being symmetric, and curvature was imposed using Cholesky decomposition (Featherstone and Moss, 1994). With input and output prices, as well as the parameters from the cost function, the optimal output quantities y⁎ were calculated using Eq. (4). The minimized costs C(W,Y⁎) were determined using the input prices and optimal output quantities. Profits were the differences between revenue and the cost: Π = P ⁎ Y⁎ − C(W,Y⁎). Costs and profits were calculated at each of the 500 data points. The unrestricted normalized quadratic profit function was estimated and the computations in Eq. (12) were used to recover the unrestricted cost function parameters. In Table 1, the first column is the estimated coefficients of the profit function, the second column are the recovered parameters of cost function using the estimated coefficients from profit function, and the third column are the assumed true parameters of the cost function used to generate the data. The results in Table 1 show that all the recovered parameters from the profit function are exactly same as parameters of the true cost function assumed for the data generating process which verifies our method. Using Hotelling's lemma, the optimal output quantities were obtained at each of the 500 data points. The mean cost was also calculated using Eq. (13) with recovered parameters of cost function, mean input prices and mean output quantities obtained by Hotelling's lemma. Then EOS at mean optimal output quantities and input prices were calculated using Eq. (14), which were 4.4%. 5. Conclusions and discussions
ð15Þ
The theoretical relationships between the intercept and linear terms of the normalized quadratic cost and unrestricted profit functions were derived. Using the theoretical relationships, parameters in the cost function can be recovered from the unrestricted profit
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
function which enables the calculation of economies of scope (EOS) using the profit function. We numerically analyzed our results using Monte-Carlo data. The recovered parameters for the cost function were identical to the true parameters. Measuring economies of scope using the profit function has a few merits that lack in methods using the cost function. First, imposing curvature on a profit function is easier than imposing curvature on a cost function (Marsh and Featherstone, 2004). Second, EOS calculated from the profit function indicates that output quantities are consistent with output prices which avoids the problem that EOS calculated from the cost function are not necessarily on the production frontier (Berger et al., 1993b). References Baumol, William J., Panzar, John C., Willig, Robert D., 1982. Contestable Markets and Industry Structure. Harcourt Brace Jovanovich, Inc., New York. Berger, Allen N., Hancock, Diana, Humphrey, David B., 1993a. Bank efficiency derived from the profit functions. Journal of Banking and Finance 17, 317–347. Berger, Allen N., Hunter, William C., Timme, Stephen G., 1993b. The efficiency of financial institutions: a review and preview of research past. Present, and Future, Journal of Banking and Finance 17, 221–249.
421
Cohn, Elchanan, Rhine, Sherrie L.W., Santos, Maria C., 1989. Institutions of higher education as multi-product firms: economies of scale and scope. The Review of Economics and Statistics 71, 284–290. Featherstone, Allen M., Moss, Charles B., 1994. Measuring economies of scale and scope in agricultural banking. American Journal of Agricultural Economics 76, 655–661. Fernandez-Cornejo, J., Gempesaw II, C.M., Elterich, J.G., Stefanou, S.E., 1992. Dynamic measure of scope and scale economies: an application to German agriculture. American Journal of Agricultural Economics 74, 329–342. Jin, Songqing, Rozelle, Scott, Alston, Julian, Huang, Jikun, 2005. Economies of scale and scope and the economic efficiency of China's agricultural research system. International Economic Review 46, 1033–1057. Lau, L.J., 1976. A characterization of the normalized restricted profit function. Journal of Economic Theory 12, 131–163. Lusk, Jayson L., Featherstone, Allen M., Mash, Thomas L., Abdulkadri, Abdullahi O., 2002. The empirical properties of duality theory. The Australian Journal of Agricultural and Resource Economics. 46, 45–68. Marsh, T.L., Featherstone, A.M., 2004. Testing and Imposing Curvature Restrictions on Restricted Profit Functions. Kansas State University. Working Paper. Mas-Colell, Andreu, Whinston, Michael D., Green, Jerry R., 1995. Microeconomic Theory. Oxford University Press, New York.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Estimating economies of scope using the profit function: A dual approach for the normalized quadratic profit function Zhifeng Gao a, Allen M. Featherstone b,⁎ a b
University of Florida, United States Kansas State University, United States
A R T I C L E
I N F O
Article history: Received 11 July 2007 Received in revised form 11 February 2008 Accepted 6 March 2008 Available online 13 March 2008
A B S T R A C T Theoretical relationships between parameters of the normalized quadratic profit and cost functions are derived allowing for economies of scope to be calculated using profit function estimates. An empirical example confirms that the cost function is recovered using the estimated profit function. © 2008 Elsevier B.V. All rights reserved.
Keywords: Duality Economies of scope Normalized quadratic cost and profit function JEL classification: D01 D2
1. Introduction Economies of scope (EOS) measure the percentage of cost savings of producing several products in a single firm compared to producing the same products separately. The sources of economies of scope lie in the complimentary property among inputs. Since Baumol et al. (1982), economies of scope have become an important concept for measuring cost savings for multiproduct firms. The common approach involves estimating a cost function, and comparing the cost of producing multiproducts jointly with the cost of producing all the products individually. The normalized quadratic functional form is often used in the study of economies of scope (Featherstone and Moss, 1994; FernandezCornejo et al., 1992; Jin et al., 2005; and Cohn et al., 1989). One of the disadvantages of the parametric approach is that the data used to estimate cost functions are not always on the efficient frontier. Because scope economies are defined only on the efficiency frontier, testing economies of scope using data off the frontier could confound scope economies with X-efficiencies (Berger et al., 1993b). In addition, imposing curvature in a profit function is easier than in a cost function. Normally, concavity in outputs and convexity in inputs are imposed for
two sub-matrices of the Hessian matrix, and off diagonal sub-matrices are not considered. Using the profit function makes it easier to impose curvature on the off diagonal sub-matrices (Marsh and Featherstone, 2004). Berger et al. (1993a) also argue that measuring scope economies from a cost function doesn't consider whether the output bundle is optimal. Therefore, they suggest that more research should concentrate on estimating economies of scope from the profit function, which includes both the revenue and cost sides of production. Berger et al. (1993a) provided a new concept of optimal scope economies, which determines “whether a firm facing a given set of prices and other exogenous factors should optimally produce the entire array of products or specialize in some of them”. Using an unrestricted profit function, the optimal quantities of outputs can be derived using Hotelling's lemma. If the optimal quantities of outputs are determined to be positive at given exogenous prices, optimal scope economies exist at that point. Following Berger et al.'s suggestion, we provide a way to estimate economies of scope using the profit function. Different from Berger et al.'s (1993a) approach, we use the classic concept of scope economies that was first provided by Baumol et al. (1982). 2. Duality and recovering cost function from unrestricted profit function
⁎ Corresponding author. Department of Agricultural Economics, Kansas State University, 313 Waters Hall, Manhattan, Kansas, 66506-4011, United States. Tel.: +1 785 532 4441; fax: +1 785 532 6925. E-mail address: [email protected] (A.M. Featherstone). 0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2008.03.013
To determine economies of scope (EOS), the cost of producing multiproducts jointly and the sum of the cost to produce these products individually are compared. Economics of scope measure the
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
savings that occur if the products are produced jointly rather than separately. Specifically, EOS is: X C ðYi Þ C ðY Þ EOS ¼
i
;
C ðY Þ
where C(Yi) is the cost of producing only Yi in a separate firm, and C(Y) is the cost of producing all outputs by a single multiproduct firm. If EOS is positive, economies of scope exist and firms are more cost efficient by diversifying production. Duality theory indicates that a profit maximizing firm also minimizes cost, and the unrestricted profit function contains the same economic information as the indirect cost function (Mas-Colell et al., 1995). Theoretically, it's possible to link the parameters of the profit function to the parameters in the cost function. Lau (1976) provides Hessian identities where under perfect competition, a restricted profit (cost) function or production function can be recovered from an unrestricted profit function and vice versa. Lusk et al. (2002) examined the relationship between the parameters of production function, unrestricted profit function and restricted profit function empirically. The Hessian identities provide a relationship to determine the quadratic effects but do not provide a mechanism for determining the intercept and the linear terms that are needed to estimate economies of scope from the profit function. We use the normalized quadratic functional form to determine that relationship. Starting with a cost function, we use the maximization process to calculate the unrestricted profit function. If the parameters of profit function can be expressed using the parameters of the cost function, an inverse relationship can be obtained, which expresses the parameters of the cost function using the parameters from the profit function.
2
a11 6 a21 6 6: AA ¼ 6 6: 4 : an1
a12 a22 : : : an2
419
3 a1m a2m 7 7 : 7 7: : 7 5 : anm
N N N N : : : N N
The Hessian matrix of the normalized quadratic cost function for input prices and output quantities are BB and CC respectively. The curvature and symmetry conditions together imply that BB and CC are negative semi-definite symmetric matrices and positive semi-definite symmetric matrices, respectively. Assume both input and output markets are perfectly competitive, the unrestricted profit function can be obtained as a result of following maximization problem: P ¼ max PTY C ðW; Y Þ:
ð2Þ
where P is a vector of exogenous output prices, P = [p1 p2 …… pm]. The first order conditions allow us to determine the optimal output AP AY ¼ W;Y Þ P V AC ðAY ¼ 0 by solving a set of equations. For the normalized quadratic cost function (1), the first order conditions are: P V¼ A Vþ CCTY þ AA VTW;
ð3Þ
and the optimal output quantities are determined by solving for Y are: Y4 ¼ CC1 TðP V A V AA VTW Þ:
ð4Þ
Plugging Y ⁎ into the original cost function (1) to solve for the cost at the optimal output quantities: C ðW; YTÞ ¼ b0 þ B TW þ A T CC1 TðP V AV AAVTW Þ þ 0:5T W VT BB T W 1Tn
nT1
1Tm
1Tn
mT1
nTn
nT1
V þ 0:54 CC1 TðP V AV AAVTW Þ T CC T CC1 TðP V AV AAVTW Þ
3. Theoretical relationship between cost and unrestricted profit functions
mTm
1Tm
mT1
ð5Þ
þ W VT AA T CC1 TðP V AV AAVTW Þ :
Suppose that we have a normalized quadratic indirect cost function C(W,Y) that is continuous in (W,Y) and differentiable in W and Y, linear homogenous and concave in W, and convex in Y. The normalized cost function with n + 1 inputs and m outputs is expressed as: C ðW; Y Þ ¼ b0 þ B T W þ A T Y þ0:5T W VT BB T W 1Tn
nT1
1Tm
mT1
1Tn
nTn
nT1
ð1Þ
þ 0:5T Y V T CC T Y þ W VT AA 4 Y ; 1Tm
mTm
mT1
1Tn
n4m
mT1
where C(W,Y) is the normalized cost, W is a vector of input normalized prices and Y is the measure of output. The cost and input prices are normalized by the n + 1 input price which imposes the homogeneity condition. Formally,
mT1
Expanding via multiplication results in: C ðW; Y4Þ ¼ b0 þ BTW þ ATCC1 TP V ATCC1 TAV A4CC1 TAAVTW h i V þ0:5TW VTBBTW þ 0:5T ðP V AV AAVTW Þ VT CC1 TðP V AV AAVTW Þ
ð6Þ
þ W VTAATCC1 TP V W VTAATCC1 TAV W VTAATCC1 TAA VTW :
Because CC and BB are symmetric matrices, (CC− 1)′ is equal to CC− 1 and (BB)′ is equal to BB. Further expanding Eq. (6), the cost function is: ð7Þ
þBTW ATCC1 TAA VTW þ 0:5TW VTAATCC1 TA V
W ¼ ½w1 w2 N N wn
þ 0:5TATCC1 TAA VTW W VTAATCC1 TA Vþ ATCC1 TP V
A ¼ ½a1 a2 N N am
0:5TATCC1 TP V 0:5TPTCC1 TA Vþ 0:5TW VTBBTW
Y ¼ ½y1 y2 N N yn b11 b12 6 b21 b22 6: : 6 BB ¼ 6 : 6: 4 : : bn1 bn2 2 c11 c12 6 c21 c22 6 : 6: CC ¼ 6 6: : 4 : : cm1 cn2
nTm
C ðW; YTÞ ¼ b0 ATCC1 TA Vþ 0:5TATCC1 TA V
B ¼ ½b1 b2 N N bn
2
1Tn
N N N N : : : N N N N N N : : : N N
3 b1n b2n 7 : 7 7 where bij ¼ bji to satisfy symmetry : 7 7 5 : bnn 3 c1m c2m 7 7 : 7 7wherecij ¼ cji to satisfy symmetry; and : 7 5 : cmm
þ 0:5TW VTAATCC1 TAAVTW W VTAATCC1 TAAVTW þ 0:5TPTCC1 TP V 0:5TW VTAATCC1 TP V 0:5TPTCC 1 TAAVTW þ W VTAATCC1 TP V: Each term in Eq. (7) is a scalar, thus we can simplify the above equation to: C ðW; YTÞ ¼ b0 0:5TATCC1 TA Vþ B ATCC1 TAA V TW 1 þ 0:5TW VT BB AATCC TAA V TW þ 0:5TPTCC1 TP V:
ð8Þ
420
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
Plugging in the optimal output Y⁎ (Eq. (4)) and the above cost function into the direct profit function (Eq. (1)) results in:
Table 1 Estimated profit function and the comparison of the true and recovered cost functions
P ¼ PTCC1 TðP V A V AA VTW Þ ½b0 0:5TATCC1 TA Vþ B ATCC1 TAA V TW þ0:5TW VT BB AATCC1 TAA V TW þ 0:5TPTCC1 TP V:
Estimated parameters in profit function
ð9Þ
By simplifying Eq. (9), we obtain P ¼ b0 þ 0:5TATCC1 TA V ATCC1 TP V þ ATCC1 TAA V B TW þ 0:5TPTCC1 TP V þ 0:5TW VT AATCC1 TAA V BB TW W VTCC1 TAAVTP V:
ð10Þ
Eq. (10) is the unrestricted profit function expressed by the parameters of a normalized cost function. The final step is to reverse the process to recover the cost function from the unrestricted profit function. The unrestricted normalized quadratic profit function is: P ¼ pb0 þ PA T P V þ PB T W þ0:5T P T PCC T P V 1Tm
mT1
1Tn
nT1
1Tm
mTm
mT1
ð11Þ
þ0:5T W VT PBB T W þ W VT PAA T P V : 1Tn
nTn
nT1
1Tn
nTm
mT1
By solving for the corresponding cost function and simplifying, we determine the following relationships: pb0 ¼ b0 þ 0:5TATCC1 TA V; PA ¼ ATCC1 ;
ð12Þ
PB ¼ ATCC1 TAA V B; PCC ¼ CC1 ; PB ¼ ATCC1 TAAV B and PBB ¼ AATCC1 TAAV BB: With these relationships between the parameters from the cost and unrestricted profit functions, the parameters for the cost function can be recovered from the profit function. 4. An empirical example: A case of three inputs and two outputs Suppose that the normalized quadratic cost/profit function has three inputs (w1 w2 w3) and two outputs (y1 y2), and the input prices w1, w2 and cost C(W,Y) are normalized on the third input price w3. The cost function is w1 y1 C ðW; Y Þ ¼ b0 þ ½ b1 b2 T þ ½ a1 a2 T w2 y2 w1 b11 b12 þ 0:5T½ w1 w2 T T b12 b22 w2 c11 c12 y1 þ 0:5T½ y1 y2 T T c12 c22 y2 a11 a12 y1 þ ½ w1 w2 T T : a21 a22 y2
ð13Þ
Economies of scope can be expressed as EOS ¼
b0 þ b1 Tw1 þ b2 Tw2 þ 0:5Tb11 Tw21 þ b12 Tw1 Tw2 þ 0:5Tb22 Tw22 c12 Ty1 Ty2 : C ðW; Y Þ
ð14Þ The unrestricted profit function corresponding for the cost function is w1 p1 þ ½ pa1 pa2 T P ¼ pb0 þ ½ pb1 pb2 T w2 p2 w1 pb11 pb12 þ 0:5T½ w1 w2 T T pb12 pb22 w2 pc11 pc12 p1 þ 0:5T½ y1 y2 T T pc12 pc22 p2 pa11 pa12 p1 þ ½ w1 w2 T T : pa21 pa22 p2
pb0 pb1 pb2 pa1 pa2 pb11 pb12 pb22 pc11 pc12 pc22 pa11 pa12 pa21 pa22
−29.340 −9.912 −34.901 −0.147 −0.616 0.101 0.080 0.749 0.250 −0.001 0.309 −0.045 −0.031 −0.032 −0.040
Recovered parameters in cost function
True parameters in cost function
b0 b1 b2 a1 a2 b11 b12 b22 c11 c12 c22 a11 a12 a21 a22
b0 b1 b2 a1 a2 b11 b12 b22 c11 c12 c22 a11 a12 a21 a22
30.000 10.000 35.000 0.600 2.000 −0.090 −0.075 −0.740 4.000 0.010 3.241 0.180 0.100 0.130 0.130
30.000 10.000 35.000 0.600 2.000 −0.090 −0.075 −0.740 4.000 0.010 3.240 0.180 0.100 0.130 0.130
Following the theoretical results from Eq. (12), the parameters in the cost function (13) can be calculated from the unrestricted profit function (15). Also, note that the optimal output y1 and y2 in Eq. (14) are calAP culated using Hotelling's lemma, Ap ¼ yi . Thus, economies of scope at i the optimal quantity can be calculated using the recovered cost function parameters and the optimal output supplies. To empirically demonstrate the results, 500 data points for the three input prices and two output prices were generated using a Monte-Carlo procedure. The parameters of the normalized cost function (true cost function) were specified so that the cost function satisfied homogeneity, symmetry, and curvature conditions. Homogeneity was satisfied by normalizing all prices and cost by the third input price, symmetry was imposed by letting the Hessian matrices of input prices and output quantities being symmetric, and curvature was imposed using Cholesky decomposition (Featherstone and Moss, 1994). With input and output prices, as well as the parameters from the cost function, the optimal output quantities y⁎ were calculated using Eq. (4). The minimized costs C(W,Y⁎) were determined using the input prices and optimal output quantities. Profits were the differences between revenue and the cost: Π = P ⁎ Y⁎ − C(W,Y⁎). Costs and profits were calculated at each of the 500 data points. The unrestricted normalized quadratic profit function was estimated and the computations in Eq. (12) were used to recover the unrestricted cost function parameters. In Table 1, the first column is the estimated coefficients of the profit function, the second column are the recovered parameters of cost function using the estimated coefficients from profit function, and the third column are the assumed true parameters of the cost function used to generate the data. The results in Table 1 show that all the recovered parameters from the profit function are exactly same as parameters of the true cost function assumed for the data generating process which verifies our method. Using Hotelling's lemma, the optimal output quantities were obtained at each of the 500 data points. The mean cost was also calculated using Eq. (13) with recovered parameters of cost function, mean input prices and mean output quantities obtained by Hotelling's lemma. Then EOS at mean optimal output quantities and input prices were calculated using Eq. (14), which were 4.4%. 5. Conclusions and discussions
ð15Þ
The theoretical relationships between the intercept and linear terms of the normalized quadratic cost and unrestricted profit functions were derived. Using the theoretical relationships, parameters in the cost function can be recovered from the unrestricted profit
Z. Gao, A.M. Featherstone / Economics Letters 100 (2008) 418–421
function which enables the calculation of economies of scope (EOS) using the profit function. We numerically analyzed our results using Monte-Carlo data. The recovered parameters for the cost function were identical to the true parameters. Measuring economies of scope using the profit function has a few merits that lack in methods using the cost function. First, imposing curvature on a profit function is easier than imposing curvature on a cost function (Marsh and Featherstone, 2004). Second, EOS calculated from the profit function indicates that output quantities are consistent with output prices which avoids the problem that EOS calculated from the cost function are not necessarily on the production frontier (Berger et al., 1993b). References Baumol, William J., Panzar, John C., Willig, Robert D., 1982. Contestable Markets and Industry Structure. Harcourt Brace Jovanovich, Inc., New York. Berger, Allen N., Hancock, Diana, Humphrey, David B., 1993a. Bank efficiency derived from the profit functions. Journal of Banking and Finance 17, 317–347. Berger, Allen N., Hunter, William C., Timme, Stephen G., 1993b. The efficiency of financial institutions: a review and preview of research past. Present, and Future, Journal of Banking and Finance 17, 221–249.
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