Lecture20-2011 - Concepts of Subadditivity Composite Cost...

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Unformatted text preview: Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Subadditity of Cost Functions: Lecture XX Charles B. Moss1 1 University of Florida November 3, 2011 Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale 1 Concepts of Subadditivity Mathematics of the Cost Function Subadditivity and the Translog Admissible Region 2 Composite Cost Functions and Subadditivity 3 Economies of Scope 4 Economies of Scale Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Evans, D. S. and J. J. Heckman. 1984. A Test for Subadditivity of the Cost Function with an Application to the Bell System. American Economic Review 74, 615-23. The issue addressed in this article involves the emergence of natural monopolies. Specifically, is it possible that a single firm is the most cost-efficient way to generate the product. In the specific application, the researchers are interested in the Bell System (the phone company before it was split up). Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Mathematics of te Cost Function Until this lecture, we have considered cost functions of single outputs. The the cost frontier is easily extended to consider multiple outputs C ( w 1 , w 2 , y1 , y2 ) = α 0 + α 1 w 1 + α 2 w 2 + 1 α11 w1 w1 + α12 w1 w2 + α22 w2 w2 + β1 y1 + β2 y2 + 2 1 1 β11 y1 y1 + β12 y1 y2 + β22 y2 y2 2 2 γ11 w1 y1 + γ12 w1 y2 + γ21 w2 y1 + γ22 w2 y2 Charles B. Moss Subadditity of Cost Functions: Lecture XX (1) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region The cost function C (y ) is subadditive at some output level if and only if C (y ) < n ￿ n ￿ C ( yi ) i =1 (2) qi = q i =1 which states that the cost function is subadditive if a single firm could produce the same output for less cost. As a mathematical nicety, the point must have at least two nonzero firms. Otherwise the cost function is by definition the same. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Developing a formal test, Evans and Heckman assume a cost function based on two input ￿ C (ai y1 , bi y2 ) > C (y1 , y2 ) i = 1, · · · n i￿ ￿ (3) ai , bi = 1, ai ≥ 0, bi ≥ 0 i i Thus, each of i firms produce ai percent of output y1 and bi percent of the output y2 . Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region A primary focus of the article is the region over which subadditivity is tested. The cost function is subadditive, and the technology implies a natural monopoly. ￿ C ( a i y1 , b i y 2 ) > C ( y 1 , y 2 ) . (4) i The cost function is superadditive, and the firm could save money by breaking itself up into two or more divisions. ￿ C ( a i y 1 , b i y 2 ) < C ( y1 , y 2 ) (5) i The cost function is additive if ￿ C ( a i y 1 , b i y 2 ) = C ( y1 , y 2 ) i Charles B. Moss Subadditity of Cost Functions: Lecture XX (6) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region The notion of additivity combines two concepts from the cost function: Economies of Scope and Economies of Scale. Under Economies of Scope, it is cheaper to produce two goods together. The example I typically give for this is the grazing cattle on winter wheat. However, we also recognize following the concepts of Coase, Williamson, and Grossman and Hart that there may diseconomies of scope. The second concept is the economies of scale argument that we have discussed before. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Subadditivity and the Translog As stated previously, a primary focus of this article is the region of subadditivity. In our discussion of cost functions, I have mentioned the concepts of Global versus local. To make the discussion more concrete, let us return to our discussion of concavity. From the properties of the cost function, we know that the cost function is concave in input price space. Thus, using the Translog form 1 ￿ ln (C ) = α0 + α￿ ln (w ) + ln (w ) ln (w ) 2 1 ￿ ￿ β ￿ ln (y ) + ln (y ) B ln (y ) + ln (w ) Γ ln (y ) 2 Charles B. Moss Subadditity of Cost Functions: Lecture XX (7) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Gradient Continued The gradient vector for the Translog cost function is then s1 . ∇w C (w , y ) = exp [ln (C )] . . sn α1 + A·q ln (w ) + Γ￿ · y 1 . . = exp [ln (C )] . αn + A·n ln (w ) + Γ￿ · y n Denoting the vector of share equations as φ (w , y ) α1 + A·q ln (w ) + Γ1· ln (y ) . . φ (w , y ) = . ￿ αn + A·n ln (w ) + Γn· ln (y ) Charles B. Moss Subadditity of Cost Functions: Lecture XX (8) (9) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Hessian α1 + A·q ln (w ) + Γ1· ln (y ) . . ∇2 C (w , y ) = exp [ln (C )] . xx ￿ αn + A·n ln (w ) + Γn· ln (y ) ￿ α1 + A·q ln (w ) + Γ1· ln (y ) A11 · · · . . .. . × + exp [ln (C )] . . . . αn + A·n ln (w ) + Γ￿ · ln (y ) A 1n · · · n ￿ ￿ C × A + φ (w , y ) φ (w , y )￿ Charles B. Moss Subadditity of Cost Functions: Lecture XX A 1n . . . Ann (10) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Given that the cost is always positive, the positive versus negative nature of the matrix is determined by A + φ (w , y ) φ (w , y )￿ (11) Comparing this results with the result for the quadratic function, we see that ∇2 C ( w , y ) = A ww (12) Thus, the Hessian of the Translog varies over input prices and output levels while the Hessian matrix for the Quadratic does not. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region In this sense, the restrictions on concavity for the Quadratic cost function are globalthey do not change with respect to output and input prices. However, the concavity restrictions on the Translog are localfixed at a specific point, because they depend on prices and output levels. Note that this is important for the Translog. Specifically, if we want the cost function to be concave in input prices ￿ ￿￿ x ￿ A + φ (w , y ) φ (w , y ) x ≤ 0∀x ￿ ⇒ x ￿ Ax + x ￿ φ (w , y ) φ (w , y ) ≤ 0 (13) ￿ ￿ ￿￿ ￿ ￿￿ ￿ ⇒ x Ax + φ (w , y ) x φ (w , y ) x ≤ 0 ￿ ￿ ￿￿ ￿ ￿￿ But φ (w , y ) x φ (w , y ) x ≥ 0. Thus, any discussion of subadditivity, especially if a Translog cost function is used (or any cost function other than a quadratic), needs to consider the region over which the cost function is to be tested. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region C q1 Admissible Region q2 Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Thus, much of the discussion in Evans and Heckman involve the choice of the region for the test. Specifically, the test region is restricted to a region of observed point. ∗ Defining y1m as the minimum amount of y1 produced by any ∗ firm and y2m as the minimum amount of y2 produced, we can define alternative production bundles ytB ∗ ∗ ytA = (φy1t + y1m , ω y2t + y2m ) ∗ ∗ = ((1 − φ) y1t + y1m , (1 − ω ) y2t + y2m ) 0 ≤ φ ≤ 1, 0 ≤ ω ≤ 1 Charles B. Moss Subadditity of Cost Functions: Lecture XX (14) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Thus, the production for any firm can be divided into two components within the observed range of output. Thus, subadditivity can be defined as ￿￿ ￿ ￿ CtA (φ, ω ) = C ￿ytA ￿ = C ￿ym + ytA ￿ CtB (φ, ω ) =￿C ytB = C ym + ytB ￿ Ct =￿C ytA + ytB = C (yt ) ￿ Ct − CtA (φ, ω ) − CtB (φ, ω ) Subt (φ, ω ) Ct (15) If Subt (φ, ω ) is less than zero, the cost function is subadditive, if it is equal to zero the cost function is additive, and if it is greater than zero, the cost function is superadditive Consistent with their concept of the region of the test, Evans and Heckman calculate the maximum and minimum Subt (φ, ω )for the region. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Composite Cost Functions and Subadditivity Pulley, L. B. and Y. M. Braunstein. 1992. A Composite Cost Function for Multiproduct Firms with an Application to Economies of Scope in Banking. Review of Economics and Statistics 74, 221-30. Building on the concept of subadditivity and the global nature of the flexible function form, it is apparent that the estimation of subadditivity is dependent on functional form Pulley and Braunstein allow for a more general form of the cost function by allowing the Box-Cox transformation to be different for the inputs and outputs. ￿ ￿ φ y −1 y (φ ) = ￿: φ ￿= 0 (16) φ = ln (y ) ￿: φ = 0 Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale The composite cost function form C (φ ) ￿ ￿ 1 = exp α0 + α q + q (π)￿ Aq (π) + q (π) Ψr 2 ￿ ￿￿ (φ) 1￿ ￿ (π ) exp β0 + β r + r Br + q Υr 2 f (φ) (q , ln (r )) ￿ (π ) ￿ (τ ) (17) Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale If φ = 0, π = 0, and τ = 1 the form yields a standard Translog with normal share equations. If φ = 0 and τ = 1 the form yields a generalized Translog 1 ln (C ) = α0 + α￿ q (π) + q (φ)￿ Aq (π) + q (π) Ψ ln (r ) + 2 1 β ￿ ln (r ) + ln (r )￿ B ln (r ) 2 s = Ψq (π) + β + B ln (r ) (18) If π = 1, τ = 0, and Ψ = 0, the specification becomes a separable quadratic specification ￿￿ ￿ 1￿ (φ ) ￿ C = α0 + α q + q Aq 2 ￿ ￿￿ (φ) (19) 1 ￿ ￿ × exp β0 + β ln (r ) + ln (r ) B ln (r ) 2 s = β + B ln (r ) Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale The demand equations for the composite function is ￿ 1 s = α0 + α q + q ￿ Aq + q Ψ ln (r ) 2 B ln (r ) + Υ￿ q ￿ Charles B. Moss ￿ −1 Ψq + Subadditity of Cost Functions: Lecture XX (20) Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Economies of Scope Given the estimates, we can then measure Economies of Scope in two ways. The first measures is a traditional measure Scope = C (q1 , 0, · · · 0, r ) + · · · C (0, 0, · · · qn , r ) − C (q1 , q2 , · · · qn , r ) C ( q1 , q2 , · · · qn , r ) (21) Another measure suggested by the article is “quasi” economies of scope Qscope = C ({ 1 − (m − 1) ￿} q1 , q2 ￿, · · · r ) · · · − C (q1 , q2 , · · · qn , r ) C ( q1 , q2 , · · · r ) (22) Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale The Economies of Scale are then defined as Scale = ￿ i Charles B. Moss C (q , r ) ∂ C (q , r ) qi ∂ qi Subadditity of Cost Functions: Lecture XX (23) ...
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