Economies of scale for multi-product firm

1. Economies of scale for multi-product firm • Concepts such as decreasing average cost and economies of scale must be redefined in a multiproduct setting. • Equating natural monopoly with decreasing average cost is meaningless when the cost function involves more than one product.

2. Definitions -Subadditivity let be the ith vector of m output vectors i= 1,…., m, where each vector contans one or more of n different outputs. A necessary and sufficient condition for natural monopoly is that the cost function exhibit strict and global subadditivity of costs, or (2.8) for any m output vectors. If (2.8) is satisfied, then the least expensive method of producing is with a single firm.

3. Economies of scale for multi-product firm Consider an input-output vector (x1, …, xr , q1 , …., qn ), where xk is input k, k = 1,…,r and scalars w> 1 and δ>0. Strict Economies of scale exist if (wx1, …, wxr, v1q1, …, vnqn) is a feasible input-output vector, where all vi = w + δ. Thus, an expansion of all inputs by w implies a greater expansion of all outputs.

4. Economies of Scope for multi-product firm • Economies of scope constitute a restricted form of subadditivity and it captures the essence of multi-product versus single product production. • It contrasts the cost of producing output q1 , …., qn all in a single firm, with the total cost of producing each output qi , I =1,…,n, in separate firms, each specializing in the production of one product.

5. Decreasing Average Cost • We cannot define decreasing average cost in the usual manner because there is no single unambiguously acceptable measure of aggregate output to divide into total cost. • However, we can consider proportionate changes in output along a ray from the origin in output space and then observe the shape of the cost function as we move along the ray. • Decreasing ray average cost is expressed as C(vq1, …, vqn)/v < C(wq1, …, wqn)/w for v>w, where v and w are measures of the scale of output along a ray through output vector q = (q1, …, qn).

6. Declining ray average cost

7. Transray-convex • A cost function is transray-convex through q*= ( ,…., ) if there exists any set of positive constants w1,…, wnsuch that for every two output vector qa =( ), qb =( ) lying in the same hyperplane through q*, we have C(q*)=C(k qa +(1-k) qb) ≤ kC(qa)+ (1-k)C(qb) for any k, 0<k<1. Transray convexity requires that a linear combination of the costs of producing qa and qbin separate firms be no less than the cost of producing the same linear combination ofqa and qbin a single firm. Or, on the cost surface, a straight line connecting point C(qa)with point C(qb) must not lie below the cost surface anywhere between these points. (See figure 2.6)

8. Transray-convexity

9. Relationships for multiproduct cost functions • Four relationships developed by Baumol (1977) First, declining ray average cost is not necessary for strict subadditivity. Second, strict concavity of a cost function is not sufficient to guarantee subadditivity. Third, scale of economies are neither necessary nor sufficient for subadditivity. Fourth, if a cost function exhibits strictly declining ray average costs and transray convexity along any one hyperplane, then the cost function is subadditive for output q.

10. Multi-product problems • Entry for some products but not others • If implementing a pricing structure, how should you charge different prices for the different products? • Such questions are addressed in applications to the communication and transportation industries.

11. Empirical Findings • Evans and Heckman (1984) developed a method of testing for subadditivity, and they applied it to the Bell System for the years 1958-77. However, they rejected local and global subadditivity for the Bell System. • Mayo (1984) reported diseconomies of scale at large output rates, as well as diseconomies of scope for the case of electricity, gas, and combination gas-electricity companies. • Chappell and Wilder (1986) reestimated the model, excluding utilities with nuclear facilities, and found economies of scale and scope.