If economic analysis is to be at all useful, then the multiproduct setting must be addressed. 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.

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.

Economies of scale for multi-product firm Consider an input-output vector (x 1, …, x r, q 1, …., q n ), where x k is input k, k = 1,…,r and scalars w> 1 and δ>0. Strict Economies of scale exist if (wx 1, …, wx r, v 1 q 1, …, v n q n ) is a feasible input-output vector, where all v i = w + δ. Thus, an expansion of all inputs by w implies a greater expansion of all outputs.

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 q 1, …., q n all in a single firm, with the total cost of producing each output q i, I =1,…,n, in separate firms, each specializing in the production of one product.

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(vq 1, …, vq n )/v < C(wq 1, …, wq n )/w for v>w, where v and w are measures of the scale of output along a ray through output vector q = (q 1, …, q n ).

Declining ray average cost

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

Transray-convexity

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.

Empirical Findings Evans and Heckman (1984) developed a method of testing for subadditivity, and they applied it to the Bell System for the years 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.

Barriers to Entry With entry barriers, marginal-cost pricing for all outputs is still necessary for economic efficiency, but the profit-maximizing monopolist seeks to equate marginal revenue and marginal cost in each output market. If ray average costs are strictly decreasing, the regulator can choose to enforce marginal-cost pricing and subsidize the firm, in which case there will be no incentive for entry unless subsidies are available to all. Or the firm may be allowed to operate free of regulation, in which case the threat of entry will force prices to a level at which the firm will just break even. If ray average cost are not strictly decreasing, the regulator may be required to protect the monopoly from attack, because marginal-cost pricing can lead to positive profits and the threat of entry.

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.