t minimization requires making sure that the repeated traffic has the least-cost set of links along its path. In contrast, suppose that the costs of links are not linear in traffic but instead exhibit economies of scale. Suppose, for example, that a link ij has a fixed cost fij as well as a cost per unit of traffic Cij. Then, depending on the relative levels of these costs, it might be worth avoiding the creation of some links in order not to bear the fixed cost. This means that longer routes could be more attractive. The result is a concentration of traffic on a smaller set of links, which lowers unit costs on those links and yields economies of scale for the cost function overall. 14 To illustrate the effects of economies of scale at the level of individual links, consider a network with four nodes, N = {1, 2, 3, 4}. The nodes can be connected directly at a constant cost per unit of traffic, C. The traffic patterns are given by city-pair outputs, Qij, with every node requiring direct or indirect connections with every other node. The solution is that a complete point-to-point network is needed because direct connections are the shortest paths. Total cost is therefore 14 For a related discussion of economies of scale and network design, see Robertazzi (1999) and Yaged (1972). The Design and Costs of Networks represented by the constant-returns-to-scale cost function C(Q12, Q13, Q14, Q23, Q24, Q34) =cQ12+cQ13+cQ14+cQ23+cQ24 Q34, where L is the set of links {(1,2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} Now suppose that establishing a link requires a fixed cost f. Suppose that the traffic requirements are evenly distributed: Qij = Q for all pairs (i,j) in the set of links. Suppose also that fixed costs are sufficiently large relative to variable costs, f>2cQ, SO that it is less costly to add more traffic on three links, 3cQ, than to add a direct link, f + cQ. Then we are back at the problem of choosing the least-cost tree that connects all the nodes. By the symmetry of the problem, any such tree will do, say for 