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Electricity transmission has become the pivotal industry segment for electricity restructuring. Yet, little is known about the shape of transmission cost functions. Reasons for this can be a lack of consensus about the definition of transmission output and the complexitity of the relationship between optimal grid expansion and output expansion. Knowledge of transmission cost functions could help firms (Transcos) and regulators plan transmission expansion and could help design regulatory incentive mechanisms. We explore transmission cost functions when the transmission output is defined as point-to-point transactions or financial transmission right (FTR) obligations and particularly explore expansion under loop-flows. We test the behavior of FTR-based cost functions for distinct network topologies and find evidence that cost functions defined as FTR outputs are piecewise differentiable and that they contain sections with negative marginal costs. Simulations, however, illustrate that such unusual properties do not stand in the way of applying price-cap incentive mechanisms to real-world transmission expansion. Key words: Electricity transmission, cost function, incentive regulation, merchant investment