To “ a Unique Costly Contemplation Representation ”

This online supplement provides additional material to accompany the printed paper, henceforth denoted ES. In Section S.1, we review some mathematical background on Fenchel duality used in the proof of the representation and uniqueness results in ES. In Section S.2, we establish the necessity of the L-continuity axiom for the signed RFCC representation defined in Appendix C. Section S.3 includes the details of the proof of Proposition 1, which constructs the function V in a signed RFCC representation. Finally, in Section S.4, we give a proof of Equation (24), which is used in the proof of Theorem 6, characterizing constant cost functions in a signed RFCC representation.