Technical Appendix to Enduring Relationships in an Economy with Capital and Private Information

In this appendix, we introduce a more general allocation and demonstrate the equal treatment property. All the proofs are in section 2. Let C ⊆ R be the domain of the utility function, so C = (0,∞) for log utility and γ < 0 and [0,∞) for γ > 0. By C − x we will mean the algebraic sum of the sets C and {−x}. Similarly, define V = u(C). At any given time t , the principal knows an agent’s initial promised utility w0, a history of reports ht−1, and vectors of allocated capital (kt−1) and consumption transfers (bt−1). Then an randomized allocation is defined as: