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The model is one of endogenous reputations. One possible outcome of the model is that types that can choose project choose bad projects to start with, gain a reputation through luckily not defaulting, have sufficient value from their reputation to behave (i.e. choose good projects) for some periods, and then when insufficient time remains they choose bad projects again. The solution concept is [http://en.wikipedia.org/wiki/Sequential_equilibrium Sequential Equilibrium]

To characterize the equilibrium we need to specific the actions of <mathGBmath>GB\,</math> types and show that the rate sequences are best responses to these actions.This is done through continuation values for the <math>GB\,</math> types in periods <math>T\,</math> (where <math>GB\,</math> types will choose the bad project) and <math>t<T\,</math> as follows (where the subscripts indicate choosing that action for period <math>t\,</math> but then continuing with optimal choices forward): <math>V_{T}=\beta \pi (B-r_{T})\,</math>  :<math>V_{t}^{b}=\beta \pi (B-r_{t}+V_{t+1})\,</math>  :<math>V_{t}^{g}=\beta (G-r_{t}+V_{t+1})\,</math>  In equilibrium it must be that: :<math>V_{t}=\max \{V_{t}^{b},V_{t}^{g}\}\,</math>  So the good project is choosen at time <math>t\,</math> iff: :<math>\beta(G-r_{t}+V_{t+1})\geq \beta \pi (B-r_{t}+V_{t+1}) \; \therefore r_{t}-V_{t+1}\leq \frac{G-\pi B}{1-\pi}\,</math>  Which leads to the lemma: Lemma: Good projects are optimal at time <math>t\,</math> if and only if <math>(1-\pi )V_{t}\geq \beta \pi (B-G)\,</math>  Notice that as <math>r_t\,</math> goes down good projects become more attractive (holding <math>V_{t+1}\,</math> constant). In addition <math>V_t\,</math> goes up, as does <math>V_{t-1}\,</math>, meaning that if a good project is choosen at time <math>t\,</math>, it will be choose \forall <math>t' < t\,</math>. However, for a good to be choosen at all reputation effects must kick in. This is in the next lemma and its proof.  Lemma: Type <math>BG\,</math> borrowers will select the safe project on some date <math>t\,</math> only if (i.e. a necessary condition): :<math>\frac{\beta (G-r)}{1-\beta }\geq \frac{\beta \pi (B-r)}{1-\beta \pi }\,</math>  The proof is as follows: Choosing a good project at time <math>t'\,</math> means that this condition is satisfied: :<math>r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\,</math>  A bad project is choosen in the last period, so choosing bad projects in all periods after <math>t'\,</math> implies: :<math>V_{t^{\prime }+1}=\sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r_{t})\leq \sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r)<\sum_{t=1}^{\infty }(\beta \pi )^{t}(B-r)=\frac{\beta \pi (B-r)}{1-\beta\pi }\,</math>  And as :<math>r_{t}\geq r\,</math>: :<math>r-\frac{\beta \pi (B-r)}{1-\beta \pi }<\frac{G-\pi B}{1-\pi }\,</math> which gives the result.  A sufficient condition on the interest rates follows (it's proof and feasibility is ignored here): Fix <math>t^{\prime }\,</math>. If for all <math>t\in \{t^{\prime },t^{\prime}+1,...,T\}\,</math>: :<math>r_{t}<\beta G+\frac{(1-\beta )(G-\pi B)}{(1-\pi )}\,</math> then there exists <math>T<\infty\,</math> such that: :<math>r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\,</math> and a safe project is chosen at <math>t^{\prime }\,</math>  We then need the Bayesian rational interest rates. These depend on the number of survivers of <math>B\,</math> types and the actions (and survivors) of <math>GB\,</math> types. For the <math>G\,</math> types: :<math>f_{Gt}=f_{G}\,</math>  For the <math>B\,</math> types: :<math>f_{Bt}=\pi ^{t-1}f_{B}\,</math>  For the <math>GB\,</math> types, if in period <math>t\,</math> they choose <math>G\,</math> with probability <math>\sigma \in [0,1]\,</math>: <math>f_{BGt}=\sigma f_{BGt-1}+(1-\sigma )\pi f_{BGt-1}\,</math>  Which has bounds: :<math>r_{t}^{b}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+\pi f_{BGt}+f_{G}}\,</math>  and  :<math>r_{t}^{g}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+f_{BGt}+f_{G}}\,</math>  using the notation from before.  We now (ab)use measure theory to get three results. The (ab)uses are by assuming that some population is of measure zero but that by continuity we can allow some positive measure into the population and the result will hold. First we solve the end game with <math>f_{BG}=0\,</math> - the end game being the part of the game from which the <math>GB\,</math> types choose bad projects forward to infinity. If the end game is bounded then the part of the game in which reputation works is unbounded. With <math>f_{BG}=0\,</math> the interest rates are deterministic and tend towards r: :<math>r_{t}=r\cdot \frac{\pi ^{t-1}f_{B}+1-f_{B}}{\pi ^{t}f_{B}+1-f_{B}}\,</math> Building on this there is the case of moral hazard without adverse selection - that is there are no bad types. In this case we get the following proposition: '''Proposition''': With no adverse selection interest rates are constant, and if reputation ever works it works immediately at <math>t=1\,</math> and stops working at some <math>t^{\prime }<T\,</math>. With adverse selection as well we get the alternative proposition: '''Proposition''': If <math>r_{t}\,</math> falls over time and a type <math>GB\,</math> borrower optimally selects good projects at time <math>t^{\prime \prime }\,</math> and bad projects at some <math>t^{\prime }<t^{\prime \prime }\,</math>, then bad projects are optimal for all <math>t<t^{\prime }\,</math>. This implies that if good projects are optimal at two dates <math>t_{1}<t_{2}\,</math>, then good projects are optimal for all <math>t\in \{t_{1},t_{1}+1,...,t_{2}\}\,</math>. For further detail see either the paper or Tadelis' write-up of the paper:*Tadelis, Steve (2007), "Topics in Contracts and Organizations: Lecture Notes", UC Berkeley, September [http://www.edegan.com/repository/Tadelis%20(2007)%20-%20Topics%20in%20Contracts%20and%20Organizations%20Lecture%20Notes.pdf pdf] [http://www.edegan.com/repository/Tadelis%20(2007)%20-%20Topics%20in%20Contracts%20and%20Organizations%20Lecture%20Notes.tex tex]