Changes

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 abuse measure theory to get three results. The abuses are by assuming that some population is of measure zero but that by continuity we can all some positive measure into the population and the result will hold.