Exercise 13.8

Reputation, Complete Asset Markets, And Reentry

⬅ Return

Problem

Extend the reputational model of Section 13.3.4 by allowing for the possibility of regaining access to international capital markets after default. Specifically, assume that with constant probability \(\delta\in(0,1)\) defaulters can reentry capital markets the next period.

  1. Derive the value function of a country in bad financial standing, \(v^b(\epsilon)\), as a function of current and future expected values of \(u(\bar{y}+\epsilon)\) and \(u(\bar{y}+\epsilon-d(\epsilon))\) only.

  2. Write down the incentive-compatibility constraint.

  3. Write down the optimization problem of the country and its associated Lagrangian.

  4. Derive the optimality conditions of the country’s problem.

  5. Show that all of the results of Section 13.3.4 pertaining to the reputation model hold under this extension.

  6. It is intuitively obvious that if \(\delta=1\), lending breaks down, since in this case lenders have no way to punish delinquent debtors. Show this result formally.

Answer

\[ \begin{eqnarray} v^b(\epsilon) &\equiv& u(\bar{y}+\epsilon) + \beta \int_{\epsilon^L}^{\epsilon^H} [(1-\delta)v^b(\epsilon') + \delta v^g(\epsilon')] \pi(\epsilon')d\epsilon' \nonumber \\ &=& u(\bar{y}+\epsilon) + \beta \int_{\epsilon^L}^{\epsilon^H} [(1-\delta)v^b(\epsilon') + \delta v^c(\epsilon')] \pi(\epsilon')d\epsilon' \nonumber \\\nonumber &=& u(\bar{y}+\epsilon) + \frac{\beta(1-\delta)}{1-\beta(1-\delta)} \int_{\epsilon^L}^{\epsilon^H} u(\bar{y}+\epsilon')\pi(\epsilon')d\epsilon' + \frac{\beta\delta}{1-\beta}\int_{\epsilon^L}^{\epsilon^H} u(\bar{y}+\epsilon'-d(\epsilon'))\pi(\epsilon')d\epsilon'. \end{eqnarray} \]