\(\displaystyle U_b = max_{\{R_b^S, R_b^F\}}\{ p_HR_b^S + (1-P_H)R_b^F-A \} \)

Subject to the entrepreneur's incentive constraint

\(\displaystyle (\Delta p)(R_b^S - R_b^F) \geq BI\)

and the investor's breakeven constraint:

\(\displaystyle p_H(R^SI -R_b^S)+(1-p_H)(R^FI-R_b^F) \geq I - A \)

My notes say that the optimal contract is \(\displaystyle \{R_b^S, R_b^F\} = \{ \frac{(p_HR +R^F -1)I +A}{p_H},0 \}\) but I have no idea how to set up the Lagrangian's to solve for this.

\(\displaystyle p_H\) - entrepreneur's effort

\(\displaystyle 1-p_H\) - no effort

\(\displaystyle R_b^S\) - return for borrower if project is success

\(\displaystyle R_b^F\) - return for borrower if project fails

\(\displaystyle BI\) - benefit per unit of investment, when no effort

\(\displaystyle I\) - investment

\(\displaystyle A\) - cash holding

\(\displaystyle R^SI\) - return on investment if success

\(\displaystyle R^FI\) - return on investment if failure

\(\displaystyle \Delta p = p_H - p_L\)