The optimal contract maximizes the entrepreneur's expected compensation:

$\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

$\displaystyle \Delta p = p_H - p_L$