The optimal contract maximizes the entrepreneur's expected compensation:

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

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

and the investor's breakeven constraint:

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 \{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.

p_H - entrepreneur's effort
1-p_H - no effort
R_b^S - return for borrower if project is success
R_b^F - return for borrower if project fails
BI - benefit per unit of investment, when no effort
I - investment
A - cash holding
R^SI - return on investment if success
R^FI - return on investment if failure
\Delta p = p_H - p_L