## Maximization problem with two constraints

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
$\Delta p = p_H - p_L$