# Maximization problem with two constraints

#### garymarkhov

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
$$\displaystyle \Delta p = p_H - p_L$$