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Optimizing Lagrangian Function Subject to 4 Input/Output Constraints:
The objective function:
$$\text{utility}=U\left(x_{c}, y_{c}\right)$$
subject to,
$x_{o}=f\left(y_{i}\right)$
$y_{o}=g\left(x_{i}, x_{o}\right)$
$x_{c}+x_{i}=x_{o}+x^{*}$
$y_{c}+y_{i}=y_{o}+y^{*}$
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