Lagrangefunktion
$$L(x_{11},x_{12},x_{21},x_{22},y_1,y_2, \lambda, \mu_{11},
\mu_{21}, \mu_2, \nu)$$
$$
= \ U_1(x_{11}, x_{12})$$
$$- \ \lambda\left(\overline{U}_2 - U_2(x_{21},x_{22})\right)$$
$$- \ \mu_{11} (x_{11} - y_1)$$
$$- \ \mu_{21} (x_{21} - y_1)$$
$$- \ \mu_2 (x_{12} + y_{22} - y_2)$$
$$- \ \nu F(y_1,y_2)$$
$$
{\partial L \over \partial x_{11}}
= {\partial U_1 \over \partial x_{11}} - \mu_{11} = 0
\Rightarrow
{\partial U_1 \over \partial x_{11}} = \mu_{11}
$$
$$
{\partial L \over \partial x_{12}} =
{\partial U_1 \over\partial x_{12}} - \mu_{2} = 0
\Rightarrow
{\partial U_1 \over\partial x_{12}} = \mu_{2}
$$
$$
{\partial L \over \partial x_{21}} = {\partial U_2 \over \partial x_{21}} - \mu_{21} = 0
\Rightarrow
{\partial U_2 \over \partial x_{21}} = \mu_{21}
$$
$$
{\partial L \over \partial x_{22}} = {\partial U_2 \over \partial x_{22}} - \mu_{2} = 0
\Rightarrow
{\partial U_2 \over \partial x_{22}} = \mu_{2}
$$
$${\partial L \over \partial y_1}
= \mu_{11} + \mu_{21} - \nu {\partial F \over \partial y_1} = 0
\Rightarrow
\mu_{11} + \mu_{21} = \nu {\partial F \over \partial y_1}
$$
$${\partial L \over \partial y_2}
= \mu_{2} - \nu {\partial F \over \partial y_2} = 0
\Rightarrow
\mu_{2} = \nu {\partial F \over \partial y_2}
$$
$${\mu_{11} + \mu_{21} \over \mu_2} = {\partial F / \partial y_1
\over \partial F / \partial y_2}$$
$${\mu_{11} + \mu_{21} \over \mu_2} = {{\partial U_1 \over \partial
x_{11}} + {\partial U_2 \over \partial x_{21}} \over \partial U_1 /
\partial x_{12}} = {{\partial U_1 \over \partial x_{11}} + {\partial
U_2 \over \partial x_{21}} \over \partial U_2 / \partial x_{22}}$$
$\mu_2$ kann als Preis des privaten Gutes interpretiert werden,
$\mu_{i1}$ ist der Preisbeitrag von Individuum $j$ zum
Produktionspreis des öffentlichen Gutes.