# Optimizing Lagrangian Function Subject to 4 Input/Output Constraints:

The objective function:

$$\text{utility}=U\left(x_{c}, y_{c}\right)$$

subject to,

1. $$x_{o}=f\left(y_{i}\right)$$
2. $$y_{o}=g\left(x_{i}, x_{o}\right)$$
3. $$x_{c}+x_{i}=x_{o}+x^{*}$$
4. $$y_{c}+y_{i}=y_{o}+y^{*}$$

where o represents output, i is inputs, c is consumed, and * represents initial quantity stocks. Lagrangian is combined and the first-order conditions are as follows: $$\begin{array}{l}{\partial \mathscr{L} / \partial x_{c}=U_{1}+\lambda_{3}=0} \\ {\partial \mathscr{L} / \partial y_{c}=U_{2}+\lambda_{4}=0} \\ {\partial \mathscr{L} / \partial x_{i}=\lambda_{2} g_{1}+\lambda_{3}=0} \\ {\partial \mathscr{L} / \partial y_{i}=\lambda_{1} f_{y}+\lambda_{4}=0} \\ {\partial \mathscr{L} / \partial x_{o}=-\lambda_{1}+\lambda_{2} g_{2}-\lambda_{3}=0} \\ {\partial \mathscr{L} / \partial y_{o}=-\lambda_{2}-\lambda_{4}=0}\end{array}$$

Simplification leads to the following expressions: $$M R S=\frac{U_{1}}{U_{2}}=\frac{\lambda_{3}}{\lambda_{4}}$$ $$M R S=\frac{\lambda_{3}}{\lambda_{4}}=\frac{\lambda_{2} g_{1}}{\lambda_{2}}=g_{1}$$ $$M R S=\frac{\lambda_{3}}{\lambda_{4}}=\frac{-\lambda_{1}+\lambda_{2} g_{2}}{\lambda_{4}}=\frac{-\lambda_{1}}{\lambda_{4}}+\frac{\lambda_{2} g_{2}}{\lambda_{4}}$$ $$=\frac{1}{f_{y}}-g_{2}$$

Q: The text states that optimality in y production requires the individual's MRS in consumption equal the marginal productivity of x in the production of y. I do not understand this statement. We are optimizing utility not the production function?

I think what it’s referring to optimal $$y$$ (i.e. what would be normally in calculus denoted as $$y^*$$- but you already use that for the initial stock).
• Does this make any sense to you: "The ﬁrst term in the expression, 1/fy represents the reciprocal of the marginal productivity of y in x production—this is the ﬁrst component of dy/dx as it relates to x production. The second term, g2, represents the negative impact that added x production has on y output—this is the second component of dy/dx as it relates to x production. This ﬁnal term occurs because of the need to consider the externality from x production. If $g_2$ were zero, then Eqns 19.11 & 19.12 be the same condition for efficient production, which would apply to both x and y." – aisync Jan 9 at 8:21