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?

1 Answer

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).

It makes sense that to maximize the utility MRS should be equal to marginal productivity since MRS should be equal to ratio of prices and in case like this where the person produces goods for themselves the prices will equal to the marginal costs of production which in turn will also equal to the marginal product (essentially producing for yourself is like being in a perfectly competitive market). The intuition is that person will produce good only to the extent the marginal utility of consumption becomes equal to marginal disutility of production, as if you are producing less you can increase utility by producing more and if you are producing more than that you can increase utility by producing less. At least that’s my understanding of the intuition behind this kind of model.

• 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
• For the first question, I more or less needed confirmation the text was poorly worded. Appreciate the help. – aisync Jan 9 at 8:22
• @Aisync I personally like to think about the first term about marginal rate of technical substitution but since only 1 good is produced one part of fraction in this case denominator is 1 instead of having the marginal product of the other good. Also yes the g gives you the negative effect of work on utility - I would hesitate to call it externality since that word has a very special meaning in public/welfare economics, I think it’s better to call it disutility from work/production (it represents the fact that most people don’t enjoy work for works sake) – 1muflon1 Jan 9 at 13:35
• Sorry I meant to say numerator – 1muflon1 Jan 9 at 16:32