# Why do we optimize utility from X1 according to itself?

In lecturer's notes we have a utility function

$$U_i = X_i^A * (1-L_i)^{1-a},$$ $$0< a< 1$$, $$i = 1,2$$

$$MP_1 = w_1$$

$$MP_2 = w_2$$

$$X_1 + X_2 = w_1 * L_1 + w_2 * L_2$$

And we need to form a Lagrangian function to optimize for these variables. In his notes our Lagrangian function is:

$$\zeta = X_1^a * (1-L_1)^{1-a} - \lambda(\bar{u} - X_2^a * (1-L_2)^{1-a} - \mu [X_1 + X_2 - w_1 * L_1 - w_2 * L_2]$$

and he differentiates according to that lagrangian function. However I suppose that:

$$X_1^a * (1-L_1)^{1-a} + X_2^a * (1-L_2)^{1-a} = \bar{u}$$

So his lagrangian functions becomes something like:

$$\zeta = X_1^a * (1-L_1)^{1-a} - \lambda(X_1^a * (1-L_1)^{1-a} - \mu [X_1 + X_2 - w_1 * L_1 - w_2 * L_2]$$

which does not make any sense for me. I feel like it should have been something like:

$$\zeta = X_1^a * (1-L_1)^{1-a} - \lambda(\bar{u} - X_1^a * (1-L_1)^{1-a} - \mu [X_1 + X_2 - w_1 * L_1 - w_2 * L_2]$$

So we can optimize it both for $$X_1$$ and $$X_2$$ I would be so happy if you could help me with the theory and logic behind it.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. Commented Apr 2, 2022 at 6:29
• @Giskard It's really difficult for me to specify the problem than I already did above, but I will give it another try. I cannot see why we used both the $X_1^a$ and the $\lambda * X_1^a - (1- L_1)^{1-a}$ in the lagrangian function. How can a variable can become its own constraint. Commented Apr 2, 2022 at 7:38
• What goal function are you trying to maximize? Commented Apr 2, 2022 at 9:20
• @Giskard $U_i$ I suppose Commented Apr 3, 2022 at 15:20
• It seems like that is indexed notation denoting two functions. Commented Apr 3, 2022 at 15:46