Utility Maximizing Problem

For consumer utility maximization, suppose a consumer wishes to buy 2 goods. Money does not bring utility directly, but he faces a budget constraint.

The consumer’s problem is

$$maxu(x,y)s.t.p_{x}x+p_{y}y\le w$$

And we have to apply to Largrangian Multiplier $\lambda$,

$$L=u(x,y)+\lambda (w-p_{x}x-p_{y}y)$$

Then take derivates with respect to the control variables in order to get the FOCs.

The results would give you $\frac{u_x(x,y)}{p_x}=\frac{u_y(x,y)}{p_y}$, which means spending \ \epsilon$moreon$x$oron$y$ should give you the same Marginal Utility