Ok, I'm running up against my deadline and am totally stuck on this utility maximization problem.
$$U=-\frac1x-\frac1y-\frac1z$$ subject to $$I=P_xx+P_yy+P_zz$$
where $P_x$, $P_y$ and $P_z$ are the prices of $X$, $Y$ and $Z$ with $I$ being some budget. I set up my Lagrangian as follows and got the following first order conditions:
$$\mathcal L=-\frac1x-\frac1y-\frac1z+\lambda[I-P_xx-P_yy-P_zz]$$
$\mathcal L_x=1/x^2-\lambda P_x=0.$
$\mathcal L_y=1/y^2-\lambda P_y=0.$
$\mathcal L_z=1/z^2-\lambda P_z=0.$
$\mathcal L_\lambda=I-P_xx-P_yy-P_zz=0.$
Any thoughts on how to proceed? I know you should set $\lambda = \lambda$ but the math is just not working. Thanks!
$$x = \sqrt{\frac1{\lambda P_x}} = \frac1{\sqrt\lambda}\cdot{1\over\sqrt{P_x}},$$ and analagous for other variables.
Substituting into the $\mathcal L_\lambda$ equality (and factoring), $$I=\frac{P_x}{\sqrt{\lambda P_x}}+\frac{P_y}{\sqrt{\lambda P_y}}+\frac{P_z}{\sqrt{\lambda P_y}} \Longrightarrow {1\over\sqrt\lambda} = {I\over \sqrt{P_x}+\sqrt{P_y}+\sqrt{P_z}}.$$
Substitute this back into the expressions for each of your variables, and you should have your answer.