Suppose that consumer's utility function for three goods is separable, that is,
$U(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ ...(i)
where $f_i$ is increasing and strictly concave, i=1,2,3. Show that
- $ df_i/dP_1 $ for i = 2,3 can be either positive or negative, but they must be of the same sign where $f_i$ is the demand function for good i and $P_1$ is price of good 1.
i reasoned using the equimarginal principle where at optimum we must have
$ MU_1/P1 = MU_2/P2 = MU_3/P3 $.
suppose good 1 is normal, if $P_1$ rises demand for good 1 falls, that is $df_1/dP_1$ < 0. Due to law of diminishing marginal utility $MU_1$ rises, however to maintain equilibrium $ MU_2, MU_3$ must also rise. So, $df_2/dp_1 < 0$ and $ df_3/dP_1 < 0$. But i couldn't do it more mathematically, taking the derivative of (i) w.r.t $P_1$ isn't of much use, please help.