I have the following utility function:
$$U(x_i)=x_1x_2+x_3$$
with budget constraint:
$$p_1x_1+p_2x_2+p_3x_3\leq I$$
I use the Kuhn-Tucker method to find the optimal choices of the Utility maximization problem. My equations are:
$$x_2-\lambda p_1+M_1=0$$ $$x_1-\lambda p_2+M_2=0$$
$$1-\lambda p_3+M_3=0$$
$$p_1M_1=0$$ $$p_2M_2=0$$ $$p_3M_3=0$$ $$p_1x_1+p_2x_2+p_3x_3-I=0$$
$$(M_1,M_2,M_3,\lambda \geq 0)$$
When I set $M_1=M_2=M_3=0$ (Lagrangian case), I got the optimal solutions for $x_1,x_2$ as:
$$x_1=\frac{p2}{p3}$$ and $$x_2=\frac{p_1}{p_3}$$
How could I construct a Marshallian demand function in this case? The optimal solutions haven't got the I (income) variable.
Is it correct to define a Marshallian demand function for good $x_1$ as: $x_1(p,I)=\frac{p_2}{p_3}$?
Marshallian demand function (named after Alfred Marshall) specifies what the consumer would buy in each price and income or wealth situation, assuming it perfectly solves the utility maximization problem