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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

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These are proper Marshallian demand functions, even though Income does not appear in them. This is due to specific form of the utility function (and the candidate solution of all goods being purchased at strictly positive quantities). It emerges that there is no income effect for goods $x_1$ and $x_2$ - optimal uncompensated demand does not depend after all on the level of income, but only on the relative prices.

The reason why Marshallian demand is defined as it is, is to make clear that it does not include any kind of "income compensation" as Hicksian demand does. But this does not preclude a case like yours, which again, depends on the form of the utility function.

Thinking economically, what goods can you think of whose demand may realistically not depend on the level of income but only on relative prices? Answering this question would be useful in order to map the mathematical expression of the utility function to real-world economic phenomena.

Also, note that the specific utility function points to "quasi-linear" frameworks, where the good that enters additively, essentially functions as Income itself.

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