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U =$X_1+X_2^aX_3^{1-a}$ $a ∈[0,1]$

$s.t. p·x≤w , x≥0 $

I have tried FOC for x1 x2 x3 and λ, but I cannot get two pairs of equalities separately in order to express two unknowns as a function of the third from the FOC. (for example, express x1 with x2 and x3) So I cannot substitute the solutions into the budget constraint so that we only have one unknown.

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Try cases:

  1. $x_2 = 0$ or $x_3 = 0$, then consume $x_1$ only.

  2. If only $x_1 = 0$, solve for the Walrasian for Cobb Douglas

  3. All $x$ are non-zeros. I have a feeling you might reach a contradiction in the Kuhn-Tucker conditions, so you will be able to rule out this case.

Anyhow, you need to work this out on your own.

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  • $\begingroup$ This is incorrect, there are solutions where all x are non-zero. They occur when $p_1 = \min_{x_2,x_3}\{p_2x_2 + p_3x_3 \lvert x_2^\alpha x_3^{1-\alpha} \geq 1 \}$. $\endgroup$ Nov 4 '21 at 17:48

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