# how to solve such UMP where utility function is quasi-linear with cobb-douglas function as the non-linear part [closed]

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.

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