Deriving a parameter in optimization problem

Question

My question:

My solution for the part (i)

Hopefully my solution is correct. Especially check the budget constraint.

If it is correct, then my actual question to you is the second part (i). I cannot derive $\alpha$ In this part, only help me derive alpha. I can do the rest of them. I cannot derive alpha. For that please give me a hint or a solution thank you.

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Answer 1

Hint for (i): The budget constraint is $$y-[\alpha(c+p_1)+(1-\alpha)p_1]x_1-p_2x_2$$

The extra cost $c$ hat to be added to $p_1$

And the partial derivatives of the lagrange function are

$$\frac{\partial \mathcal L}{\partial x_1}=\alpha\cdot qx_1+(1-\alpha)x_2-\lambda[\alpha(c+p_1)+(1-\alpha)p_1] =0$$

$$\frac{\partial \mathcal L}{\partial x_2}=\alpha\cdot qx_1+(1-\alpha)x_1-\lambda p_2 =0$$

$$\frac{\partial \mathcal L}{\partial \lambda}=y-[\alpha(c+p_1)+(1-\alpha)p_1]x_1-p_2x_2=0$$