# Quasi-linear microeconomics problem

Hey guys I need your help to solve the following problem. Here's my attempt.

Given the function: $$u(x_1,x_2)=x_1+x_2^\beta, \ \ \beta \in (0,1).$$

• Write down the UMP and solve for the Walrasian Demands of $$x_1, x_2$$.
• Is good 1 a Normal good? And good 2?
• Compute the indirect utility function for this problem and show that it can be written as $$v(w,\mathbf{p})=\tilde{v}(w,p_1)$$.

My attempt: $$\mathcal{L}(\mathbf{x},\lambda)=x_1+x_2^\beta-\lambda(p_1x_1+p_2x_2-w).$$ $$\frac{\partial\mathcal{L}}{\partial x_1}=1-\lambda p_1 = 0.$$ $$\frac{\partial\mathcal{L}}{\partial x_2}=\beta x_2^{(\beta - 1)}-\lambda p_2 = 0$$ $$\frac{\partial\mathcal{L}}{\partial \lambda}=p_1x_1+p_2x_2=w$$ By solving the system I end up with: $$x_1^*(\mathbf{p},w)=\frac{w}{p_1}-(\frac{p_2}{p_1})^{\frac{\beta}{\beta -1}}\cdot \frac{1}{\beta^{\frac{1}{\beta - 1}}} \\ x_2^*(\mathbf{p},w)=(\frac{p_2}{\beta p_1})^{\frac{1}{\beta -1}}$$ A good is defined normal if $$\frac{\partial x_i^*}{\partial w} > 0.$$ $$\frac{\partial x_1^*}{\partial w} = \frac{1}{p_1} > 0.$$ Good 1 is a normal good. Good 2 is not a function of $$w$$, implying that $$x_2$$ is a neutral good.

Indirect utility function: $$v(\mathbf{p},w) \equiv u(x_1^*,x_2^*)$$ $$v(\mathbf{p},w)= \frac{w}{p_1}-(\frac{p_2}{p_1})^{\frac{\beta}{\beta -1}}\cdot \frac{1}{\beta^{\frac{1}{\beta - 1}}}+((\frac{p_2}{\beta p_1})^{\frac{1}{\beta -1}})^\beta .$$ The indirect utility function is function of both $$p_1,p_2$$, how can I show that is function of $$p_1$$ only?

• Why would it only be a function of p1? P2 affects the indirect utility as well... Commented May 26 at 14:19
• It's what the exercise asked. It may be an error of the professor. Commented May 31 at 0:19

Here is the utility maximisation problem: $$\begin{eqnarray*}\max_{(x_1,x_2)\in\mathbb{R}^2_+} & x_1+x_2^\beta \\ \text{s.t. } & p_1x_1+p_2x_2\leq M\end{eqnarray*}$$ where $$p_1>0, p_2>0, M\geq 0, \beta\in (0,1)$$ are given. Solving this problem, we get demand as: $$\begin{eqnarray*}(x_1^d,x_2^d)(p_1,p_2,M)=\begin{cases} \left(0,\frac{M}{p_2}\right) & \text{if } M Indirect Utility function is $$v(p_1,p_2,M)=\frac{\max\left\{M-p_2\left(\frac{\beta p_1}{p_2}\right)^{\frac{1}{1-\beta}},0\right\}}{p_1}+\left(\min\left\{\left(\frac{\beta p_1}{p_2}\right)^{\frac{1}{1-\beta}},\frac{M}{p_2}\right\}\right)^\beta$$
• Thanks for your response but the indirect utility function is still function of $p_2$ Commented May 18 at 10:02