Problem. Find demand for this utility function: u(x,y,w,z)=2x^{\frac{1}{2}}y^{\frac{1}{2}}+\min(w,z)
Here we need to solve the following consumer’s problem:
\displaystyle\max_{x,y,w,z} \ 2x^{\frac{1}{2}}y^{\frac{1}{2}}+\min(w,z) \\ \text{s.t. } p_Xx+p_Yy+p_Ww+p_Zz\leq M \\ \text{and } x\geq 0, \ y\geq 0, \ w\geq 0,\ z\geq 0
where p_X> 0, \ p_Y > 0, \ p_W > 0,\ p_Z > 0, \ M\geq 0 are given.
Solution. Solving this problem, we get the demand as:
(x^d,y^d,w^d,z^d)\in\begin{cases}\left\{\left(\dfrac{M}{2p_X},\dfrac{M}{2p_Y},0,0\right)\right\} & \text{if } \sqrt{p_Xp_Y}<p_W+p_Z \\ \left\{\left(0,0,\dfrac{M}{p_W+p_Z},\dfrac{M}{p_W+p_Z}\right)\right\} & \text{if } \sqrt{p_Xp_Y}>p_W+p_Z \\ \left\{\left(\dfrac{\alpha M}{2p_X},\dfrac{\alpha M}{2p_Y},\dfrac{(1-\alpha) M}{p_W+p_Z},\dfrac{(1-\alpha)M}{p_W+p_Z}\right)|0\leq\alpha\leq 1\right\} & \text{if } \sqrt{p_Xp_Y}=p_W+p_Z \end{cases}
Indirect utility function is V(p_X,p_Y,p_W,p_Z,M)=\dfrac{M}{\min(\sqrt{p_Xp_Y},p_W+p_Z)}
Problem. Suppose the utility function is \displaystyle u(x_1,\ldots,x_L, y,z) =\left(\sum_{i=1}^{L}\alpha_i\ln x_i\right)+y^\beta z^{1-\beta}, where \displaystyle\sum_{i=1}^{L}\alpha_i = 1 and \alpha_i > 0 for all i\in\{1,2,\ldots,L\}, and \beta\in (0,1). What is the solution to this consumer’s problem?
\displaystyle\max_{x_1,x_2,\ldots,x_L,y,z} \ \left(\sum_{i=1}^{L}\alpha_i\ln x_i\right)+y^\beta z^{1-\beta}
\displaystyle\text{s.t.} \ \sum_{i=1}^{L}p_ix_i+(p_Yy+p_Zz)\leq M
\text{and } x_1> 0, \ x_2> 0, \ldots, x_L> 0, y\geq 0, z\geq 0
where L \in\mathbb{N}, p_1>0, p_2>0,\ldots, p_L>0, p_Y>0, p_Z>0 and M> 0
Solution. Solving this problem we get the demand for x_i as:
\displaystyle x_i^d(p_1,p_2,\ldots,p_L,p_Y,p_Z,M) = \begin{cases} \dfrac{\alpha_iM}{p_i} & \text{if } \displaystyle M\leq \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \\ \dfrac{\alpha_ip_Y^\beta p_Z^{1-\beta}}{p_i\beta^\beta (1-\beta)^{1-\beta}} & \text{if } \displaystyle M> \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \end{cases}
and demand for y and z as:
\displaystyle y^d(p_1,p_2,\ldots,p_L,p_Y,p_Z,M) = \begin{cases} 0 & \text{if } \displaystyle M \leq \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \\ \dfrac{\beta}{p_Y}\left(M - \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}}\right) & \text{if } \displaystyle M > \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \end{cases}
\displaystyle z^d(p_1,p_2,\ldots,p_L,p_Y,p_Z,M) = \begin{cases} 0 & \text{if } \displaystyle M \leq \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \\ \left(\dfrac{1-\beta}{p_Z}\right)\left(M - \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}}\right) & \text{if } \displaystyle M > \dfrac{p_Y^\beta p_Z^{1-\beta}}{\beta^\beta (1-\beta)^{1-\beta}} \end{cases}
respectively.