# Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$ [duplicate]

Suppose preferences are represented by the following utility function

$$u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$$

Write the

1. Walrasian demand functions
2. Hicksian demand functions
3. Indirect utility function
4. Expenditure function

Remark

I come from a math background and my only experience with economics are (so far) the first three chapters of the book Microeconomic Theory by Mas-Colell, Whinston, and Green. Following their approach, I want to solve the following utility maximization problem

$$\max_{\vec{\mathbf{x}}\in \mathbb{R}^4} \{u(\vec{\mathbf{x}})\} = \max_{x_1,x_2,x_3,x_4} \{\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \} \}$$

s.t. $p_1 x_1 + p_2 x_2 = w$

to get the Walrasian demand functions $x_i(\vec{\mathbf{p}},w)=x_i(p_1,p_2,p_3,p_4,w)$. Then I could get the indirect utility function by substituting these solutions into the utility function: \begin{equation*} v(\vec{\mathbf{p}},w)=u(x_1^*(\vec{\mathbf{p}},w),x_2^*(\vec{\mathbf{p}},w),x_3^*(\vec{\mathbf{p}},w),x_4^*(\vec{\mathbf{p}},w)) \end{equation*} Similarly, to get the Hicksian demand functions $h_i(p_1,p_2,p_3,p_4,u)$, I would solve the expenditure minimization problem \begin{equation*} \min_{\vec{\mathbf{x}}} \{\vec{\mathbf{p}} \cdot \vec{\mathbf{x}}\} =\min_{x_1,x_2,x_3,x_4} \{p_1x_1+p_2x_2+p_3x_3+p_4x_4\} \end{equation*} and then get the expenditure function by multiplying \begin{align*} e(\vec{\mathbf{p}},u) &=\vec{\mathbf{p}} \cdot \vec{\mathbf{h}}^*(\vec{\mathbf{p}},u) \\ &=p_1 h_1^*(\vec{\mathbf{p}},u) +p_2 h_2^*(\vec{\mathbf{p}},u)+p_3 h_3^*(\vec{\mathbf{p}},u) + p_4 h_4^*(\vec{\mathbf{p}},u) \end{align*} Although I can visualize the entire process, I'm stuck at the beginning - solving the constrained utility maximization and expenditure minimization problems. Starting with the utility maximization problem, I want to write the Lagrangian \begin{equation*} \mathcal{L}(x_1,x_2,x_3,\lambda)=\alpha \min\{a x_1, b x_2 \} + \beta \min \{c x_3, d x_4\}+\lambda(w-p_1x_1-p_2x_2) \end{equation*} and take the first order conditions \begin{equation*} \frac{\partial u(x_1,x_2,x_3,x_4)}{\partial x_i}=\lambda p_i \end{equation*} however I don't know how, mathematically, to take the above first order conditions with respect to $x_i$ since $\min$ isn't everywhere differentiable. (However, as this answer points out, the partial derivatives of our objective function exist almost everywhere.)

This is my first experience with solving for demand functions, aside from the Cobb-Douglas example provided in the book Microeconomic Theory. Could anyone help me complete this example?

• economics.stackexchange.com/questions/2969/… should help you. In particular the first sentence "No, you should not use Lagrange multipliers here, but sound thinking." Nov 4, 2015 at 2:43
• Great. Now you need to solve for different cases. Assume that $ax_1 > bx_2$ and find the solutions. Then assume other way around and solve again. Similarly for $x_3$ and $x_4$. You will have different solution for different prices. Nov 4, 2015 at 3:28