<p>Suppose preferences are represented by the following utility function</p>

<p>\begin{equation}
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 \}
\end{equation}</p>

<p>Write the</p>

<ol>
<li>Walrasian demand functions</li>
<li>Hicksian demand functions</li>
<li>Indirect utility function</li>
<li>Expenditure function</li>
</ol>

<hr>

<p><strong>Remark</strong></p>

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

<p>\begin{equation}
\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 \} \}
\end{equation}</p>

<p>s.t. $p_1 x_1 + p_2 x_2 = w$</p>

<p>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)
&amp;=\vec{\mathbf{p}} \cdot \vec{\mathbf{h}}^*(\vec{\mathbf{p}},u)
 \\
&amp;=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 <a href="https://math.stackexchange.com/questions/150960/derivative-of-the-fx-y-minx-y">this answer points out</a>, the partial derivatives of our objective function exist <em>almost</em> everywhere.)</p>

<p>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?</p>

CC BY-SA 3.0
2015-11-04T02:43:02.500Z
http://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function?rq=1 should help you. In particular  the first sentence "No, you should not use Lagrange multipliers here, but sound thinking."
Martin Van der Linden
CC BY-SA 3.0
2015-11-04T03:28:28.893Z
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.
Sher Afghan
<microeconomics><utility><consumer-theory><optimization>
Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$
EthanAlvaree
# Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$

(Asked by: username_0 on 2015-11-04)

Suppose preferences are represented by the following utility function

\begin{equation}
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 \}
\end{equation}

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

\begin{equation}
\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 \} \}
\end{equation}

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](https://math.stackexchange.com/questions/150960/derivative-of-the-fx-y-minx-y), 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?

**Tags:** microeconomics, utility, consumer-theory, optimization
> **Comments:**
> **username_1**: http://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function?rq=1 should help you. In particular  the first sentence "No, you should not use Lagrange multipliers here, but sound thinking."  
> **username_2**: 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.  

---

## 0 answers

