<p>Regarding the first question, @Oliv suggestion in the comments to formally write down the consumer's problem, budget constraint, <em>Kunh-Tucker</em> multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:</p>

<p>Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function</p>

<p>$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3$$</p>

<p>will take its maximum value if we purchase only good $x_3$? For any tiny amount $\Delta x_3$ of $x_3$ that you don't purchase, you lose $2\Delta x_3$ in utility terms, while you gain (in utility terms) only $\Delta x_1 + \Delta x_2 = \Delta x_3$ for any combination of $\Delta x_1$ and $\Delta x_2$. Since $2\Delta x_3 &gt; \Delta x_3$ it will be suboptimal to purchase anything else than $x_3$. That was the "plain logic" part, and I hope it answers the "<em>what I don't understand</em>..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.</p>

CC BY-SA 3.0
2015-06-19T12:21:19.303Z
2015-06-19T17:27:29.783Z
2015-06-19T17:27:29.783Z
Alecos Papadopoulos
<ol>
<li><p>I have to find the consumer's demand at prices $\textbf{p} = (1,1,1)$
and budget $m=4$ when the utility function is given by
$$u(x_1,x_2,x_3) = 3 \sqrt{x_1+x_2+2x_3}$$ or when transformed
$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3 $$</p>

<p>I know the answer is $\textbf{x} = (0,0,4)$</p></li>
</ol>

<p><em>What I don't understand:</em> For perfect substitutes the demand is given by $x_1^* = \frac{m}{p_1}$ so the demand should have been
   $\textbf{x} = (4,4,4)$. But why is $x_1^*,x_2^* = 0$?</p>

<ol start="2">
<li><p>For the same utility function I want to find the demand at prices
$p=(2,3,p_3) &gt;&gt;0$ and budget $m&gt;0$.</p>

<p>The answer is  \begin{equation} \textbf{x} = \left(0,0,
    \frac{m}{p_3} \right) \quad \text{when} \quad p_3&lt;4 \end{equation}
\begin{equation} \textbf{x} = \left(\frac{m}{2},0,0 \right) \quad
    \text{when} \quad p_3&gt;4 \end{equation} \begin{equation} \textbf{x} =
    \left(\frac{\alpha m}{2}, 0, \frac{(1-\alpha)m}{4} \right) \quad
    \text{when} \quad p_3=4 \end{equation}
where $\alpha \in [0,1]$ because the consumer is indifferent between the bundles</p></li>
</ol>

<p><em>What I don't understand:</em> Why is the demand different depending on what $p_3$ is and how do I know when the demand is different? Personally, I would've said the demand is $\textbf{x} = \left( \frac{m}{2}, \ \frac{m}{3}, \ \frac{m}{p_3} \right)$</p>

CC BY-SA 3.0
2015-06-18T16:09:00.820Z
What have you tried to do? Have you written and solved the consumer's problem? Otherwise, where does the formula $x_1^{*}=\dfrac{m}{p_1}$ come from? Hint: do you think the bundle $x=(4,4,4)$ is feasible when $p=(1,1,1)$ and $m=4$?
Oliv
CC BY-SA 3.0
2015-06-18T16:22:10.420Z
$x_1^* = \frac{m}{p_1}$ is from my book (Intermediate Microeconomics, Varian). Hm, it's true that it isn't feasible. But it might as well have been $x=(1,1,2)$.
Stranqer95
CC BY-SA 3.0
2015-06-18T16:24:19.103Z
You should examine Varian a bit more carefully, that is not the demand function for all values of $p_1$.
Giskard
CC BY-SA 3.0
2015-06-18T16:27:26.497Z
Oh right, I found this the other day $x_1 = \frac{m}{p_1}$ when $p_1 <p_2$, $x_1 =$ any number between 0 and $m/p_1$ when $p_1 = p_2$ and $x_1 = 0$ when $p_1 > p_2$, but I wasn't sure how to apply it..
Stranqer95
CC BY-SA 3.0
2015-06-18T16:58:12.137Z
@Stranqer95 now you can generalize the formula for three goods instead of two and you'll get the result.
Oliv
CC BY-SA 3.0
2015-06-18T17:23:29.863Z
I can see why I get the demands stated above for $p_3 < 4$ and $p_3>4$. But I'm can't quite figure out the ones for $x=(0,0,4)$. Why is $x_1,x_2 = 0$. The book says "any number between 0 and $m/p_1$". Then couldn't it be $x=(1,1,2)$ as well. Also, for $p_3=4$ it's the same thing. Why does $x_1 = \frac{\alpha m}{2}$ and $x_2=0$ when it might as well be something else?
Stranqer95
CC BY-SA 3.0
2015-06-18T18:41:05.157Z
You are trying to apply a formula that holds in the two-goods case to a situation where there are three goods. Instead of using the formula you should try to understand the logic behind it: where does this result come from? I suggest that you try to write the consumer's problem and solve it, and report your progress here to receive feedback and advice.
Oliv
<supply-and-demand>
Demand for utility function given prices and budget
Stranqer95
# Demand for utility function given prices and budget

(Asked by: username_0 on 2015-06-18)

1. I have to find the consumer's demand at prices $\textbf{p} = (1,1,1)$
   and budget $m=4$ when the utility function is given by
   $$u(x\_1,x\_2,x\_3) = 3 \sqrt{x\_1+x\_2+2x\_3}$$ or when transformed
   $$\hat{u}(x\_1,x\_2,x\_3) = x\_1+x\_2+2x\_3 $$

   I know the answer is $\textbf{x} = (0,0,4)$

*What I don't understand:* For perfect substitutes the demand is given by $x\_1^\* = \frac{m}{p\_1}$ so the demand should have been
$\textbf{x} = (4,4,4)$. But why is $x\_1^\*,x\_2^\* = 0$?

2. For the same utility function I want to find the demand at prices
   $p=(2,3,p\_3) >>0$ and budget $m>0$.

   The answer is \begin{equation} \textbf{x} = \left(0,0,
   \frac{m}{p\_3} \right) \quad \text{when} \quad p\_3<4 \end{equation}
   \begin{equation} \textbf{x} = \left(\frac{m}{2},0,0 \right) \quad
   \text{when} \quad p\_3>4 \end{equation} \begin{equation} \textbf{x} =
   \left(\frac{\alpha m}{2}, 0, \frac{(1-\alpha)m}{4} \right) \quad
   \text{when} \quad p\_3=4 \end{equation}
   where $\alpha \in [0,1]$ because the consumer is indifferent between the bundles

*What I don't understand:* Why is the demand different depending on what $p\_3$ is and how do I know when the demand is different? Personally, I would've said the demand is $\textbf{x} = \left( \frac{m}{2}, \ \frac{m}{3}, \ \frac{m}{p\_3} \right)$

**Tags:** supply-and-demand

> **Comments:**
> **username_1**: What have you tried to do? Have you written and solved the consumer's problem? Otherwise, where does the formula $x_1^{*}=\dfrac{m}{p_1}$ come from? Hint: do you think the bundle $x=(4,4,4)$ is feasible when $p=(1,1,1)$ and $m=4$?  
> **username_0**: $x_1^* = \frac{m}{p_1}$ is from my book (Intermediate Microeconomics, Varian). Hm, it's true that it isn't feasible. But it might as well have been $x=(1,1,2)$.  
> **username_2**: You should examine Varian a bit more carefully, that is not the demand function for all values of $p_1$.  
> **username_0**: Oh right, I found this the other day $x_1 = \frac{m}{p_1}$ when $p_1 <p_2$, $x_1 =$ any number between 0 and $m/p_1$ when $p_1 = p_2$ and $x_1 = 0$ when $p_1 > p_2$, but I wasn't sure how to apply it..  
> **username_1**: @username_0 now you can generalize the formula for three goods instead of two and you'll get the result.  
> **username_0**: I can see why I get the demands stated above for $p_3 < 4$ and $p_3>4$. But I'm can't quite figure out the ones for $x=(0,0,4)$. Why is $x_1,x_2 = 0$. The book says "any number between 0 and $m/p_1$". Then couldn't it be $x=(1,1,2)$ as well. Also, for $p_3=4$ it's the same thing. Why does $x_1 = \frac{\alpha m}{2}$ and $x_2=0$ when it might as well be something else?  
> **username_1**: You are trying to apply a formula that holds in the two-goods case to a situation where there are three goods. Instead of using the formula you should try to understand the logic behind it: where does this result come from? I suggest that you try to write the consumer's problem and solve it, and report your progress here to receive feedback and advice.  

---

## 1 answers

### Answer by username_3 on 2015-06-19. Score: 1

Regarding the first question, @username_1 suggestion in the comments to formally write down the consumer's problem, budget constraint, *Kunh-Tucker* multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:

Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function

$$\hat{u}(x\_1,x\_2,x\_3) = x\_1+x\_2+2x\_3$$

will take its maximum value if we purchase only good $x\_3$? For any tiny amount $\Delta x\_3$ of $x\_3$ that you don't purchase, you lose $2\Delta x\_3$ in utility terms, while you gain (in utility terms) only $\Delta x\_1 + \Delta x\_2 = \Delta x\_3$ for any combination of $\Delta x\_1$ and $\Delta x\_2$. Since $2\Delta x\_3 > \Delta x\_3$ it will be suboptimal to purchase anything else than $x\_3$. That was the "plain logic" part, and I hope it answers the "*what I don't understand*..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.


---

