<p>there are 3 goods. If she buys more of <span class="math-container">$x_1$</span>, she will buy less of the composite good <span class="math-container">$k$</span>.</p>
<p>Notice that as <span class="math-container">$x_1 = a-p_1$</span> and <span class="math-container">$x_2 = a - p_2$</span>, you have that:
<span class="math-container">$$
\begin{align*}
k &amp;= I - p_1 x_1 - p_2 x_2,\\
&amp;= I - p_1 a + (p_1)^2 - p_2 a + (p_2)^2,
\end{align*}
$$</span>
So, for example, an increase in <span class="math-container">$x_1$</span> due to a price decrease of <span class="math-container">$p_1$</span> will cause a decrease in the amount <span class="math-container">$k$</span>.</p>

CC BY-SA 4.0
2021-06-15T18:17:01.237Z
2021-06-15T18:47:16.267Z
2021-06-15T18:47:16.267Z
tdm
<p>Suppose a representative consumer has the following quasilinear utility function:
U_i(x_1,x_2)=ax_1+ax_2-(1/2)*[(x_1)^2+(x_2)^2] + k</p>
<p>where a&gt;0 is a utility parameter, x_1 and x_2 are the goods, and k is &quot;all other goods.&quot;</p>
<p>Maximizing U_i s.t. I=p_1<em>x_1+p_2</em>x_2+k yields the inverse demand functions
p_1=a-x_1, p_2=a-x_2.</p>
<p>My question is, how could it be that you have a representative consumer with  a budget, and yet he need not give up one of the goods when he buys more of the other one. That is, according to the demand functions, x_1 and x_2 are independent. But how is that possible when the consumer spends his entire budget? You would think that buying more of one good would necessitate giving up the other.</p>
<p>Now, I understand that this has something to do with the quasi-linearity of the utility function, but it is unclear to me what exactly.
I would be grateful any helpful comments.</p>

CC BY-SA 4.0
2021-06-15T18:28:09.920Z
Unless there is a typo, this utility function is convex and the maximization problem does not yield interior solutions.
Giskard
<utility><consumer-theory>
How come a representative consumer with quasilinear utility need not economize?
Ohad Osterreicher
# How come a representative consumer with quasilinear utility need not economize?

(Asked by: username_0 on 2021-06-15)

Suppose a representative consumer has the following quasilinear utility function:
U\_i(x\_1,x\_2)=ax\_1+ax\_2-(1/2)\*[(x\_1)^2+(x\_2)^2] + k

where a>0 is a utility parameter, x\_1 and x\_2 are the goods, and k is "all other goods."

Maximizing U\_i s.t. I=p\_1*x\_1+p\_2*x\_2+k yields the inverse demand functions
p\_1=a-x\_1, p\_2=a-x\_2.

My question is, how could it be that you have a representative consumer with a budget, and yet he need not give up one of the goods when he buys more of the other one. That is, according to the demand functions, x\_1 and x\_2 are independent. But how is that possible when the consumer spends his entire budget? You would think that buying more of one good would necessitate giving up the other.

Now, I understand that this has something to do with the quasi-linearity of the utility function, but it is unclear to me what exactly.
I would be grateful any helpful comments.

**Tags:** utility, consumer-theory
> **Comments:**
> **username_1**: Unless there is a typo, this utility function is convex and the maximization problem does not yield interior solutions.  

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## 1 answers

### Answer by username_2 on 2021-06-15. Score: 2 | Accepted answer

there are 3 goods. If she buys more of $x\_1$, she will buy less of the composite good $k$.

Notice that as $x\_1 = a-p\_1$ and $x\_2 = a - p\_2$, you have that:
$$
\begin{align\*}
k &= I - p\_1 x\_1 - p\_2 x\_2,\\
&= I - p\_1 a + (p\_1)^2 - p\_2 a + (p\_2)^2,
\end{align\*}
$$
So, for example, an increase in $x\_1$ due to a price decrease of $p\_1$ will cause a decrease in the amount $k$.


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