<blockquote>
  <p>Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?</p>
</blockquote>

<p>No, it is exactly the other way around.</p>

<p>The utility function in the two goods case will have the following form:</p>

<p><span class="math-container">$$U(x,y)=u(x)+y$$</span></p>

<p>We assume that <span class="math-container">$u'(x)&gt;0$</span> and <span class="math-container">$u''(x)&lt;0$</span> or marginal utility of <span class="math-container">$x$</span> is  decreasing in <span class="math-container">$x$</span>. <span class="math-container">$MU_y$</span> meanwhile is constant. 
Another assumption that is often not explicitly stated is that <span class="math-container">$\frac{MU_x}{p_x}&gt;\frac{MU_y}{p_y}$</span> for some <span class="math-container">$x$</span>. In words: There is some range of <span class="math-container">$x$</span> values at which the marginal utility of a dollar spent on <span class="math-container">$x$</span> exceeds the marginal utility of a dollar spent on <span class="math-container">$y$</span>. Now because we know <span class="math-container">$MU_x$</span> is decreasing in <span class="math-container">$x$</span>, we know that the preceding inequality will  hold true when the consumer is consuming a relatively small quantity of <span class="math-container">$x$</span>.</p>

<p>Now if we assume that the budget is very large, we know that the left side of the inequality gets smaller and smaller while the right side remains constant as the consumer consumes more and more of <span class="math-container">$x$</span>. Because of decreasing <span class="math-container">$MU_x$</span> a moment will come where a dollar spent on <span class="math-container">$y$</span> will generate at least as much utility as a dollar spent on <span class="math-container">$x$</span>. When this moment comes (the consumption of <span class="math-container">$x$</span> at which <span class="math-container">$\frac{MU_x}{p_x}=\frac{MU_y}{p_y}$</span> ), the consumer will switch to consuming <span class="math-container">$y$</span> with the rest of his budget.</p>

<p>When the consumer has a small budget however, he will never attain the point where the preceding equality holds true, that is, he is left consuming <span class="math-container">$x$</span> in a quantity for which <span class="math-container">$\frac{MU_x}{p_x}&gt;\frac{MU_y}{p_y}$</span> This is the case of insufficient wealth. In this case the consumer only consumes the nonlinear good, in this case denoted by <span class="math-container">$x$</span>.</p>

CC BY-SA 4.0
2019-08-07T14:32:28.390Z
Awesome, that was a very intuitive answer ;) thanks a lot
SimonDude
CC BY-SA 4.0
2019-08-08T09:31:04.997Z
happy to be of help
user18214
CC BY-SA 4.0
2019-08-07T10:42:51.400Z
2019-08-07T10:42:51.400Z
user18214
<p>This depends on the exact preferences, but usually the utility function
<span class="math-container">$$
U(x,y) = v(x) + y
$$</span>
is such that
<span class="math-container">$$
\lim_{x \to 0} |MRS(x,y)| = \lim_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.
$$</span>
In this case it is the consumption of an additional marginal unit of the nonlinear good <span class="math-container">$x$</span> that is infinitely useful compared to the consumption of an additional marginal unit of the linear good <span class="math-container">$y$</span>. Hence the consumer will consume some units of <span class="math-container">$x$</span> before buying any of <span class="math-container">$y$</span>. If the consumer is not 'wealthy enough', the corner solution will be such that <span class="math-container">$x &gt; 0$</span>, <span class="math-container">$y = 0$</span>.</p>

CC BY-SA 4.0
2019-08-07T10:04:34.363Z
2019-08-07T10:04:34.363Z
Giskard
<p>Whenever we talk about quasi-linear preferences, we assume that the consumer is sufficiently wealthy. As far as I understand is that we need that assumption in order to obtain an interior solution.
But what would the economic intuition behind that premise be? Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?</p>

<consumer-theory><preferences><wealth>
Assumption of sufficient wealth in quasi-linear preferences
SimonDude
# Assumption of sufficient wealth in quasi-linear preferences

(Asked by: username_0 on 2019-08-07)

Whenever we talk about quasi-linear preferences, we assume that the consumer is sufficiently wealthy. As far as I understand is that we need that assumption in order to obtain an interior solution.
But what would the economic intuition behind that premise be? Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?

**Tags:** consumer-theory, preferences, wealth


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## 2 answers

### Answer by username_1 on 2019-08-07. Score: 1 | Accepted answer

> Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?

No, it is exactly the other way around.

The utility function in the two goods case will have the following form:

$$U(x,y)=u(x)+y$$

We assume that $u'(x)>0$ and $u''(x)<0$ or marginal utility of $x$ is decreasing in $x$. $MU\_y$ meanwhile is constant.
Another assumption that is often not explicitly stated is that $\frac{MU\_x}{p\_x}>\frac{MU\_y}{p\_y}$ for some $x$. In words: There is some range of $x$ values at which the marginal utility of a dollar spent on $x$ exceeds the marginal utility of a dollar spent on $y$. Now because we know $MU\_x$ is decreasing in $x$, we know that the preceding inequality will hold true when the consumer is consuming a relatively small quantity of $x$.

Now if we assume that the budget is very large, we know that the left side of the inequality gets smaller and smaller while the right side remains constant as the consumer consumes more and more of $x$. Because of decreasing $MU\_x$ a moment will come where a dollar spent on $y$ will generate at least as much utility as a dollar spent on $x$. When this moment comes (the consumption of $x$ at which $\frac{MU\_x}{p\_x}=\frac{MU\_y}{p\_y}$ ), the consumer will switch to consuming $y$ with the rest of his budget.

When the consumer has a small budget however, he will never attain the point where the preceding equality holds true, that is, he is left consuming $x$ in a quantity for which $\frac{MU\_x}{p\_x}>\frac{MU\_y}{p\_y}$ This is the case of insufficient wealth. In this case the consumer only consumes the nonlinear good, in this case denoted by $x$.

> **Comments:**
> **username_0**: Awesome, that was a very intuitive answer ;) thanks a lot  
> **username_1**: happy to be of help  

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### Answer by username_2 on 2019-08-07. Score: 2

This depends on the exact preferences, but usually the utility function
$$
U(x,y) = v(x) + y
$$
is such that
$$
\lim\_{x \to 0} |MRS(x,y)| = \lim\_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.
$$
In this case it is the consumption of an additional marginal unit of the nonlinear good $x$ that is infinitely useful compared to the consumption of an additional marginal unit of the linear good $y$. Hence the consumer will consume some units of $x$ before buying any of $y$. If the consumer is not 'wealthy enough', the corner solution will be such that $x > 0$, $y = 0$.


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