<p>Consider quasi-linear utility function:
<span class="math-container">$u(x_1,x_2)=2\sqrt{x_1}+x_2$</span></p>
<p>Here demand for <span class="math-container">$x_1$</span> is <span class="math-container">$x_1^d = \min(y, 1)$</span>,  and demand for <span class="math-container">$x_2^d = \max(0, y-1)$</span>.
Clearly, <span class="math-container">$\lim_{y\rightarrow\infty} \frac{x_1^d}{y} = \lim_{y\rightarrow\infty} \frac{1}{y} = 0$</span>, and <span class="math-container">$\lim_{y\rightarrow\infty} \frac{x_2^d}{y} = \lim_{y\rightarrow\infty} \frac{y-1}{y} = 1$</span>. Also, both the demand functions are non-decreasing functions of <span class="math-container">$y$</span>. Though it does not satisfy <span class="math-container">$x_2^d &gt; 0$</span> for all <span class="math-container">$y&gt;0$</span>, but I thought it is an example worth mentioning.</p>
<p>To see how to find the demand for quasi-linear utility, please refer to this answer: <a href="https://qr.ae/pGJuvH" rel="nofollow noreferrer">https://qr.ae/pGJuvH</a></p>
<p>Another &quot;non-standard Cobb Douglas&quot; example of a utility function is:
<span class="math-container">$u(x_1,x_2;y) = x_1x_2^y$</span>. Please note that this is a non-standard consumer model where income directly enters the utility function of an individual.
Here demand for <span class="math-container">$x_1$</span> is <span class="math-container">$x_1^d = \frac{y}{1+y}$</span>,  and demand for <span class="math-container">$x_2^d = \frac{y^2}{1+y}$</span>. Clearly, <span class="math-container">$\lim_{y\rightarrow\infty} \frac{x_1^d}{y} = \lim_{y\rightarrow\infty} \frac{1}{1+y} = 0$</span>, and <span class="math-container">$\lim_{y\rightarrow\infty} \frac{x_2^d}{y} = \lim_{y\rightarrow\infty} \frac{y}{1+y} = 1$</span>. Also, both the demand functions are increasing functions of <span class="math-container">$y$</span>.</p>

CC BY-SA 4.0
2023-03-01T03:28:23.707Z
If you don't want $y$ in the utility function (in the second example), you can change $u$ to $u(x_1,x_2)=x_1x_2^{x_1+x_2}$. Although it represents different preferences compared to the one given in the answer, but optimal solution stays the same and therefore, it satisfy all the required properties.
Amit
CC BY-SA 4.0
2023-03-01T23:54:10.907Z
Thank you. The thing that most deviated from what I want is that $x_1$ is non-decreasing instead of strictly increasing function of y.
Rodrigo Miyamoto
CC BY-SA 4.0
2023-03-02T00:14:49.020Z
In the second example i.e. with $u=x_1x_2^{x_1+x_2}$, both $x_1$ and $x_2$ are increasing functions of $y$.
Amit
CC BY-SA 4.0
2023-03-01T00:34:33.297Z
2023-03-01T02:58:37.303Z
2023-03-01T02:58:37.303Z
Amit
<p><span class="math-container">$U(x_1,x_2)=\min\{x_1^2,x_2\}$</span>.</p>

CC BY-SA 4.0
2023-03-01T03:07:38.427Z
That's a nice one :)
Amit
CC BY-SA 4.0
2023-03-01T23:54:52.637Z
I think it solves what I asked. Thank you.
May I add one more thing? I intend to create a dynamic problem beginning with that. The household should be able to transfer the endowment over time time at a fix rate R. I am having some trouble to write the lagrangian of the problem. What should be the shadow price of budget constraint that include a one-period bond?
Rodrigo Miyamoto
CC BY-SA 4.0
2023-03-02T11:04:16.043Z
I'm not sure what exactly you are trying to do. Note also that this utility function is not differentiable, so the Lagrangian might not be that useful.
Michael Greinecker
CC BY-SA 4.0
2023-03-02T19:41:45.007Z
Let's say that the household receive a aleatory endownment each period with mean 1. If the utility function is what you describe, it is possible that the household try to take part of the income (endownment) and transfer to another period.
$ x_1+x_2 +\frac{B_{t+1}}{R_t} \leq y_t + B_t$
Rodrigo Miyamoto
CC BY-SA 4.0
2023-03-03T08:36:43.367Z
I think the most straightforward approach would be to calculate the (per period) indirect utility function. This gives you the value of having a certain amount of wealth in each period. Then you have the one-dimensional optimization problem of allocating wealth over the two periods.
Michael Greinecker
CC BY-SA 4.0
2023-02-28T23:43:44.407Z
2023-02-28T23:43:44.407Z
Michael Greinecker
<p>I would like an example of an utility function with 2 goods where the proportion of one goes to zero (and the other goes to one).
I am thinking of a problem where the household receive an endowment y and has to divide it in two goods <span class="math-container">$(x_1,x_2)$</span>. So what I want is:</p>
<p><span class="math-container">$U(x_1,x_2)$</span></p>
<p>such that the (real) bugdet constraint is
<span class="math-container">$x_1+x_2=y$</span></p>
<p>and
<span class="math-container">$\lim_{y\rightarrow \infty} \frac{x_1}{y} \rightarrow 0$</span>
and
<span class="math-container">$\lim_{y\rightarrow \infty} \frac{x_2}{y} \rightarrow 1$</span>
Where both <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are normal goods and <span class="math-container">$x_1&gt;0$</span> and <span class="math-container">$x_2&gt;0$</span> in the optimal allocation.</p>
<p>Would somebody help me, please?</p>

<utility><ces-function>
What is an example of utility function where the proportion of one goes to zero?
Rodrigo Miyamoto
# What is an example of utility function where the proportion of one goes to zero?

(Asked by: username_0 on 2023-02-28)

I would like an example of an utility function with 2 goods where the proportion of one goes to zero (and the other goes to one).
I am thinking of a problem where the household receive an endowment y and has to divide it in two goods $(x\_1,x\_2)$. So what I want is:

$U(x\_1,x\_2)$

such that the (real) bugdet constraint is
$x\_1+x\_2=y$

and
$\lim\_{y\rightarrow \infty} \frac{x\_1}{y} \rightarrow 0$
and
$\lim\_{y\rightarrow \infty} \frac{x\_2}{y} \rightarrow 1$
Where both $x\_1$ and $x\_2$ are normal goods and $x\_1>0$ and $x\_2>0$ in the optimal allocation.

Would somebody help me, please?

**Tags:** utility, ces-function

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## 2 answers

### Answer by username_1 on 2023-03-01. Score: 2

Consider quasi-linear utility function:
$u(x\_1,x\_2)=2\sqrt{x\_1}+x\_2$

Here demand for $x\_1$ is $x\_1^d = \min(y, 1)$, and demand for $x\_2^d = \max(0, y-1)$.
Clearly, $\lim\_{y\rightarrow\infty} \frac{x\_1^d}{y} = \lim\_{y\rightarrow\infty} \frac{1}{y} = 0$, and $\lim\_{y\rightarrow\infty} \frac{x\_2^d}{y} = \lim\_{y\rightarrow\infty} \frac{y-1}{y} = 1$. Also, both the demand functions are non-decreasing functions of $y$. Though it does not satisfy $x\_2^d > 0$ for all $y>0$, but I thought it is an example worth mentioning.

To see how to find the demand for quasi-linear utility, please refer to this answer: <https://qr.ae/pGJuvH>

Another "non-standard Cobb Douglas" example of a utility function is:
$u(x\_1,x\_2;y) = x\_1x\_2^y$. Please note that this is a non-standard consumer model where income directly enters the utility function of an individual.
Here demand for $x\_1$ is $x\_1^d = \frac{y}{1+y}$, and demand for $x\_2^d = \frac{y^2}{1+y}$. Clearly, $\lim\_{y\rightarrow\infty} \frac{x\_1^d}{y} = \lim\_{y\rightarrow\infty} \frac{1}{1+y} = 0$, and $\lim\_{y\rightarrow\infty} \frac{x\_2^d}{y} = \lim\_{y\rightarrow\infty} \frac{y}{1+y} = 1$. Also, both the demand functions are increasing functions of $y$.

> **Comments:**
> **username_1**: If you don't want $y$ in the utility function (in the second example), you can change $u$ to $u(x_1,x_2)=x_1x_2^{x_1+x_2}$. Although it represents different preferences compared to the one given in the answer, but optimal solution stays the same and therefore, it satisfy all the required properties.  
> **username_0**: Thank you. The thing that most deviated from what I want is that $x_1$ is non-decreasing instead of strictly increasing function of y.  
> **username_1**: In the second example i.e. with $u=x_1x_2^{x_1+x_2}$, both $x_1$ and $x_2$ are increasing functions of $y$.  

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### Answer by username_2 on 2023-02-28. Score: 1

$U(x\_1,x\_2)=\min\{x\_1^2,x\_2\}$.

> **Comments:**
> **username_1**: That's a nice one :)  
> **username_0**: I think it solves what I asked. Thank you. May I add one more thing? I intend to create a dynamic problem beginning with that. The household should be able to transfer the endowment over time time at a fix rate R. I am having some trouble to write the lagrangian of the problem. What should be the shadow price of budget constraint that include a one-period bond?  
> **username_2**: I'm not sure what exactly you are trying to do. Note also that this utility function is not differentiable, so the Lagrangian might not be that useful.  
> **username_0**: Let's say that the household receive a aleatory endownment each period with mean 1. If the utility function is what you describe, it is possible that the household try to take part of the income (endownment) and transfer to another period. $ x_1+x_2 +\frac{B_{t+1}}{R_t} \leq y_t + B_t$  
> **username_2**: I think the most straightforward approach would be to calculate the (per period) indirect utility function. This gives you the value of having a certain amount of wealth in each period. Then you have the one-dimensional optimization problem of allocating wealth over the two periods.  

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