<p>In order to have <span class="math-container">$x^*_j&gt;0$</span>, it is possible to impose either<br />
(i) a condition on the marginal utility: <span class="math-container">$\lim_{x_j\rightarrow0} \partial{U}/\partial{x_j}(x)=+\infty$</span><br />
(ii) or an inequality <span class="math-container">$x_j \geq a_j&gt;0$</span> where <span class="math-container">$a_j$</span> is interpreted as a subsistence level of <span class="math-container">$x_j$</span><br />
Often the utility function is reparameterized and written <span class="math-container">$U(x-a)$</span> with the constraint <span class="math-container">$X:=x-a\geq 0$</span>, which ensures that <span class="math-container">$x^*\geq a$</span> at the inner optimum.</p>
<p>Example: in the Cobb-Douglas case with subsistence levels, we have
<span class="math-container">$$ U(X_1,X_2)=U(x_1-a_1,x_2-a_2)=(x_1-a_1)^\alpha(x_2-a_2)^\beta $$</span></p>
<p>Provided that the income <span class="math-container">$m$</span> is high enough, the Marshallian demands are given by:</p>
<p><span class="math-container">$$x^*_1 = a_1 + \frac{\alpha}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_1}\\
x^*_2 = a_2 + \frac{\beta}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_2}. $$</span></p>

CC BY-SA 4.0
2021-01-11T13:21:45.677Z
Not sure I can see how the second strategy gives an interior solution, by which I mean one where MRS equals relative prices.
user270396
CC BY-SA 4.0
2021-01-11T17:21:25.907Z
I have added an example for illustration.
Bertrand
CC BY-SA 4.0
2021-01-14T12:04:44.150Z
Thank you for the example. However, if I understand it correctly there is only interior solution if income is large enough otherwise I guess there is a border solution. To repeat I am not looking for a way to ensure that consumption is positive or above some minimum threshold but instead to ensure that there is an interior solution in the sense that MRS = relative prices (which is not the case for border solutions). I am only interested in $x_1>0$ and $x_2>0$ because that implies MRS=relative prices.
user270396
CC BY-SA 4.0
2021-01-14T13:14:04.747Z
OK in this case only (i) is a suitable answer not (ii).
Bertrand
CC BY-SA 4.0
2021-01-10T11:19:47.973Z
2021-01-11T17:20:35.863Z
2021-01-11T17:20:35.863Z
Bertrand
<p>I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure an interior solution.</p>
<p>But are there not a set of conditions that can be imposed on the utility function for example - rather that stipulating it as a primitive - to insure that the solution is interior?</p>
<p>I realize that this offcourse depends on the particular program under consideration. So to be concrete I am primarily interested in the standard program where</p>
<p><span class="math-container">$$\max_x \ u(x) \\
s.t. \ \ px \leq w$$</span></p>
<p>where <span class="math-container">$p&gt;&gt;0$</span> and <span class="math-container">$w&gt;0$</span>.</p>

<utility><optimization><demand>
Essential goods: How does one restrict the utility function?
user270396
# Essential goods: How does one restrict the utility function?

(Asked by: username_0 on 2021-01-10)

I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure an interior solution.

But are there not a set of conditions that can be imposed on the utility function for example - rather that stipulating it as a primitive - to insure that the solution is interior?

I realize that this offcourse depends on the particular program under consideration. So to be concrete I am primarily interested in the standard program where

$$\max\_x \ u(x) \\
s.t. \ \ px \leq w$$

where $p>>0$ and $w>0$.

**Tags:** utility, optimization, demand


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## 1 answers

### Answer by username_1 on 2021-01-10. Score: 1

In order to have $x^\*\_j>0$, it is possible to impose either  
(i) a condition on the marginal utility: $\lim\_{x\_j\rightarrow0} \partial{U}/\partial{x\_j}(x)=+\infty$  
(ii) or an inequality $x\_j \geq a\_j>0$ where $a\_j$ is interpreted as a subsistence level of $x\_j$  
Often the utility function is reparameterized and written $U(x-a)$ with the constraint $X:=x-a\geq 0$, which ensures that $x^\*\geq a$ at the inner optimum.

Example: in the Cobb-Douglas case with subsistence levels, we have
$$ U(X\_1,X\_2)=U(x\_1-a\_1,x\_2-a\_2)=(x\_1-a\_1)^\alpha(x\_2-a\_2)^\beta $$

Provided that the income $m$ is high enough, the Marshallian demands are given by:

$$x^\*\_1 = a\_1 + \frac{\alpha}{\alpha+\beta}\frac{m-p\_1a\_1-p\_2a\_2}{p\_1}\\
x^\*\_2 = a\_2 + \frac{\beta}{\alpha+\beta}\frac{m-p\_1a\_1-p\_2a\_2}{p\_2}. $$

> **Comments:**
> **username_0**: Not sure I can see how the second strategy gives an interior solution, by which I mean one where MRS equals relative prices.  
> **username_1**: I have added an example for illustration.  
> **username_0**: Thank you for the example. However, if I understand it correctly there is only interior solution if income is large enough otherwise I guess there is a border solution. To repeat I am not looking for a way to ensure that consumption is positive or above some minimum threshold but instead to ensure that there is an interior solution in the sense that MRS = relative prices (which is not the case for border solutions). I am only interested in $x_1>0$ and $x_2>0$ because that implies MRS=relative prices.  
> **username_1**: OK in this case only (i) is a suitable answer not (ii).  

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