<p>Let <span class="math-container">$b_i := \frac{a_i}{p_i}$</span>, now choose the good that maximize <span class="math-container">$b_i$</span>, let it be <span class="math-container">$b_j$</span> i.e. the <span class="math-container">$j$</span>th good. Optimal choice is <span class="math-container">$x_j=m/p_j$</span>, and <span class="math-container">$x_i = 0$</span> for <span class="math-container">$i\neq j$</span>.</p>

CC BY-SA 4.0
2020-07-12T23:15:55.577Z
I get it! Do you mind telling me like a way to prove it? Thanks!
manifold
CC BY-SA 4.0
2020-07-12T23:23:05.243Z
And other think, I get the intuition, but should not the parameter be compared as a weight versus all other parameters (like the $\frac{a_i}{\sum_{i=1}^{n}a_i}$ part in Cobb-Douglas)? Thanks!
manifold
CC BY-SA 4.0
2020-07-13T08:08:22.820Z
(-1) Displaying formulas is not helping.
Giskard
CC BY-SA 4.0
2022-04-08T03:10:37.140Z
@Giskard In my opinion, I have not displayed just the formula. The one who posted the question already knows how to solve the problem for the case of two goods, so according to me this is sufficient.
Amit
CC BY-SA 4.0
2020-07-12T05:23:13.380Z
2020-07-12T05:23:13.380Z
Amit
<p>I have the utility function:</p>
<p><span class="math-container">$U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$</span> (maybe <span class="math-container">$a_0$</span> could be zero)</p>
<p><span class="math-container">$\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$</span> where <span class="math-container">$K&lt;\infty$</span></p>
<p>Where the problem is:</p>
<p><span class="math-container">$\max_{x_1,...,x_n}U\;\;\;\;\;\;\;\;c.t.\;\;\sum_{i=1}^{n}p_ix_i=m\;\;\;\;\;$</span> (With positive prices and income)</p>
<p>Which is the general case for perfect substitutes preferences for finding Walrasian-Marshallian demands. In the particular case <span class="math-container">$n=2$</span> the solution could be found geometrically, and the optimum turns out to be one of three cases (where two of them are corner solutions) depending on the relation between prices and parameters <span class="math-container">$a$</span>. When <span class="math-container">$p_1/p_2&gt;a_1/a_2$</span> solution is <span class="math-container">$(x_1^*=0, x_2^*=m/p_2)$</span>, and the opposite case she spends all income in <span class="math-container">$x_1$</span>, and the third case is a continuum of feasible points that satisfy the budget constraint.</p>
<p>So, here's the question: Is there a way to obtain a generalized Walrasian-Marshallian demand function (I imagine it would be by parts) for <span class="math-container">$n$</span>-goods in this case, just like in Cobb-Douglas preferences that it turns out to be <span class="math-container">$x_i^m=\frac{a_im}{p_i\sum_{i=1}^{n}a_i}$</span> (please prove).</p>

<microeconomics><mathematical-economics><consumer-theory><optimization><demand>
Generalizing demand for perfect substitutes utility function
manifold
# Generalizing demand for perfect substitutes utility function

(Asked by: username_0 on 2020-07-12)

I have the utility function:

$U(x\_1,...,x\_n)=a\_0+\sum\_{i=1}^{n}a\_ix\_i\;\;\;\;\;\;\;\;\;a\_j\in\mathbb{R}\_+ \;\;\forall j=\{0,...,n\}$ (maybe $a\_0$ could be zero)

$\sum\_{i=1}^{n}a\_i\in (0,K)\;\;\;$ where $K<\infty$

Where the problem is:

$\max\_{x\_1,...,x\_n}U\;\;\;\;\;\;\;\;c.t.\;\;\sum\_{i=1}^{n}p\_ix\_i=m\;\;\;\;\;$ (With positive prices and income)

Which is the general case for perfect substitutes preferences for finding Walrasian-Marshallian demands. In the particular case $n=2$ the solution could be found geometrically, and the optimum turns out to be one of three cases (where two of them are corner solutions) depending on the relation between prices and parameters $a$. When $p\_1/p\_2>a\_1/a\_2$ solution is $(x\_1^\*=0, x\_2^\*=m/p\_2)$, and the opposite case she spends all income in $x\_1$, and the third case is a continuum of feasible points that satisfy the budget constraint.

So, here's the question: Is there a way to obtain a generalized Walrasian-Marshallian demand function (I imagine it would be by parts) for $n$-goods in this case, just like in Cobb-Douglas preferences that it turns out to be $x\_i^m=\frac{a\_im}{p\_i\sum\_{i=1}^{n}a\_i}$ (please prove).

**Tags:** microeconomics, mathematical-economics, consumer-theory, optimization, demand

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## 1 answers

### Answer by username_1 on 2020-07-12. Score: 1

Let $b\_i := \frac{a\_i}{p\_i}$, now choose the good that maximize $b\_i$, let it be $b\_j$ i.e. the $j$th good. Optimal choice is $x\_j=m/p\_j$, and $x\_i = 0$ for $i\neq j$.

> **Comments:**
> **username_0**: I get it! Do you mind telling me like a way to prove it? Thanks!  
> **username_0**: And other think, I get the intuition, but should not the parameter be compared as a weight versus all other parameters (like the $\frac{a_i}{\sum_{i=1}^{n}a_i}$ part in Cobb-Douglas)? Thanks!  
> **username_2**: (-1) Displaying formulas is not helping.  
> **username_1**: @Giskard In my opinion, I have not displayed just the formula. The one who posted the question already knows how to solve the problem for the case of two goods, so according to me this is sufficient.  

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