<p>Demand is positive, so <span class="math-container">$A&gt;0$</span>.</p>
<p>If <span class="math-container">$p_1$</span> goes to <span class="math-container">$\infty$</span>, <span class="math-container">$x_1$</span> has to go to 0, since <span class="math-container">$p_1x_1$</span> is bounded by <span class="math-container">$M$</span>. Thus <span class="math-container">$\alpha &lt; 0$</span>.</p>
<p>If <span class="math-container">$p_2$</span> goes to 0, <span class="math-container">$x_1$</span> cannot go to <span class="math-container">$\infty$</span>, since <span class="math-container">$p_1x_1$</span> is bounded by <span class="math-container">$M$</span>. Thus <span class="math-container">$\beta\ge 0$</span>.</p>
<p>If <span class="math-container">$M$</span> goes to 0, <span class="math-container">$x_1$</span> has to go to 0, since <span class="math-container">$p_1x_1$</span> is bounded by <span class="math-container">$M$</span>. Thus <span class="math-container">$\gamma &gt; 0$</span>.</p>
<p>If both prices and income are scaled up by the same factor, then demand doesn't change, so <span class="math-container">$\alpha+\beta+\gamma=0$</span>.</p>

CC BY-SA 4.0
2021-04-16T10:16:46.387Z
2021-04-16T11:36:57.623Z
2021-04-16T11:36:57.623Z
VARulle
<p>I have a question that asks:</p>
<p>Let <span class="math-container">$x_1$</span> be the quantity of a good 1, <span class="math-container">$p_1$</span> the price of good 1, <span class="math-container">$p_2$</span> the price of good 2, and
<span class="math-container">$M$</span> is income. Let
<span class="math-container">$_1(_1, _2, ; ) = _1^_2^^$</span> Where <span class="math-container">$$</span>, <span class="math-container">$$</span>, <span class="math-container">$$</span>, and <span class="math-container">$$</span> are parameters.
If this is a demand function, what restrictions does this impose on the parameters?</p>
<p>I am a bit confused if my understanding of the question is correct, currently what I've gone and done is impose the restriction <span class="math-container">$x_1 p_1+x_2 p_2≤M$</span>, and then sub in <span class="math-container">$_1^_2^^$</span> for <span class="math-container">$x_1$</span>, resulting in <span class="math-container">$Ap_1^α p_2^β M^γ (p_1 )+x_2 p_2≤M$</span>. I have then gone and rearranged for the various parameters, resulting in:</p>
<p><span class="math-container">$A≤(M-x_2 p_2)/(p_1^(α+1) p_2^β M^γ )$</span>,</p>
<p><span class="math-container">$α≤ln⁡((M-x_2 p_2)/(Ap_2^β M^γ ))/ln⁡(p_1 ) -1,Ap_2^β M^γ$</span>,</p>
<p><span class="math-container">$β≤ln⁡((M-x_2 p_2)/(Ap_1^(α+1) M^γ ))/ln⁡(p_2 )$</span>,</p>
<p><span class="math-container">$γ≤ln⁡((M-x_2 p_2)/(Ap_1^(α+1) p_2^β ))/ln⁡(M) $</span></p>
<p>I'm not really sure if my approach is correct at all, as this seems more of a maths rearranging formulas answer rather than something related to economics. I was wondering if anyone could help me get on the right track if this is wrong. Thanks</p>

<demand><cobb-douglas>
Finding restrictions on parameters for a demand function
DoubleRainbowZ
# Finding restrictions on parameters for a demand function

(Asked by: username_0 on 2021-04-15)

I have a question that asks:

Let $x\_1$ be the quantity of a good 1, $p\_1$ the price of good 1, $p\_2$ the price of good 2, and
$M$ is income. Let
$\_1(\_1, \_2, ; ) = \_1^\_2^^$ Where $$, $$, $$, and $$ are parameters.
If this is a demand function, what restrictions does this impose on the parameters?

I am a bit confused if my understanding of the question is correct, currently what I've gone and done is impose the restriction $x\_1 p\_1+x\_2 p\_2≤M$, and then sub in $\_1^\_2^^$ for $x\_1$, resulting in $Ap\_1^α p\_2^β M^γ (p\_1 )+x\_2 p\_2≤M$. I have then gone and rearranged for the various parameters, resulting in:

$A≤(M-x\_2 p\_2)/(p\_1^(α+1) p\_2^β M^γ )$,

$α≤ln⁡((M-x\_2 p\_2)/(Ap\_2^β M^γ ))/ln⁡(p\_1 ) -1,Ap\_2^β M^γ$,

$β≤ln⁡((M-x\_2 p\_2)/(Ap\_1^(α+1) M^γ ))/ln⁡(p\_2 )$,

$γ≤ln⁡((M-x\_2 p\_2)/(Ap\_1^(α+1) p\_2^β ))/ln⁡(M) $

I'm not really sure if my approach is correct at all, as this seems more of a maths rearranging formulas answer rather than something related to economics. I was wondering if anyone could help me get on the right track if this is wrong. Thanks

**Tags:** demand, cobb-douglas


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## 1 answers

### Answer by username_1 on 2021-04-16. Score: 4 | Accepted answer

Demand is positive, so $A>0$.

If $p\_1$ goes to $\infty$, $x\_1$ has to go to 0, since $p\_1x\_1$ is bounded by $M$. Thus $\alpha < 0$.

If $p\_2$ goes to 0, $x\_1$ cannot go to $\infty$, since $p\_1x\_1$ is bounded by $M$. Thus $\beta\ge 0$.

If $M$ goes to 0, $x\_1$ has to go to 0, since $p\_1x\_1$ is bounded by $M$. Thus $\gamma > 0$.

If both prices and income are scaled up by the same factor, then demand doesn't change, so $\alpha+\beta+\gamma=0$.


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