Finding restrictions on parameters for a demand function 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
Demand is positive, so $A&gt;0$.
If $p_1$ goes to $\infty$, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\alpha &lt; 0$.
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 $\gamma &gt; 0$.
If both prices and income are scaled up by the same factor, then demand doesn't change, so $\alpha+\beta+\gamma=0$.