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 $𝑥_1(𝑝_1, 𝑝_2, 𝑀; 𝐴) = 𝐴𝑝_1^𝛼𝑝_2^𝛽𝑀^𝛾$ Where $𝐴$, $𝛼$, $𝛽$, and $𝛾$ 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 $𝐴𝑝_1^𝛼𝑝_2^𝛽𝑀^𝛾$ 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