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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

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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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