# 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 $$𝑥_1(𝑝_1, 𝑝_2, 𝑀; 𝐴) = 𝐴𝑝_1^𝛼𝑝_2^𝛽𝑀^𝛾$$ 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

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 $$\alpha+\beta+\gamma=0$$.