# Derive demand function from utility [closed]

Never encountered such a problem as I am new.

$$U(x_1,x_2)=(a\ln(x_1)+b\ln(x_2))^n$$

and $$a,b,n>0$$ with income $$w>0$$ and prices $$p_1,p_2>0$$.

• Find the demand function.

Attempt

I am thinking that I am able to use that $$h(u(x))$$ for a strictly increasing function represents the same præferences. I.e I can just work on $$a\ln(x_1)+b\ln(x_2)$$. I am also thinking about using the $$MRS$$ but not sure on how to continue.

• So what is MRS and what must MRS be equal to? Pls. add the answer to this question to the description of your attempt. Also write up income constraint, that gives you two equations in two unknowns, solve for $x_1$ and $x_2$ as function of params a,b,p1,p2,m. May 13 at 22:15

Firstly find the marginal utility of $$x_1$$ and $$x_2$$. Hint: Use the chain rule. $$\frac{\partial U}{\partial x_1} = \frac{na(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$$ $$\frac{\partial U}{\partial x_1} = \frac{nb(aln(x_1)+ bln(x_2))^{n-1}}{x_1}$$
The consumer maximises their utility when $$\frac{MU_{x_1}}{P_{x_1}}= \frac{MU_{x_2}}{P_{x_2}}$$.
From that we simply need to solve for $$x_1$$ and $$x_2$$.
Plug each expression into the budget constraint which in this case is $$w =P_{x_1}x_1 + P_{x_2}x_2$$. Then solve for $$x_1$$ and $$x_2$$, giving each demand function.