Lagrange Multiplierd for Economics

Question

An individual purchases quantities a, b, and, c of three different commodities whose prices are p, q, and r, respectively. The consumer spends m dollars, where $m\gt2p$, and the utility function of consuming a, b, and c is given by:

$U(a,b,c)=a+ln(bc)$

Find the consumer’s demand for each good as a function of prices p,q,r, and, m. Show that when utility is maximized, the expenditure on each of the second and the third good is equal to p. Find a, b, and c when utility is maximized.

So I understand that my constraint is going to be something like

$ap+bq+cr=m$

Edit: A previous version of this post falsely wrote that $ap+bq+cr\gt2p$

Answer 1

I think it is best to set the constraint that $m>2p$ aside until the end, and then possibly eliminate solutions that violate that constraint.

$F(a,b,c,\lambda) = a + \ln b + \ln c -\lambda (ap+bq+cr - m)\\ \frac {\partial F}{\partial a} = 1 - \lambda p = 0\\ \frac {\partial F}{\partial b} = \frac {1}{b} - \lambda q = 0\\ \frac {\partial F}{\partial c} = \frac {1}{c} - \lambda r = 0\\ \frac {\partial F}{\partial \lambda} = ap+bq+cr = m$

$\lambda = \frac {1}{p}\\ b = \frac {p}{q}\\ c = \frac {p}{r}\\ ap + p + p = m$

We have no solution unless $m>2p$

$a = \frac {m}{p} - 2$