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 > 2p, and the utility function is given by

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

Show that the optimal expenditure on the second and third good is equal to p (that is, 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 (you may assume Lagrange multipliers corresponds to the maximum solution).