A consumer has the utility function U(x, y, l) = ln x+ ln y+(1- - )ln(L - l), where

the individual consumes x and y quantities of two goods, and works for l hours. Assume

that and are positive parameters such that ( + ) < 1, x > 0, y > 0, and 0 l < L.

The individual faces the budget constraint px + qy = wl where w is the wage per hour of

labor. Find the expressions for U*, x*, y*, l*, and * that solves the problem.

Please show a detailed solution.