Denote by Px and Py the market prices of the two goods, and let I represent the consumer's income level per market period. This expression is then maximized by setting the partial derivatives of L with respect to X, Y, and l equal to zero and then solving for X, Y, and l in terms of the problem parameters, I, Px, and Py. First the problem is set up. Then the first order conditions are set equal to zero and the entire system of three equations is solved for the three unknowns in terms of the problem parameters. By setting the last first order condition equal to zero, the solution is assured to satisfy the constraint.
Maple then assigns the problem solutions to the variables.
Furthermore, since the quantities of both goods increase with rising levels of income, both commodities are, by definition, normal goods. Lastly, the quantities demanded of good X and good Y are both independent of the other's price. The optimal value of the Lagrangian multiplier, l, is also of economic interest. From the Lagrangian expression, l is defined to be the marginal utility of income - the extra happiness the consumer derives from one more dollar of income. That is, one more dollar of income is worth less to the consumer as prices rise.