We now solve the well-known problem of the household optimum for the utility function for a given budget , when the prices of goods are and , i.e.
We form the Lagrange function
and the first order conditions:
The FOC 3 represents the second order condition. We transform the other two by adding and , respectively
and then dividing the equations with each other. This cancels .
The resulting equation represents the central point of the solution. It represents a relationship between the marginal utility ratio and the price ratio. We now summarize this equation with the constraint, where we have slightly transformed both:
We insert and obtain
and therefore the only positive solution (Note: Consider this as a quadratic equation in !)
then results as