9.7 The dual problem

Here, the equivalence of the maximization- and minimization- problem is explained. In the previous pages we have solved the problem of the household optimum in the form of a maximum, i.e. we have looked for the combination of goods $x$ and $y$ which maximizes the utility $U\left(x,y\right)$ for a given budget $B$ when the prices of goods are $p_x$ and $p_y$. An alternative formulation would be to minimize the costs $x{p}_x+y{p}_y$ in order to achieve a certain level of utility $U_0$.
Hence,

$$\begin{array}{llll}\hfill \underset{x,y}{\max <!--\; FUNCTION\; APPLICATION\; -->}U\left(x,y\right)\text{ under the condition that }x{p}_x+y{p}_y& =B\text{ oder}& \hfill & \\ \hfill \underset{x,y}{\min <!--\; FUNCTION\; APPLICATION\; -->}x{p}_x+y{p}_y\text{ under the condition that }U\left(x,y\right)& =U_0.& \hfill & \end{array}$$

The duality principle states that the solutions to both problems are identical if the budget $B$ corresponds to the utility level $U_0$. If the maximum utility level $U_0$ is reached with the budget $B$ (maximum problem), then the minimum costs to reach the utility level $U_0$ (minimum problem) are exactly $B$ and the respective optimal combinations of $x$ and $y$ are the same. This can be shown easily by means of the Lagrange equation systems.

Thus, you can see that the central equation "marginal utility ratio = price ratio" appears identically in the solution for both problems. And if the budget $B$ corresponds to the utility level $U_0$, then the respective third FOCs are equivalent as well.