 ### 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,

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.

 Maximum problem Minimum problem Lagrange function Lagrange function $𝕃\left(x,y,\lambda \right)=U\left(x,y\right)+\lambda \left(x{p}_{x}+y{p}_{y}-B\right)$ $𝕃\left(x,y,\lambda \right)=x{p}_{x}+y{p}_{y}+\stackrel{̃}{\lambda }\left(U\left(x,y\right)-{U}_{0}\right)$ First order conditions: First order conditions: $\frac{d}{\mathit{dx}}U\left(x,y\right)+\lambda {p}_{x}=0$$\frac{d}{\mathit{dy}}U\left(x,y\right)+\lambda {p}_{y}=0$$x{p}_{x}+y{p}_{y}-B=0$ ${p}_{x}+\stackrel{̃}{\lambda }\frac{d}{\mathit{dx}}U\left(x,y\right)=0$${p}_{y}+\stackrel{̃}{\lambda }\frac{d}{\mathit{dy}}U\left(x,y\right)=0$$U\left(x,y\right)-{U}_{0}=0$ $\frac{\frac{d}{\mathit{dx}}U\left(x,y\right)}{\frac{d}{\mathit{dy}}U\left(x,y\right)}=\frac{{p}_{x}}{{p}_{y}}$ $\frac{\frac{d}{\mathit{dx}}U\left(x,y\right)}{\frac{d}{\mathit{dy}}U\left(x,y\right)}=\frac{{p}_{x}}{{p}_{y}}$
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.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de