There are two ways to solve a consumer’s choice problem. That is, we can either fix a *budget* and obtain the *maximum utility* from it (primal demand) or set a level of *utility* we want to achieve and *minimise cost* (dual demand).

The way to solve either problem is very similar: we look for the Lagrangian function and obtain first order conditions, then solve the system.

When dealing with primal demand, that is, utility maximisation, our Langrangian is as follows:
Subj. to: That is, our Lagrangian is our utility function, which depends on x |

When dealing with dual demand, that is, cost minimisation, our Lagrangian system is as follows:
Subj. to: So that: |

**Video – Consumption duality:**