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:
That is, our Lagrangian is our utility function, which depends on x1, x2 minus the restriction- our budget. The first order conditions (which we obtain from the first derivatives) give us Marshallian demand curves.
|When dealing with dual demand, that is, cost minimisation, our Lagrangian system is as follows:
Video – Consumption duality: