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Consumption duality

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:

So that:

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:

Subj. to:

So that:
That is, our Lagrangian is our cost function, which depends on x₁, x₂ minus our utility function, which must equal a constant. The first order conditions give us Hicksian demand curves.

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