Consumption II: Consumption duality

Summary

This Learning Path is a bit more of a mixed bag than the previous one, finishing off our consumer choice problem, looking at the some useful implications of this in demand theory before moving on to other types of demand theories.

Consumption duality II:

Cost minimisation
Consumption duality

Further analysis:

Reshaping theory:

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.

Video – Consumption duality:

We all know that, in theory, when the price of something goes up we buy less of it. But there are two factors at play here: one is the fact that we will look for something similar but less expensive and the second is the fact that if what goes up takes up a large proportion of our budget (a mortgage), we simply have less to spend. In the next entry, we cover the dynamics of this in more detail.

Consumption II: Cost minimisation

Consumption II: Substitution and income effects