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.
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.
Substitution and income effects
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.
Generally, if the price of something goes down, we buy more of it. This is down to two effects:
- Income effect: because it’s less expensive, we have more purchasing power because it is a smaller drain on our personal finances.
- Substitution effect: because it offers more utility per unit of money, other alternatives become less attractive.
What Eugen Slutsky managed to do was find an equation that decomposes this effect based on Hicksian and Marshallian demand curves.
Graphically:
Mathematically, it is based on the derivatives of Marshallian and Hickisan demands:
The left hand side of the equation is the total effect- that is, the derivative of x (quantity) respect p (price). It shows us how much the total quantity of x that we consume varies when we change price. The next part is the substitution effect- how much the variation is due to us finding similar options. It is obtained from the derivative of the Hicksian demand with regards price. The right hand side is the income effect, how much changes in our purchasing power affect the amount we consume of a certain good. It is the derivative of the Marshallian demand with regards wealth (multiplied by the quantity).
Whether the SE and the IE are positive or negative when prices rise depends on the type of good:
TE | SE | IE | |
+ | Substitute goods | Substitute goods | Inferior goods |
– | Complementary goods | Complementary goods | Normal goods |
It is not always possible to tell what the total effect will be- if we are talking about inferior complementary goods, for example, the SE and the IE pull in opposite directions. The TE will depend on which effect is stronger.
Video – Income and substitution effects:
Marshall and Hicks treated substitution and rent effects differently, judging whether or not they should both be included in demand functions. Let’s see why and how this affects what we would, in theory, consume.
https://cutt.ly/L00ZAs6
