Note: The crux of this post really is in talking about the Lagrange Multiplier, so scroll down to ‘The Lagrange’ for that.
What is a constrained optimisation problem?
Suppose we want to maximise our own utility (our enjoyment) but have a limited amount of resources (time and money, say) we can use to do that. Or say I want to produce the most value with a blog post but have a limited number of words I can use, e.g. no more than 300. In both cases we have a constrained optimisation problem: we want to maximise (or minimise) something like benefit to ourselves or others and have a constraint, e.g. a limited amount of resources.
Setting up the problem
Utility Function and Budget Constraint
Suppose for simplicity that all my resources are dollars and that I have ten dollars. With every dollar I can make my life slightly better by e.g. buying food to eat. I can describe the relationship between the money I spend and the benefit I get from consuming what I spend with the function
where is the amount that I spend. We expect that the more I spend, the more value I get, so we might have .
- Aside: Usually you get less marginal benefit you get from every dollar you spend so we’d have something like but that’s for another post. Think about it – you probably benefit more from spending your first $5 on food so you don’t starve to death than from spending an extra $5 when you’ve already spent $500 on food, housing, clothes and such.
Since I only have ten dollars, . Let’s assume that in the time period we care about I can’t borrow money and am not going to have additional income (nobody is going to give me money).
That is, is our utility function and is our budget constraint. Easy.
Solving the Problem: Setting up the Lagrangian
To solve this problem, we set up the Lagrangian thus:
where is the shadow value of relaxing the constraint.
The Lagrange Multiplier
This setup means that if I spend more than I have (which we don’t want in our solution because that’s not supposed to be possible), I will be penalised an amount per dollar spent that is precisely enough to make me not spend those extra dollars. That is, is the marginal cost of spending each of those dollars and will be equal to the marginal benefit of spending each of those dollars.
If I spend less than $10 when I’m maximising my utility, , so we just have . The constraint is not binding and becomes redundant in a sense.
From the above we can infer that if and if . This is called complementary slackness.
So is flexible and fluctuates to make our solution stay within the constraint set.
Written after IIA Microeconomics (Michaelmas) Lecture 6 by Dr. Elliott, where we discussed the problem of building a road between two villages.