# What does the Lagrange Multiplier mean?

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

$u = u(x)$,

where $x$ is the amount that I spend. We expect that the more I spend, the more value I get, so we might have $u(x) = x$.

• Aside: Usually you get less marginal benefit you get from every dollar you spend so we’d have something like $u(x) = log(1+x)$ 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, $x \leq 10$. 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, $u(x)$ is our utility function and $x \leq 10$ is our budget constraint. Easy. Solving the Problem: Setting up the Lagrangian To solve this problem, we set up the Lagrangian thus: $\max_x {u(x) - \lambda (x-10)}$ where $\lambda \leq 0$ 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 $\lambda$ per dollar spent that is precisely enough to make me not spend those extra dollars. That is, $\lambda$ 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, $\lambda = 0$, so we just have $\max_x u(x)$. The constraint is not binding and becomes redundant in a sense.

From the above we can infer that if $\lambda > 0, x = 10$ and if $\lambda = 0, x < 10$. This is called complementary slackness.

So $\lambda$ 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.