# Exercise where lagrangian is needed?

I teach a general equilibrium class in my university and I want to have an exercise that is not too difficult where the Lagrangian multiplier is needed. I was under the impression that with Cobb Douglas and 3 goods I could force them to use it but in the end it turned out that they could easily solve it without it. Is there a rule of thumb or an exercise for what kind of problems force students to use the Lagrangian? To be clear, I want an example with exchange.

• You could just ask them to tell you what the Langrange multiplier's value is and to provide an interpretation for it. That way, you ensure they use the Lagrangian approach and that they understand what the multiplier means. – Ubiquitous Nov 10 at 19:31

I think a situation in which corner solutions may occur would be more fruitful. Cobb-Douglass ensures an interior solution, so they can use the expenditure share trick pretty easily, for any number of goods. Perhaps a really simple CES function, like $$x^.5+y^.5+z^{.5}$$? Or, if you still want an interior solution, pick something that's not a well-known function, but where marginal utility becomes infinite as either good approaches 0, like $$\ln(x) - \frac{1}{\sqrt{y}} - \frac{1}{z}$$ You'll still get tidy derivatives, but substitution won't be possible.