# 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 '18 at 19:31

## 1 Answer

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.

• Thanks for the suggestion, i'm testing it now. – Dio Oct 11 '18 at 15:49
• I don't think the first one works, I can get demand function without using the lagrangian. Not sure where to post the latex. – Dio Oct 11 '18 at 16:24
• but anyway the procedure is quite simple, you do MRS_(y to x) and MRS_(z to x), then you solve for z and y, plug those into the budget, and solve for x. Then substitute the x back into each MRS and you got the demand for each. – Dio Oct 11 '18 at 16:29
• Well, unfortunately, if you want interior solutions , then the MRS/p rule is a fact of life- it's a characterization of any solution to the first order conditions, derived from using the Lagrangian. I misunderstood what you had meant about them avoiding a Lagrangian- I thought they had just learned the Cobb-Douglas trick. – Mathew Knudson Oct 11 '18 at 23:51