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.

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 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 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 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 at 23:51

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