<p>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 <span class="math-container">$x^.5+y^.5+z^{.5}$</span>? 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 <span class="math-container">$\ln(x) - \frac{1}{\sqrt{y}} - \frac{1}{z}$</span> You'll still get tidy derivatives, but substitution won't be possible.</p>

CC BY-SA 4.0
2018-10-11T15:49:10.210Z
Thanks for the suggestion, i'm testing it now.
Dio
CC BY-SA 4.0
2018-10-11T16:24:41.293Z
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
CC BY-SA 4.0
2018-10-11T16:29:31.590Z
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
CC BY-SA 4.0
2018-10-11T23:51:34.853Z
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
CC BY-SA 4.0
2018-10-11T14:55:29.127Z
2018-10-11T14:55:29.127Z
Mathew Knudson
<p>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. </p>

CC BY-SA 4.0
2018-11-10T19:31:25.583Z
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
<utility><general-equilibrium>
Exercise where lagrangian is needed?
Dio
# Exercise where lagrangian is needed?

(Asked by: username_0 on 2018-10-11)

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.

**Tags:** utility, general-equilibrium

> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2018-10-11. Score: 1

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.

> **Comments:**
> **username_0**: Thanks for the suggestion, i'm testing it now.  
> **username_0**: I don't think the first one works, I can get demand function without using the lagrangian. Not sure where to post the latex.  
> **username_0**: 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.  
> **username_2**: 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.  

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