<p>I want to show mathematically that the lump sum principle does not apply to perfect complements. I was able to show it applied with a specific Cobb-Douglas utility function, but I am not sure how to show it does <em>not</em> apply with a perfect complements utility function:
<span class="math-container">$$ U(x,y)=Min\{ax,by\} $$</span>
or the form I am not sure about:
<span class="math-container">$$ U(x,y)= \lim_{\delta\to-\infty} [ ax^\delta + by^\delta]^{(1/\delta)}
$$</span>
Any hints or corrections?
Thanks</p>

CC BY-SA 4.0
2021-10-05T06:59:50.283Z
The $\min$ form is fine. But this seems very straightforward; what process did you follow with Cobb-Douglas that does not work with perfect complements?
Giskard
CC BY-SA 4.0
2021-10-05T10:57:45.590Z
You can't use differentiation with a $\min$-function. Instead, draw the indifference curves and the different budget lines. It should then be easy to see why the principle doesn't apply. Finally, translate the argument into maths.
VARulle
CC BY-SA 4.0
2021-10-05T15:53:02.783Z
@Giskard This was not a generalized proof. I just found the utility difference for a specific example. See my attempt at a generalized proof for perfect substitutes here: https://math.stackexchange.com/questions/4268077/how-do-we-show-mathematically-that-the-economic-lump-sum-principle-applies-to-pe
kleinerde
<utility><perfect-complements>
How do we show mathematically that the lump sum principle does not apply to perfect complements?
kleinerde
# How do we show mathematically that the lump sum principle does not apply to perfect complements?

(Asked by: username_0 on 2021-10-04)

I want to show mathematically that the lump sum principle does not apply to perfect complements. I was able to show it applied with a specific Cobb-Douglas utility function, but I am not sure how to show it does *not* apply with a perfect complements utility function:
$$ U(x,y)=Min\{ax,by\} $$
or the form I am not sure about:
$$ U(x,y)= \lim\_{\delta\to-\infty} [ ax^\delta + by^\delta]^{(1/\delta)}
$$
Any hints or corrections?
Thanks

**Tags:** utility, perfect-complements

> **Comments:**
> **username_1**: The $\min$ form is fine. But this seems very straightforward; what process did you follow with Cobb-Douglas that does not work with perfect complements?  
> **username_2**: You can't use differentiation with a $\min$-function. Instead, draw the indifference curves and the different budget lines. It should then be easy to see why the principle doesn't apply. Finally, translate the argument into maths.  
> **username_0**: @username_1 This was not a generalized proof. I just found the utility difference for a specific example. See my attempt at a generalized proof for perfect substitutes here: https://math.stackexchange.com/questions/4268077/how-do-we-show-mathematically-that-the-economic-lump-sum-principle-applies-to-pe  

---

## 0 answers

