# How do we show mathematically that the lump sum principle does not apply to perfect complements?

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

• 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? Commented Oct 5, 2021 at 6:59
• 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. Commented Oct 5, 2021 at 10:57
• @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: math.stackexchange.com/questions/4268077/… Commented Oct 5, 2021 at 15:53