# 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? Oct 5 '21 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. Oct 5 '21 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/… Oct 5 '21 at 15:53