Demand function

[Asd1995]

They don't. Its just a way to show that if two variables are strung together (like xy,zw) in the previous problem it doesn't mean that optimization will require one of these to be zero.

[Asd1995]

Write down the bordered Hessian for this. Its relatively simple, there are a lot of zeroes. This function obeys the conditions for quasi convexity. Hence the indifference curves are concave. Hence, we will have a boundary solution. Next part is easy. (My lagrangian was also giving me a minimum, not a maximum)

[Nikkita]

What I just see is that simply suppose as an instance, px= Py= pz= Pw, then rather than taking x=Y=Z=W= 3(let), it's better if we take x= Y=6 and z=w= 0 (or vice versa).
If all= 3, then U= 9+9=18
If x,Y are 6, others 0, then U=6*6+0= 36

This was just an eg. without proceeding to tricky methods..

[Nikkita]

agree?

[Asd1995]

That's true, because if you take prices to be equal. (you've not proved this, but your observatio/intuition is correct. You have only considered two distributions, one where all quantities are equal one where two of them are zero. There are infinitely many other possibilities.)

If prices are unequal, I doubt intuition could solve the problem though.

[Nikkita]

I tried out in mind only other possibilities..
tell if u get any contradiction from this intuition taking any other case..i am not getting any..

[Asd1995]

There is no contradiction. Your intuition worked and will work for every function that is quasi-concave or quasi-convex. Otherwise you'll run into problems. (here the function is quasi convex, which implies a border solution, if it was quasi concave it would imply a 'symmetric solution")

for clarity, consider

f(x,y)=xy
which has to be maximised given
x+y=10

answer is (5,5) (symmetric solution as function is quasi-concave)

f(x,y)= x + y
which has to be maximised given

xy=20

solution is (20,1) (extreme solution as function is quasi convex)

[Nikkita]

yup