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Ok, this is an economics maths question, I'll try my best to explain
U = (x1)^3/4(x2)^3/4
Find the Marginal Rate of Substitution, (basically dx2/dx1), in terms of x1 and x2. (If it makes it easier just x1 and x2 to p and q or whatever else. I have an answer for this part, this is just more of a check to see if it's right.
And also, what does this simplify to:
-3/4(p^-1/4)(q^3/4)
3/4(p^3/4)(q^-1/4)
Thanks.
I can't help with the first bit, the the simplification:
-p^(-1/4 - 3/4) * q^(3/4 - -1/4) = -p^(-1) * q^(1) = -q/p
By the chain rule, dU/dx2*dx2/dx1 = dU/dx1, so
dx2/dx1 = (dU/dx1) / (dU/dx2)
From the product rule and partial differentiation we get
dU/dx1 = (x2)^(3/4)*(3/4)*(x1)^(-1/4)
dU/dx2 = (x1)^(3/4)*(3/4)*(x2)^(-1/4)
Substituting these into the formula above, we find
dx2/dx1 =
(x2)^(3/4)*(3/4)*(x1)^(-1/4)
----------------------------
(x1)^(3/4)*(3/4)*(x2)^(-1/4)
= x2/x1.
This should be the right answer barring algebraic mistakes, and I'm pretty sure it's the same general approach you used because of the similarity between this and your q/p formula.
Yeah that's what I got. Thanks a lot.