Determine whether or not the following utility functions represent \"Strictly Co

Question

Determine whether or not the following utility functions represent "Strictly Convex" preferences. Justify your answers very clearly and show how the indifference curve associated with each utility function looks like.
a) U(x1, x2) = ln(x14x21/2+3)

Explanation / Answer

Strictly convex preferences implies that utility is strictly concave. I am going to assume those are supposed to be exponents. Please correct me in the comments if this assumption is incorrect. Also, for easier notation, set x1 = A and x2 = B U = ln(A^4 * B^(1/2) + 3) To test for strict concavity, take the determinant of the hessian. If |H| < 0, then U is strictly concave. |H| = Uaa*Ubb - Ua*Ub Ua = 4*(A^3)*B^(1/2)/(A^4 * B^(1/2) + 3) Ub = -(A^4)/2B^(1/2)*(A^4 * B^(1/2) + 3) So, without having to actually calculate Ua*Ub (which is messy), we know Ua*Ub < 0 Uaa = [(A^4 * B^(1/2) + 3)*12*(A^2)*B^(1/2) - 16*(A^6)*B] / (A^4 * B^(1/2) + 3)^2 Ub = -(1/2)*(A^4)*B^(-1/2)/(A^4 * B^(1/2) + 3) Ubb = [(A^4 * B^(1/2) + 3)*(1/4)*(A^4)*B^(-3/2) - (A^4 * B^(1/2) + 3)^2] / (A^4 * B^(1/2) + 3)^2 Ok, so you COULD plug in all of these values and calculate: |H| = Uaa*Ubb - Ua*Ub |H| = ([(A^4 * B^(1/2) + 3)*12*(A^2)*B^(1/2) - 16*(A^6)*B] / (A^4 * B^(1/2) + 3)^2)*([(A^4 * B^(1/2) + 3)*(1/4)*(A^4)*B^(-3/2) - (A^4 * B^(1/2) + 3)^2] / (A^4 * B^(1/2) + 3)^2) - (4*(A^3)*B^(1/2)/(A^4 * B^(1/2) + 3))*(-(1/2)*(A^4)*B^(-1/2)/(A^4 * B^(1/2) + 3)) And see if you get something that is strictly negative. And maybe this is what your professor wants you to do. But it would take me all day. Or you could recognize that the function will not be concave when either of the following conditions are met: Uaa*Ubb > 0 (1) Uaa > 0 & Ubb > 0 (2) Uaa < 0 & Ubb < 0 (1) implies: :(A^4 * B^(1/2) + 3)*12*(A^2)*B^(1/2) > 16*(A^6)*B :(A^4 * B^(1/2) + 3)*(1/4)*(A^4)*B^(-3/2) > (A^4 * B^(1/2) + 3)^2] (2) implies: :(A^4 * B^(1/2) + 3)*12*(A^2)*B^(1/2) < 16*(A^6)*B :(A^4 * B^(1/2) + 3)*(1/4)*(A^4)*B^(-3/2) < (A^4 * B^(1/2) + 3)^2] You can probably simplify these down to simpler conditions. But that will take way too long for me to do here. Good luck!! :)

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