A consumer of two goods faces positive prices for both goods and has positive in

Question

A consumer of two goods faces positive prices for both goods and has positive income. Her preferences over
consumption of good 1 and good 2 are represented by the following utility function:
u(x1; x2) = e^(x1+ln(x2))^.5
Assume, the price of good 1 is 1 (p1 = 1) and the price of good two is p2 > 0. Use m to denote income.
a. What properties about utility functions will make this problem easier to solve?
b. Which of the non negativity constraints on x1; x2 will bind for small m?
c. Derive for the Marshallian demand functions and the indirect utility function.
d. Derive the expenditure function for utility level u.

Explanation / Answer

u = ln(1+e^x) du/dx = (e^x)/(1+e^x) a ln will properties about utility functions will make this problem easier to solve? y = u^5 dy/du = 5u^4 derivative = (du/dx) (dy/du) derivate of ln(x) = x'/x 5(ln(1+e^x))^4 * (e^x)/(1+e^x) This seems right. I think they're trying to trick people into bring the 5 out front, but you can only do that if is is inside the log.

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