Question: Consider a consumer with (net utility) function given by U(x;p) = 10 ln(x+1) - px, where x is the quantity of the consumption good consumed, and p > 0 is the price of x. The consumer's income is m > 0. The level of consumption cannot be negative, so x >= 0, and the consumer is bound by her budget, so px <= m; thus, the domain of U(x;p) is the set D = [0, m/p]
a. For which range of values for p will the consumer maximize her utility on the boundary 0 (namely, by NOT consuming x at all)?
b. If the price is below the range you found in part (a), for which range of values for m will the consumer maximize her utility on the boundary m/p (namely, by spending all of her income on x)?
c. If the price is below the range you found in (a), and the consumer's income is above the range you found in (b), how much will the consumer demand of good x given p to maximize her utility? Call this x(p). (don't forget to check the SOC)