Question

2 Utility Functions (2 Points) Consider the utility function u(c) where c denotes consumption of some arbitrary good and ơ (the Greek letter "sigma") is known as the "curvature parameter" because its value governs how curved the utility function is and is treated as a constant. In the following, restrict your attention to the region c > 0 (because "negative consumption" is an ill-defined concept) a. (0.50 Points) Plot the utility function for σ 0, Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? b. (0.50 Points) Plot the utility function for ơ 1/2 Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? c. (0.50 Points) Consider instead the natural-log utility function u(c) - In(c). Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? d. (0.50 Points) Determine the value of σ (if any value exists at all) that makes the general utility function presented above collapse to the natural-log utility function in part c (Hint: Examine the derivatives of the two functions)

Answer 1

a) When sigma = 0, u(c) = c It is constant utility function increases linearly as consumption increases and marginal utility is non-negative and constant. b) When sigma = 1/2, u(c) = c^1/2/(1/2) = 2c^1/2 The utility function is concave and displays diminishing Mu: which is not negative. c) If u(c) = ln c, u'(c) = 1/c as c increases Mu would diminish but not negative.