In our development of consumer theory, we made a big point about the fact that neoclassical economics does not put much stock in the idea of “cardinally” measuring utility (in terms of units of happiness or “utils”). Rather, our theory of consumer behavior is based only on the assumption that individuals can simply “order” pairs of bundles in terms of which they prefer or whether they are indifferent between the two. In this sense, we said neoclassical consumer theory was ordinal and not cardinal. We now ask whether the same continues to hold for our theory of choice in the presence of risk.
A. Return to the example from exercise 17.1 — where consumption levels differ depending on whether outcome A or outcome B occurs (where A occurs with probability δ and B with probability (1 - δ)). In the absence of insurance, these outcomes are x1 and x2 respectively (with x1 < x2).
(a) Draw a graph with consumption xA in outcome A on the horizontal and consumption xB in outcome B on the vertical axis. Then locate (x1, x2) — the consumption levels you will enjoy in the absence of insurance depending on which outcome occurs.
(b) Calculate, as you did in part A(e) of exercise 17.1, how much I would have to give you in outcome A if you agree to give me $1 in outcome B assuming we want your expected consumption level to remain unchanged.
(c) Now identify all bundles in your graph that become available if we assume that you and I are willing to make trades of this kind on these terms — i.e. on terms that keep your expected consumption unchanged. Indicate the slope (in terms of δ) of the line you have just drawn.
(d) If you are risk neutral, should you care which bundle you face on the line you just drew?
(e) We can define someone as risk averse if, when faced with two gambles that give rise to the same expected consumption level, she prefers the one that has less risk. Using this definition, which bundle on our line should a risk averse individual prefer? Could the same bundle be optimal for some one that loves risk?
(f) Now suppose that we assume individuals can make ordinal comparisons between bundles— i.e. when faced by two bundles in your graph, they can tell us which they prefer or whether they are indifferent. Suppose these rankings are “rational”, that “more is better” and that there are “no sudden jumps" as we defined these in our development of consumer theory earlier in the text. Is this sufficient to allow us to assume there exist downward-sloping indifference curves which describe an individual’s tastes over the risky gambles we are graphing?
(g) What does your answer to (d) further imply about these indifference curves when tastes are risk neutral?
(h) Now consider the case of risk aversion. Pick a bundle C that lies off the 45-degree line on the “budget line" you have drawn in your graph. In light of your answer to (e), is the point D that lies at the intersection of your “budget line” with the 45-degree line more or less preferred? What does this imply for the shape of the indifference curve that runs through C?
(i) What does your answer to (e) imply about the MRS along the 45-degreelineinyourgraph?
(j) True or False: Risk aversion implies strict convexity of indifference curves over bundles of consumption for different outcomes, with all risk averse tastes sharing the same MRS along the 45-degree line if tastes are state-independent.
(k) True or False: As the probability of each outcome changes, so do the indifference curves.
(l) Have we needed to make any appeal to being able to measure utility in “cardinal” terms? True or False: Although risk aversion appears to arise from how we measure utility in our graphs of consumption/utility relationships (such as those in exercise 17.1), the underlying theory of tastes over risky gambles does not in fact require any such cardinal measurements.
(m) Repeat (h) for the case of someone who is risk loving.
B. Consider again the case of the consumption / utility relationship described by u(x) = xa with a > 0. In exercise17.1B(a), you should have concluded that a < 1 implies risk aversion, a = 1 implies risk neutrality and a > 1 implies risk loving— because the first results in a concave relationship, the second in an upward sloping line and the third in a convex relationship.
(a) Let xA represent consumption under outcome A (which occurs with probability δ) and xB consumption under outcome B (which occurs with probability (1 - δ).) Suppose we can in fact use u(x) to express tastes over risky gambles as expected utilities. Define the expected utility function U (xA, xB).
(b) Next, consider the shape of the indifference curves that are represented by the expected utility function U. First, derive the MRS of U(xA, xB).
What is the MRS when α = 1? How does this compare to your answer to A(g)?
(d) Regardless of the size of a, what is the MRS along the 45-degree line? How does this compare to your answer to A(i) for risk averse tastes?
(e) Is the MRS diminishing — giving rise to convex tastes? Does your answer depend on what value a takes? How does your answer compare to your answer to A(h)?
(f) What do indifference curves look like when α > 1?
(g) Do the slopes of indifference curves change with δ? How does your answer compare to your answer to A(k)?
(h) Suppose we used u(x) = βxa (instead of u(x) = xa) to calculate expected utilities. Would the indifference map that arises from the expected utility function change?
(i) Suppose we used u(x) = xαβ (instead of u(x) = xα) to calculate expected utilities. Would the indifference map that arises from the expected utility function change?
(j) True or False: The tastes represented by expected utility functions are immune to linear trans-formations of the consumption/utility relationship u(x) that is used to calculate expected utility — but are not immune to all types of positive transformations.
(k) Consider the expected utility function U(xA, xB ) that uses u(x) = xa. Will the underlying indifference curves change under any order-preserving transformation?
(l) True or False: Expected utility functions can be transformed like all utility functions without changing the underlying indifference curves, but such transformations can then no longer be written as if they were the expected value of two different utility values emerging from an underlying function u.
(m) In light of all this, can you reconcile the assertion that expected utility theory is nota theory that relies on cardinal interpretations of utility?