Reference no: EM132672418

Information:

Wish to investigate the economic meaning of the signs of higher order derivatives of utility functions via a class of lotteries, such that the direction of preferences between these lotteries reveals the sign of the 3rd order derivative of a utility function. Suppose the utility function (u) considered in this assignment is infinitely differentiable and all the expectations below are finite.

We denote a lottery by vector [ε1,ε2], where ε1 and ε2 are two independent random variables. The outcomes of lottery [ε1,ε2] are ε1 with probability 0.5 and ε2 with probability 0.5, and this outcome is independent of ε1 and ε2 themselves.

For any two lotteries Y = [y1,y2] and Z = [z1,z2], we use the notation B ≥x A to denote the individual's preference relation "lottery B is at least as good as lottery A under initial wealth levels x."

The notation " ≥x" is understood in the expected utility sense. That is, Y ≥x Z if and only if

(0.5).E[u(x + y1)] + (0.5).E[u(x + y2)] ≥ (0.5).E[u(x + z1)] + (0.5).E[u(x + z2)].

For ease of exposition, we define a special lottery [a], i.e., getting a with certainty, where a is a random variable (which might be a constant).
We say [a2] ≥x [a1] if and only if E[u(x + a2)] ≥ E[u(x + a1)].

There are two building blocks in this assignment. The first is a sure reduction in wealth of size k > 0. The second is the addition of a zero-mean random variable ε, where ε is assumed not to be a constant. We let x denote the individual's initial wealth, where x > 0. We suppose that the preferences are defined over positive levels of the wealth and all changes to wealth are chosen to preserve positive wealth.

Note that ε1 and (or) ε2 may be constants.

Questions:

(i) Define lottery B1 as B1 = [0], and lottery A1 as A1 = [-k]. Show that utility u shows the property of non-satiation if and only if B1 ≥x A1, ∀k > 0, x > 0.
(ii) Define lottery B2 as B2 = [0], and lottery A2 as A2 = [ε]. Show that utility u shows the property of risk aversion if and only if B2 ≥x A2, ∀x > 0, E[ε] = 0.

(iii) Define lottery B3 as B3 = [ε, -k] and lottery A3 as A3 = [0, ε - k]. Show that the 3rd order derivative of utility u is non-negative if and only if B3 ≥x A3,∀k > 0,x > 0,E[ε] = 0.

From this question, we can see that the sign of the third order derivative of a utility function implies a type of "location" preference for a risk and a lottery. In particular, suppose that an individual's preference is non-satiated and consider the lottery [0,-k]. Now suppose the individual is told that she must accept a zero-mean random variable ε, but she must receive it only in tandem with one of the two lottery outcomes. If the individual's utility function has a non-negative third order derivative, then she will always prefer to attach the risk ε to the better outcome 0, rather than to the worse outcome -k.

(iv) We say a utility function u shows the property of prudence if and only if u′′′(x) ≥ 0,∀x > 0. Verify that the power utility and the exponential utility show the property of prudence.

(v) Construct a utility function u, such that u shows the properties of non-satiation and risk aversion but does not show the property of prudence.

(vi) Construct a utility function u, such that u shows the properties of non-satiation and prudence but does not show the property of risk aversion.

Questions (v) and (vi) illustrate that risk aversion does not implies prudence, and vice versa.