Part A

Expected utility theory with the utility function: \(u(x) = \frac{x}{10}\) with total wealth = 0.

EV= 0.2 x \(\frac{40}{10}\) + 0.6 x \(\frac{50}{10}\) + 0.2 x \(\frac{30}{10}\) = 4.4

CE: \(\frac{m}{10}\) = 4.4 \(\rightarrow\) m=44 The certainty equivalent is 44

Part B

Rank dependent utility with the utility function: \(u(x) = \frac{x}{10}\) and \(w(p) = p^{2}\) with total wealth = 0.}

Probability of outcome q or better:

q=40, p=(0.2) + (0.6) = 0.8
q=50, p=(0.6)
q=30, p=(0.2) + (0.6) + (0.2) = 1

Probability outcome is strictly better than q:

q=40, p=0.6
q=50, p=0
q=30, p=(0.2) + (0.6) = 0.8

\(w(p)=p^2\)
w(0.8) - w(0.6) = \((0.8)^2\) - \((0.6)^2\)= 0.28
w(0.6) - w(0.0) = \((0.0)^2\) - \((0.0)^2\)= 0.36
w(1) - w(0.8) = \((1)^2\) - \((0.8)^2\) = 0.36

Expected utility using these updated probability weights is:

EU = 0.28 x \(\frac{40}{10}\) + 0.36 x \(\frac{50}{10}\) + 0.36 x \(\frac{30}{10}\) = 4

CE: \(\frac{m}{10}\) = 4 \(\rightarrow\) m = 40 The certainty equivalent is 40.

Part A

Show that this choice pattern violates Expected Utility theory. Under expected utility, preferences for prospect A implies:

u(3000) \(>\) 0.8 u(4000) + 0.2 u(0)

\(\frac{1}{4}\)(u(3000) \(>\) 0.8 u(4000) + 0.2 u(0)) \(\rightarrow\) 0.25 u(3000) \(>\) 0.2 u (4000) + 0.05 u(0) +0.75 u(0)

\(\rightarrow\) 0.25 u(3000)+ 0.75 u(0) \(>\) 0.2 u(4000)+ 0.8 u (4000)

Preferring prospect D to C implies:

0.2u(4000) + 0.8 u(0) \(>\) 0.25u(3000) +0.75u(0)

Therefore, under prospect theory an individual cannot prefer A to B and D to C.

Part B

Show that Disappointment theory as presented and parameterized on the slides (i.e., with u(x) = x and θ = 0.0002) can accommodate the observed choice pattern.)

EV(A) = 3000 D(A) = 3000

EV(B) = EU = 3200

D(B)=0.8(4000+0.0002x\((4000-3200)^2\) + 0.2(0-0.0002x\((0-3200)^2\))=2892.8

Hence, D(A) \(>\) D(B)

EV(C) = EU = 750

D(C)=0.25(3000+0.0002x\((3000-750)^2\)) + 0.75(0-0.0002x\((0-750)^2\))=918.75

EV(D) = EU = 800

D(D)=0.2(4000+0.0002x\((4000-800)^2\)) + 0.8(0-0.0002x\((0-800)^2\))=1107.2

Hence, D(D) \(>\) D(C)

Part C

Show that Cumulative Prospect Theory can accommodate the choice pattern.
Using the parametrization by Tversky and Kahnemann:

CPT(A)

CPT(B)

CPT(C)

CPT(D)

\(\rightarrow\) CPT(A) \(>\) CPT(B)

\(\rightarrow\) CPT(D) \(>\) CPT(C)

Part A

(0.25, €75; 0.25, €50; 0.25, €25; 0.25, €0)}

\(u^+(x)\) = \(x^{0.88}\) if x \(\geq\) 0

\(u^-(x)\) = 2.25 * \(-(-x)^{0.88}\) if x \(<\) 0 (with \(\lambda\) = 2.25)

\(w^+(p) = \frac{p^{0.61}}{(p^{0.61}+(1-p)^{0.61})^{1/0.61}}\)

\(w^-(p) = \frac{p^{0.69}}{(p^{0.69}+(1-p)^{0.69})^{1/0.69}}\)

\(w^+(0) = \frac{0^{0.61}}{(0^{0.61}+(1)^{0.61})^{1/0.61}}\) = 0
\(w^+(0.25) = \frac{0.25^{0.61}}{(0.25^{0.61}+(0.75)^{0.61})^{1/0.61}}\) = 0.290
\(w^+(0.50) = \frac{0.50^{0.61}}{(0.50^{0.61}+(0.50)^{0.61})^{1/0.61}}\) = 0.42064
\(w^+(0.75) = \frac{0.75^{0.61}}{(0.75^{0.61}+(0.25)^{0.61})^{1/0.61}}\) = 0.56827
\(w^+(1) = \frac{1^{0.61}}{(1^{0.61}+(0)^{0.61})^{1/0.61}}\) = 1

0.29074 - 0 = 0.29074
0.42064 - 0.29074 = 0.12990
0.56827 - 0.42064 = 0.14763
1 - 0.56827 = 0.43173

\(u^+(75)\) = \(75^{0.88}\) = 44.674
\(u^+(50)\) = \(50^{0.88}\) = 31.268
\(u^+(25)\) = \(25^{0.88}\) = 16.990
\(u^+(0)\) = \(0^{0.88}\) = 0

CPT = 12.989 + 4.062 + 2.508 + 0 = 19.559

\(CE^{0.88}\) = 19.559 \(\rightarrow\) CE = 29.339

EV = \(\frac{1}{4}(75)+\frac{1}{4}(50)+\frac{1}{4}(25)+\frac{1}{4}(0)=37.5\)
RP = 37.5 - 29.339 = 8.161

Part B

2. (0.25, €0; 0.25, -€25; 0.25, -€50; 0.25, -€75)

\(w^-(0.75) = \frac{0.75^{0.69}}{(0.75^{0.69}+(0.25)^{0.69})^{1/0.69}}\) = 0.62640 0.62640-0.45399= 0.17241 \(u^-(-75)\) = 2.25 * \(-(--75)^{0.88}\) = -100.516

CPT = 0 + -6.591 + -11.289 + -29.503 = -47.383 \(CE^{0.88}\) = -47.383 \(\rightarrow\) CE = -31.909

EV = \(\frac{1}{4}(0)+\frac{1}{4}(-25)+\frac{1}{4}(-50)+\frac{1}{4}(-75)= -37.50\) RP = -37.50 - -31.909 = -5.591

Using the example from the lecture slides, people can prefer E to F and G to H without violating the expectations of an s-shaped utility curve (see the graph on the following slide). The intuition is that losing x hurts more than gaining x brings pleasure. Although people should prefer 0.75 x 3000 to 0.5 x 4500, the possible loss of 6000 with probability of 0.25 hurts too much and will outweigh the extra utility from 0.75 x 3000.

\(w^+(p)=w^-(p)=p^2\)
\(u^+(x)=x^5\)
\(u^-(x)=-2x^5\)

CPT(E) = \(w^-(0.5)u^-(-3000)+w^+(0.5)u^+(4500)\) = \(0.5^2\) x (-2) x \(3000^5\) + \(0.5^2\) x \(4500^5\) = 4.6 x \(10^6\)

CPT(F) = \(w^-(0.25)u^-(-6000)+w^+(0.75)u^+(3000)\) = \(0.25^2\) x (-2) x \(6000^5\) + \(0.75^2\) x \(3000^5\) = -8.35 x \(10^{17}\)

CPT(E) \(>\) CPT(F)

CPT(G) = \(w^-(0.5)u^-(-1500)+w^+(0.5)u^+(4500)\) = \(0.5^2\) x (-2) x \(1500^2\) + \(0.25^2\) x \(4500^5\) = 4.58 x \(10^{17}\)

CPT(H) = \(w^-(0.25)u^-(3000)+w^+(0.75)u^+(3000)\) = \(0.25^2\) x (-2) x \(3000^5\) + \(0.75^2\) x \(3000^5\) = 1.06 x \(10^{17}\)

CPT(G) \(>\) CPT(H)

Marginal Utility is Dependent on Domain

A second possible explanation that comes to the same conclusion that we do not agree with the statement by Levy and Levy (2002) is the following: Looking at the probability weighting of cumulative prospect theory, shows that small probabilities are overweighted and moderate and large probabilities are underweighted (see the second graph below).

Why do people prefer E over F? In prospect F there is a small probability of a very large negative outcome. Since it is a small probability and a negative outcome, people are pessimistic according to CPT. Prospect E on the other hand has equal moderate probabilities and is hence preferred over a prospect with a small overweighted outcome of loosing 6000 with probability 0.25.

The same logic holds for prospects G and H. In Prospect H there is a small probability of a very large negative outcome. Since it is a small probability and a negative outcome, people are pessimistic according to CPT and hence prefer prospect G over prospect H.