Expected Utility. One method for deciding among various investment involves the concept of expected utility. Economists describe the importance of various levels of wealth by using utility functions. For instance, in most case, a single dollar is more important (has greater utility ) for someone with little wealth than for someone with greater wealth Consider two investments, say investment A and B. Measured in thousand of dollars, suppose that investment A yields 0, 1, and 4 with probability 0.1 and16 with probability 0.5, 0.3 and 0.2 respectively. Let Y denote the yield of an investment. For the two investment, determine and compare.

Part (a) The mean of Y, the expected yield.

Part (b) The mean of $\sqrt{Y}$,the expected utility, using the utility function role="math" localid="1651845051902" $\upsilon \left(y\right)=\sqrt{y}$Interpret the utility function $u$

Part (c) The mean of $y^{3/2}$,the expected utility, using the utility function $v\left(y\right)=y^{3/2}$. Interpret the utility function v

The yields of two investments A and B (in thousands of dollars) are shown below, along with their probability.

Investment A

Investment B

Let Y be the yield of an investment.

The investment A's projected return is

$$\begin{gathered} {\mu }_A=\sum _{}a·P(A=a) \\ {\mu }_A=0\times 0.1+1000\times 0.5+4000\times 0.4 \\ {\mu }_A=\$2100 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mu }_B=\sum _{}b·P(B=b) \\ {\mu }_B=0\times 0.5+1000\times 0.3+16000\times 0.2 \\ {\mu }_B=\$3500 \end{gathered}$$

The predicted yield of the investment A is lower than the expected yield of the investment B, according to the aforementioned estimate.

The utility function is as follows:

$u\left(y\right)=\sqrt{Y}$

The investment A's projected return is

$$\begin{gathered} {\mu }_{\sqrt{A}}=\sum _{}\sqrt{a}·P(A=a) \\ {\mu }_{\sqrt{A}}=\sqrt{0}\times 0.1+\sqrt{1000}\times 0.5+\sqrt{4000}\times 0.4 \\ {\mu }_{\sqrt{A}}=\$41.1096 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mu }_{\sqrt{B}}=\sum _{}\sqrt{b}·P(B=b) \\ {\mu }_{\sqrt{B}}=0\times 0.5+\sqrt{1000}\times 0.3+\sqrt{16000}\times 0.2 \\ {\mu }_{\sqrt{B}}=34.7851 \end{gathered}$$

Investment A has an expected utility of 41.1096 while investment B has an expected utility of 34.7851. As a result, the predicted utility of investment A is greater than that of investment B.

The utility function is as follows:

$v\left(y\right)=y^{3/2}$

The investment A's projected return is

$$\begin{gathered} {\mu }_{A^{3/2}}=\sum _{}{a}^{3/2}·P(A=a) \\ {\mu }_{A^{3/2}}=0^{3/2}\times 0.1+{1000}^{3/2}\times 0.5+{4000}^{3/2}\times 0.4 \\ {\mu }_{A^{3/2}}=117004.2734 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mu }_{B^{3/2}}=\sum _{}{b}^{3/2}·P(B=b) \\ {\mu }_{B^{3/2}}=0^{3/2}\times 0.5+{1000}^{3/2}\times 0.3+{16000}^{3/2}\times 0.2 \\ {\mu }_{B^{3/2}}=414258.3735 \end{gathered}$$

Investment A's expected utility is 117004.2734, while investment B's estimated utility is 414258.3735. As a result, investment A's predicted utility is lower than investment B's expected utility.