Activity: Answer this problem with complete solution

Consider a person with the following utility function over wealth: u(w) ew, where e is the exponential function (approximately equal to 2.7183) and w wealth in hundreds of thousands of dollars. Suppose that this person has a 40% chance of wealth of P50,000 and a 60% chance of wealth of P1,000,000 as summarized by P(0.40, P50,000, P1,000,000).

1). What is the expected value of wealth?

2). is this person risk averse, risk neutral, or a risk seeker?

3). what isthus person's certainly equivalent for the prospect?

Let's solve this problem step by step, given the utility function \( u(w) = e^w \), where \( e \) is the exponential function and \( w \) is wealth in hundreds of thousands of dollars. We are also given that the person has a 40% chance of having P50,000 and a 60% chance of having P1,000,000.

### 1) Expected Value of Wealth

First, we need to calculate the expected value of wealth. The wealth levels are:

- \( w_1 = P50,000 = 0.5 \) (since wealth is in hundreds of thousands of dollars),
- \( w_2 = P1,000,000 = 10 \).

The probabilities associated with these wealth levels are:

- \( p_1 = 0.40 \) for \( w_1 = 0.5 \),
- \( p_2 = 0.60 \) for \( w_2 = 10 \).

The expected value \( E(W) \) of wealth is calculated using the formula for the expected value:

\[
E(W) = p_1 \cdot w_1 + p_2 \cdot w_2
\]

Substitute the values:

\[
E(W) = (0.40 \cdot 0.5) + (0.60 \cdot 10)
\]

\[
E(W) = 0.2 + 6 = 6.2
\]

Thus, the expected value of wealth is **6.2** (or P620,000).

---

### 2) Is this person risk averse, risk neutral, or a risk seeker?

To determine if the person is risk averse, risk neutral, or a risk seeker, we need to analyze the curvature of the utility function. Specifically, we look at the second derivative of the utility function \( u(w) = e^w \).

- First derivative (marginal utility): 
  \[
  u'(w) = e^w
  \]
  
- Second derivative:
  \[
  u''(w) = e^w
  \]

Since the second derivative \( u''(w) = e^w > 0 \) for all \( w \), the utility function is convex. A convex utility function indicates that the person is **risk-seeking**, as they prefer more risk to less risk at any given level of wealth.

---

### 3) What is this person's certainty equivalent?

The certainty equivalent is the amount of wealth that gives the person the same utility as the expected utility of the risky prospect. To calculate the certainty equivalent, we follow these steps:

1. **Calculate the expected utility**:

The expected utility \( E(U) \) is given by:

\[
E(U) = p_1 \cdot u(w_1) + p_2 \cdot u(w_2)
\]

Substitute the values:

\[
E(U) = (0.40 \cdot e^{0.5}) + (0.60 \cdot e^{10})
\]

Using \( e \approx 2.7183 \), we compute the values of \( e^{0.5} \) and \( e^{10} \):

\[
e^{0.5} \approx 1.6487, \quad e^{10} \approx 22026.47
\]

Substitute these into the equation:

\[
E(U) = (0.40 \cdot 1.6487) + (0.60 \cdot 22026.47)
\]

\[
E(U) = 0.6595 + 13215.882 = 13216.5415
\]

So, the expected utility \( E(U) = 13216.5415 \).

2. **Find the certainty equivalent**:

The certainty equivalent \( w_{CE} \) is the value of wealth that gives the same utility as the expected utility. This means solving for \( w_{CE} \) in the equation:

\[
u(w_{CE}) = E(U)
\]

Since \( u(w) = e^w \), we have:

\[
e^{w_{CE}} = 13216.5415
\]

Taking the natural logarithm on both sides:

\[
w_{CE} = \ln(13216.5415)
\]

\[
w_{CE} \approx 9.49
\]

Thus, the certainty equivalent is **9.49** (or P949,000).

---

### Summary of Answers:

1. The expected value of wealth is **P620,000**.
2. The person is a **risk seeker**.
3. The certainty equivalent is **P949,000**.

---

Do you want further details on any of the steps? Here are some related questions:

1. How does the shape of a utility function affect risk preferences?
2. How would the certainty equivalent change if the probabilities were different?
3. Can a person have both risk-seeking and risk-averse tendencies depending on the wealth level?
4. How is the expected utility theory applied in real-world decision-making?
5. What would happen to the certainty equivalent if the utility function was logarithmic?

Tip: In problems with probabilities, always start by calculating the expected value of the main variables.

Consider a person with the following utility function over wealth: u(w) = e^w, where e is the exponential function (approximately equal to 2.7183) and w wealth in hundreds of thousands of dollars. Suppose that this person has a 40% chance of wealth of P50,000 and a 60% chance of wealth of P1,000,000 as summarized by P(0.40, P50,000, P1,000,000). 1). What is the expected value of wealth? 2). Is this person risk averse, risk neutral, or a risk seeker? 3). What is this person's certainty equivalent for the prospect?

Solve a problem involving the exponential utility function u(w) = e^w to calculate the expected value of wealth, assess risk preferences, and find the certainty equivalent for a person with a 40% chance of P50,000 and a 60% chance of P1,000,000. This analysis uses concepts such as expected value, risk-seeking behavior, and expected utility theory.

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