91Ó°ÊÓ

You are romantically interested in Chris, but have always wanted to date the president of the Economics Club. As it turns out, Chris is battling Pat for control of the Econ Club. That battle should be decided in a year, and you estimate the odds of Chris winning at \(60 \%\). Attracting Chris and kindling a relationship will involve \(\$ 1,000\) of effort on your part; if Chris wins the presidency, you will receive benefits worth \(\$ 2,200\) (assume you receive these benefits one year after beginning the relationship). If Chris loses the election, you receive nothing. a. Assume an interest rate of \(10 \% .\) Calculate the net present value of building a relationship with Chris today. Notice that the costs of kindling a relationship today are certain, but the benefits are uncertain. b. Considering only your answer to (a), should you initiate a relationship with Chris at this time? Assume you are risk-neutral in formulating your answer. c. Calculate the net present value of waiting until the presidency is decided to build a relationship with Chris. Note that both the costs and benefits of kindling a relationship are uncertain at this point, but that two will be certain in one year. d. Based on your answers to both (a) and (c), should you initiate a relationship with Chris today, or should you wait to initiate the relationship until the presidency is determined?

Step by step solution

01

Calculate Expected Benefits if Initiating Today

If Chris wins (60% chance), the benefits will be \(2,200 one year from now. To find the expected value of these benefits, we calculate: \( 0.6 \times 2200 = 1320 \\) \). Since the benefits are not realized until next year, we need to discount them back to present value using the formula: \( \text{Present value} = \frac{\text{Future value}}{(1 + r)^n} \), where \( r = 0.10 \) and \( n = 1 \). Therefore, the present value is \( \frac{1320}{1.1} = 1200 \$.\)

02

Calculate Net Present Value of Initiating Today

The costs of initiating the relationship are \(1,000 today. We calculated the present value of the expected benefits to be \)1,200 in Step 1. The net present value (NPV) is given by: \( \text{NPV} = \text{Present value of benefits} - \text{Cost} = 1200 - 1000 = 200 \$.\)

03

Decision Based on NPV for Today

Since the NPV of initiating the relationship today is positive ($200), it would generally be advisable, in economic terms, to proceed with the relationship from a purely financial perspective, assuming risk neutrality.

04

Calculate NPV of Waiting for the Outcome

If we wait one year, we make the decision only after knowing the presidency's outcome. If Chris wins, we have benefits of $2,200 after one year, discounted back to form a present-day cost-benefit analysis. In this scenario, cost and benefit occur simultaneously, giving NPV = $2,200 - $1,000 = $1,200 directly, as they both happen in the same future period without discount.

05

Compare NPVs to Decide Timing

From Steps 2 and 4, the NPV of initiating today is $200, while waiting results in an NPV of $1,200 (if Chris wins). In expected terms weighted by probabilities, the true cost is zero when Chris does not win as no relationship is pursued; hence, waiting could result in either 0 or 1,200 based on outcomes known after one year. The averaging of probabilities and costs suggests waiting is optimal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Rate and Discounting

Understanding the interest rate is a fundamental part of evaluating investments and decision-making in economics. When we talk about the interest rate, we're essentially talking about the cost of borrowing money, or the reward for saving. In this exercise, the interest rate is set at 10%. This rate is crucial because it affects how we compare money today to money in the future.

Discounting is the process of finding the present value of a sum of money that you will receive or pay in the future. To do this, we use the formula:\[\text{Present value} = \frac{\text{Future value}}{(1 + r)^n}\]where:

• \( r \) is the interest rate (0.10 or 10% in our case).
• \( n \) is the number of periods until the money is received (one year here).

By using discounting, we determine that the value of possible future benefits reduces over time, which means you have to weigh the current cost against these future benefits. In this scenario, it helps determine if the relationship costs today are worth the potential future benefits.

Expected Value Calculation

Expected value calculation is a tool used in decision-making to assess the potential benefits of different choices under uncertainty. In this context, it calculates the weighted average of all possible outcomes, taking into account their probabilities. This means it's especially valuable when outcomes are not certain, such as in our situation with Chris.

To calculate the expected value of the benefits from a relationship with Chris, given a 60% chance of winning the presidency, we multiply the probability by the amount of benefits: \( 0.6 \times 2200 = 1320 \).

However, these benefits are expected to occur in one year, so we must discount them to find their present value. We use our discounting formula here, resulting in a present value of \$1200 for the expected benefits. In making financial decisions, always consider potential benefits and risks. The expected value is a way to quantify those potential outcomes and choose the best option.

Decision-Making under Uncertainty

Decisions under uncertainty involve evaluating choices when the outcome is unknown, often guided by expected value and net present value (NPV). In this exercise, the uncertainty springs from not knowing who will win the presidency, which affects the relationship's potential benefits.

In decision-making, NPV is a vital tool. It compares the present value of cash inflows (benefits) with outflows (costs). A positive NPV, like our \\(200 for initiating today, usually suggests a good financial move. However, you must also consider the possible outcomes if you wait. Waiting gives more definitive information. If Chris wins, the NPV would be \\)1,200, given costs and benefits happen in the same period, making it more attractive than acting on uncertainty.

Ultimately, the choice involves weighing immediate action against informed, delayed decisions. In this scenario, waiting seems optimal since it offers a higher NPV and reduces future uncertainty.