Question

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is \(\$ 20,000,\) and she knows where she can buy the television she wants for \(\$ 2,000\). She has heard the rumor that the same set can be bought at Crazy Eddie's (recently out of bankruptcy) for \(\$ 1,700,\) but is unsure if the rumor is true. Suppose this individual's utility is given by \\[\text { utility }=\ln (Y).\\] where Fis her income after buying the television. a. What is Molly's utility if she buys from the location she knows? b. What is Molly's utility if Crazy Eddie's really does offer the lower price? c. Suppose Molly believes there is a \(50-50\) chance that Crazy Eddie does offer the lowerpriced television, but it will cost her \(\$ 100\) to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip?

Short answer

a. If Molly buys the television for $2,000, her income after purchase is $18,000. Her utility for this option is: $$utility_a = \ln(18000)$$ b. If Molly buys the television from Crazy Eddie's for $1,700, her income after purchase is $18,300. Her utility for this option is: $$utility_b = \ln(18300)$$ c. If Molly spends $100 to find out about Crazy Eddie's, she has a 50% chance to save $300 or buy the television at the original price. Her expected income is $18,050, and her expected utility is: $$Expected\,Utility = \ln(18,050)$$ To determine if it's worth investing $100, compare the expected utility to the utility of buying from the known location. If the expected utility is greater than the utility of buying from the known location, it's worth investing $100 to find out about Crazy Eddie's.

Step-by-step solution

a. Calculate utility for buying television from known location

If Molly buys the television for \(2,000\), she will have an income after purchase: \(Y_a = 20000 - 2000 = 18000\). Now, we can find her utility: \(utility_a = \ln(Y_a) = \ln(18000)\).

b. Calculate utility for buying television from Crazy Eddie's

If Molly buys at Crazy Eddie's for \(1,700\), her income after purchase will be: \(Y_b = 20000 - 1700 = 18300\). Let's find her utility in this case: \(utility_b = \ln(Y_b) = \ln(18300)\).

c. Analyze if it's worth investing $100 to find out about Crazy Eddie's

If Molly spends \(100\) to find out about Crazy Eddie's, she has \(50\%\) chance to either save \(300\) or end up buying the television for the original price. The expected income from this decision is: $$Expected\,Income = 0.5*(20,000 - 100 - 1,700) + 0.5*(20,000 - 100 - 2,000)$$ $$Expected\,Income = 0.5*(18,200) + 0.5*(17,900) = 18,050$$ Now calculate the expected utility: $$Expected\,Utility = \ln(18,050)$$ Now, compare the expected utility to the utility of buying from the known location: If \(Expected\,Utility > Utility_a\), then it is worth investing the \(100\) to find out about Crazy Eddie's.

Key concepts

Consumer Decision-Making

Understanding how consumers make decisions is a fundamental aspect of microeconomic theory. It encompasses the processes consumers use to choose among different options and allocate their limited resources optimally.

In the case of Molly Jock, we observe consumer decision-making in action. Molly is faced with a choice: purchase a high-definition television from a known location or investigate a potential deal at Crazy Eddie's, which carries inherent uncertainties and costs. Her decision will be guided by two main factors: the price of the television and her resulting utility or satisfaction derived from the purchase.

This process often involves predicting the outcomes of various choices, weighing the costs and benefits, and considering the risks involved. Molly's decision to either buy the television at the known price or spend additional resources to potentially secure a better deal reflects the complexities that consumers face in everyday purchase decisions.

Utility Function

A utility function is a mathematical representation of how a consumer ranks different bundles of goods based on the level of satisfaction or happiness that they provide. Essentially, it translates personal preferences into a quantifiable form that can be analyzed and compared.

In the exercise, Molly's utility function is defined as the natural logarithm of her income after purchasing the television, \( \text{utility} = \(ln(Y) \) \). This suggests that her utility is tied directly to how much money she retains after the purchase, which is a common assumption in many microeconomic models that deal with income and consumption.

The utility function can take various forms, but the natural logarithm is frequently used in economics due to its mathematical properties, such as reflecting diminishing marginal utility – the principle that as one consumes more of a good, the added satisfaction tends to decrease.

Expected Utility Theory

Expected utility theory is an important concept in economics that deals with the decision-making under uncertainty. It involves calculating the expected value of utility across different possible outcomes, giving weight to each based on its probability.

For Molly, the expected utility theory helps her determine if it's worth the risk and cost of driving to Crazy Eddie's. She considers the possible outcomes (finding the cheaper price or not) and their probabilities (50-50 in this case). By taking the expected income from these scenarios and plugging them into her utility function, she can determine the expected utility of taking the risk.

If this expected utility is higher than the utility of buying the television at the known price, the theory suggests it is rational for her to take the chance. This principle extends to various real-life situations, guiding consumers in making choices that maximize their expected level of satisfaction based on their preferences and the uncertainties involved.