Question

An individual receives utility from daily income \((Y),\) given by \\[ U(Y)=100 Y-\frac{1}{2} Y^{2} \\] The only source of income is earnings. Hence, \(Y=w L\), where \(w\) is the hourly wage and \(L\) is hours worked per day. The individual knows of a job that pays \(\$ 5\) per hour for a certain 8 hour day. What wage must be offered for a construction job where hours of work are random with a mean of 8 hours and a standard deviation of 6 hours to get the individual to accept this more "risky" job? Hint: This problem makes use of the statistical identity \\[ E\left(X^{2}\right)=\operatorname{Var} X+E(X)^{2} \\] where \(E\) means "expected value."

Short answer

Answer: To make the individual accept the risky construction job, the wage offered must be $8 per hour.

Step-by-step solution

Calculate the utility for the stable job

To find the utility of the stable job, we need to calculate the daily income and insert it into the utility function. The wage is \(5 per hour for 8 hours a day, so the daily income for the stable job is \)Y=wL = 5\times8=40\(. Now, we just need to plug this value into the utility function. \)U(40) = 100(40) - \frac{1}{2}(40)^2 = 4000 - 800 = 3200$.

Calculate the expected value of daily income for the risky job

To find the expected value, we know that the mean of hours worked per day is 8 and the standard deviation is 6. We can use the statistical identity \(E(X^2) = \operatorname{Var}X+E(X)^2\) to find the expected value of \(Y^2\) for the risky job. First, since variance is the square of the standard deviation, the variance of working hours is \(6^2 = 36\). We can plug the variance and the mean into the equation: \(E(Y^2) = \operatorname{Var}(wL) + E(wL)^2 = w^2\operatorname{Var}(L) + w^2E(L)^2 = w^2(36 + 8^2) = w^2 100\).

Find the wage that equates the expected utility for both jobs

We want to find the wage \(w\) for the risky job such that the expected utility for the stable job and the risky job are equal (\(E[U(wL)] = U(40)\)). We need to find the expected value of the income squared term in the utility function: \(E[U(wL)] = E[100wL - \frac{1}{2}(wL)^2]\). Using the values we calculated in step 2 and substituting into the expected value of utility: \(E[U(wL)] = 100w \times 8 - \frac{1}{2}(w^2 \times 100) = 800w - 50w^2\). Now, we can set the two utilities equal and solve for w: \(3200 = 800w - 50w^2\). Simplify the equation by dividing by 100: \(32 = 8w - 0.5w^2\). Rearrange the equation: \(0.5w^2 - 8w + 32 = 0\).

Solve the quadratic equation to find the required wage

Now, we have a quadratic equation to solve for the wage (\(w\)) that makes the expected utility equal for both jobs. We can do this by using the quadratic equation formula or by factoring: In this case, the equation can be factored to \((w - 4)(w - 8) = 0\). This gives us two solutions: \(w = 4\) and \(w = 8\). However, since we know that the individual needs a higher wage for the risky job to make it worth their while, we can discard the \(w = 4\) solution. So to make the individual accept the risky job, the construction job must offer a wage of $8 per hour.

Key concepts

Utility Function

In microeconomic theory, the utility function is a mathematical representation of consumer preference. It translates the satisfaction or happiness a person derives from consuming goods and services into numbers, which can be analyzed quantitatively. Consider the utility function from our exercise:

\[ U(Y) = 100Y - \frac{1}{2}Y^2 \].

This quadratic function implies diminishing marginal utility, meaning that as income increases, each additional dollar generates less additional utility. When solving economic problems involving utility, we often replace actual 'utility' with a representation that's easier to manipulate mathematically. Here, utility is measured against income (Y). It's crucial to understand that the aim is to maximize this utility, subject to constraints like income and time.

In the provided exercise, the predictable job yields a steady income, and hence utility can be calculated directly. However, this differs for the 'risky' job since income there is variable. The candidate has to compare the sure utility from the stable job with the expected utility from the risky job, which requires the concept of expected value.

Expected Value

The expected value is the average outcome if an experiment or process were repeated a large number of times. Essentially, it's the long-term average value of repetitions of the experiment it represents. For instance, if a dice is thrown repeatedly, the expected value of the number observed would be 3.5, because that's the average of all numbers that could come up (1 through 6).

Within the context of the exercise, the expected value gives us the average daily income for the risky job, considering that the hours are random. It's calculated using the formula \( E(wL) = wE(L) \), where \( E(L) \) is the expected value of hours worked. However, because of the utility function's squared term, we also need to consider the expected value of income squared \( E(Y^2) \), which uses the statistical identity mentioned in the exercise.

Risk and Income

In microeconomics, understanding risk and income is pivotal when individuals face uncertain outcomes. Risk impacts decision-making, especially when considering job options where income can fluctuate. A 'risk-averse' individual prefers a sure amount of income over a gamble with the same expected income due to the dislike of uncertainty.

The exercise scenario is a classic example where the individual considers a sure income from a predictable job versus a risky job with random hours. To equate the utility, or satisfaction, between the assured and risky incomes, one must include a risk premium—additional compensation to make the risky job as attractive as the secure one.

Variance and Standard Deviation

Variance and standard deviation are key concepts in statistics used to measure variability or spread in a set of data. Variance is the average squared deviation from the mean, while standard deviation is the square root of variance, providing a measure of dispersion in the same units as the data. High variance and standard deviation indicate data points are spread out over a large range of values, implying higher uncertainty or risk.

In our exercise, the standard deviation indicates variability in the number of hours worked in the construction job. It's critical to consider this because higher variability leads to higher uncertainty in daily income. Therefore, understanding standard deviation helps us assess the risk associated with income, which in turn, affects the utility derived from the job.