Question

Imagine a person who makes \(\$ 400\) per week working 40 hours per week for 50 weeks of the year. She is currently eligible for a welfare program, available to people with income below \(\$ 21,000\), that gives her \(\$ 800\) a year. No such program is available to people with income above \(\$ 21,000\) per year. Her boss offers her a promotion that would increase her wage by 25 cents per hour. [LO 21.4] a. What is her total income before the promotion? b. What is her total income if she accepts the promotion? c. Should she accept the promotion if she wants to have higher income?

Short answer

She should accept the promotion as it increases her total income.

Step-by-step solution

Calculate Weekly Income Before Promotion

The person makes \( \\( 400 \) per week working 40 hours. Her annual income before the promotion is calculated by multiplying weekly income by the number of weeks she works: \( \\) 400 \times 50 = \$ 20,000. \)

Add Welfare Income to Total Income

She receives an additional \( \\( 800 \) per year from the welfare program. Thus, her total income before the promotion is \( \\) 20,000 + \\( 800 = \\) 20,800. \)

Calculate New Hourly Wage and Weekly Income

With a 25 cent per hour raise, her new wage is \( 0.25 + 10 = 10.25 \) dollars per hour (assuming the original wage was \( \\(10\) per hour), so her new weekly income is \( 10.25 \times 40 = \\) 410. \)

Calculate Annual Income After Promotion

Multiply the new weekly income by the number of weeks worked: \( \\( 410 \times 50 = \\) 20,500. \)

Determine Welfare Eligibility

Her income after the promotion is \( \\( 20,500 \), which is below the \( \\) 21,000 \) threshold, so she is still eligible for the \( \\( 800 \) welfare income, making her total income \( \\) 20,500 + \\( 800 = \\) 21,300. \)

Decision Based on Higher Income

Her total income after promotion is \( \\( 21,300 \), which is higher than her total initial income of \( \\) 20,800 \). Therefore, accepting the promotion would provide her with a higher income.

Key concepts

Welfare Program

A welfare program is a government assistance initiative designed to help individuals who earn below a certain income threshold. In this scenario, the threshold is set at $21,000 per year. For individuals whose income falls below this limit, the program provides additional financial assistance, in this case amounting to $800 annually. It's important to note that if the individual’s income rises above the $21,000 threshold, eligibility for the welfare benefit is lost. In this exercise, understanding whether the person's total income, with or without welfare, meets this criteria is central to deciding between accepting or declining a promotion. The welfare program acts as a safety net, providing vital support to ensure that individuals can meet their basic needs even if their income is insufficient. Keeping track of such thresholds and the impact of additional earnings on welfare eligibility is crucial for effective personal financial planning.

Hourly Wage Calculation

Hourly wage calculation involves determining how much an individual earns per each hour worked. In the given problem, the individual originally earns $10 per hour. The calculation of her new hourly wage involves a simple addition of the raise to her existing wage. If she's offered a raise of 25 cents per hour, her new wage becomes $10.25 per hour. This can be calculated using: - Original hourly wage: $10 - Raise per hour: $0.25 - New hourly wage: $10 + $0.25 = $10.25 Calculating the hourly wage is crucial in determining weekly and subsequently annual income, especially when considering the effect of prospective wage increases on overall earnings. As wages rise, understanding their transformations into larger sums such as weekly or yearly earnings can guide decisions about promotions and welfare benefits.

Annual Income Calculation

Annual income calculation is a key financial metric that provides an overall snapshot of a person's earnings over a year. It can be broken down into steps starting with weekly income. For the problem at hand, the individual's weekly income prior to the promotion is \(400, calculated from their hourly work multiplied by the hours worked per week. To find the annual income, multiply weekly income by the number of weeks worked in a year. Here's the breakdown:- Weekly income: \)400- Number of working weeks in a year: 50- Annual income: \(400 \times 50 = \)20,000After receiving a wage increase to \(10.25 per hour, weekly income becomes \)410. Thus, the new annual income is calculated as:- New weekly income: \(410- Annual income post-promotion: \)410 \times 50 = $20,500Understanding annual income helps determine eligibility for welfare programs and whether additional income, such as from a promotion, surpasses the existing income benefits a person receives.

Promotion Decision

The decision to accept a promotion should consider whether it results in a net income benefit after accounting for changes in salary and corresponding welfare benefits. The critical factor to evaluate is whether the increased hourly wage balances against potential reductions in welfare support. In this problem, accepting the promotion results in a new income of $21,300, which surpasses the initial $20,800 total income when considering her continued eligibility for the welfare program. Here's the benefit evaluation: - Initial total income with welfare: $20,800 - New total income post-promotion: $20,500 (salary) + $800 (welfare) = $21,300 Conclusively, calculating the net gain from the promotion, understanding eligibility for welfare based on income levels, and examining the total financial outcome aids in making an informed promotion decision. It's not just about the immediate wage increase, but about sustaining and potentially enhancing overall financial health.