Question

Maria has a \(\$ 300\) gift certificate at a spa that she can use on massages or manicures. A massage costs \(\$ 100,\) and a manicure costs \(\$ 30\). [LO 7.3, 7.4] a. Write the equation for Maria's budget constraint. Let \(x=\) massages. Let \(y=\) manicures. b. Suppose Maria decides to split her gift certificate with a friend and transfers half of the value of her gift certificate to her friend. Write the equation for her new budget constraint. c. After giving away half of her gift certificate, suppose the price of massages increases by 50 percent before Maria can use her gift certificate. Write the equation for her new budget constraint.

Short answer

a) \(100x + 30y = 300\); b) \(100x + 30y = 150\); c) \(150x + 30y = 150\).

Step-by-step solution

Define Original Budget Constraint Equation

Maria's original gift certificate value is \(300. The cost of each massage is \)100, and the cost of each manicure is $30. We denote the number of massages as \(x\) and the number of manicures as \(y\). Thus, her budget constraint equation can be written as:\[ 100x + 30y = 300 \]

Adjust for Transfer to Friend

Maria gives away half of her \\(300 gift certificate, resulting in a total of \\)150 for herself. Keeping the same costs for massages and manicures, her new budget constraint becomes:\[ 100x + 30y = 150 \]

Update for Price Increase of Massages

After giving away half of the gift certificate, the price of massages increases by 50%. Thus, the new cost of a massage becomes \(100 + 0.5 \times 100 = 150\) dollars. The manicure price remains at $30. Maria's updated budget constraint is:\[ 150x + 30y = 150 \]

Key concepts

Consumer Choice

Consumer choice is a fundamental concept in microeconomics that explores how individuals make decisions to allocate their limited resources among various goods and services. When making such decisions, consumers consider multiple factors, such as their preferences, the prices of items, and their budget constraints. In Maria's case, she needs to decide how best to spend her $300 gift certificate at a spa, either on massages or manicures.

This decision involves weighing the satisfaction, or utility, she derives from each service against her financial limitations. The choice is influenced by her preferences for massages versus manicures and how these preferences align with her overall budget. When faced with various options, consumers like Maria tend to choose combinations of goods that maximize their utility or satisfaction given their budget constraints. These choices typically form a "utility-maximizing" bundle, where the marginal utility per dollar spent on each good is equal.

In summary, consumer choice boils down to the balancing act of maximizing satisfaction while adhering to a limited budget. This leads to varied decisions based on personal preferences and perceived value of each available option.

Price Elasticity

Price elasticity refers to how sensitive the quantity demanded of a good is to a change in its price. It's an important concept when analyzing consumer behavior and how changes in price can affect demand. In Maria's scenario, even though the exercise doesn't ask for it directly, understanding price elasticity can provide deeper insights into her decision-making process.

• If massages have a high price elasticity of demand, Maria might significantly reduce the number of massages she opts for if their price increases.
• Conversely, if massages are inelastic to price changes, Maria might not change her consumption by much, even with a price increase.

Considering Maria’s gift certificate situation, the increase in the price of massages reflects a situation where price elasticity becomes relevant. If the price of massages goes up, Maria might need to rethink how she allocates her spa budget or if the increase will significantly impact her consumption changes. In real-world scenarios, this might mean shifting her preference toward more manicures or perhaps even sharing this resource differently.

Budget Constraint Equation

A budget constraint equation represents all the combinations of goods and services a consumer can purchase with their limited income, given the prices of goods. In Maria's example, she starts with a gift certificate worth \(300 and can choose between massages and manicures.

This budget constraint can be expressed mathematically, showing the trade-offs Maria faces. Initially, with massages costing \)100 and manicures $30, the budget constraint is: \[ 100x + 30y = 300 \]
Here, each combination of \(x\) (massages) and \(y\) (manicures) that satisfies the equation is an affordable option within her budget.

• When Maria shares her gift certificate with a friend, the constraint adjusts to: \[ 100x + 30y = 150 \]This new equation reflects her reduced budget, forcing her to reassess her number of purchases of massages and manicures.
• The budget constraint changes again when massage prices increase by 50%. The new constraint becomes: \[ 150x + 30y = 150 \]This illustrates how external changes (like price increases) impact the feasible options for consumers, widening the gap between their desires and financial capabilities.

Understanding budget constraints helps highlight how changes in income or prices directly affect available consumer choices, thus influencing decision-making.