Question

Use the following data to work Problems 16 to 19 Amy has \(\$ 20\) a week to spend on coffee and cake. The price of coffee is \(\$ 4\) a cup, and the price of cake is \(\$ 2\) slice. Calculate the cquation for Amy's budget line (with cups of coffec on the left side).

Short answer

C = 5 - 0.5K

Step-by-step solution

Determine the Budget Constraint

Amy has \(20 to spend on coffee and cake. Let us denote the number of cups of coffee Amy buys as C and the number of slices of cake as K. The prices given are \)4 for a cup of coffee and $2 for a slice of cake.

Set Up the Budget Equation

The budget equation can be set up based on the total amount Amy has to spend. The total expenditure on coffee is \( 4C \) and the total expenditure on cake is \( 2K \). Therefore, the budget constraint equation is given as: \[ 4C + 2K = 20 \]

Isolate the Coffee Variable

To isolate the coffee variable \( C \) on the left side, the equation \( 4C + 2K = 20 \) can be rearranged: \[ 4C = 20 - 2K \]

Solve for Coffee Cups

Divide both sides of the equation by 4 to solve for \( C \), resulting in: \[ C = 5 - 0.5K \]

Key concepts

Budget Line

A budget line represents all the combinations of two goods that a consumer can purchase given their income and the prices of the goods. For Amy, the budget line equation is derived based on her weekly budget of \(20, the price of a cup of coffee (\text{denoted as } C) which is \)4, and the price of a slice of cake (\text{denoted as } K) which is \(2.
The budget constraint equation is given by: 4C + 2K = 20 C = 5 - 0.5K
The budget line tells us that if Amy spends all her money on coffee, she can buy 5 cups (since \)20/\(4 = 5). Alternatively, if she spends all her money on cake, she can buy 10 slices (since \)20/\(2 = 10). Any point on the line represents a combination of coffee and cake that costs exactly \)20.

Expenditure

Expenditure refers to the amount of money spent on goods and services. In this context, Amy's total expenditure is the sum of her spending on coffee and cake.
Here’s how the expenditure is calculated based on her purchases:

• If Amy buys 3 cups of coffee, her expenditure on coffee is 3 * \(4 = \)12.
• If Amy buys 2 slices of cake alongside, her expenditure on cake is 2 * \(2 = \)4.
• So, her total expenditure is \(12 (coffee) + \)4 (cake) = \(16.

The budget line equation, 4C + 2K = 20, ensures that Amy's expenditure does not exceed her \)20 weekly budget.

Opportunity Cost

Opportunity cost is the value of the next best alternative that is forgone when making a choice. For Amy, every time she chooses to buy more of one good, she sacrifices the opportunity to buy some amount of the other good.
For instance:

• The opportunity cost of buying one more cup of coffee is the cake slices she gives up, which is given by the ratio of their prices. If Amy buys one more cup of coffee at \(4, she forgoes 2 slices of cake (since she could have used that \)4 to buy two \(2 slices of cake).
• The opportunity cost of buying one more slice of cake is half a cup of coffee because one slice of cake costs \)2, and she could have used that \(2 to buy half a \)4 cup of coffee.

Understanding opportunity costs helps Amy make more informed decisions about how to allocate her limited budget.

Consumer Choices

Consumer choices refer to the decisions individuals make on what goods or services to purchase using their limited resources. Amy's choices are influenced by her preferences, budget constraint, prices of coffee and cake, and her subjective valuation of their utility.

Amy might decide:

• To buy 3 cups of coffee and 2 slices of cake, if she values coffee more and desires a balance between coffee and cake.
• To buy 5 cups of coffee and no cake, if she prefers coffee significantly more and derives more satisfaction from drinking coffee.
• To buy 10 slices of cake and no coffee, if she has a sweeter tooth and gains more enjoyment from cake.

By evaluating her budget line, Amy can visualize all possible combinations of coffee and cake within her budget and make the choice that best satisfies her preferences.