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. If Amy's income increases to \(\$ 24\) a week and the prices of coffee and cake remain unchanged, describe the change in her budget line.

Short answer

Amy's budget line shifts outward from \[K = 10 - 2C\] to \[K = 12 - 2C\], allowing her to buy more of both coffee and cake.

Step-by-step solution

Understand the Initial Budget Line

The budget line shows all the combinations of two goods that a consumer can purchase with a given income, given the prices of the two goods. Initially, Amy has \$20\ to spend on coffee (C) and cake (K). The price of coffee is \$4\ per cup, and the price of cake is \$2\ per slice. The budget line equation is given by: \[4C + 2K = 20\]

Set up the Budget Line Equation

Rewrite the budget line equation to express K in terms of C: \[K = 10 - 2C\]. This represents the initial budget line where Amy can buy maximum of \$0\ items or any point on the line represents a combination of both items.

Consider the Increase in Income

Amy's income increases to \$24\ while the prices of coffee and cake remain unchanged. Update the budget line equation using the new income: \[4C + 2K = 24\].

Rearrange the New Budget Line Equation

Rewrite the new budget line equation to express K in terms of C: \[K = 12 - 2C\]. This represents the new budget line after the income increase.

Describe the Change in the Budget Line

The increase in Amy's income causes a parallel shift of the budget line outward. The slope of the budget line remains the same, but it now intersects the K-axis at 12 (when C = 0) and the C-axis at 6 (when K = 0), indicating she can purchase more of both goods under the new income.

Key concepts

consumer budget constraint

The consumer budget constraint represents all possible combinations of goods and services that a consumer can afford, given their income and the prices of those goods and services.
In Amy’s case, her budget constraint can be visualized with a budget line. Initially, she has \(20 to spend on coffee and cake, with coffee priced at \)4 per cup and cake at \(2 per slice.
The budget constraint can be expressed as the equation: 4C + 2K = 20 where \(C\) represents the number of cups of coffee and \(K\) the number of slices of cake. This equation tells us that if Amy spends all her money on coffee, she can buy 5 cups. If she spends it all on cake, she can buy 10 slices.
Any combination of coffee and cake that totals exactly \)20 lies on this budget line, showing her consumer budget constraint.

income changes

When a consumer's income changes, their budget constraint shifts.
In Amy's scenario, her income increased from \(20 to \)24, while the prices of coffee and cake remained constant. This changes her budget equation to: 4C + 2K = 24 Rearranging this equation to express \(K\) in terms of \(C\), we get: K = 12 - 2C This new equation represents a new budget line that shows Amy can now afford more coffee and cake combinations. Specifically, she can buy up to 6 cups of coffee or up to 12 slices of cake, if she spends her entire income on just one of the goods.
As her income increased, the budget line shifted outward, making more combinations of coffee and cake attainable.

price constancy

In this exercise, while Amy’s income changes, the prices of coffee and cake remain unchanged. This is known as price constancy.
When prices stay the same, but income increases, the effect is that the budget line shifts outward without changing its slope. The slope of the budget line depends on the prices of both goods. If the price of coffee remains at \(4 per cup and cake at \)2 per slice, the slope, calculated as \frac{Price \ of \ Coffee}{Price \ of \ Cake} \ = \ \frac{4}{2} remains the same. Thus, when Amy’s income rises from \(20 to \)24, with prices constant, the line shifts outward without rotating. This parallel shift indicates that Amy can now afford to buy more of both goods.

budget line shift

A budget line shift occurs when there is a change in either the consumer’s income or the prices of goods. In Amy’s case, an increase in income causes the shift.
The new budget line is derived from the updated budget constraint equation: 4C + 2K = 24 The shift is characterized by:

1. The line moving outward from the original budget constraint, indicating higher purchasing power.
2. The intercept of the budget line on the \(C\) (coffee) axis increasing from 5 to 6 (when \(K = 0\))
3. The intercept of the budget line on the \(K\) (cake) axis increasing from 10 to 12 (when \(C = 0\))

The slope stays the same because the prices of both goods are constant. This parallel, outward shift shows that Amy can now purchase more either separately or in combination, thus expanding her consumption possibilities.