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 the price of cake doubles while the price of coffee remains at \(\$ 4\) a cup and Amy's income remains at \(\$ 20,\) describe the change in her budget line.

Short answer

The budget line shifts inward, allowing fewer slices of cake (from 10 to 5), while the maximum quantity of coffee (5 cups) remains the same.

Step-by-step solution

Understanding the Initial Budget Line

Initially, Amy has \(\textdollar 20\) to spend. The price of coffee is \(\textdollar 4\) per cup, and the price of cake is \(\textdollar 2\) per slice. To establish the budget line, determine how many units of each product Amy can buy with her budget. For coffee, \(\frac{20}{4} = 5\) cups, and for cake, \(\frac{20}{2} = 10\) slices.

Calculate the New Price of Cake

If the price of cake doubles, the new price becomes \(\textdollar 4\) per slice.

Establish the New Budget Line

With the new cake price of \(\textdollar 4\) per slice and coffee remaining at \(\textdollar 4\) per cup, recalculate how many units of each good can be bought with \(\textdollar 20\). For both coffee and cake, \(\frac{20}{4} = 5\) units. Therefore, the new budget line will be determined by these quantities.

Describe the Change in the Budget Line

With the change in the price of cake doubling and the price of coffee unchanged, Amy's budget line will shift. Previously, she could buy up to 10 slices of cake if she bought no coffee. Now, she can only buy up to 5 slices of cake. Similarly, if she buys no cake, she can still buy 5 cups of coffee. The budget line thus shifts inward, reflecting fewer slices of cake for the same amount of money while the maximum quantity of coffee Amy can buy remains unchanged.

Key concepts

budget line

The budget line represents all the combinations of goods and services that a consumer can purchase given their income and the prices of those goods and services.
For Amy, her initial budget line was based on her \( \text{\$20} \) income, the price of coffee being \( \text{\$4} \) per cup, and the price of cake being \( \text{\$2} \) per slice.
This means Amy could buy up to 5 cups of coffee (if she bought no cake) or up to 10 slices of cake (if she bought no coffee), given by the equation:
\[ 20 = 4Q_c + 2Q_k \]
Here, \( Q_c \) is the quantity of coffee and \( Q_k \) is the quantity of cake.

price change

A price change occurs when the cost of a good or service shifts.
In this exercise, the price of cake doubled from \( \text{\$2} \) to \( \text{\$4} \) per slice while the price of coffee remained unchanged at \( \text{\$4} \) per cup.
When the price of cake increased, Amy's purchasing power for cake decreased because each slice of cake now costs the same as a cup of coffee.
This change is expressed in the new budget constraint equation:
\[ 20 = 4Q_c + 4Q_k \]
This implies she can now buy a maximum of 5 units of either coffee or cake.

income effect

The income effect describes how a change in the price of a good impacts the purchasing power of a consumer's income.
In Amy's case, the price of cake doubled, effectively reducing her purchasing power regarding cake.
Initially, she could buy up to 10 slices of cake with \( \text{\$20} \), but now she can only afford 5 slices with the same income.
Her real income, in terms of cake, has decreased even though her nominal income remains unchanged at \( \text{\$20} \).
Consequently, Amy must reassess her consumption choices to maximize her satisfaction.

optimization

Optimization involves making the best possible decision given the constraints.
Amy must now optimize her consumption of coffee and cake given her fixed income and the changed price of cake.
To do this, she needs to find the combination of coffee and cake that maximizes her utility.
The budget line shift due to the price change affects her options, reducing the number of slices of cake she can afford without impacting her coffee consumption capacity.
By finding the new optimal point on her budget line, Amy can allocate her \( \text{\$20} \) in a way that yields the highest possible satisfaction, considering the new prices.