Question

Joy goes bowling once and ice skating twice a month when he has \(\$ 20\) to spend on these activities. A visit to the bowling alley costs \(\$ 10,\) and an ice skating ticket costs \(\$ 5 .\) Draw Joy's budget line. If the price of an ice skating ticket falls to \(\$ 4,\) describe how Joy's consumption possibilities change.

Short answer

The decreased price of ice skating tickets allows Joy to afford more ice skating visits, increasing the maximum from 4 to 5 visits, while keeping the maximum possible bowling visits unchanged at 2.

Step-by-step solution

Define Variables and Prices

Let the cost of a visit to the bowling alley be represented as \(P_b = 10\) dollars and the cost of an ice skating ticket be represented as \(P_s = 5\) dollars. Joy's budget is \(B = 20\) dollars.

Establish the Budget Constraint with Original Prices

Joy spends \(10\) dollars per visit to the bowling alley and \(5\) dollars per visit to the ice skating rink. Let \(b\) be the number of bowling visits and \(s\) be the number of ice skating visits. The budget equation is \(10b + 5s = 20\).

Draw the Budget Line

To draw the budget line, determine the intercepts. Set \(b = 0\) to find the maximum number of ice skating visits: \(5s = 20\) which gives \(s = 4\). Set \(s = 0\) to find the maximum number of bowling visits: \(10b = 20\) which gives \(b = 2\). Plot the points (2, 0) and (0, 4) and connect them with a line.

Analyze the New Budget Line with Changed Prices

If the price of an ice skating ticket decreases to \(4\) dollars, the new budget equation becomes \(10b + 4s = 20\).

Draw the New Budget Line

Determine the new intercepts. Set \(b = 0\) to find the maximum number of ice skating visits: \(4s = 20\) which gives \(s = 5\). Set \(s = 0\) to find the maximum number of bowling visits: \(10b = 20\) which still gives \(b = 2\). Plot the points (2, 0) and (0, 5) and connect them with a line.

Describe the Change in Consumption Possibilities

With the price change, Joy can now afford more ice skating visits. This shifts the budget line outward, increasing Joy's consumption possibilities for ice skating while keeping the maximum possible bowling visits unchanged.

Key concepts

Consumer Choice

When consumers like Joy allocate their budget on different goods or activities, they aim to maximize their overall satisfaction, or utility. Choices depend on preferences and the prices of goods. For example, Joy chooses between bowling and ice skating. Given \(\$20\), he can afford combinations of these activities. To make the best choice, Joy weighs how much enjoyment he gets from each activity against their costs. This helps him decide how many times to go bowling or ice skating within his budget.

Budget Line

The budget line shows all the combinations of two goods or activities that a consumer can afford with a given budget. For Joy, his budget line is determined by his \(\$20\) and the prices of bowling \(\(\$10\)\) and ice skating \(\(\$5\)\). Mathematically, this is represented by the equation \(10b + 5s = 20\), where \[b\] is the number of bowling visits and \[s\] is the number of ice skating visits. If Joy spends all his budget on bowling, he can afford 2 visits (20 ÷ 10). If he spends it all on ice skating, he can afford 4 visits (20 ÷ 5). Points on this line represent all the possible combinations Joy can choose.

Price Change

When the price of one of the goods changes, it affects the budget line. For instance, if the cost of an ice skating ticket drops from \(\$5\) to \(\$4\), Joy's budget line changes. The new budget equation becomes \(10b + 4s = 20\). Now, if Joy spends his entire budget on ice skating, he can afford more visits. Instead of 4 visits, he can now buy 5 tickets (20 ÷ 4). This shifts the budget line outward, showing that his purchasing power for ice skating has increased, while the maximum number of bowling visits remains the same.

Consumption Possibilities

With a new budget line, Joy's consumption possibilities have expanded. He now has more combinations of bowling and ice skating to choose from. Before the price change, his budget allowed for combinations like 2 bowling visits and 0 ice skating, or 1 bowling and 2 ice skating. After the price drop, he can still choose 2 bowling visits but up to 5 ice skating visits. This means that Joy can enjoy more activities and have better consumption choices. This also showcases how a price change in one activity can affect overall consumption possibilities and ultimately enhance a consumer's utility.