Question

Sara's income is \(\$ 12\) a week. The price of popcorn rises from \(\$ 3\) to \(\$ 6\) a bag, and the price of a smoothie is unchanged at \(\$ 3 .\) Explain how Sara's budget line changes with smoothies on the \(x\) -axis.

Short answer

The budget line changes from \(P + S = 4\) to \(2P + S = 4\). Sara can now afford fewer bags of popcorn for the same income.

Step-by-step solution

Determine Initial Budget Constraint

Sara's initial income is \$12\ per week. The price of popcorn is \$3\ per bag, and the price of smoothies is \$3\ per smoothie. The budget constraint can be initially expressed as \[ 12 = 3P + 3S \] where \(P\) is the number of popcorn bags and \(S\) is the number of smoothies.

Simplify the Initial Budget Constraint

Simplify the initial budget constraint equation to: \[ P + S = 4 \] This represents Sara's initial budget line.

Determine New Budget Constraint

With the price of popcorn increasing to \$6\, the new budget constraint becomes \[ 12 = 6P + 3S \] where \(P\) is the number of popcorn bags and \(S\) is the number of smoothies.

Simplify the New Budget Constraint

Simplify the new budget constraint equation to: \[ 2P + S = 4 \] This represents Sara's new budget line.

Compare the Budget Lines

Initially, the budget line was \( P + S = 4 \). After the price increase, the budget line is \( 2P + S = 4 \).

Key concepts

Income

Income is the amount of money an individual has available to spend on goods and services. In this problem, Sara has an income of \( \$12 \) per week. Her income sets the limit on the maximum amount she can purchase within that week. When creating a budget constraint, you need to consider the total income available. This amount does not change when the prices of goods change, but it affects how much of each good Sara can afford.
For example:

• Initial income: \( \$ 12 \)
• Initial prices: Popcorn: \( \$ 3 \)/bag, Smoothies: \( \$ 3 \)/smoothie

Sara needs to allocate her \( \$12 \) such that she maximizes her utility given the prices of the items she wishes to buy.

Price Changes

Price changes impact how much of each good you can buy with your given income. In Sara's case, the price of popcorn has risen from \( \$ 3 \) to \( \$ 6 \) per bag. The price of smoothies remains at \( \$ 3 \) per smoothie.
Think of it in this way:

• Before the price change, for every \( \$3 \) Sara spends, she can buy 1 bag of popcorn, and for every \( \$3 \) spent, she can buy 1 smoothie.
• After the price increase, \( \$6 \) is needed to buy 1 bag of popcorn, meaning Sara needs twice her old budget for the same quantity of popcorn.

Price increases reduce the quantity of the product Sara can afford if her income remains constant.

Budget Line

A budget line represents all the combinations of two goods that can be purchased with a given income. To visualize: with smoothies on the \( x \)-axis, the initial budget line for Sara is \( P + S = 4 \) (Sara can purchase 4 items if all money is allocated equally between popcorn and smoothies).
After the price of popcorn increases, the new budget line becomes \( 2P + S = 4 \).

• Initial budget line: \( P + S = 4 \)
• New budget line: \( 2P + S = 4 \)

This shows Sara can now only buy fewer popcorn bags if she wants the same number of smoothies or vice versa.

Opportunity Cost

Opportunity cost refers to the value of the next best alternative that you give up when you make a choice. In Sara's context, it means how many smoothies she has to give up to afford one more bag of popcorn and vice versa.
Initially, each bag of popcorn costs one smoothie \( \left( 3P = 3S \right) \), so the opportunity cost of 1 bag of popcorn = 1 smoothie.
After the price increase, the opportunity cost changes because each bag of popcorn now costs \( 6 \) dollars or the equivalent of \( 2 \) smoothies. To express it:

• Opportunity cost of 1 bag of popcorn (initially): 1 smoothie
• Opportunity cost of 1 bag of popcorn (after price rise): 2 smoothies

This higher opportunity cost affects Sara's decision on how to allocate her income.