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Chapter 3: Problem 16

Budget constraints: Your family likes to eat fruit, but because of budget constraints, you spend only \(\$ 5\) each week on fruit. Your two choices are apples and grapes. Apples cost \(\$ 0.50\) per pound, and grapes cost \(\$ 1\) per pound. Let \(a\) denote the number of pounds of apples you buy and \(g\) the number of pounds of grapes. Because of your budget, it is possible to express \(g\) as a linear function of the variable \(a\). To find the linear formula, we need to find its slope and initial value. a. If you buy one more pound of apples, how much less money do you have available to spend on grapes? Then how many fewer pounds of grapes can you buy? b. Use your answer to part a to find the slope of \(g\) as a linear function of \(a\). (Hint: Remember that the slope is the change in the function that results from increasing the variable by 1. Should the slope of \(g\) be positive or negative?) c. To find the initial value of \(g\), determine how many pounds of grapes you can buy if you buy no apples. d. Use your answers to parts \(b\) and \(c\) to find \(a\) formula for \(g\) as a linear function of \(a\).

Short Answer

Expert verified
The linear formula is \(g = -0.5a + 5\).

Step by step solution

01

Define budget constraint equation

The total cost for apples and grapes must not exceed your budget. Let's express this constraint with the equation: \(0.50a + 1g = 5\). This equation shows that the sum of money spent on apples and grapes should be the budget of \(5\) dollars.
02

Calculate reduction in grape buying capacity when buying more apples

Buying one more pound of apples costs an additional \(0.50\) dollars. To maintain the same budget, this means you have \(0.50\) less available for grapes. Since grapes cost \(1\) dollar per pound, \(0.50\) dollars means you can buy \(0.5\) fewer pounds of grapes.
03

Determine slope of g as a function of a

Per each additional pound of apples (an increase of \(a\) by 1), you buy \(0.5\) fewer pounds of grapes. Therefore, the change in \(g\), or the slope, is \(-0.5\). The slope should be negative because buying more apples results in buying fewer grapes.
04

Find the initial value of g

When no apples are bought (\(a = 0\)), the maximum budget of \(5\) is spent solely on grapes. Solving the equation \(1g = 5\) gives \(g = 5\), meaning \(5\) pounds of grapes can be bought initially if no apples are purchased.
05

Formulate the linear equation for g as a function of a

Using the slope \(-0.5\) from Step 3 and the initial value \(g = 5\) from Step 4, the linear function is \(g = -0.5a + 5\). This equation shows how the number of grapes purchased decreases as more apples are bought.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Budget Constraints
Budget constraints are a fundamental concept in economics that reflect the limitations of spending based on available resources. In everyday life, we encounter budget constraints when we have to make choices about how we spend our money.
In the given exercise, your budget constraint is represented by the equation:
  • \(0.50a + 1g = 5\)
This equation means that the total spending on apples (\(a\) pounds at \(0.50 per pound) and grapes (\(g\) pounds at \)1 per pound) must equal or be less than the $5 budget.
The constraint requires you to decide how much of each item you can afford, necessitating a choice about allocation. Balancing this equation involves understanding how spending on one type of fruit affects the other. With each purchasing decision, consumers evaluate their preferences within the limits of their budget, thereby following the economic principle of opportunity cost.
Slope
In the context of linear equations, the slope is a measure of how one variable changes with respect to another. Specifically, it is the "rise over run" between two points on a line. In this exercise, the slope represents how buying more apples affects the number of grapes you can purchase.
When you buy one more pound of apples, you spend an extra \(0.50. This means you have \)0.50 less to spend on grapes. Therefore, you buy 0.5 fewer pounds of grapes.
  • Mathematically, the slope is represented by \(-0.5\).
The negative slope reflects an inverse relationship: more apples mean fewer grapes. In broader terms, a negative slope in economics often implies trade-offs between goods due to budget constraints, highlighting opportunity cost.
Linear Functions
Linear functions are mathematical expressions that create a straight line when graphed. They have the basic form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions are essential in economics to model relationships between different variables.
To formulate the linear function for this exercise, you identify:
  • The slope \(-0.5\, (\text{from the previous section}\))
  • The initial value of grapes if no apples are purchased: \(g = 5\)
The resulting linear function is:
  • \(g = -0.5a + 5\)
The function describes how the number of grapes you can buy changes with each additional apple purchased. Linear functions like this provide simple yet powerful ways to analyze economic behavior and interactions among budgetary choices.
Consumer Choice Model
The Consumer Choice Model is a framework used to understand how consumers make choices based on preferences, prices, and budget constraints. It operates under the assumption that consumers aim to maximize their utility - or satisfaction - given their limited income or resources.
In the exercise, the model is exemplified by your family's decision on how many pounds of apples and grapes to buy with a $5 weekly fruit budget.
By framing the decision within a linear equation, the model helps illustrate:
  • Constraints: you can't exceed a $5 budget.
  • Preferences: choosing a balance between apples and grapes based on cost and desired variety.
In a broader sense, consumer choice models help economists and businesses predict consumer behavior, set prices, and understand the demand for goods. It emphasizes the continuous balancing act consumers face when attempting to optimize satisfaction in light of financial limitations.

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