Question

Suppose that a consumer always consumes 2 spoons of sugar with each cup of coffee. If the price of sugar is \(p_{1}\) per spoonful and the price of coffee is \(p_{2}\) per cup and the consumer has \(m\) dollars to spend on coffee and sugar, how much will he or she want to purchase?

Short answer

Buy \( \frac{2m}{4p_1 + p_2} \) spoons of sugar and \( \frac{m}{4p_1 + p_2} \) cups of coffee.

Step-by-step solution

Define the Problem Mathematically

The consumer wants to purchase sugar and coffee with a constraint on their total budget. The budget constraint equation is given by:\[ 2p_1x_1 + p_2x_2 = m \]where \( x_1 \) is the number of sugar spoons and \( x_2 \) is the number of cups of coffee consumed.

Set up the Consumption Relationship

Given that the consumer always consumes 2 spoons of sugar for each cup of coffee, we can express the consumption of sugar in terms of coffee as:\[ x_1 = 2x_2 \]This relationship will help us to substitute the sugar variable in the budget constraint equation.

Substitute and Simplify the Budget Constraint

Substitute \( x_1 = 2x_2 \) into the budget constraint:\[ 2p_1(2x_2) + p_2x_2 = m \]Simplify this equation:\[ 4p_1x_2 + p_2x_2 = m \]\[ (4p_1 + p_2)x_2 = m \]

Solve for the Quantity of Coffee

Solve for \( x_2 \) (the number of cups of coffee) in the simplified equation:\[ x_2 = \frac{m}{4p_1 + p_2} \]

Solve for the Quantity of Sugar

Using the relationship \( x_1 = 2x_2 \), substitute the value of \( x_2 \):\[ x_1 = 2\left(\frac{m}{4p_1 + p_2}\right) = \frac{2m}{4p_1 + p_2} \]

Key concepts

Budget Constraint

A budget constraint is one of the basic tools used in economics to illustrate the trade-offs that a consumer faces. It represents all the combinations of goods and services a consumer can purchase, given their income level and the prices of goods. In our exercise, the consumer's budget constraint can be mathematically defined by the equation:\[ 2p_1x_1 + p_2x_2 = m \]Here, \(2p_1x_1\) is the total cost of sugar, and \(p_2x_2\) is the total cost of coffee, where \(m\) represents the total money available to spend. This equation ensures that all spending remains within the bounds of the consumer’s budget.

• The budget constraint limits choices based on available resources.
• It encourages finding the optimal balance between different goods.

Understanding budget constraints is crucial for making informed decisions in consumer economics. When prices or income change, the budget constraint shifts, influencing possible consumption choices.

Consumption Relationship

The consumption relationship in economics describes how consumption of different goods is interrelated. In our scenario, the consumer always consumes 2 spoons of sugar for each cup of coffee. This consistent pattern is key in figuring out how changes in the consumption of one good affect the other. Mathematically, this relationship is expressed as:\[ x_1 = 2x_2 \]This means that for every cup of coffee (\(x_2\)) consumed, 2 spoons of sugar (\(x_1\)) are used.

• Helps in simplifying complex consumption scenarios.
• Essential for substituting into the budget constraint for solution finding.

Recognizing these kinds of consumption patterns helps predict how consumption changes when one of the goods’ prices changes or when income levels change.

Optimization Problem

The optimization problem in this context involves maximizing the utility of the consumer, based on the given budget constraint and consumption relationship. The consumer aims to decide on the quantity of goods that provide the most benefit without exceeding their budget.To achieve this, the problem is simplified and solved by substituting the consumption relationship into the budget constraint, transforming it into:\[(4p_1 + p_2)x_2 = m\]The next steps involve solving this equation to determine the quantities of coffee and sugar that maximize the consumer’s utility.

• Optimization involves solving for the maximum utility within constraints.
• It requires balancing wants and needs with available resources.

Understanding optimization allows consumers to make the most out of their limited resources, efficiently managing their consumption decisions.

Mathematical Economics

Mathematical economics involves the application of mathematical techniques to represent economic theories and analyze problems. In this exercise, mathematical economics is used to create equations that describe consumer choice processes.

• Helps in transforming economic theories into testable models.
• Leads to precise solutions to complex economic problems.

For instance, the use of algebra to link budget constraints with consumption patterns in our problem enables us to solve for optimal purchasing choices with clarity. This approach facilitates a better understanding of economic dynamics and enhances decision-making capabilities.