Problem 35 - Applied Calculus

Question

A shopper buys \(x\) units of item \(A\) and \(y\) units of item \(B,\) obtaining satisfaction \(s(x, y)\) from the purchase. (Satisfaction is called utility by economists.) The contours \(s(x, y)=x y=c\) are called indifference curves because they show pairs of purchases that give the shopper the same satisfaction. (a) A shopper buys 8 units of \(A\) and 2 units of \(B\). What is the equation of the indifference curve showing the other purchases that give the shopper the same satisfaction? Sketch this curve. (b) After buying 4 units of item \(A\), how many units of \(B\) must the shopper buy to obtain the same satisfaction as obtained from buying 8 units of \(A\) and 2 units of \(B ?\) (c) The shopper reduces the purchase of item \(A\) by \(k,\) a fixed number of units, while increasing the purchase of \(B\) to maintain satisfaction. In which of the following cases is the increase in \(B\) largest? Initial purchase of \(A\) is 6 units Initial purchase of \(A\) is 8 units

Step-by-Step Solution

Answer

(a) Indifference curve is \(xy = 16\). (b) For 4 units of A, buy 4 units of B. (c) Increase in B is larger when initial A is 6 units.

Step 1: Define the equation for indifference curve

The indifference curve for satisfaction is given by the equation \(s(x, y) = xy = c\), where \(c\) is the level of satisfaction. The value of \(c\) can be determined from the units purchased by the shopper when satisfaction is constant.

Step 2: Determine the value of c for 8 units of A and 2 units of B

Using the equation \(xy = c\), substitute \(x = 8\) and \(y = 2\). Thus, \(c = 8 \times 2 = 16\). Therefore, the equation of the indifference curve is \(xy = 16\).

Step 3: Sketch the indifference curve

The curve \(xy = 16\) is a hyperbola. Sketch it on the Cartesian plane by choosing several values of \(x\) (such as 1, 2, 4, 8) and calculating corresponding values of \(y\) that satisfy the equation \(xy = 16\).

Step 4: Find the number of units of B for 4 units of A

Substitute \(x = 4\) in the indifference curve equation \(xy = 16\) to find \(y\). So, \(4y = 16\), solving for \(y\) gives \(y = \frac{16}{4} = 4\). Thus, the shopper must buy 4 units of \(B\) to maintain the same utility.

Step 5: Analyze the effect of reducing item A by k units

Start with one case: \(x = 6\). The initial product is \(xy = 6y = 16\), giving \(y = \frac{16}{6} \approx 2.67\). For the second case: \(x = 8\). The product is already \(8 \times 2 = 16\). When \(x\) is reduced by \(k\) units, the new \(y\) will be higher, but the change must be calculated based on initially. Therefore, reducing \(k\) when \(x = 6\) results in a higher increase in \(y\) compared to \(x = 8\) because the rate of change is more significant with less initial "A" units.

Key Concepts

Utility

Utility is a fundamental concept in economics that describes the satisfaction or pleasure that a person derives from consuming goods and services. Imagine when you eat your favorite food or buy a gadget you've been wanting — that feeling of happiness and satisfaction is what economists call utility.

Utility helps economists understand and explain consumer preferences and behavior. It allows us to model how individuals make choices, balancing their limited resources with their wants. By expressing utility mathematically, it becomes easier to predict how changes in prices or incomes affect consumer decisions.

The shopper in the exercise experiences utility from buying different combinations of items A and B. The idea is that there is a level of satisfaction or utility that they can achieve, and this is represented on a graph by the indifference curve. This concept is crucial for understanding consumer choices and how they seek to maximize their satisfaction given their budget constraints.

Hyperbola

A hyperbola is a specific type of curve on a graph, which in the context of economics, can represent an indifference curve. An indifference curve depicts combinations of two goods that provide the same level of satisfaction or utility to a consumer.

In our exercise, the equation of the indifference curve is given by \(xy = 16\). This equation forms a hyperbola when plotted on a graph. A hyperbola is characterized by its two symmetric halves, or "arms", that bend away from each other.

Understanding hyperbolas is important because they help visualize how a consumer might substitute between goods while maintaining the same level of utility. For example, in the exercise, if the shopper decreases the quantity of good A, they must increase the quantity of good B to remain on the same indifference curve, demonstrating that they are indifferent to these comparable bundles of goods.

Satisfaction

Satisfaction, in economic terms, is often interchangeable with utility. It's the fulfillment or contentment a consumer experiences from their purchasing choices. The utility function, such as \(s(x, y) = xy\), mathematically represents this satisfaction.

The exercise involves examining a consumer's satisfaction when they purchase certain amounts of items A and B. By maintaining the equation \(xy = 16\), we're ensuring that the consumer's satisfaction level doesn't change, no matter how they substitute between the two goods.

This concept of maintaining satisfaction is crucial in understanding consumer behavior. It illustrates how consumers attempt to maximize their happiness relative to the options available to them within a given budget. This balance is a key element in the broader field of consumer choice theory in economics.

Economics

Economics is the study of how individuals and societies use limited resources to satisfy unlimited wants. Within this broad field, concepts like utility and indifference curves help us understand consumer behavior on a more granular level.

The exercise presented here is a classic microeconomic problem dealing with consumer preferences and choices. By understanding how a consumer reacts to different combinations of goods that provide the same utility, economists can predict how changes in prices or income might affect purchasing decisions.

Economics provides the tools to analyze and model this behavior, which is essential for businesses setting prices, governments creating policies, and individuals making informed choices. This consumer model and indications from indifference curves contribute significantly to economic theories that shape both policy and business strategies globally.