Question

When goods \(x\) and \(y\) are perfect complements, the indifference curve is a downward sloping straight line and the \(M R S_{-}\)is constant.

Short answer

Answer: Perfect complements are goods that are consumed together in fixed proportions, such that consumers require a constant proportion of both goods to achieve their desired level of satisfaction (utility). The indifference curve for perfect complements is a downward sloping straight line, representing that the proportion between goods x and y must always remain constant. The Marginal Rate of Substitution (MRS) for perfect complements is constant, as it represents the rate at which a consumer is willing to give up good y for good x while maintaining the same level of satisfaction, and this rate is consistent due to the fixed proportion of goods required.

Step-by-step solution

Understand Perfect Complements

Perfect complements are goods that are consumed together in fixed proportions. If a consumer has one unit of good \(x\), they must have a certain constant amount of good \(y\) to achieve the desired level of satisfaction (utility). An example of perfect complements is shoes: If a consumer has one left shoe (good \(x\)), they would want one right shoe (good \(y\)) to achieve full satisfaction.

Graph the Indifference Curve

An indifference curve shows all the combinations of goods \(x\) and \(y\) that provide the same level of satisfaction (utility) to the consumer. To graph the indifference curve for perfect complements: 1. Draw the horizontal and vertical axes, representing the quantities of goods \(x\) and \(y\), respectively. 2. Use a downward sloping straight line at a constant angle to represent the indifference curve. This is because the proportion between goods \(x\) and \(y\) must always remain constant, as they are perfect complements.

Understand Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (\(M R S_{-}\)) represents the rate at which a consumer is willing to give up good \(y\) for good \(x\) while maintaining the same level of satisfaction (utility). Mathematically, it is the negative of the slope of the indifference curve, or \(MRS_{-}=-\frac{dy}{dx}\).

MRS and Perfect Complements

Since the indifference curve is a downward sloping straight line, the slope of the line is constant. Therefore, the \(M R S_{-}\) for perfect complements is constant. This is because the consumer always requires a fixed proportion of goods \(x\) and \(y\) to achieve the desired level of satisfaction (utility). To find the constant \(MRS_{-}\) value for perfect complements, calculate the slope of the indifference curve using any two points on the line.

Key concepts

Indifference Curve

An indifference curve is a graphical representation showing combinations of two goods that provide a consumer with the same level of satisfaction, or utility. When visualizing indifference curves, think of a consumer deciding between different bundles of goods that make them equally happy.
For perfect complements, these curves take on a distinct shape. Since perfect complements are consumed in fixed proportions, their indifference curves are typically L-shaped. Imagine drawing a line for goods like a pair of shoes, where one left shoe always needs to match with one right shoe.

• The right angle part of the 'L' indicates the fixed ratio - you cannot consume more of one without the other and maintain the same level of satisfaction.
• In contrast to other types of indifference curves which bow inwards, perfect complements showcase a rigid fixed consumption pattern.

Understanding this fixed relationship helps explain consumer behavior when buying goods that naturally go together.

Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) quantifies how much of one good a consumer is willing to exchange for another while keeping their satisfaction constant. In essence, it measures the trade-off and is depicted as the slope of the indifference curve.
For most goods, the MRS is variable and changes along the curve, reflecting different rates at which a consumer substitutes one good for another. However, for perfect complements, this idea becomes a bit different because these goods are not substituted but consumed together.

• When dealing with perfect complements, the MRS doesn't change, because the fixed consumption ratio means no substitution can occur without reducing utility.
• This makes the MRS constant, illustrated by the straight-line portions of the L-shaped indifference curves.

Understanding the MRS is key to grasping consumer choices and preferences accurately, particularly when goods are perfect complements.

Consumer Utility

Consumer utility refers to the satisfaction or pleasure derived from consuming goods and services. In economics, understanding utility is crucial as it underpins consumer decision-making strategies.

With perfect complements, utility is only achieved when goods are used together in specific, fixed ratios. For example, a left and right shoe together give a complete pair, providing full utility.

• Utility does not increase if only one of either goods increases – more left shoes without additional right shoes do not enhance satisfaction.
• Thus, consumer satisfaction hinges entirely on the right balance of goods, which is why their utility is at a maximum when goods align perfectly with consumer expectations.

It's the pursuit of maximizing utility that drives the purchase and use of goods in these fixed proportions. Recognizing this helps explain consumer preferences and market demand for products that are traditionally consumed as pairs.