Chapter 6: Problem 4

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

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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.

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