Question

Show that perfect substitutes are an example of homothetic preferences.

Short answer

Perfect substitutes are homothetic because their utility scales proportionally.

Step-by-step solution

Understanding Homothetic Preferences

Homothetic preferences refer to a situation where preferences are proportionate or scalable. If preferences are homothetic, the consumer's indifference curves are linear transformations of each other. This means that the consumer's preference for goods scales proportionately when their income increases.

Definition of Perfect Substitutes

Perfect substitutes are two goods that can completely replace each other in consumption. The consumer is willing to substitute one for the other at a constant rate without affecting their overall satisfaction. The indifference curves for perfect substitutes are linear, indicating a constant marginal rate of substitution.

Analyzing Perfect Substitutes Indifference Curves

For perfect substitutes, the indifference curves are straight lines. This implies that the consumer is indifferent between combinations of the two goods as long as they maintain a consistent ratio. Mathematically, this can be represented as \( u(x_1, x_2) = ax_1 + bx_2 \), where \( a \) and \( b \) are constants, and \( x_1 \) and \( x_2 \) are the quantities of the two goods.

Testing for Homothetic Preferences

To test if perfect substitutes are homothetic, we need to check if scaling the quantities of goods by a positive number \( k \) results in the utility being scaled by the same proportion. If \( (x_1, x_2) \) gives utility \( u(x_1, x_2) \), then \( (kx_1, kx_2) \) should give utility \( k \cdot u(x_1, x_2) \).

Validation of Monotonic Transformation

For perfect substitutes, \( u(x_1, x_2) = ax_1 + bx_2 \) leads to \( u(kx_1, kx_2) = a(kx_1) + b(kx_2) = k(ax_1 + bx_2) = k u(x_1, x_2) \). Since the utility function scales in the same proportion, perfect substitutes illustrate homothetic preferences.

Key concepts

Perfect Substitutes

Imagine two goods that you find identical in satisfaction, like two flavors of juice. These are called **perfect substitutes**. You would be willing to swap one of these goods for the other at a steady rate because they perform the same role for you.

• For instance, if you have orange juice, each glass can be swapped for one glass of apple juice without impacting your enjoyment.
• This means you don’t have a strong preference between them.

In economic terms, the consumer perceives these goods as exactly exchangeable in their utility, maintaining a constant substitution rate.

Indifference Curves

In the world of economics, **indifference curves** depict combinations of goods that offer the same satisfaction to a consumer. They help economists understand consumer preferences. For perfect substitutes, these curves are represented as straight lines. - This linearity shows that as long as goods are exchanged at a constant rate, the consumer's satisfaction remains the same. - For example, shifting your diet from only apples to a mixture of apples and applesauce doesn't change your happiness if they are perfect substitutes. These straight-line curves indicate that consumers will substitute goods consistently without altering their level of satisfaction.

Marginal Rate of Substitution

The **marginal rate of substitution (MRS)** is a key concept that tells us how much of one good a consumer is willing to give up for an additional unit of another good, while maintaining the same level of satisfaction. - For perfect substitutes, the MRS is constant, because these goods can replace each other seamlessly. - Imagine swapping chocolate bars with candy bars; you trade them one-to-one without any change in pleasure. This constant rate is depicted on the indifference curve as a straight line, highlighting the ease of substitution between these goods.

Utility Function

Utility functions are mathematical models that describe a consumer's preference for different bundles of goods. They measure satisfaction or happiness.For perfect substitutes, we use a linear utility function like \[ u(x_1, x_2) = ax_1 + bx_2 \]where:- \( a \) and \( b \) are constants that reflect the rate of substitution between the goods.- \( x_1 \) and \( x_2 \) are quantities of the two goods. This simple linearity illustrates that satisfaction increases steadily as more of either good is consumed, aligning perfectly with the idea of homothetic preferences, where utility scales consistently, reflecting constant willingness to substitute.