Question

Show that Cobb-Douglas preferences are homothetic preferences.

Short answer

Cobb-Douglas preferences are homothetic because they satisfy \( U(tx_1, tx_2) = t^{a+b} U(x_1, x_2) \).

Step-by-step solution

Understand Cobb-Douglas Preferences

Cobb-Douglas preferences have a utility function of the form \( U(x_1, x_2) = x_1^a x_2^b \), where \( a \) and \( b \) are positive constants that represent the exponents of goods \( x_1 \) and \( x_2 \). These preferences are everywhere differentiable and represent typical How consumer preferences might interact among goods.

Define Homothetic Preferences

Preferences are homothetic if any increase in income results in a proportionate increase in consumption that keeps the marginal rate of substitution the same. Mathematically, preferences are homothetic if \( U(tx_1, tx_2) = t^k U(x_1, x_2) \) for any scalar \( t > 0 \).

Substitute and Verify the Scalar Property

For Cobb-Douglas functions, compute \( U(tx_1, tx_2) = (tx_1)^a (tx_2)^b = t^a (x_1)^a t^b (x_2)^b = t^{a+b} (x_1)^a (x_2)^b = t^{a+b} U(x_1, x_2) \). This satisfies the definition of homothetic preferences as the utility is scaled by \( t^{a+b} \).

Evaluate the Implication

This computation means that scaling all inputs by a positive factor \( t \) scales the utility function by a factor of \( t^{a+b} \). This implies that the marginal rate of substitution remains unchanged as it is contingent purely on the ratio of \( a \) to \( b \), independent of the scale factor \( t \).

Key concepts

Homothetic Preferences

The idea of homothetic preferences is crucial in understanding consumer behavior, particularly in relation to how goods are consumed as income changes. Preferences are termed homothetic if, when a consumer's income increases, their consumption patterns scale up proportionally. This means that the shape of the indifference curves remain the same across different income levels, and the consumer spends the same proportion of their income on different goods.

Mathematically, a utility function is homothetic if scaling all inputs by a common factor results in the utility being scaled by a power of that factor. This is fundamental in economics because it implies that the consumer’s rate of substitution between goods does not change with income, making it easier to predict consumer choices under varying income conditions.

For Cobb-Douglas preferences, characterized by the function \(U(x_1, x_2) = x_1^a x_2^b\), the function retains its form upon scaling, demonstrating its homothetic nature. Thus, if the prices of goods don't change, consumers will simply buy more of each good proportionally when their income increases, keeping their spending ratios constant.

Marginal Rate of Substitution

The marginal rate of substitution (MRS) is a concept that describes how a consumer is willing to trade one good for another while maintaining the same level of utility. It reflects the consumer's willingness to substitute one good for another in response to changes in relative prices or preferences.

In the context of Cobb-Douglas preferences, the MRS between two goods, say, \(x_1\) and \(x_2\), is given by the ratio of their marginal utilities. The MRS can be expressed as:

• \(MRS = \frac{MU_{x_1}}{MU_{x_2}}\)

Where \(MU_{x_1}\) and \(MU_{x_2}\) are the marginal utilities of goods \(x_1\) and \(x_2\) respectively.

The MRS remains constant along a given indifference curve for Cobb-Douglas preferences because it only depends on the exponents \(a\) and \(b\) of the goods in the utility function. This feature of Cobb-Douglas functions means that regardless of how much income changes, the rate at which consumers are willing to swap goods remains the same, reinforcing the concept of homothetic preferences.

Utility Function

Utility functions are mathematical representations of consumer preferences. They assign a number to each possible combination of goods, reflecting the satisfaction or utility a consumer derives from that combination.

The Cobb-Douglas utility function, \(U(x_1, x_2) = x_1^a x_2^b\), is one of the most commonly used utility functions in economics due to its simplicity and insightful properties. Here, \(a\) and \(b\) are parameters that represent the relative importance or weight assigned to each good by the consumer.

Some characteristics of Cobb-Douglas utility functions include:

• They are continuously differentiable, which means they can be easily manipulated and analyzed using calculus.
• The exponents \(a\) and \(b\) add up to 1 when preferences are assumed to be strictly convex.
• They inherently assume goods are normal goods, meaning consumers purchase more as their income increases.

These features make Cobb-Douglas utility functions a staple in both theoretical and applied economic analyses, allowing for straightforward predictions about consumer demand patterns and responses to price changes.