Question

As defined in Chapter 3 , a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The \(M R S\) depends on the ratio \(y / x\) a. Prove that, in this case, \(\partial x / \partial I\) is constant. b. Prove that if an individual's tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur.

Short answer

Question: Show that in a homothetic utility function, (a) the partial derivative of x with respect to income is constant, and (b) Giffen's paradox cannot occur. Answer: (a) In a homothetic utility function, the partial derivative of x with respect to income is constant because the MRS only depends on the ratio y/x, and (b) Giffen's paradox cannot occur in a homothetic utility function as the demand for good x must decrease as its price increases, meaning that price and quantity must move in opposite directions.

Step-by-step solution

Part A: Proving that \(\partial x / \partial I\) is constant

Given a homothetic utility function where any straight line through the origin cuts all indifference curves at points of equal slope, we know that the \(MRS\) only depends on the ratio \(y/x\). We can denote the \(MRS\) function as follows: \(MRS(x,y) = f(\frac{y}{x})\) Now, consider the budget constraint of the individual with income \(I\) and prices \(p_x\) and \(p_y\): \(I = p_x x + p_y y\) To derive the demand for good \(x\), differentiate the budget constraint with respect to income \(I\): \(\frac{1}{p_x} = \frac{\partial x}{\partial I} + \frac{p_y}{p_x} \frac{\partial y}{\partial I}\) Now, using the homothetic property of the utility function, we can say: \(\frac{\partial x}{\partial y} = \frac{MRS(x,y)}{p_x/p_y}\) And replace the demand for good \(y\) in the above equation: \(\frac{1}{p_x} = \frac{\partial x}{\partial I} + \frac{p_y}{p_x} \frac{\partial x}{\partial I} \frac{MRS(x,y)}{p_x/p_y}\) Simplifying the equation: \(\frac{1}{p_x} = \frac{\partial x}{\partial I}(1 + MRS(x,y))\) Now, since \(MRS(x,y)\) depends only on the ratio \(y/x\), we can see that \(\partial x / \partial I\) is constant, proving part a of the exercise.

Part B: Proving that Giffen's paradox cannot occur

We now need to prove that if an individual's tastes are represented by a homothetic indifference map, Giffen's paradox cannot occur. For the paradox to happen, the demand for good \(x\) must increase as its price \(p_x\) increases, or \(\frac{\partial x}{\partial p_x} > 0\). Let's differentiate the demand for good \(x\) with respect to its price \(p_x\): \(\frac{\partial x}{\partial p_x} = \frac{-I}{p_x^2} + \frac{\partial x}{\partial I} \frac{-p_y y}{p_x^2} + \frac{\partial x}{\partial y} \frac{-p_y \partial y/\partial p_x}{p_x^2}\) Since we already know that \(\frac{\partial x}{\partial I}\) is constant and MRS only depends on the ratio \(y/x\), this equation can be simplified as: \(\frac{\partial x}{\partial p_x} = \frac{-I}{p_x^2} + C_1 \frac{-p_y y}{p_x^2} + C_2 \frac{-p_y \partial y/\partial p_x}{p_x^2}\) Where \(C_1\) and \(C_2\) are constants. Now, since all the terms on the right side of the equation are negative, it implies that the demand for good \(x\) must decrease as its price \(p_x\) increases: \(\frac{\partial x}{\partial p_x} < 0\) This proves that Giffen's paradox cannot occur if an individual's tastes are represented by a homothetic indifference map. Price and quantity must move in opposite directions.

Key concepts

Marginal Rate of Substitution

Understanding the Marginal Rate of Substitution (MRS) is crucial for making sense of consumer behavior in economics. MRS highlights how willing a consumer is to substitute one good for another while maintaining the same level of utility. It's calculated as the absolute value of the slope of an indifference curve, representing various combinations of goods that provide equal satisfaction to the consumer.

In a homothetic utility function scenario, for any pair of goods, the MRS is the same along a ray from the origin, which means that consumer preferences exhibit consistent proportional trade-offs regardless of their consumption levels. This property ensures constancy in the rate at which a consumer is willing to trade between goods straight from the very first unit, and it's a critical aspect in proving both the constancy of \( \partial x / \partial I \) and the impossibility of Giffen's paradox in such context.

Giffen's Paradox

Giffen's paradox is a counterintuitive economic phenomenon where a rise in the price of a 'Giffen good' leads to an increase in its quantity demanded, contrary to the law of demand. This paradox mainly occurs with inferior goods—products that consumers buy more of as their income decreases. The paradox hinges on the income effect of a price increase outweighing the substitution effect.

In the realm of homothetic utility functions, this paradox does not materialize because the substitution and income effects move in harmony due to consistent MRS. When the price of a good goes up, consumers consistently switch to other goods, ensuring the quantity demanded for the pricier good decreases, keeping the relationship between price and quantity demanded in its conventional downward slope.

Indifference Curves

Indifference curves represent a fundamental concept in understanding consumer choices. These curves graphically portray various bundles of goods that provide the same level of utility or satisfaction to a consumer. The shape and slope of these curves can reveal a lot about consumer preferences.

In a homothetic context, indifference curves are straight lines through the origin, signifying that consumer preferences remain proportional as the consumption of goods changes. The equal slope at all points along the rays from the origin means that the MRS is consistent across different levels of consumption, simplifying the analysis of consumer behavior and demand.

Budget Constraint

The budget constraint is a line that represents the combination of goods a consumer can purchase with their limited income, given the prices of the goods. This constraint reflects the trade-offs a consumer has to consider given their budgetary limitations.

With the exercise involving homothetic utility functions, we see how the budget constraint interacts closely with the constancy of MRS and ultimately simplifies the process of deriving demand for a good. The slope of the budget constraint helps inform the consumer's optimal choice by pinpointing where they can get the most satisfaction for their available income, which in homothetic preferences results in a consistent response to income and price changes for any given good.