Question

Hicks's "second law" of demand states that the predominant relationship among goods is net substitutability (see footnote 3 of Chapter 6 ). To prove this result: a. Show why compensated demand functions $$X ;=h,\left(P_{U} \ldots, P_{n}, V\right)$$ are homogeneous of degree zero in \(P_{x} \ldots P_{n}\) for a given level of \(V\) b. Use Euler's theorem for homogeneous functions (for a statement of this theorem, see footnote 5 of Chapter 7 ) to show that \(=0 \text { (for all } i=1, n)\) c. Use the "first law of demand" to conclude that is, net substitution must prevail, on average.

Short answer

Answer: Hicks's second law of demand states that the predominant relationship among goods is net substitutability. To prove it, we follow three steps: (1) show that compensated demand functions are homogeneous of degree zero in prices and expenditure, (2) apply Euler's theorem for homogeneous functions to the compensated demand functions, and (3) use the first law of demand to conclude that net substitution must prevail on average.

Step-by-step solution

Show That Compensated Demand Functions are Homogeneous of Degree Zero

To prove that compensated demand functions \(X_i=h_i(P_x,\ldots,P_n,V)\) are homogeneous of degree zero, we need to show that for any \(\lambda > 0\), $$h_i(\lambda P_x, \ldots, \lambda P_n, \lambda V) = h_i(P_x, \ldots, P_n, V).$$ Since the demand function is compensated, the consumer's utility \(U\) is constant, and we have: $$U=h^{-1}(P_x,\ldots,P_n,V).$$ Now, let's analyze the effect of scaling all prices and expenditure by a factor \(\lambda\), that is, computing the utility when all prices and expenditure are multiplied by \(\lambda\): $$U=h^{-1}(\lambda P_x, \ldots, \lambda P_n, \lambda V).$$ Since the consumer's utility does not change (\(U\) is constant), we have: $$h^{-1}(P_x, \ldots, P_n, V)=h^{-1}(\lambda P_x, \ldots, \lambda P_n, \lambda V).$$ Taking the inverse of both sides, we obtain: $$h_i(\lambda P_x, \ldots, \lambda P_n, \lambda V) = h_i(P_x, \ldots, P_n, V).$$ Therefore, the compensated demand functions are homogeneous of degree zero.

Apply Euler's Theorem to the Compensated Demand Functions

Euler's theorem for homogeneous functions states that, for a homogeneous function \(g\) of degree \(k\), we have: $$\sum_{i=1}^n x_i\frac{\partial g}{\partial x_i} = kg(x_1, \ldots, x_n).$$ Since the compensated demand functions \(h_i\) are homogeneous of degree zero, we can apply Euler's theorem with \(k=0\), and sum over all goods \(i = 1,\ldots, n\). For each \(i\), we will compute the partial derivative of \(h_i\) with respect to expenditure \(V\) and multiply it by the corresponding price \(P_i\): $$\sum_{i=1}^n P_i\frac{\partial h_i}{\partial V} = 0.$$

Use the First Law of Demand to Conclude Net Substitution Must Prevail

The first law of demand states that, as the price of a good increases, the quantity demanded of that good decreases, holding utility constant. By applying the first law of demand to the expression from step 2, we can conclude that net substitution must prevail on average. If we increase the price of one good in the set, the first law of demand implies that the quantity demanded of that specific good will decrease, while the quantities demanded of the remaining goods will increase, maintaining the overall expenditure constant. To sum up, we have proven that the compensated demand functions are homogeneous of degree zero, and we have used Euler's theorem and the first law of demand to show that net substitution must prevail in these functions on average, proving Hicks's second law of demand.

Key concepts

Compensated Demand Functions

Compensated demand functions play a pivotal role in consumer theory by holding utility constant while analyzing how demand changes. When prices shift, these functions help us understand the substitution effect without considering the income effect. For a compensated demand function, the consumer's level of satisfaction, or utility, remains unchanged even as prices and income shift together.

The key property of compensated demand functions is that they are 'homogeneous of degree zero' in prices and income. This means that if all prices and income are multiplied by the same factor, the quantity demanded remains unchanged. Think of it like doubling prices and income; your demand doesn't change because your buying power is the same. This becomes crucial in proving other economic principles, such as Hicks's second law, where net substitutability among goods can be observed.

Homogeneous Functions

In mathematics and economics, a function is termed 'homogeneous' if multiplying all inputs by a positive constant results in the output multiplied by a power of that constant. Specifically, a homogeneous function of degree zero means that changing all inputs by the same proportion doesn't affect the output.

Consider the compensated demand function, which is a perfect example. If we multiply all prices and income by the same factor, the compensated demand remains unchanged. This property is central to consumer theory because it shows that only the relative prices, not their absolute levels, affect demand. Homogeneous functions allow economists to simplify models and focus on more significant changes, like relative price shifts, crucial for understanding consumer behavior.

Euler's Theorem

Euler's theorem for homogeneous functions is a foundational concept in mathematics used extensively in economics. The theorem states that for a homogeneous function of degree k, the weighted sum of its partial derivatives equals k times the function itself.

For a degree zero function, such as compensated demand functions, Euler's theorem implies that the weighted sum of changes in input prices results in no change in demand. In simple terms, it explains why, when all prices change in the same proportion, the overall demand remains constant. This theorem provides a calculus-based framework to analyze how price changes, informed by their degree of homogeneity, impact consumer demand. It supports the analysis of substitution effects without income alterations.

First Law of Demand

The first law of demand is a basic economic principle stating that, all else being equal, when the price of a good rises, the quantity demanded falls, and vice versa. This rule is fundamental to understanding market dynamics and consumer choice.

In the context of Hicks's second law, the first law of demand implies that if the price of a single good increases, the compensated quantity demanded for that good decreases, assuming utility remains constant. This decrease in demand for one good can lead to an increase in demand for substitutes. Thus, this law is a central building block in understanding net substitutability, proving that traditionally, goods are predominantly substitutes within an economy.

Net Substitutability

Net substitutability is a concept related to how changes in the price of one good affect the demand for another. When goods are net substitutes, an increase in the price of one leads to an increase in demand for the other, as consumers switch preferences.

Hicks's second law suggests that net substitutability is prevalent among goods. Using compensated demand functions, we see this in action: when the price of one good rises, maintaining constant utility, we observe a shift in demand to other goods. It reflects the natural consumer behavior to substitute pricier items with relatively cheaper alternatives. Understanding net substitutability helps predict market responses to price changes, crucial for firms and policymakers in making informed economic decisions.