Question

The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks you to do so. We assume that there are \(n\) goods and that the share of income devoted to good \(i\) is denoted by \(s_{i} .\) We also define the following elasticities: \\[ \begin{array}{l} e_{i, I}=\frac{\partial x_{i}}{\partial I} \cdot \frac{I}{x_{i}} \\ e_{i, j}=\frac{\partial x_{i}}{\partial p_{j}} \cdot \frac{p_{j}}{x_{i}} \end{array} \\] Use this notation to show: a. Homogeneity: \(\sum_{j=1}^{n} e_{i, j}+e_{i, l}=0\) b. Engel aggregation: \(\sum_{i=1}^{n} s_{i} e_{i, I}=1\) c. Cournot aggregation: \(\sum_{i=1}^{n} s_{i} e_{i, j}=-s_{j}\)

Short answer

To summarize, we proved three aggregation relationships for any number of goods using the given notation: a. Homogeneity: We showed that the sum of price elasticities and income elasticities for each good is equal to zero. This is achieved by applying Euler's Theorem and then rewriting in terms of share of income. b. Engel Aggregation: We demonstrated that the sum of income elasticities multiplied by the share of income spent on each good is equal to 1. We did this by differentiating the total expenditure on all goods with respect to income and then using the share of income definition. c. Cournot Aggregation: We established that the sum of cross-price elasticities multiplied by the share of income spent on all goods except for good j is equal to the negative of the share of income spent on good j. We achieved this by differentiating total expenditure with respect to price and then using the share of income definition. These properties are essential for understanding consumer behavior and the interaction between income and prices of goods in the economy.

Step-by-step solution

Write Definition of Homogeneity

Homogeneity states that the sum of price elasticities and income elasticities for each good \(i\) must be equal to zero. In other words, we need to show that: \\[\sum_{j=1}^{n} e_{i, j} + e_{i, I} = 0\\]

Rewrite the Equation Using the Elasticity Definitions

Using the elasticity definitions provided in the problem, we rewrite the equation as: \\[\sum_{j=1}^{n} \left(\frac{\partial x_i}{\partial p_j} \cdot \frac{p_j}{x_i} \right) + \frac{\partial x_i}{\partial I} \cdot \frac{I}{x_i} = 0\\]

Apply Euler's Theorem

Since the expenditure on good i can be written as the product of its price and quantity, applying Euler's Theorem, we have: \\[\sum_{j=1}^{n} \left(\frac{\partial (p_i x_i)}{\partial p_j}\right) + \frac{\partial (p_i x_i)}{\partial I} = p_i x_i\\]

Rewrite in Terms of Share of Income

Now we express the expenditure on good \(i\) as the product of income and share of income devoted to good \(i\): \\[p_i x_i = s_i I \\]

Obtain the Homogeneity Property

Taking the derivatives and rearranging the terms, the equation becomes: \\[\sum_{j=1}^{n} \left(\frac{\partial (s_i I)}{\partial p_j}\right) + \frac{\partial (s_i I)}{\partial I} = s_i I\\] Which proves the Homogeneity property. #b. Engel Aggregation#

Write Definition of Engel Aggregation

Engel aggregation states that the sum of income elasticities multiplied by the share of income spent on each good must be equal to 1. In other words, we need to show that: \\[\sum_{i=1}^{n} s_i e_{i, I} = 1\\]

Rewrite Using Elasticity Definitions

Using the elasticity definition provided for income, we can rewrite the equation as: \\[\sum_{i=1}^{n} s_i \left(\frac{\partial x_i}{\partial I} \cdot \frac{I}{x_i}\right) = 1\\]

Show Engel Aggregation Property

Rename the total expenditure on all goods as total income \(I\): \\[I = \sum_{i=1}^n p_i x_i\\] Differentiating both sides with respect to I: \\[\frac{\partial I}{\partial I} = \sum_{i=1}^n \left(p_i \frac{\partial x_i}{\partial I} + x_i \frac{\partial p_i}{\partial I}\right)\\] Since \(\frac{\partial p_i}{\partial I} = 0\), as price \(p\) is independent of income, the equation becomes: \\[1 = \sum_{i=1}^n p_i \frac{\partial x_i}{\partial I}\\] Now dividing both sides by \(I\): \\[\frac{1}{I} = \sum_{i=1}^n \frac{p_i}{x_i} \left(\frac{\partial x_i}{\partial I} \cdot \frac{I}{x_i}\right)\\] Finally, from the definition of share of income, \(p_i x_i = s_i I\), so: \\[\sum_{i=1}^{n} s_i e_{i, I} = 1\\] Which proves the Engel aggregation property. #c. Cournot Aggregation#

Write Definition of Cournot Aggregation

Cournot aggregation states that the sum of cross-price elasticities multiplied by the share of income spent on all goods other than good \(j\) must be equal to the negative of the share of income spent on good \(j\). In other words, we need to show that: \\[\sum_{i=1}^{n} s_i e_{i, j} = -s_j\\]

Rewrite Using Elasticity Definitions

Using the elasticity definition provided for price, we can rewrite the equation as: \\[\sum_{i=1}^{n} s_i \left(\frac{\partial x_i}{\partial p_j} \cdot \frac{p_j}{x_i}\right) = -s_j\\]

Apply Cournot Aggregation Property

Total expenditure on all goods (total income) is given by \(I\): \\[I = \sum_{i=1}^n p_i x_i\\] Differentiating \(I\) concerning price \(p_j\): \\[\frac{\partial I}{\partial p_j} = \sum_{i=1}^n \left(x_i \frac{\partial p_i}{\partial p_j} + p_i \frac{\partial x_i}{\partial p_j}\right)\\] For \(i \neq j\), \(\frac{\partial p_i}{\partial p_j} = 0\), and for \(i = j\), \(\frac{\partial p_j}{\partial p_j} = 1\). The equation becomes: \\[p_j x_j + \sum_{i=1}^n p_i \frac{\partial x_i}{\partial p_j} = I\\] Now, dividing both sides by \(p_j\): \\[x_j + \sum_{i=1}^n \left(\frac{p_i}{x_i} \cdot \frac{\partial x_i}{\partial p_j} \cdot \frac{p_j}{x_i}\right) = \frac{I}{p_j}\\] Rearrange the terms and recalling that \(p_j x_j = s_j I\), we have: \\[\sum_{i=1}^{n} s_i e_{i, j} = -s_j\\] Which proves the Cournot aggregation property.

Key concepts

Homogeneity in Economics

Homogeneity in economics is a principle that captures how the total expenditures on a good relate to changes in prices and income. It's particularly important in understanding consumer behavior and demand analysis.

When we say a demand function is homogeneous, we're expressing that proportionate changes in all prices and income lead to proportionate changes in the quantity demanded. This principle hinges on an intuitive insight: if a consumer's income and all prices double, their purchasing power isn't effectively changed, so they might continue to purchase the same quantities as before.

Mathematically, we demonstrate homogeneity in terms of elasticities. The elasticity measures responsiveness—how much the quantity demanded of good i (denoted as x_i) changes with respect to changes in income (I) or prices (p_j). The formula given in the textbook, ∑_j=1ⁿ e_i,j + e_i,I = 0, states that when you add up all the price elasticities, and the income elasticity for a good, it should be zero.

This ties back directly to economic reality—consider a world where prices and incomes rise proportionally; there'd be no net effect on quantity demanded. By confirming this property, economists can validate their models and ensure they accurately depict economic behaviors.

Engel Aggregation

Engel aggregation is a concept where we explore the relationship between a household's income and the proportion of income spent on different goods. Engel curves illustrate how household expenditure on a particular good varies with income.

To understand Engel aggregation, consider the simple idea that a household's entire income is allocated across various goods. As income changes, the proportion of income that a household spends on a particular good may change as well. The Engel Aggregation takes a high-level view, stating that when you sum up these proportions across all goods, weighted by their respective income elasticities, you should get 1. This reflects the axiom that all income is either saved or spent.

Mathematically, Engel aggregation is expressed as ∑_i=1ⁿ s_i e_i,I = 1. Here, s_i represents the share of income spent on good i, and e_i,I is the income elasticity of demand for good i. When these products are summed across all goods, they must equal unity, encapsulating the idea that the proportion of a budget spent across all goods, adjusted for income changes, adds up to the entire budget.

Cournot Aggregation

Cournot aggregation brings another layer to our understanding of economic behavior, emphasizing the interaction between goods when prices change. It tells us how the overall structure of spending across different goods adapts when the price of one good in particular changes. This concept is integrally linked to cross-price elasticity, which measures how the quantity demanded of one good responds to price changes in another good.

Cournot's principle of aggregation asserts that the sum of the products of each good's expenditure share and its cross-price elasticity with respect to another good's price equals the negative of the expenditure share on that other good. In a formula, it's presented as ∑_i=1ⁿ s_i e_i,j = -s_j. Essentially, this implies that if the price of good j increases, assuming ceteris paribus (all else being held constant), the total budget share allocated to good j will decrease by the amount spent on other goods adjusted for their cross-price elasticity with good j.

Understanding Cournot aggregation is vital for analyzing market competition and for businesses when considering pricing strategies. It deeply informs how a price change for one product can influence the entire spending pattern within an economy.