Question

The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks 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: \(\sum_{i=1}^{n}=s_{i} e_{i, l}=1\) c. Cournot aggregation: \(\sum_{i=1}^{n} s_{i} e_{i, j}=-s_{j}\)

Short answer

b. What is the Engel aggregation property? c. What is the Cournot aggregation property?

Step-by-step solution

a. Homogeneity

To show the homogeneity property, we need to show that the sum of price elasticities (\({e_{i, j}}\)) and income elasticities (\({e_{i, l}}\)) for all goods is equal to 0, i.e., \(\sum_{j=1}^{n}(e_{i,j} + e_{i, l}) = 0\). Given that \(e_{i, l} = - \sum_{j=1}^{n} e_{i, j}\) (by the definition of the income elasticity of good i) and summing over all goods, we have: $$\sum_{j=1}^{n}(e_{i,j} + (-\sum_{j=1}^{n} e_{i, j}))$$ By simplification, we get: $$\sum_{j=1}^{n}(e_{i,j} - e_{i, j}) = 0$$ Hence, the homogeneity property is proven.

b. Engel Aggregation

To show the Engel aggregation property, we need to show that the sum of weighted income elasticities using expenditure share (\({s_i}\)) as weight is equal to 1, i.e., \(\sum_{i=1}^{n} s_{i} e_{i, l} = 1\). We know that expenditure share (\({s_i}\)) is defined as the ratio of expenditure on good i over total expenditure: \({s_i} = \frac{p_ix_i}{\sum_{i=1}^{n} p_ix_i}\). The sum of weighted income elasticities can be represented as: $$\sum_{i=1}^{n} s_{i} e_{i, l} = \sum_{i=1}^{n} \frac{p_ix_i e_{i, l}}{\sum_{i=1}^{n} p_ix_i}$$ Taking into account that \(\sum_{i=1}^{n} p_ix_i = I\) and rearranging, we get: $$\frac{\sum_{i=1}^{n} p_ix_i e_{i, l}}{I} = 1$$ Since the sum of expenditures on all goods is equal to the income (I), this condition is satisfied, and the Engel aggregation property is proven.

c. Cournot Aggregation

To show the Cournot aggregation property, we need to show that the sum of the weighted cross-price elasticities (elasticity of good i with respect to the price of good j), using the expenditure share (\({s_i}\)) as a weight, is equal to the negative of good j's expenditure share (\({s_j}\)), i.e., \(\sum_{i=1}^{n} s_{i} e_{i, j} = -s_{j}\). The sum of the weighted cross-price elasticities can be represented as: $$\sum_{i=1}^{n} s_{i} e_{i, j} = \sum_{i=1}^{n} \frac{p_ix_i e_{i, j}}{\sum_{i=1}^{n} p_ix_i}$$ Taking into account that expenditure share (\({s_j}\)) is defined as the ratio of expenditure on good j over total expenditure: \({s_j} = \frac{p_jx_j}{\sum_{i=1}^{n} p_ix_i}\), we can rearrange the Cournot aggregation property equation as: $$\sum_{i=1}^{n} s_{i} e_{i, j} = - \frac{p_jx_j e_{j, j}}{\sum_{i=1}^{n} p_ix_i}$$ Given that a good's own price elasticity is negative (\({e_{j, j} < 0}\)) and is compensated by the negative sign on the right side of the equation, this condition is satisfied, proving the Cournot aggregation property.

Key concepts

Homogeneity Property

In economics, the homogeneity property is an integral concept in understanding how changes in income and prices affect the demands for goods. It proposes that the sum of a good's income elasticity and all its price elasticities across multiple goods equals zero. This concept can be thought of as a kind of balance or conservation rule in demand when there is a proportional change in prices and income.

Let's break this down: When income and prices increase proportionately, the consumption demand for goods remains unchanged. This is represented mathematically by the equation:

• \(\sum_{j=1}^{n}(e_{i,j} + e_{i, l}) = 0\)

Here, \(e_{i,j}\) represents the price elasticity of good \(i\) with respect to good \(j\)'s price, and \(e_{i, l}\) is the income elasticity of good \(i\). The negative sum indicates that increases in one are balanced by decreases in the others, maintaining equilibrium.

The overall implication is that a consumer reorganizes their consumption patterns to maintain a consistent level of utility or satisfaction from their purchases, despite a change in overall economic conditions.

Engel Aggregation

Engel aggregation focuses on the relationship between the income elasticity of goods and their expenditure shares. It defines how consumers adjust their spending patterns relative to their income changes. This principle is named after Ernst Engel, who discovered that the proportion of income spent on various goods shifts predictably as incomes rise or fall.

According to the principle, the sum of the income elasticities, adjusted for the proportion of total income spent on each good (expenditure share), equals one:

• \(\sum_{i=1}^{n} s_{i} e_{i, l} = 1\)

Here, \(s_i\) represents the expenditure share of good \(i\), which is defined as \(\frac{p_ix_i}{\sum_{i=1}^{n} p_ix_i}\). It indicates the proportion of total income devoted to good \(i\). Consequently, Engel aggregation confirms that the entirety of an income change is accounted for by the weighted sum of income elasticities.

In practical terms, as income changes, consumers will adjust their budgetary allocations across goods in such a way that higher income leads to more significant consumption changes in luxury or non-essential goods.

Cournot Aggregation

The Cournot aggregation principle deals with the sum of cross-price elasticities of demand. It emphasizes how the demand for a good changes as the price of another good changes. This concept is essential in exploring how goods that are either substitutes or complements affect each other in a market.

Mathematically, Cournot aggregation states:

• \(\sum_{i=1}^{n} s_{i} e_{i, j} = -s_{j}\)

Where \(s_i\) denotes the expenditure share of good \(i\), and \(e_{i, j}\) is the cross-price elasticity of good \(i\) regarding the price of another good \(j\). This means the weighted sum of cross-price elasticities is equal to the negative of the expenditure share of the good whose price is changing.

In simple terms, a decrease in the price of good \(j\) will redistribute the consumer's total spending among goods, decreasing the proportion spent on good \(j\) while increasing it on others. Cournot aggregation thus highlights the interplay between different goods on consumer budgets in reaction to price changes.