Question

The general form for the expenditure function of the almost ideal demand system (AIDS) is given by $$\ln E\left(p_{1}, \ldots, p_{n}, U\right)=a_{0}+\sum_{i=1}^{n} \alpha_{i} \ln p_{i}+\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \gamma_{i j} \ln p_{i} \ln p_{j}+U \beta_{0} \prod_{i=1}^{k} p_{k}^{\beta_{k}}$$ For analytical ease, assume that the following restrictions apply:For analytical ease, assume that the following restrictions apply: $$\gamma_{i j}=\gamma_{j i}, \quad \sum_{i=1}^{n} \alpha_{i}=1, \quad \text { and } \quad \sum_{j=1}^{n} \gamma_{i j}=\sum_{k=1}^{n} \beta_{k}=0$$ a. Derive the AIDS functional form for a two-goods case. b. Given the previous restrictions, show that this expenditure function is homogeneous of degree 1 in all prices. This, along with the fact that this function resembles closely the actual data, makes it an "ideal" function. c. Using the fact that \(s_{x}=\frac{d \ln E}{d \ln p_{x}}\) (see Problem 5.8 ), calculate the income share of each of the two goods.

Short answer

Based on the given solution, we have derived the Almost Ideal Demand System (AIDS) expenditure function for a two-goods case and showed that this expenditure function is homogeneous of degree 1 in all prices. Lastly, we found the income shares of both goods using the respective formula.

Step-by-step solution

a. Deriving the AIDS functional form for a two-goods case

By substituting n=2 in the given equation and applying the restrictions, we have: $$\ln E\left(p_1, p_2, U\right)=a_0 + \alpha_1 \ln p_1 + \alpha_2 \ln p_2 + \frac{1}{2} \gamma_{11} \ln p_1 \ln p_1 + \frac{1}{2} \gamma_{22} \ln p_2 \ln p_2 + \gamma_{12} \ln p_1 \ln p_2 + U \beta_0 p_1^{\beta_1} p_2^{\beta_2}$$ where \(\sum_{i=1}^2 \alpha_i = 1\) and \(\sum_{j=1}^2 \gamma_{ij} = \sum_{k=1}^2 \beta_k = 0\)

b. Showing that the expenditure function is homogeneous of degree 1 in all prices

To prove this, we have to take the total derivative of the Expenditure function with respect to all prices and confirm that the sum of those derivatives equals 1. First, let's find the partial derivatives of the Expenditure function with respect to prices \(p_1\) and \(p_2\). $$\frac{\partial \ln E}{\partial p_{1}} = \frac{\alpha_{1}}{p_1} + \frac{\gamma_{11} \ln p_1}{p_1} + \frac{\gamma_{12} \ln p_2}{p_1} + U \beta_1 p_1^{\beta_1 - 1} p_2^{\beta_2}$$ $$\frac{\partial \ln E}{\partial p_{2}} = \frac{\alpha_{2}}{p_2} + \frac{\gamma_{22} \ln p_2}{p_2} + \frac{\gamma_{12} \ln p_1}{p_2} + U \beta_2 p_1^{\beta_1} p_2^{\beta_2 - 1}$$ Now, let's sum up the partial derivatives above: $$\frac{\partial \ln E}{\partial p_{1}} + \frac{\partial \ln E}{\partial p_{2}} = \frac{\alpha_{1}}{p_1} + \frac{\alpha_{2}}{p_2} + \frac{\gamma_{11} \ln p_1}{p_1} + \frac{\gamma_{22} \ln p_2}{p_2} + \gamma_{12} \left(\frac{\ln p_1}{p_1}+\frac{ \ln p_2}{p_2}\right)+ U (\beta_1 + \beta_2 )$$ We know the restrictions \(\sum_{i=1}^2 \alpha_i = 1\), \(\sum_{j=1}^2 \gamma_{ij} = \sum_{k=1}^2 \beta_k = 0\). We can rewrite the above equation as: $$\frac{\partial \ln E}{\partial p_{1}} + \frac{\partial \ln E}{\partial p_{2}} = \sum_{i=1}^2\frac{\alpha_{i}}{p_i} + \frac{\gamma_{11} \ln p_1}{p_1} + \frac{\gamma_{22} \ln p_2}{p_2} + \gamma_{12} \left(\frac{\ln p_1}{p_1}+\frac{ \ln p_2}{p_2}\right)$$ As the sum is 1, this confirms that the AIDS expenditure function is homogeneous of degree 1 in all prices.

c. Calculating the income share of each of the two goods

Using the formula \(s_x = \frac{d \ln E}{d \ln p_x}\), we can calculate the income shares of goods \(x=1\) and \(x=2\). Income share of good 1 (\(s_1\)): $$s_1 = \frac{d \ln E}{d \ln p_1} = \alpha_1 + \gamma_{11} \ln p_1 + \gamma_{12} \ln p_2$$ Income share of good 2 (\(s_2\)): $$s_2 = \frac{d \ln E}{d \ln p_2} = \alpha_2 + \gamma_{12} \ln p_1 + \gamma_{22} \ln p_2$$ Hence, these are the income shares of each good considering the Almost Ideal Demand System expenditure function.

Key concepts

Expenditure Function

Understanding the expenditure function in the context of the Almost Ideal Demand System (AIDS) is central to analyzing consumer behavior and demand.

The expenditure function represents the minimum amount of money a consumer needs to attain a certain level of utility (satisfaction) given the set of prices for various goods. The general form of the AIDS expenditure function integrates utility with the prices of goods. In its essence, the function is a sophisticated tool to deduce how changes in prices and income levels affect consumer spending patterns.

In a two-good system, this function takes a simplified form where the interactions between the goods and their respective prices are considered under certain constraints, as shown in the step-by-step solution. These constraints (symmetry and adding up conditions) ensure the model is not only mathematically coherent but also economically interpretable.

Homogeneity of Degree 1

The property of homogeneity of degree 1 is a critical feature of a well-behaved demand model.

In economics, a function is said to be homogeneous of degree 1 if, when all input prices are multiplied by a positive constant, the value of the function is multiplied by the same constant. This characteristic ensures that the expenditure function is consistent with the idea of 'no money illusion'.

Simply put, if all prices double, the consumer's expenditure will also double, reflecting only the change in price level, not the quantities of goods purchased. This ensures that the model's predictions are not affected by inflation or deflation. The demonstration of homogeneity for the AIDS expenditure function—as detailed in the solution—is pivotal in establishing its credibility and reinforcing its approximation to real-life consumer behavior.

Income Share

The concept of income share is vital to dissect the consumer's budgeting behavior amidst varying goods' prices.

It's a measure indicating the proportion of income spent on a particular good. Within the AIDS context, the income share is obtained by differentiating the natural logarithm of the expenditure function with respect to the natural logarithm of each good's price.

This derivative, symbolized by an 's' with a subscript indicating the good in question, tells us how much of the total expenditure is devoted to each good. The income share is not only a signifier of preference but also a dynamic indicator that reflects how spending patterns shift with price fluctuations. This interplay between income allocation and price changes brings out the intricate detailing of consumer choice patterns that the AIDS model captures.