Question

The general form for the expenditure function of the almost ideal demand system (AIDS) is given by \\[ \begin{aligned} \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_{i}} \end{aligned} \\] 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

In summary, we derived the Almost Ideal Demand System (AIDS) functional form for a two-goods case, showed that the expenditure function is homogeneous of degree 1 in all prices, and calculated the income share of each of the two goods using the given formula. The income share of good 1, \(s_1\), is given by \(\alpha_1 + \gamma_{11} \ln p_1 + \gamma_{12} \ln p_2\), whereas the income share of good 2, \(s_2\), is given by \(\alpha_2 + \gamma_{21} \ln p_1 + \gamma_{22} \ln p_2\).

Step-by-step solution

Simplify the general formula for two-goods case

Substitute n=2 (two-goods) in the given general formula and write down the simplified equation: \\[ \begin{aligned} \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)^2 + \gamma_{12} \ln p_1 \ln p_2 + \frac{1}{2} \gamma_{22} (\ln p_2)^2 \\ &+ U \beta_0 p_1^{\beta_1} p_2^{\beta_2} \end{aligned} \\] #b. Show that the expenditure function is homogeneous of degree 1 in all prices#

Apply homogeneity condition

To show the expenditure function is homogeneous of degree 1 in all prices, we need to replace \(p_i\) with \(\lambda p_i\) and show that the resulting function is \(\ln E(\lambda p_1, \lambda p_2, U) = \ln E(p_1, p_2, U) + \ln \lambda\). Make this substitution and simplify the equation: \\[ \begin{aligned} \ln E\left(\lambda p_1, \lambda p_2, U\right) &= a_0 + \alpha_1 \ln (\lambda p_1) + \alpha_2 \ln (\lambda p_2) + \frac{1}{2} \gamma_{11} (\ln (\lambda p_1))^2 + \gamma_{12} \ln (\lambda p_1) \ln (\lambda p_2) \\ &+ \frac{1}{2} \gamma_{22} (\ln (\lambda p_2))^2 + U \beta_0 (\lambda p_1)^{\beta_1} (\lambda p_2)^{\beta_2} \\ &= a_0 + \alpha_1 (\ln \lambda + \ln p_1) + \alpha_2 (\ln \lambda + \ln p_2) + \frac{1}{2} \gamma_{11} (\ln \lambda + \ln p_1)^2 \\ &+ \gamma_{12} (\ln \lambda + \ln p_1) (\ln \lambda + \ln p_2) + \frac{1}{2} \gamma_{22} (\ln \lambda + \ln p_2)^2 \\ &+ U \beta_0 (\lambda^{\beta_1} p_1^{\beta_1})(\lambda^{\beta_2} p_2^{\beta_2}) \end{aligned} \\]

Simplify the equation and apply given restrictions

Now, simplify the above equation while remembering to apply the given restrictions: \\[ \begin{aligned} \ln E\left(\lambda p_1, \lambda p_2, U\right) - \ln \lambda &= a_0 + (\alpha_1 + \alpha_2) \ln \lambda + \alpha_1 \ln p_1 + \alpha_2 \ln p_2 + \frac{1}{2} \gamma_{11} (\ln p_1)^2 \\ &+ \gamma_{12} \ln p_1 \ln p_2 + \frac{1}{2} \gamma_{22} (\ln p_2)^2 + \ln \lambda (\beta_1 + \beta_2) + U \beta_0 p_1^{\beta_1} p_2^{\beta_2} \\ &= \ln E( p_1, p_2, U) \end{aligned} \\] Thus, our expenditure function is homogeneous of degree 1 in all prices. #c. Calculate the income share of each of the two goods#

Apply the given formula for income share and find the derivatives

In order to find the income share of each good, apply the given formula \(s_x = \frac{d \ln E}{d \ln p_x}\) on the two-goods AIDS expenditure function: \\[ \begin{aligned} s_1 &= \frac{d \ln E}{d \ln p_1} = \alpha_1 + \gamma_{11} \ln p_1 + \gamma_{12} \ln p_2 \\ s_2 &= \frac{d \ln E}{d \ln p_2} = \alpha_2 + \gamma_{21} \ln p_1 + \gamma_{22} \ln p_2 \end{aligned} \\] Now we have the income share of both goods 1 and 2, \(s_1\) and \(s_2\), using the given formula.

Key concepts

Expenditure Function in the Almost Ideal Demand System

The expenditure function is a powerful concept in economics that tells us how much a consumer needs to spend to reach a certain utility level, given the prices of goods. In the context of the Almost Ideal Demand System (AIDS), it relates utility to the prices of goods and income levels in a flexible form that can be adjusted for various numbers of goods.

The AIDS expenditure function provided in the exercise demonstrates the relationship between two goods, their prices, and the utility level of the consumer. It's represented in a log-linear form which is mathematically convenient and provides insights into consumer behavior. When you are calculating the AIDS functional form for a two-goods case, you're essentially simplifying the general formula to work specifically with just two goods, by plugging in n=2 into the general formula.

Understanding the AIDS expenditure function helps in analyzing consumer budgeting and spending patterns, which are vital for studying market dynamics and formulating economic policies.

Homogeneity of Degree 1 of the Expenditure Function

Homogeneity of degree 1 is a term that refers to the characteristic of a function to maintain its proportional relationship when all its arguments are scaled by a constant factor. In simpler terms, if you double the prices of all goods, you'd expect to need to double your budget to maintain the same level of utility.

In the context of the AIDS expenditure function, proving that it's homogeneous of degree 1 involves showing that if you multiply all prices by a constant, the new level of expenditure would just be that constant multiplied by the original level of expenditure. This property is important because it ensures that the function maintains consistency with economic theory, which assumes that only relative prices matter to consumers, not the absolute levels of price.

The substitution step demonstrated in the exercise is key for proving this property. Applying the restrictions, such as the sum of the parameters being equal to 0, simplifies the equation and completes the proof. This characteristic reinforces the practicality of the AIDS model and forms the basis for its widespread use in economic analyses.

Calculating the Income Share for Goods

Income share in economics represents the proportion of total income that is spent on a particular good. When looking at two goods specifically, the income share tells us how a consumer's budget is split between those two products. It's a fundamental concept that reflects consumers' preferences and the importance of each good within their consumption pattern.

In the exercise solution, the income share of each of the two goods is determined by using partial derivatives of the log of the expenditure function with respect to the log of each good's price. This calculation is grounded in the idea that small changes in the prices of goods will impact the overall expenditure in varying degrees, corresponding to each good's share of the budget.

By applying the formula given in the exercise and using the specified functional form, you can solve for the income shares of both goods. These shares are vital for understanding market demands as well as for making pricing decisions and economic forecasting. The AIDS model offers a nuanced approach to assessing how shifts in prices influence consumer spending patterns.