*3.1. The Duality Problem of the AIDS Model*

The AIDS model assumes that the consumer behavior satisfies the price indifferent generalized logarithmic preference hypothesis, so the expenditure function is

$$
\ln(\mathcal{C}(\mu, p)) = (1 - \mu)\ln(a(p)) + \mu\ln(b(p)),
\tag{1}
$$

where *u*(0 ≤ *u* ≤ 1) is the utility index, *u* = 0 only to maintain the basic physiological needs of the utility, and *u* = 1 for the utility to achieve its maximum. *a*(*p*) and *b*(*p*) denote the minimum expenditure required by the consumers to meet their basic physiological needs and obtain the maximum utility, and the form is as follows:

$$\begin{cases} \ln(a(p)) = a\_0 + \sum\_{i=1}^n a\_i \ln(p\_i) + \frac{1}{2} \sum\_{i=1}^n \sum\_{j=1}^n r\_{ij} \ln(p\_i) \ln(p\_j), \\\quad \ln(b(p)) = \ln(a(p)) + b\_0 \prod\_{i=1}^n p\_i^{b\_i} \end{cases} \tag{2}$$

According to the principle of duality, the expenditure function and indirect utility function are inverse functions, so the indirect utility function corresponding to the expenditure function is

$$\sigma(p,m) = (\ln m - \ln a(p)) / (\ln b(p) - \ln a(p)),\tag{3}$$

As obtained from the Roy equation, the Marshallian demand function for the commodity *i* is

$$\exp(p,m) = -\left[\partial v(p,m)/\partial p\_i\right]/\left[\partial v(p,m)/\partial m\right] = (m/p\_i)(a\_i + \sum\_{j=1}^n r\_{ij}^\* \ln(p\_j) + b\_i \ln(m/a(p))),\tag{4}$$

where *r*∗ *ij* = (*rij* + *rji*)/2. Given *μ<sup>i</sup>* = *pixi*(*p*, *m*)/*m* represents the ratio of the expenditure to income for commodity *i*, we get an expanded version of the AIDS model with income *m* as follows:

$$\mu\_i = a\_i + \sum\_{j=1}^n r\_{ij}^\* \ln(p\_j) + b\_i \ln(m/a(p)),\tag{5}$$

If we take the difference between the income and expenditure as savings, this can be seen as a special commodity that buys a certain amount of vouchers for future consumption. Let the discount rate 1/(1 + *r*) be the price of the savings, where *r* is the real interest rate. To preserve the aggregation properties of AIDS, this is *<sup>n</sup>* ∑ *i*=1 *ai* <sup>=</sup> 1, *<sup>n</sup>* ∑ *i*=1 *r*∗ *ij* <sup>=</sup> 0, *<sup>n</sup>* ∑ *i*=1 *bi* = 0. Furthermore, *a*(*p*) can be understood as a price index, so it can be set to *a*(*p*) ≈ *θP*, where *P* is the total price index. We take it into the Equation (5), then

$$\mu\_i = a\_i^\* + \sum\_{j=1}^n r\_{ij}^\* \ln(p\_j) + b\_i \ln(m/P),\tag{6}$$

where *a*∗ *<sup>i</sup>* = *ai* − *bi* ln *θ*, and it also satisfies the aggregation property. Moreover, Equation (6) also satisfies the homogeneity *<sup>n</sup>* ∑ *j*=1 *r*∗ *ij* = 0 and the symmetry *r*<sup>∗</sup> *ij* = *r*<sup>∗</sup> *ji*.

#### *3.2. Counterfactual Decomposition of Income Distribution*

Before the AIDS model is dynamically expanded, we should complete the measurement of the income distribution changes. For the decomposing method of the income distribution, by using a counterfactual analysis method, Jenkins and Van Kerm [33] successfully decomposed the income distribution change into three parts, which reflect the mean, variance, and residual changes. In fact, the residual change reflects the skewness, kurtosis, and other high-order moments changes of the income distribution. Counterfactual analysis can be understood as an application of comparative static analysis, a qualitative research method in economics. The difference is that counterfactual analysis is a quantitative analysis method. It constructs a counterfactual situation in which only one factor changes but other factors remain unchanged relative to a basic fact. And it evaluates the impact of a single factor by comparing the results of counterfactual and factual situations based on specific regression models or statistical methods., so we will use the counterfactual decomposition method of Jenkins and Van Kermin in this study.

Suppose that we have and *y*<sup>2</sup> for two years of the income survey data, assuming we temporarily disregard the price factors and they follow the same distribution, *y*<sup>1</sup> ∼ *F*(*μ*1,*σ*<sup>1</sup> <sup>2</sup>) and *<sup>y</sup>*<sup>2</sup> ∼ *<sup>F</sup>*(*μ*2,*σ*<sup>2</sup> 2 ), then the decomposition process can be illustrated below:

$$y\_1(\mu\_1, \sigma\_1^2) \stackrel{\text{manchange}}{\rightarrow} \mathbb{Z}\_1 \sim F(\mu\_2, \sigma\_1^2) \stackrel{\text{varianechange}}{\rightarrow} \mathbb{Z}\_2 \sim F(\mu\_2, \sigma\_2^2) \stackrel{\text{residualchange}}{\rightarrow} y\_2(\mu\_2, \sigma\_2^2), \tag{7}$$

Assuming there is only a change in the mean between the two income samples, and then the underlying income *ξ*<sup>1</sup> relative to the base period *y*<sup>1</sup> can be denoted by

$$\vec{y}\_1 = y\_1 + \Delta y = y\_1 + (\mu\_2 - \mu\_1),\tag{8}$$

The variance change reflects the polarization of the income between the individuals around the mean. Based on the counterfactual analysis, we keep the mean of the counterfactual income *ξ*<sup>1</sup> and *ξ*<sup>2</sup> qual and only allow their variance changes. According to the statistical knowledge it is easy to see that

$$(\not\xi\_1 - \mu\_2) / \sigma\_1 = (\not\xi\_2 - \mu\_2) / \sigma\_2 \sim F(0, 1). \tag{9}$$

So we get

$$
\xi\_2 = \mu\_2 + \sigma\_2(\xi\_1 - \mu\_2) / \sigma\_1. \tag{10}
$$

The difference between *ξ*<sup>2</sup> and *y*<sup>2</sup> is the residual change.
