The idea of overlapping generations was put forward by Samuelson in 1958. Samuelson (1958) introduced an intertemporal consumption loan model to explain the social contrivance of money. His model brought the neoclassical general equilibrium theory into the studies of macroeconomics and became one of the most important theories of modern macroeconomics. In Samuelson's model, generations overlapped at any time and lasted indefinitely into the future. Each generation lived through different periods with different endowments. Based on previous research, this study also adopts the Diamond (1965) overlapping generations model as the theoretical basis, establishing a three-period lifecycle model. It assumes that each family consists of three generations: the youth (t-1 period), the middle-aged (t period), and the elderly (t + 1 period). Individuals in their youth are in the stage of receiving education, have not entered the labor market, do not have labor income, and do not participate in household decision-making. Individuals in their middle age have entered the labor market, earn labor income, and bear the responsibility of raising children and supporting parents. Middle-aged individuals are the decision-makers in household economic behavior, making decisions regarding consumption, savings, support expenditures, and education expenditures for their children. They may also receive support from the older generation in the form of financial assistance or time. As the highest generation in the family, elderly individuals have exited the labor market. In addition to receiving pensions, they can use their savings and support payments from adult children for consumption and provide support to their children in terms of finances, time, and other aspects based on the actual situation.
The main source of income for families is individuals in their middle age, who are the decision-makers in family economic behavior. They prioritize consumption in their middle and old age, as well as the utility generated by the number and quality of their children. Therefore, the individual utility function can be represented as follows:
\({U}_{t}={\alpha }_{1}ln{C}_{t}+{\alpha }_{2}ln{C}_{t+1}+{\alpha }_{3}ln{h}_{t+1}\) | (1) |
Where \({U}_{t}\) represents the utility of individuals in their middle age, \({C}_{t}\) represents consumption in the middle age, \({C}_{t+1}\) represents consumption in the old age, and \({h}_{t+1}\) represents the human capital level of their child. \({\alpha }_{1}\), \({\alpha }_{2}\), and \({\alpha }_{3}\) represent the individual's preference coefficients for consumption in the middle age, consumption in the old age, and the level of human capital of their children, respectively.
Based on the budget constraint, we can deduce the following intertemporal consumption functions:
\({C}_{t}={w}_{t}{h}_{t}-{E}_{t}-{S}_{t}-v\) | (2) |
\({C}_{t+1}={S}_{t}(1+{r}_{t+1})\) | (3) |
\(v=b-{M}_{t}\left({x}_{t+1}\right)\) | (4) |
In Eq. (2), \({S}_{t}\) represents savings, \({w}_{t}{h}_{t}\) represents wage income, where \({w}_{t}\) and \({h}_{t}\) respectively represent the wage level per unit of labor and the human capital level of individuals in their middle age. \({E}_{t}\) represents educational expenditure on their children, \({S}_{t}\) represents the individual's financial assets, and \({r}_{t+1}\) represents the market interest rate.
In Eq. (4), \(v\) represents the net value of support payments, b represents the support payments, and \({M}_{t}\) represents the financial support given by individuals in their old age to individuals in their middle age. On one hand, compared to adult children providing support payments to their parents, parents' financial support to adult children is relatively low. Therefore, \({M}_{t}\) is a function of whether wealth transfer exists, denoted as \({x}_{t+1}\), reflecting the property rights of elderly individuals. If there is no wealth transfer, \({M}_{t}\) equals 0, support payments provided by individuals in their middle age are equal to \(b\). On the other hand, the influence of financial support on educational expenditure depends on individuals in their middle age. Whether the received financial support is used to increase family educational expenditure relies on the spending decisions of individuals in their middle age. Hence, \({M}_{t}\) is used instead of \({M}_{t+1}\) to represent it. Finally, the form \({M}_{t}\left({x}_{t+1}\right)\) incorporates individuals in their old age. Under the exchange motivation, there is a mutually reinforcing relationship between downward and upward wealth transfer, financial support from elderly individuals to individuals in their middle age would lead to higher support payments from individuals in their middle age. However, considering the budget constraints, an increase in support payments could crowd out educational expenditure. To simplify the analysis, it is assumed as \(v=b-{M}_{t}\left({x}_{t+1}\right)\).
Next, let's set up the human capital function. Considering that the human capital investment of young individuals mainly depends on parental decisions and the individuals themselves have no choice, we assume that the acquisition of human capital by young individuals is influenced by both family and external environmental factors. Suppose the human capital function follows the Cobb-Douglas production function:
$${h}_{t+1}=A{{{h}_{t}}^{\phi }E}_{t}^{1-\phi }$$
Where \(A\) represents government education expenditure, \({h}_{t}\) represents the human capital of middle-aged individuals, and \({E}_{t}\) represents the educational expenditure on children by middle-aged individuals.
By combining the utility function, consumption function, and human capital function, we can construct a utility function for a typical middle-aged individual in a family. By determining appropriate values for \({S}_{t}\) ,\({E}_{t}\) and \(v\), we aim to achieve utility maximization. To simplify the analysis, we assume that the human capital function is set as \({h}_{t+1}\)=\({E}_{t}\). By substituting \({C}_{t}\), \({C}_{t+1}\), \(v\), and \({h}_{t+1}\) into the utility function \({U}_{t}\), we have:
\({U}_{t}\left({S}_{t},{E}_{t},v\right)={\alpha }_{1}\text{ln}\left({w}_{t}{h}_{t}-{E}_{t}-{S}_{t}-(b-{M}_{t}({x}_{t+1}\left)\right)\right)+{\alpha }_{2}\text{ln}\left({S}_{t}(1+{r}_{t+1})\right)+{\alpha }_{3}\text{l}\text{n}{E}_{t}\) | (5) |
Treating \({w}_{t}\), \({w}_{t}\), \({r}_{t+1}\), \({x}_{t+1}\), and \(b\) as exogenous given variables, we can take partial derivatives of \({U}_{t}\) with respect to \({S}_{t}\), \({E}_{t}\), and \(v\) to obtain:
\(\frac{\partial {U}_{t}}{\partial {S}_{t}}=-\frac{{\alpha }_{1}}{{w}_{t}{h}_{t}-{E}_{t}-{S}_{t}+{M}_{t}\left({x}_{t+1}\right)}+\frac{{\alpha }_{2}\left(1+{r}_{t+1}\right)}{{S}_{t}\left(1+{r}_{t+1}\right)}=0\) | (6) |
\(\frac{\partial {U}_{t}}{\partial {E}_{t}}=-\frac{{\alpha }_{1}}{{w}_{t}{h}_{t}-{E}_{t}-{S}_{t}+{M}_{t}\left({x}_{t+1}\right)}+\frac{{\alpha }_{3}}{{E}_{t}}=0\) | (7) |
\(\frac{\partial {U}_{t}}{\partial {M}_{t}}=\frac{{\alpha }_{1}}{{w}_{t}{h}_{t}-{E}_{t}-{S}_{t}+{M}_{t}\left({x}_{t+1}\right)}\) | (8) |
In Eq. (8), \({w}_{t}{h}_{t}-{E}_{t}-{S}_{t}>0\), preference coefficient \({0<\alpha }_{1}<1\), and \({M}_{t}\left({x}_{t+1}\right)\ge 0\). Therefore, \(\frac{\partial {U}_{t}}{\partial {M}_{t}}>0\) indicating that the economic support resulting from intergenerational wealth transfer increases the utility of middle-aged individuals.
By solving equations (6) and (5), we can obtain the solution for utility maximization:
\({S}_{t}=\frac{{\alpha }_{2}}{{\alpha }_{1}+{\alpha }_{2}}({w}_{t}{h}_{t}-{E}_{t}+{M}_{t}\left({x}_{t+1}\right))\) | (9) |
\({E}_{t}=\frac{{\alpha }_{3}}{{\alpha }_{1}+{\alpha }_{3}}({w}_{t}{h}_{t}-{S}_{t}+{M}_{t}\left({x}_{t+1}\right))\) | (10) |
Substituting Eq. (9) into Eq. (10) and taking the partial derivative with respect to \({M}_{t}\left({x}_{t+1}\right)\), we get:
\(\frac{\partial {E}_{t}}{\partial {M}_{t}\left({x}_{t+1}\right)}=\frac{{\alpha }_{3}}{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\) | (11) |
Since \({\alpha }_{1}\), \({\alpha }_{2}\), and \({\alpha }_{3}\) are all between 0 and 1, \(\frac{\partial {E}_{t}}{\partial {M}_{t}\left({x}_{t+1}\right)}>0\). This implies that household educational expenditure is positively correlated with economic support. As economic support depends on whether intergenerational wealth transfer occurs in the elderly individuals, \({M}_{t}\left({x}_{t+1}\right)>0\) if transfer happens, otherwise it is 0. Ultimately, intergenerational wealth transfer leads to economic support for middle-aged individuals from the elderly individuals, resulting in increased educational expenditure.