The factors affecting health are complex and diverse. The health ecological model provides a good perspective to analyze what factors affect the health of the elderly, and lays a foundation for the model construction and heterogeneity analysis of this paper [22]. This paper examines how Internet use affects the health of older adults from the second and third layers. In order to clarify the impact mechanism of Internet use on the health of older adults, this paper constructs a healthy production function model. It is based on the Grossman’theory [23], which has been widely used in various studies. For example, Zhang et al. [24] selected control variables based on the Grossman’theory. Aregbeshola & Khan [25] selected explanatory variables based on Grossman’s model.
Consider that individuals have the dual roles of consumer and producer. Assuming that at any given period (t), as a consumer, personal utility is determined by health ({H}_{t}) and non-health consumer goods ({Z}_{t}), the individual’s utility in period (t) is:
$${U}_{t}=U({H}_{t},{Z}_{t})$$
(1)
Faced with time constraints, an individual’s time (T) is fully allocated to four activities, namely, ({T}^{W}) Work: earning income to improve (H) and (Z); ({T}^{Z}): Time spent to improve (Z), such as leisure, travel; ({T}^{H}): Time spent to enhance (H), such as exercise; ({T}^{S}): Time of illness, during which nothing can be produced and it is entirely determined by (H). The constraint lines are:
$$T={T}^{W}+{T}^{Z}+{T}^{H}+{T}^{S}$$
(2)
As producers, individuals produce (H ,text{a}text{n}text{d},Z) that satisfy utility by combining the goods bought from the market with their own time. That is, the production of (H, text{a}text{n}text{d} ,Z) requires two inputs: market goods and personal time. (M) represents market goods that produce health (H), such as fitness equipment, and (J) represents market goods that produce non-health consumer goods (Z), such as tickets. It is worth noting that (Z) is a flow that is consumed in each period produced, while (H) is the stock, similar to capital, which is accumulated and consumed in each period. Thus, at any given (t) period, the production functions of ({H}_{t},text{a}text{n}text{d} ,{Z}_{t}) are:
$${H}_{t}=H({H}_{t-1},{T}_{t}^{H},{M}_{t})$$
(3)
$${Z}_{t}=Z({T}_{t}^{Z},{J}_{t})$$
(4)
Individuals face budget and time constraints. Assuming that in the period (t), the remuneration per unit of work is (w), then the total income is (Y_t=wcdot T_t^W), and the individual can allocate the income between the two types of goods whose prices are ({P}_{m} ,text{a}text{n}text{d} ,{P}_{j}) respectively, then the budget constraint is:
$$P_mcdot M_t+P_jcdot J_tleq wcdot T_t^W=Y_t$$
(5)
Assume that for older adults, ({T}^{W}left(text{w}text{o}text{r}text{k}text{i}text{n}text{g} ,text{h}text{o}text{u}text{r}text{s}right)=0), and their income comes from pensions, child support, etc., and ({Y}_{t}) is a fixed value. So, in the model for older people, there is no need to consider working hours. Production time ({T}^{P}) is non-sick time, that is, the time used for the production of health and non-health consumer goods. In any given period, if the individual is healthier, then the duration of illness (({T}^{S})) is shorter and the production time (({T}^{P})) is longer. The production time constraint for older adults is:
$${T}^{P}=T-{T}^{S}={T}^{Z}+{T}^{H}$$
(6)
At the same time, the marginal return of health on productive time is diminishing, i.e. (frac{partial T^P}{partial H}>0,;frac{partial^2T^P}{partial H^2}<0). When a person is healthy enough, the increase in production time brought by an extra unit of health is small. When it is very unhealthy, the duration of illness is long, and even a small increase in health can bring a larger growth in production time. And there is a minimum point of health level, ({H}_{min}), at which ({T}^{P}=0), indicating that the entire period is sick, which is effectively equal to death. Based on the model, we propose three hypotheses:
Internet use facilitates access to health information for older people, so that they can acquire more health knowledge and improve their health production skills [26]. Search sites, chat groups, short videos and other platforms provide individuals with real-time, convenient, diverse and multi-point health information. It can help disseminate health information to older adults and enhance their healthcare capacity [13]. A study by Cotten [12] also found that people who used the Internet to search for health information had a higher level of health than those who searched by other means.
A good lifestyle is an important way to stay healthy [27]. The Internet’s publicity on healthy lifestyles such as strengthening exercise, health preservation, regular work and rest is conducive to the reasonable allocation of “production time”. And the Internet has expanded the channels of sports publicity, which contributes to enhancing older adults’ awareness and skills, and even the frequency and intensity of exercise [28]. Exercise has a strong antidepressant effect, which is conducive to alleviating loneliness and improving happiness in older adults [29]. Therefore, the subtle impact of the Internet helps older adults improve their lifestyle, increase exercise time, and then enhance their health. So, this paper proposes:
Health ecology models show that multiple factors have an important impact on individual health. The new health production function is ({H}_{t}=AHleft({H}_{t-1},{T}_{t}^{H},{M}_{t},{X}_{t}right)) and ({X}_{t}) denotes other factors affecting health, such as social interactions. The use of the Internet facilitates the connection and interaction between older adults and other people [30]. This helps maintain social relationships, stay active and socially engaged, overcome loneliness and depression, and receive physical and emotional health support [31], i.e., (frac{partial H}{partial X_t}>0).