The comparison of habit formation of urban and rural households for food and non-food goods: using Envelope theorem and Euler equations approach

1. Introduction
It is important for policy makers and planners to study the consumer behavior of households and to understand the role of consumer habits in shaping their consumption pattern for consumption of various food and non-food commodity groups. Since researchers believe that habits can play an important role in consumer behavior of individuals and households, they, along with structural parameters of the utility function such as risk aversion, Elasticity of Intertemporal Substitution (EIS), are also considered habits formation of consumers as one of the parameters of the time separation utility function. Duesenberry (1949) believed that the consumption of each individual consumer was not independent of the consumption of others, and consumer preferences were determined not only on the basis of the absolute level of his expenditures, but his relative consumption of the rest of society and past consumption of the individual also influenced his consumption behavior. Therefore, the habit formation of household’s consumption shows their consumption behavior and it is an important factor in the utility function. In present research, consumption of past periods has been introduced as an indicator of the habits in the consumer utility function. In fact, the main questions that the research seeks to answer are:
How is the consumption habit of urban households for eating food compared to rural households? How is the pattern of consumption habits of urban households for non-food commodities compared to rural households? The formation of the consumption habits of urban households for food is more powerful or non-food commodities? How is the pattern of consumption habits of urban households for non-food commodities compared to rural households?
 
 2. Materials and Methods
Since the introduction of the permanent hypothesis (Friedman, 1957) and the consumption life cycle (Modigliani and Bromberg, 1954), the concept of consumption smoothing has been widely used to explain household behavior. Hall (1978), in his paper, modeled the total consumption dynamics by extracting Euler's equation from the first-order condition of the optimization problem of consumer choice. The Euler equation has been widely used in economic literature to estimate the parameters of the risk-free utility. This approach reflects the fact that consumers are also interested in determining their current consumption of their past use or consumption of others (Dayton, 1992, 16). In models where consumption habits have been introduced in the household utility function, the utility depends on the level of consumption and accumulation of habits, which are measured by the past periods of the average consumption (fakhre hoseini, 2015, 69). This means that the habits of people are formed over a long period of time. In general, habits of consumption can be divided into two types of external and internal. External habits, presented by researchers like Abel 1990, Campbell and Cochran 1999, Galli 1994, Auer 2013), suggest that household preferences are based on total consumption lags. On the other hand, the formation of the habits of consumption is based on the assumption that household consumption depends on the consumption of their own past periods (Reader & Hill, 1973; Sandrasans, 1989; Constantinides, 1990, Dreyer& Schneider& Smith, 2013, Kwan, Leung,  & Dong, 2015).
In the following, we Model the formation of consuming habits and extract the Euler equation.
Assume that each household tries to select  at time t in order to maximize the following expression (Carroll, 2000, 69):
max E_t [∑^T_s=t β^s-t u(c_s , h_s)]                                              (1)
 
Which β is the time preferences factor, h is accumulations of habits, and E is the expectation operator. Suppose that the constraints that the above maximization problem faces are:
x_t+1= R [x_t-c_t]+ y_t+1                                              (2)
h_t+1=h_t+λ(c_t-h_t)                                                     (3)
 
  Which R is the risk-free interest rate, y, the income of the workforce in the period t,x is the cash of the household (the total amount of resources available to spend in period t). The Bellman equation corresponding to the above equations is defined as:
 v_t(x_t, h_t)=max _{c_t} u(c_t,h_t)+βE_t[v_t+1(x_t+1, h_t+1)]                    (4)
The first-order optimization condition for Equation (4) relative to expenditures consumed by  is:
0=u^c_t+βE_t(λv^h_t+1 - Rv^x_t+1)                                                  (5)
uct=βE_t (Rv^x_t+1- λv^h_t+1)                                                      (6)
That  λ is a parameter of "consumption habits". To obtain the Euler equation, we use the equations (1) to (6) and envelope theorem. After some calculations, we have: 
u^c_t+ β[λv^h_t+1+ (1-λ)u^c_t+1]=Rβ[u^c_t+1+β(λv^h_t+1(1-λ)u^c_t+2]      (7)

In fact, (7) is Euler's equation for maximizing the problem of present article.
If we assume that the formation of habits is such that the level of habits in the period t is equal to the consumption of the previous period. That means:
h_t= c_{t-1                                   (8)}
 
In this case, equation (7) is simplified as follows:
u^c_t+βu^h_t+1=Rβ[u^c_t+1+βu^h_t+2]      (9)
 
Suppose that, like Melbourne (1988), for the utility function, we consider the following form:
u (c , h) = v (c - αh)                      (10)
 
Therefore:
u^c = v’        ,     uh = -αv’             (11)
 
Replacement (11) in (9) gives:
v’_t-αβv’_t-1=Rβ(v’_t+1-αβv’_t+2]            (12)
 
The stable solution corresponding to the first root (12) under the fixed asset yields is:
v’_t = RβE_t (v’_t+1)     or    1= RβE_t (v’_t+1⁄v’_t)        (13)
 
Now if we assume that the utility function is in the form:(z=z^1-ρ/1-ρ), then  (z)=z^-ρ. Therefore, according to (13), we will have:
1=RβE_t (z_t+1⁄z_t)^-ρ                 (14)
 
Considering relations (8) and (9), yield: z_t=c_t-αc_t-1 , So:
1=  RβE_t (c_t+1-αc_t⁄c_t- αc_t-1)^-ρ            (15)
1=  RβE_t ((c_t+1/c_t)/1-αc_t-1/c_t)^-ρ         (16)
 
The parameters of nonlinear Euler equation (16) are estimated using generalized moments (GMM) and suitable instrument variables.
 
3. Empirical Results and Discussion
In this paper is estimated the coefficient of habit formation for the cost of food and nonfood goods of urban and rural households in Iran during 1357-1394, using Euler equations GMM approach. Household’s consumption expenditure data is located at the Iranian Statistics Center, the results of the survey on household expenditure and household income. To further acquaint with the average annual costs of nonfood goods for a rural household (crnok), the average annual costs of food and tobacco consumption for rural household (crk), the average annual costs of nonfood goods for a urban household (cunok), the average annual costs of food goods for a urban household (cuk), they are presented in Table 1:
 
Table 1: Consumption expenditures of food and nonfood goods for urban and rural households in Iran

Consumption expenditures      Variable            Mean                Median              Min            Max
  Nonfood(urban household)         cunok             35475367          35475367           307990        200000000
      food(urban household)            cuk                11866378          86637811          175217          62431000
   Nonfood(rural household)          crnok             17181566           4167900            108888          89205000
     food(rural household)              cuk                11677346           3578014            125977          57778000
Source: Iranian Statistics Center and research calculates 
 
According to the table 1, we can say that the average consumption expenditure of urban and rural households for food goods is approximately equal, but nonfood consumption costs for each urban household is more than twice of a rural household.
  
3.1 Calculation of the coefficient of formation of consumption habits for all kinds of consumption expenditures of urban and rural households
For estimating of coefficients of habit formation for consumption expenditures In Euler equation (16), we apply GMM approach. So, it is necessary that all of the variables that mentioned to be stationary. Hence, we applied ADF test for this mean and we ensured that all of the variables are stationary. After that, we used GMM method for estimating coefficient of habit formation
 for all kinds of consumption expenditures of Iranian urban and rural households using equation (16). Results are presented in table (2):
 
Table 2: The results of calculating the coefficient of formation of consumption habits or α and relative risk aversion coefficient ρ using the formula of mole (16) and method GMM
Consumption expenditures                       α                    ρ                            J                J^*=N*J 
   Nonfood(urban household)                   1.5(0.04)         1.73(0.042)          0.00014            0.0049
     food(urban household)                      1.41(0.00)         1.7(0.014)            0.015               0.54
    Nonfood(rural household)                   1.62(0.00)         3.98(0.08)            0.021               0.74
    food(rural household)                        1.01(0.00)         0.81(0.001)           0.017               0.56
                              *numbers in parentheses () are p-values for t statistic and x² _1,%5.
 
Hansen’s J-statistic indicates the validity of all the models and instrument variables used in them, because: 
 
  J^*=N*J=35*0.00014=0.0049<χ²_{r-l,%5=x² 1,%5}=3.841
 J^*=N*J=36*0.015=0.54<χ²_{r-l,%5=x² 1,%5}=3.841
 J^*=N*J=34*0.021=0.74<χ²_{r-l,%5=x² 1,%5}=3.841
 J^*=N*J=33*0.017=0.056<χ²_{r-l,%5=x² 1,%5}=3.841    
 
It is worth noting that the kernel of all equations is Bartlett, and the fixed-bandwidth is selected from the Newey West.
 
4. Conclusion
The results of the research showed that the role of habits in the utility function of urban and rural households is significant and important for both groups, and consumption of past periods of households has an impact on the consumption of their current period; So, the coefficient of formation of consumption habits for rural household food expenditure are more than this coefficient among rural households. On the other hand, the coefficient of habits formation for urban households for non-food is about 40% more than that for rural households. On the other hand, the results of this study indicate that the relative risk aversion for consumption of rural household goods is more than this factor for urban households. So, the tendency of rural households to consume food in the present time is more than the desire of urban households. While for non-food commodity groups, relative risk aversion is higher for urban households than rural households. In other words, urban households tend to be more likely to use non-food commodities than rural households at the current time, and are less likely to postpone consumption of these types of goods. Also, descriptive statistics indicate that during survey period, the average expenditure of non food goods for every urban household is more than twice that of rural household. While the average consumption expenditure of urban and rural households for consumer goods is approximately equal. In sum, the findings of this study showed that the formation of habits in the consumption pattern of Iranian urban and rural households has a significant place.