Discrete Choice Model: An Application to the Educational Decision of Ivorian Households

2.1 Theoretical basis of discrete choice models

Suppose the consumer can compare all possible alternatives. There is a utility function U that expresses consumer preferences. Let C_n be the set of alternatives available to the n decision-makers during the decision process, where U_i is the utility of the decision-maker associated with the alternative i; the utility function can be defined in terms of attributes as follows:

where Z_i is the vector of the attributes for the alternative i. Thus, for the decision-maker n, the alternative i is chosen if and only if

In fact, when repeating the same choice test, or with the same set of choices, the same attributes, and the same socioeconomic characteristics, different individuals will choose different alternatives. The theory of probabilistic choice explains this inconsistency of the preferences of individuals. It is assumed that human behavior is intrinsically probabilistic or that more specific information about the individual decision-making process is lacking. The probabilistic mechanism can capture the effects of unobservable variations among decision-makers and the unobservable attributes of alternatives. It also considers the stochastic behavior and the error caused by the method of data collection.

Thus, the probabilistic characteristics of the choice decision make it possible to highlight the alternative that a decision-maker will choose in the decision-making process by calculating the probability that a decision-maker will choose the alternative. The hypothesis of the agent’s rationality always assumes that individuals select alternatives with the highest utility. The probability that a decision-maker selects the alternative i will be that the utility of this alternative i is greater than that of the other alternatives:

Since the utilities are not known for certain, they must be treated as random variables by decomposing the random utility function of a two-part alternative:

Since each agent has a set of choices designated by C_n, with j≺Jas the number of choices (alternatives), the probability that the alternative i in C_n is chosen can be rewritten as

where V_i denotes the systematic component of utility and εirefers to the random component of utility.

The determination of the model specification depends on the choice of the form of the utility function. This specification concerns the systematic component which is supposed to be a linear function on the parameters (acronym for “linear in parameters”). Let β, the vector of k unknown parameters, be the linear function on the parameters written as

In the equation above, the parameters β1, β2, and βkare assumed to be the same for all. But in reality, the socioeconomic characteristics are not identical for all agents. The parameters must not be fixed and must instead be variable according to the different characteristics of the individuals. This problem can be solved by treating the parameter βas a random variable that follows a probabilistic distribution.

Moreover, assuming that the chosen alternative i is the first alternative in Cn and fεiε2…εjthat the joint density function of the error terms is designated, the probability can be written in the form

The density function of the error terms εjdepends on the correlation between these error terms. Correlations internal to the observations are the correlations between the residues relative to the different alternatives for the same individual. In this case, for every individual, we have Eεjεj'=∑n, and ∑nis no longer a diagonal matrix. By making assumptions about the joint probabilistic distribution of the error terms εj, any multinomial choice model can be deduced. In the following, only the multinomial logit model and the logit model with random parameters will be processed.

2.2 Logit multinomial model

If one assumes that they are independently and identically distributed (IID), hypothesis equivalent to the hypothesis independence of irrelevant alternatives (IIA), and that what follows them is a distribution of Gumbel, one obtains the multinomial logit model (MNL model):

If the utility function is linear on the parameters, the model is written as

where Xiis explanatory variables representing the socioeconomic and demographic characteristics of individuals, their environment, or contextual characteristics and βis the parameters to estimate.

2.3 Logit model with random parameters

In the logit model, βkis constant (set for all individuals) and therefore cannot capture the effects of individual characteristics. To remove this constraint, we assume that βkis a random variable of specific or normal distribution. In this case, the probability of choice can be written in the following form:

where fβ1…βkis the density function of the parameters of the individual utility function

For example, if C is the cost of education, T is the duration of education, and X is the other explanatory variables, the linear utility function is written as

Assuming, moreover, that the coefficient of the duration of education βTtakes a random value [17], of the normal type, the function of the probabilistic density βTis written as

In this case, the probability of choice can be written as follows:

Although this model is also based on the hypothesis IIA, the fact that the coefficients of the attributes can vary among the individuals improves the specification of the logit model. In the next section, the multinomial logit model will be applied to Ivorian data to explain the choice of Ivorian households in education.

3.1 Theoretical framework of the analysis model

The economic agent who invests in training expects a return higher than the cost of his investment in terms of labor compensation. Thus, a methodology is developed from gain functions (see [18, 19, 20, 21]). Starting from Becker’s models of education (see [22, 23, 24]), the demand for education can be modeled from the utility function of the household. Let us consider a model of choice of inter-temporal education where the representative household has only one child and lives two periods (i.e., [25]). The household derives its utility from the consumption of goods and services (C) and the cognitive skills of its child (A). In period 1, the child may be in school, work, or both. In the latter case, the child goes to school first and works after school [26]. The utility function of the household can then be written as

where δis the discount factor of future consumption and σrepresents parents’ preferences for child-rearing. Children’s education can increase parental consumption. It also directly affects the usefulness of parents. The acquisition of cognitive skills can be expressed using a production function as follows:

where αis the child’s learning efficiency that encompasses a number of factors, such as the child’s learning abilities and motivation and the parents’ ability to support their child in school work, Q is the quality of the school, and S is the grade. The parents’ consumption in each period is expressed as follows:

where p is the price of education and Y1and Y2are the income of parents at periods 1 and 2. Ycis the child’s income when working and the share of that income paid to parents. 1-S is the time the child devotes to work. Income is completely exhausted at the end of each period. The household does not go into debt either. Children’s income can be modeled on cognitive skills:

where πis the productivity of cognitive skills in the labor market.

Substituting Eq. (16) in Eq. (19), Eq. (19) in Eqs. (17) and (18), and Eqs. (16)–(18) in Eq. (15), the utility function of parents is written as a function of years of schooling and the quality of the school:

If the quality of the school is considered exogenous, then the variable that determines the choice is the time of education (S). The optimal duration of education is obtained by maximizing the utility function of the household. But, parents have the opportunity to also choose the quality of the school they want for their child. Thus, the price of education will depend on the quality of the school:

where p0is the basic price of education. By replacing pby p0Qin Eq. (6), we obtain the expression of the utility function to be maximized according to the quality variables (Q) and the study time (S):

To simplify derivation calculations, one postulates that the quality function of the school and the duration of education have the following functional forms (see [27]): fQ=QβandgS=Sγwith β>0;γ>0. One can then write the functional form of the utility function of the parents as being equal to

Maximizing the utility function of parents following S and Q determines the optimal values of the length of education and the quality of the school:

S∗is positive if and only ifγ−β≥0. The sensitivity of cognitive skills to learning times must be greater than the sensitivity of cognitive skills to the quality of the school. Eq. (24) suggests that parents’ preferences for education and future consumption have a positive influence on the length of their child’s education. But when the share of his income that the child has to give back to his parents increases less he makes long studies. Also, a high productivity of cognitive skills in the labor market will encourage the child to opt for work earlier than education:

From Eq. (25), we conclude that the child’s learning abilities, preferences for future consumption, and parents’ level of education are positively related to the quality of children’s education. On the other hand, the basic price of education negatively influences the quality of education that parents are willing to choose for their child.

The level of knowledge acquisition is determined by integrating Eqs. (24) and (25) into the cognitive skill acquisition equation. The production of cognitive skills can be expressed in a linear form for the sake of simplification [25]. The functional form of this production function is

The parameter μis the vector of the coefficients to be estimated and εthe error term which captures the measurement errors of the variables Q,Sandα. The quality of the school differs according to the type of school. Also, learning efficiency is also a multidimensional notion and can be influenced by several factors. The equation of acquisition of cognitive skills can then be rewritten in the form

The level of knowledge can be validly equated with the level of education. As a result, the level of education is explained by a set of variables relating to the school, its quality and its environment, the child, the parents, and the socioeconomic context.

3.2 Econometric analysis model

The empirical application of the educational demand model will be done using a multinomial qualitative variable model as in the study for the analysis of the demand for education in rural areas of Benin (see [28]). We first estimate the probability of being schooled using a logit model in which the variable of interest is a binary variable that takes the value 1 when the child is enrolled and 0 if not:

where X_i is a vector capturing the individual, family, and community characteristics that can influence the probability of a child going to school, β is the vector of unknown parameters to estimate, and Ф (.) is the normal cumulative distribution function. It is therefore necessary to estimate the probability of being educated conditionally to the explanatory variables transformed by the distribution function;

In a second step, we estimate a multinomial model to capture the explanatory factors of the continuation of school life once children are enrolled:

with k = 1, 2, and 3 corresponding, respectively, to primary, first, and second cycles of secondary school. It is a question of estimating the function Uik=Xiβk+εikwhere εikis an independent random variable and the individual characteristics of the child, those of the household, and the place of residence. The probability of choosing a category k is given by

Household living standard survey data (ENV98 and ENV2008) will be used for applications. They provide information on the characteristics of households, their members, and their living environment. Each individual is attached to a household whose demographic structure and socioeconomic context are well-known.

4.1 Factors explaining the school decision

The analysis of the determinants of schooling will be conducted according to individual characteristics, family determinants, and contextual elements (Table 1). We also calculate odds ratios (Table 2) and marginal effects (Table 3).

4.2 Determinants of the continuation of educational life in the secondary cycle

The analysis of the determinants of the pursuit of education follows the same logic as that of the explanatory factors of schooling (Table 4).