Research Article Discrete Stochastic Dynamics of Income Inequality in Education: An Applied Stochastic Model and a Case Study

This paper develops a model for the discrete stochastic dynamics of economic inequality-induced 
 outcomes in education and demonstrates, in the context of the Turkish higher education sector, that higher 
 incomes lead to higher performance levels; and hence income inequality matters. The paper formulates 
 a stochastic subsidy policy that could help the sector to escape the low performance equilibria, and that could stabilize the sector at relatively higher performance levels.

These works explore various aspects of the service sector in general and education sector in particular.The range of the topics they cover is indeed impressive but not exhaustive.There are still a number of topics that remain underexplored.Among the topics in question is the stochastic dynamics of inequality which we will examine in this paper.We will present a model of the stochastic dynamics of economic inequality-induced outcomes in education and demonstrate, in the context of the Turkish higher education sector, that higher incomes lead to higher performance levels; and hence income inequality matters.We will formulate a stochastic subsidy policy that could help the sector to escape the low performance equilibria, and that could stabilize the sector at relatively higher performance levels.
In Section 2 of the paper, we develop the model.(The author has previously done (or participated in) work on the dynamic analysis of sectors, such as Kara et al. [34], Kara and Kurtulmus ¸ [32] and Kara [42,43], none of which did deal with the dynamics of income inequality in education, which is the focus of this paper.)Section 3 presents the empirical results.The policy implications are articulated in Section 4. The concluding remarks follow in Section 5.

The model
Consider an education sector where suppliers (such as universities) provide a service, say x, to the customers.(Educational firms could provide multiple services, in which case x could be conceived as "a composite service" representing these services.)For the sake of simplicity, we will analyze the case of a representative supplier in the market.Let D t denote a peculiarly defined concept, namely the quantity demanded for service x supplied by the firm, which indicates the degree to which customers are willing to buy the service at time t.D t depends on the relative price of the service at time t (π r t ), and customers' income at time t (M ts ), that is, Let S t denote a peculiarly defined concept, namely the quantity supplied for the service, which indicates the degree to which the supplier is willing to supply the service at time t.Suppose that S t depends on the relative price of the service (π r t ) as well as on the present and past performances (P t ,P t−1 ), that is, the peculiar supply function is S t = f s π r t ,P t ,P t−1 .
(2.3) (The demand and supply equations could be obtained through utility maximization and profit maximization, resp.).
Ahmet Kara 3 For analytical purposes, we will assume that the demand and supply functions have the following explicit forms: where u t and v t are independent normally distributed white noise stochastic terms uncorrelated over time.They have zero means and variances σ 2 u and σ 2 v , respectively.Here a peculiar feature of the supply behavior of the higher education institutions in Turkey needs to be noted.Even at low performance levels, many of these institutions do end up supplying services.The level of these services at time t depends on the level of these services at t = 0, and the growth rate of these services reflecting roughly the growth of student population in the system.Let, at the minimal performance levels, and in the absence of stochastic shocks, S t have the value of A, which grows at a rate of g over time.Thus, at P t = 1 and P t−1 = 1, S t = A(1 + g) t ⇒ lnS t = t.lnA(1+ g) = β 0 (by the argument presented in the subsection on supply behavior below, the effects of prices have been left out).
To theorize about the movements over time (i.e., the dynamic trajectory) of service performance, we will make two reasonable assumptions.First, the relative strength (or magnitude) of the demand compared to the supply provides an impetus for performance to be adjusted upwards over time.Second, productivity growth contributes to performance improvements over time.Taking these factors into account, we formulate the following adjustment dynamic for performance.
where k is the coefficient of adjustment and δ is a productivity growth at t. Taking the logarithmic transformation of both sides, we get We will call this the dynamic adjustment equation.Substituting the functional expressions (forms) for ln D t and lnS t specified above, setting the values of M t , π r t , P t , and P t−1 to their average values M avr , π ravr , P avr , and P avr −1 , and rearranging the terms in the equation, we get which is a second-order stochastic difference equation, the solution of which is provided in the appendix.
The solution in the appendix shows that the intertemporal equilibrium performance P * is where (2.9) In case where λ 1 and λ 2 are conjugate complex numbers, that is, λ 1 , λ 2 = h ± vi = r(cosθ ± isinθ), the intertemporal equilibrium performance is where r is the absolute value of the complex number, and sin θ = v/r and cosθ = h/r.
To determine the levels of intertemporal equilibrium performances associated with the low income and high income levels, and to determine whether they remain stable over time, we need to empirically estimate the parameters involved.This is done in the next section.

Empirical analysis
(a) The sample.Data for this study was gathered using a questionnaire including questions about demand, supply, incomes, prices, and performances in the higher educational services in Turkey.The questionnaire was distributed to the students in two consecutive time periods.In the first period 100 students were asked to respond to the relevant questions.66 useable questionnaires were returned giving a response rate of 66 percent, which was considered satisfactory for subsequent analysis.Some responses with considerable missing information were excluded.Each question (item) was rated on a seven-point Likert scale anchored at the numeral 1 representing the lowest score that can be assigned, and at the numeral 7 representing the highest.The same procedure has been repeated to obtain the data for the second period.The supply-side information is obtained from the educational institutions.
Ahmet Kara 5 (b) Estimation of the parameters.To estimate the parameters involved, we formulate the following regression equations: where u t * = u t + α 1 ε t and v t are disturbance terms.(i) Demand.Since the prices under consideration are fairly close to one another, the relative prices are close to one, thus lnP r t is close to zero, and as such, it drops out of the equation.This supposition and an additional assumption that minimal income induces minimal demand imply that α 0 = 0. We will estimate demand for two cases, namely for the case with low incomes and for the case with high incomes.The regression-results are as follows: The low income-case: lnD t = 0.762 ln M t (2.017) The high income-case: lnD t = 0.762 ln M t (10.407)R 2 = 0.87.t-statistic is given in parentheses.(ii) Supply.Supply is largely determined by central bureaucratic authorities whose decisions are based on certain criteria, such as the adequacy and quality of physical infrastructure and human resources, rather than prices.Thus, prices could be conveniently left out of the supply function.P r t drops out of the log-linear formulation of the supply equation.To estimate the other parameters of the supply equation, we asked officials of the relevant institutions questions, the answers of which were designed to give the values of the elasticities of supply with respect to the present and past d performances.The answers indicate that a 1% increase in the past performance would increase the quantity supplied by about 0.25%, but a 1% increase in the present performance would increase the quantity supplied by about 0.75%.However, by virtue of the enrollment constraints placed by the Higher Education Council, what the institutions under examination could supply was 90% of what they were willing to supply.Thus, β 1 = 0.9 * 0.75 = 0.675, β 2 = 0.9 * 0.25 = 0.225. (3.4) The value of A is normalized to 1. (iii) The coefficient of adjustment (k).For simplicity, we will assume that P t+1 /P t is proportional to the ratio of demand to supply, and hence, k = 1.
Given the empirical values of the parameters obtained above, we get, We will now consider a particularly interesting case where the student population growth is equal to the productivity growth, that is, g = δ.With this assumption and with all the needed parameter values at hand, the intertemporal equilibrium performance for lowincome and high-income cases, P * low and P * high , are 0.47 j sinθ( j + 1) sinθ z t− j , 0.47 j sinθ( j + 1) sinθ z t− j . (3.6) For analytical convenience, we will carry out some of our analysis in terms of logarithmically transformed performance, lnP, rather than P. Since ln function is an orderpreserving transformation, analysis in terms of ln P and P will yield the same qualitative results; and the quantitative results could be transformed into one another.The expected value of the logarithmically transformed intertemporal equilibrium performances for low-income and high-income cases are 0.47 j sinθ( j + 1) sinθ E z t− j . (3.7) Since, by virtue of the assumptions about ε t , u t , and v t , E(ε t ) = 0, E(u t ) = 0, and In view of the logarithmically transformed performance scale of ln 1 = 0 to ln7 ∼ = 1.95, an intertemporal equilibrium expected performance of 0.56 is low, and a performance of 1.5 is high.As proven in the appendix, these performance values are also stable over time in the particular sense that they have stationary distributions with constant means and variances.This indicates that low income and high income help lead to, respectively, stable low performance and stable high performance values over time.
The following section will formulate a stochastic subsidy policy, which will enable the sector to escape the low performance equilibria by helping the sector to reach a highperformance target, and which will stabilize the sector around that target.

Policy implications: an example of a stochastic subsidy policy
Suppose that the educational service providers in Turkey aim to reach a stable (sustainable) high-performance target in the presence of stochastic shocks.Consider a stochastic shock to income (ln M ts ) in the magnitude of ε t , which may have come, for instance, from an economic downturn (of the kind that took place in Turkey in 2001) leading to reductions in customers' incomes.Let us design the following demand side stochastic policy response (a subsidy rule): where η 0 and η 1 represent the nonstochastic and stochastic components, respectively.With this rule, the modified income level, in logarithmic form, will be defined as With the modified income level, the second order stochastic difference equation will be Reaching a stable (minimally varying) expected quality target in the presence of stochastic shocks turns out to be equivalent to minimizing the expected loss function of the following kind: where P * * is the performance target.We decompose the expected loss function in the following manner.For a decomposition, though in a different context, see Sargent [46]; Since E(Pt) − P * * is not random and since E(Pt − E(Pt)) = E(Pt) − E(Pt) = 0, the decomposition will boil down to The first term represents the variance of performance and the second term denotes the "squared deviation" around P * * .Thus, minimizing expected loss is equivalent to minimizing the squared deviation, which requires that expected performance to be equal to the performance target, and minimizing the variance of performance, enabling the educational service provider to reach a stable (minimally varying) performance target.
To find, for the special case where g = δ, the parameters of the stochastic policy rule which minimize the expected loss function, let us incorporate the rule into the function, The values of η 0 and η 1 that minimize the expected loss function are (4.8) For instance, the value of η 0 , which will bring the low-income-induced performance above (0.56) to the level of a performance target P * * = 1.5, is calculated to be 1.11.This represents the nonstochastic component of the subsidy.The value of the stochastic component of the subsidy, η 1 = −1, implies that, for stabilization against the kind of negative (income-reducing) shock exemplified here, income should be increased by the magnitude of the stochastic shock.
The applicability of the model developed in this paper is not restricted to the case in Turkey.Inequality-induced outcomes in education are characteristics of many developing and even developed countries.Inequality patterns tend to evolve in response to a number of factors including changes in demand and technology.This model could serve as a frame of reference to account for such changes/developments in the education sectors in various countries.Among these developments are the information-technology-based changes, which have been transforming the supply and demand in education sectors across the world.Information technology has increased productivity and quality, reduced costs, and increased the expectations for higher income associated with information-intensive skills acquired through education.The stochastic subsidy policy proposed above could be reformulated to incorporate the effects of information technology.Our conjecture is that possibilities created by the information technology may, under certain conditions, well increase the effectiveness of the stochastic subsidy policy in reducing the degree of inequality in education.

Concluding remarks
The paper develops a model for the stochastic dynamics of economic inequality-induced outcomes in education, and explores the differences in service performance induced by unequal incomes.The paper formulates a stochastic subsidy policy that could help the sector to escape the low performance equilibria and that could stabilize the sector at relatively higher performance levels.The designed policy response is one among many other stochastic resolutions which could take the form of, for instance, a demand side policy or a supply side policy.The rich array of policies in question are worthy of future research.
Using the properties of some series and lag operators and doing some algebraic manipulations, we get This is the parametric expression of x t = lnP t at the intertemporal equilibrium.Let P * denote the intertemporal equilibrium performance.Thus, (A.9) In case where λ 1 and λ 2 are conjugate complex numbers, that is, λ 1 , λ 2 = h ± vi = r(cosθ ± isinθ), the intertemporal equilibrium performance is where r is the absolute value of the complex number, and sin θ = v/r and cosθ = h/r.
( (A.23) Ahmet Kara 13 Thus, logarithmically transformed intertemporal performances in low-income and highincome cases have stationary distributions in the sense that they have constant means and variances.
b) Complementary function.To find this component of the solution, we need to consider the following reduced form of the second-order difference equation:lnP t+1 + kβ 1 − 1 lnP t + kβ 2 lnP t−1 = 0. (A.11)A possible general solution could take the form ln P t = Ay t .Hence, lnP t+1 = Ay t+1 and lnP Since, by virtue of the assumptions about ε t , u t , and v t , E(ε t ) = 0, E(u t ) = 0, andE(v t ) = 0, E(z t ) = k(α 1 E(ε t ) + E(u t ) − E(v t )) = 0. Thus, E lnP * low = 0.56, E lnP * high = 1.50, (A.21)which are nothing but the intertemporal expected equilibrium performances in lowincome and high-income cases, respectively.Note that ε t , u t , and v t are uncorrelated over time, and so is z t .They have zero covariances.Thus, the variance (V ) of lnP Please note the value of sinθ specified above.)It is straightforward to show that the variance in the case of high income is the same as well.Taking the limits of mean and variance as t →∝, t−1 = Ay t−1 .Substituting these expressions into the reduced form of the second-order equation, we getAy t+1 + kβ 1 − 1 Ay t + kβ 2 Ay t−1 = 0. * is V lnP * low = V 0.56 + t→∝ V lnP * low = lim t→∝ V lnP * high = ∝ j=00.47 2 j sinθ( j + 1) sinθ