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# Numerical Analysis

In the last chapter, we discussed the equilibrium conditions with and without private school as well as taking voucher programs into consideration by means of a general model. By setting the utility functions in the form of logarithms, we are able to discuss the problems numerically to see how the results in each case are. Therefore, in this chapter we would first like to discuss how different income endowments affect the results. Then we will discuss when private school exists, and how does the existence affect the behavior of the residents of the two communities. Finally, we would like to see if the voucher program is feasible, and how it promotes the establishment of private school.

We assume that there are one hundred people in each community, and the total number of low ability and high ability students are one hundred each, which means that N1=100, N2=100, nu=100, ns=100. The utility function is assumed to be in logarithm form. For families with child of low ability, the utility function is

1 1 2

( , ( , , )) ln ln ln(1 )

U C e Iθ a = C+β I+β +θ

And for families with child of high ability the utility function is

2 1 1 2 2

( , ( , , )) ln ln ln(1 )

U C e Iθ a = C+α β I+α β +θ

The parameter β1 falls in the range (0,1). It affects the proportion spent on educational inputs to total consumption, which means the higher β1 is the more income is spent on I. Moreover, β2 indicates the importance of peer effect to families, which means that the higher β2 is, the marginal utility of peer effects is larger. The parameter α1 falls in the range [1,∞). It is to show that families with child of higher ability obtain higher marginal benefit from educational input than what the families with child of lower ability does. And finally, the parameter α2 falls in the range (0,1) and is to capture the extent that low-ability family obtains relatively higher marginal

benefit from peer effect.

To explore the features of the equilibrium numerically, we substitute numbers into the equilibrium conditions to solve for comparable solutions. However, substituting different values to the parameters derive different outcomes, we would like to discuss whether the equilibrium exists or not, thus we have to find out the conditions for an equilibrium to occur. Also, the numerical results are shown in the appendix A through F at the end of this paper, while we will cover the interpretations in this chapter.

Result 1: We derive the following results with our setting of the numerical model for heterogeneous equilibrium of de Bartolome (1990) to exist .

1. α1 has to be a number greater than one, but cannot be too large.

2. When α2 is too large, the equilibrium does not exist.

3. The maximum value of α2 that enables the existence of equilibrium is lowered in the increase of α1.

4. The change in income does not change the allocation of peer effects, but the rent premium is different.

5. When β1 or β2 increases, there is less chance for a heterogeneous equilibrium to exist.

The most important two parameters that matters when finding the conditions for the equilibrium to exist are α1 and α2, which is because they measure the difference of the magnitude of marginal peer effects and marginal input effects between the families with child of low ability and families with child of high ability. Thus, given the value of y, β1, and β2, we try to find out how the variation of values substituted into these to parameters affect the existence of equilibrium. When α1 is a number greater than one, an increase in α2 increases the equilibrium values of θ and r. This is because when α2 increases, the amount of rent the families of high ability would

demand for them to give up a certain amount of peer effect increases, and for families of low ability to become willing to purchase peer effect at that price of rent, peer effect has to increase. To be more explicit, we discuss the problem with Figure 3 that plots the equilibrium which we have shown in the last chapter.

As we can see in the graph, AA′ curve is all the combinations of θ and r for low ability students to be indifferent between living in the city and living in the suburbs, and BB′ curve is for that of high ability students. When α1 is a number greater than one, an increase in α2 pivots the BB′ counterclockwise leaving less room for transaction, since now the rent premium offered by families with child of high-ability to sell the peer effects is now higher for any given θ, while the willingness to pay for the peer effect remains unchanged. However, when α2 is large and close to one, a mixed equilibrium does not exist anymore, since it makes AA′ parallel to BB′. This is because when α2 is too large given the value of α1, the price that the residents in the suburbs are willing to sell peer effect is too high to induce low ability students to migrate, and thus a separated equilibrium is formed.

Moreover, when α1 increases, the threshold for the value of α2 to derive a mixed equilibrium becomes lower. When α1 approaches to one, which means the two communities only differ in their preference on peer effect, then any value of α2 that is below one would cause a separated equilibrium, which implies that the rent to make residents feel indifferent of where to live does not exist. Now given the value of α2, we can see that when α1 increases, the equilibrium values increase, vice versa. This is because the increase in the value substituted to α1 increase the difference between the educational inputs Iu and Is in the two communities, which makes families of low ability prefer living in the urban area, and families of high ability prefer to live in the suburban area. When we look at the graph, an increase in α1 given the value of α2 makes AA′ curve pivot counterclockwise, and BB′ curve pivots clockwise,

intersecting at a point where both θ and r are larger in equilibrium. To offset the effect of changing the values substituted into α1, α2 has to be moved towards the opposite position. In other words, to sustain a mixed equilibrium, when α1 increases, α2 has to decrease.

Now we take a look at the common parameters of the utility functions of both types of families. An increase in the values of β1 increases the importance of educational input and decreases the relative importance of peer effect in both communities, vice versa. However, this increase also reduces the room for trade since the educational input the residents have to pay increased in both communities, which means the willingness to pay rent for a given proportion of more able students has decreased, and the decrease is more for families with child of low ability since they prefer peer effect to input effect. Thus both θ and r will increase when β1 increases.

When we look at the graph, this change in β1 pivots both AA′ and BB′

counterclockwise, with a larger pivot in BB′, and lessens the room for trade of peer effect, which is shown as the distance between the curves. Moreover, when the value of β2 increases, the room for trade increases, vice versa.. When β1 equals to β2, a simultaneous increase in the two parameters keeps the proportion of more able students in equilibrium almost unchanged but increases the rent premium.

Now we keep parameters other than income fixed to see how the values of y1 and y2 affect the equilibrium values. We find out that the value of income does not affect the outcomes, although the increase in income affects the amount of consumption and input decision, this only affects the amount of rent premium, and will not change the preference of peer effects and input effects.

Result 2: After we choose y1=10, y2=10, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case, the comparative statics of the numerical model is as following:

Other things unchanged,

1. When y increases, only r increases, θ is unchanged.

2. When α1 increases, both θ and r increase.

3. When α2 increases, both θ and r increase.

4. When β1 increases, both θ and r increase

5. When β2 increases, only θ increases, r is unchanged.

From the preceding paragraphs, we choose our benchmark values for substitution to be y1=10, y2=10, α1=2, α2=0.8, β1=0.04, β2=0.25. Since we have no explicit function of the equilibrium values, we now simulate a comparative statics by keeping other parameters fixed and see how the equilibrium is affected when each of the values of parameters increases. The results are shown in the appendix E.

In the original equilibrium, approximately 60 out of 100 families with child of high ability live in the suburbs. The rent premium is relatively low compared to the amount of consumption, while the input expenditure is about one tenth of the total consumption. An increase in income affects everything but the equilibrium, which is because income has nothing to do with the preference of families. An increase in α1

increases the equilibrium θ and r, and also the level of educational input. However, the consumption level of residents in the suburbs has decreased because they have to pay more educational input. An increase in α2 has similar effects as α1, but the decrease in educational input and consumption for suburban families is caused by the increase in relative importance of peer effects. An increase in β1 reasonably causes consumption and educational input of both communities to increase since it stands for the proportion of income spent on those expenses. An increase in β2 caused θ to decrease since it makes peer effects more expensive.

Before going through the discussion on the conditions that enables private school to exist, we need to know how our model with different income endowments affects the conditions for the equilibrium to exist.

Result 3: In our model, the conditions for the equilibrium to exist are 1 . The difference between y1 and y2 cannot be too big.

2. α1 has to be a number greater than one.

3. α2 cannot be too large.

4. β1 has to move oppositely according to the value of α1 5. β2 cannot be too small

The difference between this model and the model of de Bartolome (1990) is that there is difference in income endowment. In addition to the difference, we assumed that the input level of urban area is decided by families of low-ability low-income, and that of the suburban are is decided by families of high-ability high-income, this cause the gap of the level of educational input between the two communities to increase in size, and lowers the chance for a mixed equilibrium to exist if the gap is too big, since none of the high-ability low-income students can afford to live in the suburbs. β1 has to move oppositely according to the value of α1, because if they move in the same direction, it will make the educational input level of the suburbs become too high, and the residents there would rather move to the urban area. If β2 is too small, then the peer effect would be trivial to influence the equilibrium, and causes separation of different abilities of families to emerge.

Result 4: After we choose y1=10, y2=15, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case, the comparative statics of the numerical model is as following:

Given other things unchanged,

1. When y1 increases, both θ and r increase.

2. When y2 increases, both θ and r decrease.

3. When α1 increases, both θ and r increase.

4. When α2 increases, both θ and r increase.

5. When β1 increases, both θ and r increase.

6. When β2 increases, only θ has decreases, r is unchanged.

In equilibrium the values of θ and r are 0.577234 and 0.123756. We can know from the equilibrium values that approximately 57 out of 100 high ability students are in the suburbs. The rent premium is relatively low to the level of consumption and educational input. The level of consumption and educational input are not close to each other anymore, which is the result we want to see from adding the difference of income endowment to the model. The results are very similar to that of no difference in income, except for the fact that variation in the endowed income now affects the value of θ and r in equilibrium. The results for the comparative statics are shown in appendix F. Now that we know what the conditions for equilibrium in this model are, we can discuss the conditions for private school to exist. The definition of exist in this thesis means that given the no migration conditions are satisfied in equilibrium, students gain higher utility from attending private school than attending public schools, which implies that students are willing to attend private school. Otherwise, no student can be recruited.

Result 5: Private school exists when

1. The product of α1 and β1 is not too large.

2. The value of α2 cannot be too small.

3. The value of β2 cannot be too small.

Thus we choose y1=10, y2=15, α1=2, α2=0.8, β1=0.04, β2=0.25 to be the benchmark case for the existence of private school.

The most appealing feature of private school is that it provides high peer effects, however, without the aid of vouchers, it is somewhat expensive for students to attend private school since they not only have to pay tuition for public school but also have to pay an additional fee for attending private school. Therefore, when peer effect is much important than input effect, the product of α1 and β1 cannot be too large, and if β2 is larger, then the original θ in equilibrium should be small relative to the peer effect that private school offers to appeal students from the urban area. When α1 increases, α2 has to decrease, vice versa. This is to keep the proportion of input expenditure fixed near a certain value; otherwise private school will not exist, since if the tuition they have to pay for public school is too high, they would not pay additional tuition to attend private school, no matter how high the peer effect is. The second point is that when α2 is too small, high ability students would not necessarily be better off attending private school. The thirds point is that when β2 is too small, there will be no mixed equilibrium in the community.The last point is that the larger the gap between low income groups and high income groups, the higher α1 has to be for private school to exist. This is because if α1 is too low, the equilibrium values would violate the assumption that the majority of suburban area is families with child of high ability. It is also because this change keeps the proportion of educational input fixed at a certain value.

After we discuss the conditions of when private school exists, we would then

move on to explore the properties of the equilibrium with private school. Since the most appealing feature of private school is that it gathers students of high-ability so that it is most like that students of high-ability low-income who lives in the urban area would like to attend private school. Suppose private school recruits ten high-ability low-income students from the urban area. Then we re-calculate the equilibrium for the community model. The new equilibrium θ and r are 0.545376 and 0.123756, respectively. This means that 54 out of 100 high ability students remain in the suburbs.

We then substitute the equilibrium values into the utility functions and confirmed that the residents are indifferent of migrating. The next step is what level of input expenditure private school should set to make the families indifferent of attending private school and public school. To do so we try to find the level of input that maximizes the educational output private school creates subject to the fact that students who attend private school are at least as well of as they are when they attend public school which is the optimization problem (11) mentioned in chapter three.

In the following we are going to discuss the behavior of private school. In the preceding paragraph, we first consider the situation that there are ten students attending private school, and all of them are high-ability low-income students from the urban area. When discussing peer effect in educational economics, private school usually plays the role of a broker who facilitates the transaction of peer effects.

Therefore, we would like to see if recruiting some students with low ability but high income who would like to pay higher tuition in order to attend private school would improve the educational output it creates. Suppose now in another scenario privates school recruits nine high-ability low-income students and one low-ability high-income student instead of all high-ability low-income students, and gives a discount to high-ability low-income students while low-ability high-income student burdens the rest of the payments. This is because the attending of low-ability

high-income student lowers the peer effect, thus he has to pay more tuition to compensate for the loss. We assume students of high-ability low-income receive a 4%

discount, then after private school recruits students, the new equilibrium is 0.550895, and the rent premium is unchanged. Again we try to find the optimal level of tuition private school should charge. Finally, we compare the total educational output that private school created in each case. The following chart shows the results.

H-L/ L-H θ r Total Educational Output

10/0 0.54376 0.123756 1.74242

9/1 0.550895 0.123756 1.63375

8/2 0.556414 0.123756 1.44838

To conclude, when private school exists, it is better to have all high-ability low-income students attending private school. This is probably because the value we substituted into β1 is too small or because we considered peer effects linearly so that it could not capture the significance of peer effect in the consideration of private school.

Also we have found that when private school exists, the students that attend public school obtain lower utility. This is because when some of the students in the community attend private school, the number of students that attend public school decreases. However, the ones that remain in public school did not gain additional educational input. We assume that the government collected the redundant inputs, so that with lower peer effects and paying the same amount of tuition makes the students who still attend public school after private school recruited students are worse off.

Another reason is that private school chooses the tuition that makes students indifferent of attending public and private school and maximizes the educational input it creates, which means that private school is actually better off after recruiting

students.

The final part of this section we would like to look at the effects of school vouchers. We attempt to find out how vouchers affect the existence and the operation of private school. Also we would like to know the affect on educational output. First we would like to see how different levels of vouchers affect the existence of private school. The final part we would like to look at how different levels of vouchers affect the total educational output created by private school.

Result 6: School vouchers can promote the existence of private school, on condition

Result 6: School vouchers can promote the existence of private school, on condition

Outline