Online ppendix for "ffi rmative ction: One Size Does Not Fit ll" Kala Krishna The Pennsylvania State University, CESifo and NBER lexander Tarasov University of Munich ugust 15 ppendix For Online Publication In this ppendix, we consider a special case to illustrate the intuition in the model Specifically, we parametrize the model to derive a closed-form solution and thereby compare the magnitudes of the effort effect and the selection effect To simplify the analysis, we assume linearity so sa = Sa, fa, e = a + e, and ce = Ce, where S and C are parameters In this case, the effort put in is given by e a, = max a,, which means that agents with total ability greater than do not put in any effort The equilibrium conditions under the non-discriminating quota where the marginal agent exerts positive effort are given by γ 1 1 H 1 a + γ 1 H a = α, 1 Sa T C a =, where the first equation determines the cutoff for the total ability, a, while the second one determines the performance cutoff: = S + Ca T C Recall that is the share of group i in the total population We also assume that both types of ability are uniformly distributed across the agents That is, H N a N = a N /a N max on [, a N max] and H i a = a /a on [, a ] where a max,1 a max, Note that under this assumption, H 1 a LR H a but H 1 a 1 H a so that our assumption about the likelihood stochastic order does not hold anymore However, as shown below, with uniform 1
distributions of abilities, first-order stochastic dominance is suffi cient for all the results formulated in the main body of the paper to hold In equilibrium, the cutoff ability a is pinned down by the number of seats α see 1 In our analysis, we consider the case that a a N max and a a for i = 1, That is, the number of seats is so few that an agent needs both types of ability to get in Next, we derive explicit expressions for the effort and selection effects evaluated at the non-discriminating quota θ Proposition 1 The eff ort and selection eff ects evaluated at the non-discriminating quota are given by a max, min, a max, a max,1 min, a max,1 α S + C EE θ=θ = a max,1 a max, + SE θ=θ = α1 βs a max,1 a max,, where a max, a a = a + a N max a max,1 a, Proof In the subsection below Given the performance cutoff, the magnitude of the effort effect positively depends on the parameters describing the returns to education and the cost of effort, S and C Moreover, it is straightforward to see that the magnitude of the effort effect is increasing in the performance cutoff : ie EE θ=θ / Indeed, if < a for i = 1,, so that some of both abilities get in, then α S + C a EE θ=θ = a max,1 a max, a max, a a max,1 a, which is negative and decreasing in as we assume that a < If a max, a max,1, then α S + C EE θ=θ = a max,1 a max, a max,1 a max,1 a, which is also decreasing in Finally, if > a max,1, then the effort effect is given by α S + C EE θ=θ = a max,1 a max,, and, therefore, does not depend on Thus, the effort effect is strictly decreasing in on [a, a max,1 and then is flat Note that if > a max,1, the overall effect on social welfare is negative: W a θ = EE θ=θ + SE θ=θ = α max,1 amax, βs + C < θ=θ
In this case, the effort effect dominates the selection effect Hence, we can conclude that, for suffi ciently high values of the performance cutoff which represents the level of competition for seats, which in turn depends on tuition and the availability of seats, the effort effect is stronger than the selection effect and, as a result, a quota in favor of disadvantaged results in welfare losses Whereas, for suffi ciently low values of, the selection effect prevails over the effort effect and a reservation quota in favor of disadvantaged can increase the social welfare Note that the tuition fee T affects the effort and selection effects only through Moreover, a rise in T reduces the performance cutoff Thus, Proposition There exists a value of the tuition fee, T tr, such that a marginal increase in the quota for the disadvantaged group when it is set to the non-discriminating level, θ, raises welfare if and only if T > T tr In other words, W θ θ=θ > if and only if T > T tr Intuitively, a higher tuition level reduces the magnitude of the effort effect and, as a result, a quota in favor of the disadvantaged is more likely to be welfare improving For this reason, the model suggests that affi rmative action is likely to reduce welfare in a setting where education is subsidized In India, for example, backward castes and tribes have a share of seats given by their population share reserved for them in publicly funded higher education These reservations result in the cutoff entrance exam scores that are much lower for these groups than for the general category 1 s public higher education is much cheaper than private, and as the very best institutions are public and seats are scarce, competition to get in is extreme In such a setting, reservations are likely to be welfare reducing Higher education is also subsidized in many European countries However, supply is abundant, and as a result, effort exerted to get in is far less than in the Indian context In the US, State Universities tend to be cheaper than private ones of a similar quality However, the emphasis on need blind admissions and the availability of financial aid significantly reduces the difference in price The Proof of Proposition 1 Recall that the effort effect evaluated at the non-discriminating quota can be written as follows: amax, c e a, EE θ=θ = D e a, a dh amax,1 a c e a, e a, a h a dh 1 a h 1 a, where D = α s a c e a, e a, c e a, e a, 1 In the celebrated Indian Institutes of Technology, the entrance exam marks for the general category are in the high nineties, while they are in the low fifties for the reserved category 3
Given the assumptions, the effort effect can be rewritten in the following way recall that agents with total ability higher than put in zero effort and, therefore, the upper bound of the integrals is min, a : min,amax, h EE θ=θ = α S + C min,amax,1 ada h 1 ada a a h a h 1 a so Recall that Since H N and H i are uniform, h i a = 1 a an max H i a = h i a = a a mina,a H N a ydh i y, h N a yh i ydy dy = mina, a max, a an max max,a a N max a an max This implies that if a a max [ a max,1, a max,, a N max] as assumed, then where a = a + an max Hence, min,a a h i ada = h i a = a + an max a a an max = a a a, an max a a a min, a a an max Substituting this in the expression for the effort effect, we derive that EE θ=θ = + α S + C a max,1 a α S + C a max, a α S + C = a max,1 a max, + a max,1 min, a max,1 a max,1 a a max, min, a max, a max, a a max, min, a max, a max, a a max,1 min, a max,1 a max,1 a Remember also that the selection effect is given by SE θ=θ = α a max,1 h 1 a α h a a max, sa sa 1 β a h N a a dh 1 a sa sa 1 β a h N a a dh a 4
Using the functional forms assumed, the selection effect can be written as a max,1 a SE θ=θ = α1 βs h N a a dh 1 a a max, a h N a a dh h 1 a a h a Notice that since a > a a a h N a a dh i a h i a = a a a a N da max a + an max a = a + a a N max Therefore, a max,1 + a a N max SE θ=θ = α1 βs = α1 βs a max,1 a max, a max, + a a N max ppendix B For Online Publication In this ppendix, we consider the extension of the benchmark model with two universities of different quality The Model We assume that the universities are different in that they offer education of different qualities, which affects the payoffs from being educated s a result, in equilibrium, the performance cutoff for the better university is higher so that it needs more effort to be accepted to the higher quality university The payoffs from being educated at university i {H, L} are given by q i sa, where q i is the quality measure of university i as before, a is the total ability The net payoffs are given by q i sa T i ce a, i, where T i is the tuition fee, i is the performance cutoff at university i H is assumed to be higher than L see the discussion below, and e a, i is the effort level put in to be accepted c is weakly convex Lemma 1 shows that the difference between the net payoffs from studying in the better and worse university is increasing in ability s a result, more able agents are matched with the better university 5
Lemma 1 For any given performance cutoffs, we define the difference between the net payoffs from studying in the better and worse university as Da; H, L = q H sa T H ce a, H q L sa T L ce a, L = qsa T ce a, H + ce a, L where q = q H q L > and T = T H T L Then, Da; H, [ L = qs a + c e a, H e a, H [ = qs a c e a, H c e a, ] e L a, H +c e a, L e a, L e a, H > c e a, L e a, L ] Proof Using the fact that it is easy to see that e a, = fa, e a, =, 1 f e a, e a, > That is, for any given ability a, meeting a higher cutoff requires more effort Thus, as c is convex, c e a, H c e a, L In addition, e aa, = f aa, e a, f e a, e a, < 3 This implies Finally, [ c e a, H c e a, ] e L a, H > e a a, = f eaa, e a, + f ee a, e a, e aa, f e a, e a, <, as f ee <, e aa, <, and f ea > This in turn means that e aa, L > e aa, H, implying that c e a, L e a, L e a, H > Using the above finding and the fact that s a >, it follows that Da; H, L > 6
Figure 1: Payoffs from Education: Two Universities q H sa T H ce a, H q L sa T L ce a, L a L a H a max a It would be helpful to see this lemma in a picture like Figure 1 q H sa is steeper than q L sa, as q H > q L and sa is increasing in ability so that more able individuals earn more at any given education level, and this is more so at better institutions In order to get into the university H L, each agent must put in e a, H e a, L and this needs the cost of ce a, H ce a, L to be incurred These costs are decreasing in ability as the more able need to put in less effort to meet any given performance cutoff Moreover, they decrease faster in ability when the cutoff is higher This happens because the higher performance cutoff requires more effort from all individuals, but due to the complementarity between ability and effort in creating performance, more able agents need to put in less effort to attain the higher cutoff s they are putting in less effort to get the lower performance cutoff anyway, this increased effort to meet a higher cutoff is also less costly for them Thus, the net surplus the benefit net of the cost from going to university is increasing in ability, and more so for the better university as depicted in Figure 1 This means that when we add the tuition cost which is independent of ability, the net benefit of going to the better university rises faster than that of the worse school so that these curves can cross each other at most once and better students will opt for the better university Note this fact is independent of tuition, though a high enough tuition could make the payoff from that university lie entirely below that of the other so no one goes there Next we consider the equilibrium and the comparative statics of the model In the equilibrium, there are two total ability cutoffs: a H and a L gents with abilities lower than a L choose the outside option The cutoffs are determined such that agents with abilities more than a H fill the available seats in the 7
better university and agents with abilities between a L and a H fill the worse university seats This gives the equilibrium conditions: 1 Ha H = α H, 4 Ha H Ha L = α L, 5 where α i is the number of seats in university i and H is the cumulative distribution of total ability s before, we assume that the natural and acquired abilities are independently distributed across agents The distribution functions are given by H N a N and H a on [, a N max] and [, a max], respectively Then, the distribution function for the total ability a is Ha on [, a max ], where and a max = a N max + a max Ha = a max H N a ydh y 6 The agent at a L is indifferent between the worse university and the outside option of zero which pins down ce a L, L and therefore L The agent at a H is indifferent between the two universities which pins down ce a H, H and therefore H Thus q L sa L T L ce a L, L =, 7 qsa H T ce a H, H + ce a H, L = 8 Thus, we have four unknowns, a H, a L, H, and L which can be pinned down by four equilibrium equations 4, 5, 7, and 8 Note that the condition, H > L, is equivalent to ce a H, H ce a H, L > Therefore, from the equilibrium condition 8, we can infer that H > L if and only if qsa H T > In other words, that the difference in the tuition levels is not set too high relative to the difference in quality Next, we explore how changes in T i affect the equilibrium outcome The following lemma holds Lemma 1 The cutoffs, a H and a L, do not depend on the tuition fees, T H and T L rise in T H does not affect L and decreases H 3 rise in T L decreases L and increases H 4 rise in T H and T L such that T does not change decreases L and H Proof 1, and 4 follow directly from the equilibrium equations Let us prove the third statement in the lemma From the equilibrium condition 7, we have L T L 1 = c e a L, L e a L, L < 8
In addition, differentiating 8 with respect to T L gives H T L c e a H, L = c e a H, H e a H, L L e a H, H T L 1 c e a H, L = c e a H, H e a H, L L e a H, H T L + 1 The sign of the derivative is the same as the sign of the numerator as the denominator is positive The numerator is in turn equal to c e a H, e L a H, L e a H, L e a L, L L T L + 1 = 1 c e a H, L e a H, L c e a L, L e a L, L Note that as a L < a H, e a H, L < e a L, L implying that c e a H, L Moreover, since e a, = 1/f e a, e a,, f e a L, e a L, L = f e a H, e a H, < 1, L < c e a L, L as f ea > and f ee < That is, the sign of the numerator is positive This proves the third statement The intuition behind 1 and in the lemma is straightforward The idea behind 3 is as follows Keep the performance cutoffs fixed n increase in T L shifts the payoff curve for L down and reduces a H while raising a L s a result, more agents apply for the seats in the high-quality university and fewer for the low quality one s the number of seats remains fixed, H must rise and L must fall The intuition behind 4 is simple Suppose we increase tuition fees by the same amount t any given performance cutoffs, while this change does not affect the intersection of the two curves a H as it shifts down both curves by the same amount, it raises a L This makes the demand for the worse university seats less than its supply which in turn reduces the performance cutoff L This fall in L must shift the payoff curve for the worse university back to its original position so that it goes through the original level of a L However, the fall in L s performance cutoff has a smaller impact on the payoff for higher ability agents and so makes it flatter This reduces a H from its original level, requiring a fall in H s performance cutoff as well Next, we explore the effects of changes in the number of available seats on the equilibrium outcomes Using the equilibrium equations we see that a rise in α L does not change a H as it is pinned down by α H and decreases a L This in turn means that L and H fall Intuitively, more available seats in the low quality university reduces the performance cutoff in that university, making it more attractive compared to the high quality university This absorbs some students from the high quality university to Formally, this result comes from the impact of T H on H being stronger than that of T L 9
the low quality university but since the number of seats in the high quality university does not change, the performance cutoff, H, falls to compensate for the decrease in the demand for seats in the high quality university rise in α H decreases both the ability cutoffs, a L and a H The decrease in a L in turn results in lower L The low quality university has to reduce its performance cutoff in order to fill in all the available seats The impact on H is also straightforward The direct effect of a rise in α H is to decrease a H and consequently H In addition, the rise in α H decreases L, which further reduces H see 8 s can be seen, both effects work in the same direction s a result, H falls The following lemma summarizes the above reasoning Lemma 3 1 rise in α L does not change a H and decreases a L, L, and H rise in α H decreases a H and a L and L and H Next, we examine the welfare implications of changes in the parameters in the model Social Welfare s before, we allow the private gains from education to be different from the social gains Specifically, for an individual, natural and acquired abilities are of the same importance but for the society natural ability is more important than the acquired ability Social welfare is given by the outside option is normalized to zero W = a N +a a H + a L a N +a <a H q H sa N + βa ce a N + a, H F dh N a N dh a 9 q L sa N + βa ce a N + a, L F dh N a N dh a, where F is the social cost of education per student and β < 1 Note that as the tuition is a lump-sum transfer, T does not directly affect the welfare It only affects welfare via the effort put in by agents Next, we explore the effects of the tuition fees on the social welfare First, we examine how changes in T L and T H affect the welfare Then, we find the values of T L and T H that maximize the social welfare function It is straightforward to see that W T i = a N +a a H + a L a N +a <a H q H sa N + βa ce a N + a, H F dh N a N dh a T i q L sa N + βa ce a N + a, L F T i dh N a N dh a Here, we used the fact that the ability cutoffs do not depend on the tuition fees see Lemma 1
From the results stated in Lemma, we can conclude that recall that L /T H = W = H T H T H a N +a a H c e a N + a, H e a N + a, H dh N a N dh a >, while W = H T L T L L T L a N +a a H a L a N +a <a H c e a N + a, H e a N + a, H dh N a N dh a c e a N + a, L e a N + a, L dh N a N dh a The sign of the latter expression is ambiguous, as H /T L > and L /T L < s can be seen, the impact of T H on welfare is similar to that in the model with one university: W/T H > The intuition is similar as well rise in T H reduces the effort put in by the agents who decide to apply for the high quality university and does not change the effort of the agents who apply for the low quality university s a result, welfare rises The impact of T L is ambiguous in general rise in T L reduces the effort put in by the agents applying for the low quality university and increases the effort put in by the agents applying for the high quality university s a result, given T H, there exists a certain optimal level of T L such that W/T L = unless the condition W/T L = delivers the minimum However, if the goal is to determine the value of the pair T H, T L that delivers the maximum social welfare, the outcome will be exactly the same as in the case with one university In other words, the social welfare as a function of T H and T L is maximized when the effort put in by the marginal agent is equal to zero That is, e a H, H = and e a L, L = This can be obviously seen from the expression for the social welfare 9, which is maximized when there is no wasted effort The conditions of zero effort put in by the marginal agent can be written as follows: sa H = T H T L, 1 q sa L = T L /q L 11 Since the ability cutoffs are determined by the number of seats in the universities, we can find the optimal values for the tuition levels from the above equations Note that if we assume that T H = T L = T, then the welfare function will be increasing in T However, the optimal value of T does not elicit the zero effort put in by all agents Indeed, a rise in T reduces L and, thereby, H recall that T = In this case, it is straightforward to show that the social welfare is increasing in T Therefore, we keep increasing T till the effort put in by the marginal agent a L becomes zero a further increase in T does not affect welfare, as L is not affected anymore The 11
equilibrium conditions in this case are qsa H ce a H, H + ce a H, L =, q L sa L T = s ce a L, L is equal to zero, ce a H, L is equal to zero as well conditions can be written as follows: Therefore, the equilibrium qsa H ce a H, H =, q L sa L T = s can be seen, e a H, H is strictly positive in the equilibrium That is, the marginal agent applying for the high quality university puts in some positive effort The corresponding value of H can be found from the first equation in the last system of equations Finally, similar to the benchmark case with one university, the distortion caused by selection into education can not be completely removed, as the social gains from education are different from the private gains The Case with Quotas In this section, we introduce educational quotas in the above framework We assume there are two groups of agents indexed by i {1, }, which have identical distributions of natural ability and potentially different distributions of acquired ability The latter is motivated by the fact that agents with different social backgrounds have had different educational inputs prior to taking the exam, which in turn results in different acquired abilities In particular, we assume that H 1 N a N = H N a N H N a N, while H 1 a LR H a where LR stands for the likelihood stochastic order Hence, h 1 a h 1 x > h a h x for any a, x : a > x This means that group 1 is more favored in terms of acquired ability than group In addition, we assume that the distribution of natural ability has a log-concave density This assumption is needed to ensure the likelihood stochastic order of the distributions of total ability: ie H 1 a LR H a 3 The share of each group in the total mass of agents which is normalized to unity is denoted by, where γ 1 + γ = 1 We then define θ ik as a share of available seats reserved for group i in university k: θ 1k + θ k = 1 for k {H, L} If these quotas are binding, then the cutoffs for the two groups will differ 3 See Theorem 1C9 in Shaked and Shanthikumar 7 for the proof This assumption is not very restrictive, as a number of commonly used distributions such as the normal, uniform, Gamma, and Beta distributions satisfy it 1
Note that the quota given to a certain group can be in general different in different universities The equilibrium conditions can be then written as follows: where i {1, } 1 H i a = θ α H, H i a H i a il = θ il α L, q L sa il T L ce a il, il =, qsa T ce a, + ce a, il =, We define θ k a non-discriminating quota in university k the quota assigned to group 1 so that the corresponding quota assigned to group is 1 θ k such that the quota leads to 1k = k, ie the performance cutoffs are the same for both groups If both universities set the non-discriminating quotas, then it is straightforward to show that If in addition H 1 a H a = Ha, then This is similar to the case with only one university a 1H = a H, a 1L = a L θ H = θ L = γ 1 Next, we write down the social welfare under the presence of two groups of agents: W = q H sa N + βa ce a N + a, F dh N a N dha i 1 i a N +a a + q L sa N + βa ce a N + a, il F dh N a N dha i i a il a N +a <a = q H sa N + βa ce a N + a, T H dh N a N dha i i a N +a a + q L sa N + βa ce a N + a, il T L dh N a N dha i i a il a N +a <a +α H T H F + α L T L F In the next sections, we explore the behavior of social welfare as a function of the quotas around the non-discrimination quotas Symmetric Groups with no Selection Effect In this subsection, we assume that the groups are symmetric, ie H 1 a H a = Ha, and examine how uniform changes in the quotas set by the universities locally affect the social welfare in the case of 13
no selection effect In particular, we assume that θ 1k = µθ k, implying that θ k = 1 µθ k This specification allows us consider uniform changes in the quotas set by the universities Moreover, if µ = 1, then both universities set the non-discrimination quotas θ k If µ =, then in both universities all seats are given to the second group Finally, if µ = 1/γ 1 > 1, then all seats in both universities are given to the first group recall that if H 1 a H a, θ H = θ L = γ 1 Next, we consider the social welfare as a function of µ in the case of no selection effect, ie β = 1 Taking into account 1, the derivative of the social welfare function with respect to µ can be written as Since β = 1, we have W W = + i = i + i il il W W W + il + W il il il c e a, e a, a a dh i a c e a, il e a, il a dh i a il a<a q H sa ce a, T H h i a q L sa ce a, il T L h i a q L sa il ce a il, il T L h i a il Taking into account the equilibrium conditions for the marginal agent, these can be written as follows: W = + i il c e a, e a, a a dh i a c e a, il e a, il a dh i a il a<a q L sa ce a, il T L h i a q L sa ce a, il T L h i a il hi a il 14
as agent a is indifferent between schools so that W = il a a a il a<a c e a, e a, dh i a c e a, il e a, il dh i a Let us then find the derivative of ik with respect to µ From the equilibrium conditions, we have Taking into account that we derive il = q Ls a il c e a il, il e c e a il, il e a il, il a il, il θ α H + θ il α L = 1 H i a il, il 1L L = θ Hα H + θ Lα L γ 1 ha 1L <, = θ Hα H + θ Lα L γ ha L > This implies 1L L = q Ls a 1L c e a 1L, 1L e a 1L, 1L c e a 1L, 1L e a 1L, 1L θ Hα H + θ Lα L γ 1 ha 1L <, = q Ls a L c e a L, L e a L, L θ Hα H + θ Lα L c e a L, L e a L, L γ ha L > s can be seen, at the non-discriminating quotas when µ = 1, implying that recall that H 1 a H a = i il a il a<a 1L L γ 1 = γ, c e a, il e a, il dh i a = Next, we consider the derivative of with respect to µ From the equilibrium conditions, [ qs a c e a, e a, + c e a, il e a, il a, il c e a, e a, il + c e a, il e c e a, e a, ] 15
In addition, we have that Using all the previous results, we can see The latter follows from the fact that 1H = θ Hα H γ 1 ha 1H <, H = θ Hα H γ ha H > 1H 1L < and H > <, L qs a c e a, e a, Moreover, if µ = 1, it is straightforward to see that implying that Thus, we have i a a 1H γ 1 = γ, >, and see Lemma 1 + c e a, il e a, il H c e a, e a, dh i a = W µ=1 = That is, non-discrimination delivers a local extremum > In the case of concave social welfare, nondiscrimination is globally optimal Next, we explore the case when the groups are asymmetric in terms of the distribution of total ability Intuitively, the logic is exactly the same When the two groups are the same, the losses of one group exactly make up for the gains of the other for slight changes Thus, if welfare is concave, this is a local maximum symmetric Groups with no Selection Effect ssume now that H 1 a LR H a Using the results derived in the above section, the derivative of social welfare with respect to µ is given by W = γ i i L i a a a il a<a First, consider the second term of the derivative: L c e a, il e a, il i a dh i a il a<a q L s a 1L c e a 1L, 1L e a 1L, 1L = θ Hα H + θ c Lα L e a 1L, 1L e a 1L, 1L a L, L q Ls a L c e a L, L e c e a L, L e a L, L c e a, e a, dh i a c e a, il e a, il dh i a a 1L a<a 1H a L a<a H c e a, 1L e a, 1L c e a, L e a, L h 1 a h 1 a 1L da h a h a L da 16
t the non-discriminating quota, a L = a 1L and L = 1L This implies that q L s a L c e a L, L e a L, L c e a L, L e a L, L = q Ls a 1L c e a 1L, 1L e a 1L, 1L c e a 1L, 1L e a 1L, 1L Moreover, as H 1 a LR H a, Thus, a 1L a<a 1H implying that c e a, 1L e a, 1L h 1 a h 1 a 1L h a h a L for any a > a 1L = a L il h 1 a h 1 a 1L da > a il a<a when evaluated at the non-discriminating quota µ = 1 a L a<a H c e a, L e a, L c e a, il e a, il dh i a > Consider then the first term of the derivative, which is given by From the previous section, we have that = a a c e a, e a, dh i a [ qs a c e a, e a, + c e a, il e a, il a, il c e a, e a, il + c e a, il e c e a, e a, This means that at the non-discriminating quotas, we have h a h a L da, ] 1H γ 1 H γ = qs a 1H c e a 1H, 1H e a 1H, 1H + c e a 1H, 1L e a 1H, 1L θ Hα H c e a 1H, 1H e a 1H, 1H h 1 a 1H c e a 1H, 1L e a 1H, 1L q L s a 1L c e a 1L, 1L e a 1L, 1L c e a 1L, 1L e a 1L, 1L c e a 1H, 1H e a 1H, 1H θ Hα H + θ Lα L h 1 a 1L, = qs a H c e a H, H e a H, H + c e a H, L e a H, L θ Hα H c e a H, H e a H, H h a H + c e a H, L e a H, i q L s a L c e a L, L e a L, L c e a L, L e a L, L c e a H, H e a H, H θ Hα H + θ Lα L h a L 17
Note that at the non-discrimination quota: θ qs a 1H Hα c e a 1H, 1H e a 1H, 1H + c e a 1H, 1L e a 1H, 1L H c e a 1H, 1H e a 1H, 1H = θ qs a H Hα c e a H, H e a H, H + c e a H, L e a H, L H c e a H, H e a H, H Moreover, at the non-discriminating quota: θ Hα H + θ Lα L = θ Hα H + θ Lα L Thus, at the non-discriminating quota: c e a 1H, 1L e a 1H, 1L c e a H, L e a H, L q L s a 1L c e a 1L, 1L e a 1L, 1L c e a 1L, 1L e a 1L, 1L c e a 1H, 1H e a 1H, 1H q L s a L c e a L, L e a L, L c e a L, L e a L, L c e a H, H e a H, H > B > 1H γ 1 H γ = h 1 a H B h 1 a L, = h a H + B h a L s a result, at the non-discriminating quota: = i h 1 a H + h a H + a a B h 1 a L B h a L c e a, e a, dh i a c e a, H e a, H h 1 ada a a H a a H c e a, H e a, H h ada Taking into account the stochastic order of the distributions of total ability, it is straightforward to see that B a a H a a H c e a, H e a, H c e a, H e a, H h 1 a h 1 a H da > h 1 a h 1 a L da > B In other words, at the non-discriminating quota: a a a a H a a H c e a, H e a, H c e a, H e a, H c e a, e a, dh i a > h a h a H da, h a h a L da To sum up, we have shown that the derivative of social welfare with respect to µ evaluated at the non-discriminating quota is positive This means that discriminating in favor of group 1 locally increases 18
the social welfare This results is the same as that in the case of one university That is, the effort effect works in favor of the advantaged group EE µ=1 > This makes sense as weaker students need to put in more effort to get in and this effort is wasteful So discriminating against the less advantaged group reduces the wasteful effort and thus raises welfare Next, we explore the role of the selection effect The Selection Effect Recall that the social welfare when β < 1 is given by W = i a N +a a + i a il a N +a <a = i a N +a a + i a il a N +a <a +α H T H F + α L T L F The last expression can be written as follows: a N max W = i a a a + a i a il a +α H T H F + α L T L F q H sa N + βa ce a N + a, F dh N a N dh i a q L sa N + βa ce a N + a, il F dh N a N dha i q H sa N + βa ce a N + a, T H dh N a N dh i a a a q L sa N + βa ce a N + a, il T L dh N a N dh i a q H sa N + βa ce a N + a, T H dh N a N dh i a q L sa N + βa ce a N + a, il T L dh N a N dha i When we explore only the selection effect, by definition we look at the effect which works just through the ability cutoffs and not through the performance cutoffs Thus, the selection effect is as follows: SE = + i = il il a a a a a q H sa a + βa ce a, T H h N a a dh i a q L sa a + βa ce a, il T L h N a a dh i a q L sa il a + βa ce a il, il T L h N a il a dh i a qsa a + βa ce a, +ce a, il T h N a a dh i a q L sa il a + βa ce a il, il T L h N a il a dh i a 19
Taking into account the equilibrium conditions, we know that qsa ce a, + ce a, il T = q L sa il ce a il, il T L = Therefore, qsa a + βa ce a, + ce a, il T = q sa a + βa sa, and q L sa il a + βa ce a il, il T L = q L sa il a + βa sa il Hence, SE = q i q L i il a a sa a + βa sa h N a a dh i a 13 sa il a + βa sa il h N a il a dh i a Note that in the case of one university we have only the second component of the above expression However, we can apply the technique developed for the case with one university to both components, as they have similar functional forms Consider, for instance, the first term in the above expression given by: SE 1 = q i a sa a + βa sa h N a a dh i a Recall that 1H = θ Hα H γ 1 h 1 a 1H <, H = θ Hα H γ h a H > Hence, SE 1 evaluated at the non-discriminating quota can be written as follows: SE 1 θ=θ = qα H θ H a max,1 qα H θ H a max, sa H a + βa sa H h N a H a h 1 a h 1 a H da sa H a + βa sa H h Na H a h a h a H da s in the benchmark case, we consider the following density functions: h i h i a a h N a H a a h N a H yhi ydy Let H i a be its associated cumulative distribution function s H 1 a LR H a, h 1 a h 1 x = h1 a h N a H a h 1 xh Na H x h a h N a H a h xh N a H x = h a h x for any a, x : a x
That is, H1 a LR H a which in turn implies H 1 a 1 H a Then, SE 1 can be rewritten in the following way: SE 1 θ=θ = qα H θ H + qα H θ H a max,1 a max, Equivalently, as H 1 a LR H a implies a max,1 a max, SE 1 θ=θ = qα H θ H qα H θ H a max,1 a max, a max, Integrating the second term above by parts implies that SE 1 θ=θ = qα H θ H a max,1 a max, sa H sa H 1 β a d H 1 a sa H sa H 1 β a d H a sa H sa H 1 β a d H 1 a sa H sa H 1 β a d H1 a H a sa H sa H 1 β a d H 1 a + qα H θ H sa H sa H 1 β a max, 1 H amax, 1 a qα H θ max, H 1 β H a H a 1 s a H 1 β a da <, as s is positive and H a H 1 a for any a recall that H 1 a 1 H a and, moreover, a max,1 a max, sa H sa H 1 β a d H a 1 > sa H sa H 1 β a max, 1 H a 1 max, Similarly, we can show that the second term in 13 evaluated at the non-discriminating quota: SE θ=θ = q L i il a sa L a + βa sa L h N a L a dh i a, is negative as well the proof is exactly the same as that for SE 1 s a result, we can show that the selection effect evaluated at the non-discriminating quotas is negative, suggesting that we need to give quotas to the disadvantaged group Moreover, if the groups are symmetric, then the selection effect is equal to zero at the non-discriminating quotas 1