EFFORT INCENTIVES, ACHIEVEMENT GAPS AND AFFIRMA. AND AFFIRMATIVE ACTION POLICIES (very preliminary)

Transcription

1 EFFORT INCENTIVES, ACHIEVEMENT GAPS AND AFFIRMATIVE ACTION POLICIES (very preliminary) Brent Hickman Department of Economics, University of Iowa December 22, 2008

2 Brief Model Outline In this paper, I use an all-pay auction framework to model effort choice in competitions where outcomes do not depend solely on ones effort. MOTIVATING EXAMPLE:

3 Brief Model Outline In this paper, I use an all-pay auction framework to model effort choice in competitions where outcomes do not depend solely on ones effort. MOTIVATING EXAMPLE: High school students, the bidders, are heterogeneous with respect to the effort (i.e., utility cost) required to achieve a particular GPA or ACT/SAT score.

4 Brief Model Outline In this paper, I use an all-pay auction framework to model effort choice in competitions where outcomes do not depend solely on ones effort. MOTIVATING EXAMPLE: High school students, the bidders, are heterogeneous with respect to the effort (i.e., utility cost) required to achieve a particular GPA or ACT/SAT score. Given private costs, they choose grades and test scores as their bids.

5 Brief Model Outline In this paper, I use an all-pay auction framework to model effort choice in competitions where outcomes do not depend solely on ones effort. MOTIVATING EXAMPLE: High school students, the bidders, are heterogeneous with respect to the effort (i.e., utility cost) required to achieve a particular GPA or ACT/SAT score. Given private costs, they choose grades and test scores as their bids. College admissions boards, the auctioneer, review bids and allocate seats at various schools of differing quality.

6 Brief Model Outline In this paper, I use an all-pay auction framework to model effort choice in competitions where outcomes do not depend solely on ones effort. MOTIVATING EXAMPLE: High school students, the bidders, are heterogeneous with respect to the effort (i.e., utility cost) required to achieve a particular GPA or ACT/SAT score. Given private costs, they choose grades and test scores as their bids. College admissions boards, the auctioneer, review bids and allocate seats at various schools of differing quality. Affirmative Action initiatives require that allocations be based partially on exogenous characteristics like race.

7 Quotas vs. Bid Subsidies Related Work Bidders belong to two groups, M and W. Affirmative Action is a policy which bases allocation partially on exogenous group affiliation. TWO PROMINENT FORMS OF AA:

8 Quotas vs. Bid Subsidies Related Work Bidders belong to two groups, M and W. Affirmative Action is a policy which bases allocation partially on exogenous group affiliation. TWO PROMINENT FORMS OF AA: Quota system: Auctioneer reserves N m = M prizes for allocation only to group M India (Reservation Laws), Malaysia, Sri Lanka

9 Quotas vs. Bid Subsidies Related Work Bidders belong to two groups, M and W. Affirmative Action is a policy which bases allocation partially on exogenous group affiliation. TWO PROMINENT FORMS OF AA: Quota system: Auctioneer reserves N m = M prizes for allocation only to group M India (Reservation Laws), Malaysia, Sri Lanka Additive Bid Subsidies: Auctioneer artificially increases bids submitted by group M before comparisons are made Undergraduate admissions at the University of Michigan, pre-2004

10 Related Work AA at The University of Michigan Prior to 2004 Source:

11 THEORETICAL QUESTIONS Related Work 1 What effect do AA policies have on study effort choice? 2 What effect do AA policies have on achievement gaps? Achievement gap = avg. grade in group W avg. grade in group M 3 How do different AA policies compare to each other and to color-blind allocations in terms of the above two criteria?

12 Related Work Affirmative Action and Skills Acquisition Coate and Loury (AER, 1993): labor-search model of skills acquisition; minority workers strategically interact with employers who have tastes for racial discrimination á la Becker. Workers decide whether to forgo a fixed skills-acquisition cost; govt. mandates minimal minority employment (quota).

13 Related Work Affirmative Action and Skills Acquisition Coate and Loury (AER, 1993): labor-search model of skills acquisition; minority workers strategically interact with employers who have tastes for racial discrimination á la Becker. Workers decide whether to forgo a fixed skills-acquisition cost; govt. mandates minimal minority employment (quota). Result: Bid decisions are given by a threshold rule. Govt. can gradually increase mandate over time so that only a desirable effect is produced on minority employment and skills acquisition.

14 Related Work Affirmative Action and Skills Acquisition (cont d) DIFFERENCES: Discrete vs. Continuous Effort Choice Coate and Loury (1993) maps into a single-bidder all-pay auction with a strategic seller choosing a reserve price. Nature chooses ones bid and agents face a binary choice of whether or not to bid.

15 Related Work Affirmative Action and Skills Acquisition (cont d) DIFFERENCES: Discrete vs. Continuous Effort Choice Coate and Loury (1993) maps into a single-bidder all-pay auction with a strategic seller choosing a reserve price. Nature chooses ones bid and agents face a binary choice of whether or not to bid. In contrast, I endogenize bid levels (more flexible behavior choice). AA policies always distort effort choices

16 Related Work Affirmative Action and Skills Acquisition Quiang Fu (Economic Inquiry, 2006): Two-player, single-unit, asymmetric all-pay auction with complete information (i.e., private values are common knowledge) and bid preferences. Main Result: Bid Preferences increase average effort exerted by both groups At the optimal bid preference, the achievement gap is also maximized.

17 Related Work Affirmative Action and Skills Acquisition Quiang Fu (Economic Inquiry, 2006): Two-player, single-unit, asymmetric all-pay auction with complete information (i.e., private values are common knowledge) and bid preferences. Main Result: Bid Preferences increase average effort exerted by both groups At the optimal bid preference, the achievement gap is also maximized. IMPORTANT DIFFERENCES 1 Incomplete information (rather than a degenerate distribution) 2 Multiple players and prizes 3 Heterogeneity in bidding costs, rather than consumption utilities 4 Very different results

18 Environment INTRODUCTION

19 Environment INTRODUCTION COSTS: c [c,c], C F C (c) abs. cts.

20 Environment INTRODUCTION COSTS: c [c,c], C F C (c) abs. cts. BIDS: Each bidder submits one non-refundable bid, b Disutility of forfeiting bid b is D(b; c) D is differentiable, strictly increasing in b and c For now, let D(b; c) = cb

21 Environment INTRODUCTION COSTS: c [c,c], C F C (c) abs. cts. BIDS: Each bidder submits one non-refundable bid, b Disutility of forfeiting bid b is D(b; c) D is differentiable, strictly increasing in b and c For now, let D(b; c) = cb BIDDERS: There are N bidders w/iid costs

22 Environment INTRODUCTION COSTS: c [c,c], C F C (c) abs. cts. BIDS: Each bidder submits one non-refundable bid, b Disutility of forfeiting bid b is D(b; c) D is differentiable, strictly increasing in b and c For now, let D(b; c) = cb BIDDERS: There are N bidders w/iid costs PRIZES: K = N distinct prizes, chosen from P = [0, p] some cts natural random process generating prizes from P p k is an iid draw from F P (p) F P is abs. cts., has full support on P Prize values are known to bidders before they bid Zero marginal utility of extra units

23 Prize Allocation INTRODUCTION Standard Allocation Rule: 1 Auctioneer orders the sample of prizes 2 Auctioneer receives bids, orders the sample of bids, matches corresponding order statistics 3 Tie-breaking rule (for k 2 bidders with the same bid) Each bidder is randomly assigned an index drawn from Uniform(0, 1) Bidders are ordered according to the order statistics of observed indices

24 An equilibrium of the N-bidder game is a bidding function β N : [c,c] R + such that a bid of b = β N (c) maximizes ones expected payoff when all other players also formulate their bids according to β N. Proposition: In the N-player game there is a unique symmetric equilibrium which is monotonic and differentiable in private costs. Sketch Proof:

25 An equilibrium of the N-bidder game is a bidding function β N : [c,c] R + such that a bid of b = β N (c) maximizes ones expected payoff when all other players also formulate their bids according to β N. Proposition: In the N-player game there is a unique symmetric equilibrium which is monotonic and differentiable in private costs. Sketch Proof: 1 Jackson and Swinkels (Econometrica, 2005): Existence of a monotonic equilibrium in the N-player version of the game.

26 An equilibrium of the N-bidder game is a bidding function β N : [c,c] R + such that a bid of b = β N (c) maximizes ones expected payoff when all other players also formulate their bids according to β N. Proposition: In the N-player game there is a unique symmetric equilibrium which is monotonic and differentiable in private costs. Sketch Proof: 1 Jackson and Swinkels (Econometrica, 2005): Existence of a monotonic equilibrium in the N-player version of the game. 2 Hickman (2008, Appendix B): The N-player equilibrium is also cts., differentiable.

27 An equilibrium of the N-bidder game is a bidding function β N : [c,c] R + such that a bid of b = β N (c) maximizes ones expected payoff when all other players also formulate their bids according to β N. Proposition: In the N-player game there is a unique symmetric equilibrium which is monotonic and differentiable in private costs. Sketch Proof: 1 Jackson and Swinkels (Econometrica, 2005): Existence of a monotonic equilibrium in the N-player version of the game. 2 Hickman (2008, Appendix B): The N-player equilibrium is also cts., differentiable. 3 The FOCs of bidders decision problem have a unique solution which defines the equilibrium

28 An Asymptotic Approximation for Large N I study a game in which N is large: upon observing c, one has a very good idea of ones rank within the cohort of (N 1) opponents.

29 An Asymptotic Approximation for Large N I study a game in which N is large: upon observing c, one has a very good idea of ones rank within the cohort of (N 1) opponents. For simplicity and tractability, I will approximate this large finite model by considering the limiting decision problem as N. In the limit, the objective becomes a simple function of F C and F P.

30 An Asymptotic Approximation for Large N I study a game in which N is large: upon observing c, one has a very good idea of ones rank within the cohort of (N 1) opponents. For simplicity and tractability, I will approximate this large finite model by considering the limiting decision problem as N. In the limit, the objective becomes a simple function of F C and F P. I derive an approximate equilibrium, β(c), for the finite game: given ε > 0, N ε s.t. for N N ε, β(c) approximates β N (c) up to ε-precision.

31 An Asymptotic Approximation for Large N I study a game in which N is large: upon observing c, one has a very good idea of ones rank within the cohort of (N 1) opponents. For simplicity and tractability, I will approximate this large finite model by considering the limiting decision problem as N. In the limit, the objective becomes a simple function of F C and F P. I derive an approximate equilibrium, β(c), for the finite game: given ε > 0, N ε s.t. for N N ε, β(c) approximates β N (c) up to ε-precision. β(c) generates an ε-equilibrium: it approximates equilibrium payoffs to arbitrary precision for large enough N.

32 Henceforth, I abuse terminology by referring to the infinite game for expositional ease. When I speak of the equilibrium of the infinite game, I refer to the solution to the limit of bidders decision problem as N. STANDARD ALLOCATION RULE IN THE INFINITE GAME: Auctioneer observes the bid distribution G(b) Auctioneer maps the bid quantiles into the corresponding prize quantiles via the following payoff function: Π(b) F 1 P [G(b)].

33 Henceforth, I abuse terminology by referring to the infinite game for expositional ease. When I speak of the equilibrium of the infinite game, I refer to the solution to the limit of bidders decision problem as N. STANDARD ALLOCATION RULE IN THE INFINITE GAME: Auctioneer observes the bid distribution G(b) Auctioneer maps the bid quantiles into the corresponding prize quantiles via the following payoff function: Π(b) F 1 P [G(b)]. Note that this payoff function preserves the scarcity aspect of the finite allocation rule: there are only enough available prizes in the top r th percentile to serve the top r th percentile of bidders for any r.

34 In equilibrium, the utility for a bidder who has cost c and bids b is [G(b)] cb = F 1 ( [ 1 FC β 1 (b) ]) cb. F 1 P P Differentiating, I get the following FOC: [ f C β 1 (b) ] [ f P F 1 P (1 F C[β 1 (b)]) ] = c. (1) β [β 1 (b)] In equilibrium, β 1 (b) = c, so β f C (c) (c) = ( f P F 1 P [1 F C(c)] ) c. (2) Equation (2), in combination with the boundary condition β(c) = 0, characterizes the solution for β.

35 Theorem: For large N, it is sufficient to consider the solution to the limiting objective function, rather than the actual equilibrium: given ε > 0, N ε such that (i) equation (2) with boundary condition β(c) = D 1 (0;c) generates an ε-equilibrium of the N ε -player game and (ii) β(c) β(c;n ε ) sup < ε.

36 Theorem: For large N, it is sufficient to consider the solution to the limiting objective function, rather than the actual equilibrium: given ε > 0, N ε such that (i) equation (2) with boundary condition β(c) = D 1 (0;c) generates an ε-equilibrium of the N ε -player game and (ii) β(c) β(c;n ε ) sup < ε. Sketch Proof: Fix ε > 0 1 By monotonicity, N ε such that for n N ε the bid distribution generated by β n (c;n) is within ε of the bid distribution generated by β(c) under the sup-norm.

37 Theorem: For large N, it is sufficient to consider the solution to the limiting objective function, rather than the actual equilibrium: given ε > 0, N ε such that (i) equation (2) with boundary condition β(c) = D 1 (0;c) generates an ε-equilibrium of the N ε -player game and (ii) β(c) β(c;n ε ) sup < ε. Sketch Proof: Fix ε > 0 1 By monotonicity, N ε such that for n N ε the bid distribution generated by β n (c;n) is within ε of the bid distribution generated by β(c) under the sup-norm. 2 Given the previous point, for n N ε it is nearly optimal to optimize as if all of one s opponents adopted β(c) as their strategy (i).

38 Theorem: For large N, it is sufficient to consider the solution to the limiting objective function, rather than the actual equilibrium: given ε > 0, N ε such that (i) equation (2) with boundary condition β(c) = D 1 (0;c) generates an ε-equilibrium of the N ε -player game and (ii) β(c) β(c;n ε ) sup < ε. Sketch Proof: Fix ε > 0 1 By monotonicity, N ε such that for n N ε the bid distribution generated by β n (c;n) is within ε of the bid distribution generated by β(c) under the sup-norm. 2 Given the previous point, for n N ε it is nearly optimal to optimize as if all of one s opponents adopted β(c) as their strategy (i). 3 When the underlying distributions over prizes and private costs are well-behaved, it can also be shown that the maximizers are close too (ii).

39 Asymmetric Bidders Bidders belong to M or W, denoted by subscripts

40 Asymmetric Bidders Bidders belong to M or W, denoted by subscripts Private costs w/in each group are distributed according to F i (c), i = m,w.

41 Asymmetric Bidders Bidders belong to M or W, denoted by subscripts Private costs w/in each group are distributed according to F i (c), i = m,w. FINITE GAME: Nature assigns each of N agents to group W with probability ω [0,1], then each draws an iid cost from the appropriate distribution.

42 Asymmetric Bidders Bidders belong to M or W, denoted by subscripts Private costs w/in each group are distributed according to F i (c), i = m,w. FINITE GAME: Nature assigns each of N agents to group W with probability ω [0,1], then each draws an iid cost from the appropriate distribution. The limiting relative frequency of group-w bidders is exactly ω with probability one. In the infinite game the overall private cost distribution is a mixture of the within-group distributions F C (c) = ωf w (c) + (1 ω)f m (c).

43 Asymmetric Bidders Assume that costs in group M are stochastically higher, or F m (c) < ωf w (c) + (1 ω)f m (c) < F w (c), c (c,c). The idea here is that minority students in the United States, on average, receive less funding for their education than white students. Thus, they must put forth more personal effort to compensate for the disparity in educational quality. MAPS

44 ALTERNATIVE POLICIES (denoted by superscripts):

45 ALTERNATIVE POLICIES (denoted by superscripts): 1 Standard (henceforth, Color-Blind ) Allocation Rule: β cb i, and G cb i

46 ALTERNATIVE POLICIES (denoted by superscripts): 1 Standard (henceforth, Color-Blind ) Allocation Rule: βi cb, and Gi cb 2 Quota System: β q i, and Gq i

47 ALTERNATIVE POLICIES (denoted by superscripts): 1 Standard (henceforth, Color-Blind ) Allocation Rule: βi cb, and Gi cb 2 Quota System: β q i, and Gq i 3 Bid Subsidy System: βi bs, and Gi bs

48 ALTERNATIVE POLICIES (denoted by superscripts): 1 Standard (henceforth, Color-Blind ) Allocation Rule: βi cb, and Gi cb 2 Quota System: β q i, and Gq i 3 Bid Subsidy System: βi bs, and Gi bs 4 Ex-Post Utility Subsidies: β ep i, and G ep i

49 COLOR-BLIND GAME Under the color-blind rule, the equilibrium is the same as in equation (2), where F C (c) = ωf w (c) + (1 ω)f m (c). When allocations are color-blind, any bidder having private cost c will bid the same, regardless of his group since group affiliation is payoff-irrelevant. Thus, the equilibrium is given by (βm cb ) (c) = (βw cb ) (c) = (β cb ) ωf w (c) + (1 ω)f C (m) (c) = [ f P F 1 P (1 [ωf w(c) + (1 ω)f C (m)]) ] c, (βm cb )(c) = (βw cb )(c) = (β cb )(c) = 0. (3)

50 QUOTA SYSTEM INTRODUCTION Sets of prizes set apart for allocation to a specific group. For the seats allocated to a particular group, competition is within-group only (e.g., higher ed. admissions under Indian Reservation Law).

51 QUOTA SYSTEM INTRODUCTION Sets of prizes set apart for allocation to a specific group. For the seats allocated to a particular group, competition is within-group only (e.g., higher ed. admissions under Indian Reservation Law). FINITE GAME: On observing N m = M, the auctioneer selects a representative group of N m prizes for allocation to type-m bidders. N distributions of prizes reserved for each group will converge to F P with probability one.

52 QUOTA SYSTEM INTRODUCTION Sets of prizes set apart for allocation to a specific group. For the seats allocated to a particular group, competition is within-group only (e.g., higher ed. admissions under Indian Reservation Law). FINITE GAME: On observing N m = M, the auctioneer selects a representative group of N m prizes for allocation to type-m bidders. N distributions of prizes reserved for each group will converge to F P with probability one. INFINITE GAME: Group-specific payoff functions: Π q [ 1 i (b) = FP G q i (b) ], i = m,w.

53 NOTE: Π q i preserves the three key characteristics of real-world quotas:

54 NOTE: Π q i preserves the three key characteristics of real-world quotas: 1 It generates proportional allocation within any interval I = (p,p ) [0,p] Π q i is the limit of a scheme where the relative frequency of group-m bidders receiving prizes in I will be (1 ω) with probability approaching one as N gets large.

55 NOTE: Π q i preserves the three key characteristics of real-world quotas: 1 It generates proportional allocation within any interval I = (p,p ) [0,p] Π q i is the limit of a scheme where the relative frequency of group-m bidders receiving prizes in I will be (1 ω) with probability approaching one as N gets large. 2 Π q i implies that bidders from a given group compete only amongst themselves for prizes.

56 NOTE: Π q i preserves the three key characteristics of real-world quotas: 1 It generates proportional allocation within any interval I = (p,p ) [0,p] Π q i is the limit of a scheme where the relative frequency of group-m bidders receiving prizes in I will be (1 ω) with probability approaching one as N gets large. 2 Π q i implies that bidders from a given group compete only amongst themselves for prizes. 3 It preserves a meaningful notion of scarcity. Group-specific percentiles population percentiles AND monotonic equilibrium bidding Π q i will have a negative (positive) effect on the equilibrium payoffs of group-w (group-m) bidders, relative to color-blind allocation. Prize quality for type-m bidders cannot be improved without decreasing prize quality for type-w bidders.

57 In equilibrium, the utility for a type-i bidder who has cost c and bids b is F 1 [ P G q i (b) ] cb = F 1 ( [ P 1 Fi (β q i ) 1 (b) ]) cb. This is identical to payoffs in the symmetric case, except that F C has been replaced with F i. Thus, the equilibrium is given by (β q f i (c) i ) (c) = ( f P F 1 P [1 F i(c)] ) c, (4) β q i (c) = 0.

58 BID SUBSIDIES INTRODUCTION The auctioneer artificially augments bids from group M before comparisons are made (e.g., U of Michigan pre-2004).

59 BID SUBSIDIES INTRODUCTION The auctioneer artificially augments bids from group M before comparisons are made (e.g., U of Michigan pre-2004). FINITE GAME: The order statistics of transformed bids {b w1,...,b wnw,n,(b m1 + δ),...,(b mnm,n + δ)} are matched with the corresponding order statistics of prizes.

60 BID SUBSIDIES INTRODUCTION The auctioneer artificially augments bids from group M before comparisons are made (e.g., U of Michigan pre-2004). FINITE GAME: The order statistics of transformed bids {b w1,...,b wnw,n,(b m1 + δ),...,(b mnm,n + δ)} are matched with the corresponding order statistics of prizes. INFINITE GAME: Move type-m bidders ahead of counterparts who bid within δ: Π bs m (b) = F 1 Π bs w P (b) = F 1 P [ [ 1 (ωf w [ 1 ] [ ])] w ) 1 (b + δ) + (1 ω)f m (βm bs ) 1 (b), ] [ ])] + (1 ω)f m (βm bs ) 1 (b δ). (β bs (ωf w [ (β bs w ) 1 (b)

61 BID SUBSIDIES INTRODUCTION Important Observation: Π bs m (b) = Π bs w (b + δ). Let φ bs i (b) denote the inverse bid function for group i. Taking FOCs for each group and doing some algebra reveals φ bs m (b) = φbs w (b + δ). Substituting back into the objective function for group W and taking FOCs gives the following differential equation: ωf w (c) + (1 ω)f m (c) w ) (c) = [ f P F 1 P (1 [ωf w(c) + (1 ω)f m (c)]) ] c. (5) (β bs

62 BID SUBSIDIES INTRODUCTION Important Observation: Π bs m (b) = Π bs w (b + δ). Let φ bs i (b) denote the inverse bid function for group i. Taking FOCs for each group and doing some algebra reveals φ bs m (b) = φbs w (b + δ). Substituting back into the objective function for group W and taking FOCs gives the following differential equation: ωf w (c) + (1 ω)f m (c) w ) (c) = [ f P F 1 P (1 [ωf w(c) + (1 ω)f m (c)]) ] c. (5) (β bs NOTE: Eqn. (5) only valid where (φ bs m ) (b) exists and is negative.

63 BID SUBSIDIES INTRODUCTION Let c δ = φ bs w (δ) denote the cost for group W bidders who bid exactly δ. It can be shown that on the interval [c δ,c], we have β bs w (c) = β q w(c) and on [c,c δ [ ] the solution is Eqn. (5) with boundary condition φ q w (δ) ] = δ. For group M, the solution is β bs w β bs m (c) = max{β bs w (c) δ,0}.

64 EX-POST UTILITY SUBSIDIES Idea: Asymmetric bidding arises from asymmetric distributions. Could we implement some policy that would shift the weak distribution to lie on the strong one?

65 EX-POST UTILITY SUBSIDIES Idea: Asymmetric bidding arises from asymmetric distributions. Could we implement some policy that would shift the weak distribution to lie on the strong one? Assume the auctioneer wishes to maximize overall effort and (if possible) minimize the achievement gap. What if instead of altering allocation rules, the auctioneer implemented a bid-specific ex-post payment system in order to compensate type-m bidders for the additional disutility that they experience, relative to their W counterparts? Π ep m (b) = Π cb (b) + S(b) Π ep w (b) = Π cb

66 NORMAL (interior mode) σ = 1 µ m = 3.35, µ w = 2.65 [0,p] = [0,100], F P (p) = p 100, ω =.75, [c,c] = [1,5] PARETO (left mode) POWER (right mode) F i (c) = 1 c k i 1 5 k i, i = m,w k m = 1, k w = 4 F i (c) = ( ) c 1 θi 4, i = m,w θ m = 3, θ w = 1

67 Choosing δ I wish to compare quotas to bid subsidies, so choice of δ is important.

68 Choosing δ I wish to compare quotas to bid subsidies, so choice of δ is important. A quota rule ensures that the average prize awarded to each group is the same. Thus, choose δ so that the average prize allocated to both groups is also the same. Pareto: δ = (31.4% of maximum bid) Normal: δ = (33.1% of maximum bid) Power: δ = (29.5% of maximum bid)

69 Simulation Results (2,000,000 Bids) Table: % from Color-Blind Rule (2,000,000 simulated bids): Pareto Normal Power Color-Blind Effort (std err): 16.43(.0115) 11.06(.0067) 9.65(.0054) Minority Effort Quota: +15.0% +24.8% +27.4% Additive Bid Subsidy: -67.5% -74.3% -77.2% Color-Blind Effort (std err): 26.37(.0126) 17.21(.0076) 15.6(.0073) Non-Minority Effort Quota: -1.7% -4.4% -4.2% Additive Bid Subsidy: -6.2% -7.9% -7.6% Achievement Gaps Quota: -29.4% -56.8% -55.3% Additive Bid Subsidy: +95.3% % % Population Effort Quota: +1.1% +0.8% +1.2% Additive Bid Subsidy: -16.7% -19.6% -19.5%

70 Pareto Private Costs CDF DISTRIBUTIONS BIDS NON MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) w β bs (c) (SUBSIDY) w 0.2 F w (c) F m (c) PRIVATE COSTS PRIVATE COSTS DENSITIES f w (c) f m (c) MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) m β bs (c) (SUBSIDY) m 40 PDF 1 BIDS PRIVATE COSTS PRIVATE COSTS

71 Pareto Private Costs 1 NON MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: COLOR BLIND CDF G cb (b) (COLOR BLIND) w G q (b) (QUOTA) w G bs (b) (SUBSIDY) w BIDS CDF G cb w (b) G cb m (b) BIDS 1 MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: QUOTA vs. SUBSIDY CDF G cb (b) (COLOR BLIND) m G q (b) (QUOTA) m G bs (b) (SUBSIDY) m BIDS CDF G q w (b) G q m (b) G bs w (b) G bs m (b) BIDS

72 Normal Private Costs CDF DISTRIBUTIONS BIDS NON MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) w β bs (c) (SUBSIDY) w 0.2 F w (c) F m (c) PRIVATE COSTS PRIVATE COSTS DENSITIES f w (c) f m (c) MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) m β bs (c) (SUBSIDY) m PDF BIDS PRIVATE COSTS PRIVATE COSTS

73 Normal Private Costs 1 NON MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: COLOR BLIND CDF G cb (b) (COLOR BLIND) w G q (b) (QUOTA) w G bs (b) (SUBSIDY) w BIDS CDF G cb w (b) G cb m (b) BIDS 1 MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: QUOTA vs. SUBSIDY CDF G cb (b) (COLOR BLIND) m G q (b) (QUOTA) m G bs (b) (SUBSIDY) m BIDS CDF G q w (b) G q m (b) G bs w (b) G bs m (b) BIDS

74 Power Private Costs CDF DISTRIBUTIONS BIDS NON MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) w β bs (c) (SUBSIDY) w 0.2 F w (c) F m (c) PRIVATE COSTS PRIVATE COSTS f w (c) f m (c) DENSITIES MINORITY BIDDING β cb (c) (COLOR BLIND) β q (c) (QUOTA) m β bs (c) (SUBSIDY) m PDF BIDS PRIVATE COSTS PRIVATE COSTS

75 Power Private Costs 1 NON MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: COLOR BLIND CDF G cb (b) (COLOR BLIND) w G q (b) (QUOTA) w G bs (b) (SUBSIDY) w BIDS CDF G cb w (b) G cb m (b) BIDS 1 MINORITY BID DISTRIBUTIONS CROSS GROUP COMPARISSON: QUOTA vs. SUBSIDY CDF G cb (b) (COLOR BLIND) m G q (b) (QUOTA) m G bs (b) (SUBSIDY) m BIDS CDF G q w (b) G q m (b) G bs w (b) G bs m (b) BIDS

76 QUOTAS VS. BID SUBSIDIES OBSERVATIONS:

77 QUOTAS VS. BID SUBSIDIES OBSERVATIONS: 1 Additive bid subsidies lead to a leftward shift of the minority bid distribution and the non-minority bid distribution. 2 Additive bid subsidy rule has no allocational effect among players with costs above c δ. 3 Additive bid subsidy increases the achievement gap

78 QUOTAS VS. BID SUBSIDIES OBSERVATIONS: 1 Additive bid subsidies lead to a leftward shift of the minority bid distribution and the non-minority bid distribution. 2 Additive bid subsidy rule has no allocational effect among players with costs above c δ. 3 Additive bid subsidy increases the achievement gap 4 Quota rule increases minority effort, decreases non-minority effort slightly, and increases overall population effort 5 Quota rule decreases the achievement gap

79 QUOTAS VS. BID SUBSIDIES (cont d) OBSERVATIONS: Proposition: For either a quota or an additive bid subsidy, there exists c (c,c) such that the acheivement gap is widened for c [c,c ]. Proof: Under construction, see previous Figures for illustration.

80 QUANTITATIVE QUESTIONS 1 What is the nature of the asymmetries in the cost distributions among minorities and non-minorities? 2 What effect do actual AA policies have on observed achievement gaps? 3 Which form of AA would lead to a more favorable outcome in terms of gaps and overall average test scores? Counterfactual exercise

81 Estimation

82 Estimation GOAL: To recover F M and F W, the latent private-cost distributions

83 Estimation GOAL: To recover F M and F W, the latent private-cost distributions TASKS: 1 Estimate F P Using data and methodology developed by US News & World Report ( America s Best Colleges rankings) 2 Estimate δ Chung and Espenshade (2005) National Study of College Experience (NSCE) 3 Estimate F i with a structural model, using ε-strategy approximation Roughly 1.8Million US college applicants in 2005 Using data on SAT/ACT scores, back out cost distributions using equilibrium conditions.

84 Racial Achievement Gaps in Secondary and Higher Education Hot topic in various disciplines

85 Racial Achievement Gaps in Secondary and Higher Education Hot topic in various disciplines Several studies show that average standardized test scores in reading & math for American minorities lag significantly behind those of whites. GAP RESULTS

86 Racial Achievement Gaps in Secondary and Higher Education Hot topic in various disciplines Several studies show that average standardized test scores in reading & math for American minorities lag significantly behind those of whites. GAP RESULTS It is believed that this contributes to low college enrollment for American minorities. I assume that for minorities it is more costly on average to achieve a given standardized test score due to exogenous disparities in educational funding/quality

87 Motivation for Asymmetry Assumption in US White Population Density Black Population Density (# persons/congressional district) (proportional bins)

88 Motivation for Asymmetry Assumption in US White Population Density (# persons/congressional district) Black Population Density (proportional bins) If white and black populations in the US were distributed similarly, we would see similar coloring in the two maps...

89 Motivation for Asymmetry Assumption in US Out of 14 states that are dominated by large sections of dark orange (TX, AR, TN, LA, MS, AL, GA, FL, SC, NC, VA, MD, DE, NJ), 11 spend less per-pupil on K-12 education than the national average. ASYMMETRIC Black Population Density

90 Empirical Documentation of the Racial Test Score Gap: GAP MAP LEE Educational Researcher, 2002 Document the evolution of racial standardized test-score gaps during 70s, 80s and 90s for various age groups and various performance levels for 13-year-olds Gaps generally plummeted during 70s and 80s, but leveled or rose during 90s Gaps remain at significant levels FRYER & LEVITT Review of Economics and Statistics2004 After controlling for student-specific socioeconomic characteristics, a significant test score gap is found after 2 years of school, whereas none existed prior to schooling. Fryer and Levitt test various hypotheses which might explain the gap and find that differing school quality is the only hypothesis which is not rejected.