A characterization of "Phelpsian" statistical discrimination
aa r X i v : . [ ec on . T H ] A ug A CHARACTERIZATION OF “PHELPSIAN”STATISTICAL DISCRIMINATION
CHRISTOPHER P. CHAMBERS AND FEDERICO ECHENIQUE
Abstract.
We establish that statistical discrimination is possi-ble if and only if it is impossible to uniquely identify the signalstructure observed by an employer from a realized empirical dis-tribution of skills. The impossibility of statistical discrimination isshown to be equivalent to the existence of a fair, skill-dependent,remuneration for workers. Finally, we connect the statistical dis-crimination literature to Bayesian persuasion, establishing that ifdiscrimination is absent, then the optimal signaling problem resultsin a linear payoff function (as well as a kind of converse). Introduction
In seminal contributions, Arrow (1971; 1973) and Phelps (1972) pos-tulated that discrimination along racial lines, or gender identities, canhave a statistical explanation. In this note we focus on Phelps’ no-tion of statistical discrimination: on the idea that two populations ofworkers, who are in essence identical, may have different economic re-munerations for purely informational reasons. Phelps’ theory connects worker remuneration with the distribution ofsignals that can be observed about worker skills. Phelps assumes a firmwho observes a signal about the underlying skills of a worker. The firmobserves the signal before assigning the worker to a task. The worker (Chambers)
Department of Economics, Georgetown University (Echenique)
Division of the Humanities and Social Sciences, CaliforniaInstitute of Technology
Echenique thanks the NSF for support through the grants SES-1558757 and CNS-1518941. We are grateful to Leeat Yariv for comments on a previous draft. We follow the interpretation of Phelps’ model due to Aigner and Cain (1977).Arrow’s theory of statistical discrimination relies on a coordination failure, andis quite different from Phelps’. Statistical discrimination stands in contrast withtaste-based discrimination, as in Becker (1957). is paid her expected contribution to the firm, conditional on the firm’sobserved signal about the worker. (A competitive market ensures thatworkers are paid their contributions.) Consider now two populationsof workers: group A and group B. If the signal is more informativefor As than for Bs, then (the argument goes), a worker from group Amay be ex-ante more valuable to the firm than a B worker. This isbecause the additional information about the A worker may be usedto better assign her a task matching her skills. Even more, the signalmay be the result of a test that has been designed with a populationfrom group A in mind. The signal implemented by the test will thenbe more informative about the skills of a prospective A worker than aB worker. As a consequence of the difference in informativeness, the firm mayvalue a group A worker over a group B worker. We formulate the theoryof statistical discrimination using the language of the recent literatureon informational design. A firm observes a signal about a worker’sskills, and bases both the assigned task and the payment to the workeron the revenue it expects to gain from the action taken by the workerat the firm. A group of workers comes with a distribution over signals:an information structure. The distribution over signals of group A maybe more informative than the distribution over signals of group B. Wesay that statistical discrimination is present if two groups of workers,each group having their distinct distribution over signals, but the samedistribution of skills, receive different payments in expectation.Our contribution is to connect statistical discrimination with twoseemingly distinct properties of the economic environment: one is iden-tification (in the econometric sense) of signals from skills, and the otheris the linearity of firm revenue in “fair” skill-dependent payoffs. First,we show that the absence of statistical discrimination is equivalent to As an example, Aigner and Cain cite evidence from the education literature tothe effect that the SAT is less informative about the abilities of African-Americanstudents than it is for white students.
TATISTICAL DISCRIMINATION 3 the econometric identification of signals. Specifically, we prove that sta-tistical discrimination is not possible if and only if every given distribu-tion of skills arises from a unique distribution of signals. By definition,when discrimination is possible, the identification property must be vi-olated. Our contribution is in the converse: whenever identification isimpossible, discrimination can arise.Second, we show that identification, and therefore the absence ofdiscrimination, is equivalent to the existence of a fair skill-based remu-neration for workers. Workers’ payments are a linear combination ofthe fair remunerations. Each list of skills must be associated with avalue, which is independent of any signals, and every worker is paidthe expectation according to the distribution of skills inherent in herrealized signal.Finally, we show that the optimal information structure in the senseof Kamenica and Gentzkow (2011) achieves precisely the fair remuner-ation in our results. 2.
The model
Notation.
A set is binary it is has one or two elements. If A is aclosed subset of a Euclidean space, we denote by ∆( A ) the set of Borelprobability measures on A .2.2. The model.
The model involves a firm and a worker. The firmfaces uncertainty over the revenues it can obtain from the worker’sactions. The firm’s revenue depends on the worker’s skills, and howthose skills match up with the technology of the firm.Let Θ denote a finite set of uncertain states of the world ; thesestates represent the skill set of the worker, and are unknown to thefirm. The firm asks the worker to undertake some action, and it onlycares about the state-contingent payoff that results from the worker’saction. Formally, then, an action is an element a ∈ R Θ . Thus, thetask of the firm is to properly match a worker to an action with theappropriate skill set. CHAMBERS AND ECHENIQUE
There is a closed set of signals , or payoff-relevant types , S ⊆ ∆(Θ).Here we identify signals with the posterior distribution that they induceover Θ. The firm observes s ∈ S before asking the worker to undertakean action. Thus, the goal of the firm is to choose the appropriateaction for the appropriate worker, after a signal of worker skill hasbeen observed.The firm solves the following problem. For a given s ∈ S , and finiteset of actions A , v A ( s ) ≡ max a ∈ A X θ ∈ Θ a ( θ ) s ( θ ) . Given signal s ∈ S , v A ( s ) is the maximal expected revenue the em-ployer can achieve. We maintain the assumption that labor marketsare competitive, and therefore a worker of type s is paid the revenue v A ( s ) that she generates for the firm. This is as in Phelps (1972) andAigner and Cain (1977). Observe that v A is the “value function” of A ,as in Blackwell (1953) or Machina (1984), and is thus always convex.A probability π ∈ ∆( S ) is an information structure . It induces aprobability over Θ via: p π ( θ ) = R S s ( θ ) dπ ( s ). For a set E ⊆ S , we caninterpret π ( E ) as an empirical frequency of individuals who generatesignals s ∈ E . The empirical frequency π then generates an empiricalfrequency of skills, which is p π .We say that the set of signals S is non-discriminatory if for anyinformation structures π, π ′ ∈ ∆( S ), and any finite set A ⊆ R Θ , if p π = p π ′ , then Z S v A ( t ) dπ ( t ) = Z S v A ( t ) dπ ′ ( t ) . Interpret π ( E ) as the frequency of individuals of type E ⊆ S . Underthe competitive markets assumption, the set S being non-discriminatorymeans that the average remuneration paid to a class of workers with dis-tribution π ultimately depends only on the distribution of their skills.2.3. Motivation and a Phelpsian example.
We start by a simpleexample to recreate the point made by Phelps (1972). It is a min-imal example; the simplest we can think of that delivers the Phelp-sian message. Let Θ = { θ , θ , θ } be the set of states, and A = TATISTICAL DISCRIMINATION 5 { (1 , , , (0 , / , } be the set of available actions. Observe thatwith this specification, workers are not “high” or “low” quality, butthey simply have differing aptitudes for the available actions.Suppose that S = { (1 , , , (1 / , / , , (0 , / , / , (0 , , } is the set of signals, or worker types.Consider two information structures, π and π ′ , described in the tablebelow, together with the profit function v A resulting from our assumedΘ and A : t = (1 , , t = (1 / , / , t = (0 , / , / t = (0 , , π ( t ) 1 / / π ′ ( t ) 0 2 / / v A ( t ) 1 1 / / π , while Bs distributionover signals is π ′ . Observe that p π = p π ′ = (1 / , / , / a = (1 , ,
0) or action a = (0 , / , t = (1 / , / ,
0) tellsthe employer that a is the optimal choice given the information athand, but leaves the employer with some doubts as to whether a mayhave been the optimal action. In consequence, we have Z T v A ( t ) dπ ( t ) = 1 / / > / Z T v A ( t ) dπ ′ ( t ) . If workers are paid according to the revenues that they contribute tothe firm, as would be the case in a competitive market, then A workersare paid more than B workers in aggregate. The differences in expected
CHAMBERS AND ECHENIQUE (or population) remuneration between the two is purely a consequenceof the informational content in their corresponding signal structures.In our example of Phelpsian statistical discrimination, the two dif-ferent information structures have the same mean. This is a necessaryrequirement for the existence of statistical discrimination. It is impor-tant to point out, however, that skill can always be inferred from wages,even when there is discrimination. We present Proposition 1 to makethis point.For a set of actions A = { a , . . . , a n } , and action k , let A + k = { a + k, . . . , a n + k } . Proposition 1.
For any S and any set of actions A , if π, π ′ ∈ ∆( S ) for which p π = p π ′ , then there is k for which Z T v A + k ( t ) dπ ( t ) = Z T v A + k ( t ) dπ ′ ( t ) . When discrimination is impossible.
Our discussion suggeststhat discrimination is tied to identification. Skills are always identifiedfrom payoffs, even when there is discrimination (Proposition 1). Theproblem is the converse identification: Here we show that the absence ofdiscrimination is equivalent to the ability to estimate skills from signals.Importantly, we show that this can only happen when payments arelinear in signals. So the absence of discrimination is equivalent tothe existence of a state-dependent, signal-independent, “fair” payoff.Payments equal the expected value of such a payoff, and are called fairvaluations.We say that S• is identified if for any π, π ′ ∈ ∆( S ), if p π = p π ′ , then π = π ′ ; • admits fair valuations if for any finite subset A ⊆ R Θ , there is α A ∈ R Θ for which for all t ∈ S , v A ( t ) = X θ α A ( θ ) t ( θ ) . • admits fair valuations for binary sets if for any binary subset A ⊆ R Θ , there is α A ∈ R Θ for which for all t ∈ S , v A ( t ) = P θ α A ( θ ) t ( θ ). TATISTICAL DISCRIMINATION 7
The notion that S admits fair valuations captures the idea that anyindividual is paid according to her expected skill. Thus, for A , α A ( θ )represents the value to the firm with technology A of skill set θ ∈ Θ,and if an individual sends signal s then she is paid the expected valueof α A according to s . Importantly, if π ∈ ∆( S ), then Z v A ( s ) dπ ( s ) = α A · Z sdπ ( s ) = α A · p π . So, under fair valuations, the expected payment to a population ofagents with information structure π only depends on the distributionof skills in that population.Finally, say that S is non-discriminatory for binary sets if for any π, π ′ ∈ ∆( S ) and any binary A ⊆ R Θ , if p π = p π ′ , then Z S v A ( t ) dπ ( t ) = Z S v A ( t ) dπ ′ ( t ) . Theorem 2.
The following are equivalent. (1) S is non-discriminatory. (2) S is non-discriminatory for binary sets. (3) S is identified. (4) S admits fair valuations. (5) S admits fair valuations for binary sets. The main import of Theorem 2 is that there is a α A , independent ofthe signal s , so that the optimal contribution of the worker to the firmis the expected value of α A . The worker is therefore remunerated ac-cording to some “fundamental” value α A , and receives the expectationof α A according to the signal s . Proposition 3. If S admits fair valuations, then for each finite A ⊆ R Θ and corresponding α A ∈ R Θ , we have for every s ∗ ∈ S : X θ α A ( θ ) s ∗ ( θ ) = inf { X θ y ( θ ) s ∗ ( θ ) : y ∈ R Θ and v A ( s ) ≤ X θ y ( θ ) s ( θ ) ∀ s ∈ S} . Proposition 3 means that the value of a worker with type s ∗ to thefirm is the minimum expected payment that guarantees the workera payoff of at least v A ( s ), for all signals s ∈ S . This is a kind of CHAMBERS AND ECHENIQUE participation, or individual rationality, constraint. The worker may beable to guarantee a payment of v A ( s ) on the market, if her signal is s ,and thus a firm must guarantee at least v A ( s ) in its choice of the “fair”payoff α A ∈ R Θ .2.5. Connection to Bayesian persuasion.
The recent literature onBayesian persuasion (Kamenica and Gentzkow (2011)) deals with theoptimal design of information structures. It turns out that the valueof optimal information design is linear if and only if S admits no dis-crimination.We now focus a bit more in depth on the notion of signal structure.As in Blackwell (1953), there is a natural notion of “comparative in-formativeness” for π, π ′ ∈ ∆( S ). We say that π is more informative than π ′ if for every A , R v A ( t ) dπ ( t ) ≥ R v A ( t ) dπ ′ ( t ). Most economistswill have heard of the notion of a “mean-preserving spread;” π turnsout to be more informative than π ′ if it consists of a mean-preservingspread of π ′ .We know that optimal information design will never utilize signalstructures that are dominated according to the more informativenessorder. As a result, optimal information structures will place probabilityzero on signals that can be obtained as the mean of other signals.Formally, an optimal information structure will have support on theextreme points of the convex hull of S .Now, let T be the closed convex hull of S . An information structureis any probability measure π ∈ ∆( T ). Then define W A : T → R via W A ( s ) ≡ max { Z T v A (˜ s ) dπ (˜ s ) : π ∈ ∆( T ) and s = Z T ˜ sdπ (˜ s ) } .W A ( s ) is the value of an optimal information structure for a populationwith skill distribution s . In the following, ∂T denotes the extremepoints of T ; those points which are not convex combinations of otherpoints in T .Return to our motivating “Phelpsian” example. There, discrimina-tion was present even though S consisted of the extreme points of itsconvex hull T , and thus S was maximally informative. Phelps’ original TATISTICAL DISCRIMINATION 9 point can thus be refined: discrimination obtains because an employerhas “different” information about two classes of individuals, rather than“better” information.Let us see how this manifests itself in the choice of optimal infor-mation structure. In this case, for each s ∈ S , we have (clearly) W A ( s ) = v A ( s ), as each s is extreme in the convex hull of T . Wetherefore obtain: (2 / W A (1 / , / ,
0) + (1 / W A (0 , ,
1) = < =(1 / v A (1 , ,
0) + (2 / v A (0 , / , / ≤ W A (1 / , / , / W A is nonlinear in this case. This is a general artifact ofnon-identification and discrimination, as is evidenced by the followingresult. Corollary 4.
For any S , ∂T is non-discriminatory iff for every A , W A is affine (linear). As in Kamenica and Gentzkow (2011), W A is always weakly concave,which admits the possibility that it is affine. Corollary 4 says thatdiscrimination is possible exactly when W A exhibits strict concavities.3. Conclusion
We have formulated Phelps’ theory of statistical discrimination us-ing the modern language of information design. Our results shed newlight on the nature of discrimination, and on some of the empiricalapproaches one might take to establish the existence of statistical dis-crimination.Statistical discrimination turns out to be equivalent to the absenceof econometric identification of signals from skills. While the identifi-cation of skills from salaries is always possible, even in the presence ofdiscrimination, we show that the crucial identification property is thatof signals from skills.In second place, we connect discrimination with the source of workerremunerations. We show that identification is impossible if and only if Because the domain of W A is a set of probability measures, W A is linear if it isaffine. In fact, in this case we have W A ( s ) = P θ ∈ Θ α A ( θ ) s ( θ ), where α A is as inProposition 3. remunerations are linear in “fair” skill-dependent, signal-independent,payoffs.Our results have immediate consequences for empirical research ondiscrimination. They imply that discrimination is absent if and only ifempirical approaches to linearly estimating fair skills-based payoffs areviable. 4. Proofs
Let T be the closed convex hull of S . Recall that ∂T denotes theextreme points of T . The definition of v A extends to T . Let Y A : T → R be the concave envelope of v A , defined as the pointwise infimum of theaffine functions that dominate v A . So if A ( T ) denotes the space of allaffine functions on T , then v A ( t ) = inf { l ( t ) : l ∈ A ( T ) and v A ≤ l } .Recall the definition of W A from Section 2.5. Lemma 5. Y A = W A Proof.
Let l : T → R be an affine function and v A ≤ l . For any π ∈ ∆( T ) with R T qdπ ( q ) = p , Z T v A ( q ) dπ ( q ) ≤ Z T l ( q ) dπ ( q ) = l (cid:18)Z T qdπ ( q ) (cid:19) = l ( p ) , as l is affine. Thus W A ≤ l , as π was arbitrary. This implies that W A ≤ Y A , as l was arbitrary.Now suppose that W A ( p ) < Y A ( p ). Recall that W A is concave. Thenthe set D = { ( q, y ) ∈ T × R : y ≤ W A ( q ) } is closed and convex, so thereexists α with ( q, y ) · α ≤ ( p, W A ( p )) · α < ( p, y ′ ) · α for all ( q, y ) ∈ D and all y ′ ≥ Y A ( p ). Write α = ( α , α ) ∈ R Θ × R . Clearly we cannothave α = 0 as ( p, W A ( p ) ∈ D . Consider the affine function l : T → R defined by q (1 /α )(( p, W A ( p )) · α − α · q ) . This means that l ( p ) = W A ( p ) < Y A ( p ). Moreover, for any q ∈ T , α · ( q, W A ( q )) ≤ α · ( p, W A ( p )); hence, l ( q ) = (1 /α ) α · p + W A ( p ) − (1 /α ) α · q ≥ W A ( q ) ≥ v A ( q ) , TATISTICAL DISCRIMINATION 11 where the last inequality follows from the definition of W A . Then l ∈A ( T ), v A ≤ l , and l ( p ) < Y A ( p ); a contradiction. (cid:3) Proof of Theorem 2.
By the Choquet-Meyer Theorem (Theo-rem II.3.7 in Alfsen (2012) or p. 56-57 in Phelps (2000)), T is a simplexiff ∂T is identified.Now, to prove the theorem: it is obvious that 3 = ⇒ ⇒
2. Weshall prove that 2 = ⇒
3. To this end, let S be non-discriminatory forbinary menus. The proof that 2 = ⇒ S = ∂T . The second is that T must be a simplex.First, it is obvious by definition of T that ∂T ⊆ S . So we prove that S ⊆ ∂T . To this end, suppose by means of contradiction that there is s ∗ ∈ S for which there are t, t ′ ∈ T , t = t ′ , and γ ∈ (0 ,
1) for which s ∗ = γt + (1 − γ ) t ′ . Let f = ( s ∗ − t ) + [ t · s ∗ − s ∗ · s ∗ ] and g = − f .Observe that f · s ∗ = 0, g · t = − t · ( s ∗ − t ) − s ∗ · ( t − s ∗ ) > f · t ′ = ( s ∗ − t ) · ( t ′ − s ∗ ) = γ (1 − γ )( t ′ − s ∗ ) · ( t ′ − s ∗ ) > A ≡ { f, g } . Then we obtain that v A ( t ) ≥ g · t > v A ( t ′ ) ≥ f · t ′ >
0, while v A ( s ∗ ) = 0 (as f · s ∗ = g · s ∗ = 0).Now, for each of t, t ′ , there are finitely supported (by Caratheodory’stheorem) π t and π t ′ on ∂T (so in particular on S ) for which t = R S sdπ ( s ) and t ′ = R S sdπ ′ ( s ). This means that R S v A ( s ) dπ ( s ) ≥ v A ( t ) > R S v A ( s ) dπ ′ ( s ) ≥ v A ( t ′ ) >
0, as v A is convex. Then Z S v A ( s ) d ( γπ + (1 − γ ) π ′ )( s ) > . But this contradicts 2 as R S sd ( γπ + (1 − γ ) π ′ )( s ) = γt + (1 − γ ) t ′ = s ∗ and v A ( s ∗ ) = 0.So we have shown that S = ∂T , and we turn to the proof that T is a simplex (and thus S identified). By Alfsen (2012) Theorem II.4.1,since T is convex and compact, T is a simplex if and only if A ( T ) formsa lattice in the usual (pointwise) ordering on functions. So, suppose bymeans of contradiction that A ( T ) does not form a lattice. Then, thereare f, g ∈ A ( T ) which possess no supremum in A ( T ). Lemma 6.
Let f, g ∈ A ( T ) . For any z ∈ ∂T , if f ( z ) ≥ g ( z ) , thenthere is h ∈ A ( T ) for which h ≥ f, g and h ( z ) = f ( z ) .Proof. Let M be the subgraph of the concave envelope of v { f,g } . Ob-serve by definition that it is the convex hull of the points { ( z, v { f,g } ) : z ∈ S} , so that it is polyhedral (Corollary 19.I.2 of Rockafellar (1970)).Therefore, by definition of polyhedral concave function, there is h sup-porting it at ( z, f ( z )). (cid:3) From Lemma 6, and the fact that f and g possess no supremumin A ( T ), it follows that there is no affine function h for which forall z ∈ ∂T , h ( z ) = max { f ( z ) , g ( z ) } . Consequently, if A ≡ { f, g } ,then Y A is not affine, since for all z ∈ ∂T , it follows that Y A ( z ) =max { f ( z ) , g ( z ) } = v A ( z ). Now, Y A being concave and not affine meansthat there is ˆ π ∈ ∆( T ) with R S Y A ( q ) d ˆ π ( q ) < Y A ( p ˆ π ). Since S = ∂T ,and Y A is concave, we can in fact find (by Lemma 4.1 in Phelps (2000)) π ∈ ∆( S ) with p ˆ π = p π and Z S v A ( q ) dπ ( q ) = Z S Y A ( q ) dπ ( q ) ≤ Z S Y A ( q ) d ˆ π ( q ) < Y A ( p ˆ π ) = Y A ( p π ) , where the first equality follows from v A ( q ) = Y A ( q ) for q ∈ S , and thesecond inequality from the choice of π .Now, by Lemma 5, Y A ( p π ) = sup { R v A ( q ) d ˜ π ( q ) : ˜ π ∈ ∆( T ) and p ˜ π = p π } . Then there is π ′ ∈ ∆( S ) (as the sup is achieved for a measure withsupport in ∂T = S ) with p π = p π ′ and R S v A ( q ) dπ ( q ) < R S v A ( q ) dπ ′ ( q ),contradicting the fact that S is non-discriminatory for binary menus.4.2. Proof of Proposition 3.
The Lagrangian for the maximizationproblem in the definition of W A is L ( π, λ ) = Z T v A ( t ) dπ ( t ) + λ · (cid:20) p − Z T qdπ ( q ) (cid:21) = λ · p + Z T ( v A ( t ) − λ · p ) dπ ( t )and apply the maximin theorem (see for example Theorem 6.2.7 inAubin and Ekeland (2006), which applies here because ∆( T ) is com-pact). TATISTICAL DISCRIMINATION 13
Proof of Proposition 1.
Observe that for any A and any action l , we have v A + l ( t ) = v A ( t ) + l · t . Now, since p π = p π ′ , there is l forwhich l · p π = l · p π ′ . Consequently, there is α for which: αl · ( p π − p π ′ ) = Z T v A ( t ) dπ ′ ( t ) − Z T v A ( t ) dπ ( t ) . Let k = αl , and conclude that: Z T v A + k ( t ) dπ ( t ) = k · p π + Z T v A ( t ) dπ ( t ) = k · p π ′ + Z T v A ( t ) dπ ′ ( t ) = Z T v A + k ( t ) dπ ′ ( t ) . Proof of Corollary 4.
By the Choquet-Meyers Theorem (The-orem II.3.7 in Alfsen (2012)) T is a simplex iff the concave envelopeof every lower semicontinuous and convex function is affine. Clearly,when S is identified, T is a simplex, and since v A is convex and lowersemicontinuous, we obtain that W A = Y A , the concave envelope. So W A is affine.Conversely, suppose that W A is affine for each finite A . We willshow that T is a simplex (so that ∂T forms the vertices of a simplex,and is identified). But this again follows from the fact that W A is thesmallest concave function on T dominating each a ∈ A . Since it isaffine, it follows that A ( T ) is a lattice, and hence T is a simplex. ReferencesAigner, D. J. and G. G. Cain (1977): “Statistical theories of dis-crimination in labor markets,”
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