Decline of Class: A Group-Theoretic Approach

Transcription

1 : A Group-Theoretic Approach Y. Stephen Chiu Weifeng Zhong University of Hong Kong Northwestern University APET workshop in Bangkok June 2011

2 Motivation Composition of a society According to people s characteristics (race, religion, etc) or their views (pro-life vs pro-choice; etc) Which dimension is more important? Why?

3 Motivation Declining importance of economic classes in politics (esp. in developed countries). Roemer s (1998) two-dimensional preference model. Is the salience of non-class issues "manufactured" by the elitists to divide the working class? salience of racial/ethnic conflicts (esp. in the third world). Race "rigidity" (Fearon, 1999; Caselli & Coleman II, 2010). Capital-labor complementarity (Esteban & Ray, 2008). to provide a complementary theory to understand this "decline of class" phenomenon

4 Motivation a community populated by agents with different characteristics (both econ and non-econ) each agent is concerned about his material payoff v, as well as his psychological payoff α his total utility u = v + α his psychological payoff depends on with whom he interacts (for how long; how often) his bargaining power is affected => a material foundation of psychology allows us to discuss the incentive to change a group s psychology, or to manipulate another group s psychology, or to form an alliance

5 Main results in a framework with four groups with two dimensions: black versus white; capitalists versus workers, we find that 1 when capitalists are predominantly of the same race, say, white, alliance between W capitalists and W workers exists if the white population is below a critical proportion, and no alliances of any kind exist o.w. 2 alliance based on race is more commonplace than alliance of workers across race 3 asymmetry between alliance of (B & W) workers and alliance of (B & W) capitalists. While the former exists under some conditions, the latter never.

6 Set-up Related Literature Preliminaries Exogenous Set-up: Groups and Production n groups, denoted by 1,, n, of individuals with respective sizes s 1,, s n.

7 Set-up Related Literature Preliminaries Exogenous Set-up: Groups and Production n groups, denoted by 1,, n, of individuals with respective sizes s 1,, s n. Groups produce output according to characteristic function v (). Assumption 1: v () is strictly superadditive, i.e., v (R 1 [ R 2 ) > v (R 1 ) + v (R 2 ), where R 1, R 2 are any disjoint unions of groups.

8 Exogenous Set-up: Identity Set-up Related Literature Preliminaries a i (a i1,, a in ) 2 R n represents group i s identity. a ij measures group i s amicability (or hostility) towards group j.

9 Exogenous Set-up: Identity Set-up Related Literature Preliminaries a i (a i1,, a in ) 2 R n represents group i s identity. a ij measures group i s amicability (or hostility) towards group j. Group i in coalition R obtains psychological payoff s j α i (R) s i j2r a s ij, R where s R = j2r s j.

10 Exogenous Set-up: Identity Set-up Related Literature Preliminaries a i (a i1,, a in ) 2 R n represents group i s identity. a ij measures group i s amicability (or hostility) towards group j. Group i in coalition R obtains psychological payoff s j α i (R) s i j2r a s ij, R where s R = j2r s j. Suppose R = f1, 2g, s 1 = 100 and s 2 = 300. α 1 (R) a a 12,

11 Exogenous Set-up: Identity Set-up Related Literature Preliminaries Coalition R obtains total payoff by total characteristic function u (R) v (R) + j2r α j (R). Assumptions 1: v () is strictly superadditive, i.e., v (R 1 [ R 2 ) > v (R 1 ) + v (R 2 ), where R 1, R 2 are any disjoint unions of groups. Assumption 2: u () is also strictly superadditive. Grand coalition f1,, ng, also denoted by N, is always formed.

12 Solution Concept Set-up Related Literature Preliminaries Group i obtains its own Shapley value (1953) φ i T:i/2T jtj! (n jtj 1)! n! (u (T [ i) u(t)). Group i s material payoffs is β i φ i α i (N).

13 Solution Concept Set-up Related Literature Preliminaries Group i obtains its own Shapley value (1953) φ i T:i/2T jtj! (n jtj 1)! n! Group i s material payoffs is β i φ i α i (N). Nomenclature: groups, (sub)coalition, alliance (u (T [ i) u(t)). Comparative Statics: the effect of (a 1,, a n ) on (β 1,, β n )

14 Related Literature Set-up Related Literature Preliminaries Long-documented: "ethnocentrism" (Sumner, 1906) "homophily" (Lazarsfeld and Merton, 1954) self-categorization theory (social psychology) Economics of identity: Akerlof and Kranton (2000, 2002, 2005) Bénabou and Tirole (2010) Psychological payoffs: Bernheim (1994) Alesina and La Ferrara (2000) Luttmer (2001) Currarini, Jackson, and Pin (2009)

15 Set-up Related Literature Preliminaries Material foundation of psychology Proposition For any distinct i, j, k with s i, s j, s k > 0; 1 β i / a ii > 0; 2 β i / a jj < 0; 3 β i / a ji > 0; 4 β i / a jk < 0; 2 < > 5 β i / a ij < 0 if s n(n 1)+2 s N i + s j > 2 1 n n 1 2 s N.

16 Set-up Related Literature Preliminaries How do comparative statics work? E.g.: the sign of β i / a ii

17 Set-up Related Literature Preliminaries How do comparative statics work? E.g.: the sign of β i / a ii φ i jtj! (n jtj 1)! u (T [ i) = a ii T:i/2T n! a ii! jtj! (n jtj 1)! s = 2 i T:i/2T n! s T + s i > 0 u(t) a ii

18 Set-up Related Literature Preliminaries How do comparative statics work? E.g.: the sign of β i / a ii φ i jtj! (n jtj 1)! u (T [ i) = a ii T:i/2T n! a ii! jtj! (n jtj 1)! s = 2 i T:i/2T n! s T + s i > 0 u(t) a ii β i a ii = φ i a ii α i (N) a ii = T:i/2T jtj! (n jtj 1)! n! = T:i/2T jtj! (n jtj 1)! n!! s 2 i s 2 i s T + s i s 2 i s T + s i s 2 i s N s N! > 0

19 Set-up Related Literature Preliminaries How do comparative statics work? β i jtj! (n jtj 1)! = a ii T:i/2T n! s 2 i s T + s i s 2 i s N! > 0 The function v per se does not appear in the comparative statics, hence does not play a role The addition into or elimination from the economy of a zero-size group does not affect the comparative statics: β i / a jk remains the same for β i, increasing a ii by one unit is equal to reducing a ij for all j 6= i by one unit: β i β = a ii i. a j6=i ij

20 Set-up Related Literature Preliminaries Material foundation of psychology A "material foundation" of identity. If they could, groups have an incentive to change identities changing identity coefficients Related literature: manipulation of beliefs Mui (1999), Glaeser (2005), Bénabou and Tirole (2010).

21 Speci c Context 2 races (black B & white W) and 2 classes (capitalists K, workers L) Total population is normalized to j s j = 1. We first assume s BK = 0. Then the economy can be represented by a unit simplex. For any given s BK > 0, we can represent it by a simplex. want to study the incentives of changing/strengthening identity along either race, or class dimension

22 Simplex Representation s BK = 0 s = 1 WK s WK + s + s BL = 1 a point = a feasible population s = 1 = 1 s BL

23 A Thought Experiment We define a B WK (R) function as following: B WK (R) i,j2fwk,g β WK a ij + i,j2fbk,blg β WK a ij.

24 A Thought Experiment We define a B WK (R) function as following: B WK (R) i,j2fwk,g β WK a ij + i,j2fbk,blg β WK a ij. Interpretation: WK increases intra-racial (R) amicability with, anticipating reciprocity from BK and BL.

25 A Thought Experiment We define a B WK (R) function as following: B WK (R) i,j2fwk,g β WK a ij + i,j2fbk,blg β WK a ij. Interpretation: WK increases intra-racial (R) amicability with, anticipating reciprocity from BK and BL. B i (d) is similarly defined: i = WK,, BK, BL, and d = R, C

26 Potential Alliance De nition A potential alliance (PA) between i and j (sharing dim d) exists if B i (d) 0, B j (d) 0 (at least one is strict); B i (d) B i (d 0 ), B j (d) B j (d 0 ), where d 0 6= d.

27 Potential Alliance We could study an alliance formation game 1 proposing: groups propose to groups with same dimension 2 formation: two groups that have been proposed to each other form an alliance, the other two groups also form an alliance (reciprocity); o.w. no alliances are formed 3 payoff: each group (say i) in each alliance (with dimension d) receives B i (d); in case no alliances formed, each receives a normalized payoff of zero.

28 Potential Alliance (a less stringent de nition) A less stringent definition De nition Same as above, with the second bullet point replaced by for each t 2 fi, jg, either (i) B t (d) B t (d 0 ), or (ii) both B t (d) < B t (d 0 ) and B ft (d 0 ) (d 0 ) < 0.

29 Simple Observations multiple PAs along the same dimension. Except for knife-edged cases, PA is unique (though may not exist). If s BK = 0, PA involving BK.

30 Simple Observations multiple PAs along the same dimension. Except for knife-edged cases, PA is unique (though may not exist). If s BK = 0, PA involving BK. The last result implies that the only PAs must be WK and or and BL. Proposition Suppose s SK = 0. Suppose s WK 20%. Then the only PA that exists is one between WK and.

31 PA Between WK & Lemma Suppose s BK = 0 and s BL > 0. Then B WK (R) > 0 iff s 2 (0, min f(1 + s WK ) /2, 1 s BL g). Likewise, B (R) > 0 iff s WK 2 (0, min f(1 + s ) /2, 1 s BL g). Hence, a PA between WK and is possible only if both s WK and s are sufficiently small (hence, s BL is sufficiently large). Likewise, a PA between and BL is possible only if both s and s BL are sufficiently small (hence, s WK is sufficiently large).

32 Putting All Together s BK = 0 s = 1 WK B ( R ) > 0 BWK ( R ) > 0 BL & B ( C) > B ( R) B ( R) > B ( C) Which is more plausible? Consider s WK 20%. Only WK &! WK & s = 20% WK s = 1 B BL ( C) > 0 B ( C ) > 0 s BL = 1

33 PA Between WK & s BK = 0 s = 1 WK BWK ( R ) > 0 B WK (R) > 0 if 0 < s < 1+s WK 2. s = 1 s BL = 1

34 PA Between WK & s BK = 0 s = 1 WK B ( R ) > 0 BWK ( R ) > 0 Symmetrically for B (R) s = 1 s BL = 1

35 PA Between WK & s BK = 0 s = 1 WK B WK (C) < 0 if 0 < s < 1 s WK, which has no bite when B WK (R) 0. s = 1 s BL = 1

36 PA Between WK & s BK = 0 s = 1 WK B ( R) > B ( C) B (R) > < B < (C) if s WK > s BL. s = 1 s BL = 1

37 PA Between WK & s BK = 0 s = 1 WK B ( R ) > 0 BWK ( R ) > 0 B ( R) > B ( C) Existence of PA between WK &. WK & s = 1 s BL = 1

38 PA Between BL & GL s BK = 0 s = 1 WK BL & B ( C) > B ( R) Existence of PA between BL &. s = 1 B BL ( C) > 0 B ( C ) > 0 s BL = 1

39 Putting All Together s BK = 0 s = 1 WK B ( R ) > 0 BWK ( R ) > 0 BL & B ( C) > B ( R) B ( R) > B ( C) Which is more plausible? Consider s WK 20%. Only WK &! WK & s = 20% WK s = 1 B BL ( C) > 0 B ( C ) > 0 s BL = 1

40 Main Result Result 1 When capitalists are predominantly of the same race (called white), a PA between white capitalists and white workers exists if the white population is below a critical proportion, and no alliances of any kind exist o.w.

41 What if BK s size is not negligible? Thus far we assumed s BK = 0. The main result holds for sufficiently small s BK > 0.

42 What if BK s size is not negligible? Thus far we assumed s BK = 0. The main result holds for sufficiently small s BK > 0. We show simulations for s BK = 1%, 2.5%, 5%, and 10%, in the simplex s WK + s + s BL = 1 s BK.

43 =1% s BK = 2.5% s BK s WK 1 s BK s = 20% WK s = 20% WK PA between : BK & BL WK & BL & BK & WK = 5% s BK s = 20% WK =10% s BK s = 20% WK s = 1 sbl =1 sbk s BK

44 : less stringent def of PA =1% s BK = 2.5% s BK s WK 1 s BK s = 20% WK s = 20% WK PA between : BK & BL WK & (w/ Def. 1a) BL& (w/ Def. 1a) BK & WK = 5% s BK s = 20% WK =10% s BK s = 20% WK s = 1 sbl =1 sbk s BK

45 Result 2 Alliance based on race is more commonplace than alliance of workers across race

46 PA between WK & BK? Does smallness of WK and BK imply a PA between them? No, the fact that smallness of WK and imply a PA between them is predicated on a non-negligible third group (BL) and a negligible fourth group (BK). A PA between WK and BK requires either s /s WK or s BL /s BK too low to be plausible! Result 3 Asymmetry between PA of (B & W) workers and PA of (B & W) capitalists. While the former exists under some reasonable conditions, the latter never.

47 PA between WK & BK? s BK =1% = 2.5% s BK PA between BK and WK = 5% s BK maximum workers/capitalists s =10% ratio for the PA to be possible BK s WK (a) s BL / s BK 0.4 s / s WK = 10% s BK =1% s BK = 2.5% s BK s BK = 5% s BK =10% PA between BK and WK s WK PA between BK and WK = 5% s BK = 2.5% s BK = 1% s BK s WK (a) (b) s / s WK s BK = 10%

48 Intuition Lemma Given s WK and s BK, so s + s BL = s 1 s WK. B WK (C) is U-shaped and symmetric between s BL and s BK. From WK s perspetive, BL and BK are more fearful when they are more symmetric in group size

49 1 2 Set-up Related Literature Preliminaries 3 4

50 Key assumptions definition of PA no costs of identity changing; use reciprocity as a shortcut psychology

51 Relation to literature manipulation theory: "a view often held in Left circles that...the Right deliberately creates a certain non-economic issue... as a means of pulling working-class voters away from Left parties, thereby dirving economic policies to the right." Roemer (1998) pork theory: Fearon (1999) and Caselli & Coleman II (2010). complementarity theory: Esteban & Ray (2008)

52 Quali cations We didn t consider disenfranchised groups, group formation and/or dissolution, concrete political institutions, possibility of violence no costs of group identity changing (i.e., propaganda costs) => have examined the demand side, but not the supply side generalization to many groups, many dimensions?

53 Thank you!

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