spencenotes - Spence’s Signaling Game In Spence’s...

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Unformatted text preview: Spence’s Signaling Game In Spence’s model, workers know whether their produc— tivity is low or high, and must choose education, y. Next, firms (who do not observe a worker’s productivity) offer a wage as a function of the education level. We will as— sume that firms are risk neutral and competitive, so that the wage will be the expected productivity level. An index is an unalterable characteristic of a worker, like age, race, or gender. A signal is a characteristic subject to manipulation, like education. We assume that signaling costs are negatively correlated with productivity. This is not really an assumption, be— cause for this model the only effective signals will have this property. Type 1 workers: have productivity 6 = 1, constitute a fraction ql of the population, and have signaling cost C1(9) = 9- Type 2 workers: have productivity 9 = 2, constitute a fraction 1 —q1 of the population, and have signaling cost 02(9) = The timing is 1) Workers receive education 2) Firms choose wage contracts As part of a WPBE, firms must form beliefs about the probability that a worker with education y is type 1, even for off the equilibrium path choices of y. 3) Workers choose a contract. Note: In his classic paper, Spence talks about the timing as 1) Firms select wage contracts w(y) and select their be— liefs about which types of workers will choose which ed— ucation levels. 2) Workers choose education levels. 3) Firms’ beliefs must be rational (correct). This description does not make sense, because beliefs are not a strategic choice variable. If the timing is that firms choose w(y) and then workers choose y, then we have a screening game (like Rothschild—Stiglitz) that should be studied using game theoretic concepts like WPBE. If a firm deviates to another wage contract, this will affect workers’ incentives to acquire education, so the beliefs should not be held fixed. if [A 1; («cumming/rs c.(’\ L C ' I, \k W L égpmmfif ? V10, E2 LC , 5’ V 714. (if: jive.-- ,\ §“‘\\) I C1 92’ V! ‘ «WA ‘1‘; (A, .3, - ' ' J 1 ‘9 MCN‘MS CC‘W‘VK 54ft L0. on, an 55f Y‘ur . ‘3 GW‘E; MC'MY +o Lamar-«7%); MSfé’cAV a"? w0¥§3iiq53 I l I A I K‘ww-v. 6% Ewing) Soho/bf {giraffeSscfSp WW/ «9" /._. ., \ .n \ ‘ ‘ 4 (/4‘1‘\ 4L: Pop-waojk'm\ 5; Wm» as sciomfl‘mj in» fig} (F: VJ“YV¢('r\ 3 Sé><, (35ak\fixilfi“; C£>vk£(&‘ /7\cg.ii€;YI ‘ «axr. W ' > . ' P 7 , {,!’\L #2, ‘ 3 l0, “CUM! r45; 30 (A g! ‘ 7dr mix 0» p1!» J V, ; _/- ‘ ~ ( <. 2. YALE 5 J, F‘ I _ ‘ (C ’VJKQA P: 3‘9 Ami ‘ ’er £42258 0-K (LOI’W’KJC‘y’a vdl+llt\' 3. (L0 WM 4 paymanT Cam £12, a thcjflon of «Cth PPOdWL’HUfl'y Consider the separating equilibrium with y* z 1.9, sup— ported by beliefs that anyone with lower education is type 1. Suppose a worker deviates to y : 1.1, which is inconsistent with the equilibrium. Who is more likely to make this “mistake”? A type 1 worker prefers to choose 3/ = 0, even if she would receive a wage of 2 with y = 1.1. Under the "intuitive criterion” we rule out beliefs on information sets following a deviation that as— sign positive probability to types that would never make such a deviation (because that type is worse off for all continuation strategies). Then the only WPBE satisfy— ing the intuitive criterion is the most efficient separating equilibrium, y* = 1. Timing makes a big difference. Consider the Rothschild— Stiglitz timing, where 1) Firms choose w(y) 2) Workers choose y and accept a contract. The only candidate separating equilibria involve 111(0) 2 1 and 111(1) 2 2. The wages offered for other education levels are arbitrary, as long as no worker is induced to choose that education. If ql is small, a pooling contract will break the candidate separating equilibrium. w(y) : 2 — ql for ally will be accepted by both types if we have 1 2— >2—— Q1 2 or q1 < lf q1 > % holds, then the condidate sepa— rating equilibrium is an equilibrium (subgame perfect or WPBE). Under the Rothschild—Stiglitz timing, there is no pooling equilibrium. Consider the best pooling contract, = 2 — ql. A new firm could offer the contract, = 2 for y 2 y*. This contract is accepted by type 2 workers (who choose y = y*) if >l< y 2*—— > 2— or 2 (11 y* < 2q1. The contract is rejected by type 1 workers (who choose the pooling contract with y = 0) if 2—q1 > 2—y* or 34* > (11- Thus, any y* E [q1,2q1] breaks the candidate pooling equilibrium. ...
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spencenotes - Spence’s Signaling Game In Spence’s...

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