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Tadelis (2001) - The Market For Reputations As An Incentive Mechanism (view source)
Revision as of 22:05, 28 April 2010
, 22:05, 28 April 2010no edit summary
===Benchmark: No Reputations Markets===
Here we assume that names can not be traded and restrict attention to two periods with three generations as depicted below. <math>\vskip 5pt\hskip 200pt$t=Generation 0 dies at the end of period 1, generation 1$\hskip 57pt$t=lives for both periods, and generation 2$\hskip 80ptGeneration 0:\hskip 10pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%\TEXTsymbol{\vert}\vskip 2pt\hskip 80ptGeneration 1:\hskip 10pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_\TEXTsymbol{\vert}\vskip 2pt\hskip 80ptGeneration is born half way through and lives for period 2:\hskip 105pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%\TEXTsymbol{\vert}\,</math>.
In period 1 there is one wage as everyone has a new name. In period 2 there are three wages depending on the <math>h\,</math> (histories) observed which are either <math>S\,</math> (success), <math>F\,</math> (failure), or <math>N\,</math> (new name). <math>F\,</math> histories are irrelevant as sellers who failed are better off costly changing their name.
Fixing (for a moment) <math>\Delta w\,</math> the equilibrium <math>e\,</math> solves the FOC:
:<math>P_{G}\Delta w=c^{\prime}(e)\,</math>
The actual wage differential depends on the amount of effort that the <math> O\,</math>types exert. If all <math>O\,</math> types exert no effort (i.e. are bad) or full effort (i.e. are good) then we have: :<math>\Delta w_{B}=\frac{2(1-\gamma )P_{G}}{2-\gamma P_{G}} \mbox{ and } \Delta w_{G}=0\,</math> This leads to: '''Proposition 1''': There is a unique equilibrium of the two-period model with no reputation markets. If <math>P_{G}\Delta w_{B} \le c^{\prime }(0)\,</math> then <math>e=0\,</math> in equilibrium. If <math>P_{G}\Delta w_{B}>c^{\prime }(0)\,</math> then <math>e\in (0,1)\,</math> in equilibrium. Essentially if the opportunistic types exert no effort then the wage differential is the highest, but this provides them with an incentive to work in period 1. ===The Market for Reputations=== We now let retiring sellings from generation 0 to sell their names. Only successful names will be traded - and in fact <math>S\,</math> names will be traded in all equilibria. If no names were traded it is because they are worthless. But the supply of <math>S\,</math> names is positive, so it must be the case that <math>w_{2}(S)\leq w_{2}(N)\,</math>. But then <math>O\,</math> types would exert <math>e=0\,</math> in the first period and <math>Pr \{G|S\}=P_{G}\,</math>. However, <math>1-\gamma >0\,</math> implies that there are some <math>O\,</math> types and <math>\Pr \{S|N\}<P_{G}\,</math>, which in turn means that <math>w_{2}(S)>w_{2}(N)\,</math>, which is a contradiction. To get an equilibrium the model assumes an arbitrage condition. The value of an <math>S\,</math> name is: :<math>v(S)=w_{2}(S)-w_{2}(N)\,</math> This naturally leads all period 1 <math>O\,</math> types to exert the same effort, as they have identical incentives, whether or not they bought a name. We now need to establish the correct beliefs by buyers about effort in period 1 (again - <math>O\,</math> types will choose zero effort in period 2): :<math>w_{1}=\left[ \gamma +(1-\gamma )e\right] P_{G}\,</math> and :<math>w_{2}(h)=\Pr \{S|h\}=\Pr\{G|h\}\cdot P_{G}\,</math> Let <math>\mu\,</math> denote the proportion of <math>G\,</math> types who buy <math>S\,</math> names in <math>t=2\,</math>, and <math>\rho\,</math> the proportion of <math>O\,</math> types. Then an equilibrium is specified as a tuple <math>\left\langle \mu ,\rho ,w_{1},w_{2}(S),w_{2}(F),w_{2}(N),v(S),e\right\rangle\,</math>, with the beliefs about <math>\mu ,\rho and e\,</math> pinning it down. The equilibrium must satisfy the (supply equals demand) market clearing condition: :<math>\gamma P_{G}+(1-\gamma )eP_{G}=\mu \gamma +\rho (1-\gamma)\,</math> The beliefs must therefore be (after some algebra and plugging in market clearing): :<math>\Pr \{G|S\}=\frac{\gamma P_{G}+\gamma \mu }{2\gamma P_{G}+2(1-\gamma )eP_{G}}\,</math> :<math>\Pr \{G|N\} =\frac{2\gamma -\gamma P_{G}-\mu \gamma }{2-2\gamma P_{G}-2eP_{G}(1-\gamma)}\,</math> Proposition 4: There exist <math>\underline{\mu }<\overline{\mu }\,</math> so that <math>(\mu ,\rho ,e)\,</math> is an equilibrium if and only if the following three conditions hold:#<math>\mu \in [\underline{\mu },\overline{\mu}]\,</math>#<math>(\mu ,\rho ,e)\,</math> satisfy market clearing#<math>c^{\prime }(e)=\Delta wP_{G}\,</math> As long as the price for names reflects the wage differential that the name generates, sellers will be indifferent. At <math>\underline{\mu}\,</math> either the price of an <math>S\,</math> name is sero, or there are too few good types in the second period so that even when all <math>S\,</math> names are bought by bad types this is still better than having no history. In the interval prices for the <math>S\,</math> names are positive, and at the upper bound all good new types are buying the names without violating market clearing. In order to perform some welfare comparisons between the benchmark (which would result if trade could be made observable, say by regulation) and this model, we need to select an equilibrium from each. To do this, we need to restrict the multiple equilibria in this model - this is done by assuming that name buyers are in proportion to their population. Therefore: :<math>\mu ^{*}=\rho ^{*}=\gamma P_{G}+(1-\gamma )eP_{G}\,</math> Again we now consider when all <math>O\,</math> types are good <math>(e=1)\,</math> or bad (<math>e=0)\,</math> to get: :<math>\Delta w_{B}(\mu ^{*})=\frac{P_{G}(1-\gamma )}{2(1-\gamma P_{G})} \mbox{ and }\Delta w_{G}(\mu ^{*})=0\,</math> The market for names is bad (in the welfare sense) if: :<math>\Delta w_{B}(\mu^{*})<\Delta w_{B}\,</math> Which is true when: :<math>P_{G}<\frac{2}{3\gamma}\,</math> ===Longer Horizon and Sorting=== The paper goes on to discuss longer horizons and sorting. The derivations will not be discussed here but the following propositions are provided: Proposition 6: Names with a history consisting of only one success and no failures must be traded in all equilibria. Proposition 7: For the infinite horizon model there is no equilibrium in which only <math>S\,</math> names are traded, and these are bought only by good types and by opportunistic types who choose to be good. This sorting result is important. Suppose it were true, then it would affect beliefs - Observing an <math>S\,</math> name would lead to the inference that it is held by a good (or good opportunistic) type. Then an <math>S\,</math> name would command a premium, and <math>O\,</math> types would value it higher because they face a less attractive future without it.
