Changes

*This page is referenced in [[BPP Field Exam Papers]]

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:

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>

:<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|N\} =\frac{2\gamma -\gamma P_{G}-\mu \gamma }{2-2\gamma P_{G}-2eP_{G}(1-\gamma)}\,</math>

#<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.

The paper goes on to discuss longer horizons and sorting. The derivations will not be discussed here but the following propositions are provided:

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.