Changes

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.