Winner-take-all market

In the simplest instance a WTA market where the differences between players are unobservable and therefore the probability of each player winning is perceived as equal. Finding for the simple two sector economy; output ${\displaystyle GNP=V(K^{e})+[N-{\Bigl (}{\frac {V(K^{e})}{w}}{\Bigr )}]w=Nw}$, similar to rent seeking models and dual labour market models for developing countries.[7]

Private equilibrium versus social optimum in a winner-take all market

This model is used to demonstrate the inefficiencies of entry into the WTA market by contrasting these values with the socially optimal allocation ${\displaystyle V'(K^{o})=w}$ showing the private equilibrium allocation in a WTA market outstrips the socially optimum allocation where output is maximised.

To provide a more realistic model of the winner-take-all market – when differences in talent of the players are observable – meaning the perceived probability of each player winning varies. This model takes the factor Q which combines the selective factors ${\displaystyle T}$ talent and ${\displaystyle U}$ representing unobserved factors which influence quality such that ${\displaystyle Q_{i}=T_{i}+U_{i},}$ ${\displaystyle i=1,\ldots ,N}$ for each player where ${\displaystyle T_{i}}$ and ${\displaystyle U_{i}}$ are variables determined from some distribution and drawn randomly for the ${\displaystyle N}$ players.

While in the model of unobservable differences the market reward ${\displaystyle V}$ increases with every additional WTA player ${\displaystyle K+1}$ , in this model the value of reward ${\displaystyle V}$ increases only where the quality of the new WTA player is higher than the lowest quality of the existing player pool. The probability of this event${\displaystyle p_{1}(K)}$ then determines the marginal reward gain for new entrants ${\displaystyle V(K)-V(K-1)=p_{1}(K)Z}$

where ${\displaystyle Z}$ is the expected value gain if the new entrant wins. The model takes ${\displaystyle Z=R_{1}^{*}-X}$ where ${\displaystyle R_{1}^{*}}$ is the expected revenue gain and ${\displaystyle X}$ is a positive number as the expected gain in WTA market value is only a part of the revenue which will be received by the entrant upon winning.

This model finds that ${\displaystyle p_{1}(K^{o})R_{1}^{*}(K^{o})=w+p_{1}(K^{o})X}$ which shows that the anticipated reward (left) to new entrants exceeds the opportunity wage ${\displaystyle w}$ by ${\displaystyle p_{1}(K^{o})X}$. This model finds that observable talent narrows the gap between social optimum and equilibrium allocation of entrants though does not eliminate the inefficiency entirely.