Table 1 Results for the FB case

Subsidy

\(p\left( 1-\rho _{j}^{FB}\right) =\theta \left( z_{j}+kz_{-j}\right)\), \(j=1,2\)

Quantities

\(Q_j^{FB} = \frac{1}{2} \left( 1-\frac{p}{\beta \theta } +\frac{z_j+kz_{-j}}{\beta } \right)\), \(j=1,2\)

\(Q^{FB}=1-\frac{p}{\theta \beta }+\frac{(1+k)\left( z_{1}+z_{2}\right) }{2\beta }\)

Welfare

\(W^{FB} =Y-p+\theta \left( \frac{\beta }{2}+\frac{\left( z_{1}+z_{2}\right) \left( 1+k\right) }{2}\right) +\frac{p^{2}- \left( z_{1}+z_{2}\right) \left( 1+k\right) p\theta }{2\beta \theta } +\theta \frac{\left( z_{1}+kz_{2}\right) ^{2}+\left( z_{2}+kz_{1}\right) ^{2}}{4\beta }\)