Table 6 Welfare loss comparison

Centralisation

\(\Delta W_{\text {C}}=\left( \theta -1\right) \left( \frac{\beta +\left( z_{1}+z_{2}\right) \left( 1+k\right) }{2}-\frac{p^{2}}{2\beta \theta }\right) +\theta \frac{(z_{1}+kz_{2})^{2}+(z_{2}+kz_{1})^{2}}{4\beta } +\frac{z(z-z_{1}-z_{2})(1+k)^{2}}{2\beta }\)

Devolution with matching grant

 Devolution (Equi)

\(\Delta W_{\text {Equi}}=\frac{k^{2}\theta }{\beta }\left( \frac{1}{3}(z_{1}^{2}-z_{1}z_{2}+z_{2}^{2})+\frac{3}{10}z[z-(z_{1}+z_{2})]\right)\)

 Devolution (FR)

\(\Delta W_{\text {FR}}=\frac{k^{2}\theta }{\beta }\left( \frac{1}{3}(z_{1}^{2}-z_{1}z_{2}+z_{2}^{2})+\frac{1}{4}z[\frac{5}{6}z-(z_{1}+z_{2})]\right)\)

Devolution without matching grant

 Pure devolution

\(\Delta W_F=\frac{k^{2}\theta }{4\beta }(z_{1}^2+z_{2}^2)\)