Table 8 Coordination problem under information asymmetry
From: Devolution and grant-in-aid design for the provision of impure public goods
CG | LG | \(\Delta\)-subsidy | \(\Delta\)-quantity | \(\Delta\)-welfare | Ex-post mean welfare loss |
|---|---|---|---|---|---|
FC | FC | \({\varvec{\theta }} {\varvec{k}}({\varvec{z}}_{{\varvec{j}}}-{\varvec{z}}_{-{\varvec{j}}})\) | \(0\) | \(\frac{{\varvec{k}}^{{\varvec{2}}}{\varvec{\theta }}}{{\varvec{2}} {\varvec{\beta }}}({\varvec{z}}_{{\varvec{1}}}-{\varvec{z}}_{{\varvec{2}}})^{{\varvec{2}}}\) | \(\frac{k^{2}\theta }{3\beta }\left( z_{1}^{2}+z_{2}^{2}-z_{1}z_{2}\right)\) |
PC | \(\theta kz_{-j}\) | \(\frac{k}{\beta }\frac{z_{1}+z_{2}}{2}\) | \(\frac{k^{2}\theta }{4\beta }\left( z_{1}^{2}+z_{2}^{2}\right)\) | ||
FR | \(\theta kz_{-j}\) | \(\frac{k}{\beta }\frac{z_{1}+z_{2}}{2}\) | \(\frac{k^{2}\theta }{4\beta }\left( z_{1}^{2}+z_{2}^{2}\right)\) | ||
PC | FC | \(\theta k(z_{j}-z_{-j})\) | 0 | \(\frac{k^{2}\theta }{2\beta }(z_{1}-z_{2})^{2}\) | \(\frac{k^{2}\theta }{\beta }\left( \begin{array}{c} \frac{1}{3}\left( z_{1}^{2}+z_{2}^{2}-z_{1}z_{2}\right) \\ +\,z\left[ \frac{5}{6}z-\frac{1}{2}(z_{1}+z_{2})\right] \end{array}\right)\) |
PC | \({\varvec{\theta }} {\varvec{k(z}}_{-{\varvec{j}}} -{\varvec{z}})\) | \(\frac{{\varvec{k}}}{{\varvec{\beta }}} \left( \frac{{\varvec{z}}_{{\varvec{1}}} +{\varvec{z}}_{{\varvec{2}}}}{{\varvec{2}}}-{\varvec{z}}\right)\) | \(\frac{{\varvec{k}}^{{\varvec{2}}}{\varvec{\theta }}}{{\varvec{2\beta }}} \left[ \frac{({\varvec{z}}_{{\varvec{1}}}-{\varvec{z}})^{{\varvec{2}}}}{{\varvec{2}}}+\frac{({\varvec{z}}_{{\varvec{2}}}-{\varvec{z}})^{{\varvec{2}}}}{{\varvec{2}}} \right]\) | ||
FR | \(\theta k(z_{-j}-2z)\) | \(\frac{k}{\beta }\left( \frac{z_{1}+z_{2}}{2}-2z\right)\) | \(\frac{k^{2}\theta }{2\beta }\left[ \frac{(z_{1}-2z)^{2}}{2}+\frac{(z_{2}-2z)^{2}}{2}\right]\) | ||
FR | FC | \(\theta k(z_{j}-z_{-j})\) | 0 | \(\frac{k^{2}\theta }{2\beta }(z_{2}-z_{1})^{2}\) | \(\frac{k^{2}\theta }{\beta }\left( \begin{array}{c} \frac{1}{3}\left( z_{1}^{2}+z_{2}^{2}-z_{1}z_{2}\right) \\ +\,\frac{1}{4}z\left[ \frac{5}{6}z-(z_{1}+z_{2})\right] \end{array}\right)\) |
PC | \(\theta k(z_{-j}-\frac{z}{2})\) | \(\frac{k}{\beta }\frac{z_{1}+z_{2}-z}{2}\) | \(\frac{k^{2}\theta }{4\beta }\left[ \frac{(2z_{1}-z)^{2}}{4}+\frac{(2z_{2}-z)^{2}}{4}\right]\) | ||
FR | \({\varvec{\theta }} {\varvec{k}}({\varvec{z}}_{-{\varvec{j}}} -{\varvec{z}})\) | \(\frac{{\varvec{k}}}{{\varvec{\beta }}} \left( \frac{{\varvec{z}}_{{\varvec{1}}}+{\varvec{z}}_{{\varvec{2}}}}{{\varvec{2}}}-{\varvec{z}}\right)\) | \(\frac{{\varvec{k}}^{{\varvec{2}}}{\varvec{\theta }}}{{\varvec{2}}{\varvec{\beta }}} \left[ \frac{({\varvec{z}}_{{\varvec{1}}}-{\varvec{z}})^{{\varvec{2}}}}{{\varvec{2}}}+\frac{({\varvec{z}}_{{\varvec{2}}}-{\varvec{z}})^{{\varvec{2}}}}{{\varvec{2}}}\right]\) | ||
Equi | FC | \(\theta k(z_{j}-z_{-j})\) | 0 | \(\frac{k^{2}\theta }{2\beta }(z_{1}-z_{2})^{2}\) | \(\frac{k^{2}\theta }{\beta }\left( \begin{array}{c} \frac{1}{3}\left( z_{1}^{2}+z_{2}^{2}-z_{1}z_{2}\right) \\ +\,\frac{3}{10}z\left[ z-(z_{1}+z_{2})\right] \end{array}\right)\) |
PC | \(\theta k\left( z_{-j}-\frac{3}{5}z\right)\) | \(\frac{k}{\beta }\left( \frac{z_{1}+z_{2}}{2}-\frac{3}{5}z\right)\) | \(\frac{k^{2}\theta }{4\beta }\left[ \left( z_{1}-\frac{3}{5}z\right) ^{2}+\left( z_{2}-\frac{3}{5}z\right) ^{2}\right]\) | ||
FR | \(\theta k\left( z_{-j}-\frac{6}{5}z\right)\) | \(\frac{k}{\beta }\left( \frac{z_{1}+z_{2}}{2}-\frac{6}{5}z\right)\) | \(\frac{k^{2}\theta }{4\beta }\left[ \left( z_{1}-\frac{6}{5}z\right) ^{2}+\left( z_{2}-\frac{6}{5}z\right) ^{2}\right]\) |