On the structure of non-manipulable equilibria (Q1110426)

Consider pure exchange economies with \(\ell\) commodities and m consumers. Each consumer i has a utility function \(u_ i\), and his derived demand function \(f_ i\). Given the total endowment vector \(r\in {\mathfrak R}^{\ell}\), the set of all individual endowment vectors is given by \(\Omega (r):=\{\{\omega_ i\}_ i\in {\mathfrak R}^{\ell m}|\) \(\Sigma_ i\omega_ i=r\}\). The price domain is \(S:=p\in {\mathfrak R}^{\ell}|\) \(p_{\ell}=1\), \(p_ i>0\), \(i=1,...,\ell -1\}\). The equilibrium manifold is then the graph of the Walras correspondence, \(E:=\{(p,\omega)\in S\times \Omega (r)|\) \(\Sigma_ if_ i(p,p\cdot \omega_ i)=r\}\). Coalition G (\(\subset \{1,...,m\})\) can manipulate competitively in (p,\(\omega)\)\(\in E\), if there exists \({\bar \omega}\in \Omega (r)\) such that \({\bar \omega}{}_ i=\omega_ i\) for \(i\not\in G\), and for all \({\tilde \omega}\in [\omega,{\bar \omega}]\) and an equilibrium price \(\tilde p\) of \({\tilde \omega}\) which is on the smooth selection of p, \(u_ i(f_ i(\tilde p,\tilde p\cdot {\tilde \omega}_ i))>u_ i(f_ i(p,p\cdot \omega_ i))\) for every \(i\in G\). An equilibrium (p,\(\omega)\)\(\in E\) is called non-manipulable, if there is no such coalition G. It is shown that the set \(NM^ E\) of all non- manipulable equilibria is not necessarily arc-connected.