Equations on the Derivatives of an Initial Endowment-Competitive Equilibrium Mapping for an Exchange Economy

Consider an exchange economy with n traders and k goods. Suppose that there exists C mappings p(·) and x (·) that specify for each initial endowment w of the k goods to the n traders a price vector p(w) and a vector of net trades x(w) that define a competitive equilibrium. Three systems of determinant equations are derived that the derivatives of p(·) and x(·) necessarily satisfy regardless of the underlying preferences that determine these mappings. Each of the three systems of equations fundamentally reflects the interaction among traders in the competitive mechanism.