Note on Edgeworth’s Taxation Phenomenon and Professor Garver’s Additional Condition on Demand Functions

In a recent paper1 I derived for demand functions for n commodities conditions which, I concluded, assure the satisfaction of all further conditions which may reasonably be applied to all sets of demand functions. Letting p₁, p₂, … p _n , and q₁, q₂, …, q _n be respectively the prices and quantities, the conditions are: $$\frac{{\partial {p_i}}}{{\partial {q_j}}} = \frac{{\partial {p_j}}}{{\partial {q_i}}} < 0,\frac{{\partial \left( {{p_i},{p_j}} \right)}}{{\partial \left( {{q_i},{q_j}} \right)}} >0,\frac{{\partial \left( {{p_i},{p_j}{p_k}} \right)}}{{\partial \left( {{q_i},{q_j},{q_k}} \right)}} < 0, \ldots$$ These are equivalent to $$\frac{{\partial {q_i}}}{{\partial {p_j}}} = \frac{{\partial {q_j}}}{{\partial {p_i}}},\frac{{\partial {q_i}}}{{\partial {p_i}}} < 0,\frac{{\partial \left( {{q_i},{q_j}} \right)}}{{\partial \left( {{p_i},{p_j}} \right)}} >0,\frac{{\partial \left( {{q_i},{q_j},{q_k}} \right)}}{{\partial \left( {{p_i},{p_j},{p_k}} \right)}} < 0, \ldots ,$$ and were known, at least in part, by Edgeworth. Similar conditions in which the inequality signs are all > apply for supply functions.