Abstract: Convex function of demand,convex distribution of price and an optimal model for coexistence of multi-prices are established.At first,the quadratic function model of demand is extended to a general convex one.It is proved that the price with maximal demand increases the profits and that the price with maximal profit decreases the demand.Second,a convex distribution model for the price is proposed.The formulae are given respectively for the expectation price,the variance of the price and the income.Third,a nonlinear programming model is given for the coexistence of multi-prices on the basis of the convex function model.Finally,the results are verified by an example for sale of the purified water.