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In this paper we study a single-period two -product inventory model with stochastic demands, proportional revenues and costs, substitution. We focus on full downward substitution in our study.

We assume that the orders have to be placed before the demands are realized. And the problem is to decide the ordering quantity for each product. We develop a general profit maximization model for single-period two -product substitution problem, and show that it is concave and submodular. And we develop the optimization condition for the problem and rewrite the solution of the problem in a perfect form, which is similar to the solution of the newsboy problem. Then we compare the solution of this problem with the solution of newsboy problem, and proof that the profit can be improved using the substitution policy.

For the optimization solution, we study the effects of the parameters, such as the purchase cost, the penalty cost, the salvage value, the sale price. And we can get some interesting properties of the solution with respect to the parameters, and so we can know some instructions to adjust the order quantities while the parameters are changed. From the properties, we can find out the parameters that have stronger effects on the order quantities, and pay more attention on them while making the ordering decision.