A Theorem on Consequences of Planned Demand Changes in a Partially Fixed-Price Environment

We consider a multiproduct firm which, through price changes, plans to increase by specific amounts the demand for one or more goods, while holding constant the price of certain other goods as well as the demand for all remaining goods. It is shown under certain reasonable assumptions that when the price of a good is fixed, the demand for each other fixed-price good is either increased or unchanged and the consumer pays more for each good whose demand the firm plans to increase (relative to the case in which the price of that good is not fixed).