I think what you are looking for is a specific instance of Vertical Integration:
In microeconomics and management, vertical integration is an arrangement in which the supply chain of a company is owned by that company. Usually each member of the supply chain produces a different product or (market-specific) service, and the products combine to satisfy a common need.
per: Vertical integration - Wikipedia
Vertical integration isn't used to describe the sale of consumer products with the intent of creating (then solving) a problem, but it is the structure that a company would use if it were to sell products with that intention.
This might not completely answer your question, depending on how explicit you believe the term you're looking for is. I'll explain my reasoning to let you decide for yourself.
When you think about the idea from the perspective of the problem itself and the 'chain' of purchases a consumer makes related to that problem, the term is accurate. If one product predictably creates a problem for a consumer, then that product could be described as the 'supply' for any different product made to solve the problem. This makes sense, since the first product is the precursor step to the second product in a chain of purchases the consumer makes. The control of this chain by a single producer -via the production of both products- is the vertical integration.
Vertical integration is definitely the structure you are trying to identify, but your situation is also explicit about the second product in the chain being a solution to a problem that the first creates.
When a producer's product is created by the producer with the planning for the product to experience problems that make the product obsolete, the design is called Planned obsolescence. I think the redundancy of that description clouds it, so I'll let Wikipedia give the definition again:
Planned obsolescence, or built-in obsolescence, in industrial design and economics is a policy of planning or designing a product with an artificially limited useful life, so it will become obsolete (that is, unfashionable or no longer functional) after a certain period of time. The rationale behind the strategy is to generate long-term sales volume by reducing the time between repeat purchases (referred to as "shortening the replacement cycle").
per: Planned obsolescence - Wikipedia
This strategy creates a specific problem (the outdating of a product) where the only predictable solution is replacing the item. In this way, it the uses structure I've stated, but doesn't cover every possible problem-to-solution relationship.
You can say the situation you've provided is an application of a vertical integration structure by a company (think: a supermarket creating, then selling, store brands). Depending on the problem that was created, the specific strategy the company is using could be the application of planned obsolescence in the development of a product and the creation of its subsequent solution (think: iPhones). These terms could be used to help abbreviate the description of production strategies that fit the parameters you've set. They aren't universally applicable though, and using them to more easily describe every instance of this specific situation could slightly amend their definitions, but in a way that I feel does not detract a noteworthy amout of clarity or specificity. The use just might not always be linguistically perfect.