Some while ago, a venture capital partnership provided a small amount of initial funding to HomeSea$$ a start$-$up company who acquired a lightbulb business who went into insolvency. The intent was to integrate novel technology to make lightbulbs $$smart$$ through the Internet of Things in the private homes market. The intent was to allow the start$-$up firm to establish initial versions of its products, and to test and prepare them for viable commercial use.

HomeSea now believes that a core product is ready for market, and, from January $1$st $2023,$ it plans to begin sales in one of two potential entry markets: $($A$),$ the European Union; or $($B$),$ North America. The view is that it would be beyond the management resources available to the firm to launch the product in both markets simultaneously, even if that seemed profitable on the surface.

HomeSea chose the second of those markets $($B$),$ North America $-$ where the start$-$up firm would seek to develop market share through selling its product through major house builders. The rationale for this is not available.

The venture capital partnership is willing to commit an additional $25$ million to the start$-$up firm.$[$ For simplicity, you may assume that the $25$million is paid on January $1$st $2023$ although, in practice, some of the money would be paid earlier than that date. A venture capital partnership could be expected to provide funding in a number of blocks or tranches, with payment of each one dependent on the achievement of particular milestones relating to such as stage of product development, identification of suitable markets, acceptance of the product by opinion$-$shaping customers, market share, and early indications of profit potential. $]$ This would allow the start$-$up firm to establish a small assembly facility, buy$-$in inventories of components from suppliers, and undertake a marketing program.

The owner of the start$-$up company believes that this $25$ million in capital will be sufficient to allow them to assemble and sell a gradually increasing quantity of product, as set out below: