$4.$ Fashionable canes specializes in canes that make a statement. Three of their top selling canes are displayed above. The variable costs incurred to make each of these canes are: $$45($Purple Pearlz$),$ $$80($Arctic Blue$),$ and $$150($Swarovski$).$ The sales forecasted for each of these canes for next year are: $1,900,2,800,$ and $1,400,$ respectively.

Suppose that next year they are thinking about adding a new cane called the Lionhead, which will be priced at $$349($with variable costs of $$110).$ If they launch the Lionhead, it is forecasted to sell about $2,000$ units next year. They would incur some fixed costs such as design $$10,000$ and promotional costs of $$20,000.$ Forty percent of the Lionhead volume is incremental, while $30\%$ of the volume would be cannibalized from the Swarovski, $25\%$ from the Arctic Blue, and $5\%$ from the Tumbleweed.

a$.$ Should Fashionable Canes launch the Lionhead?

b$.$ In the second year, they forecast that the demand for Purple Pearlz will increase by $15\%,$ the Arctic Blue$$s demand will increase by $20\%,$ and the Swarovski$$s demand will increase by $25\%.$ If they launched the Lionhead, they forecast selling $2,500$ units. In the second year, the promotional spending for the Lionhead will be $$30,000.$ Assuming the cannibalization rates for the second year will be $20\%$ coming from the Swarovski, $15\%$ from the Arctic Blue and $3\%$ from the Purple Pearlz, would your decision change considering both years?

c$.$ Suppose that for year $2,$ the growth rates are as in part b$,$ but the company offered a $$25$ coupon to increase the sales of the new Lionhead cane. Assume that the discounted Lionhead volume will be $2,000$ units, and the full$-$priced volume will be $1,500$ units. Also, $25\%$ of the discounted volume will come from the Arctic Blue and $5\%$ from the Purple Pearlz, with the rest being incremental volume. Of the full$-$priced volume, $10\%$ will come from the Swarovski and the rest will be incremental volume. Based on both years, would you offer the coupon, offer it full$-$priced, or not at all?