TexasFluff is a producer of pillows. Suppose TexasFluff currently sells $7,500$ pillows a month at a price of $$10\mathrm{.}00$ per pillow. Variable Costs are $$4\mathrm{.}50$ per pillow, and $($non$-$incremental$)$ fixed costs are $$15,000.$

The company is enjoying stable demand with its current pricing, but management is looking for ways to increase profitability. One suggestion is that the company reposition their pillow as a luxury product, justifying a higher price. If successful, TexasFluff believes that it could charge $20\%$ more per pillow than it does now.

Please answer the following questions and be sure to show your work.

$($a$)[2$ points$]$ Complete the incremental breakeven analysis for the proposed $20\%$ price increase. In other words, what is the lowest sales volume that TexasFluff could tolerate before a $20\%$ increase would fail to make a positive contribution to its profitability? Express your answer in unit sales.

$($b$)[2$ points$]$ What would TexasFluff$$s contribution be if its sales declined by only $10\%$ following the $20\%$ price increase? Please state your answer in terms of dollars.

$($c$)[2$ points$]$ In order for TexasFluff to reposition as a luxury pillow provider, management believes that it will have to upgrade the fill of the pillows from synthetic material to real goose down. This change will add $$1\mathrm{.}50$ per pillow to the variable cost of sales. Now what is the break$-$even sales volume for maintaining profitability? Express your answer in unit sales.

$($d$)[2$ points$]$ To reposition as a luxury product, TexasFluff also needs to increase its monthly advertising and promotion budget by $$900,$ in addition to the increase in variable cost mentioned in part $($c$).$ Now what is the break$-$even sales volume for maintaining profitability? Express your answer in unit sales.

$($e$)[2$ points$]$ In a few sentences, explain a few pros and cons of attempting to use a price increase to raise profits.