Knight Shades Inc. is considering adding new line of sunglasses to their designer series. The glasses will sell for $$900$ per pair and have a variable cost of $$300$ per unit. The company has spent $$80,000$ for a marketing study that estimates the company will sell $55,000$ sunglasses per year for seven years.

The fixed costs each year to produce the sunglasses will be $$4,000,000.$ The company has also spent $$600,000$ on research and development for the new glasses. The new plant and equipment will cost $$10,000,000$ at start$-$up and will be depreciated on a straight$-$line basis to zero over the seven years. At the end of the seven years the plant and equipment will be worthless. The new sunglasses will also require an increase in net working capital of $$588,000$ that will be returned at the end of the project. The tax rate is $38$ percent, and the cost of capital is $13$ percent.

The marketing study also estimates that introducing the new sunglasses will reduce sales of the company's high$-$priced sunglasses by $14,000$ units. The high$-$priced glasses sell at $$1,400$ and have variable costs of $$600.$ The study also estimates that the company will increase sales of its cheap sunglasses by $8,000$ units. The cheap sunglasses sell for $$500$ and have variable costs of $$200$ per unit.

Knight Shades Inc. estimates the project's NPV to be $$47\mathrm{.}451,553\mathrm{.}75,$ but would like to know the sensitivity of NPV to changes in the price of the new sunglasses and the quantity of new sunglasses sold.

Note: you may want to verify the base$-$case NPV to make sure you're setting up the problem correctly.

Required:

$($a$)$What is the sensitivity of the NPV to changes in the price of the new sunglasses?

Note: In your economics classes you leamed that sensitivity $($elasticity$)$ is defined as the percent change in one variable due to a percent change in another variable. Hence, it is easiest to estimate the sensitivity by re$-$calculating the NPV if price increases by $1\%.($Do not round your intermediate calculations.$)$

$($b$)$What is the sensitivity of the NPV to changes in the quantity sold? Calculate the sensitivity by assuming sales increase by $1\%.($Do not round your intermediate calculations.$)$