Jacks grandparents have a $2-$bedroom cottage near Bear Mountain State Park, about an hour north of Manhattan. Their cottage is listed on Airbnb through a property management company, which charges$30\%$of rent revenue. Jacks grandfather is a retired accountant, and he has prepared a budgeted income statement for next year as follows:

Budgeted Income Statement: Revenue$($$$300$per night for$250$nights$)$$$75,000$

Variable expenses:

Utilities$($$$30$ per night for$250$nights$)7,500$

Property management fees$(30\%$of revenue$)22,500$

Contribution margin $45,000$

Fixed expenses:

Depreciation$5,000$

Property taxes$16,000$

Net operating Income$$24,000$

$*$$$300$is the average rental revenue per night after Airbnb service fees.Due to higher labor costs, the property management company recently indicated that they will only renew the service contract if the management fee rate increases to$40\%$ of revenue. Jacks grandparents think a$40\%$ charge is outrageous. They would no longer like to retain the property management company and instead ask Jack to take care of the cottage$.$

$1.$ Compute the cottage$$s break$-$even point in units $($nights$)$ for next year assuming the following scenarios:

$($Assume all other facts remain the same$)$

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a$.$ Jack$$s grandparents renew the service contract with the property management company, charging $40\%$ of revenue

b$.$ Jack$$s grandparents hire Jack to manage the cottage and pay him $$22,500$ annually $($round up to the nearest nights$)$

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$2.$ Assume that Jack$$s grandparents renew the service contract with the property management company charging $40\%$ of revenue, and the cottage is rented out for $250$ nights next year.

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a$.$ Determine the cottage$$s Contribution Margin $($CM$)$ ratio

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b$.$ If revenue increases by $10\%$ and cost behavior patterns remain unchanged, by how many dollars will the cottage$$s net operating income increase? Use the CM ratio to compute your answer

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c$.$ Determine the dollar sales required to generate the same net operating income Jack$$s grandpa predicted in the budgeted income statement for next year. Use the CM ratio to compute your answer

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$3.$ Assume that Jack$$s grandparents hire Jack to manage the cottage and pay him $$22,500$ annually, and the cottage is rented out for $250$ nights next year.

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a$.$ Determine the cottage$$s margin of safety in dollars $($round to the nearest dollar$)$

b$.$ Determine the cottage$$s degree of operating leverage $($keep two decimal places$)$

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c$.$ To better market the cottage$,$ Jack regularly shares pictures of the cottage and the surrounding area on Instagram. This leads to a $50-$night rental increase next year. By what percentage would you expect net operating income to increase? Use the degree of operating leverage to obtain your answer

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$4.$ Jack is not satisfied $$with the fixed compensation his grandparents are recommending. He wants to persuade his grandparents that he deserves a profit$-$sharing compensation plan equal to $35\%$ of revenue. He strongly believes that linking his compensation to revenue will provide him with more incentive to increase the rental income for the cottage$.$ Jack asks for your help to convince his grandparents. Jack says that he would only do the bare minimum to meet his grandparents$$ expectations of renting out $250$ nights if he is paid $$22,500$ flat. However, he would work extremely hard and boost the rental to at least $300$ nights while raising the nightly rate to $$320.$ Could you help Jack show his grandparents that they are better off by offering him compensation of $35\%$ of the revenue? Calculate the dollar increase $($or decrease$)$ in NOI of Jack$$s proposal versus the current situation