Question

A hospital has an operating room used only for eye surgery. The annual cost of heat, rent, and electricity for the operating room and its equipment is $360,000 and the annual salaries for the people who staff this room total $540,000. Each surgery performed requires the use of $760 worth of medical supplies and drugs. Every patient receives a $30 bouquet of flowers the day after surgery. Also, one-third of the patients receive a $30 pair of dark glasses. 1a. Identify the annual fixed and variable costs for running the operating room. FC=______________________________________

VC=______________________________________ Let x be the number of patients having eye surgery in a year. Write the cost as a function of x : C(x)=_______________________________________ 2a. The hospital receives $2,000 for each eye operation performed. Write the revenue function, in terms of the same variable x: R(x)= _________________________________________ 2b. Write the annual profit function for this hospital, in terms of the same variable x: P(x)= _________________________________________ 3. How many eye operations must the hospital perform each year in order to break even? Justify your answer!

4. Currently, the hospital averages 70 eye operations per month. There is a medical machine that would reduce the amount of medical supplies needed by $100 per patient; it can be leased for $100,000 annually. Keeping in mind the financial costs and benefits, advise the hospital on whether it should lease this machine. Justify completely your answer!

5. An advertising agency has proposed to the hospital administration that $30,000 per year be spent on advertising to persuade people to use the hospital. The agency estimates that this advertising would increase the business by 10 patients per month. If they are correct and if this increase is not big enough to affect variable costs or any other of the financial parameters, what impact would this advertising have on the hospital’s profits? (Hint: Compare the yearly profit without advertising with the profit using advertising.) 6. In case the advertising agency is overly optimistic, how many extra patients per year are needed to cover the cost of the proposed ads? Justify your answer! (Hint: Find the new break-even point given the additional cost of advertising and use the original break-even answer from part 3 to calculate the extra patients needed.) 7. If the ad campaign is approved and subsequently meets its projections, should the hospital purchase the machine mentioned earlier? Justify your answer! (Hint: Find a new profit function which includes advertising and the machine’s cost. Compare profits without the machine and with advertising with the profits with the machine and advertising.)

Answer 1

1a. Identify the annual fixed and variable costs for running the operating room :

Fixed Costs : FC =  Annual cost of heat, rent, and electricity for the operating room and its equipment ( $360,000), and Annual salaries for the operating room staff ($540,000)

Variable Costs per surgery performed : VC = Medical supplies and drugs ($760), Bouquet of flowers for patient ($30) and  Pair of dark glasses for one-third of the patients ($30)

If number of patients having surgery in a year = x,

Then, Annual Total Variable Costs = 760x + 30x + 1/3*x*30 (since one-third of the patients receive a pair of dark glasses)

= 760x + 30x + 10x = 800x

Annual Fixed Costs = $360,000 + $540,000 = 900,000 (these costs do not vary with the number of patients having surgery)

Cost Function : C(x) = Annual Fixed Costs + Annual Total Variable Costs = 900,000 + 800x

2a. The hospital receives $2,000 for each eye operation performed. Hence, Revenue from each eye surgery = $2,000

If number of patients having eye surgery in a year = x, then total Annual Revenue from eye surgery = 2,000x

Revenue Function : R(x) = 2,000x

2b. Annual profit = Annual Revenue - Annual Variable Costs - Annual Fixed Costs

Annual Revenue = 2,000x, Annual Variable Costs = 800x, Annual Fixed Costs = 900,000 (all calculated above)

Annual Profit Function : P(x) = 2,000x - 800x -  900,000

= 1200x - 900,000

3. How many eye operations must the hospital perform each year in order to break even? :

This means that we need to calculate number of surgeries in a year = x

Break Even Point (in units) = Annual Fixed Costs / (Revenue per unit - Variable costs per unit)

x = $900,000 / ($2,000 - $800)

= $900,000 / $1,200

= 750

Hence, 750 eye surgeries should be performed each year to break even.

Let's verify this :

Contribution per unit = Revenue - Variable costs = $2,000 - 800 = $1,200 per unit

Total Contribution from 750 units = $1,200 x 750 = $900,000 which is equal to the Annual Fixed Costs

Hence,Contribution - Fixed cost = Profit = $0, which is a break-even situation

4. Whether a medical machine should be leased :

The hospital currently averages 70 eye operations per month. Hence, average eye operations per year = 70 x 12 months = 840

The medical machine would reduce the amount of medical supplies needed by $100 per patient, hence for an average of 840 patients per year, the savings in medical supplies would be = $100 per patient x 840 patients = $84,000 per year

The annual cost of leasing the medical machine = $100,000

Hence, the benefit in terms of annual savings in amount of medical supplies needed of $84,000 would be less than the annual cost of leasing the machine of $100,000.

Hence, it is advisable that the hospital should not lease this machine.