There are two large private, for profit hospital chains in the Republic of Ivarrona. The first is a very profitable chain of private hospitals called Midas Mammon Medical Inc. (popularly known in the country as M3 which we will use to refer to them from hereon). M3 offers very high quality care to patients that are willing to pay for it. It recruits from the best medical schools in the country and employs the best doctors in the country. The average compensation that it pays its physicians in every category is at least 400% higher than average for comparable doctors outside the M3 system (for specialists in M3 this number can be more than 800% higher than the average for comparable doctors outside the M3 system). M3 charges prices roughly 12 times more than the average prices for comparable service outside the M3 system. The M3 system is the provider of choice for those in the top 5% of the income distribution of the republic of Ivarrona. As the services of M3 are very highly priced, the demand for M3s services is not very high. M3s doctors can easily see several patients more each day, if only M3 would lower its price. So M3s doctors have surplus capacity.

The second hospital chain in Ivarrona is also a private hospital chain that offers for profit care for patients. The chain is known for delivering reasonably priced care to large sections of the Republic of Ivarrona. This group of hospitals is called Ready Relief and Recovery Inc. (popularly known in the country as R3 which we will use to refer to them from hereon). R3 charges roughly 8% (one twelfth) of what M3 charges for comparable services (and procedures). R3s doctors also have surplus capacity. If R3 were to lower its prices, it would be able to attract more patients.

In addition to this there is another chain of non-profit government owned hospitals where treatment is completely free. This chain of hospitals is called Penia Patients Pennies Inc. (popularly known in the country as P3 which we will use to refer to them from hereon). P3 offers free care to anybody that walks in through its portals poor, middle income or rich.

All three hospital chains are located only in the larger islands, with moderate to high population densities. People that live in sparsely populated islands as well as in the far flung archipelagos have to make do with whatever facilities are available locally (ranging from informal medical practitioners and local healers of varying levels of expertise). The government has decided to roll out the advanced Telemedicine system all over the country (even in the far flung, sparsely populated regions) so that every resident of the country lives within 5 miles of a fully equipped and functional kiosk. To make this work, the government has tripled the capacity of P3 by hiring a lot of new doctors and equipping them with the i-MediConnect package. It has also announced a new scheme Universal Telemedicine Scheme (or UTS) to encourage patients in underserved regions to avail themselves of Telemedicine facilities.

Pricing Scheme: Under the UTS scheme all three hospital systems M3, R3, and P3 will be equipped with the i-MediConnect package at every physicians desk. The government has set the price for the three hospitals for each telemedicine consultation (each Tele-consultation session). If a patient is willing to pay the price set for a particular service, the hospital system must accept the patient, to the extent that they have capacity (assume that the government can somehow enforce this). Once they reach full capacity, they no longer need to accept any more patients.

Under the governments scheme the prices are set as follows: M3 will charge a price of $10 for each Tele consultation; R3 will charge a price of $ 3 for each Tele-consultation and P3 will offer Tele-consultations for free (price = zero dollars per Tele-consultation).

Patient Types: For simplicity, let us assume that there a total of 100 patients in the far flung areas that will use this service. Of these 100 patients, there are 10 that are Upper Income Group, 30 that are Middle Income Group and 60 that are Low Income Group.

Capacity: M3 has additional (slack) capacity to take on 10 telemedicine patients, R3 has additional capacity to taken on 30 telemedicine patients and P3 has additional capacity to take on 60 telemedicine patients.

The table below shows how much each category of patients upper, middle and low income values a single Tele consulting session service offered by each hospital group. For instance, the Upper Income Group values a Tele Consulting session offered by M3 at $12. Recall that value is the willingness to pay and if the price of a service exceeds this value, the buyer will not pay for the service. Finally, if a hospital system faces a demand greater than its capacity, it will serve the patients on a first come first served basis.

Patient Type	M3 - Tele Consult Services	R3 - Tele Consult Services	P3 - Tele Consult Services
Upper Income Group	$12.00	$10.00	$5.00
Middle Income Group	$9.00	$6.00	$2.00
Lower Income Group	$7.00	$5.00	$1.00

To answer the following questions, assume that everything happens in a single period. Do not consider multiple periods or use discount factors for time. A patient will choose a hospital and she will be either served (if the hospital has capacity) or be turned away (if the hospital has no capacity). She will not try again at another hospital. This is a simplifying assumption.

(i) How many patients and from which group will choose M3s Tele Consult services? Why?

(ii) How many patients and from which group will choose R3s Tele Consult services? Why?

(iii) How many patients and from which group will choose P3s Tele Consult services? Why?