Pricing Telemedicine Services and Capacity Utilization: The government of the Republic of Ivarrona wants to make sure that its citizens get preventive (preventative) care so as to make its people healthier and more productive. To this end, the government has decided to put in an ambitious Telemedicine infrastructure. The government's plans for infrastructure consist of the following. An advanced telecommunications network that can support upload and download speeds of 1 gigabit (roughly more than 10 times faster than today's average internet speeds). A Telemedicine service that will allow physicians to reach and treat (where possible) patients that cannot reach a doctor's physical location or a hospital easily. They are ready to roll out a special package called 'i-MediConnect' consisting of a laptop, a high-quality digital camera, and a microphone system that can be plugged into any handheld device or laptop (it can run on Windows, MacOS, Linux, etc. and works on almost all major platforms). They also plan to install hundreds of special-purpose kiosks manned by clinicians, built all over the country (Ivarrona). These Kiosks will also have devices that can capture many important elements of patient vitals - including blood pressure, A1C levels, body temperature, etc. (assume that it can do some body fluid capture and analysis too). 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 services 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 M3's services is not very high. M3's doctors can easily see several patients more each day if only M3 would lower its price. So M3's doctors have surplus capacity. The second hospital chain in Ivarrona is also a private hospital chain offering 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). R3's 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 who 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 many new doctors and equipping them with the 'i-MediConnect' package. It has also announced a new 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 physician's 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 the capacity (assume that the government can somehow enforce this). Once they reach full capacity, they no longer need to accept more patients. Under the government's 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; P3 will offer Tele-consultations for free (price = zero dollars per Tele-consultation). Patient Types: For simplicity, let us assume that there are a total of 100 patients in the far-flung areas that will use this service. Of these 100 patients, there are 10 in the Upper Income Group, 30 that are Middle Income Group, and 60 in the Low Income Group. Capacity: M3 has additional (slack) capacity to take on 10 telemedicine patients, R3 has additional capacity to take 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 service price 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/he will be either served (if the hospital has capacity) or turned away (if the hospital has no capacity). The patient will not try again at another hospital. This is a simplifying assumption. 

(1) How many patients and from which group will choose M3's Tele Consult services? Please explain Why.

(2) How many patients and from which group will choose R3's Tele Consult services? Please explain Why. 

(3) How many patients and from which group will choose P3's Tele Consult services? Please explain Why.

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