Cardinality Proof Problems Cardinality Proof Problems (1) Suppose f: S right arrow T, and that f is one to one. Prove that S ~ f(S). (2) Show that R~(a,b) for every a Solution f is a function from S to T f is given to be one to one i.e. each element in f has a unique image in T and also if f(a) = f(b) then a = b Hence if n is the no of elements in S, f(S) will have image for each of the n element. Hence n[f(S)] = n(S) S~f(S) ------------------------------------------------------------------------------------------------------- 2) b) Consider the function f(x) = sinx, where 0<=x<=pi/2 sin 0 =0 and sin pi/2 =1 Also sinx is one to one Hence (0,pi/2)~(0,1) c) Consider the interval (-pi/2, pi/2) as domain for the function f(x) = tan x As from -pi/2 to 0 tan is negative, we see that f is one to one. Hence f(x) = (-tan pi/2, tanpi/2) = (-infinity, infinity) Or (-pi/2, pi/2)~R.

1. Case Study 4.2
Investing in Cardiology Services “It’s time for us to commit to building a center of excellence in
cardiac care,” said Shea, the hospital’s chief financial officer. “Mercy and Central did it three
years ago, and they are doing extremely well. Medicare pays well, and the private insurers pay
even better. We cannot afford to miss this opportunity.”“Perhaps,” said Emerson, the hospital’s
chief medical officer. “Let me lay out a couple of issues that we need to consider. First, there’s
no guarantee that the insurers will keep on paying so well for cardiology services. Most
observers think that prices are higher than they need to be, and Medicare is facing a financial
crisis. I think that Medicare will move to a bundled payment for many cardiology services before
too long. The cardiologists will continue to do nicely, but they will have powerful incentives to
cut back on imaging, to cut back on catheteriza-tions, and to switch to less invasive
interventions. The net effect will be to slash hospital revenues. Second, there’s no guarantee that
we will be able to attract a team of top-notch cardiologists. Those guys are in short supply these
days. Without a really superb team, we will not get this off the ground.” “OK,” said Shea. “Here
is what the planning team has forecast. We will need a 24-bed unit, office space for three
cardiologists, a 64-slice CT scanner, and a cardiac catheterization lab. We estimate that this setup
will cost us $10 million. We estimate that this change will increase inpa-tient days by 1,500,
resulting in profits of $2 million. We also estimate that we will have 1,000 new outpatient
procedures, which will generate $500,000 in profits. That represents a very nice return on our
investment and leaves us some room for error in our cost and revenue forecasts. Personally, I do
not think that Medicare will make any changes fast. The ability of the federal government to
avoid taking action is unsurpassed. And I am confident that we will be able to recruit
cardiologists.”
Discussion questions:
1. How likely is a change in Medicare payment? What probability should you assign to it?
2. What will happen to hospital profits if Medicare does switch to bundled payments?
3.How likely is a failure to recruit three excellent cardiologists? What probability should you
assign to this endeavor? What will happen to profits if you are able to recruit only two excellent
cardiologists?
4. If you set up this scenario as a decision tree, which of your assumptions become clear to other
decision makers?
5.What are the advantages of making your assumptions clear? What are the disadvantages?
It chapter 4 in Economics for healthcare managers, third edition.
Solution

2. 1.How likely is a change in Medicare payment? What probability should you assign to it
Medicare Assignment Rates of Physicians: Their Responses to Changes in Reimbursement
Policy
Abstract
A physician's Medicare assignment rate is one measure of his or her willingness to participate in
the Medicare program. The assignment rate reflects the proportion of services provided to
Medicare beneficiaries for which the physician accepts the Medicare reasonable fee as payment
in full. Generally, Medicare reasonable fees are lower than the payment which a physician
receives from providing the same service to a private patient or to a Medicare patient who is not
treated on assignment. Because Medicare eligibles not treated on an assigned basis are
financially liable for the difference between the physician's charge and the Medicare reasonable
fee, the assignment rate is an indication of the out-of-pocket costs borne by Medicare eligibles.
One factor which may affect the willingness of physicians to accept patients on assignment is the
difference between the reimbursement which he or she may receive in the private market and the
fee received from treating Medicare eligibles on assignment. Throughout this paper we assume
that the physician's private price or billed charge is equivalent to the level of reimbursement
received from treating privately insured patients and Medicare non-assigned patients. Since the
level of reimbursement is generally no greater than the billed charge and may be less, this
assumption may overstate the actual reimbursement received by the physician. In all instances,
reimbursement refers to the aggregate amount received by the physician from all sources for a
given service. The lower a physician's Medicare reasonable fee relative to the private market fee
the less willing he/she may be to participate in Medicare assignment. This paper examines the
effect of changes in Medicare reimbursement on the assignment rates of physicians. It also
predicts Medicare assignment rates under a policy option which would increase Medicare
reasonable fees to the level of prevailing fees.
Medicare Reimbursement
Medicare legislation originally provided for reimbursement of physicians on the basis of their
reasonable, customary, and prevailing charges. Physicians were paid by Medicare on the basis of
their reasonable charge, which is the lowest of: (1) the actual charge for the service provided, (2)
the physician's median (customary) charge for services rendered during the calendar year
immediately preceding the fee screen year, or (3) the prevailing charge, set at the 75th percentile
of all customary charges made for similar services in the same geographic area.1
Although the CPR methodology for determining the reasonable fee put some constraints on the
amount which the Medicare program will pay physicians for their services to the elderly, it does
not necessarily constrain the out-of-pocket costs of the elderly for medical care. Physicians

3. participating in the Medicare program are permitted two billing options, assignment and non-
assignment, which they may exercise on a claim-by-claim basis. When a claim is assigned,
physicians submit their bills directly to the Medicare carrier for their area. Medicare pays the
physician 80 percent of the reasonable charge for the service, less any unpaid deductible. The
physician bills the patient for the 20 percent coinsurance and any unpaid deductible.
The second option is to bill on a non-assigned basis. Physicians choosing non-assignment must
bill the patient directly for the full amount charged. The patient in turn pays the physician
directly for the services rendered and secures reimbursement from Medicare for 80 percent of the
reasonable charge, less any unpaid deductible. Patients who are treated on a non-assigned basis
are liable for the entire difference between the billed and reasonable charges, as well as the
coinsurance and the deductible. From the physician's perspective, treating non-assigned
Medicare patients is equivalent to seeing private patients.
Physicians treating elderly patients who are also eligible for Medicaid do not have the same
billing options. Joint Medicare/Medicaid claims are in practicality only taken on an assigned
basis. The reimbursement levels are equivalent to the reasonable charge as determined by the
Medicare carrier, and Medicaid picks up the coinsurance and the deductible on behalf of the
patient. The only option the physician has in this case is to not treat the patient at all.
Because patients whose physicians do not accept assignment generally face larger out-of-pocket
expenditures than those treated on assignment, the assignment rates of physicians provide one
indication of the share of the cost borne by the beneficiaries. Increases in the assignment rates of
physicians are desirable because (ceterus paribus) they may reduce costs of medical care for the
elderly. By examining how the assignment rate changes when billed charges and reasonable fees
change, this paper will provide evidence on the extent to which changes in Medicare
reimbursement can be used to alter the assignment rate.
2.What will happen to hospital profits if Medicare does switch to bundled payments
Bundled payment
While most hospital leaders see the advantages of moving to bundled payments for an episode of
care, many are unprepared either for the mindset or the mechanics required to implement the
emerging reimbursement model. Here are the concerns and poss...
undled payment is for a specific set of services, such as an episode of care for hip or knee
replacement or all services for one year of care for a patient with asthma or diabetes. A hospital
and physicians would work together to care for a health plan's patients and may share in any
savings if costs are lower than a specified target. The providers also are required to care for
patients who have complications during the episode of care, usually 30 to 120 days for acute
conditions or a full year for chronic care.
Key Design Choices That Would Affect Savings

4. Payment bundling is a broad concept that could take many forms. The federal savings that could
result from greater bundling would depend on many design specifications, such as the types of
bundles constructed and their scope, the duration of the services covered by a bundle, the levels
at which bundled payments were set and the mechanisms used to set them, the method of
payment used, the schedule for implementing the bundling policy, and the terms of participation
(in particular, whether bundling would be voluntary or mandatory).
In general, more extensive bundles encompass more spending and may provide more
opportunities to generate savings. But they also expose health care providers to more financial
risk, particularly when the total costs of the bundle depend on services delivered by a variety of
providers who are not affiliated. Bundling payments for different providers can also raise
significant administrative challenges, and some solutions to those challenges may weaken
incentives to control costs. In addition, aggregating payments while giving doctors, hospitals,
and other providers greater leeway to share savings among themselves could encourage those
providers to generate more episodes of care.
Among the many design issues that arise, the levels of bundled payments and the rate-setting and
payment mechanisms are perhaps the most important. Fundamentally, reducing federal spending
through bundled payments would require providers to be paid less overall than they are under
current law—either because they would be delivering fewer or less complex services to enrollees
or because they would be receiving less money per service
3.How likely is a failure to recruit three excellent cardiologists
The Growing Demand for Cardiologists
Several things continue to drive demand for cardiologists. Elsewhere, I have argued that various
scientific, social, and demographic “demand catalysts” outweigh factors that might decrease
demand for cardiologists during the next decade Although the relative influence of each demand
factor will change over time, one thing is certain: the cardiovascular disease burden in the United
States is great and growing. Even if the public focuses more energy on self-preservation and
makes better choices with respect to cardiotoxic habits such as smoking or cardioprotective
habits such as exercise and healthy diets, demographers warn that we are confronting an
expanding population of older Americans that will require much more cardiovascular care.
Potential