To illustrate I create this model of NHS treatment.
In summary first. The model involves people arriving at hospital in a queue. Private patients (quite rightly) because they have paid more, get to go to the front of the queue. The remaining capacity treats public patents.
Once through the admission gate (G) all patients enjoy access to the same health-care, which is better than public funded alone because of the extra contribution of privation patients.
All looks good. However what really happens is that as more people go private, public patients have to wait longer, and since a large number do pay tax they actually end up getting a poorer service than if there was no private contribution.
Private individuals actually get a better service than they would by private means alone, since they are "stealing" resources from the public that are waiting.
The obvious question is at what rate of private admissions do public lose out. The suprising answer is that if there are any private patients, then the public lose out!
While the model may be simplistic, and even inaccurate, it illustrates how easy it is to misunderstand the logic of systems. I'm beginning to realise such misunderstanding are not the exception, but the rule. All of biology, genetics and economics is suspect due to such misunderstanding. As a result "private/public" is far too complex an issue to be handled by anyone but real professionals and has been sold to the nation either deliberately or accidentally by mistake.
The Model
========
Assuming that all patients are treated in same hospitals with money pooled from public and private sources,then:
N : size of population
i : illness rate
p : probability they go private
V : Total value of hospital services (pooling both tax and private insurance)
G : number admitted through Hospital gate per time
t : tax rate (all)
s : insurance rate (Private)
Therefore:
Nlp : patients go private in time.
Ni(1-p) : go public.
Nip : go 1st through the gate,
G-Nip : remaining spaces for public
Total amount paid by people in the queue =
Nip(t+s) : Total amount paid by ill private [they pay taxes too].
Ni(1-p)t : Total amount paid by ill public
Total value of the health service = V
V is the sum of these two + (taxes of healthy individuals)
Share of health service =
Private get : Nip/G % share of service V
public get : (G-Nip)/G % of V
The value of private is then value of treatment over cost of treatment:
(VNip/G)/(Nip(t+s)) = V/[G(s+t)]
Value public: value of those thru gate / amount paid by public
(V(G-Nip)/G)/(Ni(1-p)t) = V(G -Nip)/(GNit(1-p))
Let 'a' be the ratio of the number admitted to hospital over the number needing treatment i.e. a = G/(Ni)
V/(Nisa+Nita) == -((pV-Va)/(Nita-Nipta))
they are equal when p = (sa+ta-t)/s
Let 'b' be the ratio of tax to insurance rate: b = t/s
p -> a - b + a b
This is the proportion turning to private that gives equal value for public and private:
The contour plot for: (tax rate / private insurance cost) / (healthcare admission per ill person)
When number admitted is equal to the number ill (a > 1) it can be seen that increasing the tax rate (increasing b) bends the line of equal value to the left, i.e. fewer people must be admitted to give private a good value. This is obvious as if all public get treated, then being first in the queue has no meaning and so the private pay extra for nothing.
However if people are waiting in the queue, then increasing tax means that the curve bends the other way meaning that more people need admission to keep public and private experiecing equal value for money.
Clearly room for a better model, but it illustrates how easily the private can end up having the public pay for them if health services are shrunk, and how easily the public could have the private pay for them if the services are expanded too much. The problem is that what incentive is there to support the public? Governments and private companies will be all to happy to shrink admissions, thereby leading to a subsidy of the rich by the poor. This I suspect is the true process of society across time: the subsidy of the wealthy life style, by the poor.
