Academic Articles Hospitals International Health Care Systems Optimal Ins (Theory)

Optimal Contracts for Health Services in the Presence of Waiting Times and Asymmetric Information

Should hospitals with long waiting times have higher or lower budget transfers? Offering hospitals who have low wait times more money will increase a hospital’s incentives to decrease wait times. On the other hand, thus policy may hurt the busier hospitals and may not alleviate the wait times of those who are waiting the longest. In the case of public school transfers, if the best schools are rewarded, this encourages achievement, but may punish the worst off kids (i.e.: those at poorer schools). Transfers to low-performing school may mute incentives to increase achievement.

The issue of hospital payment structure is analyzed by Luigi Siciliani in his article on optimal contracts B.E. Journal of Economic Analysis & Policy. As with any thesis which claims to give an optimum solution, this optimum is based on some assumptions. This paper uses four major assumptions.

  1. Demand for treatment can be controlled by dumping some patients. Doctors can tell patients who wish to have medical treatment that they either a) don’t really need it or b) that they will not provide it
  2. The purchaser (i.e.: NHS, an insurance company, Medicare, etc.) can not observe the number of people dumped.
  3. Dumping is costly for the specialist. By dumping patients, the specialist receives more complaints about their service level. Thus, either the physician’s reputation is tarnished (a cost) or the physician must spent more time (another cost) convincing the patient that they do not need treatment.
  4. Hospitals differ in potential demand for treatment, either due to the catchment area of the hospital or from having a better or worse reputation.

Another key assumption is that no co-payment charges can be issued. This assumption is plausible, because it basically represents the British NHS system. Thus, the optimal solution must be seen not as the ideal optimal, but as the optimal with a centralized payer and no co-payments.

The Model

Hospitals have parameter θ which describes the public hospital type. This parameter θ indicates potential demand in the absence of a rationing system.

For each treatment, patients differ in the value they would receive from treatment. For instance, healthy patients would not benefit from heart surgery, but individuals with coronary artery blockages likely would benefit from surgery. Thus the author assumes that individuals’ value from treatment is uniformly distributed between v0 and v1.

Patients have three options:

  1. They can be treated at a public sector hospital after a wait of time w, [up(v,p)]={∫T0 v dt} -p=vT-p]
  2. They can be treated in a private sector hospital with no wait, but pay a price of p, [uNHS(v,w)]={∫Tw v*g(w) dt} =vg(w)(T-w)]
  3. or they can receive no treatment[unone=0]

There are two costs to going to the public hospital. First, the individual has to wait w weeks longer, so they do not get to enjoy the benefit of the treatment for as extended a period of time. Secondly, since 0<g(w)<1, the treatment becomes less effective or less valued the longer the patient waits.

Thus, from the math above, we can see that a person will choose a public hospital if and only if:

  • v<V(w)=p/{T-g(w)*(T-w)

The comparative statics show that longer wait times decrease the probability of using a public hospital, higher prices, p, decrease the probability of using a private hospital, and higher valuations, v, increase the probability of using a private hospital.

Demand for public hospital services is written as:

  • D(θ,w,x)=θV(w)-x
  • x is the number of patients who are dumped (i.e.: not added to the waiting list)

The number of treatment supplied by hospital θ is y(θ) and since supply must equal demand, we have:

  • θV(w)-x=y(θ)

The authors claim that providers receive disutility from dumping patients. Also, hospitals receive more disutility when they dump patients who value the treatment more (i.e.: high v, this is more likely to be the sicker patients). Thus, we are lead to our first major conclusion.

  • Conclusion 1: The patients who are dumped are the ones with the lower benefits from treatment. This means that hospitals dump the patients who don’t really need the treatment.

After some more math, the Dr. Siciliani states a second conclusion:

  • Conclusion 2: A mix of explicit rationing (through dumping) and implicit rationing (through waiting) is therefore optimal. Siciliani explains that: “Rationing by waiting alone induces excessive disutility for patients. Rationing by dumping alone generates excessive disutility for the specialists.”

The author continues to conclude that a separating or pooling equilibrium may occur.

“Under symmetric information, the optimal contract is for the purchaser simply to over a transfer in exchange for the provision of the desired level of activity and waiting time, without leaving any rent to the provider…Under asymmetric information, we found that a separating equilibrium exists when it is optimal for the purchaser of health services to contract more activity and higher waiting times to hospitals with higher demand. In this case providers with low potential demand have an incentive to mimic hospitals with high potential demand. To induce hospitals to self-select, the purchaser needs to pay a rent to hospitals with lower potential demand. [But] if it is not optimal for the purchaser to contract more activity and higher waiting time to hospitals with higher demand, then a separating equilibrium may not exist.”


One main problem with the paper is that it assumes that patients with a high value, v, cost the same to treat as low value patients. If v is a proxy for sickness, this is likely not to be the case; sicker patients with a high v are more expensive to treat. If this were the case, then conclusion 1 would not hold. Public hospitals would instead treat patients with the lowest benefit and dump patients with intermediate benefits–the high benefit patients would still go to the private sector hospital.

Also, the paper does not take into account any strategic interaction between hospitals. “If hospitals with higher potential demand are contracted higher waiting times, then patients will switch from the hospitals with high potential demand to hospitals with low potential demand, increasing excessively the amount of dumping and consequent disutility for hospitals with low potential demand.”