Scope Efficiency

Scope Efficiency in Hospital Service Configuration

Scope efficiency asks whether a hospital or health system delivers its mix of services at lower cost by keeping them together than the same services would cost in more specialised organisations. In health economics it is the efficiency question behind decisions to merge hospitals, to run emergency and planned care on the same site, or to concentrate services in specialist centres. The economics literature usually speaks of economies of scope when joint production saves cost and diseconomies of scope when separation is cheaper. Two families of measure are used, a ratio of stand-alone to joint costs taken from an estimated multiproduct cost function, and comparisons between efficiency frontiers estimated for specialised and for mixed hospitals. They answer related but different questions, and hospital evidence shows that the answer depends on which pair of services is examined.

Why joint production can save or waste resources

A hospital that treats both elective and emergency patients, or both medical and surgical ones, shares buildings, staff and support services across them. Lindaas and colleagues, studying Norwegian hospitals, give the example of sharing physical, human and digital resources across departments during surges in demand, and they set against it the cost of harmonising different competences and traditions across departments. Which effect dominates is an empirical matter for each pair of services.

The elective and emergency pair shows the mechanism. Emergency demand fluctuates, while planned care can be scheduled, so a hospital that does both can draw on planned-care resources to meet emergency peaks instead of holding costly reserve capacity, at the price of disrupting elective lists when emergencies surge. Lindaas and colleagues note that surgical and medical care need such distinct skills and physical resources that staff and equipment cannot always be shared between them.

Scope efficiency is one part of efficiency in health care. It concerns the shape of the cost frontier across output mixes, which is a different matter from how far an individual provider sits above that frontier.

The cost function measure and its extrapolation problem

The classical measure compares the cost of producing all outputs in one organisation with the summed cost of producing each output alone. Marini and Miraldo, estimating cost functions for English NHS trusts, write the global measure for three outputs, and the two-output version is shown below.

$$SC = \frac{C(y_1, 0) + C(0, y_2) - C(y_1, y_2)}{C(y_1, y_2)}$$

where $SC$ is the degree of economies of scope, $C(y_1, 0)$ and $C(0, y_2)$ are the costs of producing each output on its own, and $C(y_1, y_2)$ is the cost of producing both together. A positive value means economies of scope, a negative value diseconomies, and zero means joint and separate production cost the same. Marini and Miraldo also report product-specific scope measures, which compare producing one service alone and all the others together with producing everything jointly.

The difficulty lies in the stand-alone costs. Kittelsen and Magnussen point out that the measure presupposes knowledge of the cost of completely specialised hospitals, which their data did not contain, so any cost function estimate would need to extrapolate to a large extent. The functional form adds a further limit, because a standard translog cost function cannot be evaluated at zero output. Marini and Miraldo used a generalised multiproduct form for this reason, since it allows zero in the permissible domain of outputs.

Frontier measures of scope convexity

Frontier methods avoid extrapolating to hospitals that do not exist. Kittelsen and Magnussen used data envelopment analysis with total running costs as the only input and seven outputs. Because the method assumes a convex technology, which Lindaas and colleagues note rules out diseconomies of scope on a single frontier, the sample is split before estimation. Hospitals are ranked by the share of a given activity, such as elective care, and the top and bottom fifths are treated as specialised while the middle three fifths are treated as differentiated. A separate frontier is then estimated for each group.

Lindaas and colleagues define cost efficiency as the ratio of necessary to actual cost, and they measure the degree of economies of scope for each differentiated hospital as a cost scope convexity measure.

$$CS_i = \frac{CE_i^{S}}{CE_i^{D}} = \frac{C^{S}(\mathbf{y}_i) / c_i}{C^{D}(\mathbf{y}_i) / c_i} = \frac{C^{S}(\mathbf{y}_i)}{C^{D}(\mathbf{y}_i)}$$

where $CS_i$ is the scope convexity measure for hospital $i$, $CE_i^{S}$ and $CE_i^{D}$ are its cost efficiency against the specialised and the differentiated frontiers, $C^{S}(\mathbf{y}_i)$ and $C^{D}(\mathbf{y}_i)$ are the minimum costs of its output mix $\mathbf{y}_i$ on each frontier, and $c_i$ is its actual cost. The actual cost cancels, so the measure depends only on the two frontiers. Values above 1 mean that the differentiated frontier lies outside the specialised one and indicate economies of scope, and values below 1 indicate diseconomies.

Kittelsen and Magnussen explain that the classical measure combines a scale effect, the cost advantage of different scales of operation, with a convexity effect, the cost advantage of the output mix alone, and that their frontier measure targets the convexity effect. Specialisation in this framework is relative. In their 1992 to 1999 Norwegian data, the fifth of observations with the highest elective share still had on average less than half of their production as elective care.

What hospital studies have found

Hospital studies differ in country, method and the dimension of the service mix examined. Their results describe particular service pairs and do not amount to a general verdict on specialisation.

In the surgical and medical dimension, Kittelsen and Magnussen found on average a 7% to 8% production potential at constant cost when hospitals produced a mix of surgical and medical visits rather than a specialised set-up. They also summarise an earlier survey for the NHS by Aletras, Jones and Sheldon, which reported few findings on economies of scope and noted that merged hospitals mostly had higher costs after merging, except for small rural hospitals, while it was difficult to tell whether diseconomies of scope or of scale were responsible. Preyra and Pink observed that Ontario regulators predicted gains from consolidation with an accounting approach rather than an economic one.

Using scope estimates in service configuration decisions

A scope measure tells a planner whether a given configuration of services would cost less at best practice if it were rearranged. Kittelsen and Magnussen caution that specialisation advantages can only be harvested where two or more hospitals of similar mix and size can reallocate production between them. Lindaas and colleagues judge that their trust-level findings should inform sector-wide trends rather than decisions at lower levels, because some services within a trust may face economies of scope while others face diseconomies. They also note that an optimal hospital structure should take account of factors outside production costs, such as patient travel time.

Marini and Miraldo draw two policy points for England. They suggest that Foundation Trusts could achieve efficiency gains by offering different services on different sites, and they warn that hospital mergers are likely to reduce access to care, especially for providers in rural areas. Scope savings are provider cost savings, so in an economic evaluation of a reconfiguration they sit alongside changes in patient costs, outcomes and access rather than replacing them.

Worked example with two services and two measures

All numbers in this example are illustrative. A district provides elective surgery and emergency medical care. An estimated cost function gives a joint cost of GBP 8.5 million a year, and stand-alone costs of GBP 5.2 million for elective surgery and GBP 4.1 million for emergency medical care.

1. Degree of economies of scope. Joint production saves GBP 0.8 million against the GBP 9.3 million of separate provision.

$$SC = \frac{5.2 + 4.1 - 8.5}{8.5} = \frac{0.8}{8.5} = 0.0941$$

where the figures are the stand-alone and joint costs in GBP million. On this measure, splitting the two services would add 9.4% to the joint cost.

2. Saving measured against separate provision. Joint provision costs 0.8 / 9.3 = 0.086, or 8.6%, less than the GBP 9.3 million of separate provision. The ratio of separate to joint cost, 9.3 / 8.5 = 1.094, is simply 1 + SC, and reading it as a 9.4% cut on separate costs overstates the saving.

3. Sensitivity to the extrapolated stand-alone cost. No emergency-only unit exists in the data, so its cost is extrapolated. If the estimate were GBP 3.3 million instead of GBP 4.1 million, the measure would fall to zero.

$$SC = \frac{5.2 + 3.3 - 8.5}{8.5} = \frac{0}{8.5} = 0$$

where the figures are in GBP million. A change of GBP 0.8 million in one extrapolated cost removes the whole estimated advantage.

4. Frontier measure for a differentiated hospital. A differentiated hospital spends GBP 100 million. The frontier for differentiated hospitals says its output mix could be produced for GBP 90 million, and the frontier for specialised hospitals says GBP 98 million.

$$CS = \frac{98 / 100}{90 / 100} = \frac{0.98}{0.90} = 1.089$$

where 0.98 and 0.90 are its cost efficiency scores against the specialised and differentiated frontiers. Specialised best practice would need about 8.9% more cost than differentiated best practice for this output mix, a difference of GBP 8 million, so the mixed configuration shows economies of scope.

5. Separating scope from the hospital's own inefficiency. The same hospital spends GBP 10 million more than the differentiated frontier, a cost efficiency of 0.90. That gap is a matter of technical efficiency within the present configuration, and specialising would not remove it. At best practice, keeping the mix costs GBP 90 million and specialising costs GBP 98 million.

Scope efficiency compared with nearby concepts

Scope efficiency concerns the mix of outputs a provider produces, while scale efficiency concerns whether it operates at the right size for its technology. A merger changes both at once. Kittelsen and Magnussen isolate the convexity effect, and Lindaas and colleagues use variable returns to scale frontiers to reduce confusion between scale and scope. Cost efficiency and technical efficiency measure the distance between a provider and the frontier, and Lindaas and colleagues frame scope as a question about the shape of the frontier. Productive efficiency asks whether a given output is produced at least cost, while scope efficiency asks which organisations should produce which outputs. Allocative efficiency asks whether the services produced are the ones that best meet the health system's objectives, which no cost-based scope measure answers.

Limitations and common misreadings

Outputs in scope studies are counts of activity, usually adjusted for case mix through diagnosis-related groups, and they are not health outcomes. A configuration that looks scope efficient may still differ in quality, and Lindaas and colleagues note that DEA can include quality, geography or other cost shifters only where they can be expressed as inputs or outputs. Kittelsen and Magnussen warn that DEA makes the hospitals with the largest production of each output efficient by default, and that repeated observations of the same hospitals overstate the degrees of freedom, so nominal significance tests reject too often. Their average efficiency scores favoured specialised hospitals slightly, while the frontier pointed to economies of scope.

Several misreadings recur. Shared premises alone do not establish economies of scope, because the measure depends on whether total cost actually falls. A positive scope estimate does not show that a merger will save money, since mergers change scale and management costs as well. A global estimate for a hospital says nothing about each service pair inside it. Scores from different samples or specifications are not directly comparable, because each frontier is specific to its data.

Sources

• Kittelsen SAC, Magnussen J. Economies of scope in Norwegian hospital production: a DEA analysis. HERO Working Paper 2003:8. Oslo: Health Economics Research Programme, University of Oslo; 2003. https://med.uio.no/helsam/forskning/nettverk/hero/publikasjoner/skriftserie/2003/HERO2003_8.pdf
• Lindaas NA, Anthun KS, Kittelsen SAC, Magnussen J. Economies of scope in the Norwegian public hospital sector. European Journal of Health Economics. 2025;26(2):325-335. https://doi.org/10.1007/s10198-024-01704-z
• Marini G, Miraldo M. Economies of scale and scope in the English hospital sector. Imperial College Business School Discussion Paper 2009/05. London: Imperial College London; 2009. https://spiral.imperial.ac.uk/handle/10044/1/4202
• Preyra C, Pink G. Scale and scope efficiencies through hospital consolidations. Journal of Health Economics. 2006;25(6):1049-1068. https://doi.org/10.1016/j.jhealeco.2005.12.006
• Wang J, Zhao Z, Mahmood A. Relative efficiency, scale effect, and scope effect of public hospitals: evidence from Australia. IZA Discussion Paper No. 2520. Bonn: Institute for the Study of Labor; 2006. https://docs.iza.org/dp2520.pdf