Production Possibility Frontier

Production Possibility Frontier in Health Care Resource Allocation

A production possibility frontier marks the edge of what a health system can achieve with the money, staff and technology it has, drawn for two outputs at a time. Points inside it waste resources, and points beyond it cannot be reached without more resources or better technology. Health economists use it to make opportunity cost visible, because its slope gives how much of one output must be given up for one more unit of the other. In the health economics version the two axes measure the health of two patient groups, and the curve then shows which distributions of health a given budget can buy. That reading separates questions of efficiency, which concern whether the system is on the curve, from questions of equity, which concern where on the curve it should be.

Limited resources and the shape of the curve

The frontier starts from scarcity. The OpenStax textbook by Greenlaw, Shapiro and MacDonald explains that because society has limited labour, land, capital and raw materials at any point in time, there is a limit to the quantities of goods and services it can produce, and it illustrates the point with healthcare on one axis and education on the other. Society can choose any combination on or inside the frontier but does not have enough resources to produce outside it. More education means giving up some healthcare, and the slope measures that forgone healthcare per unit of extra education.

The same text contrasts the frontier with a consumer's budget constraint. A budget line is straight because its slope is set by relative prices, which an individual takes as fixed. The frontier is curved because of diminishing returns, so its slope changes along its length. The authors call this the law of increasing opportunity cost, which holds because some resources suit one output better than the other. When the last doctors become science teachers and the last emergency rooms become kindergartens, education gains little and health loses a great deal, which is why the curve bends outward.

The health frontier for two patient groups

Culyer, in a York Centre for Health Economics research paper, builds the health economics version from two health production functions. Each function links health care expenditure to the health of a group, rising at a diminishing rate as less productive interventions are used and harder-to-reach groups are covered. Culyer adds that a production function embodies an assumption of cost-effectiveness. With two groups, A and B, and an increase in the budget, the health frontier shows all the possible combinations of health for the two groups given the extra funding and the two production functions.

$$h_A = f_A(x_A), \qquad h_B = f_B(x_B), \qquad x_A + x_B = X$$

where $h_A$ and $h_B$ are the health gains of groups A and B (for example in QALYs), $f_A$ and $f_B$ are their health production functions, $x_A$ and $x_B$ are the amounts of the extra budget spent on each group, and $X$ is the extra budget. The frontier is traced by giving $x_A$ every value between 0 and $X$, with $x_B = X-x_A$ each time.

Differentiating along the frontier gives its slope.

$$\frac{dh_B}{dh_A} = -\frac{f_B'(x_B)}{f_A'(x_A)}$$

where $f_A'(x_A)$ and $f_B'(x_B)$ are the marginal products, the extra health bought by one more pound spent on group A or group B at the current split. The absolute value of the slope is the marginal rate of transformation, the health of group B given up per unit of health gained by group A. The marginal rate of substitution is its counterpart on the side of preferences.

Culyer states that from any point on the frontier the health of one group can be increased only at the expense of less health for the other, a property he calls Pareto efficiency, and that any point under the frontier is by definition inefficient. A larger budget shifts it outwards and a smaller one shrinks it towards the origin. Better technology also pushes it out, as the OpenStax text notes.

What the frontier says about efficiency

Textbooks do not all use the efficiency terms in the same way. Palmer and Torgerson, in a BMJ note on definitions, separate technical efficiency, using given resources to maximum advantage, from productive efficiency, the maximisation of health outcome for a given cost, and from allocative efficiency, achieving the right mixture of healthcare programmes to maximise the health of society.

On a health frontier built from production functions that already embody cost-effectiveness, a point inside the curve fails at least one of the first two tests. Every point on the curve passes both, and choosing among those points is the allocative question. The OpenStax text uses productive efficiency for any point on the frontier and allocative efficiency for the point that represents the combination society most desires.

Choosing a point on the frontier

The frontier rules out waste but does not pick an allocation. The OpenStax authors state that economics does not presume to tell a society what choice it should make along its frontier, and Culyer writes that although every point on the health frontier is efficient, every point is not equitable, so additional criteria are needed to select one. A social welfare function is the formal way to supply those criteria.

Culyer compares several equity principles on the same frontier. Equal health gain picks the point where the two groups gain the same amount. The rule that a QALY is of equal social value to whomever it accrues picks the point where the slope of the frontier equals minus one, where a small gain to one group equals the loss to the other. In the notation above this is the point where $f_A'(x_A) = f_B'(x_B)$, so the last pound buys the same health in both groups, which is the equal-margins condition used in marginal analysis and the point of maximum total health. In Culyer's example where group B lives in a remote region and is five times more costly to treat per case, the equal-health allocation, which gives most of the extra money to B, and the allocation maximising health are equally efficient, and he concludes that the choice between them is entirely a matter of equity. The equity-efficiency trade-off page covers how much total health is given up for a fairer distribution.

Worked example with GBP 10 million and two patient groups

All figures in this example are illustrative. A commissioner has an extra GBP 10 million to spend in five tranches of GBP 2 million on two groups, A and B. Each further tranche spent on a group buys fewer QALYs, as the first table shows, and tranches can be split in proportion if needed.

1. Trace the frontier. Group A reaches 400, 700, 900, 1,020 and 1,080 QALYs after one to five tranches, and group B reaches 250, 450, 600, 700 and 750.

2. Opportunity cost along the curve. Each shift of one tranche to B, beginning at 5 and 0, costs A its last tranche and gives B its next one. The five shifts cost 60 / 250 = 0.24, then 120 / 200 = 0.60, then 200 / 150 = 1.33, then 300 / 100 = 3.00 and finally 400 / 50 = 8.00 QALYs for group A per QALY gained by group B. The rising ratio is the law of increasing opportunity cost, and it makes the frontier bow outward.

3. The point of maximum total health. The split of 3 and 2 gives the largest total, 1,350 QALYs. There the last tranche in each group buys 200 QALYs, so each costs GBP 2,000,000 / 200 = GBP 10,000 per QALY. Giving B a third tranche at A's expense leaves 1,350 less 200 plus 150, or 1,300, and the reverse shift leaves 1,350 less 200 plus 120, or 1,270. The ratio in step 2 passes through 1 at this point, which is Culyer's slope of minus one.

4. Equal health gain. No whole-tranche split gives equal gains, since 2 and 3 still leaves A with 700 against B's 600. Shifting a further quarter tranche (GBP 500,000) out of A's second tranche and into B's fourth costs A 300 × 0.25 = 75 QALYs and gains B 100 × 0.25 = 25, giving 625 QALYs each. The total is 1,250, so equal gains leave the population 100 QALYs short of the maximum of 1,350. Both allocations lie on the frontier, so choosing between them is an equity judgement.

5. A point inside the frontier. Suppose the 3 and 2 split is chosen but B's two tranches are spent on low-value care that buys 300 QALYs instead of 450. The outcome, 900 for A and 300 for B with a total of 1,200, lies inside the frontier. Moving to the frontier point 3 and 2 gives B 150 more QALYs at no cost to A, a gain that needs no equity judgement.

6. A shift in the frontier. A cheaper way of reaching group B raises each B tranche by 20 per cent, to 300, 240, 180, 120 and 60 QALYs, so B's cumulative gains become 300, 540, 720, 840 and 900. The frontier moves out everywhere except at A's end point. Totals become 1,020 + 300 = 1,320 for 4 and 1, 900 + 540 = 1,440 for 3 and 2, and 700 + 720 = 1,420 for 2 and 3, so 3 and 2 remains the health-maximising split, now at 1,440 QALYs.

Step 5 is a waste problem that every equity view would fix, while steps 3 and 4 differ only in distribution and need a value judgement.

Using the frontier in economic evaluation and HTA

Health technology assessment works on the same logic with a fixed budget. A new technology funded from a fixed budget moves spending away from other services, and the health those services would have produced is the opportunity cost that a cost-effectiveness threshold is meant to represent. Net health benefit compares the gain from the technology with that displaced health.

The frontier also shows why a common outcome measure matters. Palmer and Torgerson note that productive efficiency cannot assess a reallocation of resources away from geriatric care and into mental illness services, because the health outcomes are incommensurate. A frontier with outputs in different units, such as hip operations and therapy sessions, cannot rank its points until both are valued on one scale, such as the QALY, or by explicit judgement.

Frontiers that are easy to confuse

Several frontiers in health economics share a name and a picture but answer different questions. The cost-effectiveness frontier joins the strategies that survive dominance on a plot of cost against effect, so it compares mutually exclusive options for one decision, not combinations of outputs from shared resources. The Pareto frontier is the general set of options that cannot be improved on one objective without worsening another. The production possibility frontier is the case in which the objectives are outputs and feasibility is set by resources and technology.

Empirical production frontiers are a third kind. Jacobs, in a York discussion paper on hospital efficiency, notes the ability of data envelopment analysis to manage complex production environments with multiple input and output technologies like hospitals. A frontier of that kind is built from the best observed units, while the production possibility frontier is the theoretical maximum given technology.

Limitations and common misreadings

The frontier is usually a conceptual device, and the OpenStax authors note that their curve has no numbers on the axes because the resources involved are not known. Real health systems produce many outputs, so a two-output curve is a slice through a larger set, and Culyer's version abstracts from uncertainty, noting that the axes can be read as expected health.

The outward bow is an assumption about production. Culyer derives it from diminishing marginal returns in each group's production function and describes the shape as convex, while the OpenStax text calls the same curve outward-bending. Wildman shows that where income inequality affects health, health production functions can be non-concave and the health production possibility frontier non-concave as well, in which case a point where the slope equals minus one need not give the most total health.

Three misreadings recur. The first treats any move along the frontier as an efficiency gain or loss, when every point on it is equally efficient and the move is a distributional choice. The second treats an equity goal as a reason to tolerate waste. Culyer shows that an equitable allocation inside the frontier can move towards it, raising health for both groups without any loss of equity. The third reads the opportunity cost from average ratios such as total QALYs per pound, when the slope that matters is the marginal one, which in the example climbs steadily between 0.24 and 8.00.

Sources

• Culyer AJ. Efficiency, Equity and Equality in Health and Health Care. CHE Research Paper 120. York: Centre for Health Economics, University of York; December 2015. https://www.york.ac.uk/media/che/documents/papers/researchpapers/CHERP120_Efficiency_Equity_Equality_Health_Care.pdf
• Greenlaw SA, Shapiro D, MacDonald D. Principles of Economics 3e. Houston: OpenStax; 2022. Section 2.2, The production possibilities frontier and social choices. https://openstax.org/books/principles-economics-3e/pages/2-2-the-production-possibilities-frontier-and-social-choices
• Jacobs R. Alternative Methods to Examine Hospital Efficiency: Data Envelopment Analysis and Stochastic Frontier Analysis. CHE Discussion Paper 177. York: Centre for Health Economics, University of York; 2000. https://www.york.ac.uk/media/che/documents/papers/discussionpapers/CHE%20Discussion%20Paper%20177.pdf
• Palmer S, Torgerson DJ. Definitions of efficiency. BMJ. 1999;318(7191):1136. doi:10.1136/bmj.318.7191.1136
• Wildman J. Modelling health, income and income inequality: the impact of income inequality on health and health inequality. Journal of Health Economics. 2003;22(4):521-538. doi:10.1016/S0167-6296(03)00003-1