Fundamental Cost Analysis

Concept

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Theoretically, a Cost Function is a mathematical function that describes the relationship between the total cost of producing healthcare services or interventions and the level of output or activity. It provides the theoretical basis for analysing production efficiency, economies of scale, marginal cost and resource allocation. In health economics, cost functions are used to model how healthcare expenditure changes as patient volume, service intensity or resource utilisation varies, thereby informing economic evaluation, provider efficiency analysis and healthcare planning.

Mathematically, a cost function expresses total cost as a function of one or more output variables and input prices. The simplest specification represents total cost as the sum of fixed and variable costs, while more advanced models may incorporate multiple outputs, input prices, technological factors and nonlinear relationships. Parameters are commonly estimated using econometric techniques such as ordinary least squares, maximum likelihood estimation or stochastic frontier analysis.

In practice, cost functions are estimated using observational data from hospitals, clinics, insurers or administrative databases. Estimated cost functions are used to predict healthcare expenditure, evaluate economies of scale, estimate marginal costs, compare provider efficiency and inform reimbursement policies. They also provide the mathematical basis for many cost projections used in health technology assessment and health system planning.

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Purpose

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Used to describe and estimate the relationship between healthcare costs and service production, supporting expenditure forecasting, efficiency analysis, economic evaluation and healthcare resource allocation.

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Mathematical Formulae

Primary Formula

TC(Q) = FC + VC(Q)

where:

• TC(Q) = total cost as a function of output
• FC = fixed cost
• VC(Q) = variable cost as a function of output Q

Supporting Formulae

Average Cost:

AC(Q) = TC(Q) / Q

Marginal Cost:

MC(Q) = dTC(Q) / dQ

Linear cost function:

TC(Q) = ?? + ??Q

Related Mathematical Methods

• Cost Analysis
• Average Cost
• Marginal Cost
• Economies of Scale
• Cost Estimation
• Ordinary Least Squares Regression
• Stochastic Frontier Analysis

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Example

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A hospital's annual operating costs consist of �2,000,000 in fixed costs and �350 per outpatient visit.

The cost function is:

TC(Q) = �2,000,000 + �350Q

If Q = 8,000 outpatient visits:

TC = �2,000,000 + (�350 ? 8,000)

TC = �4,800,000

This cost function can be used to estimate future expenditure as patient activity changes and to calculate average and marginal costs for economic evaluation.

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Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(B2:B101,A2:A101,TRUE,TRUE)	Estimates parameters of a linear cost function from observed cost and activity data.
FORECAST.LINEAR	=FORECAST.LINEAR(9000,CostRange,ActivityRange)	Predicts healthcare expenditure at a specified activity level.
SLOPE	=SLOPE(CostRange,ActivityRange)	Estimates the variable cost per additional unit of output.
INTERCEPT	=INTERCEPT(CostRange,ActivityRange)	Estimates fixed costs in a linear cost function.
SUM	=FixedCost+(VariableCost*Activity)	Calculates total predicted healthcare cost from the estimated cost function.

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VBA (Optional)

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VBA can automate estimation, updating and forecasting of healthcare cost functions using routinely collected provider activity and expenditure data.

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Sources

• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• Varian HR. Intermediate Microeconomics: A Modern Approach. W.W. Norton.
• Nicholson W, Snyder C. Microeconomic Theory: Basic Principles and Extensions. Cengage Learning.
• Coelli TJ, Rao DSP, O'Donnell CJ, Battese GE. An Introduction to Efficiency and Productivity Analysis. Springer.