Signature
R = a_o / a_h
| Inputs | Definition | Unit |
|---|---|---|
a_o | Output per hour of labour in the rest of the economy, the progressive sector, as a level or as an index | output per hour, or index |
a_h | Output per hour of labour in health care, measured in the same way as a_o and above zero | output per hour, or index |
R | Labour cost per unit of health care output divided by labour cost per unit of other output, c_h/c_o. When a_o and a_h are indices with a common base year, R is the relative cost as a multiple of its base-year value | ratio, without unit |
|---|
Function
Baumol unbalanced growth relative cost and labour allocation function
Maps output per hour, or its annual growth, in health care and in the rest of the economy to the labour cost of a unit of health care relative to a unit of other output, and to the share of working hours that health care needs when the mix of outputs is held fixed. With a common wage in both sectors the wage cancels, so the relative cost depends only on relative productivity. A related top-down form, used in UK fiscal projections, turns economy-wide productivity growth into an allowance for real health spending growth.
Try this function
Implementations
Excel
Relative unit labour cost of health care in one cell
With output per hour, or its index, in cells named OutputPerHourOther and OutputPerHourHealth, Excel returns the relative unit labour cost of health care.
=OutputPerHourOther/OutputPerHourHealth
Assumptions
Common hourly wage in health care and other sectors
Wages in the two sectors move together, because labour can move between them in the long run, so the same hourly wage applies to both. Baumol allowed that pay in one activity can lag behind another for a time, but not indefinitely unless the activity is disappearing.
Labour as the only cost in the Baumol ratio
Costs other than labour are ignored, a simplification Baumol himself called unrealistic. In health care, medicines, equipment and buildings follow their own price paths, so the ratio describes the labour component of unit cost only.
Worked examples
Relative unit cost after twenty years of unbalanced productivity growth
In the article's illustrative example output per hour grows by 2.0 per cent a year outside health care and 0.5 per cent a year in health care, so after twenty years the indices are about 1.485947 and 1.104896. With both sectors starting at 1, a unit of health care then costs about 1.3449 times as much, relative to other output, as in the starting year.
a_o = 1.485947; a_h = 1.104896; R = 1.3449
Relative unit cost when health care output per hour is constant
In Baumol's limiting case output per hour in the slow sector does not change, so its index stays at 1 while the other sector's index reaches about 1.485947 after twenty years at 2 per cent. Every rise in pay then feeds straight into the relative cost of health care, which rises by the full productivity gain elsewhere.
a_o = 1.485947; a_h = 1; R = 1.4859
Common errors
Inverting the output per hour ratio in the Baumol relative cost
Dividing health care output per hour by output per hour elsewhere gives about 0.7436 instead of 1.3449 in the twenty-year example, which reads as health care becoming cheaper relative to other output when it has become dearer.
Reading a rising Baumol relative cost as falling health productivity
A rising relative cost does not mean that health care output per hour has fallen or that services are run inefficiently. In the twenty-year example output per hour in health care rises by about 10.5 per cent, yet its relative unit cost rises by about 34.5 per cent, because productivity grows faster elsewhere. Baumol argued that such cost rises arise for reasons largely beyond the control of those running the services.
Sources
Baumol unit cost of output with a common wage
Baumol WJ. Macroeconomics of unbalanced growth: the anatomy of urban crisis. American Economic Review. 1967;57(3):415-426. Section 1 on the premises (common wages, labour costs only) and section 2, Proposition 1 and its proof, which gives unit cost in each sector as the wage times labour divided by output and shows that relative costs behave the same way whether or not wages follow productivity.
Unit cost growth as input cost growth less productivity growth
Nordhaus WD. Baumol's diseases: a macroeconomic perspective. NBER Working Paper 12218. Cambridge, MA: National Bureau of Economic Research; 2006. Section II, equation (2), the unit cost function in growth rates, in which unit cost grows with input costs less productivity growth, and section I on the cost and price disease of stagnant industries.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0