Signature
x_T = x_0 * ((1+g_o)/(1+g_h))^T; s_hT = x_T / (1 + x_T)
| Inputs | Definition | Unit |
|---|---|---|
x_0 | Hours in health care divided by hours elsewhere in the starting year, for example 0.10/0.90 when health care employs 10 per cent of hours | ratio, without unit |
g_o | Constant annual growth rate of output per hour in the rest of the economy, as a decimal | rate per year |
g_h | Constant annual growth rate of output per hour in health care, as a decimal | rate per year |
T | Number of years since the starting year | years |
x_T | Hours of labour in health care divided by hours in the rest of the economy, T years after the starting year | ratio, without unit |
|---|---|---|
s_hT | Hours in health care as a proportion of total hours after T years; above zero and below 1 | proportion |
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
Hours ratio and health care share of hours in two cells
With the starting hours ratio in HoursRatio0, the growth rates in GrowthOther and GrowthHealth and the horizon in Years, the first cell, named HoursRatioT, returns the hours ratio and the second returns health care's share of total hours.
=HoursRatio0*((1+GrowthOther)/(1+GrowthHealth))^Years; =HoursRatioT/(1+HoursRatioT)
Assumptions
Ratio of health output to other output held constant
The volume of health care is held in a fixed ratio to other output, because demand is price inelastic or income elastic or because government finances the service. If demand responds to the rising relative price instead, Baumol's model predicts that the slow sector's output tends to decline and this formula does not apply.
Constant productivity growth and a common wage in the hours projection
Output per hour grows at constant rates in each sector and labour is paid the same wage in both, as in the relative cost formula. The share of total spending equals the share of hours only under the common wage and the labour-only cost assumption.
Worked examples
Health care share of hours rising from 10 to 13 per cent
Health care starts with 10 per cent of hours, a ratio of 0.111111 to other hours, and productivity grows by 2.0 per cent a year elsewhere and 0.5 per cent in health care. After twenty years the ratio is about 0.1494 and health care's share of hours, and of spending, is about 0.1300, as in the article.
x_0 = 0.111111; g_o = 0.02; g_h = 0.005; T = 20; x_T = 0.1494; s_hT = 0.1300
Health care share of hours with equal productivity growth
When output per hour grows at 2.0 per cent a year in both sectors, the hours ratio and the health care share of 10 per cent stay where they started, so no labour has to move into health care to hold the output mix.
x_0 = 0.111111; g_o = 0.02; g_h = 0.02; T = 20; x_T = 0.1111; s_hT = 0.1000
Common errors
Applying the Baumol cost factor to the share of hours
Multiplying the starting share by the relative cost factor, 0.10 × 1.3449 = 0.1345, overstates health care's share after twenty years, which is about 0.1300. The factor applies to the ratio of health hours to other hours, and over long horizons the shortcut can give a share above 1.
Projecting a rising health share of hours when output is not held fixed
The rising share follows only if health output keeps its ratio to other output. Where demand falls as the relative price rises, Baumol's model predicts declining relative output in the slow sector instead, and Nordhaus found falling relative real output in stagnant United States industries, so the hours path depends on the demand assumption.
Sources
Baumol labour transfer when relative outputs are maintained
Baumol WJ. Macroeconomics of unbalanced growth: the anatomy of urban crisis. American Economic Review. 1967;57(3):415-426. Section 2, equations (4) and (5) and Proposition 3, which hold the output ratio constant and give labour in the nonprogressive sector as a share of the total labour supply that approaches the whole labour force over time, and Proposition 2 on declining output when demand is not highly inelastic.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0