Growth of the relative unit cost of health care over a horizon

Signature

R_T = R_0 * ((1+g_o)/(1+g_h))^T

Function

Baumol unbalanced growth relative cost and labour allocation function

Computational function

• Computational function: two-sector Baumol scenario from a starting share of hours

  Takes the inputs of a simple two-sector scenario, the starting share of hours in health care, annual productivity growth in each sector, annual wage growth and a horizon, and returns the unit cost indices, the relative cost, the hours ratio and share needed to hold the output mix, and real output growth in each sector. It chains HE-FM-BAUM-002 and HE-FM-BAUM-003, so the later steps use the results of the earlier ones, and its inputs differ from the formulas' symbols: it starts from a share of hours rather than a ratio and includes the wage, which drops out of the relative cost. Holding total hours fixed, it checks that both sectors still produce more.

  Inputs and outputs: s_h0: Health care share of total working hours in the starting year; required, above zero and below 1. Unit: proportion.; g_o: Annual growth of output per hour outside health care, as a decimal; required. Unit: rate per year.; g_h: Annual growth of output per hour in health care, as a decimal; required. Unit: rate per year.; g_w: Annual growth of the hourly wage common to both sectors, as a decimal; required. Unit: rate per year.; T: Horizon in years; required, zero or above. Unit: years.; C_h: Unit labour cost of health care as a multiple of its starting value. Unit: index.; C_o: Unit labour cost of other output as a multiple of its starting value. Unit: index.; R_T: Relative unit cost of health care, C_h divided by C_o. Unit: ratio.; x_0: Starting ratio of health hours to other hours. Unit: ratio.; x_T: Hours ratio needed after T years to hold the output mix. Unit: ratio.; s_hT: Health care share of total hours after T years, also its share of spending. Unit: proportion.; q_o: Real output outside health care as a multiple of its starting value. Unit: index.; q_h: Real output of health care as a multiple of its starting value. Unit: index.

  Assumption: Labour is the only cost, both sectors pay the same wage, productivity grows at constant rates, total hours are unchanged and the ratio of health output to other output is held fixed, as in the article's illustrative example.

  Worked example (Article's twenty-year example): With 10 per cent of hours in health care, productivity growth of 2.0 and 0.5 per cent and wages rising by 2.0 per cent a year, the unit cost of health care rises by a factor of about 1.3449 while other unit costs are unchanged, the health care share of hours rises to about 0.1300, and both sectors' output grows by a factor of about 1.4364. s_h0 = 0.10; g_o = 0.02; g_h = 0.005; g_w = 0.02; T = 20; C_h = 1.3449; C_o = 1; R_T = 1.3449; x_0 = 0.1111; x_T = 0.1494; s_hT = 0.1300; q_o = 1.4364; q_h = 1.4364

  Worked example (Faster wage growth leaves relative cost unchanged): With wages rising by 3.0 per cent a year instead, both unit cost indices rise, to about 1.6346 for health care and 1.2155 for other output, but their ratio is still about 1.3449 and the share of hours is unchanged. s_h0 = 0.10; g_o = 0.02; g_h = 0.005; g_w = 0.03; T = 20; C_h = 1.6346; C_o = 1.2155; R_T = 1.3449; x_0 = 0.1111; x_T = 0.1494; s_hT = 0.1300; q_o = 1.4364; q_h = 1.4364

  Worked example (Equal productivity growth in both sectors): With output per hour growing by 2.0 per cent a year in both sectors, the relative cost stays at 1, the share of hours stays at 0.1000 and both outputs grow by the full productivity factor of about 1.4859. s_h0 = 0.10; g_o = 0.02; g_h = 0.02; g_w = 0.02; T = 20; C_h = 1; C_o = 1; R_T = 1; x_0 = 0.1111; x_T = 0.1111; s_hT = 0.1000; q_o = 1.4859; q_h = 1.4859

  Excel: =((1+WageGrowth)/(1+GrowthHealth))^Years/((1+WageGrowth)/(1+GrowthOther))^Years; =ShareHealth0/(1-ShareHealth0)*RelCostT; =HoursRatioT/(1+HoursRatioT); =(1-ShareHealthT)/(1-ShareHealth0)*(1+GrowthOther)^Years; =ShareHealthT/ShareHealth0*(1+GrowthHealth)^Years Five cells that return R_T, x_T, s_hT, q_o and q_h, with the first three outputs named RelCostT, HoursRatioT and ShareHealthT.

  R: baumol_scenario <- function(s_h0, g_o, g_h, g_w, T) { C_h <- ((1+g_w)/(1+g_h))^T; C_o <- ((1+g_w)/(1+g_o))^T; R_T <- C_h/C_o; x_T <- s_h0/(1-s_h0)*R_T; s_hT <- x_T/(1+x_T); c(C_h = C_h, C_o = C_o, R_T = R_T, x_T = x_T, s_hT = s_hT, q_o = (1-s_hT)/(1-s_h0)*(1+g_o)^T, q_h = s_hT/s_h0*(1+g_h)^T) } Returns a named vector of the outputs.

  Python: def baumol_scenario(s_h0, g_o, g_h, g_w, T): C_h = ((1+g_w)/(1+g_h))**T; C_o = ((1+g_w)/(1+g_o))**T; R_T = C_h/C_o; x_T = s_h0/(1-s_h0)*R_T; s_hT = x_T/(1+x_T); return C_h, C_o, R_T, x_T, s_hT, (1-s_hT)/(1-s_h0)*(1+g_o)**T, s_hT/s_h0*(1+g_h)**T Returns the outputs as a tuple, with no imports needed.

  Test (Health and other output grow by the same factor): With the output mix held fixed and total hours unchanged, real output grows by the same factor in both sectors. Expected result: TRUE. Excel check: =ABS((1-ShareHealthT)/(1-ShareHealth0)*(1+GrowthOther)^Years-ShareHealthT/ShareHealth0*(1+GrowthHealth)^Years)<1E-9

  Test (Wage growth does not change the relative cost): The relative cost computed with any wage growth equals the productivity ratio raised to the horizon. Expected result: TRUE. Excel check: =ABS(((1+WageGrowth)/(1+GrowthHealth))^Years/((1+WageGrowth)/(1+GrowthOther))^Years-((1+GrowthOther)/(1+GrowthHealth))^Years)<1E-9

  Common error (Reading higher unit costs as lower real output): A rising health care share of spending can be read as health care crowding out other output. In the twenty-year example the share rises from 10 to 13 per cent, yet real output in both sectors is about 43.6 per cent higher, the sense in which Baumol argued that rising health costs remain affordable when productivity grows elsewhere.

  Source: Baumol WJ. Macroeconomics of unbalanced growth: the anatomy of urban crisis. American Economic Review. 1967;57(3):415-426. Section 2, equations (1) to (5), with Proposition 1 on unit costs under a common wage and Proposition 3 on the transfer of labour when the ratio of outputs is held constant.

  C_h = ((1+g_w)/(1+g_h))^T; C_o = ((1+g_w)/(1+g_o))^T; R_T = C_h / C_o; x_0 = s_h0 / (1 - s_h0); x_T = x_0 * R_T; s_hT = x_T / (1 + x_T); q_o = (1 - s_hT) / (1 - s_h0) * (1+g_o)^T; q_h = s_hT / s_h0 * (1+g_h)^T

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Implementations

• Excel

  Projected relative unit cost of health care in one cell

  With the starting ratio in RelCost0, the two growth rates as decimals in GrowthOther and GrowthHealth and the horizon in Years, Excel returns the relative unit cost after that many years.

  =RelCost0*((1+GrowthOther)/(1+GrowthHealth))^Years

Assumptions

• Constant annual productivity growth in each sector

  Output per hour grows at a constant rate in each sector and compounds once a year. The formula is a discrete-time restatement of Baumol's continuous-time model, in which the progressive sector grows at a constant compounded rate and the other sector has constant productivity.

• Productivity measured on a comparable basis in both sectors

  g_o and g_h measure the same kind of productivity, for example output per hour, with output measured comparably. Health care output is hard to count, and the ONS healthcare measure is multi-factor productivity, which the ONS says is not directly comparable with whole-economy labour productivity, so a gap between published series may partly reflect measurement.

Worked examples

• Twenty years at 2.0 and 0.5 per cent 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 in health care. Starting from a ratio of 1, the relative unit cost of health care after twenty years is about 1.3449, so health care is about 34.5 per cent dearer relative to other goods.

  R_0 = 1; g_o = 0.02; g_h = 0.005; T = 20; R_T = 1.3449

• One year of unbalanced productivity growth

  With the same growth rates the relative unit cost of health care rises by a factor of about 1.0149 in a single year, about 1.49 per cent, as in the article.

  R_0 = 1; g_o = 0.02; g_h = 0.005; T = 1; R_T = 1.0149

• Twenty years with no productivity growth in health care

  In Baumol's 2 per cent illustration, productivity is constant in the slow sector, so over twenty years at 2 per cent growth elsewhere the relative unit cost of health care rises to about 1.4859 times its starting value.

  R_0 = 1; g_o = 0.02; g_h = 0; T = 20; R_T = 1.4859

Common errors

• Adding up the Baumol productivity gap instead of compounding it

  Multiplying the gap in growth rates by the number of years, 1 + 0.015 × 20 = 1.30, understates the relative cost after twenty years, which compounds to about 1.3449. The shortfall grows with the horizon and with the size of the gap.

• Applying the Baumol labour cost factor to full unit costs

  R_T describes the labour cost of a unit of health care. Applying it to full unit costs, including medicines, equipment and buildings, assumes that those inputs follow the same path as staff costs, which the model does not claim. A projection of full costs applies the factor to the staff share and projects other inputs separately.

Sources

• Baumol relative costs of the progressive and nonprogressive sectors

  Baumol WJ. Macroeconomics of unbalanced growth: the anatomy of urban crisis. American Economic Review. 1967;57(3):415-426. Section 2, equations (1) to (3) and Proposition 1, which give the ratio of unit costs in the two sectors as rising exponentially with the productivity growth rate of the progressive sector, and section 3, with the 2 per cent illustration.

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• Rising relative prices in technologically stagnant industries

  Nordhaus WD. Baumol's diseases: a macroeconomic perspective. NBER Working Paper 12218. Cambridge, MA: National Bureau of Economic Research; 2006. Section I on the cost and price disease, under which costs and prices in industries with relatively low productivity growth rise relative to the average, and the abstract, which reports rising relative prices in stagnant sectors in United States industry data for 1948 to 2001.

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Canonical Identity