Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Baumol unbalanced growth relative cost and labour allocation function

f(g_o, g_h, T, x_0, beta, g_econ) = (R_T, x_T, s_hT, B)

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.

  • Relative unit labour cost of health care from output per hour

    R = a_o / a_h

    Gives the labour cost of one unit of health care relative to one unit of other output. Labour cost per unit is the hourly wage divided by output per hour, w/a_h in health care and w/a_o elsewhere, and with the same wage w in both sectors the ratio of the two reduces to the ratio of output per hour. Health care is therefore relatively dearer whenever output per hour is lower there, whatever the level of pay.

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

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

    Projects the relative unit labour cost of health care T years ahead when output per hour grows at a constant annual rate g_h in health care and g_o elsewhere. Each year the relative cost is multiplied by (1 + g_o)/(1 + g_h), so health care becomes relatively dearer whenever g_o exceeds g_h. The wage does not appear, so the result is the same whether pay rises with productivity in the progressive sector or not.

  • Health care share of working hours with the output mix held fixed

    x_T = x_0 * ((1+g_o)/(1+g_h))^T; s_hT = x_T / (1 + x_T)

    Gives the hours needed in health care when the ratio of health output to other output is held constant, for example because a publicly financed service keeps pace with the rest of the economy. The ratio of health hours to other hours grows by the same annual factor as the relative cost, and the share of total hours follows from that ratio. With equal wages and only labour costs, the share of hours is also health care's share of total spending.

  • Baumol component of real health spending growth in OBR projections

    B = beta * g_econ

    Gives the contribution of the Baumol effect to annual real growth in health spending as a coefficient times projected economy-wide productivity growth. The Office for Budget Responsibility (OBR) uses this top-down form, with an OECD estimate of the coefficient, within the other cost pressures that it adds to demographic and income effects.