The measurement of productivity in the nonmarket sector

Abstract

In many sectors of the economy, governments either provide various services at no cost or at highly subsidized prices. Examples are the health, education and general government sectors. The System of National Accounts 1993 recommends valuing these nonmarket outputs at their costs of production but it does not give much guidance on exactly how to do this. In this paper, an explicit methodology is developed that enables one to construct these marginal cost prices. However, in the main text, an activity analysis approach is taken in order to simplify the analysis, so in particular, constant returns to scale, no substitution production functions for the specific activities in the nonmarket sector are assumed. It is shown that it is possible to obtain meaningful measures of Total Factor Productivity growth in this framework. An “Appendix” relaxes some of the restrictive assumptions that are used in the main text.

Appendix: The measurement of output, input and productivity in the nonmarket sector for more general production functions

In this “Appendix”, we will relax the assumption that the procedure production functions are of the fixed coefficients, no input substitution variety. Theoretical output, input and productivity indexes will be defined in this more general context and we will exhibit various observable indexes that can approximate these theoretical indexes to the accuracy of at least a first order approximation.

As in Sect. 2, we assume that there are two procedures in periods 0 and 1 but now we assume that there is a production function, f ^t_i , for procedure i in period t where y ^t_i  = f ^t_i (x ^t_i1 , x ^t_i2 , …, x ^t_iN(i) ) = f ^t_i (x ^t_i ) is the amount of output for procedure i that can be produced by the input vector x ^t_i in period t for t = 0, 1 and i = 1, 2. We assume that each production function f ^t_i (x_i) is a nonnegative, increasing, continuous, concave and linearly homogeneous function in the components of its input vector x_i. As in Sect. 2, we assume that in each period t, producers in sector i face the positive vector of input prices, w ^t_i  ≡ [w ^t_i1 , w ^t_i2 , …, w ^t_iN(i) ] for i = 1, 2 and t = 0, 1. For each period t and each sector i, the sector i total cost function C ^t_i (y_i, w_i) associated with each procedure can be defined as follows:

$$ \begin{aligned} {\text{C}}_{\text{i}}^{\text{t}} ({\text{y}}_{\text{i}} ,{\text{w}}_{\text{i}} ) & \equiv { \min }_{{{\text{x}}^{\prime } {\text{s}}}} \{ {\text{w}}_{\text{i}} \cdot {\text{x}}_{\text{i}} :{\text{f}}_{\text{i}}^{\text{t}} ({\text{x}}_{\text{i}} ) \ge {\text{y}}_{\text{i}} \} ;\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1\\ & = {\text{c}}_{\text{i}}^{\text{t}} ({\text{w}}_{\text{i}} ){\text{y}}_{\text{i}} \\ \end{aligned} $$

(26)

where c ^t_i (w_i) ≡ C ^t_i (1, w_i) is the period t unit cost function for sector i; i.e., it is equal to the minimum cost of producing one unit of sector i output using the period t technology if the sector faces the vector of input prices w_i. The unit cost function c ^t_i will satisfy the same regularity conditions as the production function f ^t_i ; i.e., c ^t_i (w_i) will be a nonnegative, increasing, continuous, concave and linearly homogeneous function in the components of the input price vector w_i.¹⁹

We assume that in each period, producers minimize the cost of producing their procedure outputs.²⁰ Thus letting y ^t_i and x ^t_i denote the observed scalar output and input vector of sector i in period t, we will have the following equalities:

$$ {\text{w}}_{\text{i}}^{\text{t}} \cdot {\text{x}}_{\text{i}}^{\text{t}} = {\text{C}}_{\text{i}}^{\text{t}} \left( {{\text{y}}_{\text{i}}^{\text{t}} ,{\text{w}}_{\text{i}}^{\text{t}} } \right) = {\text{c}}_{\text{i}}^{\text{t}} \left( {{\text{w}}_{\text{i}}^{\text{t}} } \right){\text{y}}_{\text{i}}^{\text{t}} ;\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(27)

We also assume that each unit cost function is differentiable at the observed input prices for each sector and each period so that Shephard’s (1953, 11) Lemma implies the following relationships between the input quantity vectors x ^t_i and the corresponding output levels y ^t_i :

$$ {\text{x}}_{\text{i}}^{\text{t}} = \nabla_{\text{w}} {\text{C}}_{\text{i}}^{\text{t}} \left( {{\text{y}}_{\text{i}}^{\text{t}} ,{\text{w}}_{\text{i}}^{\text{t}} } \right) = \nabla_{\text{w}} {\text{c}}_{\text{i}}^{\text{t}} \left( {{\text{w}}_{\text{i}}^{\text{t}} } \right){\text{y}}_{\text{i}}^{\text{t}} ;\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(28)

The observed input–output vectors a ^t_i for each sector i and each time period can be defined as the observed input vectors x ^t_i divided by the corresponding output levels y ^t_i :

$$ {\text{a}}_{\text{i}}^{\text{t}} \equiv {{{\text{x}}_{\text{i}}^{\text{t}} } \mathord{\left/ {\vphantom {{{\text{x}}_{\text{i}}^{\text{t}} } {{\text{y}}_{\text{i}}^{\text{t}} }}} \right. \kern-\nulldelimiterspace} {{\text{y}}_{\text{i}}^{\text{t}} }};\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(29)

Comparing (28) with (29) shows that the vectors of first order partial derivatives of the unit cost functions, ∇_wc ^t_i (w ^t_i ), are also observable and are equal to the corresponding input–output vectors a ^t_i :

$$ \nabla_{\text{w}} {\text{c}}_{\text{i}}^{\text{t}} \left( {{\text{w}}_{\text{i}}^{\text{t}} } \right) = {\text{a}}_{\text{i}}^{\text{t}} ;\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(30)

The period t unit cost for sector i, c ^t_i (w ^t_i ), are also observable and can serve as our cost based output prices p ^t_i for units of output in sector i during period t; i.e., define the output prices p ^t_i as follows:

$$ {\text{p}}_{\text{i}}^{\text{t}} \equiv {\text{c}}_{\text{i}}^{\text{t}} \left( {{\text{w}}_{\text{i}}^{\text{t}} } \right) = {\text{w}}_{\text{i}}^{\text{t}} \cdot {\text{a}}_{\text{i}}^{\text{t}} = {{{\text{w}}_{\text{i}}^{\text{t}} \cdot {\text{x}}_{\text{i}}^{\text{t}} } \mathord{\left/ {\vphantom {{{\text{w}}_{\text{i}}^{\text{t}} \cdot {\text{x}}_{\text{i}}^{\text{t}} } {{\text{y}}_{\text{i}}^{\text{t}} }}} \right. \kern-\nulldelimiterspace} {{\text{y}}_{\text{i}}^{\text{t}} }};\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(31)

Definitions (31) imply the following relationships between the value of output p ^t_i y ^t_i and the value of input w ^t_i  · x ^t_i in sector i for period t:²¹

$$ {\text{p}}_{\text{i}}^{\text{t}} {\text{y}}_{\text{i}}^{\text{t}} = {\text{c}}_{\text{i}}^{\text{t}} \left( {{\text{w}}_{\text{i}}^{\text{t}} } \right){\text{y}}_{\text{i}}^{\text{t}} = {\text{w}}_{\text{i}}^{\text{t}} \cdot {\text{x}}_{\text{i}}^{\text{t}} ;\quad {\text{i}} = 1, 2;\;{\text{t}} = 0, 1. $$

(32)

We now in a position to define the health sector’s period t total cost function, C^t, but first we require some further notation. Let y ≡ [y₁, y₂] be a two dimensional reference vector of possible health sector outputs and let w ≡ [w₁, w₂] be an N(1) + N(2) dimensional vector of reference input prices. Define the health sector’s period t total cost function C^t as the sum of the period t procedure cost functions:

$$ \begin{aligned} {\text{C}}^{\text{t}} ({\text{y}},{\text{w}}) & \equiv {\text{C}}_{ 1}^{\text{t}} ({\text{y}}_{ 1} ,{\text{w}}_{ 1} ) + {\text{C}}_{ 2}^{\text{t}} ({\text{y}}_{ 2} ,{\text{w}}_{ 2} );\quad {\text{t}} = 0, 1\\ & = {\text{c}}_{ 1}^{\text{t}} ({\text{w}}_{ 1} ){\text{y}}_{ 1} + {\text{c}}_{ 2}^{\text{t}} ({\text{w}}_{ 2} ){\text{y}}_{ 2} \quad {\text{using}}\, ( 2 7 ). \\ \end{aligned} $$

(33)

We will use the sector’s total cost function C^t(y, w) in order to define indexes of health sector technical progress, output growth and input price growth going from period 0 to period 1 in what follows.²² As a preliminary step, insert the data pertaining to period t into definition (33) and we obtain the following equations:

$$ \begin{aligned} {\text{C}}^{\text{t}} \left( {{\text{y}}^{\text{t}} ,{\text{w}}^{\text{t}} } \right) & = {\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1}^{\text{t}} } \right){\text{y}}_{ 1}^{\text{t}} + {\text{c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2}^{\text{t}} } \right){\text{y}}_{ 2}^{\text{t}} ;\quad {\text{t}} = 0, 1\\ & = {\text{w}}_{ 1}^{\text{t}} \cdot {\text{x}}_{ 1}^{\text{t}} + {\text{w}}_{ 2}^{\text{t}} \cdot {\text{x}}_{ 2}^{\text{t}} \quad {\text{using}}\, ( 3 2 )\\ & = {\text{p}}_{ 1}^{\text{t}} {\text{y}}_{ 1}^{\text{t}} + {\text{p}}_{ 2}^{\text{t}} {\text{y}}_{ 2}^{\text{t}} \quad {\text{using}}\, ( 3 1 ). \\ \end{aligned} $$

(34)

We now use the total cost function in order to define a family of cost based output quantity indexes, α(y⁰, y¹, w, t), as follows:

$$ \begin{aligned} \alpha \left( {{\text{y}}^{0} ,{\text{y}}^{ 1} ,{\text{w}},{\text{t}}} \right) & \equiv {{{\text{C}}^{\text{t}} \left( {{\text{y}}^{ 1} ,{\text{w}}} \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{\text{t}} \left( {{\text{y}}^{ 1} ,{\text{w}}} \right)} {{\text{C}}^{\text{t}} \left( {{\text{y}}^{0} ,{\text{w}}} \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{\text{t}} \left( {{\text{y}}^{0} ,{\text{w}}} \right)}} \\ & = {{\left[ {{\text{c}}_{ 1}^{\text{t}} ({\text{w}}_{ 1} ){\text{y}}_{ 1}^{ 1} + {\text{c}}_{ 2}^{\text{t}} ({\text{w}}_{ 2} ){\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{\text{t}} ({\text{w}}_{ 1} ){\text{y}}_{ 1}^{ 1} + {\text{c}}_{ 2}^{\text{t}} ({\text{w}}_{ 2} ){\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 3 ). \\ \end{aligned} $$

(35)

Thus the theoretical output quantity index α(y⁰, y¹, w, t) defined by (35) is equal to the (hypothetical) total cost C^t(y¹, w) of producing the vector of observed period 1 procedure outputs, y¹ ≡ [y ¹₁ , y ¹₂ ], divided by the total cost C^t(y⁰, w) of producing the vector of observed period 0 procedure outputs, y⁰ ≡ [y ⁰₁ , y ⁰₂ ], where in both cases, we use the technology of period t and assume that the service providers face the vector of reference input prices, w ≡ [w₁, w₂], where w_i is a reference vector of input prices for sector i. Thus for each choice of technology (i.e., t could equal 0 or 1) and for each choice of a reference vector of input prices w, we obtain a (different) cost based output quantity index.

Following the example of Konüs (1939), it is natural to single out two special cases of the family of output quantity indexes defined by (35): one choice where we use the period 0 technology and set the reference prices equal to the period 0 input prices w⁰ ≡ [w ⁰₁ , w ⁰₂ ] and another choice where we use the period 1 technology and set the reference prices equal to the period 1 input prices w¹ ≡ [w ¹₁ , w ¹₂ ]. Thus define these special cases as α₀ and α₁:

$$ \begin{aligned} \alpha_{0} & \equiv \alpha \left( {{\text{y}}^{0} ,{\text{y}}^{ 1} ,{\text{w}}^{0} ,0} \right) \\ & = {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 5 )\\ & = {{\left[ {{\text{p}}_{ 1}^{0} {\text{y}}_{ 1}^{ 1} + {\text{ p}}_{ 2}^{0} {\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{p}}_{ 1}^{0} {\text{y}}_{ 1}^{ 1} + {\text{ p}}_{ 2}^{0} {\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{p}}_{ 1}^{0} {\text{y}}_{ 1}^{0} + {\text{ p}}_{ 2}^{0} {\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{p}}_{ 1}^{0} {\text{y}}_{ 1}^{0} + {\text{ p}}_{ 2}^{0} {\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 1 )\\ & = {\text{Q}}_{\text{L}} \quad {\text{using}}\, ( 1 0 ); \\ \end{aligned} $$

(36)

$$ \begin{aligned} \alpha_{ 1} & \equiv \alpha \left( {{\text{y}}^{0} ,{\text{y}}^{ 1} ,{\text{w}}^{ 1} , 1} \right) \\ & = {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 5 )\\ & = {{\left[ {{\text{p}}_{ 1}^{ 1} {\text{y}}_{ 1}^{ 1} + {\text{ p}}_{ 2}^{ 1} {\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{p}}_{ 1}^{ 1} {\text{y}}_{ 1}^{ 1} + {\text{ p}}_{ 2}^{ 1} {\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{p}}_{ 1}^{ 1} {\text{y}}_{ 1}^{0} + {\text{ p}}_{ 2}^{ 1} {\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{p}}_{ 1}^{ 1} {\text{y}}_{ 1}^{0} + {\text{ p}}_{ 2}^{ 1} {\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 1 )\\ & = {\text{Q}}_{\text{P}} \quad {\text{using}}\, ( 1 1 ). \\ \end{aligned} $$

(37)

Thus the theoretical cost based output quantity index α₀ that uses the period 0 technology and period 0 input prices w⁰ is equal to the observable Laspeyres output quantity index Q_L that was defined earlier in the main text by (10) and the theoretical cost based output quantity index α₁ that uses the period 1 technology and period 1 input prices w¹ is equal to the observable Paasche output quantity index Q_P that was defined earlier by (11). Since both theoretical output quantity indexes, α₀ and α₁, are equally representative, a single estimate of cost based output quantity growth should be set equal to a symmetric average of these two estimates. We will choose the geometric mean as our preferred symmetric average²³ and thus our preferred theoretical measure of cost based output quantity growth is the following Fisher type theoretical index, α_F:

$$ \begin{aligned} \alpha_{\text{F}} & \equiv [\alpha_{0} \alpha_{ 1} ]^{ 1/ 2} \\ & = \left[ {{\text{Q}}_{\text{L}} {\text{Q}}_{\text{P}} } \right]^{ 1/ 2} \quad {\text{using}}\, ( 3 6 )\,{\text{and}}\, ( 3 7 )\\ & = {\text{Q}}_{\text{F}} \quad {\text{using}}\,{\text{definition}}\, ( 1 2 ). \\ \end{aligned} $$

(38)

Thus our preferred measure of cost based output growth is equal to the observable Fisher quantity index, Q_F, which was defined earlier by (12) in the main text.

We now turn our attention to theoretical measures of input price growth. We now use the total cost function in order to define a family of input price indexes, β(w⁰, w¹, y, t), as follows:

$$ \begin{aligned} \beta \left( {{\text{w}}^{0} ,{\text{w}}^{ 1} ,{\text{y}},{\text{t}}} \right) & \equiv {\text{C}}^{\text{t}} {{\left( {{\text{y}},{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{y}},{\text{w}}^{ 1} } \right)} {{\text{C}}^{\text{t}} \left( {{\text{y}},{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{\text{t}} \left( {{\text{y}},{\text{w}}^{0} } \right)}} \\ & = {{\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2} } \right]} {\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{\text{t}} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{\text{t}} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2} } \right]}}\quad {\text{using}}\, ( 3 3 ). \\ \end{aligned} $$

(39)

Thus the theoretical output quantity index β(w⁰, w¹, y, t) defined by (39) is equal to the (hypothetical) total cost C^t(y, w¹) of producing the reference vector of outputs, y ≡ [y₁, y₂], when the service providers face the period 1 observed vector of input prices w¹, divided by the total cost C^t(y, w⁰) of producing the same reference vector of outputs, y, when the service providers face the period 0 observed vector of input prices w⁰, where in both cases, we use the technology of period t. Thus for each choice of technology (i.e., t could equal 0 or 1) and for each choice of a reference vector of output quantities y, we obtain a (different) input price index.

Again following the example of Konüs (1939) in his analysis of the true cost of living index, it is natural to single out two special cases of the family of input price indexes defined by (39): one choice where we use the period 0 technology and set the reference quantities equal to the period 0 quantities y⁰ ≡ [y ⁰₁ , y ⁰₂ ] and another choice where we use the period 1 technology and set the reference quantities equal to the period 1 quantities y¹ ≡ [y ¹₁ , y ¹₂ ]. Thus define these special cases as β₀ and β₁:

$$ \begin{aligned} \beta_{0} & \equiv \beta \left( {{\text{w}}^{0} ,{\text{w}}^{ 1} ,{\text{y}}^{0} ,0} \right) \\ & = {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 9 )\\ & = {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 4 ); \\ \end{aligned} $$

(40)

$$ \begin{aligned} \beta_{ 1} & \equiv \beta \left( {{\text{w}}^{0} ,{\text{w}}^{ 1} ,{\text{y}}^{ 1} , 1} \right) \\ & = {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}\quad {\text{using}}\, ( 3 9 )\\ & = {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}\quad {\text{using}}\, ( 3 4 ). \\ \end{aligned} $$

(41)

We now encounter a problem: the hypothetical unit costs c ⁰_i (w ¹_i ) and c ¹_i (w ⁰_i ) which appear in (40) and (41) are not observable so we cannot calculate the theoretical input price indexes β₀ and β₁. However, we can find bounds to these indexes as well as first order Taylor series approximations to them, which are observable, as we now show.

As mentioned above, the unit cost functions, c ^t_i (w_i) are concave functions in their input price variables w_i. It is well known that the first order Taylor series approximation to a concave function lies above (or is coincident with) the concave function²⁴ so we have the following inequalities:²⁵

$$ \begin{aligned} {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{ 1} } \right) & \le {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{0} } \right) + \nabla_{\text{w}} {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{0} } \right) \cdot \left[ {{\text{w}}_{\text{i}}^{ 1} - {\text{ w}}_{\text{i}}^{0} } \right];\quad {\text{ i}} = 1, 2\\ & = {\text{w}}_{\text{i}}^{ 1} \cdot \nabla_{\text{w}} {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{0} } \right)\quad {\text{since}}\,{\text{w}}_{\text{i}}^{ 1} \cdot \nabla_{\text{w}} {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{0} } \right) = {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{0} } \right) \\ & = {\text{w}}_{\text{i}}^{ 1} \cdot {\text{a}}_{\text{i}}^{0} \quad {\text{using}}\, ( 30 ). \\ \end{aligned} $$

(42)

The gap between the right and left hand sides of (42) represents input substitution bias. In the main text, we assumed Leontief no substitution type procedure production functions and so there was no substitution bias; i.e., under our main text assumptions, the inequalities in (42) were equalities. Now multiply both sides of inequality i in (42) by y ⁰_i and we obtain the following inequalities:

$$ \begin{aligned} {\text{c}}_{\text{i}}^{0} \left( {{\text{w}}_{\text{i}}^{ 1} } \right){\text{y}}_{\text{i}}^{0} & \le {\text{w}}_{\text{i}}^{ 1} \cdot {\text{a}}_{\text{i}}^{0} {\text{y}}_{\text{i}}^{0} ;\quad {\text{ i}} = 1, 2\\ & = {\text{w}}_{\text{i}}^{ 1} \cdot {\text{x}}_{\text{i}}^{0} \quad {\text{using}}\, ( 2 9 ). \\ \end{aligned} $$

(43)

In a similar fashion, we can establish the following inequalities:

$$ {\text{c}}_{\text{i}}^{ 1} \left( {{\text{w}}_{\text{i}}^{0} } \right){\text{y}}_{\text{i}}^{ 1} \le {\text{w}}_{\text{i}}^{0} \cdot {\text{x}}_{\text{i}}^{ 1} ;\quad {\text{i}} = 1, 2. $$

(44)

We now return to the theoretical input price indexes defined by (40) and (41). From (40), we have:

$$ \begin{aligned} \beta_{0} & = {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}} \\ & \le {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{0} } \right]} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 4 3 )\\ & \equiv {\text{P}}_{\text{L}}^{*} \\ \end{aligned} $$

(45)

where the observable Laspeyres input price index ${\text{P}}_{\text{L}}^{*}$ is defined as [w ¹₁  · x ⁰₁  + w ¹₂  · x ⁰₂ ]/[w ⁰₁  · x ⁰₁  + w ⁰₂  · x ⁰₂ ]. Similarly, from (41), we have:

$$ \begin{gathered} \beta_{ 1} = {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}} \hfill \\ \ge {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{ 1} } \right]} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{ 1} } \right]}}\quad {\text{using}}\, ( 4 4 )\hfill \\ \equiv {\text{P}}_{\text{P}}^{*} \hfill \\ \end{gathered} $$

(46)

where the observable Paasche input price index ${\text{P}}_{\text{P}}^{*}$ is defined as [w ¹₁  · x ¹₁  + w ¹₂  · x ¹₂ ]/[w ⁰₁  · x ¹₁  + w ⁰₂  · x ¹₂ ]. Thus the theoretical input price index β₀ is bounded from above by the observable Laspeyres input price index ${\text{P}}_{\text{L}}^{*}$ and the theoretical input price index β₁ is bounded from below by the observable Paasche input price index ${\text{P}}_{\text{P}}^{*}$. In both cases, the gap between the theoretical index and the observable index is due to input substitution bias, which goes in opposite directions.

Looking at the first line in (42), it can be seen that the right hand sides of (43) and (44) are first order Taylor series approximations to the corresponding left hand side entries. This means that ${\text{P}}_{\text{L}}^{*}$ is a first order approximation to the theoretical input price index β₀ and ${\text{P}}_{\text{P}}^{*}$ is a first order approximation to the theoretical input price index β₁.

Since both theoretical input price indexes, β₀ and β₁, are equally representative, a single estimate of input price change should be set equal to a symmetric average of these two estimates. We again choose the geometric mean as our preferred symmetric average and thus our preferred theoretical measure of input price growth is the following Fisher type theoretical index, β_F:

$$ \begin{aligned} \beta_{\text{F}} & \equiv [\beta_{0} \beta_{ 1} ]^{ 1/ 2} \\ & \approx \left[ {{\text{P}}_{\text{L}}^{*} {\text{P}}_{\text{P}}^{*} } \right]^{ 1/ 2} \\ & \equiv {\text{P}}_{\text{F}}^{*} \\ \end{aligned} $$

(47)

where the Fisher (1922) index of input price change, ${\text{P}}_{\text{F}}^{*}$, is defined as the geometric mean of the Laspeyres and Paasche input price indexes. Given the fact that ${\text{P}}_{\text{L}}^{*}$ is a first order approximation to β₀ and ${\text{P}}_{\text{P}}^{*}$ is a first order approximation to β₁, it is obvious that ${\text{P}}_{\text{F}}^{*}$ is at least a first order approximation to the theoretical input price index β_F. But in most cases, the approximation of ${\text{P}}_{\text{F}}^{*}$ to β_F will be much better than a first order approximation since the upward bias in ${\text{P}}_{\text{L}}^{*}$ will generally offset the downward bias in ${\text{P}}_{\text{P}}^{*}$.

We now define our last family of theoretical indexes. We again use the total cost function in order to define a family of reciprocal indexes of technical progress, γ(y, w), as follows:

$$ \begin{aligned} \gamma \left( {{\text{y}},{\text{w}}} \right) & \equiv {{{\text{C}}^{ 1} ({\text{y}},{\text{w}})} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} ({\text{y}},{\text{w}})} {{\text{C}}^{0} ({\text{y}},{\text{w}})}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} ({\text{y}},{\text{w}})}} \\ & = {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2} } \right]} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1} } \right){\text{y}}_{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2} } \right){\text{y}}_{ 2} } \right]}}\quad {\text{using}}\, ( 3 3 ). \\ \end{aligned} $$

(48)

The theoretical reciprocal technical progress index γ(y, w) defined by (48) is equal to the (hypothetical) total cost C¹(y, w) of producing the reference vector of outputs, y ≡ [y₁, y₂], when the service providers face the reference vector of input prices w using the period 1 technology, divided by the total cost C⁰(y, w) of producing the same reference vector of outputs, y, and facing the same reference vector of input prices w, where we now use the period 0 technology.²⁶ Thus γ(y, w) is a measure of the proportional reduction in costs that occurs due to technical progress between periods 0 and 1 and it can be seen that this is an inverse measure of technical progress. For each choice of a reference vector of output quantities y and reference vector of input prices w, we obtain a (different) measure of exogenous cost reduction.

Instead of singling out the reference vectors y and w that appear in the definition of γ(y, w) to be the period t quantity and price vectors (y^t, w^t) for t = 0, 1, we will choose the mixed vectors (y⁰, w¹) and (y¹, w⁰) for special attention. The reason for these rather odd looking choices will be explained below.

We want to explain the growth in total costs going from period 0 to 1, C¹(y¹, w¹)/C⁰(y⁰, w⁰), as the product of three growth factors:

• Growth in outputs; i.e., a factor of the form α(y⁰, y¹, w, t) defined above by (35);

• Growth in input prices; i.e., a factor of the form β(w⁰, w¹, y, t) defined by (39) and

• Exogenous reduction in costs due to technical progress; i.e., a factor of the form γ(y, w) defined by (48).

Simple algebra shows that we have the following decompositions of the cost ratio C¹(y¹, w¹)/C⁰(y⁰, w⁰) into explanatory factors of the above type:²⁷

$$ \begin{aligned} {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} \\ & = \left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right]\left[ {{{{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right]\left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right] \\ & = \alpha_{ 1} \beta_{0} \gamma \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)\quad {\text{using}}\,{\text{definitions}}\, ( 3 7 ) ,\, ( 4 0 )\,{\text{and}}\, ( 4 8 ); \\ \end{aligned} $$

(49)

$$ \begin{aligned} {\text{C}}^{ 1} {{\left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} \\ & = \left[ {{{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right]\left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right]\left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right] \\ & = \alpha_{0} \beta_{ 1} \gamma \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)\quad {\text{using}}\,{\text{definitions}}\, ( 3 6 ),\,(41)\,{\text{and}}\, ( 4 8 ). \\ \end{aligned} $$

(50)

The above decompositions show that the two special cases of γ(y, w) defined by (48) of particular interest are defined by (51) and (52) below:

$$ \begin{aligned} \gamma \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & \equiv {{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}\quad {\text{using}}\,(48) \\ & = {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} } \right]}}\quad {\text{using}}\, ( 3 3 ); \\ \end{aligned} $$

(51)

$$ \begin{aligned} \gamma \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & \equiv {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}\quad {\text{using}}\, ( 4 8 )\\ & = {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} } \right]}}\quad {\text{using}}\,(33). \\ \end{aligned} $$

(52)

We will now work out observable first order approximations (and observable bounds) to the two specific measures of reciprocal technical progress defined by (51) and (52). From the second equation in (51), we have:

$$ \begin{aligned} {\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & = {\text{c}}_{ 1}^{ 1} \left( {{\text{w}}_{ 1}^{ 1} } \right){\text{y}}_{ 1}^{0} + {\text{c}}_{ 2}^{ 1} \left( {{\text{w}}_{ 2}^{ 1} } \right){\text{y}}_{ 2}^{0} \\ & = {\text{p}}_{ 1}^{ 1} {\text{y}}_{ 1}^{0} + {\text{ p}}_{ 2}^{ 1} {\text{y}}_{ 2}^{0} \quad {\text{using}}\, ( 3 1 )\\ & = {\text{p}}^{ 1} \cdot {\text{y}}^{0} . \\ \end{aligned} $$

(53)

Since C⁰(y⁰, w) is concave in the components of the input price vector w, C⁰(y⁰, w) regarded as a function of w will be bounded from above by its first order Taylor series approximation around the point w⁰ so the following inequality will be satisfied:²⁸

$$ \begin{aligned} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & \le {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) + \nabla_{\text{w}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot \left[ {{\text{w}}^{ 1} - {\text{ w}}^{0} } \right] \\ & = {\text{w}}^{ 1} \cdot \nabla_{\text{w}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)\quad {\text{since}}\,{\text{w}}^{0} \cdot \nabla_{\text{w}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) = {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \\ & = {\text{w}}_{ 1}^{ 1} \cdot {\text{x}}_{ 1}^{0} + {\text{ w}}_{ 2}^{ 1} \cdot {\text{x}}_{ 2}^{0} \quad {\text{using}}\, ( 2 8 )\,{\text{and}}\, ( 3 3 )\\ & \equiv {\text{w}}^{ 1} \cdot {\text{x}}^{0} . \\ \end{aligned} $$

(54)

Thus the period 0 total cost function C⁰(y⁰, w¹), evaluated at the vector of period 0 observed outputs y⁰ and the period 1 observed input prices w¹, is bounded from above by the inner product of the period 1 input price vector w¹ and the observed vector of inputs for period 0, x⁰.²⁹ Note also from the first line of (54) that w¹ · x⁰ is a first order Taylor series approximation to the unobserved cost C⁰(y⁰, w¹). The results (53) and (54) can now be used in order to form a bound (and a first order approximation) to the measure of reciprocal technical progress γ(y⁰, w¹) defined by (51):

$$ \begin{aligned} \gamma \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & = {\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)/{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) \\ & \ge {{{\text{p}}^{ 1} \cdot {\text{y}}^{0} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{0} } {{\text{w}}^{ 1} \cdot {\text{x}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{w}}^{ 1} \cdot {\text{x}}^{0} }}\quad {\text{using}}\, ( 5 3 )\,{\text{and}}\, ( 5 4 )\\ & = {{\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{0} /{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{0} /{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \right]} {\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{0} /{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{0} /{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} } \right]}}\quad {\text{using}}\, ( 3 2 )\;{\text{ for}}\,{\text{t}} = 1\\ & = {{\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} /{\text{w}}^{ 1} \cdot {\text{x}}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} /{\text{w}}^{ 1} \cdot {\text{x}}^{0} } \right]} {\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} /{\text{p}}^{ 1} \cdot {\text{y}}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} /{\text{p}}^{ 1} \cdot {\text{y}}^{0} } \right]}}\quad {\text{rearranging}}\,{\text{terms}} \\ & = {{{\text{Q}}_{\text{P}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{P}}^{*} } {{\text{Q}}_{\text{P}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{P}} }}\quad {\text{using}}\, ( 7 )\,{\text{and}}\, ( 1 1 )\\ & = \left[ {{\text{Q}}_{\text{P}} /{\text{Q}}_{\text{P}}^{*} } \right]^{ - 1} \\ \end{aligned} $$

(55)

where Q_P is the Paasche output quantity index defined by (11) and ${\text{Q}}_{\text{P}}^{*}$ is the Paasche input quantity index defined by (7) in the main text. Note that Q_P divided by ${\text{Q}}_{\text{P}}^{*}$ is the Paasche productivity index. Thus (55) tells us that the theoretical measure of reciprocal technical progress defined by (51), γ(y⁰, w¹), is bounded from below by the reciprocal of the observable Paasche productivity index, ${\text{Q}}_{\text{P}} /{\text{Q}}_{\text{P}}^{*}$. Moreover, it can be seen that $\left[ {{\text{Q}}_{\text{P}} /{\text{Q}}_{\text{P}}^{*} } \right]^{ - 1}$ is also a first order approximation to the theoretical index γ(y⁰, w¹).

The above algebra can be repeated with minor modifications in order to derive a bound for the theoretical index γ(y¹, w⁰) defined by (52). Thus we have:

$$ \begin{aligned} {\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & = {\text{c}}_{ 1}^{0} \left( {{\text{w}}_{ 1}^{0} } \right){\text{y}}_{ 1}^{ 1} + {\text{ c}}_{ 2}^{0} \left( {{\text{w}}_{ 2}^{0} } \right){\text{y}}_{ 2}^{ 1} \\ & = {\text{p}}_{ 1}^{0} {\text{y}}_{ 1}^{ 1} + {\text{p}}_{ 2}^{0} {\text{y}}_{ 2}^{ 1} \quad {\text{using}}\, ( 3 1 )\\ & = {\text{p}}^{0} \cdot {\text{y}}^{ 1} . \\ \end{aligned} $$

(56)

Since C¹(y¹, w) is concave in the components of the input price vector w, C¹(y¹, w) regarded as a function of w will be bounded from above by its first order Taylor series approximation around the point w¹ so the following inequality will be satisfied:

$$ \begin{aligned} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & \le {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) + \nabla_{\text{w}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot \left[ {{\text{w}}^{0} - {\text{ w}}^{ 1} } \right] \\ & = {\text{w}}^{0} \cdot \nabla_{\text{w}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)\quad {\text{ since}}\,{\text{w}}^{ 1} \cdot \nabla_{\text{w}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) = {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \\ & = {\text{w}}_{ 1}^{0} \cdot {\text{x}}_{ 1}^{ 1} + {\text{ w}}_{ 2}^{0} \cdot {\text{x}}_{ 2}^{ 1} \quad {\text{using}}\, ( 2 8 )\,{\text{and}}\, ( 3 3 )\\ & \equiv {\text{w}}^{0} \cdot {\text{x}}^{ 1} . \\ \end{aligned} $$

(57)

Thus the period 1 total cost function C¹(y¹, w⁰), evaluated at the vector of period 1 observed outputs y¹ and the period 0 observed input prices w⁰, is bounded from above by the inner product of the period 0 input price vector w⁰ and the observed vector of inputs for period 1, x¹.³⁰ Note also from the first line of (57) that w⁰ · x¹ is a first order Taylor series approximation to the unobserved cost C¹(y¹, w⁰). The results (56) and (57) can now be used in order to form a bound (and a first order approximation) to the measure of reciprocal technical progress γ(y¹, w⁰) defined by (52):

$$ \begin{aligned} \gamma \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & = {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}} \\ & \le {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{ 1} \quad }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{ 1} \quad }}{\text{using}}\, ( 5 6 )\,{\text{and}}\, ( 5 7 )\\ & = {{\left[ {{\text{w}}^{0} \cdot {\text{x}}^{ 1} /{\text{w}}^{0} \cdot {\text{x}}^{0} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{w}}^{0} \cdot {\text{x}}^{ 1} /{\text{w}}^{0} \cdot {\text{x}}^{0} } \right]} {\left[ {{\text{p}}^{0} \cdot {\text{y}}^{ 1} /{\text{p}}^{0} \cdot {\text{y}}^{0} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{p}}^{0} \cdot {\text{y}}^{ 1} /{\text{p}}^{0} \cdot {\text{y}}^{0} } \right]}}\quad {\text{using}}\, ( 3 2 )\;{\text{for}}\,{\text{t}} = 0 \\ & = {{{\text{Q}}_{\text{L}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{L}}^{*} } {{\text{Q}}_{\text{L}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{L}} }}\quad {\text{using}}\, ( 6 )\,{\text{and}}\, ( 1 0 )\\ & = \left[ {{{{\text{Q}}_{\text{L}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{L}} } {{\text{Q}}_{\text{L}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{L}}^{*} }}} \right]^{ - 1} \\ \end{aligned} $$

(58)

where Q_L is the Laspeyres output quantity index defined by (10) and ${\text{Q}}_{\text{L}}^{*}$ is the Laspeyres input quantity index defined by (6) in the main text. Note that Q_L divided by ${\text{Q}}_{\text{L}}^{*}$ is the Laspeyres productivity index. Thus (58) tells us that the theoretical measure of reciprocal technical progress defined by (52), γ(y¹, w⁰), is bounded from above by the reciprocal of the observable Laspeyres productivity index, ${\text{Q}}_{\text{L}} /{\text{Q}}_{\text{L}}^{*}$. Moreover, it can be seen that $\left[ {{\text{Q}}_{\text{L}} /{\text{Q}}_{\text{L}}^{*} } \right]^{ - 1}$ is also a first order approximation to the theoretical index γ(y¹, w⁰).

Since the two cost decompositions for the rate of growth of cost, C¹(y¹, w¹)/C⁰(y⁰, w⁰), given by (49) and (50) are equally valid, we will take the geometric average of these two decompositions to obtain our preferred overall cost decomposition. This leads to the following theoretical decomposition of C¹(y¹, w¹)/C⁰(y⁰, w⁰) into explanatory factors:

$$ {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} = \alpha_{\text{F}} \beta_{\text{F}} \gamma_{\text{F}} $$

(59)

where the Fisher type output quantity growth factor α_F is defined by (38), the Fisher type input price growth factor β_F is defined by (47) and the Fisher type reciprocal measure of technical progress γ_F is defined as follows:

$$ \gamma_{\text{F}} \equiv \left[ {\gamma \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)\gamma \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \right]^{ 1/ 2} . $$

(60)

Using our first order approximations given by (55) and (58), it can be seen that an observable first order approximation to γ_F is the reciprocal of the Fisher productivity index ${\text{Q}}_{\text{F}} /{\text{Q}}_{\text{F}}^{*}$; i.e., we have:

$$ \begin{aligned} \gamma_{\text{F}} & \approx \left\{ {\left[ {{{{\text{Q}}_{\text{P}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{P}}^{*} } {{\text{Q}}_{\text{P}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{P}} }}} \right]\left[ {{{{\text{Q}}_{\text{L}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{L}}^{*} } {{\text{Q}}_{\text{L}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{L}} }}} \right]} \right\}^{ 1/ 2} \quad {\text{using}}\, ( 5 5 )\,{\text{and}}\, ( 5 8 )\\ & = \left[ {{{{\text{Q}}_{\text{F}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{F}}^{*} } {{\text{Q}}_{\text{F}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{F}} }}} \right]\quad {\text{using}}\, ( 8 )\,{\text{and}}\, ( 1 2 )\\ & = \left[ {{{{\text{Q}}_{\text{F}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{F}} } {{\text{Q}}_{\text{F}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{F}}^{*} }}} \right]^{ - 1} . \\ \end{aligned} $$

(61)

However, since the substitution biases in the first order approximations given by (55) and (58) go in opposite directions, the approximation to the theoretical index γ_F given by the right hand side of (61) will generally be much closer than a first order approximation.

We now combine the theoretical cost decomposition defined by (59) with the exact result (38) and the approximate results (47) and (61):

$$ \begin{aligned} {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} & = \alpha_{\text{F}} \beta_{\text{F}} \gamma_{\text{F}} \quad {\text{using}}\, ( 5 9 )\\ & \approx {\text{Q}}_{\text{F}} {\text{P}}_{\text{F}}^{*} \left[ {{{{\text{Q}}_{\text{F}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{F}} } {{\text{Q}}_{\text{F}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{F}}^{*} }}} \right]^{ - 1} \quad {\text{using}}\, ( 3 8 ),\,(47)\,{\text{and}}\, ( 6 1 ). \\ \end{aligned} $$

(62)

Thus (one plus) the rate of growth of industry cost, C¹(y¹, w¹)/C⁰(y⁰, w⁰), is approximately equal to (one plus) the rate of output growth defined by the Fisher output quantity index, Q_F, times (one plus) the rate of growth of input prices defined by the Fisher input price index, ${\text{P}}_{\text{F}}^{*}$, times the reciprocal of (one plus) the Fisher rate of productivity growth, ${\text{Q}}_{\text{F}} /{\text{Q}}_{\text{F}}^{*}$. However, it turns out that the left hand side of (62) is identically equal to the product of the explanatory factors on the right hand side of (62), since it can be shown that the following identity holds:³¹

$$ {{\left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right]} {{\text{P}}_{\text{F}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{P}}_{\text{F}}^{*} }} = {\text{Q}}_{\text{F}}^{*} ; $$

(63)

i.e., the Fisher implicit input quantity index, $[{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} /{\text{w}}^{0} \cdot {\text{x}}^{0} ]/{\text{P}}_{\text{F}}^{*}$, is equal to the direct Fisher input quantity index, ${\text{Q}}_{\text{F}}^{*}$.

Note that the above results are entirely nonparametric. Thus we have generalized (to a reasonable degree of approximation) the results derived in the main text under the assumption that the procedure production functions were of the no substitution variety to the case where the procedure production functions are general ones.

It is possible to further generalize our results from the case where the procedure functions are independent to the case where there are shared overheads between the procedures.³² In this more general framework, the health sector’s period t total cost function C^t(y, w) is no longer defined as the sum of the two procedure cost functions, C ^t₁ (y₁, w₁) plus C ^t₂ (y₂, w₂) as in (33), but is simply a general nonjoint cost function. The regularity conditions that we impose on each C^t(y, w) is that it is a nonnegative, jointly continuous differentiable function in its variables (y, x) and that it is linearly homogeneous,³³ nondecreasing and convex³⁴ in the components of y for fixed w and linearly homogeneous, nondecreasing and concave in the components of w for each fixed y. As usual, we assume cost minimizing behavior in periods t = 0, 1 and that we can observe the period t industry output vector, y^t ≡ [y ^t₁ , y ^t₂ ], the period t aggregate input vector x^t ≡ [x ^t₁ , x ^t₂ ] and the corresponding vector of input prices w^t ≡ [w ^t₁ , w ^t₂ ] for t = 0, 1. As usual, Shephard’s Lemma tells us that the period t vector of inputs is equal to the vector of first order partial derivatives of the period t cost function with respect to the components of the input price vector; i.e., we have:

$$ {\text{x}}^{\text{t}} = \nabla_{\text{w}} {\text{C}}^{\text{t}} \left( {{\text{y}}^{\text{t}} ,{\text{w}}^{\text{t}} } \right);\quad {\text{t}} = 0, 1. $$

(64)

The period t vector of marginal cost output prices p^t ≡ [p ^t₁ , p ^t₂ ] is defined as the vector of first order partial derivatives of the period t cost function with respect to the components of the output vector:

$$ {\text{p}}^{\text{t}} \equiv \nabla_{\text{y}} {\text{C}}^{\text{t}} \left( {{\text{y}}^{\text{t}} ,{\text{w}}^{\text{t}} } \right);\quad {\text{t}} = 0, 1. $$

(65)

It should be noted that the linear homogeneity properties of C^t(y, w) in y and w separately imply the following equalities:

$$ {\text{C}}^{\text{t}} \left( {{\text{y}}^{\text{t}} ,{\text{w}}^{\text{t}} } \right) = {\text{w}}^{\text{t}} \cdot {\text{x}}^{\text{t}} = {\text{p}}^{\text{t}} \cdot {\text{y}}^{\text{t}} ;\quad {\text{t}} = 0, 1. $$

(66)

We can now modify the above analysis in this “Appendix”, letting the marginal cost prices p^t defined by (65) replace our old unit cost prices.

In particular, we use the first line in (35) in order to define a new family of cost based output quantity indexes as α(y⁰, y¹, w, t) ≡ C^t(y¹, w)/C^t(y⁰, w) and we again use the first line in (36) and (37), to define the two specific output quantity indexes α₀ as α(y⁰, y¹, w⁰, 0) and α₁ as α(y⁰, y¹, w¹, 1). However, in our new more general model, we no longer obtain the equalities in (36) and (37); instead, we obtain the following first order approximations and bounds:³⁵

$$ \begin{aligned} \alpha_{0} & \equiv {{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} \\ & = {{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}\quad {\text{using}}\, ( 6 6 )\\ & \ge {{\left\{ {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) + \nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot \left[ {{\text{y}}^{ 1} - {\text{ y}}^{0} } \right]} \right\}} \mathord{\left/ {\vphantom {{\left\{ {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) + \nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot \left[ {{\text{y}}^{ 1} - {\text{ y}}^{0} } \right]} \right\}} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}\quad {\text{using}}\,{\text{the}}\,{\text{convexity}}\,{\text{of}}\,{\text{C}}^{0} \left( {{\text{y}},{\text{w}}^{0} } \right)\,{\text{in}}\,{\text{y}} \\ & = {{\nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{\nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot {\text{y}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}\quad {\text{using}}\,\nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot {\text{y}}^{0} = {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \\ & = {{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}\quad {\text{using}}\, ( 6 5 )\;{\text{for}}\,{\text{t}} = 0 \\ & = {\text{Q}}_{\text{L}} \quad {\text{using}}\, ( 1 0 ); \\ \end{aligned} $$

(67)

$$ \begin{aligned} \alpha_{ 1} & \equiv {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}} \\ & = {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}\quad {\text{using}}\, ( 6 6 )\\ & \le {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } {\left\{ {{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) + \nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot \left[ {{\text{y}}^{0} - {\text{y}}^{ 1} } \right]} \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) + \nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot \left[ {{\text{y}}^{0} - {\text{y}}^{ 1} } \right]} \right\}}}\quad {\text{using}}\,{\text{the}}\,{\text{convexity}}\,{\text{of}}\,{\text{C}}^{ 1} \,\left( {{\text{y}},{\text{w}}^{ 1} } \right)\,{\text{in}}\,{\text{y}} \\ & = {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } {\left\{ {\nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot {\text{y}}^{0} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {\nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot {\text{y}}^{0} } \right\}}}\quad {\text{using}}\,\nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot {\text{y}}^{ 1} = {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \\ & = {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } {{\text{p}}^{ 1} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{ 1} \cdot {\text{y}}^{0} }}\quad {\text{using}}\,(65)\quad {\text{for}}\,{\text{t}} = 1\\ & = {\text{Q}}_{\text{P}} \quad {\text{using}}\,{\text{definition}}\, ( 1 1 ). \\ \end{aligned} $$

(68)

Thus the ordinary Laspeyres output quantity index Q_L (using marginal cost prices) is no longer equal to the theoretical output index α₀ defined by the first line in (67) but is only a first order approximation and a lower bound. Similarly, ordinary Paasche output quantity index Q_P (using marginal cost prices) is no longer equal to the theoretical output index α₁ defined by the first line in (68) but is only a first order approximation and an upper bound to this theoretical output quantity index. However, as before, the Fisher theoretical output quantity index α_F defined as the geometric mean of α₀ and α₁ will be approximated by the Fisher output quantity index, Q_F ≡ [Q_LQ_P]^1/2, and due to the offsetting substitution biases in (67) and (68), Q_F will generally approximate α_F to an accuracy that is greater than a first order approximation.

The analysis associated with (39)–(47) goes through with obvious modifications using our more general model and so we will not repeat this analysis.

We can again define the family of reciprocal indexes of technical progress using the first line in (48) as γ(y, w) ≡ C¹(y, w)/C⁰(y, w). As before, the factorizations of cost growth given by (49) and (50) continue to be valid in this more general framework and so we need empirically observable approximations to the two indexes of technical progress defined by (51) and (52). The inequalities in (54) and (57) continue to be valid but the equalities in (53) and (56) are no longer valid in our more general model and need to be replaced by the following inequalities which were derived in (68) and (67):

$$ \begin{aligned} {\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & \ge {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) + \nabla_{\text{y}} {\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right) \cdot \left[ {{\text{y}}^{0} - {\text{ y}}^{ 1} } \right] \\ & = {\text{p}}^{ 1} \cdot {\text{y}}^{0} ; \\ \end{aligned} $$

(69)

$$ \begin{aligned} {\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & \ge {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) + \nabla_{\text{y}} {\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right) \cdot \left[ {{\text{y}}^{ 1} - {\text{ y}}^{0} } \right] \\ & = {\text{p}}^{0} \cdot {\text{y}}^{ 1} . \\ \end{aligned} $$

(70)

Using definition (51) and the inequalities (54) and (69) establishes the following inequality:

$$ \begin{aligned} \gamma \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right) & = {{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{ 1} } \right)}}\quad {\text{using}}\,{\text{definition}}\, ( 5 1 )\\ & \ge {{{\text{p}}^{ 1} \cdot {\text{y}}^{0} } \mathord{\left/ {\vphantom {{{\text{p}}^{ 1} \cdot {\text{y}}^{0} } {{\text{w}}^{ 1} \cdot {\text{x}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{w}}^{ 1} \cdot {\text{x}}^{0} }}\quad {\text{using}}\, ( 6 9 )\,{\text{and}}\, ( 5 4 )\\ & = {{\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{0} /{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{p}}^{ 1} \cdot {\text{y}}^{0} /{\text{p}}^{ 1} \cdot {\text{y}}^{ 1} } \right]} {\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{0} /{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{w}}^{ 1} \cdot {\text{x}}^{0} /{\text{w}}^{ 1} \cdot {\text{x}}^{ 1} } \right]}}\quad {\text{using}}\, ( 6 6 )\;{\text{for}}\,{\text{t}} = 1\\ & = {{{\text{Q}}_{\text{P}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{P}}^{*} } {{\text{Q}}_{\text{P}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{P}} }}\quad {\text{using}}\,(7)\,{\text{and}}\, ( 1 1 )\\ & = \left[ {{{{\text{Q}}_{\text{P}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{P}} } {{\text{Q}}_{\text{P}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{P}}^{*} }}} \right]^{ - 1} . \\ \end{aligned} $$

(71)

Similarly, using definition (52) and the inequalities (54) and (69) establishes the following inequality:

$$ \begin{aligned} \gamma \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right) & \equiv {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{ 1} ,{\text{w}}^{0} } \right)}}\quad {\text{using}}\,{\text{definition}}\, ( 5 2 )\\ & \le {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{ 1} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{ 1} }}\quad {\text{using}}\, ( 5 7 )\,{\text{and}}\, ( 7 0 )\\ & = {{\left[ {{{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } {{\text{w}}^{0} \cdot {\text{x}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{w}}^{0} \cdot {\text{x}}^{0} }}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{w}}^{0} \cdot {\text{x}}^{ 1} } {{\text{w}}^{0} \cdot {\text{x}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{w}}^{0} \cdot {\text{x}}^{0} }}} \right]} {\left[ {{{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } \mathord{\left/ {\vphantom {{{\text{p}}^{0} \cdot {\text{y}}^{ 1} } {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right. \kern-\nulldelimiterspace} {{\text{p}}^{0} \cdot {\text{y}}^{0} }}} \right]}}\quad {\text{using}}\, ( 6 6 )\;{\text{for}}\,{\text{t}} = 0 \\ & = {{{\text{Q}}_{\text{L}}^{*} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{L}}^{*} } {{\text{Q}}_{\text{L}} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{L}} }}\quad {\text{using}}\, ( 6 )\,{\text{and}}\, ( 1 0 )\\ & = \left[ {{{{\text{Q}}_{\text{L}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{\text{L}} } {{\text{Q}}_{\text{L}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{L}}^{*} }}} \right]^{ - 1} . \\ \end{aligned} $$

(72)

As before, define the Fisher type reciprocal measure of technical progress γ_F by (60). The rest of the above analysis goes through with minor modifications. In particular, we still obtain the cost decomposition (62) as the following exact equality:³⁶

$$ \begin{aligned} {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} \mathord{\left/ {\vphantom {{{\text{C}}^{ 1} \left( {{\text{y}}^{ 1} ,{\text{w}}^{ 1} } \right)} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}}} \right. \kern-\nulldelimiterspace} {{\text{C}}^{0} \left( {{\text{y}}^{0} ,{\text{w}}^{0} } \right)}} & = \alpha_{\text{F}} \beta_{\text{F}} \gamma_{\text{F}} \\ & = {\text{Q}}_{\text{F}} {\text{P}}_{\text{F}}^{*} \left[ {{\text{Q}}_{\text{F}} {{} \mathord{\left/ {\vphantom {{} {{\text{Q}}_{\text{F}}^{*} }}} \right. \kern-\nulldelimiterspace} {{\text{Q}}_{\text{F}}^{*} }}} \right]^{ - 1} . \\ \end{aligned} $$

(73)

Note that it is risky to use either the Laspeyres or Paasche measures of productivity growth to approximate the corresponding theoretically correct measure of productivity growth since there are two doses of substitution bias between the left hand and right hand sides of (71) and (72); i.e., the output and input substitution biases augment each other when we use Paasche or Laspeyres productivity indexes instead of offsetting each other. Thus whenever possible, we recommend the use of Fisher productivity indexes³⁷ rather than the use of their Paasche or Laspeyres counterparts.

The reader may well wonder why we did not proceed directly to our final most general model of production instead of doing the additive cost model defined by (33) where no joint costs were present. The problem is that our most general model requires estimates of marginal cost prices in order to implement the practical approximations to the theoretical indexes. Unfortunately, econometric estimation of joint cost functions will generally be required in order to estimate these marginal cost prices and econometric estimation of joint cost functions with general technical progress and the use of flexible functional forms is a generally hazardous exercise!