Resource allocation in multi-divisional multi-product firms

Abstract

This paper is concerned with specifying and estimating the productive characteristics of multidivisional multiproduct companies at the divisional level. In order to accomplish this, we augment division-level information with inputs that are imputed based on profit-maximizing allocations within each division. This study builds on work by De Loecker et al. (2016) as well as Olley and Pakes (1996), Levinsohn and Petrin (2003) and Ackerberg et al. (2015), and extends this work by lifting a key assumption that single- and multi-product/division firms have the same production technique for the same product/segment. We estimate the production function and impute input allocations simultaneously in the absence of this key assumption as well as the constant share constraint of the input portfolio. Finally, our approach is applied to estimate the division-specific productivity of firms that compete in five segments of the global oilfield market to better understand the motivation of firms’ investment, divestment, mergers and acquisitions (M&A).

Appendices

Appendix A: Imputation method

This appendix presents the imputation method including how to derive Theorem 1 under Assumptions 1–3. A simple example is introduced first to illustrate how to impute undistorted input allocations in multi-divisional firms when they maximize profits. Then, the general model is given under an “M Inputs – N Products/Segments – T periods” economy. Finally, parametric production function specification is discussed.

A.1 A simple example of the model

A.1.1 Setup

This model assumes that firms use two inputs: labor and capital. All firms can operate in a maximum of two segments following segment-specific production functions. Suppose that a total of six companies is observed: 1) firms A and B do business only in segment I; 2) firms C and D produce only in segment II; and 3) firms E and F have footprints in both segments. Therefore, firm E(F) is a multi-divisional firm with division E1(F1) in segment I and division E2(F2) in segment II. As a result, segment I has four competitors (firms A, B, E1, and F1), and segment II also has four competitors (firms C, D, E2, and F2).

A.1.2 First system of equations: segment-specific production functions

The first system of equations consists of two segment-specific production functions, one for each segment. Following Gong (2020a) and Gong et al. (2020), the production functions are assumed to be stochastic and have a Cobb-Douglas form.

$$\left\{ {\begin{array}{*{20}{c}} {\ln Y_{i1t} = \alpha _{i1t} + \beta _{1L}{\mathrm{In}}\left( {L_{i1t}} \right) + \beta _{1K}{\mathrm{In}}\left( {K_{i1t}} \right) + v_{i1t},\,\forall i,t} \\ {\ln Y_{i2t} = \alpha _{i2t} + \beta _{2L}{\mathrm{In}}\left( {L_{i2t}} \right) + \beta _{2K}{\mathrm{In}}\left( {K_{i2t}} \right) + v_{i2t},\,\forall i,t} \end{array}} \right.,$$

(8)

where Y_ijt, L_ijt, and K_ijt are production, labor, and capital, respectively, for firm i in segment/division j at time t; α_ijt indicates the Hicks-neutral productivity in logarithm for firm i in segment j at time t. The first equation is the production function of segment I and the second equation is the production function of segment II. β_mn is the coefficient of input n at segment m. v is the noise that is considered normally distributed with a mean of zero and a standard deviation of σ_v. All the coefficients β_mn can be estimated if the input allocations for firms E and F are given.

It is worth noting the observations in each of the two regressions in Eq. (8). If single-division and multi-divisional firms are assumed to have the same production technique within a segment, then the first equation includes A, B, E1, and F1 (i.e. ∀i = A, B, E1, F1), while the second equation includes C, D, E2, and F2 (i.e. ∀j = C, D, E2, F2). If single-division and multi-divisional firms are assumed to have different production techniques within a segment, then the first equation includes only E1 and F1 (i.e. ∀i = E1, F1), while the second equation includes only E2 and F2 (i.e. ∀j = E2, F2). In other words, single-division firms are removed from the regression if the latter assumption is selected.

A.1.3 Second system of equations: profit maximization

The second system of equations is the profit maximization problem for multi-divisional firms E and F. Under Assumption 1, multi-divisional firms maximize the net-present-value of discounted future profits by maximizing firm-level profits period-by-period, because firms’ decisions don’t have any intertemporal consequences. Therefore, the division managers are willing to report their true profit functions in each period so that the headquarters can better predict firm-level profit function and value function. Under Assumption 3, this study follows the framework in Olley and Pakes (1996) so that the labor used in time t is chosen when the productivity shock at time t is observed, while the capital used in time t is chosen at time t-1.

Since this study develops stochastic production functions, the profits are also stochastic, rather than deterministic. Therefore, profit-maximizing firms maximize the mathematical expectation of profit (Zellner et al. 1966). In this model, firm i first decides capital allocations to maximize the expected firm-level profit of time t knowing the production function and price information of time t-1 and then decides labor allocations to maximize the expected firm-level profit of time t knowing the production function and price information of time t.

$$\left\{ {\begin{array}{*{20}{l}} {\mathop {{\max }}\nolimits_{K_{ijt}} \,{\mathrm{E}}\left[ {\pi _{it}|I_{it - 1}} \right] = \mathop {{\max }}\nolimits_{K_{ijt}} \mathop {\sum}\nolimits_{j = 1,2} {\left[ {p_{jt - 1}A_{ijt - 1}L_{ijt - 1}^{\beta _{jL}}K_{ijt}^{\beta _{jK}}e^{\frac{{\sigma _{vj}^2}}{2}}\gamma _j - c_{ijt - 1}^L\sigma _j - c_{ijt - 1}^K\varphi _j} \right]} } \\ {\mathop {{\max }}\nolimits_{L_{ijt}} {\mathrm{E}}\left[ {\pi _{it}|I_{it}} \right] = \mathop {{\max }}\nolimits_{L_{ijt}} \mathop {\sum}\nolimits_{j = 1,2} {\left[ {p_{jt}A_{ijt}L_{ijt}^{\beta _{jL}}K_{ijt}^{\beta _{jK}}e^{\frac{{\sigma _{vj}^2}}{2}} - c_{ijt}^L - c_{ijt}^K} \right]} } \end{array}} \right.,$$

(9)

where

$$c_{ijt}^L = h_{i1t}\omega _{jt}L_{ijt}\,and\,c_{ijt}^K = h_{i2t}\rho _{jt}K_{ijt},$$

$$\gamma _j = \exp (\mu _{pj} + \mu _{Aj} + \beta _{jL}\mu _{Lj} + \frac{{\sigma _{pj}^2 + \sigma _{Aj}^2 + \sigma _{Lj}^2\beta _{jL}^2}}{2}),$$

$$\sigma _j = \exp \left( {\mu _{\omega j} + \mu _{Lj} + \frac{{\sigma _{\omega j}^2 + \sigma _{Lj}^2}}{2}} \right)\,and\,\varphi _j = \exp \left(\mu _{\rho j} + \frac{{\sigma _{\rho j}^2}}{2}\right),$$

$$h_{ist} = h_{ist}(\omega _{1t}L_{i1t} + \omega _{2t}L_{i2t} + \rho _{1t}K_{i1t} + \rho _{2t}K_{i2t})\,\forall s = 1,2.$$

π is the profit, and p is the output price; A_ijt = exp(α_ijt) indicates the Hicks-neutral productivity for firm i at segment j at time t; p_jt, ω_jt and ρ_jt are the segment average price of output, labor and capital in segment j at time t, respectively; h₁ and h₂ are the spillover effect on labor and capital price that follows Assumption 2, respectively. Firms predict $\log (x_{ijt})|\log (x_{ijt - 1})\sim N\left( {\log \left( {x_{ijt - 1}} \right) + \mu _{xj},\,\sigma _{xj}^2} \right)$, where x = p,A,L,ω,ρ and all the μ and σ are known. Therefore, γ_j, σ_j, φ_j, and $\sigma _{vj}^2$are known by companies. They are constant across firms and time, but vary across segments. Equation (9) applies to firm i = E, F.

Setting the first order condition of Eq. (9) to zero, the result is:

$$\left\{ {\begin{array}{*{20}{c}} {\beta _{1K}p_{1t - 1}A_{i1t - 1}L_{i1t - 1}^{\beta _{1L}}K_{i1t}^{\beta _{1K - 1}}\exp \left( {\frac{{\sigma _{v1}^2}}{2}} \right)\gamma _1 - \rho _{1t - 1}c_{1t - 1} = 0} \\ {\beta _{2K}p_{2t - 1}A_{i2t - 1}L_{i2t - 1}^{\beta _{2L}}K_{i2t}^{\beta _{2K - 1}}\exp \left( {\frac{{\sigma _{v2}^2}}{2}} \right)\gamma _2 - \rho _{2t - 1}c_{1t - 1} = 0} \end{array}} \right.,$$

(10)

and

$$\left\{ {\begin{array}{*{20}{c}} {\beta _{1L}p_{1t}A_{i1t}L_{i1t}^{\beta _{1L - 1}}K_{i1t}^{\beta _{1K}}\exp \left( {\frac{{\sigma _{v1}^2}}{2}} \right) - \omega _{1t}c_{2t} = 0} \\ {\beta _{2L}p_{2t}A_{i2t}L_{i2t}^{\beta _{2L - 1}}K_{i2t}^{\beta _{2K}}\exp \left( {\frac{{\sigma _{v2}^2}}{2}} \right) - \omega _{2t}c_{2t} = 0} \end{array}} \right.,$$

(11)

where

$$c_{1t} = h_{i1t}^\prime \left( {\omega _{1t}L_{i1t}\sigma _1 + \omega _{2t}L_{i2t}\sigma _2} \right) + h_{i2t}\varphi _2 + h_{i2t}^\prime (\rho _{1t}K_{i1t}\varphi _1 + \rho _{2t}K_{i2t}\varphi _2),$$

$$c_{2t} = h_{i1t}^\prime \left( {\omega _{1t}L_{i1t} + \omega _{2t}L_{i2t}} \right) + h_{i1t} + h_{i2t}^\prime (\rho _{1t}K_{i1t} + \rho _{2t}K_{i2t}).$$

This study first solves the “perfectly variable” input labor in Eq. (11). From Eq. (8), the following is known:

$$R_{ijt} = p_{jt}A_{ijt}L_{ijt}^{\beta _{jL}}K_{ijt}^{\beta _{jK}}\exp (v_{ijt})$$

$$\Leftrightarrow p_{jt}A_{ijt}L_{ijt}^{\beta _{1L - 1}}K_{ijt}^{\beta _{1K}} = R_{ijt}/\left[ {L_{ijt}\exp (v_{ijt})} \right]$$

(12)

$$\Leftrightarrow p_{jt}A_{ijt}L_{ijt}^{\beta _{1L}}K_{ijt}^{\beta _{1K - 1}} = R_{ijt}/\left[ {K_{ijt}\exp (v_{ijt})} \right].$$

(13)

Plug Eq. (12) into Eq. (11), and the result is

$$\left\{ {\begin{array}{*{20}{c}} {\frac{{\beta _{1L}\exp \left( {\frac{{\sigma _{v1}^2}}{2}} \right)R_{i1t}}}{{\left[ {L_{i1t}\exp \left( {v_{i1t}} \right)} \right]}} = \omega _{1t}c_{2t}} \\ {\frac{{\beta _{2L}\exp \left( {\frac{{\sigma _{v2}^2}}{2}} \right)R_{i2t}}}{{\left[ {L_{i2t}\exp \left( {v_{i2t}} \right)} \right]}} = \omega _{2t}c_{2t}} \end{array}} \right..$$

Then, this study divides the first by the second equation. Adding the observed firm-level labor input equation, the result is

$$\left\{ {\begin{array}{*{20}{l}} \displaystyle {\frac{{\beta _{1L}R_{i1t}L_{i2t}\exp (v_{i2t} + 0.5\sigma _{v1}^2)}}{{\beta _{2L}R_{i2t}L_{i1t}\exp (v_{i1t} + 0.5\sigma _{v2}^2)}} = \frac{{\omega _{1t}}}{{\omega _{2t}}}} \\ {L_{i1t} + L_{i2t} = L_{it}} \end{array}} \right.$$

$$\Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {L_{i1t} = \frac{{\beta _{1L}R_{i1t}\omega _{2t}}}{{\beta _{1L}R_{i1t}\omega _{2t} + \beta _{2L}R_{i2t}\omega _{1t}\exp (v_{i1t} - v_{i2t} + 0.5\sigma _{v2}^2 - 0.5\sigma _{v1}^2)}}L_{it}} \\ {L_{i2t} = \frac{{\beta _{2L}R_{i2t}\omega _{1t}}}{{\beta _{1L}R_{i1t}\omega _{2t}\exp \left( {v_{i2t} - v_{i1t} + 0.5\sigma _{v1}^2 - 0.5\sigma _{v2}^2} \right) + \beta _{2L}R_{i2t}\omega _{1t}}}L_{it}} \end{array}} \right..$$

(14)

In Eq. (14), segment actual revenues (R) and average segment labor price (ω) are observed. The rest of the variables, including the production parameters (β) and the noises (v), can be estimated in Eq. (8).

After solving the labor allocations, this study solves the capital inputs. Equation (10) can be written as

$$\left\{ {\begin{array}{*{20}{l}} \displaystyle {\beta _{1K}\frac{{p_{1t - 1}}}{{p_{1t}}}\frac{{A_{i1t - 1}}}{{A_{i1t}}}(\frac{{L_{i1t - 1}}}{{L_{i1t}}})^{\beta _{1L}} \displaystyle \left[ {p_{1t}A_{i1t}L_{i1t}^{\beta _{1L}}K_{i1t}^{\beta _{1K} - 1}} \right]\exp \left( {\frac{{\sigma _{v1}^2}}{2}} \right)\gamma _1 = \rho _{1t - 1}c_{1t - 1}} \\ \displaystyle {\beta _{2K}\frac{{p_{2t - 1}}}{{p_{2t}}}\frac{{A_{i2t - 1}}}{{A_{i2t}}}(\frac{{L_{i2t - 1}}}{{L_{i2t}}})^{\beta _{2L}}\left[ {p_{2t}A_{i2t}L_{i2t}^{\beta _{2L}}K_{i2t}^{\beta _{2K} - 1}} \right]\exp \left( {\frac{{\sigma _{v2}^2}}{2}} \right)\gamma _2 = \rho _{2t - 1}c_{1t - 1}} \end{array}} \right..$$

(15)

Plugging Eq. (13) into Eq. (15):

$$\left\{ {\begin{array}{*{20}{c}} {\beta _{1K}z_{i1t}\left[ {R_{i1t}/\left[ {K_{i1t}{\mathrm{exp}}(v_{i1t})} \right]} \right]\exp \left( {\frac{{\sigma _{v1}^2}}{2}} \right)\gamma _1 = \rho _{1t - 1}c_{1t - 1}} \\ {\beta _{2K}z_{i2t}\left[ {R_{i2t}/\left[ {K_{i2t}{\mathrm{exp}}(v_{i2t})} \right]} \right]\exp \left( {\frac{{\sigma _{v2}^2}}{2}} \right)\gamma _2 = \rho _{2t - 1}c_{1t - 1}} \end{array}} \right.,$$

where

$$z_{ijt} = \frac{{p_{jt - 1}}}{{p_{jt}}}\frac{{A_{ijt - 1}}}{{A_{ijt}}}(\frac{{L_{ijt - 1}}}{{L_{ijt}}})^{\beta _{jL}}.$$

Similar to the transformation of labor, this study divides the first equation by the second. Adding the observed firm-level capital input equation, the result is

$$\left\{ {\begin{array}{*{20}{l}} \displaystyle {\frac{{\beta _{1K}z_{i1t}R_{i1t}K_{i2t}{\mathrm{exp}}(v_{i2t} + 0.5\sigma _{v1}^2)\gamma _1}}{{\beta _{2K}z_{i2t}R_{i2t}K_{i1t}{\mathrm{exp}}(v_{i1t} + 0.5\sigma _{v2}^2)\gamma _2}} = \frac{{\rho _{1t - 1}}}{{\rho _{2t - 1}}}} \\ {K_{i1t} + K_{i2t} = K_{it}} \end{array}} \right.$$

$$\Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {K_{i1t} = \frac{{\beta _{1K}R_{i1t}\rho _{2t - 1}}}{{\beta _{1K}R_{i1t}\rho _{2t - 1} + \beta _{2K}R_{i2t}\rho _{1t - 1}\left( {\frac{{\gamma _2}}{{\gamma _1}}} \right)(\frac{{z_{i2t}}}{{z_{i1t}}}){\mathrm{exp}}(v_{i1t} - v_{i2t} + 0.5\sigma _{v2}^2 - 0.5\sigma _{v1}^2)}}K_{it}} \\ {K_{i2t} = \frac{{\beta _{2K}R_{i2t}\rho _{1t - 1}}}{{\beta _{1K}R_{i1t}\rho _{2t - 1}\left( {\frac{{\gamma _2}}{{\gamma _1}}} \right)\left( {\frac{{z_{i1t}}}{{z_{i2t}}}} \right)\exp \left( {v_{i2t} - v_{i1t} + 0.5\sigma _{v1}^2 - 0.5\sigma _{v2}^2} \right) + \beta _{2K}R_{i2t}\rho _{1t - 1}}}K_{it}} \end{array}} \right..$$

(16)

In Eq. (16), segment actual revenues (R), average segment capital price (ρ), and average segment labor price (ω) are observed. If Eq. (8) can be estimated, other variables, including the production parameters (β), the productivity shock (A)¹⁰, the noises (v) and their variances ($\sigma _v^2$), as well as z and γ, are all known. Then the capital allocations in Eq. (16) can be solved.

Equations (14) and (16) consist of the solutions of the second system of equations, which can be combined into Eq. (17)

$$\left\{ {\begin{array}{*{20}{l}} {L_{i1t} = \frac{{\beta _{1L}R_{i1t}\omega _{2t}}}{{\beta _{1L}R_{i1t}\omega _{2t} + \beta _{2L}R_{i2t}\omega _{1t}{\mathrm{exp}}(v_{i1t} - v_{i2t} + 0.5\sigma _{v2}^2 - 0.5\sigma _{v1}^2)}}L_{it}} \\ {L_{i2t} = \frac{{\beta _{2L}R_{i2t}\omega _{1t}}}{{\beta _{1L}R_{i1t}\omega _{2t}\exp \left( {v_{i2t} - v_{i1t} + 0.5\sigma _{v1}^2 - 0.5\sigma _{v2}^2} \right) + \beta _{2L}R_{i2t}\omega _{1t}}}L_{it}} \\ {K_{i1t} = \frac{{\beta _{1K}R_{i1t}\rho _{2t - 1}}}{{\beta _{1K}R_{i1t}\rho _{2t - 1} + \beta _{2K}R_{i2t}\rho _{1t - 1}\left( {\frac{{\gamma _2}}{{\gamma _1}}} \right)(\frac{{z_{i2t}}}{{z_{i1t}}}){\mathrm{exp}}(v_{i1t} - v_{i2t} + 0.5\sigma _{v2}^2 - 0.5\sigma _{v1}^2)}}K_{it}} \\ {K_{i2t} = \frac{{\beta _{2K}R_{i2t}\rho _{1t - 1}}}{{\beta _{1K}R_{i1t}\rho _{2t - 1}\left( {\frac{{\gamma _2}}{{\gamma _1}}} \right)\left( {\frac{{z_{i1t}}}{{z_{i2t}}}} \right)\exp \left( {v_{i2t} - v_{i1t} + 0.5\sigma _{v1}^2 - 0.5\sigma _{v2}^2} \right) + \beta _{2K}R_{i2t}\rho _{1t - 1}}}K_{it}} \end{array},} \right.$$

(17)

where

$$z_{ijt} = \frac{{p_{jt - 1}}}{{p_{jt}}}\frac{{A_{ijt - 1}}}{{A_{ijt}}}(\frac{{L_{ijt - 1}}}{{L_{ijt}}})^{\beta _{jL}},$$

$$\gamma _j = \exp (\mu _{pj} + \mu _{Aj} + \beta _{jL}\mu _{Lj} + \frac{{\sigma _{pj}^2 + \sigma _{Aj}^2 + \sigma _{Lj}^2\beta _{jL}^2}}{2}).$$

This system says that for each input, the ratio of expected marginal revenue equals the input price ratio across divisions. In other words, the cost for an input to generate additional expected revenue is equal across divisions at the undistorted allocations; hence, this study calls it the “equal (expected) marginal revenue per cost” condition. If the first system of equations in Eq. (8) is estimated, the input allocations for both firms E and F can be imputed in the second system of equations in Eq. (17).

The next two subsections generalize the model, allowing for more inputs, segments, and multi-divisional firms, as well as different production functional forms. The algorithm is analogous to the simple model above and can derive Theorem 1. Those who are not interested in the details can skip the rest of Appendix A.

A.2 General model

A.2.1 Setup

As an extension of the simplified model, the general model considers a generalized production function and analyzes an “M Inputs—N Products/Segments—T periods” economy. Moreover, firms can have footprints in one or multiple segments and enter or exit segments over time.

A.2.2 Single production function

The canonical stochastic production function for panel data, or what this study calls the SPF estimator, runs the regression for individual (country, region, company, etc.) i at time t¹¹:

$$Y_{it} = f\left( {X_{it};\beta _0} \right)\exp \left( {\alpha _{it}} \right)\exp \left( {v_{it}} \right),$$

(18)

where Y_it is the total output of individual i at time t; X_it = ($X_{it}^1$, $X_{it}^2$, …, $X_{it}^M$) vectors the M types of inputs; f(X_it; β₀) is the production function, β₀ = (β₀₁, β₀₂, …, β_0M) is a vector of technical parameters to be estimated. A_it = exp(α_it) is assumed to be the Hicks-neutral productivity that is log-additive and firm-specific, and exp(v_it) is the stochastic component that describes random shocks affecting the production process, where v_it is assumed to be normally distributed with a mean of zero and a standard deviation of σ_v.

A.2.3 Segment-specific production function

As has been discussed, Eq. (18) assumes a unique production function across segments and can only derive firm-level productivity without segment-specific production function concern. The SSPF analysis in an “M Inputs – N Products/Segments—T periods” economy imputes the input allocations by solving two systems of equations simultaneously.

Each equation in the first system of N Eq. (19) describes the production techniques for the corresponding segment¹².

$$\left\{ {\begin{array}{*{20}{l}} {Y_{i1t} = f_1\left( {X_{i1t};\beta _1} \right)\exp \left( {\alpha _{i1t}} \right)\exp \left( {v_{i1t}} \right)} \\ \ldots \\ {Y_{iNt} = f_N\left( {X_{iNt};\beta _N} \right)\exp \left( {\alpha _{iNt}} \right)\exp \left( {v_{iNt}} \right)} \end{array}} \right.,$$

(19)

where Y_ijt represents the observed scalar output and X_ijt vectors the unobserved inputs of individual i in segment j at time t, respectively. f_j(X_ijt; β_j) is the heterogeneous production function for segment j, where β_j = (β_j1, β_j2, …) is a vector of segment-specific technical parameters; A_ijt = exp(α_ijt) is assumed to be the Hicks-neutral productivity that is log-additive and firm-division-specific; v_ijt is the noise that is considered normally distributed with a mean of zero and a standard deviation of σ_vj. The data pooled into the j-th equation in Eq. (19) depends on the validity of the key assumption. If assuming that single-division firms and multi-divisional firms have the same production techniques, all the players in segment j are included in the regression. Otherwise, the single-division firms are removed from the regression.

The second system of equations solves the profit maximization problem of multi-divisional firms. For firm i at time t, the mathematical expectation of the profit function is

$$\left\{ {\begin{array}{*{20}{l}} {\mathop {{\max }}\nolimits_{X_{ijt}^M} \,{\mathrm{E}}\left[ {\pi _{it}|I_{it - 1}} \right] = \mathop {{\max }}\nolimits_{X_{ijt}^M} \mathop {\sum}\nolimits_{j \in W\left( {i,t} \right)} {({\mathrm{E}}\left[ {p_{jt}Y_{ijt}|I_{it - 1}} \right]} - {\mathrm{E}}\left[ {\mathop {\sum}\nolimits_{k = 1}^M {h_{ikt}c_{jt}^kX_{ijt}^k|I_{it - 1}} } \right])} \\ {\mathop {{\max }}\nolimits_{X_{ijt}^k} \,{\mathrm{E}}\left[ {\pi _{it}|I_{it}} \right] = \mathop {{\max }}\nolimits_{X_{ijt}^k} \mathop {\sum}\nolimits_{j \in W\left( {i,t} \right)} {({\mathrm{E}}\left[ {p_{jt}Y_{ijt}|I_{it}} \right]} - {\mathrm{E}}\left[ {\mathop {\sum}\nolimits_{k = 1}^M {h_{ikt}c_{jt}^kX_{ijt}^k|I_{it}} } \right]),\forall k\, < \,M} \end{array}} \right.,$$

(20)

where

$$h_{ikt} = h_{ikt}\left({\sum \limits_{k = 1}^M} {c_{jt}^k}{X_{ijt}^k}\right) ,\forall k = 1,2, \ldots ,M$$

and

$$Y_{ijt} = f_j\left( {X_{ijt};\beta _j} \right)\exp \left( {\alpha _{ijt}} \right)\exp \left( {v_{ijt}} \right),\,\forall j \in W\left( {i,t} \right),$$

(21)

where Y_ijt and $X_{ijt}^k$ represent the observed scalar output and the unobserved k-th input of individual i in segment j at time t, respectively. The last input, $X_{ijt}^M$, is the division-level capital input. h_ikt is the spillover effects on the k-th input for firm i at time t. $c_{jt}^k$is the average price of the k-th input in segment j at time t. Similar to the simple example in Subsection A.1.1.3, the general model also assumes that firms predict $\log (x_{ijt^\prime })|\log (x_{ijt})\sim N( {\log ( {x_{ijt}} ) + \mu _{xj},\,\sigma _{xj}^2} )$ where x = p, A, X^k, ω, ρ and all the μ and σ are known. W(i, t) is a subset of W = (1, 2, …, N), indicating the segments in which individual i at time t has a footprint. When estimating region or country productivity, W(i, t) is usually similar to W since most countries have production in every sector or segment. A counterexample is Singapore, which has a negligible agriculture industry. For company productivity analysis, W(i, t) is smaller than W in most cases, as only a few integrated companies do business in every segment. Take this study’s empirical study as an example; only one out of 114 firms does business in all five segments of the oilfield market.

Using the same strategy in Eq. (9), this study first solves all the “perfectly variable” inputs and then solves the capital input. By solving the first order condition of Eq. (20), the “equal (expected) marginal revenue per cost” condition can be derived. Adding the observed firm-level inputs, the first part of Theorem 1 (Eq. (3)) is derived

$$\left\{ {\begin{array}{*{20}{l}} {\displaystyle \frac{{\frac{{\partial E\left[ {R_{ijt}|I_{it}} \right]}}{{\partial X_{ijt}^k}}}}{{\frac{{\partial E\left[ {R_{ij^\prime t}\left| {I_{it}} \right.} \right]}}{{\partial X_{ij^\prime t}^k}}}} = \frac{{c_{jt}^k}}{{c_{j^\prime t}^k}} = r_{jj^\prime t}^k,\forall i,t,\forall k\, < \,M,\forall j,j^\prime \in W\left( {i,t} \right)} \\ {\displaystyle \frac{{\frac{{\partial E\left[ {R_{ijt}\left| {I_{it - 1}} \right.} \right]}}{{\partial X_{ijt}^M}}}}{{\frac{{\partial E\left[ {R_{ij^\prime t}\left| {I_{it - 1}} \right.} \right]}}{{\partial X_{ij^\prime t}^M}}}} = \frac{{c_{jt - 1}^M}}{{c_{j^\prime t - 1}^M}} = r_{jj^\prime t - 1}^M,\forall i,t,\forall j,j^\prime \in W\left( {i,t} \right)} \\ {\mathop {\sum}\nolimits_{j \in W\left( {i,t} \right)} {X_{ijt}^k = X_{it}^k,\forall i,k,t} } \end{array},} \right.$$

(22)

where $c_{jt}^k$is the average price of the k-th input in segment j at time t; $r_{jj^\prime t}^k$ represents the ratio of average price for the k-th input between segments j and j’ at time t.

Suppose N(i, t) is the number of elements (i.e., the number of segments entered by firm i at time t) in set W(i,t), and the system of Eq. (22) has ∑_i,_tN(i, t)·M equations with ∑_i,_tN(i,t)·M unknown division-level inputs (i.e. the black box of input allocations). If the first system of equations in Eq. (19) can be estimated, then system (22) is just-identified and the input allocations can be imputed.

This study looks for the undistorted allocations of segment-level inputs ($X_{ijt}^k$) that jointly solves the system of Eq. (19) and the system of Eq. (22). In order to find the undistorted input allocations and the segment-specific production functions that satisfy Eq. (19) and Eq. (22) simultaneously, this study implements the same iterative method discussed in Subsection A.1.4.

A.3 Parametric production function specification

Since the stochastic production function is a parametric approach, the mathematical form for the production function needs to be specified. This study discusses the solution of the undistorted input allocations when the production function f_j(X_ijt; β_j) has the most widely used forms: the Cobb-Douglas and Transcendental Logarithmic forms.

A.3.1 Cobb-Douglas production function

Given a C-D form, the canonical model in Eq. (18) has the form:

$$Y_{it} = {\mathrm{exp}}(\alpha _{it})\mathop {\prod}\limits_{k = 1}^2 {(X_{it}^k)^{\beta _{0k}}} {\mathrm{exp}}(v_{it})$$

$$\Leftrightarrow {\mathrm{ln}}Y_{it} = \alpha _{it} + \mathop {\sum}\limits_{{\mathrm{k}} = 1}^2 {\beta _{0k}} {\mathrm{ln}}(X_{it}^k) + v_{it}.$$

(23)

The segment-specific production function in Eq. (19) becomes:

$$\left\{ {\begin{array}{*{20}{c}} {\ln Y_{i1t} = \alpha _{i1t} + \mathop {\sum}\nolimits_{{\mathrm{k}} = 1}^{\mathrm{M}} {\beta _{1k}\ln \left( {X_{i1t}^k} \right) + v_{i1t}} } \\ {...} \\ {\ln Y_{iNt} = \alpha _{iNt} + \mathop {\sum}\nolimits_{{\mathrm{k}} = 1}^{\mathrm{M}} {\beta _{Nk}\ln \left( {X_{iNt}^k} \right) + v_{iNt}} } \end{array}} \right..$$

(24)

The “equal marginal revenue per cost” constraint is then:

$$\left\{ {\begin{array}{*{20}{l}} {\displaystyle \frac{{\frac{{\partial E\left[ {R_{ijt}\left| {I_{it}} \right.} \right]}}{{\partial X_{ijt}^k}}}}{{\frac{{\partial E\left[ {R_{ij^\prime t}\left| {I_{it}} \right.} \right]}}{{\partial X_{ij^\prime t}^k}}}} = \frac{{\beta _{jk}R_{ijt}X_{ij^\prime t}^k{\mathrm{exp}}\left( {v_{ij^\prime t} + 0.5\sigma _{vj}^2} \right)}}{{\beta _{j^\prime k}R_{ij^\prime t}X_{ijt}^k{\mathrm{exp}}\left( {v_{ijt} + 0.5\sigma _{vj^\prime }^2} \right)}} = r_{jj^\prime t}^k,\forall i,t,\forall k\, < \,M,\forall j,j^\prime \in W\left( {i,t} \right)} \\ {\displaystyle \frac{{\frac{{\partial E\left[ {R_{ijt}\left| {I_{it - 1}} \right.} \right]}}{{\partial X_{ijt}^M}}}}{{\frac{{\partial E\left[ {R_{ij^\prime t}\left| {I_{it - 1}} \right.} \right]}}{{\partial X_{ij^\prime t}^M}}}} = \frac{{\beta _{jk}z_{ijt}R_{ijt}X_{ij^\prime t}^M{\mathrm{exp}}\left( {v_{ij^\prime t} + 0.5\sigma _{vj}^2} \right)\gamma _j}}{{\beta _{j^\prime k}z_{ij^\prime t}R_{ij^\prime t}X_{ijt}^M{\mathrm{exp}}\left( {v_{ijt} + 0.5\sigma _{vj^\prime }^2} \right)\gamma _{j^\prime }}} = r_{jj^\prime t - 1}^M,\forall i,t,\forall j,j^\prime \in W\left( {i,t} \right)} \end{array}} \right.$$

where

$$z_{ijt} = \frac{{p_{jt - 1}}}{{p_{jt}}}\frac{{A_{ijt - 1}}}{{A_{ijt}}}\mathop {\prod}\limits_{k = 1, \ldots ,M - 1} {\left( {\frac{{X_{ijt - 1}^k}}{{X_{ijt}^k}}} \right)^{\beta _{jk}}} ,$$

$$A_{ijt} = {\mathrm{exp}}(\alpha _{ijt}).$$

Given that and the observed firm-level inputs, it is easy to solve the system of equations:

$$\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {X_{ijt}^k = \frac{{\beta _{jk}R_{ijt}}}{{\mathop {\sum }\nolimits_{j^\prime \in W\left( {i,t} \right)} \beta _{j^\prime k}R_{ij^\prime t}r_{jj^\prime t}^k{\mathrm{exp}}(v_{ijt} - v_{ij^\prime t} + 0.5\sigma _{vj^\prime }^2 - 0.5\sigma _{vj}^2)}}X_{it}^k,\forall k\, < \,M} \\ {X_{ijt}^M = \frac{{\beta _{jM}R_{ijt}}}{{\mathop {\sum }\nolimits_{j^\prime \in W\left( {i,t} \right)} \beta _{j^\prime M}R_{ij^\prime t}r_{jj^\prime t}^M\left( {\frac{{\gamma _{j^\prime }}}{{\gamma _j}}} \right)(\frac{{z_{ij^\prime t}}}{{z_{ijt}}}){\mathrm{exp}}(v_{ijt} - v_{ij^\prime t} + 0.5\sigma _{vj^\prime }^2 - 0.5\sigma _{vj}^2)}}X_{it}^M} \end{array}} \right.} \end{array},$$

(25)

where

$$\gamma _j = {\mathrm{exp}}(\mu _{pj} + \mu _{Aj} + \mathop {\sum }\limits_{k = 1}^{M - 1} \beta _{jk}\mu _{X^kj} + \frac{{\sigma _{pj}^2 + \sigma _{Aj}^2 + \mathop {\sum }\nolimits_{k = 1}^{M - 1} \beta _{jk}^2\sigma _{X^kj}^2}}{2}).$$

Equation (25) is the second part of Theorem 1 (Eq. (4)). The solution exists when all the parameters β_jk are positive. Finally, the iteration method is used to solve the undistorted input allocations and technical parameters jointly from Eqs. (24) and (25).

A.3.2 Transcendental logarithmic production function

Given a T-L form, the production function in Eq. (18) becomes:

$${\mathrm{ln}}Y_{it} = \alpha _{it} + \mathop {\sum }\limits_{{\mathrm{k}} = 1}^{\mathrm{M}} \beta _{0k}\ln \left( {X_{it}^k} \right) + \frac{1}{2}\mathop {\sum }\limits_{k = 1}^M \mathop {\sum }\limits_{l = 1}^M \delta _{0kl}{\mathrm{ln}}X_{it}^k{\mathrm{ln}}X_{it}^l + v_{it},$$

where

$$\delta _{0kl} = \delta _{0lk},\forall k,l \in (1,2, \ldots ,M).$$

Similarly, the segment-level production function in Eq. (19) becomes:

$$\left\{ {\begin{array}{*{20}{l}} {\ln Y_{i1t} = \alpha _{i1t} + \mathop {\sum}\nolimits_{{\mathrm{k}} = 1}^{\mathrm{M}} {\beta _{1k}} \ln \left( {X_{i1t}^k} \right) + \frac{1}{2}\mathop {\sum}\nolimits_{k = 1}^M {\mathop {\sum}\nolimits_{l = 1}^M {\delta _{1kl}} } \ln X_{i1t}^k\ln X_{i1t}^l + v_{i1t}} \\ \ldots \\ {\ln Y_{iNt} = \alpha _{iNt} + \mathop {\sum}\nolimits_{{\mathrm{k}} = 1}^{\mathrm{M}} {\beta _{Nk}} \ln \left( {X_{iNt}^k} \right) + \frac{1}{2}\mathop {\sum}\nolimits_{k = 1}^M {\mathop {\sum}\nolimits_{l = 1}^M {\delta _{Nkl}} } \ln X_{iNt}^k\ln X_{iNt}^l + v_{iNt}} \end{array}} \right.,$$

where

$$\delta _{jkl} = \delta _{jlk},\,\forall k,l \in \left( {1, \ldots ,M} \right),\,\forall j = 1, \ldots ,N.$$

The “equal marginal revenue per cost” constraint is then:

$$\left\{ {\begin{array}{*{20}{l}} {\displaystyle \frac{{\frac{{\partial E[R_{ijt}|I_{it}]}}{{\partial X_{ijt}^k}}}}{{\frac{{\partial E\left[ {R_{ij\prime t}|I_{it}} \right]}}{{\partial X_{ij\prime t}^k}}}} = \frac{{\beta _{jk}R_{ijt}X_{ij\prime t}^k{\mathrm{exp}}(v_{ij\prime t} + 0.5\sigma _{vj}^2)}}{{\beta _{j\prime k}R_{ij\prime t}X_{ijt}^k{\mathrm{exp}}(v_{ijt} + 0.5\sigma _{vj\prime }^2)}} = r_{jj\prime t}^k,\,\forall i,t,\forall k < M,\forall j,j\prime \in W\left( {i,t} \right)} \\ {\displaystyle \frac{{\frac{{\partial E[R_{ijt}|I_{it - 1}]}}{{\partial X_{ijt}^M}}}}{{\frac{{\partial E\left[ {R_{ij\prime t}|I_{it - 1}} \right]}}{{\partial X_{ij\prime t}^M}}}} = \frac{{\beta _{jk}z_{ijt}R_{ijt}X_{ij\prime t}^M{\mathrm{exp}}(v_{ij\prime t} + 0.5\sigma _{vj}^2)\gamma _j}}{{\beta _{j\prime k}z_{ij\prime t}R_{ij\prime t}X_{ijt}^M{\mathrm{exp}}(v_{ijt} + 0.5\sigma _{vj\prime }^2)\gamma _{j\prime }}} = r_{jj\prime t - 1}^M,\forall i,t,\forall j,j\prime \in W\left( {i,t} \right)} \end{array}} \right.,$$

(26)

where

$$\begin{array}{*{20}{c}} {z_{ijt} = \frac{{p_{jt - 1}}}{{p_{jt}}}\frac{{A_{ijt - 1}}}{{A_{ijt}}}\mathop {\prod}\limits_{k = 1, \ldots ,M - 1} {\left( {\frac{{X_{ijt - 1}^k}}{{X_{ijt}^k}}} \right)^{\beta _{jk}}} \,\mathop {\prod}\limits_{k,l = 1, \ldots ,M} {\frac{{\exp (\frac{1}{2}\delta _{1kl}\ln X_{i1t - 1}^k\ln X_{i1t - 1}^l)}}{{\exp (\frac{1}{2}\delta _{1kl}\ln X_{i1t}^k\ln X_{i1t}^l)}}} } \\ {A_{ijt} = \exp (\alpha _{ijt})\,and\,k + l\, < \,2M} \\ {\gamma _j = \exp \left\{ {\mu _{pj} + \mu _{Aj} + \mathop {\sum}\limits_{k = 1}^{M - 1} {\beta _{jk}\mu _{X^kj}} + \frac{{\sigma _{pj}^2 + \sigma _{Aj}^2 + \mathop {\sum}\nolimits_{k = 1}^{M - 1} {\beta _{jk}^2\sigma _{X^kj}^2} }}{2} } \right.} \\ + \,{\left. {\mathop {\sum}\limits_{k,l = 1}^M {\left[ {\left( {\beta _{jk}\mu _{X^kj} + \frac{{\beta _{jk}^2\sigma _{X^kj}^2}}{2}} \right)\left( {\beta _{jl}\mu _{X^lj} + \frac{{\beta _{jl}^2\sigma _{X^lj}^2}}{2}} \right)} \right]} } \right\}.} \end{array}$$

Compared with Eq. (24) in the C-D case, Eq. (26) in the T-L case adds the $\mathop {\sum }\nolimits_{l = 1}^M \delta _{jkl}{\mathrm{ln}}X_{ijt}^l$ part, which makes the latter case much more complicated to solve. Interestingly, this ratio itself is a C-D function.

In the T-L case, the system of equations in Eq. (22) becomes

$$\left\{ {\begin{array}{*{20}{l}} { \frac{\displaystyle{\frac{{\partial E[R_{ijt}|I_{it}]}}{{\partial X_{ijt}^k}}}}{{\frac{\displaystyle{\partial E\left[ {R_{ij^\prime t}|I_{it}} \right]}}{\displaystyle{\partial X_{ij^\prime t}^k}}}} = \frac{{(\beta _{jk} + \mathop {\sum}\nolimits_{l = 1}^M {\delta _{jkl}} \ln X_{ijt}^l)R_{ijt}X_{ij^\prime t}^k\exp (v_{ij^\prime t} + 0.5\sigma _{vj}^2)}}{{(\beta _{j^\prime k} + \mathop {\sum}\nolimits_{l = 1}^M {\delta _{j^\prime kl}} \ln X_{ij^\prime t}^l)R_{ij^\prime t}X_{ijt}^k\exp (v_{ijt} + 0.5\sigma _{vj^\prime }^2)}} = r_{jj^\prime t}^k,\forall k\, < \,M} \\ { \frac{\displaystyle{\frac{{\partial E[R_{ijt}|I_{it - 1}]}}{{\partial X_{ijt}^M}}}}{{\frac{\displaystyle{\partial E\left[ {R_{ij^\prime t}|I_{it - 1}} \right]}}{{\partial X_{ij^\prime t}^M}}}} = \frac{{z_{ijt}(\beta _{jk} + \mathop {\sum}\nolimits_{l = 1}^M {\delta _{jkl}} {\mathrm{ln}}X_{ijt}^l)R_{ijt}X_{ij^\prime t}^k{\mathrm{exp}}(v_{ij^\prime t} + 0.5\sigma _{vj}^2)\gamma _j}}{{z_{ij^\prime t}(\beta _{j^\prime k} + \mathop {\sum}\nolimits_{l = 1}^M {\delta _{j^\prime kl}} {\mathrm{ln}}X_{ij^\prime t}^l)R_{ij^\prime t}X_{ijt}^k{\mathrm{exp}}(v_{ijt} + 0.5\sigma _{vj^\prime }^2)\gamma _{j^\prime }}} = r_{jj^\prime t - 1}^M} \\ {\mathop {\sum}\nolimits_{j \in W\left( {i,t} \right)} {X_{ijt}^k} = X_{it}^k,\forall i,k,t} \end{array}.} \right.$$

(27)

Equation (27) is the third part of Theorem 1 (Eq. (5)). This system of equations is just-identified, where the number of unknown variables equals the number of equations. However, whether a solution exists depends on the initial inputs, i.e., β_jk, δ_jkl, Y_ijt, $X_{it}^k$, M, and N(i,t). The T-L production function is a flexible functional form and a generalization of the C-D production function (Allen and Hall 1997). Compared with the C-D form, the T-L form does not need the perfect substitution between inputs restriction or the linear input-output restriction (Klacek et al. 2007). However, an easy solution cannot be obtained for the T-L form in Eq. (A20) as was done for the C-D form solution in Eq. (25).

Appendix B: Accuracy test of the imputation method

In order to test the accuracy of the input allocation imputed by our method, the actual input allocation within an organization should be observed. This article uses an OECD dataset to do the test because segment-level input information is observed. It is worth noting that there is no profit maximizing manager determines input allocation across segments within a country, which makes this macro data not a perfect example. However, as we emphasized, it is difficult to observe division-level input allocation within each company for a specific industry, which motivates us to establish this imputation method. So we cannot find a better dataset to test the accuracy of our imputation method. Moreover, the main purpose of this appendix is to introduce the methodology of our accuracy test so that other scholars can apply this method when micro-level data are available.

B.1 External validation of the imputation method using OECD data

This appendix uses panel data for OECD countries to test whether the imputation method estimates accurate input allocation in order to provide some external validation of the imputation methods we will employ in our study of oilfield production processes. Many studies treat countries or regions as firms and use a production function approach to estimate the productivity of a sector or the whole economy. For example, Koop et al. (1999) decompose output change into technical, efficiency and input changes for seventeen OECD countries, Mastromarco and Ghosh (2009), derive total factor productivity for fifty-seven developing countries, and Koop et al. (1995) measure the source of growth in four regions, consisting of East Asia, Africa, Latin America, and West. Other studies focus on cross-state or cross-province rather than cross-country productivity analysis, such as the thirty provinces of China (Wei and Hao 2011; Gong 2018a) and the forty-eight contiguous US states (Puig-Junoy 2001). However, all of these studies utilize one production function for the entire economy without considering differences across segments. With regard to segment-specific production functions, the SSPF method can impute segment-level input allocations in each country.

Our external validation study uses panel data for twenty-two OECD countries¹³ from 2000 to 2006 where labor and capital are the inputs and the value-added is the output. The dataset is mainly collected from the Structural Analysis (STAN) Database, the Annual National Accounts (ANA) Database, and the Monthly Monetary and Financial Statistics (MEI) Database, all of which are in the OECD iLibrary¹⁴.

Based on the sector classifications for the OECD data, the total economy of a country can be divided into primary, secondary, and tertiary sectors and further classified into nine industries:

I) the primary sector is segmented into (1) agriculture, hunting, forestry and fishing, and (2) mining and quarrying;

II) the secondary sector is divided into three categories, consisting of (3) manufacturing, (4) electricity, gas and water supply, and (5) construction; and

III) the tertiary sector includes (6) wholesale, retail trade, restaurants and hotels, (7) transport, storage and communications, (8) finance, insurance, real estate and business services, and (9) community, social, and personal services.

This test data has segment-level output, input allocations, and price data for each country. The output is value added in US dollars by segment and country. For the labor input, total employment and compensation for employees are collected. The former is a total quantity and the latter is divided by the former, thus providing labor price information. For the capital input, net capital stock/net fixed assets, consumption of fixed assets, and interest rate are available. The consumption of fixed assets divided by the net capital stock/net fixed asset is the depreciation rate. The price of capital, or the user cost of capital, is the sum of the depreciation rate and the interest rate. Table 8 provides summary statistics for the inputs and outputs by segment.

A 3-Segment model is set up for the OECD countries, where the output is value added (Y_it), the two inputs are labor ($X_{it}^1 = L_{it}$) and capital ($X_{it}^2 = K_{it}$), and the three segments are the primary, secondary, and tertiary sectors. In order to check the robustness of the results, we further classify the economy using nine industries and impute the undistorted input allocations for a 9-Segment model. We assume a C-D technology function in this validation exercise.

B.2 Testing methods

“Equal (expected) marginal revenue per cost” method. The undistorted allocations of segment-level inputs $\widehat X_{ijt}^k$ for the OECD countries imputed from the SSPF approach are called the “equal (expected) marginal product per cost” estimation method,¹⁵ which satisfies

$$\left\{ {\begin{array}{*{20}{l}} {\displaystyle \frac{{\beta _{jk}R_{ijt}X_{ij^\prime t}^k\exp (v_{ij^\prime t} + 0.5\sigma _j^2)}}{{\beta _{j^\prime k}R_{ij^\prime t}X_{ijt}^k\exp (v_{ijt} + 0.5\sigma _{j^\prime }^2)}} = r_{jj^\prime t}^k,\forall k\, < \, M,\forall j,j^\prime \in W(i,t)} \\ {\displaystyle \frac{{z_{ijt}\beta _{jk}R_{ijt}{\mathrm{X}}_{{\mathrm{ij}}^\prime {\mathrm{t}}}^{\mathrm{M}}\exp \left( {v_{ij^\prime t} + 0.5\sigma _j^2} \right)}}{{z_{ij^\prime t}\beta _{j^\prime k}R_{ij^\prime t}X_{ijt}^M\exp (v_{ijt} + 0.5\sigma _{j^\prime }^2)}} = r_{jj^\prime t - 1}^M,\forall i,t,\forall j,j^\prime \in W(i,t)} \end{array}} \right.,$$

(28)

where

$$z_{ijt} = \frac{{p_{jt - 1}}}{{p_{jt}}}\frac{{A_{ijt - 1}}}{{A_{ijt}}}\mathop {\prod}\limits_{k = 1, \ldots ,M - 1} {\left( {\frac{{X_{ijt - 1}^k}}{{X_{ijt}^k}}} \right)^{\beta _{jk}}} .$$

We test whether method 1 (Eq. (28)) imputes allocations that are closer to the actual allocations than those based on three other methods that have been used by researchers. The three alternative methods for generating the missing allocations are presented in Eqs. (29)–(31) below.

“Equal revenue per input” method. If the differences in both price and production function across segments are ignored in Eq. (28), then the segment-level input allocations are in proportion to the actual revenue across segments:

$$\frac{{\frac{{R_{ijt}}}{{X_{ijt}^k}}}}{{\frac{{R_{ij\prime t}}}{{X_{ij^\prime t}^k}}}} = 1,\,\forall k = 1,2,\,\forall j,j^\prime \in W(i,t).$$

(29)

The input proportionality assumption is widely used, as it requires the least amount of information. For example, Foster et al. (2008) allocate inputs based on products’ revenue shares. We regard this imputation method as the benchmark since it is a reasonable first estimate and also is used in the first step of the iterative 4-step method we outline above. Financial analysts often use revenue per employee and revenue per capital as productivity ratios. This “Equal revenue per input” imputation method assumes that the productivity ratios are equal across segments within a company.

“Equal (expected) marginal revenue per input” method. In order to test whether considering the difference in price across segments in Eq. (28) improves the accuracy of the imputation, we also can base input allocations on the conditions

$$\left\{\displaystyle {\begin{array}{*{20}{l}} {\displaystyle \frac{{\beta _{jk}R_{ijt}X_{ij^\prime t}^k{\mathrm{exp}}(v_{ij^\prime t} + 0.5\sigma _j^2)}}{{\beta _{j^\prime k}R_{ij^\prime t}X_{ijt}^k{\mathrm{exp}}(v_{ijt} + 0.5\sigma _{j^\prime }^2)}} = \displaystyle 1,\forall i,t,\forall k\, < \,M,\forall j,j^\prime \in W(i,t)} \\ {\displaystyle \frac{{z_{ijt}\beta _{jk}R_{ijt}X_{ij^\prime t}^M{\mathrm{exp}}(v_{ij^\prime t} + 0.5\sigma _j^2)}}{{z_{ij^\prime t}\beta _{j^\prime k}R_{ij^\prime t}X_{ijt}^M{\mathrm{exp}}(v_{ijt} + 0.5\sigma _{j^\prime }^2)}} = \displaystyle 1,\forall i,t,\forall j,j^\prime \in W(i,t)} \end{array}} \right.,$$

(30)

where differences in the market price of an input across divisions are ignored.

“Equal revenue per cost” method. Similar to the benchmark estimation in Eq. (29), this imputation method considers average revenue rather than expected marginal revenue. However, the approach also takes input price information into consideration by assuming that average revenue per input is proportional to the price of the same input across segments:

$$\frac{{\frac{{R_{ijt}}}{{X_{ijt}^k}}}}{{\frac{{R_{ij^\prime t}}}{{X_{ij^\prime t}^k}}}} = r_{ijj^\prime t}^k,\,\forall k = 1,2,\,\forall j,j^\prime \in W(i,t).$$

(31)

This method guarantees that the average costs of value-added production are equal across segments and is therefore denoted as the “equal revenue per cost” estimation.

To sum up, only the undistorted estimation (“equal expected marginal revenue per cost”) considers both segment-specific input prices and production functions. The benchmark estimation “equal revenue per input” is the easiest to derive, as it requires no additional information. The “equal revenue per cost” estimation takes input price into account on the basis of the benchmark estimation and thus may be more likely to be close to the actual level. Both of these imputation methods ignore segment-specific technical parameters, but they are more computationally friendly than other methods, since no segment-specific production regressions and iterations are involved. The “equal marginal revenue per input” estimation, on the other hand, considers a segment-specific production function but ignores the heterogeneity in input prices across segments. This estimation can therefore substitute for the undistorted estimation if segment-level input prices are unobserved. Based on the amount of information used, this study predicts that the undistorted estimation is the most accurate and that the benchmark estimation is the least accurate. There is no measurable evidence about the relative accuracy between the “equal revenue per cost” and “equal marginal revenue per input” estimations. The advantage of the former is its lower computational burden, while the advantage of the latter is the fewer amounts of data (segment-level inputs) needed.

We base our evaluation of the accuracy of the three different imputation methods on mean squared error (MSE) and the mean absolute error (MAE). Suppose the actual segment-level value of the k-th input is $X_{ijt}^k$, while the values for the undistorted estimation (“equal expected marginal revenue per cost”), “equal revenue per input” estimation, “equal revenue per cost” estimation, and “equal marginal revenue per input” estimation are $\hat X_{ijt}^k(1)$, $\hat X_{ijt}^k\left( 2 \right)$, $\hat X_{ijt}^k(3)$, and $\hat X_{ijt}^k(4)$, respectively. Then, the mean squared error of the input allocations can be calculated using

$$MSE_{\left( p \right)} = \frac{{\mathop {\sum}\nolimits_k {\mathop {\sum}\nolimits_i {\mathop {\sum}\nolimits_j {\mathop {\sum}\nolimits_t {\left[ {\frac{{\hat X_{ijt}^k\left( p \right) - X_{ijt}^k}}{{X_{ijt}^k}}} \right]^2} } } } }}{{\mathop {\sum}\nolimits_k {\mathop {\sum}\nolimits_i {\mathop {\sum}\nolimits_j {\mathop {\sum}\nolimits_t {[{\mathrm{I}}(X_{ijt}^k \ne 0)]} } } } }}.$$

Similarly, the mean absolute error of the input allocations can be calculated using

$$MAE_{\left( p \right)} = \frac{{\mathop {\sum}\nolimits_k {\mathop {\sum}\nolimits_i {\mathop {\sum}\nolimits_j {\mathop {\sum}\nolimits_t {|(\hat X_{ijt}^k\left( p \right) - X_{ijt}^k)/X_{ijt}^k|} } } } }}{{\mathop {\sum}\nolimits_k {\mathop {\sum}\nolimits_i {\mathop {\sum}\nolimits_j {\mathop {\sum}\nolimits_t {[{\mathrm{I}}(X_{ijt}^k \ne 0)]} } } } }},$$

where I(∙) is the indicator function that takes on a value of one if the argument is correct and a value of zero otherwise. The numerator of the MSE (MAE) is the sum of the squared error (absolute error) of the segment-level estimation of the inputs to the actual level. The denominator is the number of nonzero segment-level values for the inputs.

The benchmark estimator in Eq. (29) is set as the initial starting value for the iterative method to estimate the undistorted allocations. The other two competing methods based on Eqs. (30) and (31) also can be alternative initial guesses of the iteration. We check if the iteration converges to the same allocations when given various initial allocations, which implies the robustness of the SSPF estimation.

B.3 Test Results

We utilize a stopping criterion c = 1 × 10⁻⁶ for convergence of the iteration process. Table 9 presents the error results of the 3-Segments model and the 9-Segments model. The first model takes four iterations to pass the criterion and the second model takes thirty iterations to achieve a stationary condition.

The first four columns of Table 9 show the MSE and MAE for the undistorted estimation and the other three estimations, respectively, for comparison. The fifth column shows the MSE and MAE ratios of the undistorted estimation and the benchmark estimation, while the sixth column shows the MSE and MAE ratios of the undistorted estimation and the most accurate estimation among the three competing methods.

Compared with the benchmark “equal revenue per input” estimate, the undistorted estimation significantly decreases the mean square error and hence improves the accuracy. In the 3-Segments model, the MSE of the undistorted estimation is 16% of that of the benchmark estimation. A significant improvement in estimating the input allocations is also achieved in the 9-Segments model, where the MSE of the undistorted estimation is 21% of that of the benchmark estimation. Moreover, the MAE of the undistorted estimation is around 60% of that of the benchmark estimation in both the 3-Segments model and the 9-Segments model, which also indicates that the undistorted imputation method is more accurate than the benchmark input proportionality estimation often used for imputation (see, for example, Foster et al. (2008)).

The relative MSE and MAE result in the last column of Table 9 are all smaller than one, which implies that the undistorted method is more accurate than all three competing methods. Among the three alternative methods, the “equal revenue per cost” method that is based on input price information is the most accurate, while the other two are less accurate and quite close to one another.

We also utilize all these competing imputation methods (Eqs. (29)–(31)) to derive input allocations by initializing the starting values in the iterations for our preferred equal (expected) marginal revenue per cost method based on these alternatives. In Eq. (6), it can be shown that variation in input allocations only depends on the variation of the production functions, since all of the revenues and prices in the equation are observed and fixed. Therefore, if the three initial allocations can derive similar production functions, these initial allocations can also impute similar undistorted input allocations.

Tables 10 and 11 present the estimated labor coefficient and capital coefficient in each segment for the 3-Segments model and the 9-Segments model, respectively. All coefficients derived by the three different initial allocations are quite comparable, indicating that the iterative method converges to the same undistorted input allocations using those different alternative methods as initial starting values.

In summary, we test whether our equal (expected) marginal revenue per cost method can deliver accurate imputations of input allocations using a panel of data for OECD countries where actual input allocations are observed. The results indicate that our imputation method provides estimated input allocations that are closer to the actual allocations than the other three competing imputation methods utilized in the literature and provides a level of external validity for the approach we will adopt in our empirical work below. Moreover, the observed input allocation allows us to check if labor and capital portfolios across segments are similar for the same company. In the 9-Segments setup, we find the correlation between labor and capital portfolios is only 0.34, and the average difference between labor and capital shares is over 0.10 (the average labor and capital shares is 0.11). Therefore, the assumption of constant input mix across products does not seem to be an appropriate one, which further supports our algorithm.

Table 8 OECD data summary statistics

Table 9 Accuracy test results

Table 10 Iteration results by different initial guess in 3 segments model

Table 11 Iteration results by different initial guess in 9 segments model