Quality-adjusted productivity gain in the propagation of innovation
- Satoshi Nakano^{1} and
- Kazuhiko Nishimura^{2}Email author
https://doi.org/10.1186/s40008-015-0020-9
© Nakano and Nishimura. 2015
Received: 16 December 2013
Accepted: 16 June 2015
Published: 13 August 2015
Abstract
We introduce a class of production function whose inputs and outputs constitute multiples of quality and quantity. Under the efficiency unit approach, we precisely reduce innovation regarding qualitative and quantitative improvements of production to the measurement of quality-adjusted productivity gain. We then consider a system of compatible unit cost functions inclusive of such productivity improvements in any industry, for which we can solve for the ex-post equilibrium to examine the technological structural propagation. In this way, we can evaluate any given innovation with respect to its social welfare gain. We use this framework of multi-industry multi-factor production to study effective industry-wise research and development investment allocations.
JEL classification: D24; O33; O41
Keywords
1 Background
Research and development (R&D) is considered the central driving force of national competitive advantage. Effectively using limited R&D resources is an important agenda when promoting and evaluating national level R&D programs Lee et al. (2009). In some cases, a large portion of national R&D resources are distributed to target technologies under the “selective focus and contraction strategy” Lee and Song (2007). However, we believe that this policy could impede the harmonious and sound development of the economy, because of mutual interdependencies in current and potential industrial production technologies. With this in mind, we are interested in investigating what allocations of public R&D investment could promote effective innovations and gain more welfare.
Contrary to R&D investment, innovation (the consequence of R&D) has been postulated as an intermediate stage of gaining welfare, rather than a measurement in terms of monetary value. Innovation is commonly defined as the implementation of a new or significantly improved product (good or service), process, marketing method, or organizational method in business practices, workplace organization, or external relations OECD/Eurostat (2005). Most work on innovation (as reviewed in Hall (2011)) has used surveys based on a version of this definition, typically using dummy variables that represent a product and/or process innovation (e.g., Griffith et al. (2006)), innovative sales shares (e.g., van Leeuwen and Klomp (2006)), or patent counts (e.g., Crépon et al. (1998)) as proxies to explain the growth of productivity.
Productivity growth fills the gap between growths in the quantitative inputs and outputs of production. Hence, a productivity growth signifies welfare increase that can be attributed to innovation. In other words, the productivity growth (or gain) can be used to evaluate the economic significance of an innovation that occurred within some interval of time. Additionally, because R&D is considered to inflate productivity, many researchers have investigated the connection between R&D investment (including R&D capital stock) and productivity growth (among others Griliches (1994), Hall and Mairesse (1995), Sakurai et al. (1997), Kuroda and Nomura (2004), Parisi et al. (2006), Coccia (2009), and Hall et al. (2009)).
This study also takes the position that R&D’s direct achievement is the gain in productivity. We are interested in the consequential influence of R&D investment, rather than the mechanism behind R&D’s promotion of innovation. Therefore, we directly use productivity gain as a measure of innovation, and then relate R&D investments with that measure. ^{1} More importantly, in contrast to previous research, we are concerned with the economy-wide propagation of innovation. Productivity gain reduces the marginal cost of production, and the corresponding price change influences the selection of technologies (substitution of inputs) for other industries, because of technological interdependencies between industrial sectors. A system of unit cost functions (which are compatible with constant returns to scale) can be used to model the economy-wide technological structural equilibrium and the propagation triggered by any exogenous productivity gain.
We can measure proportionality shifts using hedonic regression, where the set of observed prices are regressed against the Lancasterian attributes Lancaster (1966) of goods. This is a commonly used approach for estimating quality-adjusted consumer price indices (CPI). ^{2} Proportionality shifts can be measured via the price change (in the form of a deflator) of a quality-standardized commodity. Because such a deflator includes qualitative and quantitative improvements, we call it a quality-adjusted deflator. Monetary accounts of different periods are typically realized using deflators that only consider the price change. The quality-adjusted deflator considers the price and quality changes. ^{3}
For simplicity, we assume that any production is subject to constant returns to scale in effective quantities, whereas any assessment of quality (MRS) is universal, meaning that the same quality-adjusted deflators are applicable to any production or consumer. Then, the combined qualitative and quantitative innovation exclusively within an industry can be measured by applying quality-adjusted deflators to both inputs and outputs of production. We call this local innovation measure the quality-adjusted productivity gain of an industry. ^{4} Naturally, we relate industry-wise R&D investments with industry-wise quality-adjusted productivity gains. We use quality-adjusted unit cost functions (which map the costs of effective unit outputs) to evaluate the propagative effects of industry-wise R&D investments with respect to the gain in social welfare.
In Section 2, we introduce the concept of quality-adjusted productivity, which reflects the innovation that is exclusive to an industry (i.e., local innovation). This is closely related to the industry-wise R&D. In Section 3, we introduce the concept of structural propagation, which is initiated by the introduction of innovation involving qualitative and quantitative improvements, and connect the ex-post equilibrium state with social welfare. In addition, R&D investment allocation optimization is demonstrated via a small prototype model. Section 4 contains our concluding remarks.
2 Measurement
2.1 Illustrative example
The following example demonstrates the essence of our task. We assume that a new type of paint has been invented, which means that car manufacturers can use less paint during production. Additionally, they can also produce cars with less metal because the new paint enhances the strength of the materials on which it is applied. The new paint is easier to apply, requiring less labor in the manufacturing process. Moreover, consumers can derive more utility from the cars because the new paint looks good. As a result, the new paint affects technology selections (i.e., paint–metal and paint–labor substitutions), the quality (i.e., an attractive car), and the cost of the car.
The first assumption we make is that any two commodities of the same kind with different qualities are perfect substitutes. Suppose that there are two cans of paint named Sirius and Vega, such that 1 L of Sirius is as effective as 2 L of Vega. Then, the MRS of Sirius against Vega is 2. We can use this MRS as the measure of quality for Sirius, relative to the reference standard paint Vega. The second assumption is that this quality measure (MRS) is universal, meaning that the same MRS is applicable to any industry or individual who is willing to consume the substituting commodities. The third assumption is that quality and price are proportional in the static equilibrium. If we keep time stationary to eliminate any innovations, coexisting perfect substitutes must have price ratios equal to the MRS. Hence, Sirius must be twice as expensive as Vega if these paints were to coexist in the market. Conversely, the more cost-efficient model must dominate the market if they are not proportional.
Car paint example
Output | Inputs | |||
---|---|---|---|---|
Car | Paint | Metal | Labor | |
(a) Benchmark input-output accounts for the car industry. | ||||
Price (standard) | 1130 | 25 | 500 | 350 |
Quality (standard) | 1 | 1 | 1 | 1 |
Quantity | 1 | 15 | 0.6 | 1.3 |
Unit cost (standard) | 1130 | 375 | 300 | 455 |
Cost share | 1 | 0.332 | 0.265 | 0.403 |
Productivity | 0.419 | |||
(b) Input-output accounts for the car industry ex-post of paint innovation. | ||||
Price (sample) | 1169 | 32 | 500 | 350 |
Quality (sample) | 1.2 | 2 | 1 | 1 |
Price (standardized) | 974 | 16 | 500 | 350 |
Quantity | 1 | 12.94 | 0.517 | 1.121 |
Unit cost (standardized) | 974 | 323 | 259 | 392 |
Productivity | 0.486 (gain: 1.160) | |||
Productivity (quality-adjusted) | 0.419 (gain: 1.000) | |||
(c) Input-output accounts for the car industry ex-post of paint and car innovations. | ||||
Price (sample) | 1300 | 32 | 500 | 350 |
Quality (sample) | 1.65 | 2 | 1 | 1 |
Price (standardized) | 788 | 16 | 500 | 350 |
Quantity | 1.237 | 12.94 | 0.517 | 1.121 |
Unit Cost (standardized) | 974 | 323 | 259 | 392 |
Productivity | 0.602 (gain: 1.434) | |||
Productivity (quality-adjusted) | 0.519 (gain: 1.237) |
Suppose that, after some time, a new paint called Capella enters the market (Table 1b). Capella’s quality (MRS with respect to Vega) is the same as Sirius’s (2), and it costs 32 Kyen/liter (less than twice Vega’s). In this case, Vega and Sirius are eliminated and only Capella (and paints that are proportional to Capella) remains in the market. Note that the standardized price of paint is now 16 Kyen/liter. This means that we may now acquire 1 L of Vega equivalent paint for 16 Kyens. This standardized price change in paint affects the standardized input–output quantities for producing one standardized car, which can be calculated using the production function. We now have the new monetary inputs that combine to a unit cost of 974 Kyens for producing one Rabbit. ^{6}
We may then consider the industry’s productivity using the per-yen standardized output (measured by the number of Rabbits). Suppose that we observe a very attractive car (quality level of 1.2) called Weasel, which costs 1169 Kyens. Because the ex-post standardized price of a car is 1169/1.2=974 Kyens, Weasel is Rabbit-proportionate. The productivity gain can be calculated by the ratio of reciprocals of the standardized prices for two periods, i.e., (1/974)/(1/1130)=1.160. However, because the standardized price of a car is reduced to the standardized unit cost, we know that there have been no innovations within the car industry. A relevant measure of innovation should not change in this case. When assessing the productivity gain for only the car industry, we should eliminate any contribution from the factor inputs. In this regard, we should only consider the per-yen output change between the standardized unit cost and the standardized price, which is (1/974)/(1/974)=1.000. We distinguish the productivity gain that reflects the local innovation of an industry (in this case, the car industry) from the ordinary productivity gain estimated via the standardized price change of the output. ^{7}
Alternatively, suppose that there is some local innovation in the car industry, and a new car called Sable enters the market (Table 1c). A Sable costs 1300 Kyens, and its quality (MRS with respect to a Rabbit) is 1.65. The standardized price of a car is now 1300/1.65=788 Kyens per Rabbit, and Rabbit and Fox must both exit the market. Only the Sable-proportional cars can exist in the ex-post market. The productivity gain is then (1/788)/(1/1300)=1.434 with respect to the benchmark, but obviously this number is inclusive of the contribution of the new paint, Sirius. The relevant productivity gain is (1/788)/(1/974)=1.237, which should reflect the innovation from the Rabbit-proportional to Sable-proportional cars. We call this measure of the local innovation the quality-adjusted productivity gain, because we use standardized figures for both the input and output sides of production. Furthermore, note that this measure should reflect the outcome of the R&D in the car industry. This is how we estimate quality-adjusted productivities for different industries in this study.
2.2 Quality-adjusted productivity
where the dot operator represents element-wise multiplication. Here, z denotes the absolute productivity, which reflects the technology level of the industry in question. Any R&D investment allocated to this industry is assumed to affect z in an exclusive manner. Accordingly, we assume that the frame (i.e., the parameters) of the remaining part f(⋯) in the above formula does not change over time. ^{8}
where Δ indicates the observed differences between two periods. This measurement of local innovation can be viewed as the quality-adjusted productivity growth. We may obtain ordinary productivity growth by ignoring the differences in the observed qualities, i.e., λ=1.
2.3 Measurement of quality-adjusted productivity growth
Superscripts indicate that variables are either for the benchmark period (0) or the ex-post period (1).
where X _{ i } is the monetary input from the ith industry and μ _{ i } is the quality-adjusted deflator.
3 Propagation
3.1 Local innovation and welfare gain
The interpretation of (9) is quite simple. The marginal quality of local deflation (thus, the proportionality) is equal to the gain in productivity z defined in (1). Moreover, from a measurement perspective, this productivity gain must be equal to the quality-adjusted productivity gain that reflects the same local innovation. Considering the previous car paint example, with quality gain λ=1.61/1 and local deflation ρ=1,300/974, (9) can be used to derive the correct quality-adjusted productivity gain, i.e., z=(1.61/1)/(1,300/974)=1.27.
Hence, we may write \(\boldsymbol {\pi } \left (\mathbf {z} \right)\) in light of (11) indicating that π only depends on z.
Hence, welfare gain is the commodity-wise ratio of the benchmark and ex-post welfares. Note that p is the benchmark equilibrium price under z=1, such that \(\mathbf {p} = \mathbf {h}\left (\mathbf {p}\right)\), which is available from the benchmark state onwards.
3.2 Structural propagation
Technological structure refers to the physical input–output structure given by an n×n matrix \(\boldsymbol {\Xi } = \left (\boldsymbol {\xi }_{1}, \cdots, \boldsymbol {\xi }_{n} \right)'\), where \(\boldsymbol {\xi }_{i} = \left (x_{i1}/y_{1}, \cdots, x_{\textit {in}}/y_{n}\right)\), along with the primary input coefficients vector \(\boldsymbol {\xi }_{0} =\left (x_{01}/y_{1}, \cdots, x_{0n}/y_{n}\right)\). The technological structure, ex-post of exogenous local innovation (given as productivity gain) z is Ξ, under the equilibrium solution of (11). The ex-post π can be obtained recursively via (11) for a given z, because any unit cost function is strictly concave with respect to its argument (in this case, π). Solving for π, we normalize the standard wage (ρ _{0} π _{0}=1) according to the benchmark (p _{0}), so that ρ _{0}=π _{0}=p _{0}=1. We let π and ρ be n dimensional vectors. Let us also assume that z _{0}=1, for convenience, so that we can redefine z as an n dimensional vector.
where the angled brackets represent diagonalization. A denotes the ex-post cost share matrix. Note that the coefficients of the primary input are \(\boldsymbol {\xi }_{0} =\nabla \mathbf {h}_{0}\left (\boldsymbol {\pi }/\mathbf {z}\right) \left < \boldsymbol {\rho } \right > = \mathbf {a}_{0} \left < \boldsymbol {\pi } \cdot \boldsymbol {\rho } \right >\), because we normalize the standard wages of different periods to unity. Equation (14) shows that a bundle of local innovations z propagates through the economy recursively with respect to (11), until it finds an equilibrium technological structure.
Consider the case when social welfare is assessed by benefits and costs, where benefits constitute the benchmark final demand denoted \(\mathbf {d} = \left (d_{1}, \cdots, d_{n} \right)\), and costs are the sum of ex-post sector-wise primary inputs denoted by L=L _{1}+⋯+L _{ n }. Note that L depends upon d and z.
3.3 Propagation under different functional forms
Note that the benchmark price \(\mathbf {p}=\left (p_{1}, \cdots, p_{n} \right)\) is the equilibrium price \(\boldsymbol {\rho \pi } = \left (\rho _{1} \pi _{1}, \cdots, \rho _{n} \pi _{n} \right)\) at the benchmark state, i.e., \(\mathbf {z} = \left (1, \cdots, 1\right)\).
Primary input savings by port operation productivity doubling in different functional forms. (unit: Million JPY)
Cobb–Douglas | Leontief | CES | |
---|---|---|---|
Δ L 1 ^{′} | 927,494 | 726,101 | 875,729 |
Kurtosis | (114) | (342) | (182) |
3.4 Example: R&D investment allocation
where k denotes a parameter that can be measured by observing the actual values of r and z. Note that no local R&D investment (r=0) implies no local innovation (z=1).
where B denotes the budget constraint for R&D investments. The allocation of R&D investment is \(\mathbf {r} = \left (r_{1}, r_{2}, \dots, r_{n} \right)\).
These signs indicate that L is indeed convex with respect to r _{1} and r _{2}, which means that problem (30) has a unique solution (although the solution may be a corner).
we get the observed r as the solution of the optimization. Note that we can easily solve the nonlinear optimization problem in (29) using affordable computation equipment, if the number of sectors are limited (as in this example).
4 Conclusions
In this article, we developed a relevant link between R&D investment and its final outcome, social welfare gain. The first stage of this link corresponds to R&D and innovation that involves qualitative and quantitative improvements in the production of commodities. Because we considered industry-wise R&D investment, we focused on the industry-wise local innovation that contributes to each industry, while eliminating any foreign contribution from the input factor. We found that local innovation can be measured by the quality-adjusted productivity gain, which considers quantitative and qualitative improvements in the inputs and the outputs of production, under the efficiency unit approach. Moreover, we showed how quality-adjusted productivity gain can be measured using two cost share accounts (input–output tables) of an economy and a quality-adjusted deflator.
The second stage of the link corresponds to local innovation and its social welfare gain. In an opposite manner to the first stage, we considered the entire feedback by not avoiding the technological interdependencies among industries, initiated by the exogenous productivity gain that reflects local innovation. We call this feedback structural propagation, because the productivity gain alters the prices of the outputs of industries who can alter their technologies in response to price changes. When modeling the structural propagation, we naturally considered both qualitative and quantitative aspects of productivity. The ex-post structure allows us to assess the social welfare gain initiated by the introduction of local innovation.
Although we have demonstrated how the welfare maximizing allocation of R&D investment could be obtained under a Cobb–Douglas technology (where the structural propagation can be considered using a closed form), we used ad hoc parameters because we did not have quality-adjusted productivity measurements. Clearly, an important future task will be to measure industry-wise quality-adjusted deflators to obtain relevant productivities. We could then estimate reliable CES parameters for more factual (and less restricted) technologies, and perhaps further econometric assessments of innovation, using the structural propagation analyses outlined in this paper.
5 Appendix A: Measurement of quality-adjusted deflator
where p _{ i } is the price of the ith commodity, q _{ li } is the lth Lancasterian attribute of the ith commodity, β _{ l } is the hedonic marginal price of the lth attribute, and ϵ _{ i } is the disturbance term. Let \(\mathbf {b}^{0} = \left ({b^{0}_{1}}, {b^{0}_{2}}, \cdots \right)\) and \(\mathbf {b}^{1} = \left ({b^{1}_{1}}, {b^{1}_{2}}, \cdots \right)\) denote the estimated hedonic marginal prices for the benchmark and the ex-post, respectively. Now, for the same (standard) set of attributes \(\bar {\mathbf {q}} = (\bar {q}_{1}, \bar {q}_{2}, \cdots)'\), we can obtain the benchmark and ex-post price estimators for a standard commodity (with a set of standard attributes), i.e., \(\hat {p}^{0} = \mathbf {b}^{0} \bar {\mathbf {q}}\) and \(\hat {p}^{1} = \mathbf {b}^{1} \bar {\mathbf {q}}\). A quality-adjusted deflator can then be evaluated by \(\bar {\mu } = \hat {p}^{1}/\hat {p}^{0}\). ^{13}
where we normalize the mean utility of the outside good to zero (V _{0}=0)
We may set the marginal utility of the income of a representative consumer to unity (γ=1), as in Bresnahan (1987), or use the estimated values, as in Layard et al. (2008).
6 Appendix B: Multi-factor CES production functions
Note that the parameters δ _{ i } and σ are assumed to be constant over time, but there is only a small chance that these identities are simultaneously true.
We quantify the significance of innovation using the gain in total factor productivity (e.g., Fukao et al. (2007), Park (2012), and Sheng and Song (2013)), ignoring various associated spillover effects (e.g., Dietzenbacher (2000), Hanel (2000), and Jacobs et al. (2002)).
Triplett Triplett (2006) is an encompassing review of this subject. Jonker Jonker (2002) compares the hedonic with the discrete choice approach. Nakano and Nishimura Nakano and Nishimura (2012) provides a welfare compatible assessment of qualitative change, based on the discrete choice approach.
CPI for cameras and personal computers are quality-adjusted via the hedonic method and are used in recent official price deflators Statistics Japan (). Otherwise, deflators are not quality-adjusted.
Note that ordinary productivity gain may include innovative contributions embodied in the factor inputs, which we call foreign innovations. The basic idea is to exclude all the indirect (foreign) contributions from the gross measurement of innovation and focus on the internally established portion.
The absolute productivity, z, should satisfy 1=z(15)^{0.332}(0.6)^{0.265}(1.3)^{0.403}, because the cost shares coincide with the output elasticities of the underlying Cobb–Douglas production function.
Note that quantities are measured by the standard goods in Table 1a–c. Input quantity of paint in Table 1b, for example, is hence 0.332×974/25=12.94 L (of Vega).
The ex-post absolute productivity satisfies 1=z(12.94)^{0.332}(0.517)^{0.265}(1.21)^{0.403}, whereas the absolute quality-adjusted productivity satisfies 1=z(323/16)^{0.332}(0.517)^{0.265}(1.21)^{0.403}.
Without spillover, R&D investment can only promote local innovation, so we use z as the measure of local innovation.
Note that we only obtain ordinary productivity if we use the deflator that measures changes in the quantity-weighted nominal prices.
The underlying price π is the equilibrium price when all commodities are standardized (ρ=1) and quantified in efficiency units.
The first order condition for (19) indicates that a _{ ij } agrees with the cost share of the ith input in the jth industry and also with the monetary-based input–output coefficient. Although this coefficient remains fixed, changes to the relative factor price change the physical input–output coefficient, ξ _{ ij } (the technological structure), such that \(a_{\textit {ij}} = \left (p_{i}/ p_{j} \right) \xi _{\textit {ij}}\), where p indicates price.
We ignore (assume null) the common R&D investment for labor in industries such as education and training, because it is difficult to divide aggregated R&D investments into sector-specific and common parts.
The quality-adjusted deflator \(\bar {\mu }\) depends on the standard commodity, and we may use the average to calculate the quality-adjusted productivity gains.
Declarations
Acknowledgements
This piece is dedicated to Nori Sakurai who passed away on February 3, 2015. The authors thank the anonymous referees for their comments on the earlier version of the paper.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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