Estimation of R&D Depreciation Rate for
US Manufacturing and four Knowledge
Intensive Industries
ABSTRACT
Economists have long been interested in measuring the contribution of R&D to economic
growth. The increasing importance of R&D expenditures makes necessary the
re-examination of some important measurement issues related to R&D expenditures. A
recommendation has been put forth to capitalize R&D expenditures in the next version of
the SNA. In order to implement the idea of capitalizing R&D expenditure, we need to
construct an appropriate measure for the stock of R&D capital, which in turn requires a
scientific method to determine the depreciation rate of R&D capital. In this paper, we
develop a simple model based on the production function method in the framework of
monopolistic competition to estimate the annual depreciation rate of R&D capital. Instead
of treating R&D capital as an explicit input factor, we propose to treat R&D capital as a
technology shifter. Both the R&D stock and the time variable are used to capture the
technological progress in the model. In our view, modeling R&D capital in this manner
better represents the role that R&D plays in facilitating economic growth. The R&D
depreciation rates along with markup factors for the US total manufacturing sector and
four US knowledge intensive industries, namely chemical products (SIC 28),
non-electrical machinery (SIC 35), electrical products (SIC 36) and transportation
equipment (SIC 37), are estimated in the paper.
1. Introduction
Although research and development (R&D) expenditures account for only a small portion
of GDP, their importance in creating new technologies and promoting productivity
growth is widely recognized. Economists have long been interested in measuring the
contribution of R&D to economic growth. As the knowledge-based economy developed,
R&D expenditures played an increasingly important role in modern economic growth.
This increasing importance of R&D expenditures makes necessary the re-examination of
some important measurement issues related to these expenditures. Two of these issues are
whether the current measure of R&D expenditure is appropriate for addressing many
important concerns about economic growth and how R&D expenditures should be
measured. The answers to these questions have broad implications for economists
conducting analysis on the contribution of knowledge capital to endogenous growth and
to the nation’
s wealth, as well as for policy makers who may want to subsidize R&D
investments in order to maximize economic growth.
A widely used definition of R&D is given in the Frascati Manual (1993), which
originated as an OECD document. In this document, R&D was defined as “creative work
undertaken on a systematic basis in order to increase the stock of knowledge, including
knowledge of man, culture and society, and the use of this stock of knowledge to devise
new applications”. It covers three activities: basic research, applied research, and
experimental development. The important criteria that distinguish R&D from the other
related activities are “the presence in R&D of an appreciable element of novelty and the
resolution of scientific and/or technological uncertainty”.
The current System of National Accounts (SNA 1993) does not treat R&D expenditures
1
as investment. For example, in the National Income and Product Accounts (NIPA’s) of
US, R&D expenditures are treated as an intermediate input for business and current
consumption for the non-profit institutions and government. In current financial
accounting systems, R&D expenditures are recognized as current expenses. We know
that R&D uses resources to develop new knowledge that can be a new process or that can
be used to create new varieties of products or quality- improved outputs for future
consumption. Successful R&D adds to the stock of knowledge, and this stock in turn
provides a flow of services over time, rather than in one period. In this respect, R&D
resembles investment more closely than intermediate inputs or current consumption. In
general, current expenses are regarded as expenditures that generate revenues for the
current period, whereas capital expenditures are regarded as expenditures that provide
benefits for multiple periods. Since the benefits of R&D expenditures generally show up
as increased profits (or as lower costs and prices) in future periods, the current treatment
is not appropriate and will generally understate current period income and overstate
income in future periods. This existing treatment of R&D expenditures will generally
underestimate R&D’s contribution to national savings and the country’s stock of
knowledge. Due to the importance of accurate measurement of R&D to economic
analysis and the policy implications that may flow from the analysis, the possibility of
capitalizing R&D expenditures in the next revision of SNA has been raised again. It is
likely that the next international version of the SNA will contain recommendations about
the capitalization of R&D expenditures.
While, in principle, the idea of capitalizing R&D expenditures has been accepted by
many economists and interested groups, such as many national statistic agencies, it has
not been implemented due to several difficulties. The determination of depreciation rates
for R&D capital stock is one of the major hurdles that have kept economic statisticians
from capitalizing R&D expenditures. Measuring the R&D capital stock is the essential
2
part of capitalizing R&D expenditure. When macro-economists conduct analysis on
endogenous growth, they may choose to use the R&D stock to approximate the stock of
knowledge capital and they then investigate the contribution of knowledge capital or
technological progress to economic growth. When policy makers decide how much a
government should invest in industrial R&D, they need to know what the rate of return is
on these investments in order to maximize economic growth. Although we may assume
some arbitrary depreciation rate to construct the stock of R&D capital, which is what
people usually do currently, we need a scientific method for determining the depreciation
rate. If the new SNA recommends the capitalization of R&D expenditures, it should
provide some practical guidelines for the determination of the depreciation rate instead of
leaving it as a open question to statistical agencies.
Crucial to an accurate measurement of R&D stock, correctly measuring the R&D capital
depreciation rate is an important and urgent research topic in the technology-promoted
economy. However, measuring this depreciation rate poses a challenge, which arises
partly from the properties of this intangible asset, and partly from the lack of
observability of the asset depreciation. Incomplete markets for the R&D capital “asset”,
lack of knowledge about the “correct” method to write off the intangible asset and data
limitations are only a few of the obstacles facing the researcher in this area. Due to the
difficulties in measuring R&D capital depreciation rates, it is not surprising to see that
choosing an arbitrary depreciation rate turns out to be the popular strategy for
constructing R&D stocks in applied work. As Nadiri and Prucha (1996) pointed out,
“researchers doing applied work typically assume an arbitrary depreciation rate of 10 to
15 percent to construct the stock of R&D capital using the perpetual inventory method”.
There have been some scientific or evidence based methods that attempt to measure R&D
depreciation rates. Pakes and Shankerman (1978) estimated the decay rate of knowledge
capital using patent renewal fees; Nadiri and Prucha (1996) measured the depreciation
3
rate of the R&D stock using a factor requirements function and restricted cost function,
but they treated the R&D capital as a “normal” reproducible capital inputs as did
Bernstein and Mamuneas (2005), who estimated R&D depreciation rates in an
intertemporal cost minimization framework.
There are some problems with the above approaches. For example, patent information is
not representative for the entire range of R&D investments. The production or cost
function based estimates for R&D depreciation rates generally do not allow for non-R&D
effects that improve the technology and thus all of the technological improvement is
inappropriately attributed to the R&D investment, leading to exaggerated rates of return
to R&D investments1 . One deficiency associated with many R&D econometric efforts is
that the estimation is conducted within a framework of competitive pricing behaviour that
is not suitable in the R&D investment setting; i.e., private R&D investments are usually
undertaken with the goal of achieving some kind of monopolistic advantage over
competitors and thus competitive behaviour on output market is not a suitable assumption
in this context. Another weakness with most studies is that R&D investments are treated
in a similar manner as investments in ordinary physical capital, while in fact, R&D
investments are quite different in their effects as we will explain in more detail later.
Here, we list some of the R&D depreciation rates that have been obtained in the literature.
The Pakes and Shank erman (1978) knowledge decay rate was in the range of 18% to 36%;
Nadiri and Prucha (1996) obtained a 12 percent R&D depreciation rate for US
manufacturing and Bernstein and Mamuneas (2005) obtained 18 percent for chemical
products, 26 percent for non-electrical machinery, 29 percent for electrical products, and
21 percent for transportation equipment.
1
The work of Bernstein and Mamuneas (2005) is not subject to this criticism.
4
In this paper, we propose not to treat the stock of R&D capital as an explicit input factor;
instead we define the stock of R&D capital to be the technology index that indicates the
location of the economy’s production frontier. In other words, an increase in the stock of
R&D shifts the production frontier outwards. In this model, the R&D capital depreciation
rate is estimated within a monopolistic competition framework using gross R&D
investment data. R&D depreciation rates for the US total manufacturing sector and four
US knowledge intensive industries, including chemical products (SIC 28), non-electric
machinery (SIC 35), electric products (SIC 36) and transportation equipment (SIC 37),
are estimated in the paper.
The rest of the paper is organized in as follows. Section 2 explains the construction of the
R&D stock and the reasons for R&D depreciation. Section 3 develops the basic model for
estimating the rate of depreciation for R&D capital. Section 4 presents the estimating
methodology and the estimation results. We conclude the paper in the last section.
Appendix A shows how the estimating equations are derived from industry’s profit
maximization problem. Different sets of break points used in the linear spline model and
quadratic spline model are given in Appendix B.
2. Construction of the R&D Stock
If an R&D expenditure is capitalized, then it is an addition to the stock of R&D capital.
But what exactly is an R&D expenditure? According to the definition of R&D given by
the Frascati Manual (1993), the objective of conducting R&D is to increase the stock of
knowledge. Therefore, we believe that the stock of R&D capital should be regarded as a
proxy of the knowledge stock, representing society’s technological level and indicating
the position of production frontier. R&D investments essentially act as a mechanism for
shifting outward society’s production possibility frontier. This treatment is the major
5
difference of our approach from those in the current econometric literature, in which
R&D capital is treated as one explicit factor input in a manner that is similar to the
treatment of ordinary physical capital. We all know that R&D expenditures share some
common features of ordinary capital. In both cases, the flow of services provided by these
two types of capital stock for the production of other goods and services may last for
several periods. However, these two flows play different roles in the production process.
The fundamental differences between these two types of “assets”lead us to believe that
treating them in the same manner may be not appropriate and would result in unreliable
estimates of R&D deprecation rates. The following sub-section discusses the differences
between R&D capital and ordinary physical capital and explains why we treat R&D
stocks in a different manner from ordinary physical capital.
2.1 R&D Capital and Ordinary Physical Capital
Physical capital used to facilitate production, such as machinery, equipment and buildings,
gradually wears out through use and their efficiency tends to decline over time. As an
example of “intangible asset”, essentially, R&D investment can be viewed as the creation
of knowledge. Successful R&D ventures create new knowledge for the conducting firm.
This new knowledge can be either a new cost-saving process, or a new technology for an
innovative or quality- improved product. Unlike physical capital, R&D capital does not
wear out because of its utilization and its absolute efficiency level will be maintained the
same as time goes on. However, R&D capital is subject to a type of depreciation due to
the fact that the new processes or products created may become obsolete over time; i.e.,
old processes are replaced by ever more efficient newer processes and old products are
replaced by newer more efficient or desirable products as time marches on.
Although physical capital and R&D capital are both called “capital assets”, they have
6
some fundamental differences. First, reproducible physical capital, such as machines and
equipment, can be reproduced over multiple periods and then becomes completely
worthless while R&D capital does not have this feature. The reproducibility of the
physical capital and its durability makes it possible for us to observe either the rental
prices or the used asset prices of the different vintage capital assets at the same time and
this in turn allows us to estimate depreciation fairly accurately. For R&D capital, once the
new blueprint has been produced, it can be made available to many economic units
without further productive activity. Typically, we cannot collect price information for
different vintages of R&D capital at the same time 2 . Second ly, physical capital and R&D
capital support production in a different manner. As Pitzer (2004) pointed out, R&D
capital acts as if it were producing “recipes”while physical capital is regarded as one of
the “productive”inputs 3 that are “consumed”during the production process. Thirdly, the
reasons for depreciation are different for these two types of capital assets. We will discuss
this issue in more detail later on. Because of all these differences, we believe that the
treatment of R&D assets should necessarily be different in some respects from the
treatment of reproducible capital assets.
2.2 Constructing the Stock of R&D Capital
In this paper, we treat R&D capital in a new light. To better document the role of R&D
within the economy and lay the foundation for economic analysis and policy implications,
2
However, we can sometime observe the market price for the rights to some new technology. If the same technology is
again sold in a future period, then we could collect price information for different periods and infer a depreciation rate
for the R&D investment.
3
Pitzer (2004) defines a “productive input” as follows: “Fundamental to production is the notion that
inputs are proportional in some sense to outputs. Outputs are created by combining a particular collection of
inputs in a particular manner. … If more outputs are desired, then more inputs are necessary, and usually,
more of all inputs. It may not be necessary to double the inputs to double the outputs, but more inputs are
necessary to produce more outputs.”
7
the ideal way would be to measure the output of R&D— new knowledge or understanding.
However, lacking a good measure of R&D output, we turn to gross (real) R&D
expenditures as a proxy measure. Note that the stock of R&D capital, in our view,
represents the technology level of the economy, that is, the R&D stock at period t
represents the average technology level used by the economic entity, including the firm,
industry or even the whole economy. This average technology level is formed by the
most up to date new technology and the depreciated old technology. Because all the
technologies, new technologies or old technologies, are created by R&D investment in
each year, eventually, the R&D stock can be written as a function of a series of past R&D
investments. Therefore, the period t R&D stock can be defined as follows:
(1) R (t ) = θ1t I t −1 + θ 2t I t −2 + θ3t I t −3 + θ 4t I t − 4 + θ 5t I t −5 + ...
where
θ nt is the period t efficiency index representing how much the R&D investment,
which is made at n periods before the current period, contributes to the technology or
knowledge stock at period t. The R&D lags and the weights are all incorporated in these
efficiency indexes. It-n is the R&D investment made in year t- n. Generally speaking, the
further the R&D investment away from the current period, the closer it is related to the
old technology and hence it contributes relatively less to the prevailing knowledge. Thus
the following relationships should hold among all the efficiency indexes4 :
(2) θ1t ≥ θ 2t ≥ θ 3t ≥ θ 4t ≥ θ 5t ...
4
Another way to justify the inequality (2) is follows. Suppose that the industry or firm output is not homogeneous but
consists of a mix of new and old products. Over time, industry output shifts away from the older more obsolete
products and towards the newer improved products. Hence the use of the old technology that produces the older
obsolete products diminishes over time and hence the initial good effects of investments in technological improvements
made many periods ago gradually wither away. Thus we have a justification for the initial good efficiency effects of
R&D investments to diminish or depreciate over time.
8
Because these efficiency indexes may vary with time, we add the superscripts t in the
efficiency indexes. The speed of the technological upgrading is one factor which help to
determine the size of the efficiency indexes. If the new technology is created quickly, the
past investment may become irrelevant to the newly updated-knowledge at a fast speed.
For example, if there were more new knowledge created in year t compared to year t-1,
we would expect the following inequalities to hold:
θ1t > θ1t −1
(3) and
θ 2t < θ 2t −1 ;
θ3t < θ 3t −1 ;
θ 4t < θ 4t −1 ; ...
These inequalities show us that the previous investments become less important at a
faster speed as the technology improvement speeds up. In summary, the series of R&D
stocks can be written as follows:
R (t ) = θ1t I t −1 + θ 2t I t −2 + θ 3t I t −3 + ... + θ tt−1 I 1 + θ tt R (0)
(4)
R (t − 1) = θ1t −1 I t −2 + θ 2t −1 I t −3 + θ 3t −1 I t −4 + ... + θ tt−−21 I 1 + θ tt−−11 R (0)
R (t − 2) = θ1t −2 I t −3 + θ 2t −2 I t −4 + θ 3t − 2 I t −5 + ... + θ tt−−32 I 1 + θ tt−−22 R (0)
...
Where R(0) is the initial knowledge stock. To simplify our analysis, we assume that the
efficiency indexes do not depend on the current period and the following relationships
hold among the efficiency indexes:
(5) θ n = (1 − δ ) n−1
and
0 ≤ d ≤1
Based on the above simplifications, the stock of R&D capital can be constructed in the
following way:
(6) R (t ) = I t −1 + (1 − δ ) R (t − 1)
9
where δ can be regarded as the R&D depreciation rate that is assumed to be constant
over time. From equation (6), we can see that the stock of R&D capital at period t is
constructed from the previous R&D investment (It-1 ) and the depreciated R&D stock of
period t-1. This is a widely used method to construct capital stocks in the literature.
However, it can be clearly seen that we need a restrictive assumption about the efficiency
index to get this simple equation for constructing R&D capital stocks. According to this
equation, R&D capital accumulation depends on two opposite forces: the addition of the
new knowledge stock, which is created by the current period R&D investments 5 , and the
depreciation of the old knowledge stock. As we have pointed out previously, we will use
the stock of R&D capital as a technology index, indicating the position of the production
frontier. If the newly created knowledge stock is at least as large as the depreciation of the
old knowledge stock, we would not have a backward shifting of the production frontier. It
is possible to argue that the R&D depreciation rate should be zero, since it seems
plausible that old knowledge is preserved and hence the stock of “blueprints”never
decline. However the problem is that new innovations tend to make the innovations of
pervious periods obsolete and hence the older R&D investments suffer depreciation. Also,
consumer preference may shift over time, causing a drop in the demand for “old”
products, and hence leading to obsolescence of past R&D investments. It is possible that
the depreciation of old technologies outweighs the incremental effects of the new
technologies and hence results in a decrease of R&D capital stock.
The following figure shows the R&D investments in the US aggregate manufacturing
sector and four knowledge intensive U.S. industries over the period 1953-2000:
5
Equation (6) implies that one unit R&D investment can create one unit knowledge. This can be regarded as the
simplest functional form for knowledge production function.
10
Figure 1: R&D Investment (1953-2000)
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From the above figure, we can see how new technologies are created during each period .
The change pattern of R&D investments for the aggregate manufacturing sector and the
Transportation Equipment sector (SIC 37) are very similar. This may be due to the large
proportion of R&D investments in manufacturing that is accounted by the Transportation.
Among the four industries, R&D investments in the Chemical Product (SIC 28) and the
Non-electrical Machinery (SIC 35) increase smoothly in the past 40 years; while R&D
investments in Transportation Equipment (SIC 37) fluctuate the most. These different
trends in the R&D investment s lead to different frequencies of creating new technology
and result in different depreciation rates for the four industries. In the following part of
this section, we discuss the reasons for R&D capital depreciation in more detail.
11
2.3 Reasons for R&D Capital Depreciation
It is well known that the value of physical capital declines over time due to depreciation
and obsolescence. As time goes on, the machine will not work as efficiently as it used to
be. With the emergence of a new machine, the old one becomes out of date. The
depreciation of an asset can be estimated by using the information on the price of the
used asset (provided that it is not a uniquely constructed tangible asset). If we treat R&D
expenditures as investment, there exists a similar problem— the decline in utilization and
efficiency of the knowledge capital. Although there is no apparent wear or tear of the
intangible asset, we cannot imply that the knowledge capital has an infinite service life.
Two factors explain the changes in the value of R&D capital: price changes and quantity
changes. R&D capital depreciation, under our investigation, means the quantity change
due to the change in efficiency and utilization of the knowledge asset. The relative
efficiency of the knowledge capital will decline over time due to the following reasons:
•
The obsolescence of the technology. When newer technologies are created by R&D
investment, the old technology may be partly or entirely replaced by the newly
created technologies and conseque ntly, the relative efficiency and the utilization of
the old knowledge would decline.
•
The changing preferences of consumers. Consumer’
s tastes may change for different
reasons, such as the implementation of new health restrictions, the emergence of the
new products and the change of consumer’s personalities. Changing tastes may shift
away the demand of some products that heavily rely on older technologies. The
related market utilization of the older technology may contract because of this
shifting demand.
From the above discussion, we know that technological stock may become obsolete due
12
to the technological upgrading and changes in consumer preferences. However, due to the
fact that there are very few observed market prices for old technologies, we cannot
observe the depreciation on R&D capital as we can for a tangible asset, which trades on
second hand markets. In the following section, we will set up a framework for estimating
the R&D capital depreciation rate.
3. The Estimation Framework
We use basic production function methodology as our starting point. Each industry is
facing a monopolistic competition environment. In the production function, the R&D
stock is treated as a technology index indicating the position of production frontier. We
use an extension of a model due to Diewert and Lawrence (2005).
3.1 The Basic Framework
If there are increasing returns to scale in the industry, the assumption of competitive
profit maximizing behaviour is not suitable for the modelling of the industry’s behaviour,
since the competitive behaviour is not consistent with this situation. Hence, we consider
industry’s optimal behaviour under monopolistic profit maximisation setting.
In this setting, we assume that each industry has an aggregate production function f, with
the form of y t = f ( xt , Rt , t ) , so that f is a function that depends on the usual input
vector x, the R&D stock R, and the time variable t, which represents non R&D sources of
technical change for the production function. Thus both R and t shift the production
function over time. Defining the production function in this way, we can avoid the
overestimation of the effects of R&D capital on technological improvement, compared to
13
production functions that have only the R&D variable R as a shift variable. The aggregate
demand function for industry’
s output in year t is represented by an inverse demand
function: p t = P ( y t , t ) . Under this situation, each industry solves the following
monopolistic profit maximization problem at each period, by choosing inputs and the
next period’
s technological level.
∞
∑ β t { pt ( yt , t ) y t − wt ⋅ xt − Pr ,t I r ,t }
Max
x ,R
t
t =0
t +1
(8) subject to : y t = f ( xt , Rt , t )
t = 0,1,2....
Rt = I r ,t −1 + (1 − δ ) Rt −1
t = 0,1,2...
where β t is the period t discount factor, w is an input price vector, I r ,t is R&D
investment at period t and Pr ,t is the corresponding price index. In this simple model,
we assume that each industry tries to maximize the discounted future monopolistic profits
with full information about future prices. Ignoring the uncertainty of the future price is
not realistic, ho wever, it dramatically simplifies the problem. The first order necessary
conditions for solving the above maximization problem are:
(9) pt ∇ x f ( xt , Rt ,t ) + [ ∂P ( y t , t ) / ∂y ] yt ∇ x f ( xt , Rt , t ) = wt

(10) β t +1  yt +1

t = 0,1,...T

∂P (⋅) ∂ f (⋅)
∂f (⋅)
+ p t +1
+ Pr ,t +1 (1 − δ )  = β t Pr ,t
∂y ∂Rt +1
∂Rt +1

where pt is the output price and ∇ x f ( xt , Rt , t ) is the vector of first order partial
derivatives of the period t production function with respect to the components of the input
vector x. Factoring the output price pt and ∇ x f ( xt , Rt , t ) out from the left hand side of
equation (9), we have the following simplified form for equation (9):
(11) pt × (1 +
∂P ( y t , t ) / ∂yt
) × ∇ x f ( x t , Rt , t ) = wt
pt / y t
14
Applying similar algebraic rearrangements to equation (10), we have:

(12) pt +1 × 1 +

∂Pt +1 (⋅) / ∂y t +1  ∂f ( x t +1 , Rt +1,t + 1) β t
=
Pr ,t − (1 − δ ) Pr ,t +1 ×
p t +1 / yt +1 
∂ Rt +1
β t +1
where we assume that ( β t / β t +1 ) = 1 + rt where rt is the nominal interest rate prevailing
at time t.
Note that
∂P(⋅) / ∂y
is the inverse of the price elasticity of demand that reflects how
p/ y
output (demand) changes with respect to the price change. If we use ε to denote the price
elasticity, we can define the inverse as the period t nonnegative markup, denoted by mt ,
that is:
(13) m t ≡ −
∂P ( y, t ) / ∂yt
1
=− ≥0
pt / y t
ε
and the mark-up factor Mt can be defined as follows:
(14) M t = 1 − mt = 1 +
∂P ( y, t ) / ∂y t
1
=1 +
ε
pt / y t
If we assume that the markup factor is constant over time, then we can rewrite equation
(11) and (12) in the following ways:
(15) wt ,n = p t × M × [∂ f ( xt , Rt , t ) / ∂x n ];
n=1,2,…,N
(16) (1 + rt ) Pr ,t − (1 − δ ) Pr ,t +1 = pt +1 × M × [∂f ( x t +1 , Rt +1 , t + 1) / ∂Rt +1 ]
where n denotes the n-th input in the input vector x. More details about the derivation of
the above equations are given in Appendix A.
The left hand side of equation (16) actually gives us the user cost of a unit of R&D
15
investment purchased in period t6 . Equations (15) and (16) form our system of estimating
equations. Including equation (16) as an extra estimating equation is helpful for
distinguishing the trend variable R and t. However we may also introduce some problems
to the estimation by using some anticipated variables in this equation, where the
anticipations are formed in period t when we purchase Ir,t at the price Pr,t . To simplify our
analysis, we use the actual data at period t+1 to approximate the predicted variables.
Note that the left hand side user cost in (16) depends on the depreciation rate δ. Hence the
maximum likelihood strategy breaks down under these conditions; in order to use
maximum likelihood estimation which requires the left hand side variable to be constant
across models, we need to rewrite equation (16) in the following form:
(16a)
(1 + rt ) Pr ,t = (1 − δ ) Pr ,t +1 + pt +1 × M × [ ∂f ( xt +1 , Rt +1 , t + 1) / ∂Rt +1 ]
Equation (16a) is not very suitable for econometric estimation because it has a lagged
dependent variable problem. To solve this problem, we divide both sides of equation (16a)
by Pr,t which leads to the following estimating equation:
(16b) 1 + rt = (1 − δ )( Pr ,t +1 / Pr ,t ) + ( pt +1 / Pr ,t ) × M × [ ∂f ( xt +1 , Rt +1 , t + 1) / ∂Rt +1 ]
From equations (15), we can see that the role assumed for R&D capital in the production
function determines our econometric estimation system. If we treat R&D capital as an
ordinary input, we would obtain a four input, one output aggregate production function
model for each industry. The four inputs would be:
6
It may be a be surprising initially that the net effect of purchasing an R&D investment in period t can be expressed in
such a simple manner as is given in equation (16). However, under our perfect foresight assumptions, the producer
needs to purchase units of R&D in period t in order to adjust the stock of R&D to precisely the “right”level in period
t+1; the R&D stock for period t+2 can be adjusted to the “right”level by purchasing additional units of R&D in period
t+1 and so on.
16
•
labour;
•
Intermediates;
•
Non-R&D capital service inputs;
•
R&D capital service inputs
However if we believe that R&D capital acts as a technology shifter, there would only be
3 equations in (15). If we treat the stock of R&D capital as a technology shifter, then
R&D capital is treated in a similar manner as the time variable in that the R&D stock
shifts the technology just as other sources of productivity improvement (proxied by the
time variable) shift the technology. 7 . As we have pointed out previously, we would like to
treat R&D capital as a technology index that indicates the position of production frontier.
Hence, in our model, we will have labour, intermediate, and non-R&D capital service
inputs. In order to help identify some parameters in the model, we add the production
function to the estimating system. Thus our final estimating system includes the
following five equations:
(17)
wt ,1
∂f ( xt , Rt , t )
=M
pt
∂x t ,1
(18)
wt ,2
∂f ( xt , Rt , t )
=M
pt
∂xt , 2
(19)
wt ,3
∂f ( x t , Rt , t )
=M
pt
∂ xt ,3
 Pr ,t +1   pt +1 


+
 × M ×  ∂f ( x t +1 , Rt +1 , t + 1) 


 P  P 
∂Rt +1


 r ,t   r ,t 
(20) 1 + rt = (1 − δ )
(21) y = f ( x t , Rt , t ) .
7
However, the R&D variable is different from the time variable because we regard the time effect as being entirely
exogenous in our model whereas the R&D stock is endogenously determined by producers.
17
To specify the above estimating equations, we need to choose a functional form for the
production function. We discuss this issue in the following part of this section.
3.2 The Choice of the Functional Form for the Production Function
A starting point for the functional form for the production function that we use is the
following variant of a normalized quadratic functional form:
(22)
f ( x, R , t ) ≡ b + c1 x1 + c 2 x 2 + c 3 x3 + g 1 x1t + g 2 x2 t + g 3 x3 t + h1 x1 R + h2 x 2 R + h3 x3 R
+ e1t + e2 R − (1 / 2) x T Sx /(φ1 x1 + φ 2 x2 + φ3 x 3 )
where x1 is the labour input, x2 is the intermediate input, x3 is the non-R&D capital input,
and R is the stock of R&D capital. In addition, S = [ s ij ] is a 3 by 3 symmetric positive
semi-definite substitution matrix of unknown parameters and the φi are predetermined
positive parameters. In our empirical work, we calculate the sample means of the xi, call
these means xi* , and then set φi equal to xi*/( x1 *+ x2 *+ x3 *). The unknown parameters
b determines the degree of return to scale: if b = 0, we have constant returns to scale in
production; if b is less than 0, then there are increasing returns to scale; and if b is greater
than 0, there are decreasing returns to scale. The two parameters e1 and e 2 are
technical progress parameters.
In order to identify all of the parameters and to reduce multi-collinearity, it is necessary
to impose some linear restrictions on the matrix S. Our linear restrictions are as follows:
(23)
∑ j =1 s nj = 0 ;
3
n = 1, 2, 3.
The normalized quadratic production function defined by (22) and (23), with the
parameters b, e1 , and e 2 equal to 0, is flexible in the class of constant return to scale
18
production functions. The additional parameter b allows us to test the degree of returns to
scale locally. The main advantage of choosing this flexible functional form is that the
flexibility properties would not be destroyed by imposing curvature conditions; see
Diewert and Wales (1988). In our example, imposing positive semi-definiteness
conditions on the symmetric matrix S means that we can write S in term of the following
matrix product:
(24) S = UU T
where U ≡ [u ij ] is a 3 by 3 lower triangular matrix and UT is the transpose of U. The
linear restrictions (23) on S can be imposed on U too, that is:
u11 + u 21 + u 31 = 0
(25) u 22 + u 32 = 0
u 33 = 0
Investigating the above equations, we can find out that there are only 3 independent
parameters in the U matrix: u21 , u31 and u32 . The main diagonal parameters uii can be
represented in terms of the off diagonal parameters uij.
Partially differentiating the production function defined by equation (22) with respect to
inputs, x1 to x3, and to the next period’s R&D stock, Rt+1, and substituting these
derivatives into the estimating equations (17) to (20), we can rewrite the estimating
equations as follows:
3

∑ j=1 snj xt , j 1
xtT Sxt 
(26) wt ,n / pt = M c n + g n t + hn r −
+ φn × T 2 
φ T xt
2
(φ xt ) 


(27) 1 + rt =
P
p t +1
M {h1 x1 + h2 x 2 + h3 x 3 + e 2 } + (1 − δ ) r ,t +1
Pr ,t
Pr ,t
19
n = 1, 2, 3
where φ T xt ≡ ∑ j =1φ j xt , j .
3
Trending elasticities and non-smooth technological progress are two possible problems
with our present model. In subsequent sections, we will discuss how we model these
additional complexities.
3.3 Problems of Trending Elasticities
Diewert and Lawrence (2002) pointed out that the normalized quadratic functional form
has a problem: the elasticity estimates that this functional formal generates often have
strong trends when there are strong trends in prices and quantities. They also provided a
way to solve this problem. In this paper, we use their technique to deal wit h the problem
of possible trends in elasticities.
In the initial functional form, the substitution matrix S is constant over time. In order to
handle the trending elasticity problem, we allow the production function to be flexible at
two sample points, which means the matrix S is allowed to change over time. To do this,
we use the following weighted average substitution matrix instead:
(28) S = (1 − (t / T )) × A + (t / T ) × B
t = 0, 1, 2,…,T
where t is the time variable. T is the total periods in the estimation. In our sample, we
have data from 1953-2000 and thus T = 47. Using this weighted average substitution
matrix, essentially, the technological progress captured by time variable does not only
affect constant terms in the estimating system, but the substitution possibilities as well.
20
As in the basic case, we can impose the curvature conditions by setting A and B equal to
the product of UUT and VVT respectively, where U and V are lower triangular matrices;
that is, we set:
(29) A = UU T
and
B = VV T ;
Similarly, we can impose the following normalisations on these two matrices U and V:
(30) U T 13 = 0;
V T 13 = 0.
where 13 and 03 are vectors of 1’s and 0’s with 3 dimensions respectively. With these
constraints, we only add 3 additional independent parameters to the initial model; these
new parameters are v 21 , v31 , and v 32 . Making these changes, our production function
can be written as follows:
(31)
f ( xt , Rt , t ) ≡ b + c T xt + g T xt t + h T xt Rt + e1t + e2 Rt
− (1 / 2) xtT [(1 − (t / T )) × UU T + (t / T ) × VV T ] xt /(φ T xt )
Here we write the production function in matrix form in order to simplify our notation. In
equation (31), c T = (c1 , c 2 , c3 ) , g T = ( g 1 , g 2 , g 3 ) and h T = ( h1 , h2 , h3 ) . In the
estimation, if the trending elasticity problem exists, then we would see that the value of
likelihood function increases significantly when we add the parameters in the V matrix to
those in the U matrix. We do see these increases in our estimations.
Another problem related to the initial production function model is that it does not allow
the non-smooth change in technical progress. In the final part of this section, we will add
more features to the model in order to capture changes in the direction of technical
progress over time.
21
3.4 Problems due to Non-Smooth Technical Progress
It is natural to think that the technological progress does not take place in a very smooth
way. It changes very rapidly at some periods and does not change at all at other periods.
To model the complexity of technological progress change, we will add splines in the
production function. In our model, we add linear splines or quadratic splines in the time
variable to allow the different change patterns of technological progress at different
periods. For the linear spline model, the modified production function can be written as
follows:
f ( x t , Rt , t ) ≡ b + c T xt + ∑ j =1 g j (t )x j ,t + h T x t Rt + e1 (t ) + e2 Rt
3
(32)
− (1 / 2) x [(1 − (t / 47)) × UU + (t / 47) × VV ]xt /(φ xt )
T
t
T
T
T
where e1 (t ) and g j (t ) are linear spline functions of time t. The number of splines
depends on the break points chosen by investigating the plots of preliminary estimations.
The break point is a positive integer that is less than the maximum number of the time
variable. In our case, it is less than 47. We will illustrate how to define e1 (t) if we choose
three break points, 0<t1 <t2 <t3 <47:
(33)
e1 (t ) ≡ e11t
= e11t1 + e12 (t − t1 )
= e11t1 + e12 (t 2 − t1 ) + e13 (t − t 2 )
for t = 0,1,2,...t1
for t = t 1 + 1, t1 + 2,...t 2
for t = t 2 + 1, t 2 + 2,...t3
= e11t1 + e12 (t 2 − t1 ) + e13 (t 3 − t 2 ) + e14 (t − t 3 )
for t = t 3 + 1, t 3 + 2,...47
In equation (33), the e1j are unknown parameters to be estimated. From the above
example, we know that with n break points, there are n+1 parameters that need to be
22
estimated. In a similar manner, we can generate linear splines for the functions g j (t ) .
The subscript j means that we allow different splines for different inputs j.
By adding linear splines to our model, we induce, perhaps artificially, kinks in the
direction of technical change. If it is thought to be desirable to have smooth technological
change, then the linear splines can be replaced by quadratic splines. 8 . In this case, e1 (t )
and g j (t ) are quadratic spline functions of time t. If we choose three break points, the
quadratic spline functions can be defined as follows:
(34)
e1 (t ) = e1a (t ) = e11t + (1 / 2)e12 t 2
t ≤ t1
= e1b (t ) = e1a (t1 ) + (t − t1 )(∂e1 a ( t1 ) / ∂t ) + (1/ 2 )e13 (t − t1 ) 2
t1 < t ≤ t 2
= e11t1 + (1/ 2 )e12 t1 + (t − t1 )( e11 + e12 t1 ) + (1 / 2)e13 (t − t1 ) 2
2
= e1c (t ) = e1b (t 2 ) + (t − t 2 )(∂e1b (t 2 ) / ∂t ) + (1 / 2)e14 (t − t 2 ) 2
t 2 < t ≤ t3
= e11t1 + (1/ 2 )e12 t1 2 + (t 2 − t1 )(e11 + e12 t1 ) + (1 / 2) e13 (t 2 − t1 ) 2
+ (t − t 2 )(e11 + e12 t1 + e13 t 2 ) + (1/ 2 )e14 (t − t 2 ) 2
= e1d (t ) = e1c (t3 ) + (t − t 2 )(∂e1c (t 3 ) / ∂t ) + (1 / 2)e15 (t − t 3 ) 2
t 3 < t ≤ 47
= e11t1 + (1/ 2 )e12 t1 2 + (t 2 − t1 )(e11 + e12 t1 ) + (1 / 2) e13 (t 2 − t1 ) 2
+ (t 3 − t 2 )(e11 + e12 t1 + e13 t 2 ) + (1 / 2) e14 (t3 − t 2 ) 2 + (t − t 3 )(e11 + e12 t1 + e13 t 2 + e14 t 3 )
+ (1 / 2)e15 (t − t 3 ) 2
As in the linear spline case, the e1j are unknown parameters to be estimated. If we
choose n break points, then there are n+2 additional parameters that need to be estimated
for each equation.
8
To see how to set up quadratic splines in a normalized quadratic model, see Diewert and Wales (1992). However,
normalized quadratic cost functions are modeled there whereas we are modeling normalized quadratic production
functions.
23
Adding splines in our initial model increases the flexibility of the functional form but at
the cost of estimating more technical change parameters. In addition, choosing different
break points will result in different estimates.
The framework we use to estimate the depreciation rate of R&D capital is a simple
model. The production function method and its modified model have been used in
economic research for a long time. What distinguishing our model from the previous
literature are the following features:
•
Instead of treating R&D capital as one explicit factor input like ordinary
physical capital, we treat R&D capital as a technological index indicating the
position of the production frontier. We believe that treating the R&D stock in
this manner better represents the role that R&D stocks play in the production
process. R&D capital, which is the knowledge asset created by the R&D
investment, will not be “consumed”like the physical capital in the production;
it just indicates the technology level. Holding all the usual flow inputs
constant, an increase of the stock of R&D capital would shift the production
frontier outwards. Thus the R&D stock variable is treated in a manner similar
to the time variable.
•
Both the time variable and R&D capital stock variable are present in the
production function model. Of course, the problem with this type of model is
that the R&D stock frequently grows in a roughly linear fashion and so the
inclusion of the variables R and t in the regression equations can lead to a
multi-collinearity problem. However, if the time variable, t, is dropped from
the model, then the R variable becomes the only technical change shift
variable, and frequently, the resulting rate of return to R&D investments is
24
unrealistically large. Therefore, including both the time variable and the R&D
variable in the model, we will generally not attribute all of the technological
progress to R&D investments, and avoid overstatement of the effects of R&D
investments on both technological progress and on productivity growth to
some extent.
•
The model allows for the possibility of monopolistically competitive
behaviour that is consistent with increasing return to scale. 9 Thus our model
estimates the mark- up factor and the degree of returns to scale along with the
R&D depreciation rate.
Because of these different features of our model, we expect to see different results from
those in the previous literature. In the following section, we present our empirical
estimation and the main results.
4. Empirical Estimation and Results
We have discussed the estimating equations in the previous section. In this section, we
describe our data and our empirical results. In this paper, we estimate R&D depreciation
rates for US manufacturing and four US technology intensive industries, namely
chemical and applied product (SIC 28), non-electrical machinery (SIC 35), electrical
product (SIC 36) and transportation equipment (SIC 37), for the period of 1953-2000. In
1998, the R&D expenditures of these four industries accounted for 54.35% of the R&D
expenditures of all industries and 76.37% of the manufacturing R&D expenditures. The
high percentage of these four industries accounts in total R&D expenditures partly
9
If the estimated markup factor M turns out to equal 1, then we have competitive behavior.
25
explains why we are particularly interested in these four industries. We will discuss our
estimation methodology, data construction and estimation results respectively in this
section.
4.1 Estimation Methodology
The basic estimation system, with the curvature conditions (24) and the linear restrictions
(25), is defined by equations (22), (26) and (27). To specify these estimating equations,
we need to define the normalized quantity and the difference in these normalized
quantities. The nth normalized quantity, q n , can be defined as follows:
(35) q n =
xn
φT x
n = 1, 2, 3
The differences between the normalized quantities can be defined in the following way:
(36) q 21 = q 2 − q1 ;
q 31 = q 3 − q1 ;
q 32 = q 3 − q 2
Using the above definitions of normalized quantities and their differences, and
substituting restrictions (24) and (25) into equation (26), we can express the first order
necessary conditions for profit maximization with respect to the choice of the different
inputs in the following way:
c1 + g 1t + h1 R + (u 21 + u 31 )( u 21q 21 + u 31q31 ) 

2
2
 + 0.5φ1 (u 21q 21 + u 31q 31 ) + (u 32 q32 )

(37) wt ,1 / p t = M × 
[
]
26
c 2 + g 2 t + h2 R − u 21 (u 21q 21 + u 31q31 ) + (u 32 ) 2 q32 
(38) wt ,2 / p t = M × 

 + 0.5φ2 (u 21q 21 + u 31q 31 ) 2 + (u 32 q32 ) 2

[
]
c3 + g 3 t + h3 R − u 31 ( u 21q 21 + u 31q 31 ) − (u 32 ) 2 q32 
(39) wt ,3 / p t = M × 

+ 0.5φ3 ( u 21q 21 + u 31q 31 ) 2 + (u 32 q 32 ) 2

[
]
The production function is:
(40)
y = b + c1 x1 + c 2 x 2 + c3 x3 + g 1 x1t + g 2 x2 t + g 3 x3t + h1 x1 R + h2 x 2 R + h3 x3 R
[
+ e1t + e2 R − 0 .5(φ1 x1 + φ 2 x 2 + φ3 x 3 ) ( u 21q 21 + u31q 31 ) 2 + ( u32 q 32 ) 2
]
The above 4 equations plus equation (27) describe our basic estimating system. In total,
we need to estimate 16 parameters in our basic model. Due to the non- linearity of the
equations, we use non- linear option in the SHAZAM to do the estimation. This option
uses maximum likelihood estimation. To find the estimates for the R&D depreciation
rates, we construct a grid of depreciation rates: δ = 0, δ = 0.01, δ = 0.02, δ = 0.03, …and
δ = 1. Based on these depreciation rates, we build the corresponding stock of R&D
capital. The initial R&D stock is constructed by using the following formula:
(41) R(0 ) = I r ,0 /(δ + γ r )
δ = 0, 0.01, 0.02, 0.03, 0.04,…1
where Ir,0 denotes the constant R&D investment at the first period; γr denotes the
geometric growth rate of R&D investment over the sample period. It can be calculated
using the following equation:
(42) γ r = ( I r ,47 / I r ,0 ) ( 1 / 47 )
For the remaining periods, the R&D stock is calculated using equation (6) in the second
section.
All together, we have 101 sets of R&D stocks, corresponding to the 101 possible choices
for an R&D depreciation rate. Using these alternative R&D stock series, we can estimate
27
the five estimating equations. For each depreciation rate, we obtain the value of the
log- likelihood function. Comparing all these values of the log- likelihood function, we can
locate the depreciation rate corresponding to the maximum value of the likelihood
function. According to our estimating procedure, we believe that the depreciation rate that
maximizes the value of log-likelihood is the best estimator for the R&D depreciation rate.
4.2 Data Construction
To conduct the estimation, we need quantity series and price series for industrial input
and output, and price and quantity series for R&D investments. Industrial input and
output data other than R&D related data are obtained from the Multifactor Productivity
data sets provided by Bureau of Labour Statistics (BLS). R&D related data are derived
from the website of National Science Foundation (NSF).
From the BLS, we obtain value series in current dollar and price index series in 1996
constant dollar 10 . For each industry, we have data on sectoral output, labour input (L),
capital service input (K), energy input (E), non-energy materials (M) input, and
purchased business services (S) input. The BLS uses the Törnqvist index number formula
to construct the aggregate data. Labour is measured as the hours worked by all persons
engaged in a sector. Capital input is defined as the flow of services from physical assets,
which include equipment, structures, inventories and land. Service flows are assumed to
be proportional to stocks. The description of the measures and the methodology for
constructing all of these data sets are given in Chapter 10 and 11 of “BLS Hand Book of
Methods”, and Gullickson and Harper (1987). Our measure of intermediate input is a
Törnqvist aggregate of energy, material and purchased services. With value series and
price series, we can construct the implicit quantity series. Industrial input and output data
10
We thank Mike Harper for providing us manufacturing data base that is not available on BLS’
s website.
28
sets are relatively well constructed over the R&D data sets, but, we face a double
counting problem when we try to explicitly model the role of R&D capital, because R&D
expenditures have already been included in the initial (BLS) input data. Without good
information on the cost components of the R&D expenditure, the adjustment for R&D
expenditures on the initial input data may cause some measurement error.
The industrial R&D expenditure data are obtained from the website of NSF. For the years
1953-1998, the data are taken from the Industrial R&D Information System (IRIS) that
uses the Standard Industrial Classification (SIC) to classify the industry. For other years,
the data are obtained from “Research and Development in Industry”, for the year 2000
and 2001. These two publications use the North American Industrial Classification
System (NAICS). We use the 1998 data as the bridge to link the series based on the
different industrial classification and reclassify the data to conform to the SIC. If we use
R(Syear) denote the data based on SIC and R(Nyear) denote the data based on NAICS,
then the constructed data that we will use in the estimation can be obtained by using the
following formula:
(43) R ( S1999 ) = R ( N1999 ) ×
R( S1998 )
R( N1998 )
Equation (43) is a particular example for the year 1999. Similar adjustments can be done
for the years 2000 and 2001.
The data on the cost components of R&D expenditures come from various issues of
Research and Development in Industry. According to the NSF’s classification, the type of
R&D expenditure includes: wage and salaries, materials, R&D depreciation, and other
costs. There are two important problems associated with this data sets: one is that the
NSF does not use the above classification consistently over years; another problem is that
data are not available for quite a few years. To deal with these problems, we have to
29
make reasonable assumptions to get approximate data. For example, for the year 1953 to
1961, we do not have data related to the type of cost. Thus, we assume that the cost
structure for these years was the same as that in 1962. Similarly, for the period 1977 to
1997, we have data every two years. To obtain data for the missing years, we use moving
average methods to determine the missing data. Finally, we group the R&D expenditures
into three categories: Wage and Salaries (labour), Materials (intermediates) and Capital
expenditure. Unfortunately, the cost category, overhead or other costs, accounts for a big
portion of the expenditure. We allocate these expenditures to wage and salaries, materials
and capital expenditure according to the BLS cost share of each industry. From the above
description, we can see that our assumptions made to fill in gaps in the R&D data could
have a direct effect on the quality of the data sets.
After constructing R&D cost component information, we can make adjustments to the
initial BLS input data sets. That is, R&D labour cost, wage and salaries, is subtracted
from the total labour cost; the material component of R&D expenditure is subtracted from
the total intermediate input cost and the capital expenditure part of R&D is subtracted
from the total capital service cost. Dividing these adjusted value series by implicit price
index, we can have quantity series for labour input, intermediate input and capital service
input. The quantity series of R&D investment is constructed by using the Törnqvist index
formula. Price indexes for the three components of R&D expenditure are assumed to be
same as industrial input price indexes.
Finally, we need nominal interest rates for the year 1953 to 2000 to construct the discount
factors. Nominal interest rates are obtained from the on- line data of Federal Reserve
System. We choose long-term nominal interest rate: market yield on U.S. Treasury
securities at 10-year constant maturity, quoted on investment basis, to construct the series
of discount factor.
30
It is well known that data limitations are a bottleneck for conducting many research
projects. Data sets with poor quality will make the research result meaningless. Our
research work is designed to use the existing data sets related to R&D expenditure. In our
future research we would like to improve our estimates of R&D investments.
4.3 Estimation Results
In this final part of this section, we present some econometric estimates for the R&D
model described earlier and some variants of it.
The maximum likelihood estimation used by Non- linear option of SHAZAM is very
sensitive to the choice of starting point, which also affects the number of iterations. In our
estimation, we start from a very simple regression, in which M, b, e1 and e2 in equation
(27) and equation (36) to (40) and are assumed to be zero. The estimated parameters from
this regression are used as the starting point for the next regression, which adds additional
parameters. In order to make sure that our estimates are reasonable, we also check for
“big jumps”in the estimates.
In order to get sensible deprecation rates for R&D capital, we try the following four
different models:
•
Model I: This is the basic model defined by equation (27) and equations (36)
to (40). Without splines and without a weighted substitution matrix, this model
may not be able to reflect the complexity of the real world. We would expect
relatively low values of the log-likelihood for this class of models.
•
Model II: This model adds a weighted substitution matrix to equations (36) to
(40). It can deal with the possible trending elasticity problem. If our model
31
does have this trending elasticity problem, we would expect to see a big
increase in the value of the log-likelihood function.
•
Model III: This model adds linear splines in the time variable based on the last
model. Non-smooth change in technical progress can be captured by adding
these splines.
•
Model IV: This model adds quadratic splines in the time variable to Model II
(rather than linear splines as in Model III).
The following table lists the estimates of depreciation rate, the value of maximum
log- likelihood, and the value of markup factor for the above 4 models for the US
manufacturing and the four knowledge intensive industries.11
Table1: The Depreciation Rate Maximizing the Log-likelihoood Function
Ｍｏｄｅｌ　１　Ｍｏｄｅｌ　２　Ｍｏｄｅｌ　３（ａ）　11
ＳＩＣ　２８　ＳＩＣ　３５　ＳＩＣ　３６　ＳＩＣ　３７　Ｍａｎｕｆ　Ｄｅｐ　Ｒａｔｅ　０　０．０９　０　０．１５　０．５　Ｍａｒｋｕｐ　Ｆａｃｔｏｒ　０．９２５０８　０．　９２２２３　０．９６３３１　０．９７６９４　０．９５６１８　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　２６６．１１５　２０８．２７１　１１３．９０５　３０６．４６４　３７３．５６１　Ｄｅｐ　Ｒａｔｅ　０．１４　０．０６　０．１８　０．１２　Ｍａｒｋｕｐ　Ｆａｃｔｏｒ　０．９０００５　０．９０４６５　０．９２６７２　０．９８４１７　０．９８３７２　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　２７９．６５７　２２１．７００　１４８．２２９　３０８．４７３　３９４．４４１　Ｄｅｐ　Ｒａｔｅ　０．０１　０．０３＊＊　０．０７　０．０６＊＊　Ｍａｒｋ－ｕｐ　０．９７０９４　０．８４８６３　０．９６３８１　０．９０３９５　０．９８０４１　The break proints for the spline models are reported in Appendix B.
32
０．４９　０．２６　Ｍｏｄｅｌ　３（ｂ）　Ｍｏｄｅｌ　３（ｃ）　Ｍｏｄｅｌ　４（ａ）　Ｍｏｄｅｌ　４（ｂ）　Ｍｏｄｅｌ　４（ｃ）　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　４００．９３４　３４３．０５７　２８０．２３９　４１６．２０４　５３０．１６８　Ｄｅｐ　Ｒａｔｅ　０．０２　０　０．１２　０．２７　Ｍａｒｋｕｐ　Ｆａｃｔｏｒ　０．９８５　０．９４３９　０．９０８７　０．９６２５１　０．９７２５２　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　３９５．７２７　４００．３３１　２７３．３７３　４０７．４１５　５３２．８２７　Ｄｅｐ　Ｒａｔｅ　０．０１＊＊　０　０．０９　０．０１　Ｍａｒｋｕｐ　Ｆａｃｔｏｒ　０．９６８７２　０．９３６９８　０．８６４５５　０．８９８４２　０．９４７２３　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　３９５．９０７　３９５．５７８　２３６．８２８　４２０．１７１　５４９．８０６　Ｄｅｐ　Ｒａｔｅ　０　０　０．１４＊＊　０．３４　Ｍａｒｋ－ｕｐ　１．０１２４　１．１２４１　０．９１８９７　０．８８００６　０．９３３３８　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　３７８．７９５　３８６．０３４　３３４．５４７　４０６．４５１　５４１．７２３　Ｄｅｐ　Ｒａｔｅ　０　０　０．１　０．３　Ｍａｒｋ－ｕｐ　１．　０１７９　１．０９５５　０．９３４３１　０．８８４３７　０．９３５３２　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　３８０．３９７　３９５．８２２　２８５．８０８　３８１．８４２　５３９．３５６３　Ｄｅｐ　Ｒａｔｅ　０　０　０．０５　０．３３　Ｍａｒｋ－ｕｐ　１．００６　１．０８９８　０．９４７４３　０．８８７５６　０．９４８２４　Ｌｏｇ－ｌｉｋｅｌｉｈｏｏｄ　３７９．０４４　３９７．９５４　２７１．８２２　３９１．４４８　５４４．６８７　０．０８＊＊　０．２９　０．３２　０．３８　０．２８　From the above table, we can see the value of the maximum log- likelihood generally
improves a lot from Model I to Model II. This implies that we do have trending elasticity
problems. When we add linear splines or quadratic splines in the model, the value of
maximum log- likelihood function increases dramatically, which indicates that adding
33
splines in the model really captures some characteristics of technological progress.
Comparing the results obtained from the linear spline model (Model III) and the quadratic
spline model (Model IV), the value of log- likelihood function increases in Model IV for
Electric Products Industry, and decreases for Chemical Products Industry and
Transportation Equipment Industry. For Non-electric Machinery Industry and the
manufacturing sector, the changes of the log- likelihood value from Model III to Model IV
are mixed. We also try different break points for Model III and Model IV 12 . Unfortunately,
it turns out that the estimates are sensitive to the choice of the break points. Consequently,
in order to choose an appropriate value for the depreciation rate of R&D capital, we have
to use our subjective judgement. This is one drawback of our modeling strategy. In our
example, we choose the depreciation rate for R&D based on value of both the markup
factor and the log- likelihood function. According to the definition of mark-up factor
given in the second section, its reasonable value should be less than 1. Our final choice of
the depreciation rates for the manufacturing and the four selected industries is give n by
the following table:
Table 2: Depreciation Rates and Markup Factor
ＳＩＣ　２８　ＳＩＣ　３５　ＳＩＣ　３６　ＳＩＣ　３７　Ｍａｎｕｆａｃｔｕｒｉｎｇ　Ｄｅｐｒｅｃｉａｔｉｏｎ　Ｒａｔｅ　０．０１　０．０３　０．１４　０．０６　０．０８　Ｍａｒｋｕｐ　Ｆａｃｔｏｒ　０．９６８７２　０．８４８６３　０．９１８９７　０．９０３９５　０．９７２５２　From the estimated markup factors, we can derive the markup for the total manufacturing
sector and the four industries. The markup is 2.7 percent for the total manufacturing
sector, 3.1 percent for chemical products, 15.1 percent for non-electrical machinery, 8.1
12
Different break points corresponding to the estimations listed in Table 1 are given in the Appendix B.
34
percent for electrical products, and 9.6 percent for transportation equipment.
The following table compares our estimates with other estimates in the literature:
Table 3: Depreciation Rates
Pakes and Schankerman
Decay rate of knowledge: 18%-36%
SIC 28
SIC 35
SIC 36
SIC 37
Nadiri and Prucha
Bernstein
Manufacturing
12%
and
18%
26%
29%
21%
1%
3%
14%
6%
Mamuneas
Our Work
8%
In general, our estimates are relatively lower than those found in the existing literature.
The estimated depreciation rates for SIC 28 and SIC 35 are close to zero. We may
interpret this as the verification of some economists’belief that the knowledge asset
should not depreciate over time. Another possible explanation for the small depreciation
rate is that our model does not fit the data well in these industries. In order to estimate
accurately the R&D depreciation rate, we need some fluctuations in R&D investments.
As we have pointed our in Section 2, R&D investments in SIC 28 and SIC 35 increase
relatively smoothly in our sample period. The lack of fluctuations in R&D investments
may cause our model fail to find the appropriate depreciation rate. Our other estimates
fall in the range of R&D depreciation rates that are found in the literature.
Using our estimates of the depreciation rate for R&D capital, we can construct the series
of R&D capital stock for manufacturing and the four knowledge intensive industries.
Figure 2 shows how R&D stocks change over the period of 1953-2000.
35
Figure 2: R&D Stocks (1953-2000)
Ｒ＆Ｄ　Ｓｔｏｃｋｓ　（１９５３－２０００）
１８０
１６０
１４０
ＳＩＣ２８
ＳＩＣ３５
ＳＩＣ３６
ＳＩＣ３７
Ｍａｎｕｆａｃｔｕｒｉｎｇ
１２０
１００
８０
６０
４０
２０
１９
５３
１９
５６
１９
５９
１９
６２
１９
６５
１９
６８
１９
７１
１９
７４
１９
７７
１９
８０
１９
８３
１９
８６
１９
８９
１９
９２
１９
９５
１９
９８
０
The geome trical growth rate of R&D stock for manufacturing is 3.6%, and the geometric
growth rates of the R&D stocks for the four knowledge intensive industries, namely SIC
28, SIC 35, SIC 36 and SIC 37, are 4.5%, 5.2%, 3.9% and 3.6%, respectively.
5. Conclusion
In this paper, we have developed a simple model based on a production function to
estimate the depreciation rates of R&D capital for the US total manufacturing sector and
36
the four knowledge intensive industries, including chemical products (SIC 28),
non-electric machinery (SIC 35), electric products (SIC 36) and transportation equipment
(SIC 37). We treat R&D capital as a technology shifter instead of an ordinary input factor
in the model. Using both the R&D stock variable and time variable as technology shifters
can avoid the overestimation of R&D capital’s contribution to the productivity growth.
Along with the estimation of the depreciation rate, we have estimated the mark- up factor
for the US manufacturing and the four selected industries. The estimated R&D
depreciation rate is 8 percent for US total manufacturing sector, 1 percent for chemical
products, 3 percent for non-electrical machinery, 14 percent for electrical products and 6
percent for transportation equipment. The corresponding markup is 2.7 percent for the
total manufacturing sector, 3.1 percent for chemical products, 15.1 percent for
non-electrical machinery, 8.1 percent for electrical products, and 9.6 percent for
transportation equipment. Based on the estimated depreciation rate, the geometrical
growth rate of the R&D stock is 3.6 percent for the total manufacturing sector, 4.5
percent for chemical products, 5.2 percent for non-electrical machinery, 3.9 percent for
electrical products, and 3.6 percent for transportation equipment.
The result s reported here are just preliminary. We have not incorporated some important
features associated with R&D investment, such as the uncertainty of R&D investment
and the externality of the created knowledge. Also, we have imposed some restrictive
assumptions to simplify the problem, such as constant depreciation rates over years,
constant mark-up factor, and full information about the future price. In addition, the
robustness of the results should be checked against alternative functional forms for the
production function and against alternative ways of constructing the stock of R&D capital.
It would also be of interest to conduct the similar work at firm level, if we could have
access to the required data sets.
37
Appendix A:
Industry’
s Profit Maximization Problem
Each industry’
s profit maximization problem can be written as follows:
{
(1A) Maxxt ' s ,Rt +1 's ∑ t =0 β t Pt ( y t , t ) y t − wt x t − Pr ,t I t
∞
}
y t = f ( xt , Rt , t )
subject to:
Rt = I r ,t −1 + (1 − δ ) Rt −1
Substituting the constraints into the objective function, we have:
Maxx ' s, R 's β t {Pt ( f ( x t , Rt , t ), t ) f ( xt , Rt , t ) − wt x t − Pr ,t ( Rt +1 − (1 − δ ) Rt )}
t
t +1
+ β t +1{Pt +1 ( f ( xt +1 , Rt +1 , t + 1), t + 1) f ( xt +1 , Rt +1 , t + 1) − wt +1 x t +1
(2A)
− Pr ,t +1 ( Rt +2 − (1 − δ ) Rt +1 )}
+ β t +2 {Pt + 2 ( f ( xt + 2 , Rt +3 , t + 2), t + 2 ) f ( xt + 2 , Rt + 2 , t + 2) − wt + 2 xt + 2
− Pr ,t + 2 ( Rt +3 − (1 − δ ) Rt + 2 )}
+ ...
Therefore the first order necessary conditions with respect to vector xt can be written as:
(3A) pt ∇ x f ( xt , Rt ,t ) + [ ∂P ( y t , t ) / ∂y ] yt ∇ x f ( xt , Rt , t ) = wt
t = 0,1,...T
where pt is the output price and ∇ x f ( xt , Rt , t ) is the vector of first order partial
derivatives of the period t production function with respect to the components of the input
vector x. Factoring the output price pt and ∇ x f ( xt , Rt , t ) out from the left hand side of
equation (3A), we have the following simplified form for equation (3A):
(4A) pt × (1 +
∂P ( y t , t ) / ∂yt
) × ∇ x f ( x t , Rt , t ) = wt
pt / y t
The first order necessary conditions with respect to R&D stock variable Rt+1 are as
follows:
38

(5A) β t +1  yt +1


∂P (⋅) ∂ f (⋅)
∂f (⋅)
+ p t +1
+ Pr ,t +1 (1 − δ )  = β t Pr ,t
∂y ∂Rt +1
∂Rt +1

After some rearrangement, we can rewrite the above equation as follows:

(6A) pt +1 × 1 +

∂Pt +1 (⋅) / ∂y t +1  ∂f ( x t +1 , Rt +1,t + 1) β t
=
Pr ,t − (1 − δ ) Pr ,t +1
×
Pt +1 / y t +1 
∂ Rt +1
β t +1
Assuming that ( β t / β t +1 ) = 1 + rt where rt is the nominal interest rate prevailing at time
t, we can write the above equation like this:

(7A) pt +1 × 1 +

∂Pt +1 (⋅) / ∂y t +1  ∂f ( xt +1 , Rt +1,t + 1)
= (1 + rt ) Pr ,t − (1 − δ ) Pr ,t +1
×
Pt +1 / y t +1 
∂Rt +1
Define period t non-negative markup as follows:
(8A) m t ≡ −
∂P ( y, t ) / ∂yt
1
=− ≥0
pt / y t
ε
The corresponding mark-up factor Mt can be defined as:
(9A) M t = 1 − mt = 1 +
∂P ( y, t ) / ∂y t
1
=1 +
ε
pt / y t
If we assume that markup factors are constant over time, we can rewrite our system of
first order conditions in the following way:
(10A) wt = p t M∇ x f ( x t , Rt , t )
and
(11A) (1 + rt ) Pr ,t − (1 − δ ) Pr ,t +1 = pt +1 M
∂f ( xt +1 , Rt +1 , t + 1)
∂Rt +1
Moving the δ term to the right hand side of equation (11A) and dividing both sides of it
by Pr,t , we can derive one of our estimating equations:
39
(12A) 1 + rt =
P
p t +1 ∂f ( x t +1 , Rt +1 , t + 1)
M
+ (1 − δ ) r ,t +1
Pr ,t
∂Rt +1
Pr ,t
Equations (10A) and (12A) form our final system of estimating equations.
Appendix B: Break Points
Table: Break Points for Model III and Model IV
Equation1
Model
3(a) and
Model
4(a)
Equation2
Equation3
Equation4
Equation1
Model
3(b) and
Model
4(b)
Equation2
Equation3
Equation4
Equation1
Model
3(c) and
Model
4(c)
Equation2
Equation3
Equation4
Note:
SIC 28
9, 21, 29,
34, 44
15, 21, 32,
40
13, 28, 35,
41
21, 28, 35,
41
15, 21, 29,
34, 44
15, 21, 43
13, 28, 35,
41
21, 28, 35,
41
9, 21, 29,
34, 44
15, 21, 30,
39
13, 28, 35,
45
21, 28, 35,
41
SIC 35
4, 31, 39
SIC 36
20, 38, 46
29, 40
9, 19, 27,
39
12, 22, 38
8, 13, 22,
32, 42
30, 39
26, 40
31, 39
20, 32, 38
10, 19, 30,
38
8, 13, 22,
38, 42
19, 30, 39
9, 22, 36,
41
12, 22, 30,
40, 45
24, 37
31, 39
20, 38
10, 19, 30,
38
8, 14, 22,
37, 42
19, 30, 39
8, 22, 41
12, 22, 30,
45
24, 37
SIC 37
13, 27, 32,
42
12, 27, 30,
38
12, 28, 32,
38
12, 28, 35,
40
9, 21, 30,
43
6, 22,
30,39
13, 27, 32,
38
6, 12, 24,
28, 33, 40
9, 21, 30,
43
9, 22,
30,38
11, 27, 32,
38
12, 28, 35,
40
Equations 1-3 are estimating equations associated with the three inputs;
Equation 4 is the production function.
40
MANUFA
14, 21, 26,
29, 34, 38
9, 21, 23,
36, 40
13, 28, 36,
41, 47
11, 21, 29,
36, 40
14, 21, 29,
34, 38
9, 21, 23,
36, 40
13, 22, 36,
41, 47
11, 21, 29,
36, 40
15, 21, 29,
34, 39, 44
9, 21, 23,
28, 39, 46
13, 28, 36,
41, 45
9, 21, 29,
36, 40
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