ity gains are limited. The whole economy thus drifts to a lower productivity performance, unless the growth of services is offset by input savings in manufactur-
ing. Oulten 1997 shows how resource shifts to service sectors with sluggish productivity may increase aggregate productivity if it concerns intermediate busi-
ness rather than final personal services. Can the slowdown in total factor productivity TFP that we have experienced since the mid-seventies be ascribed to
the increasing importance of services, or has this drag been offset by big savings of other inputs in manufacturing? Have services suffered from the Baumol disease at
all, or do we instead observe an improvement of productivity in the services sectors by way of learning-by-doing or specialization? We feel that such questions are best
answered within a general equilibrium analysis of the whole economy, i.e. a structural view of the whole economy. Our approach does not belong to the class
of general equilibrium models, which model supply and demand functions and aim at finding prices, which sustain observed data as equilibrium outcomes. Our
position is to start from the fundamentals of the economy to establish the production frontier and its shift over time, and to compute competitive prices,
which sustain that frontier. We do not capture the variations of the economy about its frontier in this paper. The full theory of fundamentals based productivity
measurement is presented in ten Raa and Mohnen 2000. Also, in this paper we focus on the service sectors and assume that capital is sector specific and not
differentiated by type.
The fundamentals are the usual ones — endowments, technology, and prefer- ences. Endowments are represented by a labor force and stocks of capital. Technol-
ogy is given by the combined inputs and outputs of the sectors of the economy. Preferences are reflected by the pattern of domestic final demand. All the informa-
tion can be extracted from input and output tables in real terms, that is constant prices. The productivities are determined as follows. We maximize the level of
domestic consumption subject to commodity balances and endowment constraints. Now, as is known from the theory of mathematical programming, the Lagrange
multipliers associated with the endowment constraints measure the marginal pro- ductivities of labor and capital — the consumption increments per units of
additional labor or capital. In economics, these Lagrange multipliers are shadow prices that would reign under idealized conditions of perfect competition. We
declare these shadow prices to be the factor productivities.
The paper is organized as follows. Factor productivities and TFP are defined by means of a linear program in the next section. In Section 3 we present the data of
the Canadian economy from 1962 to 1991. In Section 4 we present our results. The last section concludes.
2. Productivities
We find the economy’s frontier by maximization of the level of domestic final demand, which excludes trade by definition. Exports and imports are endogenous,
controlled by the balance of payments. We make no distinction between competi-
tive and non-competitive imports the latter are indicated by zeros in the make table. Domestic final demand comprises consumption and investment. Investment
is merely a means to advance consumption, albeit in the future. We include it in the objective function to account for future consumption. In fact, Weitzman 1976
shows that for competitive economies domestic final demand measures the present discounted value of future consumption.
Productivity growth will be defined as the measure of the shift of the frontier. Suppose an economy with only two commodities. Instead of comparing observa-
tions of the economy in subsequent periods represented by the dots in Fig. 1, we will compare the projections on the respective frontiers the arrows.
The frontier of the economy in a particular year is determined by the resolution of a linear program. The primal program reads
max
s,g,g
e
T
f + w lg
V
T
− Us ] fc + Jg = F
c
j
K
j
s
j
5 K
j
Ls + lg 5 N − p
g 5 − pg
t
= D
s ] 0 where S is activity vector c of sectors; g stands for level of domestic final
demand scalar; G is the vector of net exports c of tradeable commodities; E denotes unit vector of all components one; Ý is transposition symbol; f represents
domestic final demand c of commodities; w is base year price for non-business
labor scalar; l signifies non-business labor employment scalar; V is the make table c of sectors by c of commodities U the use table c of commodities by
c of sectors; J is 0 – 1 matrix placing tradeables c of commodities by c of
tradeables; F is the final demand c of commodities; c
j
denotes capacity utilization rate of sector j scalar between 0 and 1; K
j
stands for capital stock of sector j scalar; L denotes labor employment row vector c of sectors; N is the
labor force scalar; p denotes US row price vector c of tradeables; g
t
is vector of net exports observed at time t c of tradeables; and D is observed trade deficit
scalar.
Fig. 1.
The linear program determines the activity levels of the various sectors of the economy s, the expansion of final demand y, and the net export vector g that
together maximize the expansion of final demand. Those are the endogenous variables of the primal. The expression in front of g in the objective function only
serves to normalize the commodity prices, which sustain the optimal solution essentially around unity, as we shall see when we interpret the dual program. The
production structure of the economy is given by the net output matrix V
Ý
− U,
the capital stock used in each sector c
j
K
j
, and the labor used in each business sector L
j
and in the non-business sector l. The structure of preferences is given by the domestic final demand vector f. The capital, labor and allowed trade deficit
endowments of the economy are given by the available capital stock in each sector K
j
, the total labor force N, and the observed trade deficit D. The competitive world prices are given by sector p. Labor is supposed to be mobile across sectors
but capital is sector-specific. The sum of net imports for tradeable commodities valued at world prices may not exceed the observed trade deficit.
The program thus activates to a certain extent the various sectors of the economy, thereby producing a certain amount of all commodities, which is given by
the observed sectoral production vectors blown up by the respective optimal activity levels s
j
. Likewise, the observed domestic final demand vector for all commodities is multiplied by the expansion factor of domestic final demand. If the
activity levels were unity, we would reproduce the observed resource allocation in the economy. The first constraint of the linear program forces production for each
commodity to be sufficient to accommodate domestic final demand and the optimal net exports in case of tradeable commodities. The second set of constraints simply
limits the activity levels of all sectors to be smaller than the inverse of the sector capacity utilization rates. The labor constraint stipulates that the sum of the labor
used in all sectors may not exceed the available labor force. Labor is perfectly mobile across sectors in the economy, but totally immobile across countries. The
last constraint, before the positivity constraint of sectoral activity levels, puts a limit to total net imports.
It should be noted that we do not set up the problem in terms of input – output coefficients, because of rectangular input – output tables, but in terms of given
sectoral production and total demand structures. If we had a square input – output matrix, we could write the first commodity balance constraint, the first constraint of
the primal, as [I − A]x ] fc + Jg, where A is the matrix of input – output coefficients and x is the vector of sectoral gross output. The role of x is played by s and the role
of A by the pair U, V.
Productivities are not measured using market prices, but are determined by the dual program, which, as is well-known, solves for the Lagrange multipliers of the
primal program. The latter measure the marginal products of the objective value with respect to the constraining entities, unlike observed factor rewards with all
their distortions. The dual program reads,
denoting a diagonal matrix, min
p,r,w,o ] 0
rK + wN + oD pV
Ý
− U 5 rcˆK
. +wL
pf + wl = e
Ý
f + w l
pJ = op. The endogenous variables in the dual program are shadow prices — p of
commodities, r of capital c of sectors, w of labor and o of foreign debt the exchange rate. Since the commodity constraint in the primal program has a zero
bound, p does not show up in the objective function of the dual program. The commodity prices p are normalized by the second dual constraint, essentially about
unity. They are not deflatiors that convert nominal values into real values, but a price vector that sustains the optimal allocation of resources in the linear program.
We now introduce the concept of productivity growth. Since labor productivity is the Lagrange multiplier or shadow price associated with the labor constraint, w,
labor productivity growth is the growth of w, w ; =dwdt. Similarly, r is the vector
of marginal productivities for each sectoral capital stock and o the marginal productivity of the trade deficit
1
. TFP-growth is obtained by summing all factor productivity growth figures over endowments, r
; K+w;N+o;D, and normalizing by the level of productivity, rK + wN + oD. Formally,
Definition. TFP-growth =
r ; K+w;N+o;DrK+wN+oD.
Remark. Replacement of f, l by lf, ll in the primal program with l \ 0 yields solution s, gl, g. The value of the objective function is not affected. By the main
theorem of linear programming, rK + wn + oD is not either. In fact, the productiv- ities are unaffected, as is, by extension, TFP-growth. The replacement does affect
the commodity prices, as to preserve the identity between national product and national income, which we present next.
This straightforward definition of TFP-growth is now related to the commonly used Solow residual SR. By the main theorem of linear programming, substituting
the price normalization equation, we obtain the macro-economic identity of na- tional product and income apart from the net exports on either side:
pfg + wlg = rK + wN + oD. By total differentiation:
TFP-growth = [pfg + wlg
: −rK:−wN:−oD:] pfg + wlg
. To establish the link with the Solow residual SR, focus on the numerator,
substituting the assumed binding labor and balance of payment constraints:
1
In fact, there is also a non-business capital stock. Its value enters the objective function. In principle, its level constrains the expansion of domestic final demand. In practice, the capital constraint in the
non-business sector is never binding at reasonable rates of capacity utilization, and hence its shadow price is zero. For notational simplicity, we have not included the non-business capital stock in the
formulation of the program.
pF − pJg + wlg : −rK:−wLs+lg:+opg:.
Differentiating products, rearranging terms, and using the second dual constraint and the definition of F presented in the primal program
pF : −rK:−wLs:−pJg;+opg:+p;F−Jg+wlg:−wlg:
= pF
: −rK:−wLs:+op;g+p;fg+w;lg. Now normalize again, dividing by rK + wN + oD. Then the first term yields the
technical change effect or Solow residual. It corresponds to the traditional Solow residual, except that here it is evaluated at shadow commodity prices and optimal
sectoral activity levels. The second term, with numerator op ; g, is the terms-of-trade
effect. Proportional changes in p are offset by a change in o. Only relative international price changes matter. The last two terms are the preference-shift
effect. By the remark, pf + wl may be held constant, so that the preference-shift effect reads − pf
: +wl:g. If demand f, l shifts to commodities with low opportu- nity costs, it is relatively easy to satisfy domestic final demand and TFP gets a
boost. The terms-of-trade and preference-shift effects disappear when there is only one commodity and no non-business labor. Under these circumstances, p is unity
and p also by the second dual constraint, hence their derivatives vanish. In other words, in a macro-economic setting, TFP-growth reduces to the Solow residual. It
should be mentioned, however, that a tiny difference remains in the denominators. We divide by pfg + wlg = pF − pJg + wlg = pF − opg + wlg = pF + oD + wlg. In
other words, we account for the deficit and non-business labor income.
2
.
1
. Remarks 1. The TFP measure used in Mohnen et al. 1997 is confined to the Solow residual
without the terms-of-trade and preference-shift effects. There is also a slight normalization difference. In this paper, we normalize with respect to rK + wN +
o D = pfg + wlg, whereas Mohnen et al. 1997 normalize with respect to pF =
pfg + pJg. 2. In discrete time, the differentials are approximated using the identity x
t
y
t
− x
t – 1
y
t – 1
= x˜
t
x
t
y
t
+ y˜
t
x
t
y
t
, where x˜
t
= x
t
− x
t – 1
x
t
and x
t
= x
t
+ x
t – 1
2, and simi- larly for y˜
t
and y
t
. 3. By Domar’s aggregation we can decompose the aggregate Solow residual into
sectoral and group-sectoral Solow residuals. Let j index the sectors, i the commodities, and k the sector groups. Define the Solow residual of group-sector
k as.
SR
k
=
j k i
p
i
s
j
6
ji
:
j k i
p
i
6
ji
s
j
−
j k i
p
i
s
j
u
ij
:
j k i
p
i
6
ji
s
j
−
j k
ws
j
L
j
:
j k i
p
i
6
ji
s
j
−
j k
r
j
K :
j j k
i
p
i
6
ji
s
j
.
Notice that if k = j, we get the Solow residual for sector j. Also notice that it is defined with respect to the available factor inputs rather than the utilized factor
inputs. It can be shown that our aggregate Solow residual expression can be written as a Domar weighted average of sectoral Solow residuals with weights adding to
more than unity, see ten Raa, 1995:
SR =
k j k
i
p
i
6
ji
s
j i
p
i
F
i
SR
k
.
In continuous time this is exact. In discrete time one must involve full employ- ment of labor at the frontier point. Otherwise a slack change term emerges that
cannot be allocated to the sectors. With capital there is no such complication, since we assume it is sector specific.
3. Data
