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Technological change and embodiment hypothesis
Economics Letters North-Holland
119
30 (1989) 119-123
TECHNOLOGICAL
CHANGE AND EMBODIMENT
HYPOTHESIS
Y.J. YOON Universityof Missouri-Columbia,Cohmbia, MO 652I1, USA Received Accepted
18 July 1988 7 September 1988
We introduce vintage capital into optimal growth models growth. Our model predicts a positive association between some empirical observations.
to explore the implications of technological change and economic growth rates in GNP and the savings rate, which is consistent with
1. Introduction
Studies by Abramovitz (1956) Solow (1957), and Kendrick (1976) confirm that progress in technology, not accumulation of capital, is the most important determinant of the growth in per capita output. A premise of this paper is that technological progress is made through the use of new machines (vintage capital). The vintage capital model was first formally described by Johansen (1959), and the most comprehensive work in economic growth with vintage capital is by Solow et al. (1966). Our approach is to introduce vintage capital into optimal growth models to explore the implications of technological change and economic growth. A new implication of our model is that a faster rate of technical progress induces a higher savings rate, unlike the neoclassical optimal growth model where the steady state savings rate is a constant, independent of the growth rate. This prediction is consistent with empirical findings by Wolff (1987), who reports a strong positive correlation between the rate of technical progress and the speed of investment across countries [see also Solow (1988)].
2. The economy Capital goods in this economy consist of different vintages built at different points in time. A type-8 machine costs 6 units of the current consumption good to build and, once installed, can produce output in the amount of 8 min(z, n), where z is the number of type-B machines utilized, and n is units of labor employed. The level of embodied technology B can be interpreted as the productivity of the machine in a fixed coefficient technology. In each period t, new capital goods can be built embodying the best available technology, 0,. We assume that 0, changes over time according to a deterministic rule
e,=xe,_,,
(1)
where parameter x 2 1 is the constant 016%1765/89/$3.50
innovation
0 1989, Elsevier Science Publishers
(or growth rate of technology)
B.V. (North-Holland)
in each period.
120
Y.J. Yoon / Technological change and embodiment
hypothesis
Production possibilities at period r are determined by the vector of existing vintages (k,-,, k,-,, . . . ) and the embodied technology levels (8,-,, Btp2,. . .), where k,_, is the number of machines built at period t - i with an embodied technology level 8,_, (i = 1, 2,. . .). We let K, denote the vector capital {(k,_,, O,_,), (krP2, 6,_,> ,... }. The production function is therefore given by
F(K,, n) = max x0,
min(z,,
(2)
n,),
21,n,
subject to E n, I n, 0 < z, I k, for all i, where n, is the labor input assigned to machine type 8,, z, is the number of machines of this type utilized, and n is the total labor supply. The summations are from t - 1 to - co. Investment in new capital goods is 0,k, and the production constraint is c, + 6,k, 5 F(K,, n), where c, is the period t consumption. The problem of optimal growth for the economy is to maximize
subject to c, + 0,k, I F(K,, n) and eq. (l), where the initial stock of vintages (k_,, k_,, . . .) and the embodied technology levels (8 _ 1, O_*, . .) are given. The period utility U( .) is given by the constant relative risk aversion (CRRA) function U(c, y) = (cl-‘l)/(l - y), 1 i y < cc, and p is the discount factor, 0 < p < 1. The individual is endowed with one unit of labor in perpetuity and owns all the capital goods in the economy. Since the problem described by (3) is recursive, we can formulate it as a dynamic program so that the time-invariant policy function generates optimal resource allocations. Slate variables for the economy are stocks of various vintages of capital (k,-,, klpz,. . . ), the embodied technology levels (8,-i, 0,_,, _. . ), and the current innovation in technology x. For each period t, the decision variable is the number of new machines (of type 6,) z,. The return function r( K, x, k) is defined by
r(K,, x, k,) = U(F(K,) - xklk,),
(4)
where the labor input (n = 1) is suppressed in notation for the production function motion of vector capital is K,, 1 = ((k,, 00, K,). By Bellman’s optimality condition, now expressed as a dynamic program, u(K,
x) = min{r(K, z
x, z) +Po(K’,
x’)},
F. The law of problem (3) is
(5)
subject to 0 5 8xz 5 F(K) and (1). The prime indicates the next period’s values. It can be shown [by Bertsekas (1979), Strauch (1966), or Lucas et al. (1985)J that there exists an optimal value function for (5) which is continuous and strictly concave in K, and there exists a unique stationary policy k = g(K) that maximizes the right side of (5). Optimal policy g is continuous in K.
3. Steady state growth path We let k’ denote the vintage capital built i periods ago (i = 1, 2,. . .); the embodied level of technology for k’ is 0x-i+1, and K = (k’, k2,. . . ) is the vector capital consisting of those vintages. The steady state vector capital will be of the form K” = (k’, k2,. . ., k”, 0, 0.. .) with k’ = l/m
Y.J. Yoon / Technological
change and embodiment
121
hypothesis
2 ,...> m), while the embodied technological level of each component will grow by x each period, and m is the economic lifetime of capital in the steady state growth path. We assume the existence of a steady state growth path. We denote by f, the productivity of vintage k’ normalized by /j: f, = ~XP’+‘/~ = XPI+‘.
(i = 1,
Let g( K, x) be the optimal investment Theorem. xy < 2j3. Then, (i) there exists an integer m satisfying
u( f,
-f,)
+
a2(f2
-fm)
+
. . . +“m-1(fm4
policy for (5), and let parameters
x, p and y satisfy
-fm) (6)
IX
I
a(f,
-f,+,>
+
a’(f2
-fm+,)
+
. . . -ta”(fm
-fm+l)>
where cy = /Ix’ pY, so that K” is a unique stationary point or g( K”, x) = l/m, (ii) for some critical value X, 1 < X < 2p, m decreases as x increases as long as x < X. Proof (i) We first show lim m _ ~ f, = 0, we obtain lim
5
m-m
&(f, -f,)
=
z
i-1
that
there
CY’X-‘+’
=
exists
a unique
x/3(x’-
fi)-’
>
integer
m
satisfying
x,
condition
(6). Since
(7)
I=’
where the inequality obtains because LXX-’ = /3x-’ -C 1 and p > xY - j3 by assumption. Since each term in the series (7) is positive, there exists a unique integer m satisfying the inequality (6). We now turn to show that (6) is a necessary condition for the steady state growth path. The envelope condition for the maximization problem (5) is
x>=
v,(K,
u’(f(K)
-xk)(f,
-fm)
+
Pu,+l(K’,
x2),
i=l,
2 ,...,
m,
(8)
where K=(k’, k2 ,..., k”, 0, 0, . ..) with k’ + . . . + k” = 1, and v, is the partial derivative of v with respect to the ith vintage k’. Since the period utility is u(c) = (cl-’ - l)/(l - y), we can write Pu,+,(K’, ~*)=/3x’-~v,+~(K’, x)=(~v,+~(K’, x), because u(F(K’)-x2k)=x1-Y~(F(K’)/xxk) + constant and v(K’, x2) = x’-~u(K’, x) + constant. The notation v,( K ‘, x) implies that embodied technology for K’ is normalized so that K and K’ can be compared term by term. We denote u,/ for u,( K ‘, x) and U’ for u’(F( K) - xk). The system of eq. (8) can then be written for right derivatives as 0: - lYv*‘+ = (fi -fm)u’, v: - q ‘+ =
(f2
-fm)u’,
(9) I
Vi,?I
1 -
!
a”??,‘+
=
(f,
-fm)u’,
and u,’ = 0; since the labor is supplied inelastically, more units of the least productive vintage k” would make no difference. We can obtain similar results for left derivatives with urn+, = 0. For K”, a vector for the steady state growth path, (9) yields u:(K”,
x)=(f’-fm)u’+a(f2-fnl)u’+
0,
x)=(f,-f,+,)u’+
(K”‘,
... ...
+am~‘(fm-fm+,)U’.
+a.m~2(fm_‘-fm)u’, (10)
122
Y.J. Yom
/ Technological change and embodiment
hypothesrs
1
x
1
X
Fig. 1. T, and Tn. (Note: For n > M, T,,(x)
> T,(x)
On the other hand, the first-order k = g( K”, x):
au:(K’,
a) rxu’(f(K)
-Xk)
for all x; if T,(x,,) = T,,,(x,,,) = 1, then
condition
Iau;(K’,
for the maximization
xn < X”,.)
problem
(5) is satisfied by
(11)
x).
Substituting the envelope condition (10) into the first-order condition (ll), we obtain the necessary condition (6). (ii) To examine the behavior of m (which determines the age structure of vintages), we return to (7) and consider the sequence of functions {T,, } defined by T,(x)
= l/x 5
i=l
(~‘(fi-f~)=
(a/x)(1
-xPn+‘)
+ ...
+(a/~)“-‘(1
-x-i)
Each function T, is continuous and concave with a maximum. Given x, condition (6) is equivalent to we denote this relationship by m = m(x). finding an integer m satisfying T,(x) I 1 < T,+,(x); Since { T, } is a monotone increasing sequence which converges to lim, _ ra T,,(x) = /3(x y - ,L3)~ ’ > 1, and since T,(l) = 0 and T,(2p) < 1 for each n, there exists a critical number X such that m = m(x) is decreasing in x if x < X. [7 The theorem provides a complete characterization of the steady state growth path of the economy under perfect foresight (or rational expectations). Output and consumption grow at the constant gross rate x. The economic lifetime of capital is m periods, and in each period there are m different vintages active in production. In each period, l/m units of new machines are built, while l/m units of the oldest vintage are thrown out (obsolescence). The savings rate is (investment)/(output) = x(1 + x-’ + . . . +x-~+~)~‘. An implication of the theorem is that, for an economy with a higher rate of technological growth (x), capital goods have shorter economic lives, and the investment-output ratio (the saving ratio) is higher. That is, countries with higher output growth invest in new plants and equipment at a higher rate than slow growers do. In his empirical studies on the long-run growth of seven countries (Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States), Wolff (1987) reports a strong positive correlation between the rate of technical progress and the speed of investment. Wolff interprets this as a confirmation of the embodiment hypothesis, and Solow (1988) is in favor of this interpretation.
Y.J. Yoon / Technological change and embodiment
hypothesis
123
References Abramovitz, M.. 1956, Resource and output trends in the United States since 1870, American Economic Review 46. 5-23. Bertsekas, D., 1978, Dynamic programming and stochastic control (Academic Press, New York). Johansen, L., 1959, Substitutability versus fixed production coefficients in the theory of economic growth: A synthesis, Econometrica 27, 157-176. Kendrick, J., 1976, The formation and stocks of total capital (Columbia University Press for NBER, New York). Lucas, E. Prescott and N. Stokey, 1985, Recursive methods in dynamic economics, unpublished manuscript. Solow, R., 1957, Technical change and the aggregate production function, Review of Economics and Statistics 39, 312-320. Solow. R., 1988, Growth theory and after, American Economic Review 78, 307-317. Solow, R., J. Tobin, C. van Weizsacker and M. Yaari, 1966, Neoclassical growth with fixed factor proportions, Review of Economic Studies XxX111, April, 79-115. Strauch, R., 1966, Negative dynamic programming, Annals of Mathematical Statistics 37, 871-890. Wolff, E., 1987, Capital formation and long-term productivity growth, Working paper (C.V. Starr Center for Applied Economics, New York University, New York).
119
30 (1989) 119-123
TECHNOLOGICAL
CHANGE AND EMBODIMENT
HYPOTHESIS
Y.J. YOON Universityof Missouri-Columbia,Cohmbia, MO 652I1, USA Received Accepted
18 July 1988 7 September 1988
We introduce vintage capital into optimal growth models growth. Our model predicts a positive association between some empirical observations.
to explore the implications of technological change and economic growth rates in GNP and the savings rate, which is consistent with
1. Introduction
Studies by Abramovitz (1956) Solow (1957), and Kendrick (1976) confirm that progress in technology, not accumulation of capital, is the most important determinant of the growth in per capita output. A premise of this paper is that technological progress is made through the use of new machines (vintage capital). The vintage capital model was first formally described by Johansen (1959), and the most comprehensive work in economic growth with vintage capital is by Solow et al. (1966). Our approach is to introduce vintage capital into optimal growth models to explore the implications of technological change and economic growth. A new implication of our model is that a faster rate of technical progress induces a higher savings rate, unlike the neoclassical optimal growth model where the steady state savings rate is a constant, independent of the growth rate. This prediction is consistent with empirical findings by Wolff (1987), who reports a strong positive correlation between the rate of technical progress and the speed of investment across countries [see also Solow (1988)].
2. The economy Capital goods in this economy consist of different vintages built at different points in time. A type-8 machine costs 6 units of the current consumption good to build and, once installed, can produce output in the amount of 8 min(z, n), where z is the number of type-B machines utilized, and n is units of labor employed. The level of embodied technology B can be interpreted as the productivity of the machine in a fixed coefficient technology. In each period t, new capital goods can be built embodying the best available technology, 0,. We assume that 0, changes over time according to a deterministic rule
e,=xe,_,,
(1)
where parameter x 2 1 is the constant 016%1765/89/$3.50
innovation
0 1989, Elsevier Science Publishers
(or growth rate of technology)
B.V. (North-Holland)
in each period.
120
Y.J. Yoon / Technological change and embodiment
hypothesis
Production possibilities at period r are determined by the vector of existing vintages (k,-,, k,-,, . . . ) and the embodied technology levels (8,-,, Btp2,. . .), where k,_, is the number of machines built at period t - i with an embodied technology level 8,_, (i = 1, 2,. . .). We let K, denote the vector capital {(k,_,, O,_,), (krP2, 6,_,> ,... }. The production function is therefore given by
F(K,, n) = max x0,
min(z,,
(2)
n,),
21,n,
subject to E n, I n, 0 < z, I k, for all i, where n, is the labor input assigned to machine type 8,, z, is the number of machines of this type utilized, and n is the total labor supply. The summations are from t - 1 to - co. Investment in new capital goods is 0,k, and the production constraint is c, + 6,k, 5 F(K,, n), where c, is the period t consumption. The problem of optimal growth for the economy is to maximize
subject to c, + 0,k, I F(K,, n) and eq. (l), where the initial stock of vintages (k_,, k_,, . . .) and the embodied technology levels (8 _ 1, O_*, . .) are given. The period utility U( .) is given by the constant relative risk aversion (CRRA) function U(c, y) = (cl-‘l)/(l - y), 1 i y < cc, and p is the discount factor, 0 < p < 1. The individual is endowed with one unit of labor in perpetuity and owns all the capital goods in the economy. Since the problem described by (3) is recursive, we can formulate it as a dynamic program so that the time-invariant policy function generates optimal resource allocations. Slate variables for the economy are stocks of various vintages of capital (k,-,, klpz,. . . ), the embodied technology levels (8,-i, 0,_,, _. . ), and the current innovation in technology x. For each period t, the decision variable is the number of new machines (of type 6,) z,. The return function r( K, x, k) is defined by
r(K,, x, k,) = U(F(K,) - xklk,),
(4)
where the labor input (n = 1) is suppressed in notation for the production function motion of vector capital is K,, 1 = ((k,, 00, K,). By Bellman’s optimality condition, now expressed as a dynamic program, u(K,
x) = min{r(K, z
x, z) +Po(K’,
x’)},
F. The law of problem (3) is
(5)
subject to 0 5 8xz 5 F(K) and (1). The prime indicates the next period’s values. It can be shown [by Bertsekas (1979), Strauch (1966), or Lucas et al. (1985)J that there exists an optimal value function for (5) which is continuous and strictly concave in K, and there exists a unique stationary policy k = g(K) that maximizes the right side of (5). Optimal policy g is continuous in K.
3. Steady state growth path We let k’ denote the vintage capital built i periods ago (i = 1, 2,. . .); the embodied level of technology for k’ is 0x-i+1, and K = (k’, k2,. . . ) is the vector capital consisting of those vintages. The steady state vector capital will be of the form K” = (k’, k2,. . ., k”, 0, 0.. .) with k’ = l/m
Y.J. Yoon / Technological
change and embodiment
121
hypothesis
2 ,...> m), while the embodied technological level of each component will grow by x each period, and m is the economic lifetime of capital in the steady state growth path. We assume the existence of a steady state growth path. We denote by f, the productivity of vintage k’ normalized by /j: f, = ~XP’+‘/~ = XPI+‘.
(i = 1,
Let g( K, x) be the optimal investment Theorem. xy < 2j3. Then, (i) there exists an integer m satisfying
u( f,
-f,)
+
a2(f2
-fm)
+
. . . +“m-1(fm4
policy for (5), and let parameters
x, p and y satisfy
-fm) (6)
IX
I
a(f,
-f,+,>
+
a’(f2
-fm+,)
+
. . . -ta”(fm
-fm+l)>
where cy = /Ix’ pY, so that K” is a unique stationary point or g( K”, x) = l/m, (ii) for some critical value X, 1 < X < 2p, m decreases as x increases as long as x < X. Proof (i) We first show lim m _ ~ f, = 0, we obtain lim
5
m-m
&(f, -f,)
=
z
i-1
that
there
CY’X-‘+’
=
exists
a unique
x/3(x’-
fi)-’
>
integer
m
satisfying
x,
condition
(6). Since
(7)
I=’
where the inequality obtains because LXX-’ = /3x-’ -C 1 and p > xY - j3 by assumption. Since each term in the series (7) is positive, there exists a unique integer m satisfying the inequality (6). We now turn to show that (6) is a necessary condition for the steady state growth path. The envelope condition for the maximization problem (5) is
x>=
v,(K,
u’(f(K)
-xk)(f,
-fm)
+
Pu,+l(K’,
x2),
i=l,
2 ,...,
m,
(8)
where K=(k’, k2 ,..., k”, 0, 0, . ..) with k’ + . . . + k” = 1, and v, is the partial derivative of v with respect to the ith vintage k’. Since the period utility is u(c) = (cl-’ - l)/(l - y), we can write Pu,+,(K’, ~*)=/3x’-~v,+~(K’, x)=(~v,+~(K’, x), because u(F(K’)-x2k)=x1-Y~(F(K’)/xxk) + constant and v(K’, x2) = x’-~u(K’, x) + constant. The notation v,( K ‘, x) implies that embodied technology for K’ is normalized so that K and K’ can be compared term by term. We denote u,/ for u,( K ‘, x) and U’ for u’(F( K) - xk). The system of eq. (8) can then be written for right derivatives as 0: - lYv*‘+ = (fi -fm)u’, v: - q ‘+ =
(f2
-fm)u’,
(9) I
Vi,?I
1 -
!
a”??,‘+
=
(f,
-fm)u’,
and u,’ = 0; since the labor is supplied inelastically, more units of the least productive vintage k” would make no difference. We can obtain similar results for left derivatives with urn+, = 0. For K”, a vector for the steady state growth path, (9) yields u:(K”,
x)=(f’-fm)u’+a(f2-fnl)u’+
0,
x)=(f,-f,+,)u’+
(K”‘,
... ...
+am~‘(fm-fm+,)U’.
+a.m~2(fm_‘-fm)u’, (10)
122
Y.J. Yom
/ Technological change and embodiment
hypothesrs
1
x
1
X
Fig. 1. T, and Tn. (Note: For n > M, T,,(x)
> T,(x)
On the other hand, the first-order k = g( K”, x):
au:(K’,
a) rxu’(f(K)
-Xk)
for all x; if T,(x,,) = T,,,(x,,,) = 1, then
condition
Iau;(K’,
for the maximization
xn < X”,.)
problem
(5) is satisfied by
(11)
x).
Substituting the envelope condition (10) into the first-order condition (ll), we obtain the necessary condition (6). (ii) To examine the behavior of m (which determines the age structure of vintages), we return to (7) and consider the sequence of functions {T,, } defined by T,(x)
= l/x 5
i=l
(~‘(fi-f~)=
(a/x)(1
-xPn+‘)
+ ...
+(a/~)“-‘(1
-x-i)
Each function T, is continuous and concave with a maximum. Given x, condition (6) is equivalent to we denote this relationship by m = m(x). finding an integer m satisfying T,(x) I 1 < T,+,(x); Since { T, } is a monotone increasing sequence which converges to lim, _ ra T,,(x) = /3(x y - ,L3)~ ’ > 1, and since T,(l) = 0 and T,(2p) < 1 for each n, there exists a critical number X such that m = m(x) is decreasing in x if x < X. [7 The theorem provides a complete characterization of the steady state growth path of the economy under perfect foresight (or rational expectations). Output and consumption grow at the constant gross rate x. The economic lifetime of capital is m periods, and in each period there are m different vintages active in production. In each period, l/m units of new machines are built, while l/m units of the oldest vintage are thrown out (obsolescence). The savings rate is (investment)/(output) = x(1 + x-’ + . . . +x-~+~)~‘. An implication of the theorem is that, for an economy with a higher rate of technological growth (x), capital goods have shorter economic lives, and the investment-output ratio (the saving ratio) is higher. That is, countries with higher output growth invest in new plants and equipment at a higher rate than slow growers do. In his empirical studies on the long-run growth of seven countries (Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States), Wolff (1987) reports a strong positive correlation between the rate of technical progress and the speed of investment. Wolff interprets this as a confirmation of the embodiment hypothesis, and Solow (1988) is in favor of this interpretation.
Y.J. Yoon / Technological change and embodiment
hypothesis
123
References Abramovitz, M.. 1956, Resource and output trends in the United States since 1870, American Economic Review 46. 5-23. Bertsekas, D., 1978, Dynamic programming and stochastic control (Academic Press, New York). Johansen, L., 1959, Substitutability versus fixed production coefficients in the theory of economic growth: A synthesis, Econometrica 27, 157-176. Kendrick, J., 1976, The formation and stocks of total capital (Columbia University Press for NBER, New York). Lucas, E. Prescott and N. Stokey, 1985, Recursive methods in dynamic economics, unpublished manuscript. Solow, R., 1957, Technical change and the aggregate production function, Review of Economics and Statistics 39, 312-320. Solow. R., 1988, Growth theory and after, American Economic Review 78, 307-317. Solow, R., J. Tobin, C. van Weizsacker and M. Yaari, 1966, Neoclassical growth with fixed factor proportions, Review of Economic Studies XxX111, April, 79-115. Strauch, R., 1966, Negative dynamic programming, Annals of Mathematical Statistics 37, 871-890. Wolff, E., 1987, Capital formation and long-term productivity growth, Working paper (C.V. Starr Center for Applied Economics, New York University, New York).