- Email: [email protected]
On endogenous growth with productivity shocks
Journal
of Monetary
Economics
On endogenous shocks
Received
August
30 11992) 17-56.
growth
1991. final version
received
North-Holland
with productivity
June
1992
Recent evidence on output convergence across economies has been widely interpreted as falsifying the predictions of endogenous growth theory. This paper shows. however. that nonconvergence is an artifact of the deterministic structure of endogenous growth models: when stochastic factor productivity is Introduced. convergence tends to occur despite nondiminishing returns to capital and persistent growth. For the case of constant returns to capital, it is shown that output paths of different economies converge with probability one in single-sector economies. For multi-sectoral economies convergence occurs for most plausible specifications of the stochastic process driving productivity. Increasing returns reduce the probability of convergence.
1. Introduction
Contrary to the predictions of deterministic endogenous growth models such as Romer (1986) Lucas (1988), or Rebel0 (1991), there is substantial empirical evidence of income convergence across economies. Notable recent studies include Barro (1991) and Mankiw et al. (1992), who use international data, and Barro and Sala-i-Martin (1992), who study convergence within the United States. More recently Durlauf and Johnson (1992) have used a regression tree approach to argue for local convergence across nations: Burkina Faso and Rwanda seem to be on a similar growth trajectory, but not on the same one as the United States and Japan. A common reaction to these findings is to reject endogenous growth in favor of the older neoclassical model with diminishing returns to capital and exogenous technological change.
Correspondencr IO: Morgan Ithaca, NY 14853. USA.
Kelly,
‘I would like to thank William discussions. All errors are mine.
0304-3392,‘92,‘505.00
G
1992-Elsevier
Department Brainard,
of Economics, Steven
Science Publishers
Durlauf,
Uris
Hall.
and
Bruce
Cornell Smith
B.V. All rights reserved
University, for helpful
48
M. Kelly. Endogrnous
growrh
with produclitiry
shocks
This reaction seems excessive. In this paper we add a slight wrinkle to a textbook endogenous growth model: in the spirit of the real business cycle literature we allow for stochastic factor productivity. It will be shown that this tends to result in output convergence, while still allowing continuous output growth without ad hoc technological change.’ For the case of an economy with constant returns to capital, we find that output paths of economies starting at different output levels converge with probability one if the economy has a single sector. For multi-sectoral economies, convergence occurs for most plausible specifications of the process for productivity shocks. Increasin g returns to capital lower the probability of convergence. Under certain circumstances outlined below, this convergence of output paths is associated with convergence to zero growth, but we emphasize that this is a special case and does not hold in general. It is easy to specify a model with convergent output paths but persistent growth. The model we consider in this paper is a variant of the log-utility Cobb-Douglas technology multi-sectoral model used by Long and Plosser (1983). This is the standard workhorse in this area and it allows us to use very straightforward mathematical results as a consequence. In case it might be objected that this is an example and, as such, proves nothing about endogenous growth models in general, we refer the interested reader to Dobrushin (1968). In that paper it is shown in general that systems with a single element (good in our case) and positive transition probabilities converge to the same distribution regardless of their starting points. Systems with many elements converge if there is sufficient noise so that each element exerts a sufficiently weak influence on the evolution of the other elements. On an empirical level, this approach may lead to a test for output convergence by seeing if the estimated variance of productivity shocks satisfies the bounds derived below. The rest of the paper is as follows. Section 2 outlines the endogenous growth/real business cycle we will examine. Sections 3 and 4 examine an economy with constant returns to capital for the case of one and many sectors, and section 5 gives examples of types of productivity shocks which do and do not lead to convergence of output paths. Section 6 shows how these results are modified in the presence of increasing returns to capital, and section 7 concludes.
2. Outline
of the model
In this section we set up and solve a simple discrete time stochastic optimal growth or real business cycle model with constant or increasing returns to
‘Convergence on very special
in a deterministic endogenous assumptions about the return
growth model occurs in Tamura to human capital investment.
(1991) but relies
.Lf. Kelly.
Endogenorts
grobr rh u.ith producricir?
shocks
49
capital. We will study its ergodic behavior in later sections. The example here takes the familiar form of logarithmic preferences and Cobb-Douglas technology and differs only slightly from the model of Long and Plosser (1983). Labor has no role in our analysis and is omitted to keep things as simple as possible. The social planner is assumed to maximize the expected utility of an infinitely-lived representative person:
. . + d, log C,,),
E,, t fl’(d, log C,, + & log C?, + I=0 where /3 is the (constant) discount modity i in period t. The production Yi, = l&K;;:;
Kf::;
rate and function
. K;;y,
(1)
C, is the consumption of comfor each commodity is given by
i=
I,2 )...(
n,
(2)
where Yi, is the output of commodity i in period t and Kji., is the quantity of capital good j allocated to activity i in period t. Bi, is a Markovian productivity shock independent of the other components of the production function. The matrix of Cobb-Douglas coefficients [rji.,] is independent and identically distributed and will be denoted by A,. None of the elements of this matrix can be negative. We will denote its expected value by E A,. Let p(EA,) denote the value of the largest eigenvalue of Eil,. It is assumed that the economy exhibits stochastic increasing returns to capital: p(EA,) 2 1. A sufficient condition for this to occur is if each column of A, has sum greater than or equal to one, so each sector exhibits increasing returns to capital. The economy must satisfy the resource constraint
Yit
ct,+ 1
2
Kij.r
+ 1
9
i = 1,2, . . . , n.
The output of each commodity must at all times be sufficient to meet final consumption and also inputs for next period’s production. Each capital good Kij lasts for a single period and its quantity must be chosen in the period before it will be used. The maximization problem will be well defined if /I x.y=, iii < 1 where Xij = E(rij.,). A straightforward application of Bellman’s prmciple shows that
Kij.
I f
I
=
rij
P Yir
I
i,j = 1,2,.
. . ,n.
(4)
From
eqs. (2) and (4). the evolution
of gross output
=(l”gei[
+ %li.rlOg~S,i
+
yli.l-
r*i.riog
1
Denoting the vector with components matrix form as
+
+ ”
‘..
’ +
+
IXni.r
is given
by
r”i,,lOg13r,i)
log Yni.f-
1.
(log Y,,) by log Y,, (5) may be written
log Y, = A, log Y,_ 1 + rr,
(5) in
(6)
where fI is a vector with components given by the bracketed term in (5). If each firm exhibits constant returns to scale with increasing returns due to Marshallian externalities depending on the aggregate capital stock of its sector, then the economy can support a competitive equilibrium, albeit with a lower rate of investment, in the manner of Romer (1986). The competitive economy will have the same A, matrix but a lower value of TI. The question we address is whether such an economy can have growth paths which do not converge through time. We can think of this problem as follows: given two different starting levels of output Y’ and Y’, will the trajectories followed eventually converge to the same probability distribution, or will economies which start at different levels of output tend to remain permanently on different paths’? From (6), the stochastic difference equation log Y: - log Yj = A, (log r:_ 1 - log Yj- 1) describes the dynamic behavior of the difference of the logs of output for two economies starting at different levels of production. To simplify notation define X, = (log Y: - log Y,‘). This difference equation may then be rewritten as
x, =
A,X,_,.
(7)
It is evident that the behavior of this equation evidently depends on the convergence in distribution of the product of the random matrices of Cobb-Douglas coefficients A,. If n,Ar tends to the zero matrix, then -Y, tends to zero. Output paths of economies with differing initial stocks of capital will converge. On the other hand, if n,AI converges to a nonnegative matrix, then different output paths will not converge and the economy will behave like a deterministic endogenous growth model. We first analyze the case of
constant social returns to capital, and then look at the case of increasing returns. The growth of an individual economy through time depends on the term r in (6) as well as on the products of the At terms. If this term is sufficiently small relative to YI. growth paths will converge to zero if output paths converge. However. if the Markov shock terms 0il are sufficiently large, economies can grow through time although output paths converge.
3. Convergence of output paths: The single-sector
case
Path dependence of output in a single-sector rather than the rule as the following proposition
economy is the exception demonstrates.
Proposition I. capital, output Proof:
Taking log
It1 a single-sector economy bvith stochastic paths concerge with probability one.
constant
returns
to
logs of eq. (7) gives us
x, =
log
x-
1
+
log .4,.
(8)
In this case A, is a random scalar, such that log EA, = 0 because of stochastic constant returns to scale. It follows that the evolution of log X, in eq. (8) is a random walk with a downkvard drift. This is because Jensen’s inequality implies that E(log A,) < log E A, = 0. The result of this is that log X, tends to minus infinity with probability one. In other words, X, tends to zero. Output paths converge with probability one. Alternatively, we may show that the product of nonnegative random scalars fl,A! does not converge to a distribution concentrated on positive scalars unless A, = 1 with probability one. Taking logs of n,A,, we see that the process converges if and only if 1, log rlr converges. The latter term converges if and only if x,E (log A,)’ converges: see, for example, Ash (1972, theorem 7.8.7). W The only exception to this is the unit root case where A, = 1 with probability one, but this possibility has measure zero and so is of little interest.
4. Convergence of output paths: The multi-sectoral
case
The previous section established that path dependence is an exceptional and fragile result in single-sector models. Once any stochastic variability is
allowed in the productivity of capital, output paths converge with probability one. The purpose of our analysis is to show that this result need not carry over to the multi-sectoral case where the output of sectors is interdependent. We denote the product AI, = Al AI . . . A,. The matrices A, are nonnegative i.i.d. n x n matrices where n > 1. Let A,(i, j) denote the i, jth element of A,. We make the following assumptions: Assumption I (irreducibility). A, is independently identically distributed (i.i.d.) through time. A,(i, j) 2 0 for all i and j with probability one. There exists some t such that P [M,(i, j) > 0, Vi, ji. > 0. P {A, has a zero row or column] = 0. This irreducibility assumption holds if and only if every commodity is used, either directly or indirectly, in the production of all other goods. It ensures the existence of a unique largest eigenvalue which is real. Assumption 2 (stochastic constant returns to scale). the largest eigenvalue of E A, and E(A,(i. j)) < x .
p(E A,) = 1 where
p is
We consider increasing returns to capital in the last section. The following proposition is the main one of the paper and says that output paths will not converge if and only if the productivity matrix A, is not too variable. Proposition 2. Under Assumptions I and 2, if economies do not decay into autarky for any initial level of output, output paths will not converge if and only if M, is bounded.
(9)
Proof. There are two cases to consider. In the first, the vector T, in (6) is sufficiently negative so that output tends to zero regardless of its initial level. In this case, output paths converge trivially. In the second, the economy survives. Under Assumptions 1 and 2, Kesten and Spitzer (1984) show that the product of the matrices A, will tend in distribution to a strictly positive matrix if and only if M, holds. There will then be a continuum of possible output paths, each depending on the starting point of the economy. If (9) does not hold, output paths converge with probability one. n
.\I. Kelly.
Endogenous
growth
wilh producririrx
shocks
The relation between high variance and convergence of output given by the next result, again due to Kesten and Spitzer (1984). Corollary.
M, is bounded
if and only
(E(A, @ A ))’ is finite I
53
paths
is
if
and bounded
in t.
(10) pEt.4, @A,) = 1, where
@
denotes
a Kronecker
product.
Hopenhayn and Prescott (1986) and Bhattacharya and Lee (1988) have shown that convergence occurs if at some time t there is a value X* such that for any starting point there is a positive probability that the process is above X* and a positive probability that it is below X*. In our context, this is of course equivalent to saying that M, is unbounded.
5. Examples
of random matrices
In this section we consider the convergence properties of some plausible types of coefficient matrix and see that the conditions (9) for nonconvergence of output paths can be quite strict. Example
I [Kesten and Spitzer (1984)-J. Let Y be a family of positive row and 2 a family of positive column vectors, such that lr = 1 for all 1 E 9’ and r E .&‘. Let ,U be any probability measure on Y x W and (lit ri) an i.i.d. sequence with each element having distribution ,u. Suppose that Ai = rili. Then
vectors
M, = r,l,r212.
. . rllr = r,(l,r,)12.
..(lr_lrl)lr
= rllt.
So, for all t 2 1, the distribution
of M, equals the distribution of the product with distributions given by the first and of p, respectively.
rl I,, where rl and I, are independent
second
marginals
Example
2. Let Ai = Bi + Ci, where Bi is a matrix whose rows sum to unity and Ci is a matrix whose rows sum to zero. Letting Bi = rbi, where r is a unit vector and bi is i.i.d. with elements which sum to unity, and Ci = rci, where ci is i.i.d. with elements which sum to zero, convergence follows as in the previous example. Example 3. Let B be a constant matrix whose rows sum to unity. Suppose that o‘t is a random scalar whose expectation is unity. Define A, = a,& Then
5-I
.W. Kelly.
by Proposition probability one.
1, output
Endogenous
paths
growth
wilh produclirity
will converge
unless
shocks
cI is constant
with
6. Convergence of output paths: Increasing returns to scale 6.1.
The single-sector
cue
To analyze the case of increasing returns only two values. More general cases may
A, =
l/o
with probability
cl,
i C!J
with probability
p,
it will be assumed that A, assumes be handled analogously.
where p > (I and o > 1. Log X, in (8) is a random walk with upward drift for p sufficiently large. We say that convergence has occurred if X, attains some arbitrarily small, positive value E. If the absolute minimum number of steps in which random walk can attain the value of log E is i, then the probability that the two output paths will ever converge equals (q/p);. [See Shiryayev (1984, p. 547) for a proof of this result.] So, in the presence of increasing returns to capital, there is a positive probability that output paths do not converge in distribution. The weaker the increasing returns (i.e., the closer p and 4 are to equality) and the closer the starting levels of output (i.e., the smaller i is), the less this probability of nonconvergence will be.
6.2. The multisectoral
case
In this case, pE(A,) > 1, so we normalize by multiplying A, by some constant q c 1 so that the resulting matrix B, = VA, has pE(B,) = 1. The difference between output paths now evolves as
Again, output paths converge with probability zero if B, satisfies the conditions of Proposition 2. Otherwise, convergence occurs with positive probability which increases with the variability of B and falls with the magnitude of the increasing returns parameter ‘1-i.
7. Conclusions
The purpose of this paper was to analyze the dynamic behavior of a simple endogenous growth model when variable factor productivity was considered. It was shown that the nonconvergence of output paths, which is frequently cited as evidence against endogenous growth models, no longer holds in general. Although the concern of this paper is with analyzing the effect of productivity shocks on the behavior of endogenous growth models. it may be worthwhile to relate our results to the older stochastic turnpike literature. Convergence to a unique steady state in single-sector models is a well-known result dating back to Brock and Mirman (1972) whose proof relies on the ergodic theorem of Markov chains. Multi-sectoral turnpike theory is a less welldeveloped area. The discrete time results of Majumdar and Zilcha (1987) are not applicable here since our model does not satisfy their Assumption 3. The Brock and Magi11 (1979) modification of the Cass-Shell (1976) Hamiltonian curvature conditions for stochastic, continuous time models produces results analogous to ours. They show convergence to a unique steady state if the economy decays deterministically to a zero capital stock or if the noise term is sufficiently variable.
References Ash. Robert. 1972, Real analysis and probability (Academic Press, New York, SYI. Barre, Robert, 1991. Economic growth in a cross section of countries. QuarterI> Journal of Economics 106. 407113. Barro, Robert and Xavier Sala-i-Martin. 1992. Convergence, Journal of Political Economy 100, 223-251. Bhattacharya. Rabi and Oesook Lee, 1988. Asymptotics of a class of Markov processes which are not in general irreducible, Annals of Probability 16, 1333-1347. Brock, William and Michael Magill. 1979, Dynamics under uncertainty, Econometrica 47, 845-868. Brock, William and Leonard Mirman, 1972, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory 4, 479-513. Cass. David and Karl Shell, 1976. The structure and stability of competitive dynamic systems, Journal of Economic Theory 12, 31-70. Dobrushin, R.L.. 1968, The description of a random field by means of conditional probabilities and conditions of its regularity, Theory of Probability and its Applications 13. 197-224. Durlauf, Steven and Paul Johnson, 1992. Local versus global convergence across national economies, NBER working paper no. 3996. Hopenhayn, Hugo and Edward Prescott, 1986, Invariant distributions of monotone Markov processes. Working paper (University of Minnesota. Minneapolis, MN). Kesten, Harry and Frank Spitzer, 1984. Convergence in distribution of products of random matrices, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 67. 363-386. Long, John and Charles Plosser, 1983, Real business cycles, Journal of Political Economy 91, 39-69. Lucas, Robert, 1988, On the mechanics and economic development, Journal of Monetary Economics 22, 312.
56
M. Kelly. Endogenous growth with producriciry shocks
Majumdar, Mukul and Itzhak Zilcha, 1987. Optimal growth in a stochastic environment: Some sensitivity and turnpike results, Journal of Economic Theory 43, 116-133. Mankiw, Gregory. David Romer. and David Weil, 1992. A contribution to the empirics of economic growth, Quarterly Journal of Economics 107. 407-437. Rebelo, Sergio, 1991, Long run policy analysis and long run growth, Journal of Political Economy 99. 500-521. Romer, Paul, 1986, increasing returns and long run growth, Journal of Political Economy 94, 1002-1037. Shiryayev. A.N., 1984, Probability (Springer-Verlag, New York, NY). Tamura. Robert, 1991, Income convergence in an endogenous growth model. Journal of Political Economy 99. 522-540.
of Monetary
Economics
On endogenous shocks
Received
August
30 11992) 17-56.
growth
1991. final version
received
North-Holland
with productivity
June
1992
Recent evidence on output convergence across economies has been widely interpreted as falsifying the predictions of endogenous growth theory. This paper shows. however. that nonconvergence is an artifact of the deterministic structure of endogenous growth models: when stochastic factor productivity is Introduced. convergence tends to occur despite nondiminishing returns to capital and persistent growth. For the case of constant returns to capital, it is shown that output paths of different economies converge with probability one in single-sector economies. For multi-sectoral economies convergence occurs for most plausible specifications of the stochastic process driving productivity. Increasing returns reduce the probability of convergence.
1. Introduction
Contrary to the predictions of deterministic endogenous growth models such as Romer (1986) Lucas (1988), or Rebel0 (1991), there is substantial empirical evidence of income convergence across economies. Notable recent studies include Barro (1991) and Mankiw et al. (1992), who use international data, and Barro and Sala-i-Martin (1992), who study convergence within the United States. More recently Durlauf and Johnson (1992) have used a regression tree approach to argue for local convergence across nations: Burkina Faso and Rwanda seem to be on a similar growth trajectory, but not on the same one as the United States and Japan. A common reaction to these findings is to reject endogenous growth in favor of the older neoclassical model with diminishing returns to capital and exogenous technological change.
Correspondencr IO: Morgan Ithaca, NY 14853. USA.
Kelly,
‘I would like to thank William discussions. All errors are mine.
0304-3392,‘92,‘505.00
G
1992-Elsevier
Department Brainard,
of Economics, Steven
Science Publishers
Durlauf,
Uris
Hall.
and
Bruce
Cornell Smith
B.V. All rights reserved
University, for helpful
48
M. Kelly. Endogrnous
growrh
with produclitiry
shocks
This reaction seems excessive. In this paper we add a slight wrinkle to a textbook endogenous growth model: in the spirit of the real business cycle literature we allow for stochastic factor productivity. It will be shown that this tends to result in output convergence, while still allowing continuous output growth without ad hoc technological change.’ For the case of an economy with constant returns to capital, we find that output paths of economies starting at different output levels converge with probability one if the economy has a single sector. For multi-sectoral economies, convergence occurs for most plausible specifications of the process for productivity shocks. Increasin g returns to capital lower the probability of convergence. Under certain circumstances outlined below, this convergence of output paths is associated with convergence to zero growth, but we emphasize that this is a special case and does not hold in general. It is easy to specify a model with convergent output paths but persistent growth. The model we consider in this paper is a variant of the log-utility Cobb-Douglas technology multi-sectoral model used by Long and Plosser (1983). This is the standard workhorse in this area and it allows us to use very straightforward mathematical results as a consequence. In case it might be objected that this is an example and, as such, proves nothing about endogenous growth models in general, we refer the interested reader to Dobrushin (1968). In that paper it is shown in general that systems with a single element (good in our case) and positive transition probabilities converge to the same distribution regardless of their starting points. Systems with many elements converge if there is sufficient noise so that each element exerts a sufficiently weak influence on the evolution of the other elements. On an empirical level, this approach may lead to a test for output convergence by seeing if the estimated variance of productivity shocks satisfies the bounds derived below. The rest of the paper is as follows. Section 2 outlines the endogenous growth/real business cycle we will examine. Sections 3 and 4 examine an economy with constant returns to capital for the case of one and many sectors, and section 5 gives examples of types of productivity shocks which do and do not lead to convergence of output paths. Section 6 shows how these results are modified in the presence of increasing returns to capital, and section 7 concludes.
2. Outline
of the model
In this section we set up and solve a simple discrete time stochastic optimal growth or real business cycle model with constant or increasing returns to
‘Convergence on very special
in a deterministic endogenous assumptions about the return
growth model occurs in Tamura to human capital investment.
(1991) but relies
.Lf. Kelly.
Endogenorts
grobr rh u.ith producricir?
shocks
49
capital. We will study its ergodic behavior in later sections. The example here takes the familiar form of logarithmic preferences and Cobb-Douglas technology and differs only slightly from the model of Long and Plosser (1983). Labor has no role in our analysis and is omitted to keep things as simple as possible. The social planner is assumed to maximize the expected utility of an infinitely-lived representative person:
. . + d, log C,,),
E,, t fl’(d, log C,, + & log C?, + I=0 where /3 is the (constant) discount modity i in period t. The production Yi, = l&K;;:;
Kf::;
rate and function
. K;;y,
(1)
C, is the consumption of comfor each commodity is given by
i=
I,2 )...(
n,
(2)
where Yi, is the output of commodity i in period t and Kji., is the quantity of capital good j allocated to activity i in period t. Bi, is a Markovian productivity shock independent of the other components of the production function. The matrix of Cobb-Douglas coefficients [rji.,] is independent and identically distributed and will be denoted by A,. None of the elements of this matrix can be negative. We will denote its expected value by E A,. Let p(EA,) denote the value of the largest eigenvalue of Eil,. It is assumed that the economy exhibits stochastic increasing returns to capital: p(EA,) 2 1. A sufficient condition for this to occur is if each column of A, has sum greater than or equal to one, so each sector exhibits increasing returns to capital. The economy must satisfy the resource constraint
Yit
ct,+ 1
2
Kij.r
+ 1
9
i = 1,2, . . . , n.
The output of each commodity must at all times be sufficient to meet final consumption and also inputs for next period’s production. Each capital good Kij lasts for a single period and its quantity must be chosen in the period before it will be used. The maximization problem will be well defined if /I x.y=, iii < 1 where Xij = E(rij.,). A straightforward application of Bellman’s prmciple shows that
Kij.
I f
I
=
rij
P Yir
I
i,j = 1,2,.
. . ,n.
(4)
From
eqs. (2) and (4). the evolution
of gross output
=(l”gei[
+ %li.rlOg~S,i
+
yli.l-
r*i.riog
1
Denoting the vector with components matrix form as
+
+ ”
‘..
’ +
+
IXni.r
is given
by
r”i,,lOg13r,i)
log Yni.f-
1.
(log Y,,) by log Y,, (5) may be written
log Y, = A, log Y,_ 1 + rr,
(5) in
(6)
where fI is a vector with components given by the bracketed term in (5). If each firm exhibits constant returns to scale with increasing returns due to Marshallian externalities depending on the aggregate capital stock of its sector, then the economy can support a competitive equilibrium, albeit with a lower rate of investment, in the manner of Romer (1986). The competitive economy will have the same A, matrix but a lower value of TI. The question we address is whether such an economy can have growth paths which do not converge through time. We can think of this problem as follows: given two different starting levels of output Y’ and Y’, will the trajectories followed eventually converge to the same probability distribution, or will economies which start at different levels of output tend to remain permanently on different paths’? From (6), the stochastic difference equation log Y: - log Yj = A, (log r:_ 1 - log Yj- 1) describes the dynamic behavior of the difference of the logs of output for two economies starting at different levels of production. To simplify notation define X, = (log Y: - log Y,‘). This difference equation may then be rewritten as
x, =
A,X,_,.
(7)
It is evident that the behavior of this equation evidently depends on the convergence in distribution of the product of the random matrices of Cobb-Douglas coefficients A,. If n,Ar tends to the zero matrix, then -Y, tends to zero. Output paths of economies with differing initial stocks of capital will converge. On the other hand, if n,AI converges to a nonnegative matrix, then different output paths will not converge and the economy will behave like a deterministic endogenous growth model. We first analyze the case of
constant social returns to capital, and then look at the case of increasing returns. The growth of an individual economy through time depends on the term r in (6) as well as on the products of the At terms. If this term is sufficiently small relative to YI. growth paths will converge to zero if output paths converge. However. if the Markov shock terms 0il are sufficiently large, economies can grow through time although output paths converge.
3. Convergence of output paths: The single-sector
case
Path dependence of output in a single-sector rather than the rule as the following proposition
economy is the exception demonstrates.
Proposition I. capital, output Proof:
Taking log
It1 a single-sector economy bvith stochastic paths concerge with probability one.
constant
returns
to
logs of eq. (7) gives us
x, =
log
x-
1
+
log .4,.
(8)
In this case A, is a random scalar, such that log EA, = 0 because of stochastic constant returns to scale. It follows that the evolution of log X, in eq. (8) is a random walk with a downkvard drift. This is because Jensen’s inequality implies that E(log A,) < log E A, = 0. The result of this is that log X, tends to minus infinity with probability one. In other words, X, tends to zero. Output paths converge with probability one. Alternatively, we may show that the product of nonnegative random scalars fl,A! does not converge to a distribution concentrated on positive scalars unless A, = 1 with probability one. Taking logs of n,A,, we see that the process converges if and only if 1, log rlr converges. The latter term converges if and only if x,E (log A,)’ converges: see, for example, Ash (1972, theorem 7.8.7). W The only exception to this is the unit root case where A, = 1 with probability one, but this possibility has measure zero and so is of little interest.
4. Convergence of output paths: The multi-sectoral
case
The previous section established that path dependence is an exceptional and fragile result in single-sector models. Once any stochastic variability is
allowed in the productivity of capital, output paths converge with probability one. The purpose of our analysis is to show that this result need not carry over to the multi-sectoral case where the output of sectors is interdependent. We denote the product AI, = Al AI . . . A,. The matrices A, are nonnegative i.i.d. n x n matrices where n > 1. Let A,(i, j) denote the i, jth element of A,. We make the following assumptions: Assumption I (irreducibility). A, is independently identically distributed (i.i.d.) through time. A,(i, j) 2 0 for all i and j with probability one. There exists some t such that P [M,(i, j) > 0, Vi, ji. > 0. P {A, has a zero row or column] = 0. This irreducibility assumption holds if and only if every commodity is used, either directly or indirectly, in the production of all other goods. It ensures the existence of a unique largest eigenvalue which is real. Assumption 2 (stochastic constant returns to scale). the largest eigenvalue of E A, and E(A,(i. j)) < x .
p(E A,) = 1 where
p is
We consider increasing returns to capital in the last section. The following proposition is the main one of the paper and says that output paths will not converge if and only if the productivity matrix A, is not too variable. Proposition 2. Under Assumptions I and 2, if economies do not decay into autarky for any initial level of output, output paths will not converge if and only if M, is bounded.
(9)
Proof. There are two cases to consider. In the first, the vector T, in (6) is sufficiently negative so that output tends to zero regardless of its initial level. In this case, output paths converge trivially. In the second, the economy survives. Under Assumptions 1 and 2, Kesten and Spitzer (1984) show that the product of the matrices A, will tend in distribution to a strictly positive matrix if and only if M, holds. There will then be a continuum of possible output paths, each depending on the starting point of the economy. If (9) does not hold, output paths converge with probability one. n
.\I. Kelly.
Endogenous
growth
wilh producririrx
shocks
The relation between high variance and convergence of output given by the next result, again due to Kesten and Spitzer (1984). Corollary.
M, is bounded
if and only
(E(A, @ A ))’ is finite I
53
paths
is
if
and bounded
in t.
(10) pEt.4, @A,) = 1, where
@
denotes
a Kronecker
product.
Hopenhayn and Prescott (1986) and Bhattacharya and Lee (1988) have shown that convergence occurs if at some time t there is a value X* such that for any starting point there is a positive probability that the process is above X* and a positive probability that it is below X*. In our context, this is of course equivalent to saying that M, is unbounded.
5. Examples
of random matrices
In this section we consider the convergence properties of some plausible types of coefficient matrix and see that the conditions (9) for nonconvergence of output paths can be quite strict. Example
I [Kesten and Spitzer (1984)-J. Let Y be a family of positive row and 2 a family of positive column vectors, such that lr = 1 for all 1 E 9’ and r E .&‘. Let ,U be any probability measure on Y x W and (lit ri) an i.i.d. sequence with each element having distribution ,u. Suppose that Ai = rili. Then
vectors
M, = r,l,r212.
. . rllr = r,(l,r,)12.
..(lr_lrl)lr
= rllt.
So, for all t 2 1, the distribution
of M, equals the distribution of the product with distributions given by the first and of p, respectively.
rl I,, where rl and I, are independent
second
marginals
Example
2. Let Ai = Bi + Ci, where Bi is a matrix whose rows sum to unity and Ci is a matrix whose rows sum to zero. Letting Bi = rbi, where r is a unit vector and bi is i.i.d. with elements which sum to unity, and Ci = rci, where ci is i.i.d. with elements which sum to zero, convergence follows as in the previous example. Example 3. Let B be a constant matrix whose rows sum to unity. Suppose that o‘t is a random scalar whose expectation is unity. Define A, = a,& Then
5-I
.W. Kelly.
by Proposition probability one.
1, output
Endogenous
paths
growth
wilh produclirity
will converge
unless
shocks
cI is constant
with
6. Convergence of output paths: Increasing returns to scale 6.1.
The single-sector
cue
To analyze the case of increasing returns only two values. More general cases may
A, =
l/o
with probability
cl,
i C!J
with probability
p,
it will be assumed that A, assumes be handled analogously.
where p > (I and o > 1. Log X, in (8) is a random walk with upward drift for p sufficiently large. We say that convergence has occurred if X, attains some arbitrarily small, positive value E. If the absolute minimum number of steps in which random walk can attain the value of log E is i, then the probability that the two output paths will ever converge equals (q/p);. [See Shiryayev (1984, p. 547) for a proof of this result.] So, in the presence of increasing returns to capital, there is a positive probability that output paths do not converge in distribution. The weaker the increasing returns (i.e., the closer p and 4 are to equality) and the closer the starting levels of output (i.e., the smaller i is), the less this probability of nonconvergence will be.
6.2. The multisectoral
case
In this case, pE(A,) > 1, so we normalize by multiplying A, by some constant q c 1 so that the resulting matrix B, = VA, has pE(B,) = 1. The difference between output paths now evolves as
Again, output paths converge with probability zero if B, satisfies the conditions of Proposition 2. Otherwise, convergence occurs with positive probability which increases with the variability of B and falls with the magnitude of the increasing returns parameter ‘1-i.
7. Conclusions
The purpose of this paper was to analyze the dynamic behavior of a simple endogenous growth model when variable factor productivity was considered. It was shown that the nonconvergence of output paths, which is frequently cited as evidence against endogenous growth models, no longer holds in general. Although the concern of this paper is with analyzing the effect of productivity shocks on the behavior of endogenous growth models. it may be worthwhile to relate our results to the older stochastic turnpike literature. Convergence to a unique steady state in single-sector models is a well-known result dating back to Brock and Mirman (1972) whose proof relies on the ergodic theorem of Markov chains. Multi-sectoral turnpike theory is a less welldeveloped area. The discrete time results of Majumdar and Zilcha (1987) are not applicable here since our model does not satisfy their Assumption 3. The Brock and Magi11 (1979) modification of the Cass-Shell (1976) Hamiltonian curvature conditions for stochastic, continuous time models produces results analogous to ours. They show convergence to a unique steady state if the economy decays deterministically to a zero capital stock or if the noise term is sufficiently variable.
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M. Kelly. Endogenous growth with producriciry shocks
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