Knowledge, Diffusion and Reallocation
Hugo A. Hopenhayn
UCLA
May 9, 2012

1

Introduction

Knowledge is costly to replicate. It is often embodied, so there is limited
capacity and opportunity costs involved in its transmission. Even in the caseof learning by doing, it still takes time to learn and attempting to speed up
this process may carry substantial costs. We identify in this paper knowledgetransmission with the reproduction of its capacity. As a consequence the
incentives and costs of knowledge reproduction play a prominent role in thedeterminants and gains from reallocation. This is a central point that hasremained largely unnoticed in the literature of reallocation and the focus of
this paper.
There is a growing literature emphasizing the gains from reallocation,
both on the empirical and theoretical front. Bertola and Bentolila andHopenhayn and Rogerson -among others- analyze the effect of firing costs,
which tax reallocation of workers from less to more productive firms. The
latter paper finds a sizable impact of these policies on productivity. Recent papers in international trade (e.g. Eaton-Kortum, Alvarez and Lucas)
consider the impact of trade barriers on productivity and welfare. Tariffs
are viewed as barriers to the reallocation of resources to most productive
firms and constitute important wedges in the optimal allocation of production across countries. These papers also identify potentially large gains fromreducing trade barriers. Finally, recent papers (Ramondo, McGrattan andPrescott and Burstein and Monge) have considered the gains from eliminating barriers to foreign direct investment, where the first two papers find very
substantial gains.
All these papers make either explicit or implicit -mostly the latter- assumptions about the process of knowledge replication. In Eaton-Kortum
and Alvarez-Lucas, firms face a constant marginal cost of production. The
marginal cost is idiosyncratic to a firm -or a country in the latter paper- andcontinuously distributed, so in the absence of any barriers -including costs
to trade- all world production of each differentiated good would be produced
by a single producer (or country.) In contrast, in Hopenhayn-Rogerson firms
also have idiosyncratic differences but face decreasing returns to scale, as
in Lucas’ (1977) classic paper on span of control. In absence of barriers to
the reallocation of labor, marginal product is equalized across all productive
units, but more than one producer will remain active. The gains to eliminating barriers to reallocation are mitigated by the strength of decreasing
returns.
1

What lies behind assumptions about returns to scale? We believe thata key element is the process of transmission and replication of knowledge.
With returns to scale -and no adjustment costs- knowledge can be costlessly
and immediately replicated. But then, what prevents others from imitating
the more advantaged firm? What is the source of this superior productivity
that can be costlessly replicated inside but not outside of the limits of the
firm? To the contrary, in most cases knowledge is embodied and as suchcostly to replicate. This gives an advantage and quasi-rents to the owner ofthis capital but at the same time places a limit on the ability to extend theassociated knowledge. An extreme version of this appears in the Lucas spanof control model, where the superior knowledge of a firm is embedded in its
managerial ability, a non reproducible factor.
In this paper we attempt to go one step further in the microfoundationsof returns to scale by providing a specific model of knowledge transmission
and replication. The model puts together a cost of adjustment technology ina vintage capital structure. The basic component of the model is knowledge
capital, a pair (z, k) where z represents the quality or productivity of theknowledge and k a measure of its capacity. This capacity can be accumulated
according to a cost of adjustment technology. There is also a technology fordeveloping new knowledge from scratch (e.g. R&D) that is drawn from a
distribution with a support that shifts over time at a constant rate. There isa fixed endowment of labor that can be used either for production or as an
input to R&D. The equilibrium determines -along a balanced growth paththe optimal degree of replication of existing knowledge capital and its life
expectancy and the amount of resources devoted to outside R&D. Along thisconstant growth path, as new knowledge arrives and wages rise old knowledge
becomes obsolete and unprofitable to use.
In order to provide a link to the gains from reallocation literature, we
consider a very simple numerical exercise to evaluate the impact of a tax onknowledge accumulation We consider three alternative scenarios: a benchmark case, a high cost of adjustment case and a high rate of technological improvement (and consequently high rate of obsolescence). In the lasttwo scenarios the rate of reproduction of knowledge capital is lower and thedifferences in the rate across different z 0 s much less pronounced. In otherwords, compared to the benchmark these are scenarios where reallocation
plays a much less prominent role. Our numerical results show that the negative impact of the tax on average productivity is considerable higher in thebenchmark economy - the other two scenarios behave much like a case where2returns to scale decrease more strongly. This is in spite the fact that thetechnology we consider has actually constant returns to scale. The culprit
lies in the interaction of adjustment costs and the limited durability and obsolescence of knowledge: when it is costly to invest in expanding the stock ofexisting knowledge and an alternative to invest in developing a new superior
one, knowledge reproduction will be limited.
The paper models knowledge accumulation in the abstract. In practice,reproduction of knowledge takes places in multiple ways: learning by doing in a firm, transfer of knowledge through mobility in the labor market,
spin-offs, chains stores opening new establishments, etc. In an attempt toconnect better the theory to the data, the last section of the paper provides
an aggregation procedure in the spirit of Klette and Kortum, where a firm is
a collection of knowledge capital and can also draw stochastically new knowledge through R&D. The model generates patterns of entry, growth and exitof firms and productivity decompositions that match remarkably well the
documented empirical evidence.
There are many papers that relate to our story. Vintage models havebeen widely used and our model draws heavily on that literature. In particular the paper by Chari and Hopenhayn considers explicitly the problem
of reproduction and diffusion of new knowledge by the interaction of learning by doing and complementarities between skilled and unskilled workers.
Franco and Filson develop a model of reproduction of knowledge and spinoffs. The importance of costly reproducible embodied knowledge has beenused by Boldrin and Levine to suggest that intellectual property protection(patents) are not necessary in order for innovators to capture rents from theirinnovations and R&D. (references to be completed).
The paper is organized as follows. Section 2 develops the model and its
comparative statics. Section3 considers the policy experiment. Section 4
provides the link to firm level dynamics and finally Section 5 concludes.

2

A model of learning and diffusion

The basic unit of analysis is a knowledge capital pair (z, k) interpreted as
an amount of capital k of knowledge type z. Production is CRS of the formzf (k, n) , where k is the capital of type z. The state of knowledge is a measureµ on (z, k) and given an aggregate amount of labor L efficient production

3

solves:max
subjectto:ZZ

zf (k, l (z, k)) dµl (z, k) dµ ≤ L.

There is a technology for replication knowledge capital with adjustment costs.
Letting  k̇ (z) denote the new flow of capital of type z, adjustment cost is
C kk̇ zk. In addition capital depreciates at an exogenous rate δ. Assume
C (0) = 0 and C is strictly increasing and strictly
It follows that convex.
adjustment cost is homogenous of degree one in k, k̇ and linear in z, thelatter reflecting that it is costlier to replicate better knowledge.
In the analysis below we focus on a balanced growth path where the economy grows at a constant rate g. In a general equilibrium framework and aclosed economy, this requires preferences that are consistent with balanced
growth. We assume there is a representative agent with CES separable preferences:Z1−θ−ρt c (t)U= e udt.
1−θThis implies that in a balanced growth path r = ρ + θg.

2.1

Optimal replication

Let v (z, t) denote the value of one unit of k (z) at time t and let w (t) represent wages, taking output as a numeraire. Total output can be used forinvestment, as part of the adjustment cost or for consumption. It is standard
to show that the value of k units of capital of type z is simply v (z, t) k. The
Bellman equation for v (z, t) is given by: !

(r + δ) v (z, t) = max
zf (1, n)−w (t) n+ max v (z, t) k̇ − C k̇ z +v2 (z, t)nk̇≥0where r is the interest rate. The interpretation is as follows: the net flowvalue is the sum of the current flow of profits plus the value of expanding
the knowledge capital and the third term captures changes in the value thatarise from the time variation in w and r.
For the rest of the paper we consider a balanced growth path where
consumption and wages grow at a constant rate g and the interest rate is4constant. The above value function can be replaced by: !

(r + δ) v (z, w) = max zf (1, n)−wn+ max v (z, w) k̇ − C k̇ z +v2 (z, w) wg
n

k̇≥0

(1)
Proposition 1 The value v (z, w) is homogenous of degree one in (z, w) .
Proof. Assume this property applies in the right hand side of the Bellman
equation. It follows immediately that the left hand side value function is also
homogeneous of degree one.
An immediate corollary to this proposition is that the choice variables n
and k̇ are also homogenous of degree one in (z, w) and thus only depend -andare increasing- on (z/w) .
It is easy to characterize the time path of a knowledge capital (z, k)starting in a period with wage w. If z/w is sufficiently large so that initiallyk̇ > δ, the stock of knowledge capital will grow. After some time, z/w reaches
a value such that k̇ = δ so the knowledge capital will start to shrink. Finally,
if f2 (1, 0) < ∞, when z/w reaches a critical value z ∗ , v (z, w) = 0 and thatknowledge capital stops being used.
Homogeneity of degree one implies that:v2 (z, w)wg = v (z, w) g − v1 (z, w) zg
so equation (1) may be rewritten as:
  



(r + δ − g) v (z, w) = max zf (1, n)−wn+ max v (z, w) k̇ − C k̇ z −v1 (z, w) zg.
nk̇

Together with the condition that v (z, w) = 0 whenever the corresponding
profits are zero, the solution to this ODE gives the value function and the
optimal accumulation rule.

2.2

Equilibrium with exogenous knowledge frontier andexogenous new arrivals

Suppose each period there is a constant flow m of units of knowledge z =xγ (t) , where 0 ≤ x ≤ 1 is distributed according to a cdf F (x) . The frontier
γ (t) = egt grows at a constant rate g. These m units of knowledge constitute5

a heterogenous cohort entering at time t. Conjecture that in the balanced
growth path wages w (t) = w0 egt , where w0 is to be determined. The value ofz measured relative to the frontier, i.e. z/γ (t) is sufficient to determine theemployment and growth rate of a unit of knowledge capital z at time t. In
terms of this state variable, the stationary equilibrium involves a distributionµ over capital, with support in [z ∗ , 1] , where v (z ∗ , 1) = 0. This measure ischaracterized by equating the flow out of a given z to the flow into this z.
Noting that








µt+∆ (z) ≈ µt zeg∆ + ∆ k̇ − δ µ (z) + ∆f (z) m
it follows that

µt zeg∆ − µt (z)µt+∆ (z) − µt (z)= lim+ k̇ − δ µ (z) + f (z) m
∆→0
∆→0
∆

∆
0
= µ (z) gz + k̇ − δ µ (z) + f (z) mlimwhich must equal zero for a stationary distribution. This gives an ordinary
differential equation on [z ∗ , 1] with boundary condition µ (1) = mf (1) .
To complete the equilibrium first note that the stationary measure µ will
be decreasing in w0 , since k̇ (z) is also decreasing in w0 . Moreover, for each(z, k) demand for labor must also be decreasing in w0 . The two effects workin the same direction, so labor demand strictly and continuously decreases in
w0 . There is thus a unique equilibrium value w0 that equates labor demand
to L.

2.3

Equilibrium with exogenous knowledge frontier change
and endogenous arrivals

To endogenize the steady state arrival rate m, consider the following alternative technology for developing new knowledge capital. Suppose that a unitof labor generates a flow mR of new technologies as described above. Thenif LR units of labor endowment are used in this research sector, m = mR LR .
The cost of a unit flow of new knowledge capital is m1R w0 egt so in equilibriumR
this must also be the expected value created v e = egt v (z, w0 ) dF (z) . This
free entry condition determines uniquely the equilibrium wages that solves:
Z1
w0 = v (z, w0 ) dF (z)
mR

6

(2)

or dividing by w0

Z
1
z
= v
, 1 dF (z)mR
w0
Labor demand for production LP (w0 ) is a strictly decreasing function of thiswage and LR is determined by

LR = L − LPwhich we assume is strictly positive.1 This will be the equilibrium consideredin the numerical analysis of section ??.
2.3.1

TurnoverIt is possible to trace any knowledge capital to a time when it was originatedand started to be replicated. Starting from an initial (z, k0 ) , we will callthe corresponding sequence {z, k (t)} a knowledge chain. A possible notionof turnover, is then the turnover of knowledge chains. This is the analogueof firm turnover in entry/exit models. Consequently, the rate of turnover(entry/exit) of chains along the balanced growth path is equal to the inverseof the expected lifetime of the average initial draw z and is thus monotonicallydecreasing in w0 and the growth rate g.
The measure of turnover of chains is unweighted, i.e. it does not take into
account that chains with different initial z 0 s will grow at different rates andthus will become overly represented in the economy’s stock of knowledge.
An alternative measure of turnover is obtained by measuring entry and exit
(discontinuance) of
knowledge capital. The flow of entry is proportional toR
LR . The stock is µ (dz) which in general is of the form LR φ(w0 ), where φis a decreasing function. For the special case where f (k, n) = min (k, n) the
total stock of active capital equals employment in production so turnover ofknowledge capital is LR / (1 − LR ) . In our analysis below we report these
two measures.
2.3.2

Comparative Statics

We consider here the effect of changes in parameter values. An increase in L
is neutral, just changing proportionally entry and the density of the distribution µ (z) . An increase in mR reduces the cost of introducing new knowledge
1

In case this were negative, in equilibrium m = 0 and it can be shown that wages will
increase (not at a constant rate) until ultimately all labor is employed in the highest z.

7

capital and leads to an increase in the equilibrium wage w0 . This implies
a reduction in v (z), lower investment in replication and a shorter lifespanfor knowledge capital. In turn, LR increases and LP decreases. Intuitively,
as it becomes cheaper to introduce new knowledge capital relative to reproducing existing one, labor in research sector substitutes for reproduction ofknowledge capital. Both measures of turnover discussed above increase.
An increase in g implies a faster wage growth and thus a faster depreciation (relative to the frontier) of existing knowledge capital. This implies alower v (z, w) and by equation (2) a lower value w0 . The rise of g and fallin w0 have opposite effects on the initial value v(z, w0 ) for new knowledge
capital. From the free entry condition, it follows that as w0 decreases the
expected value of new knowledge capital must also decrease. However since
w0 decreases, for values of z close to the initial exit threshold value must
increase and thus must decrease in a range corresponding to higher values.
Though we have not proof, we conjecture the following relationship
itbpF 3.045in1.7832in1.8957inF igure
so that the value increases up to some z (the intersection point) but islower for higher z’s. The intuition for this results is as follows. In the initiallife of a vintage of knowledge capital the wage is lower in the second scenario(higher g case.) Since wage grows faster, a point will be reached where someof these vintages will confront a higher relative wage and consequently alower value. This higher wage will affect those initial knowledge stocks thatgrow faster and live longer, which are precisely the ones with higher initialz.
An implication of the flattening of this profile is that accumulation ofknowledge capital will be less sensitive to differences in productivity z andas a consequence less reallocation will take place from lower to higher z 0 s.
On the other hand, the higher drift in productivity caused by the higher
exogenous growth g will increase the relative importance of higher z 0 s, withan obvious countervailing effect.
The net effect on turnover is not at all obvious and is likely to depend on
the distribution F. In the standard vintage model where all new entrants get
the highest z so F has all its mass at one, the expected lifetime of new knowledge capital must decrease so our first measure of turnover rises. Moreover,since in this case a lower w0 implies -through entry equilibrium condition- alower initial value v(1, w0 ) for the entrant, total knowledge accumulation ofincumbents decreases pointwise and thus our second measure of turnover also
increases. This has a natural interpretation as in the environment with higher
8g,de novo creation of knowledge capital through entry has an advantage overreplication of existing knowledge.2
Consider now a proportional increase in the knowledge accumulation costγC(k̇)z, where γ > 1.R The equilibrium wage w0 decreases and so does the
expected initial value v (z, w0 ) F (dz). The effect is similar to an increase ing. On the one hand, lower wages increase the value of knowledge capital forall z 0 s. But the increase in g has an opposite effect, which is likely to impact
higher z firms more because those are the ones that invest more. So a picture
similar to the one obtained for our previous comparative statics exercise islikely to emerge, with a consequent flattening of profiles. The overall effectis more likely to be negative on total knowledge accumulation. As for theeffect on turnover, the expected life of new knowledge increases, decreasing
our first measure of turnover. On the other hand total turnover of knowledge
capital is likely to rise as a consequence of reduced rates of replication.
Finally consider an improvement on the initial distribution F in first orderstochastic dominance. This raises the expected value of de novo knowledge
and thus raises w0 , lowering the values v(z, w0 ) and knowledge accumulationfor each z. At the same time it also increases the threshold at exit, lowering the life for a given initial z. The overall effect on expected lifetime isambiguous as the shift in F puts more weight on higher z 0 s which are more
durable and have higher accumulation rates. For example, if the original F
has all its mass at some z < 1 and the new one on z 0 > z, the effect will
be neutral as z/w0 = z 0 /w00 , implying that both durability and accumulation
are unchanged. On the other hand, suppose that the initial distribution has
two points of mass, x close to w0 and 1 and the new distribution moves someof the mass resulting in w00 > x. Selecting out this lower x may lead to anincrease in expected life and thus lower our first measure of turnover.

3

Policy experiment

In this section we evaluate the effect of a tax on knowledge replication as abarrier to reallocation. More specifically we consider  a tax that increases thecost of reproduction of knowledge capital by tC k̇ z. As seen above, this tax
decreases investment k̇ (z) and thus reallocation from less to more productive2

One might conjecture that if the distribution F has a large mass at lower levels ofz that are very short lived and some mass at one, an increase in g might increase theexpected lifetime and thus decrease our first measure of turnover.

9

capital. The impact of this tax depends intuitively on the importance ofreallocation. We consider three scenarios, a baseline case, a high adjustment
cost case and a high growth g in the knowledge frontier. Parameter values
used in the simulations are as follows:
r
5%
δ
5%
mR 0.5g
3%
The value of mR is chosen so that in equilibrium the lowest surviving z
in the benchmark is approximately 1/2. This is consistent with results from
Bartelsman and Domes where the ratio of the productivity of the top of
top 5% vs. low 5% is 1.8. The rate of growth of 3% in the benchmark
corresponds to the observed 3% growth in total factor productivity. In thehigh growth scenario the rate is twice as high.
The distribution F on z ∈ [0, 1] is given by:
F (z) =

1 − exp (−λz)
1 − exp (−λ)

with a value of λ = 2. The adjustment cost function has the following form:
c0 k 22where c0 = 15 in the base line scenario and twice as high in the high adjustment cost case. Finally the production function used is f (k, n) = min (k, n) .
The following figure give the value functions for each of three base scenarios and the corresponding replication rates. Note that as suggested by
the theory, the value function of the benchmark case crosses from below theother two value functions and replication rates are flatter (and in average
lower) under the high growth and high adjustment scenarios.
ftbpF5.6853in1.7858in0ptFigure
In the benchmark scenario replication is high, particularly for the high0z s and as a consequence reallocation is also high. The following table gives
the corresponding rates of reallocation to growing z 0 s in a unit time period
as % of total capital/employment.
Reallocation to growing z 0 s
baseline
3.4%

high g0.9%high c
1.5%
10

 

The effect of a tax to the adjustment cost function of the form tC k̇ isconsidered, where t ∈ {0, 0.5, 1} . This tax affects productivity through several channels. In the first place, it is likely to reduce the rate of replicationof high z 0 s and as a consequence lower the extent of reallocation from low tohigh z 0 s. This is similar to the effect of a hiring (or firing) tax that reduces reallocation from low to high marginal product firms. A second channel is thatas the wage w0 decreases, knowledge capital lasts longer and consequently
there is less selection. As reallocation plays a lower effect on productivity inthe high g nand high c case, it is natural to expect that the effect of taxing
investment should be lower in these two cases compared to the benchmark.
The results are given in the following table. Percentage changes of wages
and productivity relative to the case of no taxation are given for the three
scenarios. Note that the negative effect of taxation on productivity is twice
as high in the benchmark scenario. A high cost of adjustment or obsolescenceeconomy responds to this tax similarly to an economy with low returns toscale would respond to a tax on reallocation.

t=0
t = 0.5
t = 1.0

4

Base case
w0
prod
100
100
−6.5% −2.1%−9.8% −4.1%

high adj. cost
w0
prod
93.7
93.7
−3.5% −0.9%−6.1% −2%high g
w0
83.9
−3.9%
−6.4%

prod
80.7
−0.8%
−1.7%

Firms and Knowledge Capital

We have modelled knowledge accumulation in the abstract. In practice, reproduction of knowledge takes places in multiple ways: learning by doing ina firm, transfer of knowledge through mobility in the labor market, spin-offs,
chains stores opening new establishments, etc. In an attempt to connect
better the theory to the data, in this section we identify a firm as a portfolio of knowledge capital. The state of a firm can thus be described by avectorframetitle (z1 , k1 , z2 , k2 ..., zn , kn ) . At each instant a firm receives new
arrivals at an exogenously given Poisson rate m, expands its knowledge capital for higher z 0 s (and reduces it for lower ones) and discontinues knowledge
capital that becomes obsolete. The conjunction of these three sources leads
to a process of growth and contraction of a firm. In the event that all theknowledge capital of a firm becomes obsolete (for the absence of new draws),
we will say that the firm exits. A new potential entrants replaces the firm as
11

it gets a new draw. Notice that we are identifying firms through a somewhat
arbitrary aggregation process -in spirit similar to the one used in Klette andKortum- that is neutral to the equilibrium accumulation process discussed
above. It just serves the purpose of providing a framework to compare modeland data.
We simulate the process of birth-growth-death as follows. At birth, a firmgets mR units of a randomly drawn z (0) from distribution F . This capitalgrows over time according to the optimal replication rule k̇ (z/w) where w
starts initially at w0 and grows exponentially at rate g. The following arrival
is exponentially distributed with parameter m, the Poisson arrival rate. If thearrival occurs before z (0) becomes obsolete, the firm survives and otherwise
it exits and this arrival will be attributed to a new entrant. A surviving firm’s
state at any point in time can be conveniently expressed by a vector {(zj , kj )}with maximal element z̄. Letting T (z) denote the life of an initial knowledge
capital z, the life expectancy of this firm can be obtained as follows. The
probability of exit before T (z̄) is zero. Let P (t, z) denote the probability thata firm gets an arrival before t periods that is greater than z. The probabilityof exit at T (z̄) is then 1 − P (T, z̄) . The probability of exit after T (z̄) isdifficult to characterize.
The process of birth and death depends critically on the arrival rate m:higher values of m make it more likely that a firm gets draws that extend its
lifetime, thus decreasing the rate of entry and exit in the stationary equilibrium. There are very few free parameters left to match the data. These arebasically: the parameter λ for the distribution of new entrants, the wage rate
w0 (determined by wR ) and the value of m. In our simulations we set m = 1/3,
corresponding to an average time for a new draw of three years, λ = 2 andw0 = 0.5 to match the range of total factor productivity as explained above.
In spite of having few free parameters, the model fits remarkably well many
features in the data.
The following table shows the results of a regression of firm growth rate
on ln size and age using the simulated data. The results are close to Gibrat’s
law of proportionate growth, though our estimates imply growth rates thatare slightly increasing with firm size. This is somewhat mitigated by thenegative effect of age.
ftbpF2.8963in1.4218in0ptFigureThe next table provides a series of statistics of firm dynamics for the model and US economy.

12

model US
size of entrants/incumbents
45%
35%
Rate of entry/exit (annual)
5%
7%
job creation/destruction rate 7%
10%
Share of entry/exit
30%
20%
Productivity growth in the economy results from the growth of productivity within a firm, the reallocation from less to more productive firms andthrough entry and exit. Bailey, Bartelsman and Haltiwanger proposed andanalyzed the following decomposition. Let sit denote the share of firm i inperiod t, let πit be its productivity and Πt aggregate productivity. Then
∆Πt =

X

sit−1 ∆πit

i∈C

+

X

(πit − Πt−1 ) ∆sit +X∆πit ∆sit

i∈C

i∈C

+

X

sit (πit − Πt−1 )

i∈N

−

X

sit−1 (πit−1 − Πit−1 )i∈X

The first line corresponds to the within establishment productivity growthwhich in the model is the result of the change in the knowledge capital composition of a firm; the second term is the within firm change in productivity
due to reallocation of shares from less to more productive firms; the third
term measures the interaction of the first two and the two last components
measure the effect of entry and exit. The following table gives this productivity decomposition as applied to the simulations from the model and thedata.
Productivity decomposition
model data
within establishments term
47.2
48
between establishments term 7.5-8interaction term
26.7
34
net entry
15.5
26
Except for the between establishments term, our numbers are in the ballpark. The model predicts a positive within establishment term because inside
the firm better knowledge replicates faster It predicts a positive between establishment term as firms that lose shares are to those that have lower z 0 s

13while those that gain shares have higher z 0 s.3 It predicts a positive interaction term because firms that have higher z 0 s will exhibit both reallocation ofshares of its capital towards more productive knowledge and higher growth
rates that increase their shares. The net entry effect is obviously positive, as
firms that exit are always in the lower end of the distribution of productivity
and new firms are the result of drawing from the frontier.
Exogenous technological change, in the form of new vintages of knowledge, spreads in this economy through two different channels: 1) as the resultof draws taken by incumbent firms and draws made by new entrants. New
plants in the model draw from the frontier technology and are thus likely to
exhibit higher productivity. As firms age, two effects take place. On the one
hand, existing knowledge capital looses its edge relative to the technologicalfrontier. On the other hand, there is selection since survival is associatedwith success in getting draws from newer knowledge capital.
Jensen, McGuckin and Stiroh have proposed the following decomposition
for US manufactures.

Average Labor Productivity (Value added / hours worked in 1987 dollars)
1963 1967 1972 1977 1982 1987 1992
Industry
25.2
New Plants 25.2

28.3
23.7

33.1
28.3

a=1
a=2
a=3
a=4
a=5
a=6

33.7

34.5
38.9

35.2 38.3 48.8 54.0
28.3 30.5 41.4 45.0
Surviving Cohorts
34.6 36.7 44.6 51.4
36.6 38.4 45.5 53.6
41.0 39.4 48.4 49.3
44.3 50.2 52.7
55.6 53.3
60.8

The data is calculated as follows. The cohort age 1 is the cohort thatentered between the first and second census years (63-67) and survived up to
the last period. Cohort age 2 entered between the next two census dates (6772) and survived up to the end of the period...and so on. The calculation is
3
It is actually very difficult to understand why it is negative in the data, suggestingthat in average expanding establishments are less productive than those contracting.

14

not the ideal one, for it introduces an important selectivity bias which is more
pronounced for older vintages. The last column is perhaps the most interesting one. It indicates a little difference in productivity between new entrants
and the cohorts of past entrants, where the latter actually show higher levelsof productivity. Again, this is not very surprising if one considers the effect of
selection. On the other hand, the differences in productivity between cohortsof existing firms are not large and do not follow any predictable pattern.
The following table reproduces numbers obtained from our simulations. It
should be noted that numbers are a bit higher, since the above data shows arate of growth of 2.5%, while our numbers where generated under a 3% steady
growth rate of total factor productivity (which is also labor productivity in
our calibration.) The patterns obtained are quite similar.

a=1
a=2
a=3
a=4
a=5
a=6

5

29.6

Surviving Cohorts
33.8 40.2 49.5 60.9
34.4 39.3 46.7 57.6
39.9 45.6 54.2
46.4 53.0
53.9

62.6
70.7
66.9
63.0
61.6
62.6

Final Remarks

We developed a model that explicitly considers the process of replication ofknowledge. In particular we use the model to analyze the impact of policies
that tax replication and thus the reallocation of resources from less to moreproductive knowledge. Our numerical analysis suggests that the impact ofthese policies depends critically on the incentives for replication: if the rateof obsolescence is high (due to high exogenous vintage specific progress) or
the cost of replication of knowledge high, such a tax is likely to have muchless impact on the economy’s productivity.
In an attempt to match the model with the data, we considered a simple
aggregation procedure identifying firms as collections of knowledge capital.
Preliminary calculations with the model shows a remarkably good fit withfirm dynamics data and productivity decompositions.
Several extensions come immediately to mind. Knowledge is transmitted
also through different locations. The experience of some retail chains such15

as Walmart (see Holmes and Jia) and results from gravity equations forforeign direct investment suggest that proximity plays an important role inthe spatial transfer of knowledge. Ramondo’s (2007) estimates countries
that are twice as far have 45% higher cost to FDI. We propose here a simple
extension that may capture the role of distance in replication..This can bedone by allowing the use of capital at a location i to build capital in location jwith a cost of adjustment that increases with the distance of a location. There
are good reasons to justify this assumption.. In the first place, knowledge
might be to a large extent location specific and hard to replicate in distantor dissimilar locations. Such is the case of knowledge and links to highly
productive local service suppliers. In addition, if knowledge is embodied inpeople that need to travel to transfer it, long distances may impose a large
penalty on replication. These may be reasons why many retail chains tend
to develop locally prior to extending to other regions.
In a similar vein, knowledge that is used to produce certain products
may be costly transferrable for the production of other products. Such replication can be linked to firm product diversification, where the analogue togeographic distance is some measure of technological or marketing distance
between products.

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