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D
Equilibrium and calibration of model examples
In the body of the paper we presented a partial characterization of the equilibrium conditions
that we use to derive our analytic results. Here we provide additional details of the equilibrium,
calibration procedure, and quantitative results described in Section 6. We first consider consider
model example 2 (which nests model example 1) in great detail. We then consider, with less detail
since the key steps are quite similar across models, example 4 (which nests model example 3) and
model example 5.
D.1
Quality ladders model with innovation by incumbents (example
2)
Equilibrium In all the models we consider, variable profits that an incumbent firm earns in period t from production of a product with productivity exp (z) are given by [ pt (z)yt (z) Wt lt (z) Rkt k t (z)].
Marginal cost is MCt exp ( zi ( j)), where MCt = a a (1 a)a 1 Rakt Wt1 a . The incumbent firm that
owns this product chooses price and quantity, pt (z) and yt (z), to maximize these variable profits
subject to the demand for its product and the production function (3). With Bertrand competition
and limit pricing, the gross markup µ charged by the incumbent producer of each product is the
minimum of the monopoly markup, r/ (r
1), and the technology gap between the leader and
any potential second most productive n
producer of the
o good, exp (Dl ), which potentially depends
r
r 1 , exp ( Dl )
on the patent system. That is µ = min
be written as Pt
with k0 = µ
exp(z)r 1 ,
r (µ
. Variable profits from production can then
with the constant in variable profits Pt defined by
h
1) a a (1
⇣
Pt = k0 Rakt Wt1
a )1
innovation by incumbent firms as
a
ir
ttI
1
a
⌘1
r
Yt ,
(51)
. In all the models we consider, we denote the subsidy to
and the subsidy to innovation by entering firms by ttE . We
refer to policies in which t I = t E as uniform innovation subsidies.
The expected discounted stream of profits associated with selling a product with productivity
index z is given by the solution to the following Bellman equation which takes into account the
probability that the firm loses its ownership of this product to another innovating firm
Vt (z) = Pt exp(z)r
1
+
(1 dt )
Vt+1 (z) ,
1 + Rt
where dt denotes the total measure of products innovated on at time t (which every firm takes as
given), Rt denotes the interest rate denominated in the final consumption good, which with CRRA
1
1
preferences is given by 1 + Rt = b
(Ct+1 /Ct )h . Integrating this expression across z and using
the definition of Zt , we have
r 1
Vt Zt
r 1
= Pt Zt
+
(1 dt )
r 1
Vt+1 Zt ,
1 + Rt
(52)
where Vt = Vt (0). The second component of the value to a firm of owning a product is given
by the expected present value of dividends the incumbent firm expects to earn on new products
it gains through innovation minus the cost of that innovation. Given that this value depends
on the number of products owned by the firm, n, and not on its productivities, we denote it by
I n (we verify this below), the value U is
Ut (n). Conjecturing that Ut (n) = Ut n and yrt (n) = yrt
t
determined by the Bellman equation
Ut = max
I
yrt
+
⇣
1
1 + Rt
⇣ ⌘
1
r
I
sd yrt
Vt+1 Zt
1 + Rt
⇣ ⇣ ⌘
⌘
I
sd yrt
+ 1 dt Ut+1 .
1
⌘
I
ttI yrt
Prt +
1
exp (Dz )r
1
(53)
The first term on the right side indicates the firm’s investment in innovation. The second term
indicates the discounted present value of variable profits the firm expects to gain from the innovations that result from this investment. The third term denotes the expected value of the firm’s
innovative capacity from next period on, taking into account both the gain in products it expects
I n products)
to obtain from its innovative effort (i.e., a firm with n products expects to gain sd yrt
and the loss of products it expects as a result of innovative effort from other firms (i.e., a firm with
n products expects to lose dn products). Note that in equilibrium, if the incumbents’ innovation
technology is used, we must have Ut
0. Otherwise, incumbents would choose not to use their
innovation technology at all. Hence, given sufficiently low values of the function d (.) such that
incumbents choose not to innovate, this model nests the standard quality ladders model.
The first-order condition from equation (53) for optimal innovative investment per product by
I (taking d as given) is given by
incumbent firms, yrt
t
⇣
1
ttI
⌘
Prt =
I
sd0 yrt
1 + Rt
!
⇣
r 1
Vt+1 Zt
exp (Dz )r
1
⌘
+ Ut+1 .
(54)
Since none of the terms depend on the incumbent firm’s number of products n or the productivities with which the firm can produce those products, we verify our conjecture that that per-product
I , is the same for all products and firms.31 The total measure of products
innovation spending, yrt
31 One
I extends in an alternacan show that the property that all incumbent firms choose the same yrt
tive specification of our model in which each product that is innovated on draws a random markup and
innovation step size that is independent of the identity of the innovator. Under this extension, consistent
with the data as emphasized in Klette and Kortum (2004), the model generates persistent variation in labor
revenue productivity across firms and in research intensity (defined as innovation expenditures relative to
revenues). In particular firms that have, on average, products with higher markups have higher measured
2
I +N 0 .
innovated on is dt = s d yrt
t( )
The total value of an incumbent firm with n products with frontier technologies z1 , ..., zn is
the sum of the values over its current products, Âin=1 Vt exp (zi )r
1
plus the value of its innovative
capacity, Ut n. The free entry condition for new firms is given by
⇣
1
✓
⌘
ttE Prt ȳ (0)
s
1 + Rt
◆⇣
r 1
Vt+1 Zt
exp (Dz )r
1
⌘
+ Ut+1 ,
(55)
with this condition holding with equality if the measure of entering firms, Nt (0), is greater than
zero. If the equilibrium has entering firms, then the zero-profit condition for entry (55) and equation (54) imply that incumbents’ innovation per product is determined from
⇣
1 ttI
=
1
I
d0 yrt
⌘
ttE ȳr (0).
(56)
This result implies that in any equilibrium with positive entry and uniform innovation subsidies,
I ȳ (0) = 1, which coincides with the condition to maximize the current growth rate subject
d0 yrt
r
to a given level of research good (i.e. Assumption 2).
It is useful to present rescaled Bellman equations describing the value of firms. Defining vt =
r 1
Vt Zt
Prt
, ut =
Ut
Prt ,
we have,
r 1
vt =
⇣
ut =
+
1
1
1 + rt
Pt Zt
Prt
+
(1 dt )
v t +1
1 + rt exp ( gzt )r
⌘
I
ttI yrt
+
1
1
1 + rt exp ( gzt )r
⇣ ⇣ ⌘
⌘
I
sd yrt
+ 1 dt ut+1
1
,
(57)
⇣ ⌘
I
sd
yrt
vt+1 exp (Dz )r
1
1
(58)
and the free entry condition is
⇣
1
⌘
ttE y¯r (0)
where 1 + rt = (1 + Rt )
Prt
Prt+1 .
✓
s
1 + rt
◆
v t +1
exp ( gzt )r
1
exp (Dz )r
1
!
+ u t +1 ,
(59)
1+ R̄
1 = b̃ 1 1 (since Prt rises at a rate ḡy ).
exp( ḡy )
1 /P , and values VZ r 1 /P and U/P are constant over time
r
r
r
In a BGP, r̄ =
In a BGP, rescaled profits PZ r
(as well as r, gz and d as discussed above). In the semi-endogenous growth case (f < 1) the
rate at which innovations occur, d̄, is pinned down from equation (45). Innovation by incumbent
firms ȳrI , if there is positive firm entry, is determined as the solution to (56), and entry is given
labor productivity. Moreover, since all firms choose the same level of innovative investment per product, measured labor productivity and research intensity are not correlated with firm growth in terms of its
number of products. This extension of the model does not change substantially our main results.
3
by N̄ (0) =
ȳrI . In what follows we focus on parameter values such that on the BGP, new
d̄
s
firms enter ( N̄ (0) > 0 when calculated as described above) and incumbents choose to use their
innovation technology. Incumbent firms find it optimal to use their innovation technology in a
ȳ I
BGP with firm entry (ū > 0) if and only if 1 t I d(ȳrI )  1 t E ȳr (0). By equation (56), this
r
d(ȳrI )
0
I
condition is equivalent to d ȳr < ȳ I , which is satisfied if d(.) is concave and d (.) = 0. The
r
constant P̄ Z̄/ P̄r is pinned down by the free entry condition as follows. Manipulating equations
(57) and (58) and imposing BGP with positive entry, we can write the free-entry condition (59) as
⇣
1
t
E
⌘
ȳr (0) N̄ (0) =
where ȳ =
1+
r̄
d̄
⇣
1
d(ȳrI )
N̄ (0)
+1

⌘ ȳ
P̄ Z̄ r
P̄r
1
r̄ + d̄
r̄ exp ( ḡz )
r 1
/exp (Dz )r
1
+ d̄
⇣
1
⌘
t I ȳr ,
(60)
1
¯ P̄r that solves this equation, the allocation of labor L̄ p / L̄r is determined as
Given the value of P̄ Z/
a function of parameters using
L̄ p
=
L̄r
✓
1
µ
a
1
◆
P̄ Z̄
1
.
I
P̄r ȳr + ȳr (0) N̄ (0)
(61)
The level of aggregate productivity Z̄, for a given current value of scientific knowledge Ar , is
determined using equation (8). When f = 1, one can use the same procedure but instead of
solving for Z̄ (the BGP level of Z, given Ar , is not pinned down), one must solve for the growth
rate ḡz (and the corresponding values of d̄, ȳrI and N̄ (0)). The innovation intensity of the economy,
s̄r , is calculated as a function of parameters using expression (12). Finally, we solve for aggregate
output, Ȳt , using (4), the stock of physical capital, K̄t , using the factor shares of physical capital
and production-labor, and consumption, C̄t , using (1).
In Sections 4 and 6 we approximated the aggregate transition taking as given the transition
path for the innovation intensity of the economy and the ratio of physical capital to output. To
solve for the path of these two variables for a given change in policies or other parameters, we
solve the model numerically. Specifically, we solve for the path of Zt , Kt , vt , and
the four
⇣ ut using
⌘h
C
following Euler equations: (57), (58), (59) and Rkt = dk + Rt where Rt = b 1 Ct+t 1
1. Recall
that ȳrI is solved for using equation (56) assuming that there is positive firm entry (which must be
checked). Given a path of Zt and Kt we can solve for all other equilibrium outcomes using static
equations. We solve for the 4 Euler equations using either standard linearization methods or a
shooting algorithm, and we obtain very similar results.
Calibration We now describe a model calibration that rationalizes our parameter choices in
Section 6. We use a similar calibration procedure for the other model examples.
Policies: t I , t E . In order to satisfy Assumption 2, we assume uniform innovation subsidies on
4
the initial BGP, so that t̄ I = t̄ E . As a baseline calibration, we set t̄ I = t̄ E = 0, but we indicate in
the calibration formulas below where these policies enter.
Interest rate minus growth rate: The consumers’ adjusted discount factor b̃ = bexp (1
h ) ḡy
is equal to the gap between the consumption interest rate and the growth rate of consumption in
the BGP, which we set to 0.01. Therefore, b̃ = 0.99 and the interest rate in terms of the research
good in the BGP is given by r̄ = b̃
1
1 = 0.01. We set the growth rate of consumption to
ḡy = .02, which implies a consumption interest rate of R̄ = 0.03. For the exercises in which we
calculate transition dynamics exactly, we assume h = 1.
Final Consumption Good Production Function: The production function for the final consumption
good is parameterized by r, which controls the elasticity of demand curve faced by intermediate
goods producers and establishes an upper bound on the markup µ that can be chosen. We set
r = 4.
Equilibrium Markup: Defining NIPA profits to intangible capital (exclusive of taxes and subsidies)
as GDPt
Rkt Kt
Wt Lt , the share of these profits in GDP, denoted by pt , is given by
pt =
✓
1
µ
1
◆
srt .
(62)
From equation (62), the choice of the markup µ is disciplined by data on the innovation intensity
of the economy, sr , and the share of NIPA profits paid to intangible capital relative to GDP. In our
baseline calibration, we target a share of NIPA profits paid to intangible capital in GDP, p, of 1%
from McGrattan and Prescott (2005) and an innovation intensity of the economy, sr , of 11% similar
to the levels estimated by Corrado et al. (2009) for the United States over the last few years. This
implies a markup of 13.6%, µ = 1.136.
Factor shares: We set a = 0.37 to match the observation that the share of rental payments to
physical capital on the BGP, given by
a
µ,
is equal to 0.33. With this choice of a, we also have
that the share of labor compensation (including production and research) in GDP is given by
1 a
µ
+ sr = 0.66. The rest of GDP corresponds to profits paid to intangible capital, p = 1%.
The Allocation of Labor: The equilibrium allocation of labor between production and research is
pinned down in equation (12) by the choices of parameters above and our calibrated innovation
intensity of the economy, sr . In our baseline calibration, with sr = .11, we have L p = .833.
BGP Growth Rates for Scientific Knowledge and Aggregate Productivity: Given our calibration of
per capita GDP growth of 2% and our physical capital share of a, we calibrate the growth rate
of aggregate productivity in the BGP, ḡz , to 1.25%. For a given choice of f, the growth rate of
scientific knowledge consistent with these productivity growth rates is given by g Ar = (1 f) ḡz .
We do not make assumptions about this growth rate directly since we do not observe it. Instead,
we alter this parameter as we vary f.
Innovation Step Size and BGP Innovation Rate: Our choices of innovation step size Dz and the
BGP innovation rate d̄ must be consistent with the BGP growth rate of aggregate productivity for
5
intermediate firms, gz given in equation (45). We must also have that the innovation step size
exceeds the markup (Dz
µ) to ensure that an incumbent firm has a technological advantage
over its latent competitor consistent with its assumed markup. In our baseline calibration, we set
Dz = Dl = µ. Given our choice of elasticity r and the implied value of ḡz , from equation (45), we
obtain d = 0.08 in our baseline calibration.
Ratio Ȳr /(Ȳr
Yr0 ): Under the assumptions on policies made above, the ratio Ȳ ȲrY0 =
r
r
ȳrI +ȳr (0) N̄ (0)
ȳr (0)(d(ȳrI )+ N̄ (0))
is already determined by parameters that we have previously calibrated and by the choice of targets for sr and p that have been already discussed. To see this, combining the free-entry condition
(60) and the following expression for p̄/s̄r (which holds in all of our model examples)
p̄
=
s̄r
P̄ Z̄ r
P̄r
we obtain
ȳrI
=
ȳr (0) N̄ (0)
1 t̄ E
1 t̄ I
1
1
ȳr (0) N̄ (0)
⇣
✓
1+
ȳrI
ȳr (0) N̄ (0)
r̄
d̄
✓
d(ȳrI )
N̄ (0)
ȳ
1 t̄ I
⇣
ȳrI
+1
ȳr (0) N̄ (0)
d(ȳrI )
+1
N̄ (0)
=
Setting t̄ I = t̄ E = 0, we obtain
Ȳr
Ȳr
Yr0
=
⇣
ȳrI
ȳr (0) N (0)
+1
⌘
+1
◆◆
1+
ȳ
p̄
s̄r
⇣
⌘
+1
⌘
x̄
1 t̄ I
1
r̄
d̄ ⌘
1 + s̄p̄r
(63)
⇣
1
1+
p̄
s̄r
Ȳr
Ȳr Yr0
Ȳr
Ȳr Yr0
.
.
If there is no innovation by incumbents (as in our model example 1), then
which implies
⌘
(64)
ȳrI
ȳr (0) N̄ (0)
=
d(ȳrI )
N̄ (0)
= 0,
= 1. In our baseline calibration with incumbents’ innovation, we obtain
= 0.966 — a value close to its upper bound of one, which means that the average cost of
innovation by incumbents is close to the marginal cost of innovation by incumbents. With this
value of Ȳ ȲrY0 , the impact elasticity G0 calculated using equation (36) is equal to 0.0099, which is
r
r
very close to the value of 0.0102 that we use in our numerical examples. If we lower our target s̄r ,
increase p̄, or reduce r̄ (the gap between the interest rate and the growth rate), then Ȳ ȲrY0 and G0
r
r
fall.
Incumbents’ innovation cost function:
Our analytic results showed that the shape of d (.) does not matter for the first order aggregate
effects of changes in innovation policies except through Ȳ ȲrY0 , which has already been determined
r
r
in our baseline calibration as discussed above. For the numerical evaluation of our results (either non-uniform policy changes or relaxing the assumption of conditional efficiency in the initial
d
BGP), we assume that d yrI = d0 yrI 1 . We choose the level of d (.) in the initial equilibrium,
6
d ȳrI , as follows. The share of employment by entering firms is given by
s N̄ (0) exp (Dz )r
se =
exp( ḡz )r 1
1
.
Given a choice of se (we pick se = 0.03, consistent with data from the LBD), we solve for s N̄ (0) and
then solve for sd ȳrI using d̄ = s d ȳrI + N̄ (0) . In our baseline parameterization, the fraction
of research expenditures carried out by entrants (calculated using equation (64)) is
ȳr (0) N̄ (0)
ȳrI +ȳr (0) N̄ (0)
=
0.27. Using (56) we can solve for
. In any calibration of our model, one must check that
incumbents want to use their innovation technology (i.e. U > 0).
Other parameters: The parameters ȳr (0), s, and Ar at time 0 can all be normalized to 1 without
affecting our results.
ȳ (0) d0
ȳrI
Full numerical solution In the body of the paper we have presented an analytical characterization of the transition dynamics of our model, up to a first-order approximation, following a
policy-induced change in the innovation intensity of the economy. In that approximation, we took
as given the transition path for the innovation intensity of the economy and the ratio of physical
capital to output. In this section we use the calibrated baseline Klette-Kortum model described
above to solve numerically for the full transition dynamics of the economy following a permanent
change in innovation innovation subsidies. We use this solution to evaluate the usefulness of our
analytical approximation.
We first consider an unanticipated and permanent increase in uniform innovation subsidies
applied to entering and incumbent firms from t ( j) = 0 to t ( j) = t̄ 0 for all j
0. With such
a policy change, the level of investments in innovation per product by incumbents, ȳrI , remains
constant over time (see equation (56)) and the increase in innovation along the transition arises
from an increase in entry only (this is the variation considered when introducing Assumption 1a).
We choose the level of t̄ 0 so that, in the long run, the innovation intensity of the economy rises from
s̄r = 0.11 to s̄r = 0.14, so that log sr0
log s̄r = .24. In Appendix F we show that under a number of
assumptions (satisfied in all of our model examples) changes in innovation subsidies uniformly
applied to entering and incumbent firms result in the long-run in a change in fiscal expenditures
Ē0
¯ 0
GDP
Ē
¯
GDP
= t̄ 0 s̄r0 t̄ s̄r = s̄r0 s̄r . Therefore, the policy-induced increase in the
innovation intensity that we consider would require a recurring fiscal expenditure on innovation
subsidies equal to 3 percent of GDP in the long-run, roughly equal to the total revenue collected
from corporate profit taxes relative to GDP in 2007 in the U.S. In this sense, we regard this as a
large change in the innovation intensity of the economy.
We report results from this experiment in Figures 1 and 2 for the transition dynamics of our
model economy over the first 100 years and 20 years, respectively. We display in the left panels of
these figures the dynamics resulting from our first order approximation in Section 6, assuming that
on the transition path the physical capital to output ratio is constant at its BGP level, and that the
relative to GDP of
7
Figure 1: 100-year Transition Dynamics to a Permanent Increase in Uniform Innovation
Subsidies, Approximation and Full Numerical Solution, baseline Klette-Kortum model
(example 2)
0
innovation intensity of the economy jumps to its new BGP level immediately, i.e. log srt
.24 for all t
log s̄r =
0. The right panels of these figures display the dynamics resulting from the full
numerical solution of the model.
In panel B of both figures we show the dynamics of log srt over the first 100 years of the transition. Here we see that there are mild intertemporal substitution effects in innovation expenditures
in that this innovation intensity rises by a bit more than .24 in log terms in the early phase of the
transition, particularly for the case with low intertemporal knowledge spillovers. The intertemporal substitution of innovation expenditures shown in panel B is more pronounced with low f
because in this case the price of the research good is expected to rise quickly during the transition.
From Proposition 3, we have that the intertemporal substitution in the path for the innovation
intensity of the economy shown in Panel B impacts the model-implied transition for aggregate
8
productivity. By comparing the approximated path for aggregate productivity shown in panel E
to the fully simulated path shown in panel F, we can see that the approximation is fairly accurate
despite the dynamics of the innovation intensity of the economy shown in Panel B.
Figure 2: 20-year Transition Dynamics to a Permanent Increase in Uniform Innovation
Subsidies, Approximation and Full Numerical Solution, baseline Klette-Kortum model
(example 2)
In panel D we show the evolution of the log of the physical capital to output ratio over the
transition. This transition path corresponds to the negative of the transition path for the log of
the rental rate on physical capital. Clearly, there are dynamics of this ratio that we ignored in
our analytical approximation. From expression (30), we have that the full dynamics of GDP are
impacted both by the dynamics of the innovation intensity of the economy and of the ratio of
physical capital to output. On impact, these two factors have opposite effects on GDP — the
initial increase in the ratio of physical capital to output raises GDP while the initial increase in the
innovation intensity of the economy above its new long-run level lowers GDP. By comparing the
9
approximated path for GDP shown in panel G to the fully simulated path shown in panel H, we
see that the increase in GDP over the first 20 years of the transition is roughly 1.5% lower when we
take into account the dynamics in the innovation intensity of the economy and the capital-output
ratio.
Figure 2 shows that a large and persistent increase in innovation subsidies has a relatively
small impact on aggregate productivity and GDP over a 20 year horizon, and the response of
aggregates does not vary much with the extent of intertemporal knowledge spillovers assumed
in the model. These results suggest that it would be hard to verify whether innovation policies
yield large output and welfare gains using medium term data on the response of aggregates to
changes in innovation policies. We illustrate this point in Figure 3. In that figure we show results
obtained from simulating the response of aggregates in our model to our baseline increase in
innovation subsidies in an extended version of our model with Hicks neutral AR1 productivity
shocks with a persistence of 0.9 and an annual standard deviation of 2%. We introduce these
shocks as a proxy for business cycle shocks around the BGP. We show histograms generated from
3000 simulations of the model for the first 20 years of the transition. The units on the horizontal
axis show the log of the ratio of detrended GDP at the end of the 20th year of transition to initial
GDP and the vertical axis shows the frequency of the corresponding outcome for GDP. In panel A
of the figure, we show results for GDP excluding innovation expenditures. In panel B, we show
results for GDP including innovation expenditures, GDPt ⇥ (1 + srt ). The red bars show results
for the model with low intertemporal knowledge spillovers and the blue bars show the results
with high spillovers. We can observe in each panel that the distribution represented by the blue
bars is slightly to the right of that represented by the red bars. The histograms in panel B are
shifted to the right relative to those in panel A reflecting the fact that GDP including innovation
expenditures has a higher elasticity of changes in the innovation intensity of the economy. But
it is also clear in each panel that, using either measure of GDP, that it would be very hard to
distinguish the degree of intertemporal knowledge spillovers (and, hence, the long term effects
from this innovation subsidy) in aggregate time series data even if we had the benefit of a true
policy experiment.
Up to this point, we have considered policy experiments in which the economy starts on a
BGP with uniform innovation subsidies and transitions to a new BGP with a new rate of uniform
innovation subsidies. The assumption that innovation subsidies are uniform in the initial BGP
(Assumption 2) implies that it does not matter, up to a first-order approximation, whether the
additional units of the research good are allocated to entrants or incumbent firms (Lemma 1). The
assumption that innovation subsidies are uniform along the transition implies that the temporary
increase in innovation is achieved through an increase in entry, and innovation per product by
incumbents remains unchanged (see equation (56)). We now evaluate whether there are important second-order effects that arise when large non-uniform changes in innovation policies are
considered that change the level of innovation by incumbents and entry along the transition.
10
Figure 3: Histogram 20-year Increase in GDP to a Permanent Increase in Innovation Subsidies, Including Productivity Shocks, Klette-Kortum model (example 2)
Specifically, we consider a permanent increase in the innovation subsidy to incumbents that
results in a long run increase in the innovation intensity of the economy from s̄r = .11 to sr0 = .14.32
The curvature parameter of the incumbents’ innovation technology, d(.), which shapes the change
in investments by incumbents yr along the transition, is set at 0.4 consistent with the estimates in
Acemoglu et al. (2013). In Figure 4 we show the evolution over the first 20 years of the log of the
innovation intensity of the economy, the log of the physical capital output ratio, aggregate productivity, and GDP. We display results from the baseline transition with uniform policies (left column)
and the transition with non-uniform policies (right panel). We can observe that the dynamics of
aggregate productivity and GDP in the first 20 years of the transition are not substantially different
in these two cases.
32 In the long run, this subsidy requires fiscal expenditures of 3.3% relative to GDP rather than 3% under
our baseline experiment with uniform innovation policies).
11
Figure 4: 20-year Transition Dynamics to a Uniform and Non-Uniform Innovation Policy,
Full Numerical Solution Baseline Klette-Kortum model (example 2)
D.2
Simple Expanding Varieties model with innovation by incumbents
to improve their own products (example 4)
We describe some key steps in solving the equilibrium and calibrating this model. The value of a
firm with productivity exp (z) is given by
Vt (z) = Pt exp (z)
r 1
⇣
1
1 df
+
1 + Rt
ttI
1
⌘
1 df
Prt yrt +
1 + Rt
d0
d
yrt
✓
d0 + d
Zt
exp (z)
12
◆r
yrt
1
!!
✓
Zt
exp (z)
Vt+1 (z
◆r
1
Dz )
!!
Vt+1 (z + Dz )
where Pt is defined as in (51) where µ = r/ (r
⇣
⌘
exp(z) r 1
I
Vt exp (z)r 1 and yrt (z) = yrt
where
Zt
Vt = Pt
⇣
ttI
1
⌘ P yI
rt rt
r 1
Zt
+
1). It is straightforward to show that Vt (z) =
h⇣
⇣ ⌘⌘
1 df
I
Vt+1 d0 + d yrt
exp (Dz )r
1 + Rt
1
⇣
+ 1
d0
⇣ ⌘⌘
I
d yrt
exp ( Dz )r
I satisfies
and yrt
⇣
1
⌘
1 df 0 ⇣ I ⌘ r
ttI Prt =
d yrt Zt
1 + Rt
1
⇣
Vt+1 exp (Dz )r
1
exp ( Dz )r
1
⌘
With positive firm entry, we must have
⇣
In a BGP, yrI ,
r 1
Vt Zt
Prt
r 1
⌘
sZt Vt+1
ttE Prt ȳr (0) =
1 + Rt
1
(which we denote by vt ), and
✓
1
1
t̄ E
t̄ I
v̄ =
⇣
1
◆
P̄ Z̄ r
P̄r
1
r 1
Pt Zt
Prt
are constant over time, satisfying
⇣ ⌘
H 0 ȳrI ȳr (0) = a1
1
1
t̄ I ȳrI
a0 + F (ȳrI )
(1+r̄ )exp( g¯z )r
⌘
t̄ E ȳr (0) =
(65)
(66)
1
a1 v̄
(1 + r̄ ) exp ( g¯z )r
1
,
(67)
where we use the definitions above when describing this model example,
a0 = 1
⇣
df
⇣
d0 exp (Dz )r
1
1
d0 ) exp ( Dz )r
+ (1
1
⌘
1
⌘
,
(1+ R̄)
1 = b̃ 1
exp( ḡy )
1. With uniform innovation policies in the initial BGP, we obtain condition (47), which is the
a1 = s, F ȳrI
= 1
df
exp (Dz )r
exp ( Dz )r
d ȳrI , and r̄ =
condition that satisfies Assumption 2.
In order toh solve the model’s BGP given
all parameter values and f < 1, we obtain ȳrI from
i
r 1
r 1
(65), N̄ (0) = exp ( ḡz )
a0 F ȳrI /a1 (where ḡz = g Ar / (1 f)), P̄Z̄P̄ from (65) and (66),
r
L̄ p / L̄r from (61), and Z̄ (for a given current value of scientific knowledge Ar ) from (8).
The calibration procedure is very similar to that of example 2 described above, but given the
discussion in Section 6 that in our expanding varieties model examples the impact elasticity is
shaped by the share of entering firms in employment relative to their share in research expenditures (equation (37)), we now target these two shares in our calibration. There is a simple equilibrium relationship between this statistic and the ratio of the innovation intensity of the economy,
NIPA profits to intangible capital as a share of GDP and the gap between the interest rate and the
13
1
i
growth rate r̄ (in Example 2 we targeted these three statistics). Specifically, combining (63), (65),
and (66), the definition of the share of employment in entering firms
s̄e =
a1 N̄t (0)
exp ( ḡz )r
1
a0 + F ȳrI
=1
exp ( ḡz )r
1
and imposing t̄ I = t̄ E = 0 we obtain
s̄e
r̄ s̄r
=
.
p̄
ȳr (0) N̄ (0) /Ȳr
(68)
ȳr (0) N̄ (0)
Ȳr
Given our choice of s̄r = 0.11, r̄ = 0.01, s̄e = 0.03, and
= 0.2, equation (68) implies that
p̄ = 0.0073, which is slightly lower than our target p̄ = 0.01 in the calibration of example 2.
In order to evaluate the impact elasticity using the alternative (but equivalent) formula (36),
our calibration procedure determines the levels of Ȳ ȲrY0 and G00 as follows given targets for ḡz , s̄e
r
r
and s̄re = ȳr (0) N̄ (0) /Ȳr . We solve for a0 + F ȳrI and a1 N̄ (0) from
⇣ ⌘
a0 + F ȳrI = exp ( ḡz )r
1
a1 N̄ (0) = exp ( ḡz )r
We solve for
a1
ȳ I
ȳr (0) r
as
a1 I
ȳ =
ȳr (0) r
Using (65), we have F 0 ȳrI ȳrI =
are given parameters, we have F
✓
1
s̄re
s̄re
◆
a1
ȳ I . Assuming F
ȳr (0) r
ȳrI = f11 F 0 ȳrI ȳrI .
(1
1
s̄e )
s̄e .
a1 N̄ (0) .
yrI = f 0 yrI
f1
, where f 0 > 0 and f 1  1
Therefore, a0 , which determines the social
depreciation of innovation expenditures, is given by
a0 = exp ( ḡz )
r 1

1
✓
1
1+
f1
✓
1
s̄re
s̄re
◆◆
s̄e .
Social depreciation is larger (i.e. a0 is lower) the higher is s̄e , the lower is s̄re , and the lower is the
curvature parameter f 1 . Finally, we have
Ȳr
Ȳr
As f 1 ! 1,
Ȳr
Ȳr Ȳr0
Ȳr0
=
1
! 1.
14
f1
s̄re (1
f1 )
.
D.3
Simple Expanding Varieties model with innovation by incumbents
to create new products (example 5)
The solution and calibration of this model is very similar to Example 4. The value of a firm with n
products is Vt (n) = Vt n, where
⇣
Vt = Pt
⇣
and Pt = k0 Rakt Wt1
a
⌘1
r
⌘
I
ttI Prt yrt
+
1
1
Yt exp (qt)r
level of innovation per product,
I ,
yrt
⇣
The free-entry condition is
⇣ ⌘⌘
1 ⇣
I
d0 + d yrt
Vt+1 ,
1 + Rt
denotes variable profits per product. The FOC for the
is
⌘
ttI Prt =
1
⇣ ⌘
1
I
d0 yrt
Vt+1 .
1 + Rt
⇣
1
⌘
sV̄t+1
ttE Prt ȳr (0) =
.
1 + Rt
✓
1
1
ttI
ttE
Combining (69) and (70) we obtain
◆
(69)
(70)
⇣ ⌘
s
= d0 yrtI .
ȳr (0)
= d0 ȳrtI , satisfying Assumption 2.
We define vt = PVrtt , which in the BGP is constant and given by
With uniform policies in the initial BGP, we obtain
v̄ =
where 1 + r̄ =
P̄
P̄r
1
s
ȳr (0)
t̄ I ȳrI
1
I
d0 +d(ȳrt
)
(1+r̄ )exp( ḡz q )r
1
(1+ R̄)
. In deriving this expression we used the fact that in the BGP Prt+1 /Prt =
exp( ḡy )
exp ḡy exp ( ḡz q )1 r since srt = PrtYỸt rt is constant and Ỹrt = Yrt Mt grows at the rate (r 1) ( gZ
q ). Using these results and gz = q +
1
r 1
log d0 + d(ȳrI ) + s N̄ (0) , we can write the free entry
condition in BGP as
⇣
1
t̄
E
⌘
ȳr (0) N̄ (0) =
exp ( ḡz
exp ( ḡz
q )r
q )r
1
1
d0
(1 + r̄ )
d ȳrI
d0
I
d ȳrt
✓
P̄
P̄r
⇣
1
t̄
I
⌘
ȳrI
◆
.
(71)
Given these expressions, we solve for the equilibrium following very similar steps to those in
example 2.
The procedure to calibrate this model example is also very similar to that of model example
3. To calibrate the model, we make use of equation (68) relating the share of entering firms in
employment relative to their share in research expenditures (the right hand side of equation 37)
15
with the ratio of the innovation intensity of the economy, NIPA profits to intangible capital as a
share of GDP and the gap between the interest rate and the growth rate r̄. To obtain equation (68),
we use the free entry condition (71) and the definitions of the share of profits to intangible capital
relative to GDP,
pt =
I M P
yrt
ȳr (0) Ñt (0) Prt
t rt
,
GDPt
P t Mt
the share of innovation expenditures relative to GDP,
srt =
I M P + ȳ 0 Ñ 0 P
yrt
t rt
r ( ) t ( ) rt
,
GDPt
and the share of entering firms in employment,
set = 1
E
E.1
I )
d0 + d(yrt
q )r
exp ( gzt
1
.
Model extensions
Occupation choice
Suppose that workers draw a productivity x to work in the research sector, where x is drawn from
a Pareto with minimum 1 and slope coefficient c > 1. There are two wages, Wpt and Wrt . For the
marginal agent,
x̄t Wrt = Wpt
Given that the minimum value of x is 1, any interior equilibrium with positive production requires
Wrt  Wpt . The aggregate supplies of production and research labor are (having normalized the
labor force to 1),
c
L pt = F ( x̄t ) = 1-x̄t
ˆ •
c
1
Lrt =
x f ( x ) dx =
x̄t
c
1
x̄t
c
.
The equilibrium allocation of labor is determined by (as in equation (12) in the baseline model)
Wpt L pt
(1 a ) 1
=
Wrt Lrt
µt srt
and
L pt
Lrt
=
11
c
c
16
⇣
⌘ c
Wpt
Wrt
⇣ ⌘1 c .
Wpt
Wrt
Note that as c goes to infinity, Wpt /Wrt must converge to 1 in order for L pt /Lrt to be finite. The
elasticity of the aggregate labor allocation with respect to the innovation intensity of the economy
(given a constant average markup) is
L0pt
log 0
Lrt
log
(c
L̄ p
=
L̄r
(c
⇣
⌘
W̄r L̄r
1) 1 + W̄
p L̄ p
0
⇣
⌘
logsrt
W̄r L̄r
1) 1 + W̄ L̄ + 1
logs̄r
p p
and the elasticity of research labor with respect to the innovation intensity of the economy is
0
logLrt
log L̄r =
(c
( c 1)
0
⇣
⌘
logsrt
W̄r L̄r
1) 1 + W̄ L̄ + 1
logs̄r
p p
When c converges to 1 (high worker heterogeneity), the elasticity of L pt /Lrt and Lrt with respect
to srt converges to 0. When c converges to infinity (no worker heterogeneity), the elasticity of
L pt /Lrt and Lrt with respect to srt converges to
1 and L p , respectively, as in our baseline model.
In Proposition 3, equation (16) becomes
log Yrt0
where
log Ȳr =
✓(c 1) ◆
W̄r Lr
+1
(c 1) 1+ W̄
L
(c
( c 1)
0
⇣
⌘
log srt
W̄r L̄r
1) 1 + W̄ L̄ + 1
log s̄r
(1
f) logZt0
log Z̄t
p p
is bounded between 0 and L̄ p . In Proposition 3, coefficient G0 is now given
p p
by
G0 =
(c
D g0
( c 1)
⇣
⌘
,
W̄r L̄r
DYr0
1) 1 + W̄
+
1
L̄
p p
which is increasing in c (the smaller is the extent of worker heterogeneity, the higher is G0 ).
E.2
Goods and Labor used as inputs in research
We consider an extension in which research production uses both labor and consumption good,
as in the lab-equipment model of Rivera-Batiz and Romer (1991), and discuss the central changes
to our analytic results. Specifically, the production of the research good is given by
f 1 l 1 l
Lrt Xt
Yrt = Art Zt
,
and the resource constraint of the final consumption good is
Ct + Kt+1
(1
dk ) Kt + Xt = Yt .
17
Given this production technology, the BGP growth rate of aggregate productivity gz is given by
ḡz =
g Ar
q ,
where q = 1
f
1 l
1 a.
The condition for semi-endogenous growth is q > 0. The
knife-edge condition for endogenous growth is q = 0 and g Ar = 0, which can hold even if f < 1.
Revenues from the production of the research good are divided as follows
Wt Lrt = lPrt Yrt , and Xt = (1
l) Prt Yrt .
(72)
The allocation of labor between production and research (the analogous to equation (12) in our
baseline model) is related to the innovation intensity of the economy by
L pt
(1 a) 1 Yt
=
,
Lrt
lµt srt GDPt
where GDPt = Yt
(73)
(1 dk ) Kt when innovation expenditures are excluded in
GDP. Factor payments are a constant shares of Yt .
Our analytical results need to be modified for two reasons. First, the role that the term 1
f played in shaping the dynamics of the economy is now played by q. Second, several of our
t
analytical elasticities need to be modified by the ratio of GDP to Y, which is equal to GDP
=
Yt
1
(1 + (1 l)srt )  1.
In Proposition 3, the elasticity of research output Yr with respect to a change in the innovation
intensity of the economy sr , presented in equation (16), is now given by
log Yrt0
Xt = Ct + Kt+1
log Ȳr = L̄ p
¯
GDP
0
log srt
Ȳ
q logZt0
log s̄r
log Z̄t
(1 l ) a
log R0kt
1 a
log R̄k
The third term in the right hand side reflects the change in research output Yrt that result from
changes in Yt relative to Kt when l < 1. The coefficients Gk in Proposition 3 are now given by
G0 = L̄ p
"
G k +1 = 1
¯ D g0
GDP
Ȳ DYr0
(1
f)
G0
¯
L̄ p GDP
Ȳ
#
Gk .
The result in corollary 1 is now stated as
¯
log Zt0
log Z̄t =
L̄ p GDP
Ȳ
log sr0
q
log s̄r .
Finally, the result in corollary 2 is now adjusted to account for the change in GDP/Y,
GDPt
log
Yt
¯
GDP
log
=
Ȳ
✓
1
18
GDP
Y
◆
log sr0
log s̄r .
F
Fiscal expenditures and innovation intensity
In this appendix we derive the elasticity across BGPs of the innovation intensity of the economy
and fiscal expenditures on innovation policies with respect to a uniform change in innovation
policies, where uniform innovation policies are such that tt ( j) = tt . In this case, aggregate fiscal
expenditures on innovation policies are given by Et = tt Prt Yrt and
Et
Yt
= tt srt . In order to derive
our results, we specify additional details of the equilibrium that are not required in the analytic
results of Section 4. We also make use of a number of assumptions, which are satisfied in all of our
model examples described above.
Suppose we can write the value of a firm of type j at time t as
Vt ( j) = maxyrt ( j) Pt ( j)
for j
(1
tt ) Prt yrt ( j) +
1 and
Vt (0) =
(1
tt ) ȳr (0) Prt +
1
E 0
Vt+1 j0
1 + Rt tjj |yrt ( j)
1
E t0j0 Vt+1 j0
1 + Rt
where Pt ( j) is the static profits of a firm of type j at time t, and E tjj0 |yrt ( j) denotes the expectation
operator of a type j firm over future types j0 given yrt ( j) and all other variables at time t. Assume
that static profits can be written as Pt ( j) = Bt H ( j) and that in the BGP, B̄t = kȲt .33
Define vt ( j) =
Vt ( j)
.
(1 tt ) Prt
In BGP, we have
v̄t ( j) = maxyrt ( j)
for j
yrt ( j) +
v̄t (0) =
1
E t0j0 v̄t+1 j0
1 + r̄t
1 and
0
where 1 + rt =
1
P
(1+ Rt ) P rt
1
E 0
v̄t+1 j0
1 + r̄t tjj |yrt ( j)
kȲt
H ( j)
P̄rt (1 t̄ )
ȳr (0) +
. Assume that the free entry condition binds, v̄t (0) = 0.
rt+1
We now compare two BGPs which are subject to a different innovation subsidy t̄. With semiendogenous growth (f < 1), r̄ is unchanged with t̄. By the free-entry condition, ȳrt ( j) for j
and
Ȳt
P̄rt (1 t )
1
are also invariant with t̄. Comparing two BGPs, we have
Ȳt
Ȳt0
= 0
,
P̄rt (1 t̄ )
P̄rt (1 t̄ 0 )
which can be re-written as
Ȳrt
Ȳrt0
= 0
.
s̄rt (1 t̄ )
s̄rt (1 t̄ 0 )
We now provide a condition (satisfied in all of our model examples) such that output of the
33 Note
that in Neo-Schumpeterian models, k includes the rate of product destruction, which is constant
across BGPs with semi-endogenous growth.
19
research good, Yr , is unchanged between BGPs. The research good constraint (7) can be written as
"
#
N̄t ( j)
N̄t (0) Â ȳrt ( j)
+ ȳr (0) = Ȳrt
N̄t (0)
j 1
(74)
and the growth rate
ḡzt ⌘ G {ȳrt ( j)} j 1 , N̄t (0) ; { N̄t ( j)} j 1
✓
N̄t ( j)
= G {ȳrt ( j)} j 1 , N̄t (0) ; N̄t (0) {
}j
N̄t (0)
1
◆
(75)
.
0
N̄t ( j)
N̄t ( j)
0 ( j )}
= {ȳrt
j 1 , then { N̄t (0) } j 1 = { N̄t0 (0) } j 1 .
0 ), we must have N̄ (0) = N̄ 0 (0)
By equations (74), (75), and semi-endogenous growth ( ḡzt = ḡzt
t
t
0
and Ȳrt = Ȳrt . This implies that
0
s̄rt (1 t̄ ) = s̄rt
1 t̄ 0
Suppose that the T operator is such that if {ȳrt ( j)} j
and
Ē0 t
Ȳ 0 t
Ēt
0
= t 0 s̄rt
Ȳ t
1
0
t s̄rt = s̄rt
s̄rt .
We summarize this result in the following proposition.
Proposition 4. Consider our model on a BGP with semi-endogenous growth and positive firm-entry that
satisfies the assumptions described in this section. Suppose that uniform innovation policies change permanently from t̄ to t̄ 0 . Then, across the old and new BGPs the innovation intensity of the economy changes
¯ to Ē0 / GDP
¯ 0 , with these changes
from s̄r to s̄r0 , and fiscal expenditures relative to GDP change from Ē/ GDP
given by
log s̄r0
and
log s̄r = log(1
Ē0
¯ 0
GDP
t̄ )
Ē
0
¯ = s̄r
GDP
log(1
t̄ 0 )
s̄r .
This result implies that in the long-run, uniform changes in innovation subsidies result in a
change in the innovation intensity of the economy determined only by the change in the innovation subsidy rate independent of the other parameters of the model. At short and medium
horizons, however, this policy will result in a change in the path of the innovation intensity of the
•
0
economy from {s̄rt }•
t=0 (which is constant on the initial BGP) to { srt }t=0 that we will have to solve
for numerically. In our analytic results, we take this path as given.
20
G
Model examples violating baseline assumptions
Quality ladders model with variable markups
We first present a quality ladders model with variable markups based on Peters (2013). This
model violates Assumption 1 regarding concavity of the growth rate of aggregate productivity
with respect to entry and Assumption 2 regarding the allocation of innovation across heterogeneous firms. Intuitively, in the presence of misallocation due to markup variation across firms,
there may be aggregate productivity gains from reallocating innovation across firms in order to
improve the distribution of markups. In this sense, innovation policy can partly substitute antitrust policies.
Incumbent firms each produce one good and are indexed by the productivity with which they
can produce this good, exp(z). Aggregate productivity is given by (5) where Nt ( j) denotes the
mass of products with productivity z( j) and markup µ ( j). With r ! 1, the expression for aggregate productivity is Zt = Z1t Z2t where
 z( j) Nt ( j)
Z1t = exp
j 1
Z2t =
⇣
exp  j
Âj
⇣
log
µ ( j)
1
1
µ ( j)
1
1
⌘
!
Nt ( j)
Nt ( j)
⌘
.
Each entrant expends ȳr (0) units of the research good to be matched with probability s to
an existing intermediate good (produced by some other firm) raising the frontier productivity
for producing that good from exp(z) to exp(z + Dz ) and charging a markup of µ = exp (Dz ).
I units of the research good to innovate with probability sd y I ,
Each incumbent firm invests yrt
rt
raising the productivity of its own intermediate good from exp(z) to exp(z + Dz ) and increasing
the markup by a factor of exp (Dz ), where d (.) is increasing and concave. We denote by n ( j) the
number of steps ahead by a product of type j relative to the second lowest cost supplier of that
I
good, so that µ ( j) = exp (n ( j) Dz ). The total measure of products innovated is s Nt (0) + d yrt
of which a fraction Nt (0) / Nt (0) + d
I
yrt
is by entrants.
The law of motion for the first component of aggregate productivity, Z1t , is
logZ1t+1
⇣
⇣ ⌘⌘
I
logZ1t = Dz s Nt (0) + d yrt
.
The second component of aggregate productivity, Z2t , can be written as
"
Z2t = exp
!
 nDz Mt (n)  exp (nDz )
n 1
n 1
21
1
Mt ( n )
#
1
,
,
where Mt (n) denotes the measure of products that is n steps ahead, and satisfies Ân
1
Mt (n) = 1.
The law of motion of Mt (n) is
⇣
Mt + 1 ( 1 ) = 1
and for n > 1
⇣
⇣
Mt + 1 ( n ) = 1
In the BGP, M̄ (1) =
⇣
⇣ ⌘⌘⌘
I
s Nt (0) + d yrt
Mt (1) + sNt (0)
s Nt (0) + d
N̄ (0)
N̄ (0)+d(ȳrI )
and M̄ (n) =
⇣
I
yrt
⌘⌘⌘
Mt (n) + sd
d(ȳrI )
M̄ (n
N̄ (0)+d(ȳrI )
⇣
I
yrt
⌘
Mt ( n
1) .
1) for n > 1. Peters (2013) shows
that an increase in entry increases Z2t by raising the measure of products with low markups. On
the other hand, an increase in innovation by incumbents reduces Z2t and can reduce Zt . Therefore,
this model does not satisfy Assumption 2.
Moreover, this model can violate Assumption 1a regarding concavity of the log growth rate of
aggregate productivity with respect to an increase in entry. To see this, note that
logZt+1
The term log ( Z1t+1 )
logZt = (logZ1t+1
logZ1t ) + (logZ2t+1
logZ2t )
log ( Z1t ) is linearly increasing in Nt (0). The term logZ2t+1
logZ2t is
increasing in entry and can be convex (for example, set s = 0.1, Dz = 0.1, ḡz = 0.01 and in the
initial BGP
logZt+1
N̄ (0)
N̄ (0)+d(ȳrI )
= 0.05). Since the sum of a linear function and a convex function is convex,
logZt is convex, violating Assumption 1a.
Two model specifications that violate Assumptions 2 and 3 but satisfy
conditional efficiency in initial BGP
We now present two specifications of our model that fail to satisfy assumptions 2 and 3 but, however, satisfy the assumption of conditional efficiency introduced in Section 7, on the initial BGP.
That is, in these models taking as given any aggregate allocation of labor between research and
production, Lrt = L̄rt and L pt = L̄ pt , the equilibrium and social planning allocations coincide on
the BGP. Whereas the equilibrium aggregate allocation of labor, L̄rt , may not solve the social planning problem (in which case there is a role for welfare-enhancing innovation policies that increase
or reduce L̄rt ) there are no welfare gains from reallocating innovative activities across heterogeneous firms. To simplify the presentation, we abstract from physical capital accumulation (i.e. we
set a ! 0). This does not alter the result that the equilibrium is conditional efficient as long as the
Euler equation of physical capital is undistorted (which requires a production subsidy to undo the
markup).
22
Quality ladders model with heterogeneous innovative technologies
In the quality ladders model developed in Klette and Kortum (2004), product leadership and
research capacity are tied together in the sense that if a product is innovated on, the firm owning
this product loses product leadership as well as the research capacity associated with this product.
In a version of that model extended to have firm heterogeneity in terms of innovation technology,
Lentz and Mortensen (2014) show that there is a welfare enhancing role for policies that reallocate
innovation expenditures across firms. The distortion arises because individual firms do not take
into account that their innovation decisions destroy research capacity of other firms. Hence, the
equilibrium is not conditionally efficient. The model example we present here is similar to that in
Lentz and Mortensen (2014) except that product leadership and research capacity are separated.
That is, if a product is innovated on, the firm owning this product loses product leadership (i.e.
there is business stealing) but not the research capacity associated with this product. We show
that such a model specification satisfies conditional efficiency.
A firm in this model is characterized by its technology to innovate, its research capacity, the
number of products it has leadership on and the productivity of each of these products. As we will
see, in order to characterize the aggregate dynamics we only need to record the firm’s innovative
technology type, which we denote by j. Underlying this aggregate economy there is a continuum
of firms whose full types including the number of goods that each produces evolve stochastically.
We describe the environment by setting up the aggregate constrained social planning problem:34
t
max •
t=0 b u (Ct ) subject to
Ct = Zt (1
h ⇣
Zt+1
= dt exp (Dz )r
Zt
dt = s
L̄rt )
1
⌘
i r11
1 +1
!
(77)
⌘
d f ( j) + g j (irt ( j)) Nt ( j)
(78)
 d j (yrt ( j)) Nt ( j)
j 1
⇣
Nt+1 ( j) = y j Nt (0) + 1
(76)
f 1
 (yrt ( j) + irt ( j)) Nt ( j) + ȳr (0) Nt (0) = Art Zt
L̄rt
(79)
j 1
Z0 , N0 ( j)and { L̄rt }are given.
Here, Nt ( j) denotes the aggregate mass of research capacity operated by firms of type j, which
depreciates at the exogenous rate d f ( j) and that can be increased in expectation by a fraction
g j (irt ( j)) if an investment of irt ( j) Nt ( j) units of the research good is undertaken, where g j (.) is an
34 Assumption
2 that the growth rate is maximized at time t requires irt ( j) = Nt (0) = 0, which is not the
case in the social planning problem or in the equilibrium.
23
increasing and concave function. yrt ( j) denotes investment per unit of research capacity of a firm
of type j in order to acquire a product with probability sd j (yrt ( j)) per unit of research capacity
and improve its productivity by a step size Dz ,35 where d j (.) is an
⇣ increasing and concave
⌘ function.
The measure of products that receive an innovation is dt = s  j 1 d j (yrt ( j)) Nt ( j)
probability that each of the Nt (0) entrants is of type j is yj , with  j yj = 1.36
 1. The
We now characterize the solution to this constrained (given L̄rt ) social planning problem. Assuming it exists, the Lagrangean is
•
 bt u (Zt (1
L̄rt ))
t =0
subject to (including the Lagrange multipliers)
lt bt :
 (yrt ( j) + irt ( j)) Nt ( j) + ȳr (0) Nt (0)
j 1
"
ct bt : Zt+1 = s
 d j (yrt ( j)) Nt ( j)
j 1
wt ( j) bt : Nt+1 ( j)
yj Nt (0)
The FOC with respect to Zt+1 is
ct + bu0 (Ct+1 ) (1
⇣
!
⇣
1
L̄rt+1 ) + lt+1 b (f
exp (Dz )r
f 1
Art Zt
1
L̄rt
⌘
1 +1
# r11
Zt
⌘
d f ( j) + g j (irt ( j)) Nt ( j)
1)
Yrt+1
Zt+2
+ c t +1 b
=0
Zt+1
Zt+1
with respect to Nt+1 ( j)
wt ( j ) =
⇣
blt+1 (yrt+1 ( j) + irt+1 ( j)) + b 1
+
b
r
1
ct+1 Zt+1
✓
Zt+2
Zt+1
with respect to irt ( j)
◆2
r
⌘
d f ( j) + g j (irt+1 ( j)) wt+1 ( j) +
⇣
sd j (yrt+1 ( j)) exp (Dz )r
1
1
⌘
lt = wt ( j) g0j (irt ( j))
35 Our
36 In
results go through if step size varies by firm type, as long as markups are equal across products.
the alternative modelhin which product market leadership
and research capacity are combined, coni
straint (77) is instead dt = s Nt (0) + Â j
Nt+1 ( j) = (1
1
d j (yrt ( j)) Nt ( j) , constraint (77) is instead
⇥
⇤
dt ) Nt ( j) + s yj Nt (0) + d j (yrt ( j)) Nt ( j)
and constraint (79) does not include irt ( j). It is straightforward to show in this case that the equilibrium is
not conditionally efficient because the equilibrium does not take into account that changes in Nt ( j) change
dt and thus Nt+1 ( j0 ).
24
with respect to yrt ( j)
lt =
1
1
r
✓
Zt+1
Zt
◆2
r
⇣
Zt ct sd0j (yrt ( j)) exp (Dz )r
1
1
⌘
and with respect to Nt (0)
lt ȳr (0) =
 y j wt ( j )
j 1
We re-write these first order conditions as
u0 (Ct+1 ) Ct+1
ct Zt lt Zt+1
+b
+ (f
lt lt+1 Zt
l t +1
l t wt ( j )
=
l t +1 l t
+
⇣
b (yrt+1 ( j) + irt+1 ( j)) + b 1
b
r
ct+1 Zt+1
1 l t +1
✓
Zt+2
Zt+1
◆2
1=
1=
1
r
1
✓
Zt+1
Zt
◆2
r
r
⇣
Zt ct 0
sd j (yrt ( j)) exp (Dz )r
lt
 yj
We consider a BGP in which Nt ( j) = N̄ ( j), Yrt = Ȳr ,
⇣
⌘
c̄ = Zlt ct t , v̄ = r c̄ 1 exp ( ḡz )2 r exp (Dz )r 1 1 , and
(h
1) ḡy
d f ( j) + g j (irt+1 ( j))
⌘w
t +1
( j) +
l t +1
1
1
wt ( j ) 0
g (irt ( j))
lt j
j 1
1 exp
ct+1 Zt+1 Zt+2
=0
lt+1 Zt+1
⇣
sd j (yrt+1 ( j)) exp (Dz )r
ȳr (0) =
b
1) bYrt+1 + b
1
wt ( j )
lt
wt ( j )
lt
lt
l t +1
Zt+1
Zt
u0 (Ct )Ct
u0 (Ct+1 )Ct+1
= w̄ ( j),
1
⌘
⌘
= exp( ḡz ) where ḡz =
g Ar
1 f,
=
= (1 + r̄ ) b, where 1 + r̄ =
= b̃ 1 . We can write the FOC conditions in the BGP as
(1 + r̄ ) w̄ ( j) =
⇣
ȳr ( j) + īr ( j) + 1
d f ( j) + g j īr ( j)
1 = g0j īr ( j) w̄ ( j)
⌘
w̄ ( j) + sd j (ȳr ( j)) v̄
1 = sd0j (ȳr ( j)) v̄
ȳr (0) =
 yj w̄ ( j)
(80)
(81)
(82)
(83)
j 1
We can use this system of equations to solve for v̄, w̄ ( j), ȳr ( j) and īr ( j). Given ȳr ( j) and īr ( j), we
use equation (76) to solve for N̄ (0) and given a level of L̄r and Art we use equation (79) to solve
for the level of Z̄t and the corresponding level of consumption in the BGP.
In what follows we show that ȳr ( j), īr ( j) and N̄ (0) coincide in the social planning problem
and in the equilibrium and when innovation policies are uniform for entrants and incumbents,
25
tt ( j) = t̄. Given L̄r , this also implies the same level of aggregate productivity and consumption.
Therefore, the BGP equilibrium is conditionally efficient.
The equilibrium value associated to a unit of research capacity owned by a firm of type j is
Vt ( j) = max
(1
yrt( j) ,irt ( j)
t̄ ) Prt (yrt ( j) + irt ( j)) +
d f ( j) + g j (irt ( j))
sd j (yrt ( j))
Vt+1 ( j) +
Ṽt+1
1 + Rt
1 + Rt
1
where Ṽt+1 denotes the expected value of acquiring a product,
Ṽt+1 =
ˆ
Ṽt+1 (z + Dz ) dMt (z)
and Ṽt (z) denotes the value of being a leader in product z,
Ṽt (z) = Pt exp (z)r
where Pt = µ
1 r
1)Wt
r (µ
(1 + R t )
(1+r t
Prt
Prt+1 ,
Vt ( j)
,
1 )(1 t̄ ) Prt
t̄ ) Prt ȳr (0) =
vt (z) =
where
(1 + r t
1 ) wt
(1+r t
1
1 + Rt
Ṽt (z)
,
1 )(1 t̄ ) Prt
 yj Vt+1 ( j)
vt =
(1+r t
Ṽt
1 )(1
t̄ ) Prt
, pt =
Pt
(1 t̄ ) Prt
and 1 + rt =
(z) = pt exp (z)r 1 + (1 dt ) vt+1 (z)
ˆ
vt+1 = vt+1 (z + Dz ) dMt (z)
1 ) vt
ȳr (0) =
1
(1 dt )
Ṽt+1 (z)
1 + Rt
( j) = max
(yrt ( j) + irt ( j)) + ...
yrt( j) ,irt ( j)
⇣
⌘
+ 1 d f ( j) + g j (irt ( j)) wt+1 ( j) + sd j (yrt ( j)) vt+1
(1 + r t
In the BGP, 1 + r̄ = b̃
+
Yt . The free entry condition is
(1
Let wt ( j) =
1
 y j w t +1 ( j )
and the rescaled value functions are constant over time. The equilibrium
levels of ȳr ( j) , īr ( j), w̄ ( j), and v̄ are the solution to
(1 + r̄ ) w̄ ( j) = max
yr ( j),ir ( j)
⇣
(yr ( j) + ir ( j)) + 1
ȳr (0) =
⌘
d f ( j) + g j (ir ( j)) w̄ ( j) + sd j (yr ( j)) v̄
 yj w̄ ( j)
This system of equations gives the same solution for ȳr ( j) and īr ( j) as the system of equations
in the social planning problem given by (80)-(83). Moreover, given the same aggregate alloca-
26
tion of labor, we obtain the same BGP levels of aggregate productivity and consumption in the
equilibrium and social planning problems. Of course, the equilibrium allocation of labor between
production and research may be sub-optimal, which can be fixed with uniform innovation subsidies.
Expanding varieties model with heterogeneous innovation technologies
This model example builds on our model example 5 above, adding cross-firm heterogeneity in
innovation technologies, as in the second model in Luttmer (2011). Firms are characterized by
their innovative technology j
1 and the number of products they own and operate all of which
have productivity exp (z) = j j . Consider a firm that in period t has innovative technology j, owns
n products all of which have productivity exp (z) = j j , and invests yrt n units of the research good.
In period t + 1, this firm will have innovative technology j0 with probability yjj0 and will own an
expected number of d j (yrt )n products all of which have productivity exp (z) = j j0 . The function
d (.) is increasing and concave, and we do not impose here that d (0) = 0. Firms exit when the
number of products they own falls to zero. We denote by yrt ( j) the investment per product of
firms with innovative technology j, and by Nt ( j) the measure of products operated by firms with
innovative technology j. A measure Nt (0) of entrants expends ȳr (0) units of the research good
each to have innovative technology j with probability y0j and create s new products in expectation
each operated with productivity exp (z)r
1
= jj.
The social planning problem is
t
max •
t=0 b u (Ct ) subject to
Ct = Zt (1
Zt =
"
L̄rt )
 j j Nt ( j)
j 1
# r11
(84)
Nt+1 ( j) = y0j Nt (0) + Â yj0 j d j0 (yrt ( j0 )) Nt j0
(85)
j0
 yrt ( j) Nt ( j) + ȳr (0) Nt (0) = Art Zt L̄rt
f
(86)
j 1
N0 ( j) and { L̄rt } are given.
In the BGP, Nt ( j) grows at the rate (r
1) ḡz for all j , and ḡz =
37 It
g Ar
37
r 1 f.
We now characterize the
is straightforward to transform variables so that in the BGP Yrt is constant and redefine parameters
so that in the BGP grows at the rate g Ar / (1 f) as in our baseline framework. Our results are unchanged
under this redefinition of variables.
27
solution to the social planning problem. Assuming it exists, the Lagrangean is
•
 bt u (Zt (1
t =0
L̄rt ))
subject to (including the Lagrange multipliers)
lt bt :
 yrt ( j) Nt ( j) + ȳr (0) Nt (0) = Art Zt L̄rt
f
j 1
ct bt : Zt
wt ( j) bt : Nt+1 ( j)
"
 j j Nt ( j)
j 1
# r11
Â0 yj0 j d j0 (yrt ( j0 )) Nt
y0j Nt (0)
j0
j
The FOC with respect to Zt is
ct = u0 (Ct ) (1
L̄rt ) + lt f
Yrt
Zt
with respect to Nt+1 ( j)
wt ( j ) =
bZt+1
c t +1 j j
r 1
blt+1 yrt+1 ( j) + bd j (yrt ( j)) Â yjj0 wt+1 j0
j0
with respect to yrt ( j)
lt = d0 j (yrt ( j)) Â yjj0 wt j0
j0
and with respect to Nt (0)38
lt ȳr (0) =
 y0j wt ( j)
j 1
We re-write these first order conditions as
ct Zt
u0 (Ct ) Ct
Yrt
=
+f
lt
lt
lt Zt
l t wt ( j )
bZt+1 ct+1
=
jj
l t +1 l t
r 1 l t +1
byrt+1 ( j) + bd j (yrt ( j)) Â yjj0
j0
w t +1 ( j 0 )
l t +1
yjj0 wt ( j0 ) = ȳ 1(0) Â j0 1 y0j wt ( j0 ). Assumption
r
2 at time t requires that {yrt ( j)} and Nt (0) maximize Zt+1 subject to the research good constraint, which
requires d0j (yrt ( j)) Â j0 1 yjj0 j j0 = ȳ 1(0) Â j0 1 y0j0 j j0 . This condition is violated both in the planners problem
38 Combining the last two conditions implies d0
r
and in the equilibrium unless
wt ( j0 )
wt ( j )
=
j j0
jj
j ( yrt ( j )) Â j0
which in general does not hold.
28
1 = d0 j (yrt ( j)) Â yjj0
j0
ȳr (0) =
 y0j
j 1
In a BGP,
1 + r̄ = b
ct Zt
=
( r 1) l t
1 exp h
(
p̄,
wt ( j )
lt
1) ḡy
wt ( j 0 )
lt
wt ( j )
lt
= w̄ ( j), yrt ( j) = ȳr ( j) and
= b̃ 1 , where
(1 + r̄ ) w̄ ( j) = p̄ j j
lt
l t +1
=
u0 (Ct )Ct
u0 (Ct+1 )Ct+1
= (1 + r̄ ) b, where
ȳr ( j) + d j (ȳr ( j)) Â yjj0 w̄ j0
j0
1 = d0 j (ȳr ( j)) Â yjj0 w̄ j0
j0
ȳr (0) =
 y0j w̄ ( j)
j 1
We can use this system of equations to solve for w̄ ( j) and ȳr ( j). Given ȳr ( j) we can solve for
the mass of products by type normalized by the level of entry, N̄ ( j) / N̄ (0), using equation (85),
which can be expressed as
exp ( ḡz )r
1
N̄ ( j)
N̄ ( j0 )
= y0j + Â yj0 j d j0 (ȳr ( j0 ))
N̄ (0)
N̄ (0)
j0
Given a level of L̄r and Art we solve for the level of Z̄t and N̄t (0) re-writing equations (84) and
(86) as
N̄ ( j)
 ȳr ( j) N̄ (0) + ȳr (0) =
j 1
1
f
Art Z̄t L̄rt
N̄ (0)
"
N̄ ( j)
Z̄t = N̄t (0) Â j j
N̄ (0)
j 1
# r11
and finally we solve for consumption.
In what follows we show that the allocations level of ȳr ( j) for all j coincide in the equilibrium
and social planning problem when innovation policies are uniform, tt ( j) = t̄. Given L̄r , this also
implies the same level of entry, aggregate productivity and consumption. The equilibrium value
associated with a product owned by a firm with innovation technology j is
Vt ( j) = maxPt j j
yrt ( j)
(1
t̄ ) Prt yrt ( j) +
29
d j (yrt ( j))
1 + Rt
Â0 yjj0 Vt+1
j
j0
where Pt = µ
1 r
1)Wt
r (µ
Yt and uniform innovation policies tt ( j) = t̄. The associated FOC is
(1
t̄ ) Prt =
d0j (yrt ( j))
1 + Rt
Â0 yjj0 Vt+1
j0
j
and the free entry condition is
(1
Defining wt ( j) =
functions as
(1+r t
Vt ( j)
, pt
1 )(1 t̄ ) Prt
(1 + r t
1 ) wt
1
1 + Rt
t̄ ) Prt ȳ (0) =
=
Pt
(1 t̄ ) Prt
( j) = pt j j
 y0j Vt+1 ( j)
j
and 1 + rt = (1 + Rt )
Prt
Prt+1 ,
we can write the value
yrt ( j) + d j (yrt ( j)) Â yjj0 wt+1 j0
j0
where
1 = d0j (yrt ( j)) Â yjj0 wt+1 j0
j0
and the free entry condition
ȳ (0) =
 y0j wt+1 ( j)
j
In a BGP, 1 + r̄ = b̃
1
and the rescaled value functions are constant over time.
(1 + r̄ ) w̄ ( j) = p̄ j j
1=
yrt+1 ( j) + d j (ȳr ( j)) Â yjj0 w̄ j0
j0
1
d0 (ȳr ( j)) Â yjj0 w̄ j0
(1 + r̄ ) j
j0
ȳr (0) =
 y0j w̄ ( j)
j 1
This system of equations is identical to the respective one in the social planning problem and hence
both give the same solution for ȳr ( j). Moreover, given the same aggregate allocation of labor, we
obtain the same BGP levels of entry, aggregate productivity and consumption in the equilibrium
and social planning problems. Of course, the equilibrium allocation of labor between production
and research may be sub-optimal, which can be fixed with uniform innovation subsidies.
30