Online Appendices

D Equilibrium and calibration of model examples

In the body of the paper we presented a partial characterization of the equilibrium conditionsthat 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 considermodel example 2 (which nests model example 1) in great detail. We then consider, with less detailsince the key steps are quite similar across models, example 4 (which nests model example 3) andmodel example 5.

D.1 Quality ladders model with innovation by incumbents (example2)

Equilibrium In all the models we consider, variable profits that an incumbent firm earns in pe-riod t from production of a product with productivity exp (z) are given by [pt(z)yt(z)�Wtlt(z)�Rktkt (z)].Marginal cost is MCtexp (�zi (j)), where MCt = a

�a (1 � a)a�1 Ra

ktW1�a

t . The incumbent firm thatowns this product chooses price and quantity, pt (z) and yt (z), to maximize these variable profitssubject to the demand for its product and the production function (3). With Bertrand competitionand limit pricing, the gross markup µ charged by the incumbent producer of each product is theminimum of the monopoly markup, r/ (r � 1), and the technology gap between the leader andany potential second most productive producer of the good, exp (Dl), which potentially dependson the patent system. That is µ = min

n

r

r�1 , exp (Dl)o

. Variable profits from production can thenbe written as Pt exp(z)r�1, with the constant in variable profits Pt defined by

Pt = k0

⇣

Ra

ktW1�a

t

⌘1�r

Yt, (51)

with k0 = µ

�r(µ � 1)h

a

a (1 � a)1�a

i

r�1. In all the models we consider, we denote the subsidy to

innovation by incumbent firms as t

It and the subsidy to innovation by entering firms by t

Et . 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 theprobability that the firm loses its ownership of this product to another innovating firm

Vt(z) = Pt exp(z)r�1 +(1 � dt)1 + Rt

Vt+1 (z) ,

where dt denotes the total measure of products innovated on at time t (which every firm takes asgiven), Rt denotes the interest rate denominated in the final consumption good, which with CRRA

1

preferences is given by 1 + Rt = b

�1 (Ct+1/Ct)h . Integrating this expression across z and using

the definition of Zt, we have

VtZr�1t = PtZ

r�1t +

(1 � dt)1 + Rt

Vt+1Zr�1t , (52)

where Vt = Vt (0). The second component of the value to a firm of owning a product is givenby the expected present value of dividends the incumbent firm expects to earn on new productsit gains through innovation minus the cost of that innovation. Given that this value dependson the number of products owned by the firm, n, and not on its productivities, we denote it byUt (n). Conjecturing that Ut (n) = Utn and yrt (n) = yI

rtn (we verify this below), the value Ut isdetermined by the Bellman equation

Ut = maxyI

rt

�⇣

1 � t

It

⌘

yIrtPrt +

11 + Rt

sd⇣

yIrt

⌘

Vt+1Zr�1t exp (Dz)

r�1 (53)

+1

1 + Rt

⇣

sd⇣

yIrt

⌘

+ 1 � dt

⌘

Ut+1.

The first term on the right side indicates the firm’s investment in innovation. The second termindicates the discounted present value of variable profits the firm expects to gain from the inno-vations that result from this investment. The third term denotes the expected value of the firm’sinnovative capacity from next period on, taking into account both the gain in products it expectsto obtain from its innovative effort (i.e., a firm with n products expects to gain sd

�

yIrt�

n products)and the loss of products it expects as a result of innovative effort from other firms (i.e., a firm withn products expects to lose dn products). Note that in equilibrium, if the incumbents’ innovationtechnology is used, we must have Ut � 0. Otherwise, incumbents would choose not to use theirinnovation technology at all. Hence, given sufficiently low values of the function d (.) such thatincumbents 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 byincumbent firms, yI

rt (taking dt as given) is given by

⇣

1 � t

It

⌘

Prt =

sd0�

yIrt�

1 + Rt

!

⇣

Vt+1Zr�1t exp (Dz)

r�1 + Ut+1

⌘

. (54)

Since none of the terms depend on the incumbent firm’s number of products n or the productivi-ties with which the firm can produce those products, we verify our conjecture that that per-productinnovation spending, yI

rt, is the same for all products and firms.31 The total measure of products

31One can show that the property that all incumbent firms choose the same yIrt extends in an alterna-

tive specification of our model in which each product that is innovated on draws a random markup andinnovation step size that is independent of the identity of the innovator. Under this extension, consistentwith the data as emphasized in Klette and Kortum (2004), the model generates persistent variation in laborrevenue productivity across firms and in research intensity (defined as innovation expenditures relative torevenues). In particular firms that have, on average, products with higher markups have higher measured

2

innovated on is dt = s

�

d�

yIrt�

+ Nt (0)�

.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, Âni=1 Vtexp (zi)

r�1 plus the value of its innovativecapacity, Utn. The free entry condition for new firms is given by

⇣

1 � t

Et

⌘

Prty (0) �✓

s

1 + Rt

◆

⇣

Vt+1Zr�1t exp (Dz)

r�1 + Ut+1

⌘

, (55)

with this condition holding with equality if the measure of entering firms, Nt (0), is greater thanzero. If the equilibrium has entering firms, then the zero-profit condition for entry (55) and equa-tion (54) imply that incumbents’ innovation per product is determined from

�

1 � t

It�

d0�

yIrt� =

⇣

1 � t

Et

⌘

yr(0). (56)

This result implies that in any equilibrium with positive entry and uniform innovation subsidies,d0�

yIrt�

yr(0) = 1, which coincides with the condition to maximize the current growth rate subjectto 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 =VtZ

r�1t

Prt, ut =

UtPrt

, we have,

vt =PtZ

r�1t

Prt+

(1 � dt)1 + rt

vt+1

exp (gzt)r�1 , (57)

ut = �⇣

1 � t

It

⌘

yIrt +

11 + rt

1exp (gzt)

r�1 sd⇣

yIrt

⌘

vt+1 exp (Dz)r�1 (58)

+1

1 + rt

⇣

sd⇣

yIrt

⌘

+ 1 � dt

⌘

ut+1

and the free entry condition is

⇣

1 � t

Et

⌘

yr (0) �✓

s

1 + rt

◆

vt+1

exp (gzt)r�1 exp (Dz)

r�1 + ut+1

!

, (59)

where 1 + rt = (1 + Rt)Prt

Prt+1. In a BGP, r = 1+R

exp(gy)� 1 = b

�1 � 1 (since Prt rises at a rate gy).

In a BGP, rescaled profits PZr�1/Pr, and values VZr�1/Pr and U/Pr are constant over time(as well as r, gz and d as discussed above). In the semi-endogenous growth case (f < 1) therate at which innovations occur, d, is pinned down from equation (45). Innovation by incumbentfirms yI

r , 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 prod-uct, measured labor productivity and research intensity are not correlated with firm growth in terms of itsnumber of products. This extension of the model does not change substantially our main results.

3

by N (0) = d

s

� yIr . In what follows we focus on parameter values such that on the BGP, new

firms enter (N (0) > 0 when calculated as described above) and incumbents choose to use theirinnovation technology. Incumbent firms find it optimal to use their innovation technology in aBGP with firm entry (u > 0) if and only if

�

1 � t

I� yIr

d(yIr)

�

1 � t

E� yr (0). By equation (56), this

condition is equivalent to d0�

yIr�

<d(yI

r)yI

r, which is satisfied if d(.) is concave and d (.) = 0. The

constant PZ/Pr 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⌘

yr (0) N (0) =1

1 + rd

⇣

d(yIr)

N(0) + 1⌘

y

PZr�1

Pr�⇣

1 � t

I⌘

yr

�

, (60)

where y =r + d

r exp (gz)r�1 /exp (Dz)

r�1 + d

� 1

Given the value of PZ/Pr that solves this equation, the allocation of labor Lp/Lr is determined asa function of parameters using

Lp

Lr=

✓

1 � a

µ � 1

◆

PZPr

1yI

r + yr (0) N (0). (61)

The level of aggregate productivity Z, for a given current value of scientific knowledge Ar, isdetermined using equation (8). When f = 1, one can use the same procedure but instead ofsolving for Z (the BGP level of Z, given Ar, is not pinned down), one must solve for the growthrate gz (and the corresponding values of d, yI

r and N (0)). The innovation intensity of the economy,sr, is calculated as a function of parameters using expression (12). Finally, we solve for aggregateoutput, Yt, using (4), the stock of physical capital, Kt, using the factor shares of physical capitaland production-labor, and consumption, Ct, using (1).

In Sections 4 and 6 we approximated the aggregate transition taking as given the transitionpath for the innovation intensity of the economy and the ratio of physical capital to output. Tosolve for the path of these two variables for a given change in policies or other parameters, wesolve the model numerically. Specifically, we solve for the path of Zt, Kt, vt, and ut using the fourfollowing Euler equations: (57), (58), (59) and Rkt = dk + Rt where Rt = b

�1⇣

Ct+1Ct

⌘

h

� 1. Recallthat yI

r is solved for using equation (56) assuming that there is positive firm entry (which must bechecked). Given a path of Zt and Kt we can solve for all other equilibrium outcomes using staticequations. We solve for the 4 Euler equations using either standard linearization methods or ashooting algorithm, and we obtain very similar results.

Calibration We now describe a model calibration that rationalizes our parameter choices inSection 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 t

I = t

E. As a baseline calibration, we set t

I = t

E = 0, but we indicate inthe calibration formulas below where these policies enter.

Interest rate minus growth rate: The consumers’ adjusted discount factor b = bexp�

(1 � h) gy�

is equal to the gap between the consumption interest rate and the growth rate of consumption inthe BGP, which we set to 0.01. Therefore, b = 0.99 and the interest rate in terms of the researchgood in the BGP is given by r = b

�1 � 1 = 0.01. We set the growth rate of consumption togy = .02, which implies a consumption interest rate of R = 0.03. For the exercises in which wecalculate transition dynamics exactly, we assume h = 1.

Final Consumption Good Production Function: The production function for the final consumptiongood is parameterized by r, which controls the elasticity of demand curve faced by intermediategoods producers and establishes an upper bound on the markup µ that can be chosen. We setr = 4.

Equilibrium Markup: Defining NIPA profits to intangible capital (exclusive of taxes and subsidies)as GDPt � RktKt � WtLt, the share of these profits in GDP, denoted by pt, is given by

pt =

✓

1 � 1µ

◆

� srt. (62)

From equation (62), the choice of the markup µ is disciplined by data on the innovation intensityof the economy, sr, and the share of NIPA profits paid to intangible capital relative to GDP. In ourbaseline 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% similarto the levels estimated by Corrado et al. (2009) for the United States over the last few years. Thisimplies 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 tophysical capital on the BGP, given by a

µ

, is equal to 0.33. With this choice of a, we also havethat the share of labor compensation (including production and research) in GDP is given by1�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 innovationintensity of the economy, sr. In our baseline calibration, with sr = .11, we have Lp = .833.

BGP Growth Rates for Scientific Knowledge and Aggregate Productivity: Given our calibration ofper capita GDP growth of 2% and our physical capital share of a, we calibrate the growth rateof aggregate productivity in the BGP, gz, to 1.25%. For a given choice of f, the growth rate ofscientific knowledge consistent with these productivity growth rates is given by gAr = (1 � f) gz.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 theBGP innovation rate d 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 sizeexceeds the markup (Dz � µ) to ensure that an incumbent firm has a technological advantageover its latent competitor consistent with its assumed markup. In our baseline calibration, we setDz = Dl = µ. Given our choice of elasticity r and the implied value of gz, from equation (45), weobtain d = 0.08 in our baseline calibration.

Ratio Yr/(Yr �Y0r ): Under the assumptions on policies made above, the ratio Yr

Yr�Y0r= yI

r+yr(0)N(0)yr(0)(d(yI

r)+N(0))is already determined by parameters that we have previously calibrated and by the choice of tar-gets for sr and p that have been already discussed. To see this, combining the free-entry condition(60) and the following expression for p/sr (which holds in all of our model examples)

p

sr=

PZr�1

Pr

1yr(0)N(0) �

⇣

yIr

yr(0)N(0) + 1⌘

⇣

yIr

yr(0)N(0) + 1⌘ (63)

we obtain

yIr

yr (0) N(0)=

1�t

E

1�t

I

✓

1 + rd

✓

d(yIr)

N(0) + 1◆◆

� x

1�t

I

⇣

1 + p

sr

⌘

y

1�t

I

⇣

1 + p

sr

⌘

� 1.

Setting t

I = t

E = 0, we obtain

Yr

Yr � Y0r=

yIr

yr(0)N(0) + 1d(yI

r)N(0) + 1

=rd

y

⇣

1 + p

sr

⌘

� 1. (64)

If there is no innovation by incumbents (as in our model example 1), then yIr

yr(0)N(0) =d(yI

r)N(0) = 0,

which implies YrYr�Y0

r= 1. In our baseline calibration with incumbents’ innovation, we obtain

YrYr�Y0

r= 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 thisvalue of Yr

Yr�Y0r

, the impact elasticity G0 calculated using equation (36) is equal to 0.0099, which isvery close to the value of 0.0102 that we use in our numerical examples. If we lower our target sr,increase p, or reduce r (the gap between the interest rate and the growth rate), then Yr

Yr�Y0r

and G0

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 YrYr�Y0

r, which has already been determined

in our baseline calibration as discussed above. For the numerical evaluation of our results (ei-ther non-uniform policy changes or relaxing the assumption of conditional efficiency in the initialBGP), we assume that d

�

yIr�

= d0�

yIr�d1 . We choose the level of d (.) in the initial equilibrium,

6

d�

yIr�

, as follows. The share of employment by entering firms is given by

se =sN (0) exp (Dz)

r�1

exp(gz)r�1 .

Given a choice of se (we pick se = 0.03, consistent with data from the LBD), we solve for sN (0) andthen solve for sd

�

yIr�

using d = s

�

d�

yIr�

+ N (0)�

. In our baseline parameterization, the fractionof research expenditures carried out by entrants (calculated using equation (64)) is yr(0)N(0)

yIr+yr(0)N(0) =

0.27. Using (56) we can solve for y (0) d0�

yIr�

. In any calibration of our model, one must check thatincumbents want to use their innovation technology (i.e. U > 0).

Other parameters: The parameters yr (0), s, and Ar at time 0 can all be normalized to 1 withoutaffecting our results.

Full numerical solution In the body of the paper we have presented an analytical character-ization of the transition dynamics of our model, up to a first-order approximation, following apolicy-induced change in the innovation intensity of the economy. In that approximation, we tookas given the transition path for the innovation intensity of the economy and the ratio of physicalcapital to output. In this section we use the calibrated baseline Klette-Kortum model describedabove to solve numerically for the full transition dynamics of the economy following a permanentchange in innovation innovation subsidies. We use this solution to evaluate the usefulness of ouranalytical approximation.

We first consider an unanticipated and permanent increase in uniform innovation subsidiesapplied to entering and incumbent firms from t (j) = 0 to t (j) = t

0 for all j � 0. With sucha policy change, the level of investments in innovation per product by incumbents, yI

r , remainsconstant over time (see equation (56)) and the increase in innovation along the transition arisesfrom an increase in entry only (this is the variation considered when introducing Assumption 1a).We choose the level of t

0 so that, in the long run, the innovation intensity of the economy rises fromsr = 0.11 to sr = 0.14, so that log s0r � log sr = .24. In Appendix F we show that under a number ofassumptions (satisfied in all of our model examples) changes in innovation subsidies uniformlyapplied to entering and incumbent firms result in the long-run in a change in fiscal expendituresrelative to GDP of E0

¯GDP0 � E¯GDP = t

0 s0r � tsr = s0r � sr. Therefore, the policy-induced increase in theinnovation intensity that we consider would require a recurring fiscal expenditure on innovationsubsidies equal to 3 percent of GDP in the long-run, roughly equal to the total revenue collectedfrom corporate profit taxes relative to GDP in 2007 in the U.S. In this sense, we regard this as alarge change in the innovation intensity of the economy.

We report results from this experiment in Figures 1 and 2 for the transition dynamics of ourmodel economy over the first 100 years and 20 years, respectively. We display in the left panels ofthese figures the dynamics resulting from our first order approximation in Section 6, assuming thaton the transition path the physical capital to output ratio is constant at its BGP level, and that the

7

Figure 1: 100-year Transition Dynamics to a Permanent Increase in Uniform InnovationSubsidies, Approximation and Full Numerical Solution, baseline Klette-Kortum model(example 2)

innovation intensity of the economy jumps to its new BGP level immediately, i.e. log s0rt � log sr =

.24 for all t � 0. The right panels of these figures display the dynamics resulting from the fullnumerical solution of the model.

In panel B of both figures we show the dynamics of log srt over the first 100 years of the transi-tion. Here we see that there are mild intertemporal substitution effects in innovation expendituresin that this innovation intensity rises by a bit more than .24 in log terms in the early phase of thetransition, particularly for the case with low intertemporal knowledge spillovers. The intertem-poral 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 Eto the fully simulated path shown in panel F, we can see that the approximation is fairly accuratedespite 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 InnovationSubsidies, 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 thetransition. This transition path corresponds to the negative of the transition path for the log ofthe rental rate on physical capital. Clearly, there are dynamics of this ratio that we ignored inour analytical approximation. From expression (30), we have that the full dynamics of GDP areimpacted both by the dynamics of the innovation intensity of the economy and of the ratio ofphysical capital to output. On impact, these two factors have opposite effects on GDP — theinitial increase in the ratio of physical capital to output raises GDP while the initial increase in theinnovation 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, wesee that the increase in GDP over the first 20 years of the transition is roughly 1.5% lower when wetake into account the dynamics in the innovation intensity of the economy and the capital-outputratio.

Figure 2 shows that a large and persistent increase in innovation subsidies has a relativelysmall impact on aggregate productivity and GDP over a 20 year horizon, and the response ofaggregates does not vary much with the extent of intertemporal knowledge spillovers assumedin the model. These results suggest that it would be hard to verify whether innovation policiesyield large output and welfare gains using medium term data on the response of aggregates tochanges in innovation policies. We illustrate this point in Figure 3. In that figure we show resultsobtained from simulating the response of aggregates in our model to our baseline increase ininnovation subsidies in an extended version of our model with Hicks neutral AR1 productivityshocks with a persistence of 0.9 and an annual standard deviation of 2%. We introduce theseshocks as a proxy for business cycle shocks around the BGP. We show histograms generated from3000 simulations of the model for the first 20 years of the transition. The units on the horizontalaxis show the log of the ratio of detrended GDP at the end of the 20th year of transition to initialGDP and the vertical axis shows the frequency of the corresponding outcome for GDP. In panel Aof the figure, we show results for GDP excluding innovation expenditures. In panel B, we showresults for GDP including innovation expenditures, GDPt ⇥ (1 + srt). The red bars show resultsfor the model with low intertemporal knowledge spillovers and the blue bars show the resultswith high spillovers. We can observe in each panel that the distribution represented by the bluebars is slightly to the right of that represented by the red bars. The histograms in panel B areshifted to the right relative to those in panel A reflecting the fact that GDP including innovationexpenditures has a higher elasticity of changes in the innovation intensity of the economy. Butit is also clear in each panel that, using either measure of GDP, that it would be very hard todistinguish the degree of intertemporal knowledge spillovers (and, hence, the long term effectsfrom this innovation subsidy) in aggregate time series data even if we had the benefit of a truepolicy experiment.

Up to this point, we have considered policy experiments in which the economy starts on aBGP with uniform innovation subsidies and transitions to a new BGP with a new rate of uniforminnovation 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 theadditional units of the research good are allocated to entrants or incumbent firms (Lemma 1). Theassumption that innovation subsidies are uniform along the transition implies that the temporaryincrease in innovation is achieved through an increase in entry, and innovation per product byincumbents remains unchanged (see equation (56)). We now evaluate whether there are impor-tant second-order effects that arise when large non-uniform changes in innovation policies areconsidered 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 Sub-sidies, Including Productivity Shocks, Klette-Kortum model (example 2)

Specifically, we consider a permanent increase in the innovation subsidy to incumbents thatresults in a long run increase in the innovation intensity of the economy from sr = .11 to s0r = .14.32

The curvature parameter of the incumbents’ innovation technology, d(.), which shapes the changein investments by incumbents yr along the transition, is set at 0.4 consistent with the estimates inAcemoglu et al. (2013). In Figure 4 we show the evolution over the first 20 years of the log of theinnovation intensity of the economy, the log of the physical capital output ratio, aggregate produc-tivity, 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 ofaggregate productivity and GDP in the first 20 years of the transition are not substantially differentin these two cases.

32In the long run, this subsidy requires fiscal expenditures of 3.3% relative to GDP rather than 3% underour 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 incumbentsto improve their own products (example 4)

We describe some key steps in solving the equilibrium and calibrating this model. The value of afirm with productivity exp (z) is given by

Vt (z) = Ptexp (z)r�1 �⇣

1 � t

It

⌘

Prtyrt +1 � d f

1 + Rt

d0 + d

yrt

✓

Zt

exp (z)

◆

r�1!!

Vt+1 (z + Dz)

+1 � d f

1 + Rt

1 � d0 � d

yrt

✓

Zt

exp (z)

◆

r�1!!

Vt+1 (z � Dz)

12

where Pt is defined as in (51) where µ = r/ (r � 1). It is straightforward to show that Vt (z) =

Vtexp (z)r�1 and yrt (z) = yIrt

⇣exp(z)Zt

⌘

r�1where

Vt = Pt �⇣

1 � t

It

⌘ PrtyIrt

Zr�1t

+1 � d f

1 + RtVt+1

h⇣

d0 + d⇣

yIrt

⌘⌘

exp (Dz)r�1 +

⇣

1 � d0 � d⇣

yIrt

⌘⌘

exp (�Dz)r�1i

and yIrt satisfies

⇣

1 � t

It

⌘

Prt =1 � d f

1 + Rtd0⇣

yIrt

⌘

Zr�1t Vt+1

⇣

exp (Dz)r�1 � exp (�Dz)

r�1⌘

With positive firm entry, we must have

⇣

1 � t

Et

⌘

Prtyr (0) =sZr�1

t Vt+1

1 + Rt

In a BGP, yIr , VtZ

r�1t

Prt(which we denote by vt), and PtZ

r�1t

Prtare constant over time, satisfying

✓

1 � t

E

1 � t

I

◆

H0⇣

yIr

⌘

yr (0) = a1 (65)

v =PZr�1

Pr��

1 � t

I� yIr

1 � a0+F(yIr)

(1+r)exp(gz)r�1

(66)

⇣

1 � t

E⌘

yr (0) =a1v

(1 + r) exp (gz)r�1 , (67)

where we use the definitions above when describing this model example,

a0 =�

1 � d f�

⇣

d0exp (Dz)r�1 + (1 � d0) exp (�Dz)

r�1⌘

,

a1 = s, F�

yIr�

=�

1 � d f�

⇣

exp (Dz)r�1 � exp (�Dz)

r�1⌘

d�

yIr�

, and r = (1+R)exp(gy)

� 1 = b

�1 �1. With uniform innovation policies in the initial BGP, we obtain condition (47), which is thecondition that satisfies Assumption 2.

In order to solve the model’s BGP given all parameter values and f < 1, we obtain yIr from

(65), N (0) =h

exp (gz)r�1 � a0 � F

�

yIr�

i

/a1 (where gz = gAr / (1 � f)), PZr�1

Prfrom (65) and (66),

Lp/Lr from (61), and Z (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 isshaped by the share of entering firms in employment relative to their share in research expendi-tures (equation (37)), we now target these two shares in our calibration. There is a simple equilib-rium 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

growth rate r (in Example 2 we targeted these three statistics). Specifically, combining (63), (65),and (66), the definition of the share of employment in entering firms

se =a1Nt (0)

exp (gz)r�1 = 1 �

�

a0 + F�

yIr��

exp (gz)r�1

and imposing t

I = t

E = 0 we obtain

se

yr (0) N (0) /Yr=

rsr

p

. (68)

Given our choice of sr = 0.11, r = 0.01, se = 0.03, and yr(0)N(0)Yr

= 0.2, equation (68) implies thatp = 0.0073, which is slightly lower than our target p = 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 Yr

Yr�Y0r

and G00 as follows given targets for gz, se

and sre = yr (0) N (0) /Yr. We solve for a0 + F�

yIr�

and a1N (0) from

a0 + F⇣

yIr

⌘

= exp (gz)r�1 (1 � se)

a1N (0) = exp (gz)r�1 se.

We solve for a1yr(0)

yIr as

a1

yr (0)yI

r =

✓

1 � sre

sre

◆

a1N (0) .

Using (65), we have F0 �yIr�

yIr = a1

yr(0)yI

r . Assuming F�

yIr�

= f0�

yIr� f1 , where f0 > 0 and f1 1

are given parameters, we have F�

yIr�

= 1f1

F0 �yIr�

yIr . Therefore, a0, which determines the social

depreciation of innovation expenditures, is given by

a0 = exp (gz)r�1

1 �✓

1 +1f1

✓

1 � sre

sre

◆◆

se

�

.

Social depreciation is larger (i.e. a0 is lower) the higher is se, the lower is sre, and the lower is thecurvature parameter f1. Finally, we have

Yr

Yr � Y0r=

f1

1 � sre (1 � f1).

As f1 ! 1, YrYr�Y0

r! 1.

14

D.3 Simple Expanding Varieties model with innovation by incumbentsto create new products (example 5)

The solution and calibration of this model is very similar to Example 4. The value of a firm with nproducts is Vt (n) = Vtn, where

Vt = Pt �⇣

1 � t

It

⌘

PrtyIrt +

11 + Rt

⇣

d0 + d⇣

yIrt

⌘⌘

Vt+1,

and Pt = k0

⇣

Ra

ktW1�a

t

⌘1�r

Ytexp (qt)r�1 denotes variable profits per product. The FOC for thelevel of innovation per product, yI

rt, is

⇣

1 � t

It

⌘

Prt =1

1 + Rtd0⇣

yIrt

⌘

Vt+1. (69)

The free-entry condition is⇣

1 � t

Et

⌘

Prtyr (0) =sVt+1

1 + Rt. (70)

Combining (69) and (70) we obtain

✓

1 � t

It

1 � t

Et

◆

s

yr (0)= d0

⇣

yIrt

⌘

.

With uniform policies in the initial BGP, we obtain s

yr(0)= d0

�

yIrt�

, satisfying Assumption 2.We define vt =

VtPrt

, which in the BGP is constant and given by

v =PPr��

1 � t

I� yIr

1 � d0+d(yIrt)

(1+r)exp(gz�q)r�1

where 1 + r = (1+R)exp(gy)

. In deriving this expression we used the fact that in the BGP Prt+1/Prt =

exp�

gy�

exp (gz � q)1�r since srt =PrtYrt

Ytis constant and Yrt = Yrt Mt grows at the rate (r � 1) (gZ �

q). Using these results and gz = q + 1r�1 log

�

d0 + d(yIr) + sN(0)

�

, we can write the free entrycondition in BGP as

⇣

1 � t

E⌘

yr (0) N (0) =exp (gz � q)r�1 � d0 � d

�

yIr�

exp (gz � q)r�1 (1 + r)� d0 � d�

yIrt�

✓

PPr

�⇣

1 � t

I⌘

yIr

◆

. (71)

Given these expressions, we solve for the equilibrium following very similar steps to those inexample 2.

The procedure to calibrate this model example is also very similar to that of model example3. To calibrate the model, we make use of equation (68) relating the share of entering firms inemployment 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 ashare of GDP and the gap between the interest rate and the growth rate r. To obtain equation (68),we use the free entry condition (71) and the definitions of the share of profits to intangible capitalrelative to GDP,

pt =Pt Mt � yI

rt MtPrt � yr (0) Nt (0) Prt

GDPt,

the share of innovation expenditures relative to GDP,

srt =yI

rt MtPrt + yr (0) Nt (0) Prt

GDPt,

and the share of entering firms in employment,

set = 1 � d0 + d(yIrt)

exp (gzt � q)r�1 .

E Model extensions

E.1 Occupation choice

Suppose that workers draw a productivity x to work in the research sector, where x is drawn froma Pareto with minimum 1 and slope coefficient c > 1. There are two wages, Wpt and Wrt. For themarginal agent,

xtWrt = Wpt

Given that the minimum value of x is 1, any interior equilibrium with positive production requiresWrt Wpt. The aggregate supplies of production and research labor are (having normalized thelabor force to 1),

Lpt = F (xt) = 1-x�c

t

Lrt =ˆ •

xt

x f (x) dx =c

c � 1x1�c

t .

The equilibrium allocation of labor is determined by (as in equation (12) in the baseline model)

WptLpt

WrtLrt=

(1 � a)µt

1srt

and

Lpt

Lrt=

c � 1c

1 �⇣

WptWrt

⌘�c

⇣

WptWrt

⌘1�c

.

16

Note that as c goes to infinity, Wpt/Wrt must converge to 1 in order for Lpt/Lrt to be finite. Theelasticity of the aggregate labor allocation with respect to the innovation intensity of the economy(given a constant average markup) is

logL0

pt

L0rt� log

Lp

Lr= �

(c � 1)⇣

1 + Wr LrWp Lp

⌘

(c � 1)⇣

1 + Wr LrWp Lp

⌘

+ 1

�

logs0rt � logsr�

and the elasticity of research labor with respect to the innovation intensity of the economy is

�

logL0rt � logLr

�

=(c � 1)

(c � 1)⇣

1 + Wr LrWp Lp

⌘

+ 1

�

logs0rt � logsr�

When c converges to 1 (high worker heterogeneity), the elasticity of Lpt/Lrt and Lrt with respectto srt converges to 0. When c converges to infinity (no worker heterogeneity), the elasticity ofLpt/Lrt and Lrt with respect to srt converges to �1 and Lp, respectively, as in our baseline model.In Proposition 3, equation (16) becomes

log Y0rt � log Yr =

(c � 1)

(c � 1)⇣

1 + Wr LrWp Lp

⌘

+ 1

�

log s0rt � log sr�

� (1 � f)�

logZ0t � log Zt

�

where (c�1)

(c�1)✓

1+ Wr LrWp Lp

◆

+1is bounded between 0 and Lp. In Proposition 3, coefficient G0 is now given

by

G0 =(c � 1)

(c � 1)⇣

1 + Wr LrWp Lp

⌘

+ 1

Dg0

DYr0,

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 changesto our analytic results. Specifically, the production of the research good is given by

Yrt = ArtZf�1t Ll

rtX1�l

t ,

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 bygz = gAr

q

, where q = 1 � f � 1�l

1�a

. The condition for semi-endogenous growth is q > 0. Theknife-edge condition for endogenous growth is q = 0 and gAr = 0, which can hold even if f < 1.

Revenues from the production of the research good are divided as follows

WtLrt = lPrtYrt , and Xt = (1 � l) PrtYrt . (72)

The allocation of labor between production and research (the analogous to equation (12) in ourbaseline model) is related to the innovation intensity of the economy by

Lpt

Lrt=

(1 � a)lµt

1srt

Yt

GDPt, (73)

where GDPt = Yt � Xt = Ct + Kt+1 � (1 � dk)Kt when innovation expenditures are excluded inGDP. 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 ouranalytical elasticities need to be modified by the ratio of GDP to Y, which is equal to GDPt

Yt=

(1 + (1 � l)srt)�1 1.

In Proposition 3, the elasticity of research output Yr with respect to a change in the innovationintensity of the economy sr, presented in equation (16), is now given by

log Y0rt � log Yr = Lp

¯GDPY

�

log s0rt � log sr�

� q

�

logZ0t � log Zt

�

� (1 � l) a

1 � a

�

log R0kt � log Rk

�

The third term in the right hand side reflects the change in research output Yrt that result fromchanges in Yt relative to Kt when l < 1. The coefficients Gk in Proposition 3 are now given by

G0 = Lp¯GDPY

Dg0

DYr0

Gk+1 =

"

1 � (1 � f)G0

Lp¯GDPY

#

Gk.

The result in corollary 1 is now stated as

log Z0t � log Zt =

Lp¯GDPY

q

�

log s0r � log sr�

.

Finally, the result in corollary 2 is now adjusted to account for the change in GDP/Y,

logGDPt

Yt� log

¯GDPY

= �✓

1 � GDPY

◆

�

log s0r � log sr�

.

18

F Fiscal expenditures and innovation intensity

In this appendix we derive the elasticity across BGPs of the innovation intensity of the economyand fiscal expenditures on innovation policies with respect to a uniform change in innovationpolicies, where uniform innovation policies are such that tt (j) = tt. In this case, aggregate fiscalexpenditures on innovation policies are given by Et = ttPrtYrt and Et

Yt= ttsrt. In order to derive

our results, we specify additional details of the equilibrium that are not required in the analyticresults of Section 4. We also make use of a number of assumptions, which are satisfied in all of ourmodel 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)� (1 � tt) Prtyrt (j) +1

1 + RtEtjj0|yrt(j)Vt+1

�

j0�

for j � 1 and

Vt (0) = � (1 � tt) yr (0) Prt +1

1 + RtEt0j0Vt+1

�

j0�

where Pt (j) is the static profits of a firm of type j at time t, and Etjj0|yrt(j) denotes the expectationoperator of a type j firm over future types j0 given yrt (j) and all other variables at time t. Assumethat static profits can be written as Pt (j) = BtH (j) and that in the BGP, Bt = kYt.33

Define vt (j) = Vt(j)(1�tt)Prt

. In BGP, we have

vt (j) = maxyrt(j)kYt

Prt (1 � t)H (j)� yrt (j) +

11 + rt

Etjj0|yrt(j)vt+1�

j0�

for j � 1 and

0 � vt (0) = �yr (0) +1

1 + rtEt0j0 vt+1

�

j0�

where 1 + rt = 1(1+Rt)

PrtPrt+1

. Assume that the free entry condition binds, vt (0) = 0.

We now compare two BGPs which are subject to a different innovation subsidy t. With semi-endogenous growth (f < 1), r is unchanged with t. By the free-entry condition, yrt (j) for j � 1and Yt

Prt(1�t)are also invariant with t. Comparing two BGPs, we have

Yt

Prt (1 � t)=

Y0t

P0rt (1 � t

0),

which can be re-written asYrt

srt (1 � t)=

Y0rt

s0rt (1 � t

0).

We now provide a condition (satisfied in all of our model examples) such that output of the

33Note that in Neo-Schumpeterian models, k includes the rate of product destruction, which is constantacross BGPs with semi-endogenous growth.

19

research good, Yr, is unchanged between BGPs. The research good constraint (7) can be written as

Nt(0)

"

Âj�1

yrt(j)Nt(j)Nt(0)

+ yr(0)

#

= Yrt (74)

and the growth rate

gzt ⌘ G�

{yrt (j)}j�1, Nt (0) ; {Nt (j)}j�1�

(75)

= G✓

{yrt (j)}j�1, Nt (0) ; Nt (0) {Nt (j)Nt (0)

}j�1

◆

.

Suppose that the T operator is such that if {yrt (j)}j�1 = {y0rt (j)}j�1, then { Nt(j)Nt(0)

}j�1 = { N0t (j)

N0t (0)

}j�1.By equations (74), (75), and semi-endogenous growth (gzt = g0zt), we must have Nt (0) = N0

t (0)and Yrt = Y0

rt. This implies thatsrt (1 � t) = s0rt

�

1 � t

0�

andE0

t

Y0t� Et

Yt= t

0 s0rt � tsrt = s0rt � srt.

We summarize this result in the following proposition.

Proposition 4. Consider our model on a BGP with semi-endogenous growth and positive firm-entry thatsatisfies the assumptions described in this section. Suppose that uniform innovation policies change perma-nently from t to t

0. Then, across the old and new BGPs the innovation intensity of the economy changesfrom sr to s0r, and fiscal expenditures relative to GDP change from E/ ¯GDP to E0/ ¯GDP0, with these changesgiven by

log s0r � log sr = log(1 � t)� log(1 � t

0)

andE0

¯GDP0 �E¯GDP

= s0r � sr.

This result implies that in the long-run, uniform changes in innovation subsidies result in achange in the innovation intensity of the economy determined only by the change in the inno-vation subsidy rate independent of the other parameters of the model. At short and mediumhorizons, however, this policy will result in a change in the path of the innovation intensity of theeconomy from {srt}•

t=0 (which is constant on the initial BGP) to {s0rt}•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). Thismodel violates Assumption 1 regarding concavity of the growth rate of aggregate productivitywith respect to entry and Assumption 2 regarding the allocation of innovation across heteroge-neous 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 toimprove the distribution of markups. In this sense, innovation policy can partly substitute anti-trust policies.

Incumbent firms each produce one good and are indexed by the productivity with which theycan produce this good, exp(z). Aggregate productivity is given by (5) where Nt (j) denotes themass of products with productivity z(j) and markup µ (j). With r ! 1, the expression for aggre-gate productivity is Zt = Z1tZ2t where

Z1t = exp

Âj�1

z(j)Nt(j)

!

Z2t =exp

⇣

Âj�1 log⇣

µ (j)�1⌘

Nt (j)⌘

Âj�1 µ (j)�1 Nt (j).

Each entrant expends yr (0) units of the research good to be matched with probability s toan existing intermediate good (produced by some other firm) raising the frontier productivityfor producing that good from exp(z) to exp(z + Dz) and charging a markup of µ = exp (Dz).Each incumbent firm invests yI

rt units of the research good to innovate with probability sd�

yIrt�

,raising the productivity of its own intermediate good from exp(z) to exp(z + Dz) and increasingthe markup by a factor of exp (Dz), where d (.) is increasing and concave. We denote by n (j) thenumber of steps ahead by a product of type j relative to the second lowest cost supplier of thatgood, so that µ (j) = exp (n (j)Dz). The total measure of products innovated is s

�

Nt (0) + d�

yIrt��

,of which a fraction Nt (0) /

�

Nt (0) + d�

yIrt��

is by entrants.The law of motion for the first component of aggregate productivity, Z1t, is

logZ1t+1 � logZ1t = Dzs

⇣

Nt (0) + d⇣

yIrt

⌘⌘

.

The second component of aggregate productivity, Z2t, can be written as

Z2t =

"

exp

Ân�1

nDz Mt (n)

!

Ân�1

exp (nDz)�1 Mt (n)

#�1

,

21

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 � s

⇣

Nt (0) + d⇣

yIrt

⌘⌘⌘

Mt (1) + sNt (0)

and for n > 1

Mt+1 (n) =⇣

1 � s

⇣

Nt (0) + d⇣

yIrt

⌘⌘⌘

Mt (n) + sd⇣

yIrt

⌘

Mt (n � 1) .

In the BGP, M (1) = N(0)N(0)+d(yI

r)and M (n) =

d(yIr)

N(0)+d(yIr)

M (n � 1) for n > 1. Peters (2013) showsthat an increase in entry increases Z2t by raising the measure of products with low markups. Onthe 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 ofaggregate productivity with respect to an increase in entry. To see this, note that

logZt+1 � logZt = (logZ1t+1 � logZ1t) + (logZ2t+1 � logZ2t)

The term log (Z1t+1) � log (Z1t) is linearly increasing in Nt (0). The term logZ2t+1 � logZ2t isincreasing in entry and can be convex (for example, set s = 0.1, Dz = 0.1, gz = 0.01 and in theinitial BGP N(0)

N(0)+d(yIr)= 0.05). Since the sum of a linear function and a convex function is convex,

logZt+1 � logZt is convex, violating Assumption 1a.

Two model specifications that violate Assumptions 2 and 3 but satisfyconditional efficiency in initial BGP

We now present two specifications of our model that fail to satisfy assumptions 2 and 3 but, how-ever, 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 andproduction, Lrt = Lrt and Lpt = Lpt, the equilibrium and social planning allocations coincide onthe BGP. Whereas the equilibrium aggregate allocation of labor, Lrt, may not solve the social plan-ning problem (in which case there is a role for welfare-enhancing innovation policies that increaseor reduce Lrt) there are no welfare gains from reallocating innovative activities across heteroge-neous firms. To simplify the presentation, we abstract from physical capital accumulation (i.e. weset a ! 0). This does not alter the result that the equilibrium is conditional efficient as long as theEuler equation of physical capital is undistorted (which requires a production subsidy to undo themarkup).

22

Quality ladders model with heterogeneous innovative technologies

In the quality ladders model developed in Klette and Kortum (2004), product leadership andresearch capacity are tied together in the sense that if a product is innovated on, the firm owningthis 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 reallocateinnovation expenditures across firms. The distortion arises because individual firms do not takeinto account that their innovation decisions destroy research capacity of other firms. Hence, theequilibrium is not conditionally efficient. The model example we present here is similar to that inLentz 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 showthat such a model specification satisfies conditional efficiency.

A firm in this model is characterized by its technology to innovate, its research capacity, thenumber of products it has leadership on and the productivity of each of these products. As we willsee, in order to characterize the aggregate dynamics we only need to record the firm’s innovativetechnology type, which we denote by j. Underlying this aggregate economy there is a continuumof 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

max •t=0 b

tu (Ct) subject to

Ct = Zt (1 � Lrt)

Zt+1

Zt=h

dt

⇣

exp (Dz)r�1 � 1

⌘

+ 1i

1r�1 (76)

dt = s

Âj�1

dj (yrt (j)) Nt (j)

!

(77)

Nt+1 (j) = yjNt (0) +⇣

1 � d

f (j) + gj (irt (j))⌘

Nt (j) (78)

Âj�1

(yrt(j) + irt (j)) Nt(j) + yr(0)Nt(0) = ArtZf�1t Lrt (79)

Z0, N0 (j)and {Lrt}are given.

Here, Nt (j) denotes the aggregate mass of research capacity operated by firms of type j, whichdepreciates at the exogenous rate d

f (j) and that can be increased in expectation by a fractiongj (irt (j)) if an investment of irt (j) Nt (j) units of the research good is undertaken, where gj (.) is an

34Assumption 2 that the growth rate is maximized at time t requires irt (j) = Nt (0) = 0, which is not thecase 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 firmof type j in order to acquire a product with probability sdj (yrt (j)) per unit of research capacityand improve its productivity by a step size Dz,35 where dj (.) is an increasing and concave function.The measure of products that receive an innovation is dt = s

⇣

Âj�1 dj (yrt (j)) Nt (j)⌘

1. Theprobability that each of the Nt (0) entrants is of type j is yj, with Âj yj = 1.36

We now characterize the solution to this constrained (given Lrt) social planning problem. As-suming it exists, the Lagrangean is

•

Ât=0

b

tu (Zt (1 � Lrt))

subject to (including the Lagrange multipliers)

ltbt : Â

j�1(yrt(j) + irt (j)) Nt(j) + yr(0)Nt(0)� ArtZ

f�1t Lrt

ctbt : Zt+1 =

"

s

Âj�1

dj (yrt (j)) Nt (j)

!

⇣

exp (Dz)r�1 � 1

⌘

+ 1

#

1r�1

Zt

wt (j) b

t : Nt+1 (j)� yjNt (0)�⇣

1 � d

f (j) + gj (irt (j))⌘

Nt (j)

The FOC with respect to Zt+1 is

�ct + bu0 (Ct+1) (1 � Lrt+1) + lt+1b (f � 1)Yrt+1

Zt+1+ ct+1b

Zt+2

Zt+1= 0

with respect to Nt+1 (j)

wt (j) = �blt+1 (yrt+1(j) + irt+1 (j)) + b

⇣

1 � d

f (j) + gj (irt+1 (j))⌘

wt+1 (j) +

+b

r � 1ct+1Zt+1

✓

Zt+2

Zt+1

◆2�r

sdj (yrt+1 (j))⇣

exp (Dz)r�1 � 1

⌘

with respect to irt (j)lt = wt (j) g0j (irt (j))

35Our results go through if step size varies by firm type, as long as markups are equal across products.36In the alternative model in which product market leadership and research capacity are combined, con-

straint (77) is instead dt = s

h

Nt (0) + Âj�1 dj (yrt (j)) Nt (j)i

, constraint (77) is instead

Nt+1 (j) = (1 � dt) Nt (j) + s

⇥

yjNt (0) + dj (yrt (j)) Nt (j)⇤

and constraint (79) does not include irt (j). It is straightforward to show in this case that the equilibrium isnot conditionally efficient because the equilibrium does not take into account that changes in Nt (j) changedt and thus Nt+1 (j0).

24

with respect to yrt (j)

lt =1

r � 1

✓

Zt+1

Zt

◆2�r

Ztctsd0j (yrt (j))⇣

exp (Dz)r�1 � 1

⌘

and with respect to Nt (0)ltyr (0) = Â

j�1yjwt (j)

We re-write these first order conditions as

�ctZt

lt

lt

lt+1

Zt+1

Zt+ b

u0 (Ct+1)Ct+1

lt+1+ (f � 1) bYrt+1 + b

ct+1Zt+1

lt+1

Zt+2

Zt+1= 0

lt

lt+1

wt (j)lt

= �b (yrt+1(j) + irt+1 (j)) + b

⇣

1 � d

f (j) + gj (irt+1 (j))⌘

wt+1

lt+1(j) +

+b

r � 1ct+1Zt+1

lt+1

✓

Zt+2

Zt+1

◆2�r

sdj (yrt+1 (j))⇣

exp (Dz)r�1 � 1

⌘

1 =wt (j)

ltg0j (irt (j))

1 =1

r � 1

✓

Zt+1

Zt

◆2�r Ztct

ltsd0j (yrt (j))

⇣

exp (Dz)r�1 � 1

⌘

yr (0) = Âj�1

yjwt (j)

lt

We consider a BGP in which Nt (j) = N (j), Yrt = Yr,wt(j)

lt= w (j), Zt+1

Zt= exp(gz) where gz =

gAr1�f

,

c = Ztctlt

, v = c

r�1 exp (gz)2�r

⇣

exp (Dz)r�1 � 1

⌘

, and ltlt+1

= u0(Ct)Ctu0(Ct+1)Ct+1

= (1 + r) b, where 1 + r =

b

�1exp�

(h � 1) gy�

= b

�1. We can write the FOC conditions in the BGP as

(1 + r) w (j) = ��

yr(j) + ir (j)�

+⇣

1 � d

f (j) + gj�

ir (j)�

⌘

w (j) + sdj (yr (j)) v (80)

1 = g0j�

ir (j)�

w (j) (81)

1 = sd0j (yr (j)) v (82)

yr (0) = Âj�1

yjw (j) (83)

We can use this system of equations to solve for v, w (j), yr(j) and ir (j). Given yr(j) and ir (j), weuse equation (76) to solve for N (0) and given a level of Lr and Art we use equation (79) to solvefor the level of Zt and the corresponding level of consumption in the BGP.

In what follows we show that yr(j), ir (j) and N(0) coincide in the social planning problemand in the equilibrium and when innovation policies are uniform for entrants and incumbents,

25

tt (j) = t. Given Lr, 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) = maxyrt(j),irt(j)

� (1 � t) Prt (yrt (j) + irt (j)) +1 � d

f (j) + gj (irt (j))1 + Rt

Vt+1 (j) +sdj (yrt (j))

1 + RtVt+1

where Vt+1 denotes the expected value of acquiring a product,

Vt+1 =ˆ

Vt+1 (z + Dz) dMt (z)

and Vt (z) denotes the value of being a leader in product z,

Vt (z) = Ptexp (z)r�1 +(1 � dt)1 + Rt

Vt+1 (z)

where Pt = µ

�r(µ � 1)W1�r

t Yt. The free entry condition is

(1 � t) Prtyr (0) =1

1 + Rt yjVt+1 (j)

Let wt (j) = Vt(j)(1+rt�1)(1�t)Prt

, vt (z) =Vt(z)

(1+rt�1)(1�t)Prt, vt =

Vt(1+rt�1)(1�t)Prt

, pt =Pt

(1�t)Prtand 1 + rt =

(1 + Rt)Prt

Prt+1, where

(1 + rt�1)wt (j) = maxyrt(j),irt(j)

� (yrt (j) + irt (j)) + ...

+⇣

1 � d

f (j) + gj (irt (j))⌘

wt+1 (j) + sdj (yrt (j)) vt+1

(1 + rt�1) vt (z) = ptexp (z)r�1 + (1 � dt) vt+1 (z)

vt+1 =ˆ

vt+1 (z + Dz) dMt (z)

yr (0) = Â yjwt+1 (j)

In the BGP, 1 + r = b

�1 and the rescaled value functions are constant over time. The equilibriumlevels of yr (j) , ir (j), w (j), and v are the solution to

(1 + r) w (j) = maxyr(j),ir(j)

� (yr (j) + ir (j)) +⇣

1 � d

f (j) + gj (ir (j))⌘

w (j) + sdj (yr (j)) v

yr (0) = Â yjw (j)

This system of equations gives the same solution for yr (j) and ir (j) as the system of equationsin 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 theequilibrium and social planning problems. Of course, the equilibrium allocation of labor betweenproduction and research may be sub-optimal, which can be fixed with uniform innovation subsi-dies.

Expanding varieties model with heterogeneous innovation technologies

This model example builds on our model example 5 above, adding cross-firm heterogeneity ininnovation technologies, as in the second model in Luttmer (2011). Firms are characterized bytheir innovative technology j � 1 and the number of products they own and operate all of whichhave productivity exp (z) = jj. Consider a firm that in period t has innovative technology j, ownsn products all of which have productivity exp (z) = jj, and invests yrtn units of the research good.In period t + 1, this firm will have innovative technology j0 with probability yjj0 and will own anexpected number of dj(yrt)n products all of which have productivity exp (z) = jj0 . The functiond (.) is increasing and concave, and we do not impose here that d (0) = 0. Firms exit when thenumber of products they own falls to zero. We denote by yrt (j) the investment per product offirms with innovative technology j, and by Nt (j) the measure of products operated by firms withinnovative technology j. A measure Nt (0) of entrants expends yr (0) units of the research goodeach to have innovative technology j with probability y0j and create s new products in expectationeach operated with productivity exp (z)r�1 = jj.

The social planning problem is

max •t=0 b

tu (Ct) subject to

Ct = Zt (1 � Lrt)

Zt =

"

Âj�1

jjNt(j)

#

1r�1

(84)

Nt+1 (j) = y0jNt (0) + Âj0

yj0 jdj0(yrt(j0))Nt�

j0�

(85)

Âj�1

yrt(j)Nt(j) + yr(0)Nt(0) = ArtZf

t Lrt (86)

N0 (j) and {Lrt} are given.

In the BGP, Nt(j) grows at the rate (r � 1) gz for all j , and gz =gAr

r�1�f

.37 We now characterize the

37It is straightforward to transform variables so that in the BGP Yrt is constant and redefine parametersso that in the BGP grows at the rate gAr / (1 � f) as in our baseline framework. Our results are unchangedunder this redefinition of variables.

27

solution to the social planning problem. Assuming it exists, the Lagrangean is

•

Ât=0

b

tu (Zt (1 � Lrt))

subject to (including the Lagrange multipliers)

ltbt : Â

j�1yrt(j)Nt(j) + yr(0)Nt(0) = ArtZ

f

t Lrt

ctbt : Zt �

"

Âj�1

jjNt(j)

#

1r�1

wt (j) b

t : Nt+1 (j)� y0jNt (0)� Âj0

yj0 jdj0(yrt(j0))Nt�

j0�

The FOC with respect to Zt is

ct = u0 (Ct) (1 � Lrt) + ltfYrt

Zt

with respect to Nt+1 (j)

wt (j) =bZt+1

r � 1ct+1jj � blt+1yrt+1(j) + bdj(yrt(j))Â

j0yjj0wt+1

�

j0�

with respect to yrt (j)lt = d0 j(yrt(j))Â

j0yjj0wt

�

j0�

and with respect to Nt (0)38

ltyr (0) = Âj�1

y0jwt (j)

We re-write these first order conditions as

ctZt

lt=

u0 (Ct)Ct

lt+ f

Yrt

ltZt

lt

lt+1

wt (j)lt

=bZt+1

r � 1ct+1

lt+1jj � byrt+1(j) + bdj(yrt(j))Â

j0yjj0

wt+1 (j0)lt+1

38Combining the last two conditions implies d0 j(yrt(j))Âj0 yjj0wt (j0) = 1yr(0) Âj0�1 y0jwt (j0). Assumption

2 at time t requires that {yrt(j)} and Nt (0) maximize Zt+1 subject to the research good constraint, whichrequires d0j(yrt(j))Âj0�1 yjj0 jj0 =

1yr(0) Âj0�1 y0j0 jj0 . This condition is violated both in the planners problem

and in the equilibrium unless wt(j0)wt(j) =

jj0jj

which in general does not hold.

28

1 = d0 j(yrt(j))Âj0

yjj0wt (j0)

lt

yr (0) = Âj�1

y0jwt (j)

lt

In a BGP, ctZt(r�1)lt

= p, wt(j)lt

= w (j), yrt (j) = yr (j) and ltlt+1

= u0(Ct)Ctu0(Ct+1)Ct+1

= (1 + r) b, where1 + r = b

�1exp�

(h � 1) gy�

= b

�1, where

(1 + r) w (j) = pjj � yr(j) + dj(yr (j))Âj0

yjj0w�

j0�

1 = d0 j(yr (j))Âj0

yjj0w�

j0�

yr (0) = Âj�1

y0jw (j)

We can use this system of equations to solve for w (j) and yr(j). Given yr(j) we can solve forthe mass of products by type normalized by the level of entry, N (j) /N (0), using equation (85),which can be expressed as

exp (gz)r�1 N (j)

N (0)= y0j + Â

j0yj0 jdj0(yr(j0))

N (j0)N (0)

Given a level of Lr and Art we solve for the level of Zt and Nt (0) re-writing equations (84) and(86) as

Âj�1

yr(j)N (j)N (0)

+ yr(0) =1

N(0)ArtZ

f

t Lrt

Zt = Nt (0)

"

Âj�1

jjN (j)N (0)

#

1r�1

and finally we solve for consumption.In what follows we show that the allocations level of yr (j) for all j coincide in the equilibrium

and social planning problem when innovation policies are uniform, tt (j) = t. Given Lr, this alsoimplies the same level of entry, aggregate productivity and consumption. The equilibrium valueassociated with a product owned by a firm with innovation technology j is

Vt (j) = maxyrt(j)

Pt jj � (1 � t) Prtyrt (j) +dj (yrt (j))

1 + RtÂj0

yjj0Vt+1�

j0�

29

where Pt = µ

�r(µ � 1)W1�r

t Yt and uniform innovation policies tt (j) = t. The associated FOC is

(1 � t) Prt =d0j (yrt (j))

1 + RtÂj0

yjj0Vt+1�

j0�

and the free entry condition is

(1 � t) Prty (0) =1

1 + RtÂ

jy0jVt+1 (j)

Defining wt (j) = Vt(j)(1+rt�1)(1�t)Prt

, pt =Pt

(1�t)Prtand 1 + rt = (1 + Rt)

PrtPrt+1

, we can write the valuefunctions as

(1 + rt�1)wt (j) = pt jj � yrt (j) + dj (yrt (j))Âj0

yjj0wt+1�

j0�

where1 = d0j (yrt (j))Â

j0yjj0wt+1

�

j0�

and the free entry conditiony (0) = Â

jy0jwt+1 (j)

In a BGP, 1 + r = b

�1 and the rescaled value functions are constant over time.

(1 + r) w (j) = pjj � yrt+1(j) + dj(yr (j))Âj0

yjj0w�

j0�

1 =1

(1 + r)d0 j(yr (j))Â

j0yjj0w

�

j0�

yr (0) = Âj�1

y0jw (j)

This system of equations is identical to the respective one in the social planning problem and henceboth give the same solution for yr (j). Moreover, given the same aggregate allocation of labor, weobtain the same BGP levels of entry, aggregate productivity and consumption in the equilibriumand social planning problems. Of course, the equilibrium allocation of labor between productionand research may be sub-optimal, which can be fixed with uniform innovation subsidies.

30