Download Slides

Aggregate Implications of Innovation Policy
Andy Atkeson and Ariel Burstein
UCLA, FRB MPLS, and NBER
April 2016
Intro
I
I
Firm’s investments in innovation (and intangibles):
I
BEA: private sector investments in intellectual property
products, 3.8% of U.S. GDP in 2012 (private R&D, 1.7%).
I
Corrado et. al. (2009): broader measure of firms’ innovation
investments, 13% of U.S. GDP in 2005
Taxes & subsidies to encourage investments, stimulate growth
Intro
I
I
Firm’s investments in innovation (and intangibles):
I
BEA: private sector investments in intellectual property
products, 3.8% of U.S. GDP in 2012 (private R&D, 1.7%).
I
Corrado et. al. (2009): broader measure of firms’ innovation
investments, 13% of U.S. GDP in 2005
Taxes & subsidies to encourage investments, stimulate growth
1. To what extent can we change the path of macroeconomic
growth at different time horizons by inducing firms to increase
their investments in innovation?
2. What is the optimal level of these investments?
3. What policies implement optimal investment level?
Intro
I
I
Firm’s investments in innovation (and intangibles):
I
BEA: private sector investments in intellectual property
products, 3.8% of U.S. GDP in 2012 (private R&D, 1.7%).
I
Corrado et. al. (2009): broader measure of firms’ innovation
investments, 13% of U.S. GDP in 2005
Taxes & subsidies to encourage investments, stimulate growth
1. To what extent can we change the path of macroeconomic
growth at different time horizons by inducing firms to increase
their investments in innovation?
2. What is the optimal level of these investments?
3. What policies implement optimal investment level?
I
Model of growth through investments by firms to examine 1., 2
Model of growth through innovative investments by firms
I
Nests several of the widely-used models in the literature:
I
Neo-Schumpeterian and expanding varieties
I
Innovative investment by entrant and incumbent firms
I
I
I
Facts on firm-level innovation and firm dynamics
Tractable aggregation
Private and social returns differ
Model of growth through innovative investments by firms
I
Nests several of the widely-used models in the literature:
I
Neo-Schumpeterian and expanding varieties
I
Innovative investment by entrant and incumbent firms
I
I
I
I
Facts on firm-level innovation and firm dynamics
Tractable aggregation
Private and social returns differ
Three baseline assumptions: simple analytical approximation of
dynamics of aggregate productivity, GDP to policy-driven
change in innovation investments
I
Shapes calculation of socially optimal level of innovation
Key Features Shaping Aggregate Quantitative Implications
I
I
Social Depreciation of Innovation Expenditures
I
Important in shaping medium term (e.g. 20 years) implications
if model calibrated to low baseline growth rate
I
Quality-ladders and expanding varieties models implicitly make
very different assumptions and hence have very different
quantitative implications
Intertemporal Knowledge Spillovers
I
I
Important for long run implications and optimal innovation
intensity, less so for medium term if model calibrated to low
baseline growth rate and low social depreciation
Cost of innovation incumbents versus entering firms
Related literature
I
Models of innovation by firms and firm-level data
I
I
Changes in innovation intensity and long-term trends
I
I
Pakes and Schankerman (1984), Griliches (1998)
Misallocation of innovation across firms
I
I
Jones (2005), Bloom, Schankerman, and Van Reenen (2013)
Social depreciation of innovation
I
I
Jones and Williams (1998), Jones (2002)
Knowledge spillovers
I
I
Klette and Kortum (2004), Luttmer (2007, 2011), Lentz and
Mortensen (2008), Acemoglu (2008, chapter 13,14), Ackigit
and Kerr (2010), Atkeson and Burstein (2010), Ackigit and
Aghion (2014), Garcia-Macia, Hsieh and Klenow (2015)
Acemoglu, Akcigit, Bloom and Kerr (2013), Peters (2013),
Lentz and Mortensen (2014)
Sufficient statistics
I
Arkolakis, Costinot, Rodriguez-Clare (2012)
Outline of Talk and Paper
I
General model
I
I
Appendix: mapping to specific models
Simple quality ladders, expanding varieties, Klette Kortum
(2004), Atkeson Burstein (2010), Luttmer (2011)
I
Analytical characterization of transition dynamics under three
key assumptions
I
Implications for socially optimal level of innovative investments
I
Quantitative examples
I
Relation to more complex recent models
I
e.g. gains from reallocating given innovation resources across
heterogeneous firms
Production by Intermediate good firms
I
I
Firm type j = 1, 2, 3, . . .,
I
n (j) products
I
z1 (j), z2 (j), ... , zn(j ) (j) productivities
I
µ1 (j), µ2 (j), ... , µn(j ) (j) markups
I
q (j) heterogeneity in innovation technologies
Production of intermediate good m by firm of type j
ymt (j) = exp(zm (j))kmt (j) a lmt (j)1
a
Aggregate Productivity
I
I
State: measure of incumbents {Nt (j)}j
⇣
Final good: Yt = Âj
⌘r/(r 1)
n(j)
(r 1)/r N (j)
t
1 Âm=1 ymt (j)
Yt = Zt L1pt a Kta = Ct + Kt+1
I
⇣
Zt ⌘ Z {Nt (j)}j
I
1
1
(1
d k ) Kt
⌘
example with constant markups
n (j )
Zt ⌘ Z ({Nt (j)} j
Alternative
1)
=
  exp(zm (j))
j 1 m=1
(r 1)
Nt (j)
!1/(r
1)
Innovation by Intermediate Good Firms
I
Measure of incumbents {Nt (j)}j
I
I
1
innovative investment
Measure of entrants Nt (0)
I
I
{yrt (j)}j
1
ȳr (0) entry cost parameter
Transition law
{Nt+1 (j)}j
⇣
{yrt (j)}j
=
T
1
1 , Nt (0); {Nt (j)}j
1
⌘
Innovation by Intermediate Good Firms
I
Measure of incumbents {Nt (j)}j
I
I
1
innovative investment
Measure of entrants Nt (0)
I
I
{yrt (j)}j
ȳr (0) entry cost parameter
Transition law
{Nt+1 (j)}j
I
I
1
⇣
{yrt (j)}j
=
T
1
1 , Nt (0); {Nt (j)}j
1
⌘
Functions Z and T give aggregate productivity growth
⇣
gzt ⌘ log Zt+1 log Zt = G {yrt (j)}j 1 , Nt (0); {Nt (j)}j
Social Depreciation of Innovation Expenditures
⇣
⌘
⇣
Gt0 {Nt (j)}j 1 ⌘ G {0}j 1 , 0; {Nt (j)}j
Examples G
1
⌘
1
⌘
Research good
I
Research good used as input for innovation by firms
f 1
 yrt (j)Nt (j) + ȳr (0)Nt (0) = Yrt = Art Zt
j 1
I
I
I
Art freely-available scientific knowledge
f  1 intertemporal knowledge spillovers
1 f
Price of research good Prt ⌘ Art1 Wt Zt
Lrt
Research good
I
Research good used as input for innovation by firms
f 1
 yrt (j)Nt (j) + ȳr (0)Nt (0) = Yrt = Art Zt
Lrt
j 1
I
I
I
I
Art freely-available scientific knowledge
f  1 intertemporal knowledge spillovers
1 f
Price of research good Prt ⌘ Art1 Wt Zt
Innovation subsidies tt (j) Prt yrt (j), lump sum taxes
I
Welfare results: production subsidy to undo markup
Research good
I
Research good used as input for innovation by firms
f 1
 yrt (j)Nt (j) + ȳr (0)Nt (0) = Yrt = Art Zt
Lrt
j 1
I
I
I
I
Innovation subsidies tt (j) Prt yrt (j), lump sum taxes
I
I
Art freely-available scientific knowledge
f  1 intertemporal knowledge spillovers
1 f
Price of research good Prt ⌘ Art1 Wt Zt
Welfare results: production subsidy to undo markup
Innovation intensity of economy
srt ⌘
Prt Yrt
Yt
Research good
I
Research good used as input for innovation by firms
f 1
 yrt (j)Nt (j) + ȳr (0)Nt (0) = Yrt = Art Zt
Lrt
j 1
I
I
I
I
Innovation subsidies tt (j) Prt yrt (j), lump sum taxes
I
I
Art freely-available scientific knowledge
f  1 intertemporal knowledge spillovers
1 f
Price of research good Prt ⌘ Art1 Wt Zt
Welfare results: production subsidy to undo markup
Innovation intensity of economy
srt ⌘
I
Prt Yrt
Yt
Equilibrium labor allocation: srt =
µt
Lrt
1 a Lpt
Positive implications of model
I
Baseline allocations in BGP: ḡz , s̄r , Ȳr , {ȳ rt (j)}, N̄ t (j)
Positive implications of model
I
Baseline allocations in BGP: ḡz , s̄r , Ȳr , {ȳ rt (j)}, N̄ t (j)
I
At t = 0, change in innovation subsidies from t(j) to t 0 (j)
I
New equilibrium path: srt0 , Yrt0 , {yrt0 (j)}j
1,
{Nt0 (j)}j
0,
0
gzt
Positive implications of model
I
Baseline allocations in BGP: ḡz , s̄r , Ȳr , {ȳ rt (j)}, N̄ t (j)
I
At t = 0, change in innovation subsidies from t(j) to t 0 (j)
I
New equilibrium path: srt0 , Yrt0 , {yrt0 (j)}j
I
Impact elasticity
0
gz0
I
ḡz = log Z10
log Z̄1 ⇡
0
1,
{Nt0 (j)}j
log sr0 0
log s̄r 0
Dynamics
0
log Zt+1
log Z̄t+1 ⇡
t
Â
k=0
k
log srt0
k
log s̄r
0,
0
gzt
Change in growth rate: characterizing dynamics
I
0
gzt
∂G
 ∂ yrt (j)
j 1
yrt0 (j)
ȳrt (j) +
∂G
N 0 (0)
∂ Nt (0) t
∂G
Nt0 (j)
∂
N
(j)
t
1
+Â
j
ḡz ⇡
N̄t (j)
N̄t (0) +
Change in growth rate: characterizing dynamics
I
0
gzt
∂G
 ∂ yrt (j)
j 1
yrt0 (j)
ḡz ⇡
ȳrt (j) +
∂G
Nt0 (j)
∂
N
(j)
t
1
+Â
j
I
∂G
N 0 (0)
∂ Nt (0) t
N̄t (0) +
N̄t (j)
Impact elasticity N00 (j) = N¯0 (j)
I
I
Consider two specific variations. If baseline innovation
expenditures “efficient” (A2), all variations ) same 0
Concavity of G (A1) ) simple upper bound on 0
Change in growth rate: characterizing dynamics
I
0
gzt
∂G
 ∂ yrt (j)
j 1
yrt0 (j)
ḡz ⇡
ȳrt (j) +
∂G
Nt0 (j)
∂
N
(j)
t
1
+Â
j
I
N̄t (0) +
N̄t (j)
Impact elasticity N00 (j) = N¯0 (j)
I
I
I
∂G
N 0 (0)
∂ Nt (0) t
Consider two specific variations. If baseline innovation
expenditures “efficient” (A2), all variations ) same 0
Concavity of G (A1) ) simple upper bound on 0
Dynamics simplify if
Dgt
DYrt
constant over time (A3)
Proportional increase in all firms’ investments
I
yr0 0 (j) = kyr 0 (j) and N00 (0) = kN0 (0)
I
A1 Concavity
I
Growth rate (in logs) increases less than proportionally
Proportional increase in all firms’ investments
I
yr0 0 (j) = kyr 0 (j) and N00 (0) = kN0 (0)
I
A1 Concavity
I
I
Growth rate (in logs) increases less than proportionally
Impact elasticity bounded by
0
 ḡz
where G00 = G 0 {N̄0 (j)}j
1
G00 L̄p
Intuition: Impact elasticity
I
I
I
0
 ḡz
G00 L̄p
Invest Ȳr ! growth ḡz
Invest 0 ! growth G00
Concavity implies marginal increase < average increase
Increase in entry only
I
yr0 0 (j) = yr 0 (j) and N00 (0) = kN0 (0)
I
A1’ Concavity with respect to entry
I
Alternative bound:
I
I
0
 ḡz
G00
Ȳr
L̄
Ȳr Yr00 p
where Yr00 is cost of implementing G00 at fixed {ȳr 0 (j)}j
Ȳr
<1
Ȳr Yr00
1
(tighter bound) iff incumbents have lower average
social cost of innovation than entrants
Ȳr 0
Ȳr
<
0
ḡz G0
ḡz
Yr00
G00
A2: Growth maximized given current research output
I
Given Ȳrt , then {ȳrt (j)}j
⇣
max G {yrt (j)}j
1 , N̄t (0)
solves
1 , Nt (0); N̄t (j)
j 1
⌘
subject to
 yrt (j)N̄t (j) + ȳr (0)Nt (0) = Ȳrt
j 1
I
If baseline allocation satisfies A2, then 0 is independent of
how change in output of research good is allocated across
incumbent and entering firms
I
Previous bounds hold
Aggregate Dynamics
I
Productivity
0
log Zt+1
+
I
I
Dgt
logYrt0
DYrt
where Dg 0 /DYr 0 =
log Z̄t
log Ȳrt
0 /L̄p
Research good
logYrt0
log Ȳrt = logsrt0
(1
I
log Z̄t+1 ⇡ log Zt0
log s̄rt + log µt0
f ) logZt0
log µ̄t L̄p
log Z̄t
Assumption 3: Dgt /DYrt = Dg 0 /DYr 0 and µt0 = µ̄0
Productivity Dynamics
0
log Zt+1
log Z̄t+1 =
t 1
Â
k
log srt0
k
log s̄r
k=0
Impact elasticity:
h
Decay: k+1 = 1
(upper bound under A1 and A2)
i
(1 f ) L̄0 k
0
I
Higher f , lower decay
I
Higher
0,
p
larger response at every finite horizon
Long-term impact permanent change in sr : L̄p / (1
f)
Welfare: Optimal Innovation Intensity
I
Consumption equivalent change in welfare if capital undistorted
•
¯ t⇥
GDP
log x = (1 b̃ ) Â b̃ t
log Zt0 log Z¯t
C̄t
t=0
(1
I
a)
L̄r
log L0rt
L̄p
log L̄r
Tradeoff: consumption " with productivity, # with innovation
investment
Welfare: Optimal Innovation Intensity
I
Consumption equivalent change in welfare if capital undistorted
•
¯ t⇥
GDP
log x = (1 b̃ ) Â b̃ t
log Zt0 log Z¯t
C̄t
t=0
a)
(1
L̄r
log L0rt
L̄p
log L̄r
I
Tradeoff: consumption " with productivity, # with innovation
investment
I
Perturb L0r 0 , set log x = 0, use Assumption 3
sr⇤ = (1
I
a)
can use bound on
L⇤r
=
L⇤p
1
⇤
0 /L̄p
b̃
b̃ 1
⇤
0 /L̄p
(1
f ) 0 /L̄⇤p
under Assumption 1
Welfare: Optimal Innovation Intensity
I
Consumption equivalent change in welfare if capital undistorted
•
¯ t⇥
GDP
log x = (1 b̃ ) Â b̃ t
log Zt0 log Z¯t
C̄t
t=0
a)
(1
L̄r
log L0rt
L̄p
log L̄r
I
Tradeoff: consumption " with productivity, # with innovation
investment
I
Perturb L0r 0 , set log x = 0, use Assumption 3
sr⇤ = (1
I
I
a)
can use bound on
L⇤r
=
L⇤p
1
⇤
0 /L̄p
b̃
b̃ 1
⇤
0 /L̄p
(1
f ) 0 /L̄⇤p
under Assumption 1
Level of optimal subsidy depends on additional, model-specific
parameters e.g. markup, rate of production destruction Rate
Using these results: numerical examples
I
Assume:
G=
I
I
I
I
1
r
1
⇣
⌘
log a0 + a1 Nt (0) + F (yrtI )
yrtI aggregate use of research goods by incumbents
F concave, F(0)=0, A1 holds
A2 holds if a1 /ȳr (0) = F 0 (ȳrtI )
A3 holds when A2 holds
I
Appendix: Simple quality ladders, expanding varieties, Klette
Kortum (2004), Atkeson Burstein (2010), Luttmer (2011)
Using these results: numerical examples
I
Assume:
G=
I
I
I
I
r
1
⇣
⌘
log a0 + a1 Nt (0) + F (yrtI )
yrtI aggregate use of research goods by incumbents
F concave, F(0)=0, A1 holds
A2 holds if a1 /ȳr (0) = F 0 (ȳrtI )
A3 holds when A2 holds
I
I
1
Appendix: Simple quality ladders, expanding varieties, Klette
Kortum (2004), Atkeson Burstein (2010), Luttmer (2011)
Exact impact elasticity with respect to any variation
0=
I
I
1
r
exp(ḡz )r 1 exp(G00 )r
1
exp(ḡz )r 1
A1 and A2 imply
0
Ȳr
Ȳr Yr00
1
attains bound as r ! 1
1
Ȳr
L̄p
Ȳr Yr00
Neo-schumpeterian models
I
Assume G 0 = 0
I
ḡz = 0.0125, L̄p = 0.83, r = 4
I
Assume
Ȳr
Ȳr Yr00
= 1 (upper bound)
0
' L̄p ḡz = 0.0102,
I
Vary spillover f = 2, f = 0, f = 0.99
I
Long run aggregate productivity elasticity =
between 0.28 and 83
L̄p
1 f
, ranges
Aggregate productivity elasticity: 100 years
Aggregate productivity elasticity: 20 years
GDP elasticity: 100 years
Suppose K /Y fixed
Welfare: Optimal Innovation Intensity
I
Optimal BGP allocation satisfies
sr⇤ = (1
a)
L⇤r
=
L⇤p
1
⇥
b̃ 1
b̃
⇤ /L̄⇤
0 p
(1
f ) ⇤0 /L̄⇤p
I
Huge range of implications depending on b̃ and f
I
b̃ = .99, f = .99 =) sr⇤ = 1.26;
I
b̃ = .96, f = 2 =) sr⇤ = 0.16;
I
If small
0,
⇤
then disconnect between sr⇤ and 20 year response
Expanding varieties
I
Do not impose G00 = 0
I
Firms exit, firms shrink if don’t innovate
I
Entrants do not directly displace existing products
I
Exact impact elasticity (linear entry) in terms of observables
0
=
L̄p
r
(Share entrants in employment)
1 (Share entrants in innovation investments)
Expanding varieties
I
Do not impose G00 = 0
I
Firms exit, firms shrink if don’t innovate
I
Entrants do not directly displace existing products
I
Exact impact elasticity (linear entry) in terms of observables
0
=
L̄p
r
(Share entrants in employment)
1 (Share entrants in innovation investments)
I
r = 4, L̄p = 0.83, share entrants in employment (3%), share
entrants in innovation investment (20%)
I
Impact effect
I
Implies G00  0.04
0
increases from 0.01 to 0.04
4% social depreciation: Aggregate productivity 20 years
4% social depreciation: GDP 20 years
Conclusion
I
Wide class of growth models
I
Simple approximation to transition dynamics
I
Two key sufficient statistics:
I
Impact elasticity
I
I
I
I
baseline growth rate
social depreciation of innovation expenditures
incumbents’ cost of innovation relative to entrants
Intertemporal knowledge spillovers
I
important for long run and welfare, less so for 20 yrs if low
growth and depreciation
I
Useful benchmark for evaluating quantitative implications of
richer new growth models
I
Challenge: quantifying gains from reallocation, find reliable
metrics for evaluating which firms should be doing relatively
more innovation spending and by how much
Alternative formulation (Acemoglu’s textbook)
I
I
⇣
Final good: Yt = Âj
n(j)
1 r N (j)
t
1 Âm=1 ymt (j)
⌘
r(1 a)
Lpt
Intermediate good: ymt (j) = exp(zm (j))xmt (j)
n(j)
1 Âm=1 xmt (j)Nt (j)
I
Aggregate input: Xt = Âj
I
Use of final good: Yt = Ct + Xt + Kt+1
I
Aggregate productivity with constant markups
Zt ⌘ k Â
n(j)
 exp(zm (j))
(1
1 r
r
d k ) Kt
Nt (j)
j 1 m=1
I
Define Ỹt = Yt
Xt
Baseline model
Ỹt = Zt L1pt a Kta = Ct + Kt+1
(1
d k ) Kt
ra
Kt
Example 1: Simple Quality Ladders
I
Innovation only by entrants, Nt (0)
I
Fraction s Nt (0) of products: z 0 = z +
r 1
r 1
Zt+1 = Zt
I
I
+ s Nt (0) exp(
z
z)
r 1
G function:
✓
◆
Zt+1
1
G = log
=
log 1 + s Nt (0) exp(
Zt
r 1
No social depreciation G 0 = 0
r 1
1 Zt
z)
r 1
1
Example 2: Simple Quality Ladders with innovation by
incumbents
I
Based on Klette and Kortum (2004)
I
All incumbents have = investment per product, yrtI
I
s Nt (0) + d(yrtI products: z 0 = z +
concave
I
Aggregate productivity
⇣
⇣ ⌘⌘
r 1
r 1
Zt+1 = Zt + s Nt (0) + d yrtI
exp(
I
, d (.) increasing and
z)
r 1
r 1
1 Zt
G function
G=
I
z
1
r
1
⇣
⇣
⌘
log 1 + s Nt (0) + d(yrtI ) exp(
No social depreciation, G 0 = 0
r
z)
1
1
⌘
Example 3: Simple Expanding Varieties
I
Innovation only by entrants, Nt (0)
I
Entrants z = 0, incumbents exit df
(Zt+1 )r
I
r 1
df ) exp(
z)
+ Nt (0)
r 1
1
r 1
1
= s Zt
(Zt )r
1
, incumbents
r 1
+ Nt (0) s Zt
G function
G = log
I
df ) (Zt )r
= (1
Add: Entrants average exp (z)r
exogenous growth z
Zt+1 = (1
I
1
⇣
Zt+1
1
=
log (1
Zt
r 1
G 0 < 0 if (1
df ) exp (
z)
r 1
df ) exp (
<1
z)
r 1
+ s Nt (0)
⌘
Example 4: Expanding Varieties with innovation by
incumbents to improve own products
I
Based on Atkeson and Burstein (2010)
Expanding varieties with endogenous growth by incumbents
I
Incumbent with productivity z invests yrtI exp(z)
,
r 1
I
r 1
z 0 = z + r 1 1 logd yrtI
I
Zt
Entrants average exp (z)r
r 1
Zt+1 = (1
1
I
d (.) increasing and concave
I
G function
G=
1
r
1
r 1
= s Zt
⇣ ⌘
r
df ) d yrtI Zt
⇣
log (1
1
r 1
+ Nt (0) s Zt
⇣ ⌘
⌘
df ) d yrtI + s Nt (0)
I
G 0 < 0 if (1
I
Can add df (irt ) decreasing in irt as in Luttmer (Restud 2011)
df ) d (0) < 1
Example 5: Expanding Varieties with innovation by
incumbents to create new products
I
Luttmer (2011) with one type of incumbents innovation
technology
I
All products have same productivity z = 0
I
Incumbent with n products invests yrtI n units of research good
I
Expected number of products next period:
[d0 + d(yrt (j)/n(j))] n
⇥
⇤
Total measure of products Mt+1 = d0 + d(yrtI ) Mt + s Ñt (0)
I
I
I
I
Aggregate productivity Zt = Mtr
1
1
G function, defining Nt (0) = Ñt (0)/Mt ,
⇣
⌘
1
G=
log d0 + d(yrtI ) + s Nt (0)
r 1
G0 =
1
r 1 log (d0 )
General model
Change in innovation subsidies and innovation intensity
I
Up to here: aggregate impact of policy induced increase in srt
I
Now: What is the impact of uniform subsidies on srt ?
Change in innovation subsidies and innovation intensity
I
Up to here: aggregate impact of policy induced increase in srt
I
Now: What is the impact of uniform subsidies on srt ?
I
Assumptions (satisfied in our model examples)
I
I
I
I
I
I
Static profits, ⇧t (j) = B t H (j), in BGP B̄ t = k Ȳt
Growth, interest rate unchanged between BGPs (g < 1)
Positive entry in BGP, N̄ (0) > 0
Given yr (j), unique invariant distribution N (j) /N (0), in BGP
Uniform innovation policies, t (j) = t
Across old and new BGP:
s̄r (1
Ē 0
Ȳ 0
I
Welfare
t) = s̄r0 1
Ē
= s̄r0
Ȳ
t0
s̄ r
Given initial t, sr , can solve for change in t to implement sr⇤
Sketch of proof
I
Equilibrium value function:
Vt (j) = maxyrt (j) B t H (j) (1
I
Etjj 0 |yrt (j) Vt+1 (j 0 )
1 + Rt
Entrants:
Vt (0) =
I
tt ) Prt yrt (j)+
Define vt (j) =
(1
Vt (j)
(1 tt )Prt .
tt ) ȳr (0) Prt +
In BGP with entry
k Ȳt
H (j)
P̄rt (1 t)
v̄t (j) = maxyrt (j)
0 = ȳr (0) +
Ȳt
,
P̄rt (1 t)
Nt (j)
,
N̄t (0)
Et0j 0 Vt+1 (j 0 )
1 + Rt
yrt (j) +
Etjj 0 |yrt (j) v̄t+1 (j 0 )
1 + r¯t
Et0j 0 v̄t+1 (j 0 )
1 + r¯t
I
r¯,
I
Using srt = Prt Yrt /Yt and Et = tt srt implies result
ȳrt (j),
Ȳrt unchanged
Gains from reallocating innovation across firms
I
Many models violate A2 and/or A3
I
Distribution N (j) enters G
I
Starting in equilibrium BGP, there exists reallocation of
innovation investment across firms, given Ȳrt , that increases
current growth rate
I
Can imply dynamic gains from favoring innovation of one type
of firms versus another, changing firm distribution in long run
Gains from reallocating innovation across firms
I
I
Many models violate A2 and/or A3
I
Distribution N (j) enters G
I
Starting in equilibrium BGP, there exists reallocation of
innovation investment across firms, given Ȳrt , that increases
current growth rate
I
Can imply dynamic gains from favoring innovation of one type
of firms versus another, changing firm distribution in long run
Conditional efficiency
I
Given any L̄r , allocation maximizes welfare
I
Starting from conditional efficient BGP, potential welfare gains
from changing L̄r , but no first-order welfare gains from
reallocating Yr across firms
Conditional efficiency
I
Model examples that do not satisfy A2 and A3, but BGP
equilibrium conditional efficient:
1. Expanding varieties with heterogeneity in innovation
technology (Luttmer 2011)
2. Klette-Kortum with heterogeneity in innovation technology,
separate innovation capacity from product leadership, constant
markups (variation of Lentz and Mortensen 2014)
I
Starting from conditional efficient BGP, no first-order welfare
gains from reallocating Yr across firms
I
Can approximate transition dynamics of change in {srt } by
considering any variation of innovation investment by
incumbents or entrants, e.g. vary only entry, without having to
re-solve dynamic optimization
I
Different variations can imply different aggregate dynamics,
same first-order change in welfare
Expanding varieties w/ heterog. innov. techn., Luttmer 11
I
Social planner
t
max •
t=0 b u Zt 1
r 1
Zt
=
L̄rt
subject to
 jj Nt (j)
j 1
⇣
Nt+1 (j) = f0j Nt (0) + Â fj 0 j 1
j0
d f j 0 + gj 0 yrt j 0
g 1
 yrt (j)Nt (j) + ȳr (0)Nt (0) = Art Zt
L̄rt
j 1
I
Equilibrium value functions
Vt (j) = maxyrt (j) ⇧t jj
+
1
d f (j) + gj (yrt (j))
Â0 fjj 0 Vt+1 j 0
1 + Rt
j
Prt ȳ (0) =
I
Prt yrt (j) + ...
1
1 + Rt
 f0j Vt+1 (j)
j
BGP satisfies conditional efficiency
⌘
Nt j 0
Quality ladders with heterogeneity in innovation technologies
I
Social planner
t
max •
t=0 b u Zt 1
I
◆r
j 1
1
Zt+1
= dt exp rz 1
1 +1
Zt
Nt+1 (j) = (1 dt ) Nt (j) + s [fj Nt (0) + dj (yrt (j)) Nt (j)]
Equilibrium value functions:
Vt (z; j) = ⇧t exp (z)r
1
Prt yrt (j)+
V̄t+1 (j) =
Z
(1
dt ) Vt+1 (z; j) + s dj (yrt (j)) V̄t+1 (j)
1 + Rt
Vt+1 (z +
z ; j) dMt (z)
s  fj V̄t+1 (j)
1 + Rt
Does not satisfy conditional efficiency: firms do not consider
ȳr (0) =
I
subject to
!
Nt (0) + Â dj (yrt (j)) Nt (j)
dt = s
✓
L̄rt
Separate research capacity from product market leadership
I
Social planner
t
max •
t=0 b u Zt 1
✓
Zt+1
Zt
◆r
1
=s
L̄rt
 dj (irt (j)) Nt (j)
j 1
⇣
Nt+1 (j) = fj Nt (0) + 1
!
subject to
exp
r 1
z
1 +1
⌘
d f (j) + gj (yrt (j)) Nt (j)
g 1
 (yrt (j) + irt (j)) Nt (j) + ȳr (0)Nt (0) = Art Zt
L̄rt
j 1
I
Equilibrium value functions
Vt (j) =
d f (j) + gj (yrt (j))
s dj (irt (j))
Vt+1 (j) +
V̄t+1
1 + Rt
1 + Rt
1
Prt ȳr (0) =
fj Vt+1 (j)
1 + Rt Â
BGP satisfies conditional efficiency
+
I
Prt (yrt (j) + irt (j)) + ...
1
Failure of Concavity of G, Peters (2014)
I
Quality ladders, r ! 1, own-product innovation by incumbents
increases markup
I
Aggregate productivity Zt = Vt Mt ,
logVt+1
I
logVt = s (d(yrt (1)) + Nt (0))
V implies linear G, but G not only driven by technology
Mt =
I
z
exp
Rh
⇣
log µ (j)
R
µ (j)
1
1
⌘i
dNt (j)
dNt (j)
Rise in entry, reduction in markup dispersion, increase in M t
I
G can violate A1 and A1’ (also violate A2 and A3)