Quality Ladders, Competition and Endogenous Growthpublic.econ.duke.edu/~staff/wrkshop_papers/2006-07Papers/Boldrin.… · Quality Ladders, Competition and Endogenous Growth Michele

Quality Ladders, Competition and Endogenous Growth

Michele Boldrin and David K. Levine

Department of Economics, Washington University in St. Louis

First Draft: July 10, 2006

This Draft: October 1, 2006

Abstract: We develop a competitive model of innovation along a quality ladder that

mimicks the features of more traditional models based on increasing returns and

monopoly power. First we show that, in the absence of fixed cost, the competitive

equilibrium implies alternating phases of capital widening and deepening, during which a

discrete amount of resorces is invested, at regular intervals in time, to move up the ladder.

Next we introduce technological fixed cost and show that, when these are not too large,

the same first-best efficient allocation obtains in competitive equilibrium. Finally we look

at the case in which fixed cost are large and competitive equilibrium is no longer first-

best efficient. We introduce two notions of competitive equilibrium, which we deem

appropriate for handling these circumstances, and show that such equilibria exist, there

are quite a few of them and they can be pareto ranked.

1

1. Introduction

We develop a competitive model of innovation along a quality ladder that

mimicks the features of more traditional models based on increasing returns and

monopoly power, but in which neither increasing nor monopoly power. There are,

nevertheless, fixed costs, non-atomic competitive innovators, and even schumpeterian

entrepreneurs, that is to say large agents that, by forecasting the impact that their solitary

actions will have on equilibrium prices, go alone in front of the fixed cost and innovate in

order to make profits. We show that in all these cases competitive equilibrium can be

defined by slightly improving on the standard definition, it exists – in fact, many of them

exist- it is not necessarily unique, and it does not necessarily achieve first best when the

fixed cost is very large.

The analysis, still incomplete, proceeds as follows in the current version of this

paper. We begin by summarizing the main properties of the elegant model of Grossman

and Helpman, in which a fixed cost is required to move up at each step on an infinite

ladder of goods of increasing quality, and in which the creator of a new good has a

monopoly power until a better good is created. Next we introduce a model of quality

ladders under condition of competition and without any fixed cost. We compute its

competitive equilibrium paths, and compare their similarities and differences with those

that obtain in the model of Grossman and Helpman. The key result here is that growth

takes place via innovation cycles, in which a deepening and a widening period alternate

in each cycle. Further, along the competitive equilibria so derived innovators pay a

(pseudo) fixed cost insofar as they invest a discrete amount of resources every time the

introduce a new quality good. Next we modify the model to allow for technological fixed

cost, and study its equilibria. We define two different notions of competitive equilibrium,

with non-atomic innovators and with an (atomic) entrepreneur, respectively. In both cases

we compute (some of) the equilibria, characterize their trajectories and begin analyzing

their welfare properties.

2. The Grossman-Helpman Model

There is a variety of models of quality ladders with fixed costs. increasing returns,

and external effects, most notably those of Romer [1990], Grossman and Helpman [1991]

2

and Aghion and Howitt [1992]. We adopt the model of Grossman and Helpman [1991] as

a particularly clean example that leads to a simple closed form solution and includes a

straightforward welfare analysis. Here we summarize their results, employing their

notation throughout.

Goods come in different qualities.1 Denote by the consumption (demand) for

goods of quality , let be the subjective interest rate, and let be a constant

measuring the increase in quality as we move one step up the quality ladder. Utility of the

representative consumer is

jd

j ρ 1λ >

0

logt jjtj

U e dρ λ∞ − dt⎡ ⎤= ⎢ ⎥⎣ ⎦∑∫ .

One unit of output of each quality requires just a unit of labor to obtain. The first

firm to reach step on the quality ladder is awarded a legal monopoly over that

technology. This monopoly lasts only until there is a new innovation and technology

is introduced, at which time all firms have access to technology . This is the

same device used by Aghion and Howitt, and has an obvious convenience for solving the

model. Taking labor to be the numeraire, the implication is that the price of output of

technology relative to that of technology is given by the limit pricing formula

.

j

1j + j

1j + j

p λ=The intensity of R&D for a firm is denoted by ι , and the probability of

successfully achieving the next step during a period of length dt is at a cost of . dtι Ia dtι Let E denote the steady state flow of consumer spending. Since the wage rate is

numeraire and price is λ the monopolist get a share (1 of these expenditures.

Since the cost of getting the monopoly is , the rate of return is (1 .

However, there is a chance ι of losing the monopoly, reducing the rate of return by this

amount. Since in steady state consumer expenditure is constant, the interest rate in

expenditure units is equal to the subjective discount factor. Equating the rate of return to

the interest rate gives the Grossman and Helpman equation determining research intensity

1/ )λ−

Ia 1/ ) / IE aλ−

(1 1/ )I

Ea

λ ι ρ− − = .

1 In the original Grossman-Helpman paper there were a continuum of identical sectors indexed by . Since this plays little role in the analysis, and for notational simplicity, we omit it here.

ω

3

There is a single unit of labor, the demand for which comes from the units

used in R&D and the units used to produce output.Ia ι

/E λ 2 Consequently, the resource

constraint is

. / 1Ia Eι λ+ =

Notice that the same labor is used for R&D as is used to produce output. This captures

the sensible idea that there is increasing cost of R&D. That the increasing cost is due to

resources being sucked out of the output sector is analytically convenient. It implies that

the cost of R&D, measured in units of output, is proportional to the current rung on the

quality ladder, making possible steady state analysis.

These two equations can be solved for the steady state research intensity

(1 1/ )Ia

λ ριλ

−= − .

By contrast the social optimum research intensity is derived by calculating steady

state utility to be [log . Since the optimal plan in a steady

state maximizes the steady state utility subject to the resource constraint, simple algebra

gives the optimum

log ( / )log ]/E λ ι ρ λ− + ρ

1*logIaριλ

= − .

3. Climbing the Ladder under Competition

The account given by Grossman and Helpman is one in which fixed costs and

monopoly power play a key role. In particular, if the fixed cost of research intensity

goes to zero, both the equilibrium and the optimal research intensities go to infinity.

Conversely, if monopoly power goes to zero so does research intensity, and economic

growth along with it.

Ia

Diminishing returns also play a role in the model. As R&D increases, it becomes

more costly relative to output, while its benefits do not increase correspondingly. These

diminishing returns determine the equilibrium and optimal research intensities, while it is

2 We have simplified Grossman and Helpman here by normalizing the stock of fixed labor to one.

4

monopoly power that, by overcoming the obstacle imposed by the fixed cost, makes such

intensity positive, hence innovation and growth altogether possible.

The story told is one in which the benefits of moving up the quality ladder never

decrease over time as one moves higher and higher – in which case the goal is to move to

better technologies as quickly as possible. It is the presence of fixed costs increasing with

R&D effort that provides a brake and helps determining a finite rate of advance up the

quality ladder per unit of time.

No doubt, fixed costs in the R&D sector are part of the reason that we do not

move up the quality ladder more quickly. But is it the only reason, or even the most

important reason? Suppose for example, that the technology of radios had just been

invented. Do we expect that, in a competitive setting, if the fixed cost of inventing

television were replaced by a linear one, the market would be first instantly flooded with

radios and then an immediate effort put forward to invent TVs? Or is it more reasonable

to suppose that initially only a few radios are produced and that their marginal value is

quite high since there are only a few of them, and it is costly to increase their production

quickly? Because the market value of radios is high, labor is allocated to produce more of

them and not to invent television right away: the higher cost of the latter enterprise is not

compensated by a proportionally higher marginal utility relative to that of the still scarce

radios. In this story, as radio production ramps up, the marginal value of additional radios

diminishes and eventually becomes so low that it makes sense to invest resources to

inventing and producing TVs.

The story of endogenous innovation on a quality ladder we want to tell is one

where the incentives to the introduction of innovations is determined by diminishing

returns to making copies of existing goods, that is to say of previous inventions. Because

inventing a new good always costs more than making a copy of an old one, it becomes

profitable to invent the new only when enough copies of the old good are around to make

its competitive price low enough relative to that of the new one. That there is, or there is

not, a fixed cost in the innovation process matters for the detail of the dynamics and for

the appropriate notion of competitive equilibrium one needs to adopt, but does not change

the substance of either the economic intuition or the qualitative properties of the

equilibrium growth. Because of this, we study a model without fixed cost first, and then

(Sections 6 and 7) move on to address this more technical aspect of the issue.

5

Before starting with a model of endogenous innovation due to diminishing

returns, it is useful to look at a typical example of how real quality ladders work,

borrowed from Irwin and Klenow [1994]. The good in question is the DRAM memory

chip. The different qualities correspond to the capacity of a single chip. The figure below,

showing shipments of different quality chips, is reproduced from that paper. The key fact

is that production of a particular quality does not ramp up istantaneously but gradually,

and that a new quality is introduced when the stock of the old one sitting around is fairly

large. Further, the old vintage is phased out gradually as the new vintage is introduced.

The figure vividly portrays our story for the semiconductors industry. Overwhelming

evidence suggests this is the usual pattern in most industries.3 The question this evidence

poses, and the intuition upon which a competitive model of endogenous growth with

quality ladders can be built upon, is: why introduce a new product if the old one is still

doing so well?

3 See, e.g., Hannan and McDowell [1987], Manuelli and Seshadri [2003], Rose and Joskow [1990], Sarkar [1998]

6

4. Innovation with Knowledge Capital

We adopt the same demand structure as Grossman and Helpman, that is we

assume the preferences for the representative consumer are

0

logt jjtj

U e dρ λ∞ − dt⎡ ⎤= ⎢ ⎥⎣ ⎦∑∫ .

However, we assume that output is produced both from labor and the existing stock of

productive capacity of producing that kind of output. For simplicity we identify

“productive capacity” with knowledge and assume that different rungs on the quality

ladder rather than representing just different qualities of consumption, correspond to

different qualities of capital and knowledge used to produce that particular output. We

denote by the combined stock of capital and embedded knowledge that goes about

producing quality output. By explicitly modeling the stock of knowledge, we can

distinguish between investment on a given rung – spreading and adopting knowledge of a

given type through learning, imitation or copying - and investment that moves between

rungs – innovation or the creation of new knowledge. We refer to as quality

knowledge capital. In practice knowledge capital can have many forms – it can be in the

form of human knowledge or human capital, but it can also be embodied in physical

form, such as books, or factories and machines of a certain design.

jk

j

jk j

Knowledge capital has two uses: it can be used either to generate more knowledge

capital or to produce consumption. More knowledge means either increasing the stock of

its own type of knowledge capital or creating a new type. If quality knowledge capital

is used to produce more knowledge of the same type, it does so at a fixed rate b per

unit of capital input. In other words, on any given rung of the quality ladder, the

production function is linear in the knowledge capital. More radios can be produced as

the existing technology for producing radios is imitated and additional capital of that kind

accumulated. We refer to this as capital widening, which may be thought of as

competitive imitation.

j

ρ>

4

We also allow for innovation – that is, the production of a higher quality of capital

from an existing quality. This represents capital deepening. Specifically, a unit of capital

4 We begin by assuming that no external effect due to imitation is present; the impact of externalities is taken up later. As in Grossman-Helpman, the assumption of linearity is purely for algebraic convenience: any sufficiently productive concave function would also do.

7

of quality can be produced from units of quality . This represents the

conversion of capital from, say, the production of radios to the production of TVs. In the

strict interpretation of as a stock of knowledge about how to produce, capital widening

corresponds to the spread of existing knowledge, and capital deepening to the creation of

new knowledge from old one. Our retained assumption is that, when measured in units of

current consumption, creation of new knowledge is costlier than spreading knowledge

already in existence, that is . This implies that, as long as it is not needed for

expanding consumption – i.e. until all labor is endowed with the most advanced kind of

knowledge capital – it is not socially efficient to introduce a new kind of knowledge

capital. The implications of

1j + 1a > j

jk

/aλ < 1

1/ aλ ≥ are briefly considered later.

Alternatively, knowledge capital of quality can be employed in the production

of quality consumption on a one-to-one basis. As in the Grossman and Helpman

model, output of consumption also requires labor – leading to diminishing returns for

each quality of knowledge capital. Specifically, each unit of quality knowledge capital

employed in the production of consumption requires a single unit of labor, and this

produces a unit flow of quality consumption. As before, we normalize the fixed labor

supply to one.

j

j

j

j

Let denote investment of knowledge capital of quality in the production of

knowledge capital of quality . Under our assumptions the motion of quality stock

of knowledge capital is then given by

jh j

1j + j

1( ) jj j j j

hk b k d h

a−= − − + ,

where represents the knowledge capital left over from the production of

consumption. We require that . Note that over a short period of time,

there is no limit on how much quality knowledge capital can be converted to quality

, so we allow also discrete conversion . What this

means is that the path solving the differential equation above is required to be just

upper-semicontinuous in t to allow to “jump up” at a countable number of points

in time. The assumption is needed because, as we show in the analytical part, even in the

absence of fixed cost the first best competitive equilibrium path may involve discrete

jumps in the stock of each quality of capital when this is first introduced.

jk d− j

jh

/j a

max{ , }j jk d≥

j

1j + 1 /j jk k a h+Δ = −Δ =

( )jk t

( )jk t

8

The latter observation is related to the existence of “multiple” equilibria. In fact,

from every initial condition there is a continuum of consumption&welfare-equivalent

competitive equilibrium paths; they are parameterized by the timing of conversion of

capital of type j into capital of type 1j + during the “deepening” period, a concept we

define more precisely later. In the case without fixed costs, it is always possible to build

the next type of capital gradually: continuously turn a certain quantity of old capital into

new at each point in time during the deepening period while accumulating both old and

new capital at the common rate ; when enough capital of the new type has been

accumulated, begin the widening phase during which old capital is retired from the

production of consumption and “dismissed” by being turned into additional new capital,

while the latter accumulates to finally become the only type of capital around. This is

payoff-equivalent to accumulating only old capital at the rate b for some time, turning an

appropriate amount of into all at once, and then let the latter grow at the rate b

for the rest of the remaining deepening time, if any is left. Hence, there are multiple

equilibria – some have jumps and some do not; the (time) limit of this set of equilibria is

the unique equilibrium with one large jump at the end of the deepening period, upon

which we focus in this section. Said differently, even in the absence of a fixed cost our

model predicts that, at discrete points in time, competitive innovators may invest a large

amount of resources in the innovation process.

b

jk 1jk +

The key technical fact is that this economy is an ordinary diminishing return

economy: both the first and second welfare theorems hold, so efficient allocations can be

decentralized as competitive equilibria and vice versa.5 Efficient paths must also employ

at most two adjacent qualities of knowledge capital in the production of

consumption at any moment of time. If two non-adjacent qualities of knowledge capital

are used to produce consumption, a calculation in the Appendix shows that we can

produce the same consumption output, while increasing the growth of knowledge capital,

by using more intermediate level knowledge capital and less of the two extreme qualities

in the production of consumption. It is also the case that once some quality of knowledge

1,j − j

5 This is shown for the discrete time version of this model by Boldrin and Levine [2001]. Note that in that paper we assumed that one unit of capital of type j and jλ− units of labor were required to produce a unit of quality consumption, and that the capital is used up in the process of producing consumption. This simply changes the units in which knowledge capital is measured. The assumptions here are chosen for maximum compatibility with Grossman and Helpman.

j

9

capital beyond the initial quality is used to produce output, labor must be fully employed.

If not, a calculation in the Appendix shows, we could produce the same amount of

consumption while increasing knowledge capital by having produced less of the leading

edge quality of knowledge capital and instead using more low quality knowledge capital,

and some of the previously unemployed labor, to produce consumption.

From these facts, it is relatively easy to find the optimal innovation intensity.

When only qualities of knowledge capital are used, full employment of labor is

achieved if

,j j + 1

j

. 1 1j jd d ++ =

The latter and the resource constraint imply that, when there is no need to introduce a

new type of knowledge capital, the technological trade off between consumption now and

consumption later is equal to b . A simple calculation shows that if we reduce

consumption by a small amount by using more low quality and less high quality

knowledge capital to produce current consumption and then, moments later, we use the

additional high quality knowledge capital, obtained through this investment to produce

additional consumption, we can get an increase of b consumption and have the

original stock of capital.

cΔτ

cΔ

To determine the equilibrium path of this economy, we first conjecture that the

optimal plan follows a particular cycle, then verify this conjecture and characterize the

details of the cycle. Our conjecture is that this cycle alternates between deepening and

widening. Initially, during the deepening phase, only knowledge capital of the lower

quality is used to produce consumption, and any quality knowledge capital is

simply allowed to grow more of the same kind at a rate of . After enough knowledge

capital of quality has been accumulated, it becomes efficient to use it in the

production of consumption. We start by analyzing the knowledge widening phase.

j 1j +b

1j +

Widening

While widening takes place the newer kind of capital is accumulated to fully

replace the old one, and in the meanwhile positive quantities of both stocks of capital are

used to produce consumption. At the beginning of this phase, all consumption is still

produced from quality knowledge capital, so that and . Output of

final consumption,

j 1jd = 1 0jd + =1

1j j

jd dλ λ +++ , is increased by shifting labor from the less advanced

10

to the more advanced quality of knowledge capital as the latter is progressively

accumulated. As is usual with logarithmic preferences and linear production functions,

the implication of intertemporal optimality is that consumption should grow at the

constant rate b . Since we already know that , at the start of the widening

phase consumption equals

ρ− (0) 1jd =jλ . After an amount of time then, it must be that τ

. 1 (1( ) ( )j j j b

j jd d e ρ τλ τ λ τ λ+ −++ = )

Using the full employment condition, we solve to find

( )

( )1

b

jed

ρ τ λτλ

− −=−

.

This can continue only until and , at which time quality

knowledge capital need to be introduced if consumption has to keep growing. Solving

we find the amount of time after which the widening phase ends

1( ) 0jd τ = 1 1( ) 1jd τ+ = 2j +

1( ) 0jd τ = 1τ

1logb

λτρ

=−

.

Deepening

Now move one step back and imagine we are at the end of the previous widening

phase, when only type knowledge capital was used to produce consumption. How

much capital of quality should we pile up before starting the new widening phase?

Until we do so, full employment implies that consumption is constant

j

1j +

, ( ) 1jd τ =

so that this phase corresponds to a growth recession, at least in this particular industry.6

We let denote the length of this deepening phase. 0τConsider a small reduction of consumption at the very end of the previous

widening phase in order to jump-start the production of quality knowledge capital

– noting that during widening consumption may be either increased or reduced. A unit

reduction in consumption enables the building of units of knowledge

1j +

0 /be τ a

1j +

6 Constant consumption during this phase is an implication of the simplifying assumptions we have been making. With more general technologies consumption growth during the deepening phase will be strictly smaller than during the widening phase, but it does not have to be zero. In fact, it may be positive or even negative but always less than b . ρ−

11

capital by the end of the deepening period. At that time, when this extra capital is

converted to consumption on a one-to-one basis, consumption is still the same as at the

beginning of the phase, so marginal utility has remained constant, and this future

consumption is worth units of current consumption. In addition, the consumption

produced by quality knowledge is worth λ times that produced from quality

knowledge. At the social optimum, this production shift must be neutral, so when quality

is first used to produce consumption, it must be that

0e ρτ−

1j + j

1j +

. 0 01 ( be e aρτ τλ −= / )

From this we can solve to find the length of the capital deepening period

0log log

ρab

λτ −=−

,

which is positive as long as .a λ> When a λ≤ we have what in Boldrin and Levine

[2001] we termed the “Solow balanced growth path”, in which innovation takes place

instantaneously at a smooth pace: at every instant in time a small amount of a new quality

is introduced that is soon to be replaced by a better one, and so on, while full employment

is maintained. Aggregate consumption grows at a constant rate of λ per unit of time.

The very same flow of consumption service can be obtained through a smooth

innovation process turning 0 1μ< < units of type j capital into / aμ units of type 1j +

capital at each instant 0[0, ]t τ∈ to let it grow into 0(( / ) b ta e τμ )− units of the same by the

end of the deepening period. That this is the case can be checked by solving

0

0 0( )

0

b t be dt eτ

τ τμ − =∫

for μ , to get 0/(1 )bb e τμ −= − . This clarifies our earlier remark on the presence of a

continuum of payoff-equivalent equilibria, the limit of which – i.e. the equilibrium in

which the transition from type j to type 1j + takes place once and for all in a lump-sum

form at the start of the deepening period – will be the one we focus upon in the rest of the

paper.

12

Intensity of innovation

The intensity of innovation is the rate at which we move up the capital ladder.

This is just the inverse of the length of the cycle, i.e. of the sum of the two parts

of the cycle 0τ τ+ 1

*logbjaρ−= .

Remarks

There are several aspects of the model worth highlighting. First is the fact that

when new knowledge is created from old, less of the new knowledge is made available.

In other words, an additional unit of new knowledge requires more than one unit of the

old knowledge. This is a natural assumption: the stock of knowledge capital is measured

by the amount of labor it can employ. If the creation of new knowledge capital, for

example, is in the form of upgrading old machines to a new technology, there will

generally be some wastage in the conversion process. But more broadly, the fact that

when labor embodying old knowledge capital is converted into labor embodying the new

knowledge capital, this “new” labor will not have the benefit of time and experience that

had accrued to the old. For example, if the knowledge capital is human capital, then the

expertise of engineers and others in the new knowledge capital will naturally be less than

in the old.

One question is why we assume that new knowledge capital loses the productive

capacity of the old. That is, simply because a new skill is acquired does not mean that the

old one is lost – although for physical capital it may mean exactly that. However, the

critical assumption is that only one of the two skills can be employed at a moment of time

by a given individual, not that the old skill is lost: while you may still be capable of

flying a Cessna after learning to fly a Stealth bomber, at any given moment in time you

must be acting as one of the two kinds of pilot, but not as both. If we introduced a

technology for converting quality knowledge capital back to quality at the same

ratio as the forward conversion, then this would precisely capture knowledge that was not

lost, but knowledge capital as a resource that could be deployed at only one level at a

time. However, there would be no reason to ever use the backward conversion

technology, so the equilibrium consumption path would not change. Note however, that

1j + j

13

the possibility of converting back at the same ratio as forward would have an important

implication in the next section where we consider a fixed cost of conversion. The

implication is that the fixed cost would not matter, as it could be “undone” by a

subsequent backward conversion. This may partially explain why, when we consider the

empirics of fixed costs, they do not seem to matter a great deal: most electrical engineers,

if not most economists, can still change a light bulb by themselves.

Finally, there is an aspect of the conversion technology that may strike the reader

as unrealistic: the conversion takes place instantly – while in fact the development of new

knowledge generally takes time. This however turns out to be a matter of convenience

rather than substance. Suppose that a units of quality knowledge capital converted at

time do not become available as a unit of quality knowledge capital until time

. Since the model is one of perfect foresight, the need for conversion can be

foreseen, so if we imagine the conversion taking place at time t then the a units of

quality knowledge capital not being used to produce consumption would have grown

to . Hence if we consider a model with no delay but cost of conversion ,

this is equivalent to the model with delay and cost of conversion a . In effect, all we have

done is to assume that any time delay is capitalized into the cost of conversion.

j

t 1j +

t +Δ

+Δ

jbae Δ ' ba ae Δ=

Evolution of the stocks

We now compute how the stocks of knowledge capital evolve during the cycle.

We begin with the deepening phase and normalize time so it begins at and ends at 0t =

0t τ= , at which point widening begins to end at 0 1t τ τ τ= + = .

During deepening the growth rate of consumption is zero and, for all 00 t τ≤ ≤

and . What matters is the value at ( ) 1jd t = 1( ) 0jd t+ = 0t = of the total capital of type

j converted into type 1j + during this phase, which we denote with . Conversion

takes place at , hence

F

0t = (0) 1jk F= + , and ( ) 1jk t = for 0(0, ]t τ∈ . Write /k F a= , so

1( ) btjk t ke+ = and for 1( ) 0jd t+ = 0[0, ]t τ∈ .

Move now to widening, during which and shrink in parallel from 1 at ( )jk t ( )jd t

0t τ= , to 0 at 0t 1τ τ= + . From the fact that 1jd d j+= − and that /c c b ρ= − , one gets

( ) ( )1j jbd dρ λ ρ

λ−= + −−

b ,

14

a linear differential equation on the interval , with given initial conditions.

This has the unique solution 0 0 1[ ,τ τ τ+ ]

0( )( )

( )1 1

b t

jed t

ρ τ λλ λ

− −

= −− −

.

Because , during widening the amount of new capital producing

consumption expands as 1( ) 1 ( )jd t d t+ = − j

0( )( )

11( )

1 1

b t

jed t

ρ τ

λ λ

− −

+ = −− −

,

which also satisfies the initial and terminal conditions imposed by zero profits. Now use

the fact that for all to derive the time path of , the

shrinking stock, during this phase. Plugging the results in the law of motion for ,

the expanding stock, we have

( ) ( )j jk t d t= 0 0 1[ , ]t τ τ τ∈ + ( )jk t

1( )jk t+

[ ]0( )( )

1 1( ) ( ) ( 1)1 (1 )

b t

j jb ek t bk t b a

a

ρ τρ

λ λ

− −

+ += − + − +− −

for 0 0 1( , ]t τ τ τ∈ + . Solving this we find that

0( )( )

1( 1) 1( )

1 1

b t

jb a ek t C

a

ρ τρρ λ λ

− −

+− +

= − + +− −

The initial condition 01 0( ) ( / ) b

jk F a e ττ+ = can be used to eliminate the constant of

integration C to get

0 0( )( )1

( 1)( ) [1 ](1 )

b t bj

b a Fk t e ea a

ρ τ τρρ λ

− −+

− += − +

−.

For the cycle to repeat itself, at the end of the widening period the stock of capital of

quality must equal 1 again. This requires solving 1j + F+

01( )( )1 0 1

( 1)( ) [1 ] 1(1 )

bbj

b a Fk ea a

τρ τ e Fρτ τρ λ

−+

− ++ = − + = +

−

to find the value of the equilibrium initial investment in innovation, , as a function of the model’s parameters.

*F

15

Prices

The zero profit conditions can be exploited to derive the equilibrium prices

expressed in units of current consumption. Logarithmic utility implies the latter is

proportional to , while the inelastic labor supply allows us to normalize the

wage rate to zero.

( ) 1/ ( )p t c t=

Consider first prices during deepening, when consumption is constant and it is

being produced only by the capital stock of quality j . Denoting with the price of

quality

( )jq t

j knowledge capital, the zero profit condition reads

( ) ( ) 0bj jq t q t e ττ− + + = ,

which has the simple solution for ( ) (0) btj jq t q e= 0[0, ].t τ∈ One also has

1( ) ( ) 0bj jaq t q t e ττ+− + + =

during the same phase.

During widening [TO BE ADDED]

5. Comparison of the Models

We have, now, three possible models explaining endogenous growth. One is the

Grossman-Helpman model, in which the innovation rate is given by

(1 1/ )Ia

λ ριλ

−= − .

Another is the Grossman-Helpman efficient solution – which as we will argue below may

correspond better to real institutions than the particular model of monopolistic

competition they propose. Here the innovation rate is given by

1*logIaριλ

= − .

Finally, we have the model of competitive knowledge-capital accumulation, in which the

innovation rate is given by

*logbjaρ−= .

16

Each of these models is contrived to get a closed form solution. Qualitatively,

each is similar in the cost of innovating and in the assumption of impatience, summarized

by the discount factor. As consumers are more patient, the frequency of innovation goes

up. As it becomes more costly, innovation goes down. There are of course minor

differences in the functional forms between these solutions. But the functional forms

depend on a variety of assumptions – log utility, exponentially improving steps, and so

forth, that were contrived to make the models easy to solve, so the particular functional

forms have no strong claim of correctness. Moreover, the models differ in ways that are

designed to ease the solution. For example, it is technically convenient for Grossman and

Helpman to assume the same labor is used in producing knowledge as in producing

output. However, it is technically convenient for us to assume that knowledge capital is

produced only from knowledge capital.

What are then the substantive, as opposed to the technically convenient,

differences? First, the parameterλ (how high one step of the ladder is) has no effect in

our model – this is due to the presence of two offsetting effects, increasing the intensity

of innovation during the first, deepening, part of the cycle – this effect being present also

in Grossman-Helpman – and decreasing it during the second part, widening. Second, the

labor saving model has the extra widening parameter b , representing the rate (fixed only

for algebraic convenience) at which productive capacity increases or is turned into usable

output (which is the same thing.) In a certain sense the Grossman-Helpman model, like

all models of this class, assumes that b . This is because once the fixed cost is paid

and a new rung has been introduced the technology per se allows one to make an infinite

number of copies of the

= ∞

j th good: a finite number of copies is made only because the

monopoly power of the first innovator is used to prevent competitive imitation. Put it

differently, the Grossman-Helpman model assumes that the movement from one

particular vintage to the next can be infinitely rapid (once discovered knowledge is a

public good and everyone has it); the presence of a fixed cost at each step explains why

things do not go haywire. The latter seems to be the crucial difference, which should be

considered carefully.

17

6. Competitive Equilibrium with Fixed Cost: Non- Atomic Innovators

We now assume that there is a technologically determined fixed cost of

introducing new knowledge for the first time. Recall that, in the competitive equilibrium

of the model without any technological fixed cost, at the end of the deepening phase a

discrete amount of old capital, call it , is invested in the innovation process to yield *F* * /k F= a units of new capital. Retain the same notation, using a star to distinguish

the equilibrium quantities of the model without fixed cost from the exogenous

technological parameters and F k of the model with fixed costs. Specifically, we

assume that to create quality capital from quality capital requires a fixed cost of

units of quality capital. This results in the creation of

1j + j

F j /k F a= < F initial units

of quality knowledge capital. For notational convenience, once the fixed cost is

incurred, we assume it is possible to convert additional units of quality knowledge

capital to quality knowledge capital at the same rate a . Because it simplifies the

analysis and makes sense, we also assume that if quality knowledge capital is

introduced for the first time at time then quality knowledge capital cannot also

be introduced at time ; that is, the times of introduction must satisfy ,

although the gap may be arbitrarily small. We show later that, as long as innovation is

feasible and the cycle length is positive, the sum total of the deepening and the widening

periods is a constant, independent of . Together with the latter assumption this implies

that the distance in time between

1j +

j

1j +

j

jt 1j +

jt 1j jt t+ >

F

jt and 1jt + is either constant or infinity.

We are interested in studying both the case in which and the fixed cost is

therefore not binding, labelled the small fixed cost, and the opposite one in which

and the fixed cost is binding, which we call the case of a large fixed cost. All

remaining parameters are as defined before.

*F F≤

*F F>

Before proceeding further, we need to explain how invention takes place.

Equilibrium with respect to pre-existing knowledge capital accumulation, savings, and so

forth, is to be defined by means of the usual competitive notion: given prices, consumers

optimize utility, producers maximize profits, and markets clear. However, the decision to

invent now involves a discrete change in the economy and in the technology set, meaning

that if invention takes place prices necessarily change. Let us imagine that there is a large

pool of potential non-atomic innovators, which we identify with the unit measure of

18

representative consumers. We consider separately two different definitions of competitive

equilibrium that are supported by two different behavioral assumptions.

In the first, that we label competitive equilibrium with non-atomic innovators, all

agents are infinitesimaly small, hence their demand for consumption and any kind of

capital is infinitesimal relative to the size of the economy. Such agents are always

potential innovators but are also always price takers: they do not expect their actions to

affect the equilibrium prices, even when they are contributing to the innovation

enterprise. In the latter circumstances, the fixed cost is paid because, at the given

prices, each non-atomic agent demands a quantity of the stock of capital of quality

F

F j

and invests it in the invention process, and this action yields non-negative profits at the

ensuing equilibrium prices.

In the second definition, called entrepreneurial competitive equilibrium and

examined in the following section, agents are allowed to be, at least for a fleeting

moment in time, of non-atomic size insofar as a single individual may actually demand

the whole quantity , or even more. To fix intuition imagine the same pool of

individuals as before and assume that, at each moment in time, one of these people is

randomly chosen to have an “innovation opportunity” while the other are not. This

(potential) entrepreneur understands the equilibrium – and prices – that will result if he

chooses to purchase the amount and carry out the invention. Once he has created new

knowledge capital, we abstract from any (even instantaneous) monopoly power he

may have acquired and he takes the prices as given. We assume the entire amount of

newly produced knowledge capital –

F

F

1j +

k , at a minimum – must be sold, and that, once this

is done, it trades in a perfectly competitive market. The difference from the previous case

is that the entrepreneur understands the innovation will modify the state of the economy

and change the equilibrium prices, hence he takes action only when his profits are non-

negative at the new equilibrium prices. Since the opportunity to invent is fleeting –

another entrepreneur will be chosen in the next moment – the innovator innovates the

first moment it is profitable to do so.

To define equilibrium with fixed costs, we start by considering viable

initial stocks of knowledge capital. These are finite sequences of stocks of knowledge

capital of different qualities that satisfy the feasibility restriction 0 1( , , , )Jk k k k= … J

Jk ≥ k . Associated to such initial stocks, we also define feasible paths, which are

19

infinite sequences of time paths , 0 1( ) [ ( ), ( ),..., ( ),...]jk t k t k t k t= [0, )t∈ ∞ , and of initial

stocks and innovation times,

feasible from the initial conditions. More precisely, the list collects the entire path of

each kind of capital stock, while

1 1 2 2( ) [ ( ), ( ),..., ( ),...]J J J J J n J nk t k t k t k t+ + + + + +Δ = Δ Δ Δ( )k t

( )k tΔ reports the time jt at which capital of quality j

is first introduced, in the amount ( )j jk tΔ , by investing 1( )j ja k t F−− Δ ≥ units of the

previous quality of capital.

Definition 1. A competitive equilibrium with atomic innovators E with respect to

a viable consists of: Jk

(i) a non-decreasing sequence of times , with for and, for

, either , or ; 0 1( , , )t t … 0jt = j J≤

j J> 1j jt t −> jt = ∞

(ii) a path of capital and capital prices for , and a path of

consumption and consumption prices that satisfy the conditions

( ) 0jk t ≥ ( ) 0jq t ≥ jt t≥( ) 0c t ≥ ( ) 0p t ≥

(1) [Consumer Optimality] maximizes ( )c t

subject to 0

log( ( ))te c tρ∞ −∫ dt0 0

( ) ( ) ( ) ( )t te p t c t dt e p t c t dtρ ρ∞ ∞− −≤∫ ∫ .

(2) [Optimal Production Plans at ] jt

1( ) ( )/j j j jk t k t a k−Δ = −Δ ≥

1( ) ( )j j j jq t aq t−=

(3) [Optimal Production Plans for ] , and jt t> 1( ) 0, ( ) 0j jd t h t−≥ ≥

1( )( ) ( ( ) ( )) ( ) jj j j j

h tk t b k t d t h t

a−= − − + ,

, ( ) max{ ( ), ( )}j jk t d t h t≥ j

j

and if , 1( ) ( )j jq t aq t−≤ 1( ) ( )j jq t aq t−= 1( ) 0jh t− >

maximize ( ), ( )j jk t d t ( ) ( ) ( ) [ ] ( ) jj j j j j j jq t k q t k q t b k d p t dρ λ− + + − +

(4) [Social Feasibility]

( ) ( )jjtj

c t d tλ=∑(5) [Boundedness] for some number 0K >

20

, at all t . 0

( )jjk t K∞

=<∑

A few remarks on the definition are in order. The first requisite is the standard

definition of utility maximization under an intertemporal budget constraint. The second

says that the fixed cost and zero profit restrictions are satisfied each time a new quality of

capital is introduced, and that the latter takes place with a discrete upward (respectively

downward) jump at the time the new capital is invented by using a discrete amount of the

old capital. This implies equilibrium time paths of capital and prices that are only

continuous from the right. The third is standard as it requires profit maximization at each

point in time after a new kind of capital is created, and that its time path obeys the

technological law of motion. The fourth and fifth are standard material balances and

boundedness conditions. Notice what these restrictions imply: along an equilibrium path,

at a designated time , a discrete amount of old capital, jt 1jk − FΔ ≥ is transformed into

1 /j jk a k−Δ = Δ units of new capital determining a new set of initial conditions from

which the equilibrium path evolves. As there is perfect foresight, and the investment

decision of each non-atomic agent takes for the price paths as given, such jump is also

expected and incorporated into the price paths. At those prices, the continuum of

innovators find it profit maximizing to purchase an amount 1jk − FΔ ≥ of capital of type

at the price 1j − 1( )j jq t− , produce capital of type j , and sell its entire amount ( )j jk tΔ at

the price ( )j jq t .

Implications of the zero profit condition

As shown in the Appendix, the equilibrium paths in the presence of a fixed cost

resemble those in the economy without fixed cost, insofar as they are composed of

sequences alternating capital deepening and capital widening sequences. Retain the same

notation as before for denoting the lengths of such phases and note the following

remarkable result.

At time type knowledge capital can be purchased for , while at

type capital can be sold, in time 0 consumption units, for .

Also, units of quality knowledge capital purchased at

0t = j / (0)j cλ

0τ 1j + 0 10/ ( )je cρτ λ τ− +

k j 0t = will result in 0bk ea

τ units

of knowledge capital at . Consequently, zero profit for innovation implies 1j + 0τ

0( )

0( )(0)

bc ec a

ρ ττ λ−= .

21

In other words, a triple 0( , , )k Fτ is a candidate for equilibrium if, along the path

associated to it, this zero profit condition and the physical constraint are both

satisfied.

0bke Fτ ≥

As shown in the Appendix, it remain true that there are only two adjacent

qualities of knowledge capital in use, hence , and, after

some initial period, there must be full employment, so . While there

are positive stocks of both qualities of capital, widening must take place as before, so, for

11( ) ( ) ( )j j

j jc t d t d tλ λ + += +

1( ) ( ) 1j jd t d t++ =

0 0 1[ , ]t τ τ τ∈ +

. 0( )(0( ) ( ) b tc t c e ρ ττ − −= )

1When , only quality knowledge capital is used to produce

consumption hence consumption is . This gives 0t τ τ= + 1j +

1( ) jc t λ +=

, 11 ( )0( ) j bc e ρ ττ λ + − −=

and, because , the zero profit condition simplifies to (0) jc λ=

. 0 1( )( )be aρ τ τ− + =

The latter is a valuable finding: given the model’s parameters, at the associated

competitive equilibria the total length of the innovation cycle remains constant even in

the presence of a fixed cost.

Because the increase in consumption level over a full cycle is also fixed at λ , as

we move over the admissible triplets 0( , , )k Fτ , what changes is only the ratio 0 1/τ τ ,

ranging between (log log ) / loga λ λ− and 1. As the fixed cost increases or, which is

the same, the parameter 1/ decreases, the deepening phase increases in length while the

time spent widening shrinks. In the most extreme case, the consumption path is constant

at

F

a

jλ for the whole length of the cycle, i.e. for 0 1[0, )t τ τ∈ + , and then jump to 1jλ + at

time 1 0jt 1τ τ+ = + .

The Case of Small Fixed Cost

The best way to get a handle on the intuition underlying the competitive

equilibrium with non-atomic innovators is to consider first the case where the fixed cost

is small, and therefore not binding in a sense that will be made precise momentarily.

Recall that /a F k= and recall the economy without a fixed cost but with the cost of

conversion a . That economy has a unique first best equilibrium consumption path, which

22

we studied earlier and along which the periods of deepening and widening are uniquely

well defined. We want to make sure the same equilibrium obtains when a small fixed cost

applies; that exactly the same equilibrium is technologically feasible is what we mean by

saying that the fixed cost is not binding.

In the economy without fixed cost deepening lasts for periods, at the end of

which the new capital is obtained by converting units of type capital

into

0τ1j + *F ≥ F j

* * /k F a= ≥ k units of . Then deepening ends and widening starts, part of the

stock of new capital is shifted to the production of consumption, while the rest is still

used to accumulate more of itself at the rate b in order to fully replace the old capital

stock in the production of consumption. Widening ends, and a new deepening phase

begins, when only capital of type

1j +

1j + remains and consumption has reached its new

level of 1jλ + .

The simple observation to be made here is that the equilibrium in the economy

without a fixed cost satisfies all the conditions that an equilibrium with non-atomic

innovators must satisfy in the economy with a fixed cost. Because of this, it is also an

equilibrium for the economy with a fixed cost. Because the economy without fixed cost

has a unique (in the sense previously clarified) equilibrium that is also efficient, and

, the same is true of the economy with a fixed cost. We summarize *F ≥ F

Theorem 1: In the economy with a small fixed cost, for given initial conditions,

there exists a unique competitive equilibrium with non-atomic innovators. This

equilibrium is efficient.

The Case of Large Fixed Cost

When, on the other hand, , things are different as a considerable leeway is

introduced about the time at which the innovation may take place. First of all, the

competitive equilibrium of the economy without fixed cost is no longer feasible. Second,

it is also clear that the new competitive equilibrium is in general not efficient, as there is

no reason to expect that at the time at which it would be socially optimal to introduce

capital of quality

*F F>

*jt

j the zero profit condition 2 above is satisfied, i.e. there is no reason to

expect that obtains if *1( ) ( )j j j jq t aq t−= * * *

1( ) ( )/j j j jk t k t a k−Δ = −Δ ≥ is chosen at the

prescribed socially optimal time. Hence, in general, in the economy with a large fixed

23

cost, for given initial conditions, the competitive equilibrium with non-atomic innovators

is not efficient.

The computation given earlier for the total length of the innovation cycle still

applies, and a calculation given in the Appendix shows that the length of the two phases

and the associated amount of capital stock invested in the innovation process can still be

computed as a function of the parameters. Hence, equilibrium as defined here exists; in

fact: many such equilibria exist as all that is required for a feasible path from viable

initial conditions to be an equilibrium with non-atomic innovators is that, along the

candidate path, for all j J> innovation takes place when

holds, and 1 1 1( ) ( )j j j j j jk q t q t k+ + +−Δ = Δ 1+ jk F−Δ ≥ .

A complication emerges as there exist a large number of paths all satisfying such

conditions for appropriately chosen { } 1j j Jt

∞

= +, { } 1j j J

k∞

= +−Δ , and associated { }0 1( ) j Jjτ ∞

= +.

This set of equilibria is parameterized by the amount of capital of type j invested (and

relative amount of capital of type 1j + obtained), and the induced lengths of the

deepening and widening phases.

Let the pair ,0( ,j jk )τΔ denote one such equilibrium, where both the jkΔ and the

and ,0jτ may vary with j along a given equilibrium path – i.e. there are equilibria that

are not stationary but cyclic and, probably, chaotic, as shown in a calculation in the

Appendix. Let us describe how one such equilibrium unfolds. By definition widening

starts as soon as the fixed cost is paid and /jk F aΔ ≥ units of the new kind of capital

appear. Because the fixed cost is binding, this will take place after the innovation time in

the competitive equilibrium of the associated economy without fixed cost. Here, the

deepening period consists of the times 0[0, ]t τ∈ during which consumption is constant at

( ) jc t λ= , only capital of type j is used to produce it and, finally, capital of type j in

excess of is accumulated at the rate b to eventually pay the fixed cost 1jk = jk−Δ

needed to create the new knowledge capital 1j + . After the fixed cost is paid the

widening phase begins, along which consumption grows at the constant rate ( )b ρ− .

Because consumption always grows at the same rate during widening, is always constant

during deepening and always jumps up of a factor λ as its production shifts completely

from using type j to type 1j + capital, it is clear that, among the equilibria starting at the

same viable initial condition, those with a higher jkΔ will have a longer ,0jτ , less

consumption (as widening is shorter) and a higher stock of capital along the overall cycle,

24

than those with a lower jkΔ . This immediately implies that the equilibrium with the

shortest ,0jτ is also the most efficient, as it delivers more total utility.

Theorem 2: The earliest competitive equilibrium with non-atomic innovators pareto

dominates all other steady state competitive equilibria with non-atomic innovators, but is

not first best.

Other two relevant properties of the set of competitive equilibria with non-atomic

innovators are that, for every given /jk F aΔ ≥ , there is a unique stationary equilibrium

and at least one cyclical equilibrium of period two. Again, notice that in general

, hence none of these equilibria is efficient, as the efficient allocation has

. Finally, there are also equilibria, an infinite number of them in fact, in

which for , and hence

/jk FΔ > a

a/jk FΔ =

0jkΔ = *j J> * 1,0Jτ + = +∞ [TO BE ADDED: HOW TO

SELECT THE EARLIEST EQUILIBRIUM, WHICH IS THE CLOSEST TO THE

EFFICIENT ONE. HOW TO GET RID OF THE BAD EQUILIBRIA IN WHICH

AGENTS STOP INNOVATING FOREVER]

7. Competitive Equilibrium with Fixed Cost: Entrepreneurial Innovators

Now we turn to the issue of entrepreneurship – that is, TO the profitability of

introducing an innovation through an investement of a strictly positive size, into a

competitive equilibrium with non-atomic innovators. Fix one such competitive

equilibrium . A -innovation is a pair ( composed of the time

at which a single agent purchases units of capital of quality

E j , ( ))jt k t 1ˆ ˆj jt t− < < tF 1j − and turns them into

units of capital of quality /F a j . It should be interpreted as introducing quality

knowledge capital at a time t earlier than the equilibrium time when all agents behave

as non-atomistic ones. Hence the terminology: the Schumpeterian “entrepreneur” shows

up in an economy of competitive agents, and innovates all alone even if he is forced to

behave competitively all the times and, in particular, he is prevented from actively using

the monopoly power his single innovation may provide him with.

j

jt

Because an innovation causes a discrete change in the economy, we must be

explicit about what happens following an innovation. We say that a competitive

equilibrium with respect to is a feasible continuation for the -innovation ( , if E jk j )t F

(a) , 1 1ˆ( ) ( )j jk t k t F− −= − 1 1

ˆ( ) ( '), for ;j jk t k t t t− −= <

(b) for and all t . ' 'ˆ( ) ( )j jk t k t= 'j j< − 1

25

A continuation is Markov if . That is, it is

Markov if both equilibria reach the same state of the economy when the next innovation

takes place. If we consider that plans for introducing innovations are conditioned on the

current state, that is, the vector of capital stocks, then this assumption is conceptually the

same as the game theoretic notion of Markov perfection. It means that the innovation

does not cause changes in plans concerning the introduction of higher quality knowledge

capital. It rules out the possibility that entrepreneurship is “punished” by changing the

nature of the equilibrium in the future. Such “punishment” equilibria seem implausible in

a competitive setting.

1ˆ ˆˆ ˆ( ) ( ) and ( ) (jk t k t k t k t+= Δ = Δ 1)j+

We say that a -innovation is profitable with respect to a feasible

continuation if and . What this condition means

is the following: the entrepreneur that is introducing the capital of type

j ( , )t F

E 1( ) ( )j j j jq t aq t−≥ 1ˆ( ) ( )j j j jq t aq t−>

j can buy for a

shade lower than the new price if it is higher than the old price. Notice that the

introduction of an atomic entrepreneur in the competitive economy brings about capital

gains and losses at the new entrepreneurial equilibrium. [TO BE ADDED]

This issue of capital gains and losses is not a new one for the theory of

innovation: it was first explored in Hirshleifer [1971] who pointed out the possibility of

very great profits from prior trading based on inside information about a soon to occur

innovation. Here knowledge is common, so such a “secret” deviation is not possible – in

a model where innovation was less competitive it would be a crucial issue, and a

“competitive” theory could well result in too much innovation rather than too little, as

Hirshleifer himself pointed out. We should also remark that a similar issue arises in

ordinary competitive equilibrium theory and in the models of the class to which the

Grossman-Helpman model considered above belong to, although it is not widely

recognized.

In our model the potential for creating capital gains comes about through

effectively destroying part of the capital stock through premature innovation. [TO BE

ADDED] Alternatively, it could be that no trader has access to adequate resources to buy

out the market. In this setting, that would imply that innovation was impossible because

of financial constraints. While there may be less investment in innovation, or any other

type of capital investment, due to financial constraints, that is not the usual theory of

26

competition and has been well studied elsewhere, so, as the problem has little to with

innovation, we do not pursue it here.

Our general assumption concerning entrepreneurial deviations is that they are

“small” from an innovation perspective. With this we mean that they take place close in

time to the instant in which the same innovation would have taken place in the original

equilibrium with non-atomic innovators only, i.e. that , and that the innovation

consists only of obtaining existing knowledge capital and converting it. However, when

innovation stops altogether, there is an alternative class of innovations that are “small.”

Consider the class of equilibria with non-atomic innovators at which innovation ceases

after knowledge capital of quality is introduced. For , then it must be that

. In such a situation it becomes possible to

acquire a tiny bit of capital ε at time , by sacrificing consumption over an interval

of time

1jt t −>

*J *jt t≥*

*( ) 1, ( ) 0, *, ( ) JJ jk t k t j J c t λ= = ≠ =

*Jt t>ε . This capital can then be allowed to grow for a long period of time, for

example, until the time t at which is exactly the steady state capital required for

the minimal innovation constrained equilibrium. This is necessarily greater than

/bte aεk . Then

at time t all quality knowledge capital is converted to quality . The only

continuation innovation constrained equilibrium possible is now the minimal innovation

constrained equilibrium. Since there is no unemployment in the continuation equilibrium,

. To complete the argument we need to show that in

addition, because so much time has elapsed since the “seed” of innovation was planted,

and consumption, after all, simply jumps to from , it must be that the prices of

capital satisfy . It follows that holds, and the

deviations is profitable. In other words, a schumpeterian entrepreneur is enough to

destroy the bad equilibria with non-atomic innovators in which innovation stops forever

after a certain time.

*J * 1J +

* 1 * 1 * * 1( ) ( )J J J Jq t aq t +=

)+

+ +

* 1Jλ + *Jλ

* * 1 * * 1ˆ( ) (J J J Jq t q t+ > 1ˆ( ) ( )j j j jq t aq t−>

With this in mind, we now define an entrepreneurial competitive equilibrium

Definition 2. An entrepreneurial competitive equilibrium is defined as a

competitive equilibrium with non-atomic innovators that

(1) does not admit innovations that are profitable with respect to feasible Markov

continuations,

(2) does not stop innovating, that is, for all . jt <∞ j

27

Theorem 3: There is a unique entrepreneurial competitive equilibrium: it is the earliest

competitive equilibrium with non-atomic innovators.

Proof sketch:

1. If the innovation takes place “late”, then there must be some time at which one must

convert more than F units of capital. So one could go a moment earlier and convert only

units. Any new equilibrium has slightly less consumption (because of the Markov

assumption: you have to hit the same target at the end of widening) – but because you

converted so little capital, you can achieve this reduction without unemploying labor.

This means zero profits at new equilibrium prices, and since the price of capital went up

(consumption fell) positive profits if we can buy the input at the old equilibrium price.

F

2. if you innovate “on time” then you convert exactly F units of capital. That means if

you go early, to get back to the same equilibrium you have to unemploy capital. That

means the price of new capital is only lambda time the old in the new equilibrium,

implying strict loss at the new equilibrium, contradicting that we require you can’t lose

money at the new prices.

8. Conclusions

[ TO BE ADDED]

Appendix

[TO BE ADDED]

28

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