The Technology Cycle and Inequalityby

Boyan JovanovicJuly 4, 2008

ABSTRACT

Motivated by the observed rise in the trade of technology, I analyze how technologywould spread in a frictionless market. In such a world, low-skilled agents prefer touse old technology because it costs less; their skills do not justify the use of frontiertechnology. The model generates a technology-life cycle of somewhere between 68 and124 years and per-capita income differential factors between 2.3 and 4.5. The modelmatches fairly well the cross-section relation between a country’s income per capitaand the average age of the technologies that its residents use. It is also consistent withaspects of the observed positive relation between income and imports of technology.

Keywords: Product cycle, patents.

JEL Classification number: O33

Author information:

Boyan JovanovicDepartment of EconomicsNew York University19 W. 4th St.New York, N.Y. 10012USAbj2@nyu.edu

Acknowledgement : A previous version circulated as “The Product Cycle and In-equality.” I thank P. Aghion, M. Boldrin, S. Braguinsky, F. Caselli, W. Easterly, J.Eaton, J. Giummo, D. Harhoff, B. Hobijn, H. Hopenhayn, P. Howitt, S. Kortum, A.Pakes, M. Schankerman, C. Syverson, and J. Tybout for comments, A. Gavazza, M.Kredler, A. Santacreu and V. Tsyrennikov for help with the research, J. Giummo forthe Patent data, and the NSF for support.

1

1 Introduction

Recent work on the skill premium has stressed the role that technological progressplays in generating inequality — e.g., Krusell et al. (2000). This useful line of workusually takes as given the time series of either technology, or of skills, or both. Thepresent paper endogenizes both technology and skills and generates an inequalitythat persists for ever. The force generating inequality is the need for a wide varietyof intermediate inputs, and the operation of more efficient ones offers a higher returnto the accumulation of skills.In the model, each technology is of a different vintage. The appearance of new

technologies derives in part from learning-by-doing in the research sector. Comple-mentarity between skills and technology yields an assignment in which less skilledagents use older vintages. An inventor of new technology enjoys a perpetual patentwhich he licenses out to users while the vintage is in use. New technologies are betterthan old technologies, and are used by skilled agents. There are no costs to switch-ing technologies and so, as it ages, a technology moves down the skill chain, and iseventually abandoned. We may call this process the technology cycle. In the modelunskilled agents optimally choose not to catch up with the skilled agents.The model explains the cross section relation between per-capita income and the

age of the technology employed (Figure 5) documented recently by Comin and Hobijn(2004). It also matches some aspects of the relation between income and importsof technology that Caselli and Coleman (2001) document. It generates a uniquedistribution of technologies and incomes compatible with constant growth (Figures1 and 2A). The model has no exogenous heterogeneity such as skills, tastes, or luck:Inequality is the result of the vintage structure, endogenous skills, and assignment ofagents to vintages. We shall solve for the steady-state distribution of vintages andskills in use, for the cross-section distribution of income, for the world’s growth rateand for the length of the technology cycle.In the empirics we shall interpret agents as countries and derive the predicted long-

run inequality among them, inequality in which there are no reversals so that incomedifferences persist indefinitely. To infer the model’s ability to explain income gaps, wetake two alternative approaches. The first approach relies on the model’s predictionthat the relation between an agent’s income and the age of the technology that heuses is linear. I estimate this relation using the Comin-Hobijn sample of technologiesused in a set of rich countries, and extrapolates it to the poor countries and obtainsa differential of 4.5 — the factor by which the income of the richest agent exceeds theincome of the poorest agent — deriving from a technology-age usage difference of 124years.The second approach use an implication of the model for the income flow from

a patent as a function of the patent’s age. Both in the model and in the data, thisincome grows to a peak and then declines. In the model there is a one-to-one relationbetween this peak and the oldest technology, and so the latter can be estimated by

2

matching the model’s peak to the data peak (10 years). This yields 68 years as theage of the oldest technology. In turn, however, this implies a richest-to-poorest ratioof only 2.3.Under either estimate, the model’s ability to explain income gaps is modest. The

reason is roughly this: Since the productivity of the best technology improves atabout 1.2 percent per year, even a 100-year diffusion lag adds up to only a 120 percenttechnology differential. A highly elastic response of the complementary factors wouldbe needed for such a differential to produce a large effect, and the model has onlya modest elasticity. Thus the model explains observed technology-diffusion lags, butnot the bulk of the world’s income inequality.Motivation for modeling frictionless trade of technology.–The technology transfer

literature (e.g., Benhabib and Spiegel (1994), Barro and Sala-i-Martin (1997), Eeck-hout and Jovanovic (2002)) has shown that many features of the cross-country datacan be explained without markets for technology. Yet such licensing is becomingmore important, as are other forms of payments for knowledge. Some measures ofthe rise of activity in knowledge markets are:(i) Licensing revenues.–Worldwide, international licensing revenues have risen

from $15 billion in 1980 to $110 billion in 2004 (OECD 2006, Figure 7). Royaltyreceipts from abroad for patents, licenses, and copyrights in 2001 were (as a per-centage of R&D costs) 64 (U.K.), 36 (Italy), 31 (Germany), 15 (U.S.), 11 (France)and 8 (Japan) (OECD 2004, tables 69-71). Serrano (2006) finds that 18 percent ofpatents granted to small inventors are traded at least once in their lives, and thatthe citations-weighted percentage is even higher. Also, patent-sharing agreements arefairly common.(ii) International patenting.–Eaton and Kortum (1999, Table 1) document that

the U.S., the U.K., France, Germany and Japan patent abroad about one fifth of thepatents that they take out domestically, probably the most valuable fifth. Kortumand Lerner (1999, Figures 3 and 4) document a five-fold rise in U.S. patenting abroadbetween 1955 and 1993, and a comparable rise in foreign patenting in the U.S.(iii) Cross-border mergers.–Work on mergers has viewed technology transfer as

an important function of cross-border mergers. The number of cross-border dealsrose from 862 in 1987 to 6,134 in 2005 and over the same period, the dollar value ofsuch acquisitions rose by a factor of ten (UNCTAD 2006).Such highly active technology markets presumably affect technology transfer, but

how? This is the question we shall study here. We shall assume, first, that agentshave access to all the existing technologies — this we may call ‘perfect technologytransfer’ — and, second, that they must pay a license fee for any technology that theyuse — this we may call ‘perfect patent protection.’ The results of this paper shouldbe relevant to studies of optimal IP protection, such as that of Boldrin and Levine(2005).Related work.–The growth side of the model combines a Lucas (1988) type of tech-

nology for investment in skills with an Arrow (1962) type of technology for invention.

3

The necessity of inequality in the steady state is an application of asymmetric equi-librium as explained in Matsuyama (1996). The assignment of skills to technologies isfrictionless and continually changing as in Jovanovic (1998), but he does not explainthe relation between income and technology. The technology cycle relates to the fol-lowing two papers: Krugman (1979) modeled the product cycle but restricted agentsin the South from producing frontier products. Rather than impose a restriction ofthis sort, my model will derive such lagged adoption. Matsuyama (2002) shows thatincome inequality speeds up technological adoption, that each new product is boughtfirst by the rich and then by the poor, and that diffusion lags depend positively onincome dispersion; Antras (2005) argues that as a product ages it needs less manage-ment and will then be produced offshore. Finally, while the allocative role of pricesin markets for technology is as yet unexplored, four other reasons have been offeredfor why agents do not all use the same frontier technology: Technology-specific skills(Chari and Hopenhayn 1991), physical capital endowments (Basu and Weil 1998),incentives to free-ride (Eeckhout and Jovanovic 2002), and policy differences (Jones1994).Section 2 presents the model, and Section 3 discusses the model in general terms.

Section 4 estimates some parameters of the model and Section 5 concludes the paper.

2 The Model

The economy — the world economy — has two types of goods and two reproduciblefactors of production. The goods are a homogeneous consumption good y, and acollection intermediate goods xi where i ∈ [0, N ] is the list of the available technologiesand where N is the number of technologies hitherto invented. The first factor ofproduction is N , the state of technology, the second factor is homogeneous humancapital or ‘skill’ H. The production function for the final good is

y =

µZ N

0

x1/2i di

¶2, (1)

where xi is the quantity of the i’th intermediate good used. The production functionfor the i’th intermediate good is

xi = ziHi, (2)

where Hi is the amount of skill devoted to the production of intermediate good i.Skill H is measured in efficiency units and is the sum of the skills of the agentsemployed in operating technology i. This skill is general and can be combined withany technology with no fixed cost and no training cost. An intermediate-goods firmcan hire any number of agents, and employ them all in the same technology. Thereare no diminishing returns and so, in this sense, technology is replicable withoutcost. We shall assume that zi is increasing in i, so that skill is more productive ontechnologies with a higher index.

4

Leisure is not valued. Skills are divided between production, research, and theprocess of augmenting the skills. The economywide H constraint is

H =

Z N

Nmin

Hidi+HR +HA, (3)

where HR is the amount of skill devoted to research into new products, and HA isthe amount devoted to investment in H.

The measure of agents is constant and equal to unity. We assume that N > 1,so that there are more technologies than there are agents. We shall also assume,however, that an agent can work on at most one intermediate-goods technology. Thenumber of technologies in use then cannot exceed the number of agents. Since thehigher-i technologies have higher values of z, the technologies in use will be thosewith i ∈ [Nmin, N ] , where

N −Nmin ≤ 1. (4)

If the inequality in (4) is strict, some technologies employ more than one agent.

2.1 A planner’s problem

The number of technologies, N, and the stock of skill, H, grow over time and thetechnologies for accumulating N and H will be described in Section 2.2. But in orderto clarify the assumptions let us first solve the problem of allocating a given amountof H to maximize output in (1), taking account of the constraints (2), (3), and (4).That is, a static planning problem that ignores the constraint that an agent cannotoperate more than one technology simultaneously. The latter constraint involves thedistribution of skills in the population and is best considered separately later.

Let the amount of H available for intermediate-goods production, HI ≡ H −HR −HA, be given. Since the xi enter symmetrically into (1), the planner will usethe most productive technologies, i.e., those with the highest index i. In this case hewill choose the technologies from the interval [N − 1, N ], but not necessarily all ofthem. Thus the problem can be posed as follows: Given HI and N , choose (Hi)

Ni=N−1

to maximize y subject to (2) and (3). The Lagrangian is

L =

µZ N

N−1z1/2i H

1/2i di

¶2+ η

µHI −

Z N

N−1Hidi

¶, (5)

where η is the multiplier on (2). The first-order conditions are

Hi :

µziHi

y

¶1/2− η ≤ 0, i ∈ [N − 1, N ] (6)

where if the inequality is strict at i, then Hi = 0i.

5

Analysis.–From (6),Hi = η−2yzi. (7)

Substituting from (7) into (3) which must bind, we have η−2yR NNmin

zidi = HI whichthen allows us to eliminate η−2y from (7) and obtain

Hi =

ÃHIR N

Nminzidi

!zi (8)

Thus to maximize y, the planner would use as many technologies as there are agents,and would allocate to technology i the skill level Hi given by (8).Discussion.–The model does not have comparative advantage — the linearity of

(2) implies that between any pair of goods the ratio of marginal and average productsof does not depend on the quantity of H. And yet the output-maximizing allocationassigns those agents endowed with more H to produce the most advanced products.The reason stems not from comparative advantage in the traditional sense but fromthe constraint that we have so far ignored: An agent can operate only one technology.On any technology, the planner can get the same output from two agents each of whomhas a unit of H as he can from a single agent who has two units of H. But if he wereto use the two mediocre agents on that technology, this would reduce the number oftechnologies that the planner could operate overall.In general, then, how H can be allocated over technologies depends on the distri-

bution of H in the population. Since the planner chooses to use the maximal numberof technologies, to be able to implement (8), the distribution of H over agents needsto coincide with its distribution over the index i implied by the distribution of zi andby (8). We shall show that such a coincidence will indeed arise and be maintainedunder steady state growth so that the constraint holds and so that the equilibriumimplements (8). Moreover, the highest-skilled agents will operate the best technolo-gies, and so on down the technology ladder. The next subsection will describe themarket decentralization of (8).

2.2 The four markets

There are four markets: (i) final goods, (ii) intermediate goods, (iii) labor, and (iv)technology. This section has three subsections. The first subsection will describe eachof the four markets in turn, and then it will derive some properties of the instantaneousequilibrium that arises when we put these markets together and how it will shift overtime given that growth is steady. The second subsection will explain accumulation— of savings, of skill, and of technology. The third subsection will define and discussthe equilibrium. Each market clears at every instant. As a reference point we shallchoose t = 0. Because we study only constant growth, the distribution of quantitiesand prices will then be the same as that of any other date, except for a scale factor.We now discuss each market in turn.

6

2.2.1 The market for final goods

Consumers buy the sole final good in a competitive market, and the price of this goodis normalized to one. The final good is produced using perishable intermediate goodsonly. Let Pi be the date-zero price of good i in units of the final good. A final-goodsproducer’s problem is

max(xi)

N0

½y −

Z N

0

Pixidi

¾,

taking the prices Pi as given. The first-order condition for xi is

y1/2x−1/2i − Pi = 0. (9)

This equation yields the final-goods producers’ demand function for intermediategood i.

2.2.2 The market for intermediate goods

Using the technology (2) and his own skill-efficiency units s, an agent can make

xi = zis (10)

units of good i. Suppose for now that the agent chooses not to augment s by hiringon the outside market, and that there are no other producers of good i. Section 2.2.4(the paragraph on group licensing) will show that both of these assumptions followfrom optimal behavior. Being faced with an elastic demand for his product, havingproduced xi units the agent will choose to sell them all and obtain the revenue

Pixi = y1/2z1/2i s1/2,

where the equality follows after substitution from (9) and (10).

2.2.3 The market for labor

Each period an agent has a unit of labor time. The agent’s skill s devoted tointermediate-goods production depends on the agent’s human capital h and on thefraction uP of time that he devotes to production, so that s = uPh. The rest of thetime he can spend investing in raising h or he can sell it on the labor market. Thefraction of time spent investing is denoted by uI , and the fraction sold on the labormarket is denoted by uR to sell on the labor market. The time constraint thus reads

uP + uR + uI = 1. (11)

and the skill that he supplies to the research market is uRh. The wage per unit ofskill will be denoted by w so that labor earnings are wuRh.The demand for labor is exclusively that of research firms which are competitive.

The demand for labor at a given wage will be determined by the free-entry conditionfor research which we shall state in Section 2.2.4.

7

2.2.4 The market for technology

This section takes the distributions of s and z as given and derives the prices foreach z that will clear the technology market. The distributions of s and z will beendogenized later.Each technology is associated with a different good. It now is simpler to call a

good by its productivity, zi in eq. (10), instead of by its name, i, and we shall switchto this notation from now on. Taking as given the density of s, denoted bym (s) , andthe density of z, denoted by n (z), let us determine the equilibrium license fee of thetechnologies and the equilibrium assignment. We shall have a one-to-one assignmentwith side payments — the ‘transferable-utility’ case.Each technology has an infinite and fully enforced patent. The pricing of tech-

nology is competitive. Let p (z) be the per-period license fee for technology z. Bychoosing the shape of the licensing fees p(z) appropriately, one can induce agentssupplying different amounts of s to choose different technologies z. Pricing of thetechnologies is competitive.The technology-adoption decision: The intermediate-goods producer treats the

entire schedule p (·) as given, and he chooses the technology that will maximize hisprofit:

π(s) = maxz{y1/2z1/2s1/2 − p (z)}. (12)

and he maximizes this by selecting the technology, z, to license. This decision isstatic; in the next instant the agent can switch to another technology at zero cost.The first-order condition reads

1

2

³syz

´1/2− p0 (z) = 0. (13)

The problem in (12) assumes that the licensee of technology z will use only his ownlabor to operate that technology. But (2) states that many people should be ableto use the same technology and pay just one license fee. When we have derived theequilibrium price function p (z) we shall then verify that the option for several agentsto pool their effort under and to operate the same technology will be rejected.Definition: A positive-assignment equilibrium in the market for technology is a

price function p (z) and an assignment s = ψ (z) of skill to technology such that(i) ψ is increasing,(ii) z is the optimal choice of agent ψ (z) which requires (13) to hold, i.e.,

1

2

µψ (z) y

z

¶1/2− p0 (z) = 0, and (14)

(iii) the market clears, i.e., for all ‘active’ technologies z,Z 1

z

n(v)dv =

Z smax

ψ(z)

m(s)ds. (15)

8

Here we have normalized the date-zero frontier technology to zmax = 1, and wedenoted by smax the maximal date-zero skill.

Solving for equilibrium takes two steps. Step 1 is to solve (15) for ψ. Step 2is to solve the first-order ordinary differential equation (14) for p. This requires acorner condition which we get from the assumption that in steady state the numberof available technologies exceeds the number of agents; i.e., there are some inactivetechnologies. A technology is active if some agent uses it, and ψ is defined only on theset of active z’s. There is a unit measure of agents, each of whom operates a differenttechnology. If the process of technological invention has gone on for a long time, therewill be many more technologies than agents, and all those technologies with z < zminwill have been abandoned. The measure of active technologies used is therefore unity,and therefore the worst operated technology, zmin, satisfies the equationZ 1

zmin

n (z) dz = 1. (16)

Inactive technologies are not demanded and must sell at a zero price: p (z) = 0 forz < zmin. This means, that

p (zmin) = 0, (17)

for if p (zmin) were positive, technology zmin− ε would, for ε small enough, be strictlypreferred.

The distribution of existing inventions, n.–So far we have discussed the assign-ment problem taking n (z) and m (s) as given, and for simplicity we discussed onlythe situation at t = 0. The passage of time will only scale all the variables up, but itnow is necessary to introduce the time subscript. To proceed to Section 2.2, we shallneed to conjecture the shape of the steady-state distributions of z and s and derivethe properties of the resulting equilibrium assignment ψ. Suppose that zmax (t) ≡ egt,and that each technology is retired after T periods of life, where T is time-invariant.Suppose that the flow of new technologies is also constant over time. Then n (z) mustbe of a special form: ln z is then uniformly distributed for ln z ≤ gt with density g,and the distribution of z at date t is 1

gTzfor z ≤ egt. However, only a unit measure

of technologies will be employed at each date. Then if the distribution shifts to theright at the rate g, then for t ≥ 0, ln zt is uniform on [g (t− T ) , gt]; the interval is offixed length (product variety is not expanding) and shifts to the right at the rate g.The density of z is

nt(z) =

µ1

gT

¶1

z, for z ∈ [eg(t−T ), egt], (18)

as illustrated in Figure 1.

A conjectured market-clearing assignment and license fee p (z).–If nt (z) looks asdrawn in Figure 1, then we may conjecture that s and z are proportional to each

9

zgTz

tn 11)( ⎟

⎠

⎞⎜⎝

⎛=

gTe − 1 )( Ttge − gte

Technology distribution at t = 0

Technology distribution at t > 0

Den

sity

of

z

z

Figure 1: The movement of nt (z) over time

other and that their distributions move together in lockstep. Section 2.2 will thenvalidate these conjectures. For some constant θ > 0, we conjecture the time-invariantproportional assignment

ψ (z) =1

θz, (19)

the price function

pt (z) =1

2

³ytθ

´1/2 ¡z − eg(t−T )

¢for z ∈ [eg(t−T ), egt], (20)

and the distribution of s

mt(s) =

µ1

gT

¶1

s, for s ∈ [1

θeg(t−T ),

1

θegt], (21)

so that mt (s) is simply a scaled version of nt (z) as shown in Figure 2A.

Note that (19) implements the solution to the problem (5) of maximizing outputsubject to a given distribution of technologies and to a given stock of human capital.

Verifying the conjectured assignment equilibrium.–Now let us check that when nand m are as given in (18) and (21), the assignment in (19) is indeed an equilibrium.(i) Market clearing at t holds because the scaled-up version of (15)Z egt

z

nt(v)dv =

Z egt/θ

z/θ

mt(s)ds (22)

is seen to hold for all z ∈ [eg(t−T ), egt] after an integration of both sides.(ii) The first-order condition (14) evidently holds.

10

(iii) The corner condition (17) holds because zmin = eg(t−T ).

So far we have shown that the conjectured assignment is an equilibrium as long asthe distributions n in (18) and m in (21) can be made consistent with the incentivesto invent new technologies and to accumulate skill. We now pause to describe thisequilibrium. The price function pt (z) in (20) depends on three variables. First,higher-z technologies cost more. Second, there is a positive aggregate-demand effectthat works through y, which grows over time. Third, a negative obsolescence effectworks through the rise in eg(t−T ), the lowest technology in use. Since its z is fixed, theprice of a technology reaches zero exactly when it is T periods old. The assignmentin (19) is indexed by the parameter θ. The variables (g, T, θ) will be endogenized inSection 2.2.

The movement of n and m over time.–Figure 2A illustrates the relation betweenm and n and their movement over time. At t = 0, n (z) is on the interval

£e−gT , 1

¤,

marked by the heavy (blue) segment on the vertical axis. Since z = θs, m (s) mustthen be on the interval

£1θe−gT , 1

θ

¤, as marked by the heavy line segment on the

horizontal axis. We then shift to date t, when both distributions have been scaled upby a factor of egt.

The Technology Cycle.–Figure 2B describes the technology cycle, which arisesbecause of the different ways in which the distributions of s and z shift. As weshall shortly see, each agent’s s grows at the same rate g and the distribution of stherefore exhibits no rank reversals. By contrast, the distribution of z shifts entirelythrough replacement, and each good has a z that is fixed over time. Put differently,the technology cycle arises because s grows entirely on the intensive margin whilez grows entirely on the extensive margin. Thus the assignment z = θs can hold ateach t only if products move down the skill distribution. Figure 2B describes howthe distribution of z grows. At any date t, the support of the distribution of ln z is[g (t− T ) , gt]. The upper and lower bounds of ln z are also drawn. Now considerthe technology that is introduced at date t0. Its efficiency is ln zmax (t0) = gt0, whereit remains for the duration of the technology’s lifetime, which ends at date t0 + T .The technology’s efficiency rank declines continuously over this period. As its rankdeclines, so does the relative quality of its match. The absolute quality of its matchremains unchanged at smax (t0) which, at date t0, is the highest skill around butwhich, by date t0 + T , is the lowest skill. This movement of a given z down the skilldistribution is what we shall understand to be the technology cycle.

Profits of intermediate-goods producers.–From (19), z = θs. Substituting for z

into (12) yields π (s) =n(θy)1/2 s− p (θs)

o. From (20) evaluated at t = 0, since

e−gT = zmin = θsmin, p (θs) =12

¡yθ

¢1/2 ¡θs− e−gT

¢= 1

2(θy)1/2 (s− smin). Substitut-

ing into the expression for π (s) we find that it is linear in s:

π(s) =1

2(θy)1/2 (smin + s) , (23)

11

Figure 2A: Assignment at date zero and at date t

gTe−

θ1

θ1 )(1 Ttge −

θgte

θ1

t T

s

z

gTe−

)( Ttge −

1

gteθs

ln z

Assignment at date t

Assignment at date zero

Figure 2B: The Technology Cycle

ln zmax(t) = gt

ln zmin(t) = g(t-T)

Frontier technology comes in at t0

it ages

it dies at t0 + T

t0 t0 +T 0

0

12

Total Revenue = y1/2 θ1/2s

s smax smin

π(s)

Licensing cost

Net revenue

0

Figure 3: Optimized profits as a function of s

Output, (θy)1/2 s, and license fees, p (θs) = 12(θy)1/2 (s− smin) are also linear in s.

Figure 3 illustrates the breakdown of revenues into costs and profits. The cross-section(marginal) return to skill is 1

2(θy)1/2, the slope of the blue line, and it is constant.

The constancy of this return stems from the positive association between s and z —a leverage effect that rising skill has in raising z. Without the accompanying rise inz, the marginal returns to s would diminish.

Group licensing.–Once he pays the license fee, an agent can use the technologyat any scale. Any number of agents can get together, pay the license fee once, andthen all use the technology. Suppose that two agents with skills s and s0, respectively,were to form a firm and license technology zi. Since (2) has constant returns in H,they would then produce xi = zi (s+ s0) , and in this sense two mediocre people addup to one smart person. The optimal technology for them to license would then beθ (s+ s0) as long as that technology is offered in equilibrium, which is true as longas s + s0 < 1/θ (we deal with t = 0 as our reference point — see Figure 2A). Thecondition that rules this out is

π (s+ s0) ≤ π (s) + π(s0) (24)

for any (s, s0) such that s+ s0 ≤ 1/θ. If s+ s0 > 1/θ, the LHS of (24) is not defined,and the cooperative payoff would be smaller than if it were defined. Substitution into(24) fromwith (23) and division of both by 1

2(θy)1/2 transforms (24) into smin+s+s0 ≤

2smin+s+s0, which always holds. The proof extends to any numbers of agents. If nogroup of agents finds it profitable to cooperate using one technology, it also followsthat each product will be monopolized with no threat of non-cooperative entry by

13

a duopolist. Duopolists would each pay a license fee and would then set the priceof the product noncooperatively. Thus a price-taking equilibrium will result in themonopolization of each product by a single agent who uses his own labor alone.1

2.3 Accumulation

So far we have been assuming that technology and skills would grow in a particularway that would permit a proportional positive assignment, with a constant of pro-portionality, θ, that would be constant over time. Specifically, Section 2.2.4 assumedthat invention would proceed at the constant rate denoted by g, that the lifetime ofeach technology would be constant and denoted by T , resulting in a log-uniform dis-tribution of z. We then assumed that the distribution of s would also be log uniformand that it would keep up with the shifts in the distribution of z so that (19) wouldbe the result. We also took θ, g, and T as given. To determine these quantities, weshall now analyze the three forms of accumulation: Savings, skill accumulation, andresearch.

2.3.1 Saving

Lifetime utility is Z ∞

0

e−ρtc1−σt

1− σdt. (25)

Utility is homothetic and capital markets are perfect. Then the distribution of incomedoes not affect the total amount saved and consumed. We therefore imagine an agentthat at date zero owns the average wealth, call it W, in the economy, and we shalldetermine conditions for the optimal time path of per capita consumption of the finalgood, ct. Capital markets are perfect, with the interest rate being r. Our fictitiousagent maximizes (25) subject to the lifetime constraintZ ∞

0

e−rtctdt ≤W.

Let gc be the growth rate of consumption. Optimality then requires that gc =r−ρσ.

In any equilibrium, however, c must grow at the same rate as y, i.e., at the rate 2g.From (10) and (19), xs grows at the rate 2g and by (1) so does y. That is, yt = y0e

2gt.We therefore must have

2g =r − ρ

σ. (26)

1The conclusion expressed in (24) is still conditional on the truth of (18), (21) which leads to(19). We still need to verify that (18), (21) will in fact arise as a result of the investments in smkillsand in research that agents will wish to make. This will be done in Section 2.3

14

2.3.2 Investment in h

We shall now study the accumulation of h of an agent who lives in a world in whichother agents accumulate s at some hypothetical rate g and in which the distributionof z shifts to the right also at the rate g. We shall then derive condition (28) whichis necessary for the agent to be willing to also accumulate skill at the rate g.

Human-capital investment uses time and human capital, as in Lucas (1988):

h = ηuIh. (27)

All agents must choose the same uI and, if s is to grow at the same rate as h, theymust choose the same uP and, hence, the same uR.

Wealth maximization: We shall now solve the accumulation problem of someonewho is forced to set uR,t = 0 for all t. The solution will be the same as for peoplewho can set uR,t > 0 because the research wage per unit of h will be the same asthe return of h in production. Let ut ≡ uP,t + uR,t. The expression in (23) pertainsto period zero, but smin grows at the rate g. An agent that at date t supplies skillst = utht will receive an income

πt(utht) =1

2(θyt)

1/2 ¡egtsmin + utht¢.

He maximizesR∞0

e−rtπt(utht)dt, but he cannot influence the term 12(θyt)

1/2 egtsmin.

Since y1/2t = y

1/20 egt, he picks ut to maximize 1

2(θy0)

1/2 R∞0

e−(r−g)tuthtdt, which isequivalent to the problem

max(ut,ht)∞0

Z ∞

0

e−(r−g)tuthtdt, s.t. ht = η(1− ut)ht,

with h0 given. The Hamiltonian is

H = e−(r−g)tuh+ μη(1− u)h.

Let μ = e−(r−g)tμ be the current-value multiplier so that the current-value Hamil-tonian is just uh+ μη(1− u)h. We shall analyze only constant-growth paths. Evalu-ated at a point at which μ = 0, the FOC’s are

1− μη = 0,

andμη (1− u) + u = (r − g)μ.

Eliminating μ we haver = η + g. (28)

15

This is an arbitrage condition equating the rate of interest to the rate of return toinvesting in h. Together with (26) this implies that

g =η − ρ

2σ − 1 . (29)

For the present value of utility to be bounded we need that e−ρtc1−σt → 0 as t→∞which, as long as η > ρ, requires that σ > 1.

2.3.3 Research and learning by doing

Invention of new technologies in this framework is similar to Romer (1990) in fourrespects: Human capital is the only private input, there is also a vertical spilloverfrom previous inventions, N , there is free entry into research, and patents have infinitelives. There are two major differences, however. First, Romer assumed that there wasfree entry into the implementation of innovations and this allowed inventors to extractall the rents from the intermediate-goods producers. In contrast, the price functionp (z) is a market clearing price between heterogeneous technologies and heterogeneoususers, and it is (the inferior) technologies rather than their potential users that arein excess supply. The second difference is that Romer’s inventions are all the same,whereas here new inventions are better than old ones. Thus the spillover mechanismmust be different too; Romer’s spillovers facilitate the invention of a larger numberof technologies all of the same quality, whereas here spillovers will raise the qualityof future inventions relative to that of earlier inventions just as in Arrow (1962).

Learning by doing.–The quality of an invention depends on the cumulative num-ber, N, of previous inventions (see eq. [1]), i.e., on the invention’s ‘serial number.’Arrow’s eq. (8) relates ‘labor requirement’ to experience. Here, we can also call itthe ‘skill requirement’ because in Arrow’s model labor is homogeneous. In our model,(10) implies that the skill needed to produce a unit of intermediate output is 1

z. Thus

Arrow’s eq. (8) would imply a skill requirement of 1/zmax = aN−α. We shall use theform

zmax = eαN , (30)

which implies a skill requirement of 1/zmax = e−αN . Figure 4 compares Arrow’sfunctional form to ours, where the two are normalized so that they coincide at N = 1.Arrow’s functional form is probably better suited to narrower activities in which, afterinitial learning possibilities are exhausted, further improvement is relatively hard toattain. Our assumption probably fits better the transfer of knowledge among differenttypes of research where returns may not diminish as rapidly.2 Finally, (30) implies

2There is an alternative interpretation of α: In the quality-ladder literature, α would be theladder-step size; e.g., Grossman and Helpman (1991, p. 560) and Aghion and Howitt (1992, p. 328).In Howitt (2000, Section IV), e.g., the frontier grows at a rate assumed to be proportional to theoverall flow of innovations; this factor of proportionality is the spillover coefficient analogous to α.The product window is fixed in these models, whereas here it moves to the right over time.

16

6543210

1.5

1.25

1

0.75

0.5

0.25

0

NN

(Arrow) (this paper)

N

Skill requirement of frontier technology, 1/zmax

Cumulative experience,

Figure 4: Comparing (30) to Arrow’s mechanism when α = 1/3

that1

zmax

dzmaxdt≡ gt = αNt. (31)

Birth of technologies.–The number of new products is proportional to the quan-tity of skill employed in research. Let HR

t be the aggregate skill employed in research.The flow of new products is then

Nt =λ

zmax (t)HR

t . (32)

The presence of zmax in the denominator implies a ‘fishing out’ external effect in thediscovery of new products; the discovery of the first product takes fewer resourcesthan the discovery of the second, and so on.Birth = death of technologies.–Each technology lives for T periods, and the num-

ber dying, 1/T must equal the number being born, N :

N =1

T. (33)

Combining (31) with (33) gives a reduced-form relation between two endogenousvariables g and T ,

gT = α. (34)

Therefore α completely determines the range of technological quality, gT .Research Labor Supply = Demand.–Each agent must be indifferent between de-

voting the marginal unit of time to his business and working as a research worker.

17

By (23), the profit foregone from a unit reduction in s is 12(θy)1/2 = 1

2θ1/2egt. Hence

the research wage must equal

wt =θ1/2

2egt. (35)

Since h = suP, (21) implies that

HR =uRuP

Z smax(t)

smin(t)

smt (s) ds = egtuRθuP

¡1− e−gT

¢gT

.

Substituting for HR into the RHS of (32), the growth of HR exactly offsets thefishing-out externality, and the flow of new ideas is constant:

N =λ

θ

uRuP

¡1− e−gT

¢gT

. (36)

Free entry.–The value of an invention is the discounted flow of license fees. Theperiod-t frontier technology is zmax (t) = egt. Using (20), the license fee of thattechnology τ periods after its invention is

pt+τ (egτ) = e2gt

θ−1/2

2egτ¡1− eg(τ−T )

¢. (37)

for τ ∈ [0, T ]. The present value of the license income from a frontier technology is

Vt =

Z T

0

e−rτpt+τ (egτ) dτ = e2gt

θ−1/2

2

µ1− e−(r−g)T

r − g− e−gT

1− e−(r−2g)T

r − 2g

¶. (38)

The RHS of (32) implies that the free-entry condition is

wt =λ

zmax (t)Vt. (39)

Upon dividing (39) equality by λ and using (35) and (28), we get

θ

λ=1− e−ηT

η− e−gT

1− e−(η−g)T

η − g. (40)

Thus θ is determined by the free-entry condition (39). Since T = α/g and since g

is given by (29), the RHS of (40) does not depend on λ. Thus θ is proportional toλ. The intuition is that demand for technology, as summarized by θ per unit of skill,must equal the supply, which comes forth at the speed λ.

18

2.4 Equilibrium

Definition: A stationary equilibrium consists of 8 real numbers g, T , θ, uP , uR, uI ,w and N that solve (11), (27), (29), (31), (33), (35), (36), and (40).

Existence of equilibrium.–Existence of the equilibrium follows from using g from(29), uI = g/η via (27). Then uP is a function only of uR alone via (11), and (40)gives us θ uniquely in terms of T . The remaining equations are linear and easilyshown to have a unique solution. We have thus shown that there is a steady statewith inequality, but we did not prove the stability of that steady state. Even if it isstable, if T is 68 years or more, the transition dynamics are likely to be quite long.

Non-stationary equilibria may exist; I have not been able to rule them out. Inparticular, both asynchronous and synchronous cycles in h and z may exist. Asyn-chronous cycles in a related model were found by Matsuyama (1999), and synchronouscycles may possibly exist, at least if the complementarity between z and s is strongenough, along the lines of Boucekkine, Germain and Licandro (1997) who studyinvestment-echo effects in a vintage model when utility is linear.

In the stationary equilibrium, product variety is not expanding. Rather, theeconomy grows because new, more productive varieties replace the obsolete ones.That is on the technology side. On the human capital side, accumulation is standard,as in Lucas (1988). The two mechanisms combine to yield long-run growth.

3 Discussion of the model

As in many vintage-capital models — see Boucekkine, de la Croix and Licandro (2008)for a survey — the equilibrium of this model entails a coexistence of technologies ofdifferent quality. The reason for the technological heterogeneity is, however, differ-ent. In the putty-clay models of Johansen (1959) Arrow (1962) and Calvo (1976),for instance, inferior technologies (those with a higher labor requirement) survivetemporarily because new technology is costly to implement, since physical capital isneeded to embody them. In Chari and Hopenhayn (1991) it is human, not physi-cal capital, that is specific to a technology, and accumulating such capital is a costincurred not per machine but per agent. But there too, old technologies survive be-cause new technologies are costly to learn; in other words, because implementationcosts are sunk. By contrast, in my model each technology is costless to replicate, andrequires no adoption cost. Rather, inferior technologies survive because productionof the final good, described by (1), requires input variety as in Romer’s (1990) model,for example, and different inputs are produced with different technologies. Thereforethe model remains different from the putty-clay models even in the case of identicalagents with exogenous skills.

We go further and trace how such presence of these heterogenous technologiesinduces differences in skill acquisition and inequality. Chari and Hopenhayn (1991)

19

also do this, but there the members of each generation have the same lifetime utility.In contrast, in my model an agent’s rank in the income distribution never changes.Technology quality complements skill, and equilibrium entails positive sorting be-tween the two. This positive assignment amplifies the return to acquiring skill. Thisreturn depends on prices, and the prices of technology in my model are set competi-tively. This contrasts with Romer (1990) where each intermediate-goods technologyis priced monopolistically and where an infinitely elastic supply of potential users ofeach technology allows the inventor to extract all the rents. In my model, the rentsare shared; when owners of human capital can appropriate a part of the rents fromtheir skills, they will invest in their skills and this creates inequality among the usersof the technologies. In Romer’s model, by contrast, a patent licensee does not needto make a prior investment, and all agents can use any technology equally well.A positive-sorting equilibrium between physical and human capital was modeled

by Jovanovic (1998), but he did not deal with technology-cycle issues. He was not ableto solve for inequality in closed form, and was not able to estimate the counterpart ofα. In contrast, here we can solve for inequality in h explicitly, and we also can solvefor income inequality if we assume that firm ownership is fully diversified so thateach agent holds a fraction of the world portfolio proportional to his or her wealthwhich, in turn, is proportional to his or her h. In that case an agent’s income fromall sources is proportional to h. Moreover, all agents choose the same ui’s so thatincome differentials coincide with wealth differentials. Taking the ratio of richest topoorest,

YmaxYmin

=hmaxhmin

= eα. (41)

Thus α alone determines inequality, because α alone governs the dispersion in tech-nological quality among the measure 1 of the most recent vintages, thereby dictatingthe dispersion of h that will arise in steady state. The parameter α is a verticalspillover in the research sector and therefore related to Romer (1990), but technicallycloser to Arrow (1962), as Figure 4 shows.It would seem that the quantitative effects on inequality may be larger in a non-

linear model. As we shall see below, the estimated α will not be very large, and themodel will be unable to account for very much inequality. A possible way to raise theamount of inequality explained would be to follow Kremer (1993) and assume thatfirm size is fixed, and to change (10) to xi = (zis)

m for m > 1/2. This would amplifythe return to human capital and would also produce more inequality in response toa given variation in z and s.3 But this is where the supply side of z and s beginsto matter and where we have to consult the rest of the model. For m > 1/2, p (z)would be convex in z (see Kremer’s equation [10], e.g.), and π (s) in (12) would beconvex in s. Given the linear technology that the model uses for generating growth in

3On the other hand, with larger teams of workers, the number of technologies used would haveto be smaller and this would reduce equilibrium technological heterogeneity. Sorting out the variouseffects would be interesting.

20

s (Section 2.2.2) and z (Section 2.2.3), second-order conditions would fail. To restorethem, we would need returns in the skill-augmenting technology to diminish morerapidly. Although the analysis would be harder, the main argument for generatingskill inequality would work even without linearity, since it depends on there being acoexistence of different-vintage technologies and on technology-skill complementarity.

4 Empirics

A single parameter α governs the range of technologies in use and hence, immediatelygoverns the range of inequality in profits implied by the model. The empirical workfocuses on this implication — a country with human capital corresponding to thehuman capital level of the world leader t periods ago uses a technology invented tperiods ago and is halfway between where that leader country was t periods ago andwhere it is now (halfway because of an aggregate demand effect coming from the factthat this country lives in a richer world than the world leader did t periods ago). Twoapproaches are taken to estimating that parameter α. From these estimates we inferthe implied the range of technologies in use and, hence, the extent of inequality thatcan be accounted for by technology choice.

4.1 Estimating T using the relation between income and ageof technology

Substituting for Vt into (39) we find that wages grow at the rate g. Incomes (which arewages multiplied by skills) grow at the rate 2g. Define an agent’s income, Y , to be thesum of his gross entrepreneurial income and his wage income: Y = y1/2 (zs)1/2+wuRh.Using (19) and the fact that s = uPh, we have

Y =

µuP

³yθ

´1/2+ uR

w

θ

¶z

Since uP and uR are constants, and since w grows at the rate g and y at the rate 2g,the equilibrium cross-section relation between agent’s income, Y , and the age, τ , ofthe technology that he uses at date t is

Y ≡ e2gte−gτ ,

for τ ≤ T . Then the relation between relative income and the age of the technologyused is linear:

lnYmax − lnY = gτ. (42)

Data from Comin and Hobijn (2004) support such linearity — see Figure 5. Replacing‘max’ by ‘USA,’ we have

ti − tUSA = −1

g(lnYUS − lnYi) . (43)

21

A plot of the two sides of (43) is in Figure 5. The slope is negative and significant.The regression line should pass through the point (0, 0) which it does almost exactly.The Appendix reports more detail about how the data were constructed.

Now we compare the slope of the regression line with the slope implied by themodel. This requires that we map agents into countries. The simplest assumption isto assume one country per agent; i.e., that the agents in each country are homoge-neous in h. In that case, the regression’s slope would, in theory, be −1

g. We shall use

the estimate 2g = 0.0235.4 Then g = 0.012, in which case 1/g = 83.3. But in reality,agents in any country are not homogeneous: Table 5 of Sala-i-Martin (2006) showsthat within-country inequality is between 28% and 38% of inequality worldwide. Inthat case, cross-country inequality should be about two thirds of cross-agent inequal-ity. The regression slope should therefore have been −

¡23

¢83.3 = −55.5. In fact it

is significantly smaller than that. This is another way of saying that there are otherreasons for per-capita income differences other than the ones put forth in this model.

Greece, for example, is fairly close to the empirical regression line and is thereforea good candidate for comparing to the U.S.. If the model was fully consistent with theper-capita income differential between Greece and the U.S. (by a factor of e−0.79 =0.45), (43) tells us that TGreece −TUS should have equalled 0.79

0.012= 65.8 years whereas

in fact it was only 34 years. Adoption lags, in other words, are much shorter thanthey would need to be for a full accounting of the inequality.Estimating T .–According to the model, the oldest technologies are used in the

poorest countries and they are not in the sample. We can, however, extrapolate theestimate (43) to the entire range of incomes by assuming that the linearity of Figure 5holds throughout the range of the data. Assuming a 30:1 differential (Ymax/Ymin = 30)between richest and poorest, the regression yields T = 36.52 ln 30 = 124.21, and ifwe again use g = 0.012, we obtain α = 1.49 and the predicted income inequality ofrichest to poorest of eα = 4.44.

The point estimate of T = 124 is an out-of-sample forecast. The constant hasa standard error of 2.1 and the slope coefficient has a standard error of 5.59, whichtranslates into a standard error of 21.1 for T . Thus is is quite likely that T is acentury or more. Direct measures of the speed of diffusion of technology suggestthat for major technologies this estimate is reasonable. Many people in the worldstill use animal power for plowing, e.g., even though the tractor was commercializedby 1910. Many also still have no access to electricity, a technology that was widely

4This is the midpoint of 2.37%, which is the estimate of world real GNI per-capita growth 1962-2005 from the World Bank (2007), and 2.33%, the estimate of US real GDP per-capita growth fromHeston et al. (2006). This way of calculating g bypasses the practical problem of how to measurethe contribution of measured vs. unmeasured inputs to growth in income. In the model yt = e2gt,where one half of the growth comes from measured inputs, h, and half comes from unmeasuredinputs, i.e., technology. But because of the problems in identifying the role of each component, itseems safest to identify g by calculating growth of income and dividing by two, as we have donehere.

22

Figure 5: Plot of (43) and the data

implemented 100 years ago; the Comin and Hobijn (2004) data, in Figure 6, showthe long diffusion lags for electricity by region.

We also can express (20) in terms of income, Y . The cross-section relation, at datet, between income and the amount p paid for technology by an agent with income Yis

p = Ce2gtµ1− Ymin

Y

¶, (44)

where C is a constant. Thus the share of technology in income rises with incomeoverall, starting at zero when Y = Ymin, and ending up at a positive level (see Figure3). This is consistent with the finding by Caselli and Coleman (2001) that the incomeelasticity of technology imports exceeds unity.5

4.2 Estimating T using patent-income data

Patents in the U.S., Europe and Japan are protected for only 20 years. Where renewalfees are required, patent owners pay them in some cases for the full 20-year term.This implies that T > 20, and now we seek to make this estimate more precise usingdata on income derived from patents.

5The univariate results in their Table A.2 are of interest here because s varies with z and cannotbe held constant as z varies.

23

1920 to 19211921 to 19361936 to 19521952 to 19681968 to 19841984 to 2001No data

Electricity

• When did U.S. use KWHr per capita that countries were using in 1995?

Figure 6: The diffusion of electricity

The identifying piece of information will be the age at which the income from apatent peaks. Consider a technology that was at the frontier at date t and, hence,was of quality zmax (t). At that point, its license fee was pt (egt). Then let pt+τ (egt)denote its license fee at date t+ τ . Then (20) implies that

pt+τ¡egt¢=1

2

³yt+τθ

´1/2 ¡egt − eg(t+τ−T )

¢Its license fee at t relative to its license fee at age zero is

pt+τ (egt)

pt (egt)= egτ

1− eg(τ−T )

1− e−gT. (45)

because³yt+τyt

´1/2= egτ .

On the RHS of (45), for small τ the aggregate demand effect (i.e., the growth ofA) is stronger than the obsolescence effect, and patent revenues rise until they peakat the point t∗ = T − ln 2

g. This peak will exist as long as gT > ln 2. Since ln 2 = 0.7,

we get the estimator T = 0.7g+ t∗

T =0.7

g+ t∗ and α = 0.7 + gt∗. (46)

Both calculations require an assumed value for g, for which, once again, we shalluse g = 0.012. As for t∗, we shall seek to measure it using the information inFigure 7. This is the empirical counterpart of (45). The data used in Figure 7 ,provided by and described in Giummo (2005), contain information on 1172 inventions

24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

T

Series1

Age of patent

.

.

.

Income from patent relative to first-year income

1

2

4

6

8

Figure 7: Income from patents as a function of age

originally patented in Germany in 1977-1982 and their patents all renewed for the20 subsequent years. Each was also patented in the U.S., so they indeed were alllicensed internationally, and so these are the types of technologies that this modelis about. German law requires that firms estimate incomes that they derive fromtheir employees’ inventions for each year until the patent expires, and this allows usto infer income derived as a function of patent age. Figure 7 indicates that t∗ = 10which, when substituted into (46), implies that T = 0.7

0.012+ 10 = 68.3.

4.3 Implied income inequality

The ratio of richest to poorest income is eα, where α = gT . We have two separateestimations of T and α, and now we wish to summarize the estimates and present anidea of their precision. Table 1 is a summary of these numbers for the two cases.The first row of Table 1 presents the estimates based on the Comin and Hobijn

data on income and diffusion of technology. To compute these standard errors in thiscase, for T I used the forecast error from the extrapolation of the estimate of (43) topoor countries. Thus obtaining T in this first procedure does not require informationabout g. But to calculate α we need an estimate of g. For α and eα I added a standarderror to g of one-tenth of a percentage point so that the one standard-error band forg was [0.011, 0.013]. I took the worst-case scenario in which the errors were perfectlypositively correlated, as this would maximize the standard error of the estimates.That is, when T is at its mean minus one standard error, g was assumed to be at itslower bound of 0.011 so that α = (0.011) 102.9 = 1.13, and when T is at its meanplus one standard error, g was assumed to be at its higher bound of 0.013 so thatα = (0.013) 145.1 = 1.89. With the band for α thus being [1.13, 1.89] , one standarderror is half the distance between the edges, i.e., 0.38. A similar calculation

25

Data Estimate(standard error)

T α eα

Comin & Hobijn 124(21.1)

1.49(0.38)

4.44(1.76)

Giummo 68.3(4.9)

0.82(0.01)

2.27(0.02)

Table 1 : Summary of the estimates

was made to obtain the standard error of eα, its bands being [3.10, 6.62]. With someindependence between the errors the precision of the estimates would be higher thanthe first line of Table 1 indicates.

The second row of Table 1 presents the estimates based on Giummo’s data onincome derived from patents of various ages. Each data point in Figure 7 is anaverage of 1172 observations. If these observations are independent, the standarderror of the differences between the means is negligible, and those differences highlysignificant. Therefore the maximum income is precisely estimated at age 10. For g, weagain assume the band [0.011, 0.013]. Then from (46), the one-standard-error bandfor T is [63.8, 73.6], the band for α is [0.81, 0.83], and the band for eα is [2.25, 2.29] .Evidently, in this second procedure T is highly sensitive to the assumed value of g,but α is not. The situation is therefore the reverse from what it was in the firstprocedure.

Note that the two-standard-deviation bands around the estimates of α intersect,but the bands for the estimates for T do not. The T based on Giummo’s data issignificantly lower than the estimate based on Comin and Hobijn’s data. The Comin-Hobijn data also imply that the model can explain development differentials of 4.44,twice as high as the 2.25 differential implied by the Giummo data.

5 Conclusion

In light of the rising trade in technology, in this paper we studied what would happenif markets in technology were frictionless. We derived the technology assignment andthe diffusion lags that would arise. We studied incentives for skill accumulation andproduct innovation. The calibrated model generated a long technology cycle, but notvery much income inequality, even though it fits well the relation between agents’income and the age of the technologies that they use. The model also explains theinverted-U shape of revenue from patents by a rising aggregate-demand effect that atfirst dominates patent obsolescence, and that is later dominated by it.

The model is that of a single economy. If skill is unevenly distributed in geo-graphical space, then the model also has implications for the pattern of trade — new

26

intermediate goods would be produced in rich locations, and old intermediate goodsin poor areas. Therefore the rich would export new products and import old prod-ucts. The poor would import new products and export old ones. In other words, aproduct cycle.

References

[1] Aghion, Philippe and Peter Howitt. “A Model of Growth Through CreativeDestruction.” Econometrica 60, no. 2 (March 1992): 323-351.

[2] Antras, Paul. “Incomplete Contracts and the Product Cycle.” American Eco-nomic Review 95, no. 4 (September 2005): 1054-1073.

[3] Arrow, Kenneth. “The Economic Implications of Learning by Doing.” Review ofEconomic Studies 29, no. 3 (June 1962): 155-173.

[4] Barro, Robert and Xavier Sala-i-Martin. “Technological Diffusion, Convergence,and Growth.” Journal of Economic Growth 2, no. 1 (March 1997): 1-26.

[5] Basu, Susanto and David Weil. “Appropriate Technology and Growth.” Quar-terly Journal of Economics 113, no. 4 (November 1998): 1025-1054.

[6] Benhabib, Jess and Mark Spiegel. “The Role of Human Capital in EconomicDevelopment: Evidence from Aggregate Cross-Country Data.” Journal of Mon-etary Economics 34 (October 1994): 143-174.

[7] Boldrin, Michele and David Levine. “IP and Market Size.” Washington Univer-sity, 2005.

[8] Boucekkine, Raouf, Marc Germain, and Omar Licandro. “Replacement Echoesin the Vintage Capital Growth Model.” Journal of Economic Theory 74 (1997):333-48.

[9] Boucekkine, Rauf, David de la Croix and Omar Licandro. “Vintage capital.”New Palgrave Dictionary in Economics 2nd ed. (2008).

[10] Calvo, Guillermo. “Optimal Growth in a Putty-Clay Model.” Econometrica 44,no. 5 (September 1976): 867-878.

[11] Caselli, Francesco and John Coleman. “Cross-Country Technol-ogy Diffusion: The Case of Computers.” AEA Papers and Pro-ceedings 91, no. 2 (May 2001): 3-335. Long version posted athttp://personal.lse.ac.uk/CASELLIF/papers/cctd.pdf

[12] Chari, V.V. and Hugo Hopenhayn. “Vintage Human Capital.” Journal of Polit-ical Economy 99, no. 6 (December 1991): 1142-1165.

27

[13] Comin, Diego and Bart Hobijn. “Cross-country Technology Adoption: Makingthe Theories Face the Facts.” Journal of Monetary Economics 51, no. 1 (2004):39-83.

[14] Eaton, Jonathan and Samuel Kortum. “International Technology Diffusion: The-ory and Measurement.” International Economic Review 40, no. 3 (August 1999):537-70.

[15] Eeckhout, Jan and Boyan Jovanovic. “Knowledge Spillovers and Inequality.”American Economic Review 92, no. 5 (December 2002): 1290-1307.

[16] Giummo, Jesse. “German Employee Inventor Compensation Records: AWindowinto the Returns to Patented Invention.” Dartmouth College, July 2005.

[17] Grossman, Gene and Elhanan Helpman. “Quality Ladders and Product Cycles.”Quarterly Journal of Economics 106, no. 2 (May, 1991): 557-586.

[18] Heston, Alan, Robert Summers and Bettina Aten. “Penn World Table Version6.2.” Center for International Comparisons of Production, Income and Prices atthe University of Pennsylvania. September 2006.

[19] Howitt, Peter. “Endogenous Growth and Cross-Country Income Differences.”American Economic Review 90, no. 4 (September 2000): 829-46.

[20] Johansen, Leif. “Substitution versis Fixed Proportions Coefficients in the Theoryof Economic Growth: A Synthesis.” Econometrica 27 (1959): 157-76.

[21] Jones, Charles. “Economic Growth and the Relative Price of Capital.” Journalof Monetary Economics 34, no. 3 (December 1994): 359-382.

[22] Jovanovic, Boyan. “Vintage Capital and Inequality.” Review of Economic Dy-namics 1, no. 2 (April 1998): 497 - 530.

[23] Kortum, Samuel and Joshua Lerner. “What is Behind the Recent Surge inPatenting?” Research Policy 28, no. 1 (January 1999): 1-22.

[24] Kremer, Michael. “The O-Ring Theory of Economic Development.” QuarterlyJournal of Economics 108, no. 3 (August 1993): 551-76.

[25] Krugman, Paul. “A Model of Innovation, Technology Transfer, and the WorldDistribution of Income.” Journal of Political Economy 87, no. 2 (April 1979):253-266.

[26] Krusell, Per, Lee Ohanian, José-Victor Ríos-Rull and Giovanni Violante.“Capital-Skill Complementarity and Inequality.” Econometrica 68:5 (2000):1029-1053.

28

[27] Lucas, Robert E., Jr. “On the Mechanics of Economic Development.” Journalof Monetary Economics 1988.

[28] Matsuyama, Kiminori. “Why Are There Rich and Poor Countries?: Symmetry-Breaking in the World Economy.” Journal of the Japanese and InternationalEconomies 10 (December 1996): 419-439.

[29] Matsuyama, Kiminori. “Growing Through Cycles.” Econometrica 67, no. 2.(March 1999): 335-347.

[30] Matsuyama, Kiminori. “The Rise of Mass-Consumption Societies.” Journal ofPolitical Economy 110, no. 5 (October 2002): 1035-70.

[31] OECD Main Science and Technology Indicators. 2004.

[32] OECD. Science, Technology and Industry Outlook. 2006.

[33] Romer, Paul. “Endogenous Technological Change.” Journal of Political Economy98, No. 5, Part 2 (October 1990): S71-102.

[34] Sala-i-Martin, Xavier. “The World Distribution of Income.” Quarterly Journalof Economics 121, no 2 (May 2006): 351-397.

[35] Serrano, Carlos. “The Dynamics of the Transfer and Renewal of Patents.” Uni-versity of Toronto, March 2006.

[36] UNCTAD. World Investment Report. FDI from Developing and TransitionEconomies: Implications for Development. 2006.

[37] World Bank. “World Development Indicators 2007.” April 2007.

29

Time

Percent adopting

t1 t2 t4 t3 t5 t6

0.1

country 1

country 2

country 3country 4

country 5

country 6

Figure 8: Average diffusion rates and the calculation of the ti

6 Appendix

Calculating the diffusion lags reported in Figure 5.–Comin and Hobijn (2004) havecompiled data that cover 20 advanced countries and eleven technologies over the pasttwo hundred years.6 The variable ti was defined to be the average of the dates thatthe eleven technologies spread to ten percent of country i’s population. Figure 8illustrates how the ti were calculated. Ten percent is low enough that nine of eleventechnologies have reached it in all the countries covered. The eleven technologiesare private cars, radios, phones, television, personal computers, aviation passengers,telegraph, newspapers, mail, mobile phones, and rail.

6See the "Historical Cross-Country Technological Adoption: Dataset" at www.nber.org/data/

30