** 4 Robustness of the empirical results**

** ACCUMULATION EQUATION (YEARS)**

### 5

### Conclusion

Technological progress is a dual phenomenon which makes a diﬀerent use of labor inputs at diﬀerent levels of development. Far from the technological fron- tier, imitation of technologies is the main engine of total factor productivity growth. As a country gets closer to the frontier, it relies more and more on innovation, which implies reallocating labor from one activity to the other. Us- ing an endogenous growth model, we demonstrate that this reallocation process can create a complementarity between skilled labor and proximity to the fron- tier. We use this theoretical insight to revisit the empirical relationship between schooling level and growth in rich countries, which previous research had found to be slightly negative. Using a panel of 19 OECD countries between 1960 and 2000, we obtain two main results. First, the growth-enhancing margin in OECD

countries is that of skilled human capital rather than that of total human cap- ital. Second, skilled human capital has a stronger growth-enhancing eﬀect in economies which are closer to the technological frontier.

Our model is stylized and could be enriched in several dimensions. First, we have taken the skill composition of the labor force to be exogenous. A further step would be to endogeneize it by modeling explicitly schooling decisions and by allowing for cross-country migrations. The current research by Aghion-Boustan- Hoxby-Vandenbussche (2004) on Education, Migration, and Growth across US

states, is a first attempt in that direction27. Our empirical estimates in this

paper can be used to derive policy prescriptions on the optimal composition of

education spending in developed countries28_{, however endogenizing educational}

achievement would make this objective more rigorously reachable. Second, for simplicity we abstracted from international trade considerations. Developing a dynamic Ricardian model around the core idea presented in this paper would make it possible to study cross-sectoral allocation of skilled and unskilled labor in a context of international specialization. Such analysis would certainly yield further insights on the relationship between distance to frontier, composition of human capital and economic growth.

### 6

### Appendix 1: An extension of the model

In Section 2, we assume that productivity improvements result from the addition of two separate components, imitation and innovation. This specification bears the underlying assumption of the activities being perfect substitutes. Yet, one might think that technological progress necessitates a combination of imitation

2 7_{Focusing on the US States, that paper is able to both use a finer set of instruments and}

construct more precise measures of educational achievement.

2 8_{See for example Sapir (2003), which builds on the present paper to argue that EU countries}

should invest more in higher education in order to reduce the productivity growth deficit vis- à-vis the US. Currently, EU countries invest 1.1% of their GDP in higher education, compared to 3% in the US.

and innovation, i.e. that these activities are not perfect substitutes. In this appendix, we show first that for an elasticity of substitution strictly smaller than one the solution to the problem is always interior for 0 < a < 1. Second, we show by means of a numerical simulation that for an elasticity of substitution suﬃciently close to one, Proposition 1 still holds.

We now contemplate the more general productivity growth function: Ai,t = Ai,t−1+ λ[(um,i,tσs1m,i,t−σ)ρ( ¯At−1− At−1)ρ+ γ(uφn,i,ts

1−φ n,i,t)ρA

ρ

t−1]1/ρ (16)

where ρ ∈ [0, 1] measures the substitutability between imitation and innovation activities in generating productivity growth and other variables and parameters are the same as in (3).

First order conditions are now:

σsρ(1_{m} −σ)_{(1 − a)}ρuσρ_{m}−1= γφsρ(1_{n} −φ)aρuφρ_{n} −1 (17)

and

(1 − σ)uρσm(1 − a)ρs(1m−σ)ρ−1= γ(1 − φ)uρφn aρs(1n−φ)ρ−1 (18)

Dividing across equations, we find again (6), and so:

ψun
sn
= um
sm
and
um=
ψU sm
S + (ψ − 1)sm
Replacing in (17), we get:
[σ(1 − a)
ρ
φaρ ψ
σρ−1_{U}ρ(σ−φ)_{]} 1
ρ−1_{s}
m[S + (ψ − 1)sm]
ρ
1−ρ(σ−ρ)_{= S − s}
m

This equation makes it clear that when ρ < 1, we can rule out corner solutions.

side goes to S. Conversely, when smgoes to S, the right hand side goes to zero

while the left hand side does not. This guaranties that if a solution exists, it has to be an interior solution.

The above equation cannot be solved analytically. However, whenever _{1}ρ

−ρ(σ−

ρ) is a strictly positive integer, i.e. for ρ = _{N +σ}N_{−φ} _{(where N ∈ N}+_{), it is a}

polynomial equation, thus allowing the application of numerical simulation. We do so for the arbitrary vector of parameters (λ, γ, σ, φ, N, U ) = (1, 1, .8, .2, 2, 1). This choice of parameters is without loss of generality. Figure 2a plots the growth rate as a function of proximity to the frontier on the horizontal axis (we let proximity vary between .4 and .98) and the stock of skilled human cap- ital on the vertical axis (we let it vary between 0.02 and .9). Figure 2b plots the derivative of the growth rate with respect to proximity, and Figure 2c the cross-derivative of the growth rate with respect to proximity and skilled human capital. On this last figure, the black (resp. white) area represents the domain where the cross-derivative is positive (resp. negative) and we observe that in the domain close to the frontier, Proposition 1 indeed holds.

FIGURE 2a: GROWTH RATE
0.4
0.98
a
0.02
0.9
S
0
2
g
0.4
0.98
a
FIGURE 2b: dg_{da}
0.4
0.98
a
0.02
0.9
S
-0.1
0
dg
da
0.4
0.98
a

FIGURE 2c: _{dSda}d2g
0.4
0.98
a
0.02
S
-0.01
0.01
d2_{g}
dSda
0.4
0.98
a

**Proximity** **YearsPS** **YearsT** **Prox*YPS** **Prox*YT**
**Lagged proximity** 0.1 5.7 0.59 -7.7 -0.34
(.33) (4.8) (.56) (2.6)*** (.21)
**Lagged PS-exp./capita** 0.091 -0.98 -0.26 1.35 0.08
(.06) (.9) (.13)* (.54)** (.048)
**Lagged T-exp./capita** -0.1 0.77 0.12 -0.89 -0.019
(.029)*** (.5) (.07) (.27)*** (.023)
**Lagged Prox*PS-exp/capita** 0.25 -1.1 -0.27 3.1 0.067
(.1)** (1.7) (.21) (.76)*** (.07)
**Lagged Prox*T-exp/capita** -0.24 0.66 0.29 -1.7 0.017
(.06)*** (1) (.18)** (.52)*** (.054)
**R2** 0.95 0.91 0.94 0.89 0.87
**Number of observations** 122 122 122 122 122

One, two and three * indicate significance at the 10, 5 and 1% level respectively.

**TABLE A2a**
**REDUCED FORM (YEARS I BL)**

Note: standard errors in parentheses. Time and country dummies not reported. All independent variables are in logs. Expenditures are in dollars. A test for the joint significance of country dummies yields a p-value of 0. The same is true for time dummies.

**Proximity** **YearsPS** **YearsT** **Prox*YPS** **Prox*YT**
**Lagged Proximity** 0.1 -1.49 0.94 -5.5 -0.44
(.33) (3.55) (.81) (3.8) (.3)
**Lagged PS-exp./capita** 0.091 0.39 -0.37 1.3 0.1
(.06) (.72) (.175)** (.75)* (.067)
**Lagged T-exp./capita** -0.1 -0.01 0.21 -0.92 -0.045
(.029)*** (.36) (.11)* (.38)** (.031)
**Lagged Prox*PS-exp/capita** 0.25 0.67 -0.51 3.3 0.14
(.1)** (1.04) (.3) (1.2)** (.1)
**Lagged Prox*T-exp./capita** -0.24 -0.49 0.53 -2.15 -0.06
(.06)*** (.64) (.23)** (.83)** (.08)
**R2** 0.95 0.99 0.96 0.9 0.88
**Number of observations** 118 118 118 118 118

One, two and three * indicate significance at the 10, 5 and 1% level respectively.

**TABLE A2b**
**REDUCED FORM (YEARS I DD)**

Note: standard errors in parentheses. Time and country dummies not reported. All independent variables are in logs. A test for the joint significance of country dummies yields a p-value of 0. The same is true for time dummies.

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