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Structural change and non-constant biased technical change

  • Edgar Cruz EMAIL logo


Empirical evidence suggests that the differences in rates of technical progress across sectors are time-variant, implying that the bias in technological change is not constant. In this paper, we analyze the implications of this non-constant sectoral biased technical change for structural change and we assess whether this is an important factor behind structural transformations. To this end, we develop a multi-sectoral growth model where TFP growth rates across sectors are non-constant. We calibrate our model to match the development of the U.S. economy during the twentieth century. Our findings show that, by assuming non-constant biased technical change, a purely technological approach is able to replicate the sectoral transformations in the U.S. economy not only after but also prior to World War II.

JEL Classification: O41; O47


I thank my thesis advisor, Xavier Raurich, for his helpful suggestions. I thank Berthold Herrendorf for helpful comments and suggestions on an earlier version of the paper. I am grateful to Akos Valentinyi, Marc Teignier, Marti Mestieri, and Miguel Leon-Ledesma for valuable comments and suggestions. I acknowledge financial support from the Universitat de Barcelona through the grant APIF and from CONACYT grant 383793.


A Equilibrium properties

Solution to the representative consumer optimization problem

The Hamiltonian function associated with the maximization of (15) subject to (9), (10), (11), and (12) is


where μa, μs and μm are the co-state variables corresponding to the constraints (11) and (12), respectively. The first order conditions are




The sectoral allocation of capital and relative prices

From combining (34) and (36), we obtain


and combining (35) and (37), we obtain


We substitute υa and υs in (9), and taking into account (10), the optimal capital share in the manufacturing sector is


which implies


By assuming that the manufacturing good is the numerarie and dividing equations (34) by (35), and combining (10), (39), (40) and (41), we obtain the relative prices


Note that the relative prices are the ratio between the co-state variables.

The GDP is obtained by substitution of (16) and (17) in (45). Firstly, we substitute (16) in (8) to obtain


and GDP


By combining with (17), (44) and (45), we obtain


and, given (10), we obtain that


The Euler equation

From (19), we obtain Ca and Cs as functions of Cm and the relative prices

Ci=(ηiηm)ϵpiϵCm for i=a,s.

We then substitute these equation in (14) and combining with (33), we obtain


Substituting (46) in (20) we obtain


where total expenditure is a function of the co-state variable corresponding to the constraint (12). By substituting (16) in (8), and combining with (38), we obtain the growth rate of the co-state variables μm as follows


We then log-differentiate C=μm1 and combine with (47), we obtain (24).

Proof of Proposition 4.1. If ωm > 0, then the dynamic system is


From equation v^m, it follows that there is a unique steady value such that


and substituting in z^ and ĉ, we obtain


The steady state, it must be satisfied that z^ = ĉ = 0, implying that


Solving the system for z and c, we obtain


If ϖm = 0, then the dynamic system at the steady state is


At the steady state, we obtain


Proof of Proposition 4.2. If ωm > 0, there are two state variables and one variable control. Using (26), (27), and (28), we obtain the following Jacobian matrix evaluated at the steady state




It is immediate to see that the eigenvalues are λ1=ωm, and the two roots λ2 and λ3 are the solution of the following equation


where the solutions are


Insofar as a21 < 0 and ρ > 0, it follows that one of the roots, for example λ2 is always negative and the other one, λ3, is positive. So, λ1, λ2 < 0 and λ3 > 0. This result implies that there is a two-dimensional stable manifold in (z, c, v3) space.

On the other hand, If ωm = 0, there is one state variable and one control variable. Using (26), (27), we obtain the following Jacobian matrix evaluated at the steady state






the eigenvalues of the system are real numbers of opposite signs, and the steady state is saddle path stable.

B Estimation of the technology

Solution of differential equation

We then pose the law of motion of productivity in the i sector as follows


where we define the inverse of the distance across sectors and the frontier as follows


By taking the log-derivative of (48), we obtain that the law of motion of technological gaps is


We rewrite (49) as follows


then (50) can be integrated after a single substitution. Let




Substituting in (50), integrating and solving the integral equation,


we obtain


We substitute back into the mi to obtain


and substituting (51) in (48), we finally obtain


Estimation procedures: using EUKLEMS (1970–2005)

From the definition of (52), we can estimate ωi and ϕi using the data on the TFP growth rates of the agriculture, manufacturing, and services sectors from the EUKLEMS database that covers 1970–2005. To this end, we estimate the following equation which derives from (52) and our assumption that the technology frontier grows at a constant rate in equation (1). Taking logs in (52) we obtain


and normalizing the initial stock in the frontier to one, A0 = 1, we can estimate the following system of equations



αi=ϕiγωi;βi=γ; and δi=ωi.

We estimate the parameters in (53) constrained to βi = γ for all sector using non-linear squares. We report the results in Table 1 and Table 2 reports the estimated values of ωa, and ωs and the values of ϕa and ϕs, which are obtained by nonlinear combinations of the estimated parameters α^i and β^i using nlcom (nonlinear combination) command in STATA.


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Published Online: 2017-7-4

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