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 μ_a, μ_s and μ_m are the co-state variables corresponding to the constraints (11) and (12), respectively. The first order conditions are

and

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 C_a and C_s as functions of C_m and the relative prices

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 μ_m as follows

We then log-differentiate C=μm−1 and combine with (47), we obtain (24).

Proof of Proposition 4.1. If ω_m > 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 ϖ_m = 0, then the dynamic system at the steady state is

At the steady state, we obtain

Proof of Proposition 4.2. If ω_m > 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

where

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

where

As

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

where

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

we obtain

We substitute back into the m_i to obtain

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

Estimation procedures: using EUKLEMS (1970–2005)

From the definition of (52), we can estimate ω_i and ϕ_i 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

where

We estimate the parameters in (53) constrained to β_i = γ for all sector using non-linear squares. We report the results in Table 1 and Table 2 reports the estimated values of ω_a, and ω_s and the values of ϕ_a and ϕ_s, which are obtained by nonlinear combinations of the estimated parameters α^_i and β^_i using nlcom (nonlinear combination) command in STATA.