
6
B. The Model Equations
The model includes the following differential equations. The dependent variable in each equation is the rate of growth of the variable so that each variable x grows at a rate Dlogx a c c o r d i n g t o t h e d i f f e r e n c e b e t w e e n t h e a c t u a l ( x ) a n d t h e p a r t i a l e q u i l i b r i u m v a l u e ( x d ) . stands for the derivative with respect to time. The superscript d defines the partial equilibrium or desired value, in the sense that it expresses the motion of the variable under consideration of the endogenous variable as a function of endogenous and exogenous variables.^{3 }Solutions for the steady state growth rates are presented in the Appendix and depend as usual on the rates of growth of the exogenous variables. Endogenous variables include output (Y), business services,^{4 }both domestic and imported (Sh, Sm) and technology (T). α, γ, and δ are parameters to be estimated. In continuous time the speed of adjustment can be interpreted in terms of the mean time lag, as its reciprocal represents the time required for about the 63% of the difference between the observed and the desired variables to be eliminated (see Gandolfo, 1981). The model is a panel, hence each equation refers to a number of countries. To better clarify this point and explain how we model technology diffusion a model with many countries is discussed in Section II.C. For simplicity’s sake we omit the residual terms and refer the reader to Gandolfo (1981) for an analysis of the stochastic proprieties of residuals in continuous time. D
Output (1)
( ⎪ ⎨ ⎧ + = l l o g l o g 1 Y Y Y D d d α α og = α_{0 }
− logY )
l
og
T
+α_{sh2 }
l
og
S
h
+α_{mh2 }
log S_{m }+α_{3 }log K +α_{4 }log L
Services domestic (2)
( ⎩ ⎨ ⎧ + + = l o g l o g l o g l − log S_{h }) d o g 2 1 Y S S S D s h s h d h h s h h γ γ γ = γ _{sh0 }
logT + γ _{sh3 }
log STR + γ _{sh4 }
log ICT + γ _{sh5 }
log REG
Services imported (2’)
( ⎩ ⎨ ⎧ + + = l o g l o g l o g l − log S_{m }) d o g 2 1 Y S S S D s m s m d m m s m m γ γ γ = γ _{sm0 }
logT + γ _{sm3 }
log STR + γ _{sm4 }
log ICT + γ _{sm5 }
log REG
Technology (3)
3
For an application of the same methodology to a trade and growth model at the sectoral level see Padoan (1998).
4 Business services include also Communication services and Finance and Insurance. These sectors have been chosen as qualitative studies have shown their relevance in the diffusion of technology (for a review see Guerrieri and Meliciani, 2004).