Comparative Dynamics
Comparative Dynamics
Comparative dynamics does for dynamics what comparative statics does for statics. The difference lies in the fact that comparative dynamics is concerned with the effects of changes in the data (parameters, exogenous variables, initial conditions) on the whole motion over time of a dynamic economic model. This motion will usually be some sort of dynamic equilibrium path, such as, for example, a steady-state growth path where all variables grow at constant rates, or an optimal path deriving from a dynamic optimization problem.
The method of comparative dynamics can be summarized as follows. We have a set of dynamic functional equations, whose solution gives the time path of the economic system. In this solution, parameters, exogenous variables, and initial conditions also appear. Therefore, a different solution (time path) corresponds to a different set of data. It is then possible to ascertain the effect on the solution of a change in any one of the data.
Comparative dynamics, as such, does not say anything about the transition from one equilibrium growth path to another: The study of this transition belongs to stability analysis. Additionally, the conditions of stability of the equilibrium path may be useful in obtaining information on comparative dynamics: This can be regarded as the dynamic analogue of Paul Samuelson’s correspondence principle in comparative statics, with dynamic equilibrium path replacing static equilibrium point.
This is about as far as one can go with intuition. Formally, take the case of steady-state growth and consider a differential equation system
where x is a vector of endogenous variables, f is a vector of functions, ω is a vector of exogenous variables (given functions of time, also called forcing functions), and θ is a vector of parameters. We are interested in a particular solution of this system that serves as the reference path for comparative dynamics. If we assume that all the exogenous variables grow at a constant proportional rate (which may be equal or different across variables), we have
where ωi(0) and γ i are given. A steady-state (or balanced-growth) path is a particular solution to system (1) having the form
where the initial values xi (0) and the growth rates ρ i are to be determined. This particular solution is usually obtained by the method of undetermined coefficients: equations (2) and (3) are substituted into system (1) and the values of xi (0) and ρi are determined so that the system is identically satisfied. This will give rise to a set of equations in the unknowns xi (0) and ρi : Typically, the ρi are obtained by solving the equations derived from equating to zero the coefficients of the terms containing t, whereas the xi (0) are obtained by solving the equations derived from equating to zero all the other terms not containing t.
The solution will express the unknowns in terms of the data; usually we shall obtain
where υ⊂θ, that is, υ is a vector containing just a few parameters of the full set of the model’s parameters. The functions hi are typically fairly simple, while the functions (φi are often very complicated. All results from comparative dynamics are obtained calculating the partial derivatives of the functions hi and φi with respect to the element we are interested in.
SEE ALSO Comparative Statics; Eigen-Values and Eigen-Vectors, Perron-Frobenius Theorem: Economic Applications; Phase Diagrams; Stability in Economics; Steady State
BIBLIOGRAPHY
Gandolfo, Giancarlo. 1996. Economic Dynamics (chap. 20). 3rd ed. Berlin and New York: Springer.
Kamien, Morton I., and Nancy L. Schwartz. 1991. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management (pt. 2, sect. 8). 2nd ed. Amsterdam: North-Holland.
Samuelson, Paul A. 1947. Foundations of Economic Analysis (chap. 12). Cambridge, MA: Harvard University Press.
Giancarlo Gandolfo