Abstract
In this paper we introduce stochastic parameters into the network game model with production and knowledge externalities. This model was proposed by V. Matveenko and A. Korolev as a generalization of the two-period Romer model. Agents differ in their productivities which have deterministic and stochastic (Wiener) components. We study the dynamics of a single agent and the dynamics of a dyad where two agents are aggregated. We derive explicit expressions for the dynamics of a single agent and dyad dynamics in the form of Brownian random processes, and qualitatively analyze the solutions of stochastic equations and systems of stochastic equations.
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Appendix
Appendix
The proof of Proposition 6.4.
Proof
The system of differential equations in the deterministic case has the form
The characteristic equation for system (6.11) is as follows
therefore eigenvalues are
where \(\bar {A}=\frac {A_1+A_2}{2}\). Obviously, we can choose as the eigenvectors of the matrix A the vectors
So the transition matrix is
then
where
The general solution of system (6.11) is as follows
We find the constants D 1 and D 2 by solving the system of equations
It is easy to verify that they are determined by the expression (6.18). We find the integration constants C 1 and C 2 from the initial conditions:
so
where
Then
Thus, the solution is determined by expression (6.17). ■
The proof of Theorem 6.3.
Proof
It is clear that the matrices A and α commute; therefore, for the matrix exponentials, the relation
holds and we can solve the matrix equation (6.16) by multiplying from the left by the matrix exponent
Denote, as in the one-dimensional case, for brevity
Then we have
Thus, Eq. (6.16) takes the form
therefore, the solution of matrix equation (6.8) can be written as
Notice, that
The eigenvalues of the matrix
are obviously λ 1 = −1 and \(\lambda _2=\frac {\bar {A}}{a}-\frac {(\alpha _1+\alpha _2)^2}{8a^2}-1\). As eigenvectors we can take
and
or in view of (6.14)
The eigenvalues of the matrix α are λ 1 = 0 and λ 2 = α 1 + α 2, and obviously we can choose the same e 1 and e 2 as eigenvectors as for the matrix \(A-\frac {\alpha ^2}{2}\). Therefore, to reduce to the diagonal form of the matrices \(\left (A-\frac {\alpha ^2}{2}\right )t\) and αW t we can use the same transition matrices
so we get
where
and correspondingly
Substituting (6.26) into (6.25) we obtain
Calculating expression (6.27) we get expressions (6.19)–(6.20). ■
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Korolev, A. (2020). Adjustment Dynamics in Network Games with Stochastic Parameters. In: Petrosyan, L.A., Mazalov, V.V., Zenkevich, N.A. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51941-4_6
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DOI: https://doi.org/10.1007/978-3-030-51941-4_6
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