,Economic Dynamics: Theory and Computation
Second Edition
Additional Exercises
John Stachurski
(Solutions available on request)
Page 1
, P RELIMINARIES
The questions below use the material in the textbook plus the following two additional
results.
Theorem 1 (Neumann Series Lemma). Let A be n × n, let b be n × 1 and let I be the n × n
identity matrix. If r ( A) < 1, then I − A is nonsingular and x = Ax + b has the unique
solution
x ∗ = ( I − A)−1 b = ∑ Ak b.
k ≥0
In the theorem above, r ( A) is the spectral radius of A, defined by
r ( A) := max{|λ| : λ is an eigenvalue of A} (1)
Here |λ| indicates the modulus of the complex number λ.
In the next result, we take S and T to be self-maps on M ⊂ Rn and let ≤ be the point-
wise partial order on Rn . We say that T dominates S on M if Sx ≤ Tx for all x ∈ M.
Theorem 2. If T dominates S on M and, in addition, T is monotone increasing and globally
stable on M, then its unique fixed point dominates any fixed point of S.
Also, for a given pair of distributions ϕ, ψ on R, we say that ψ first order stochastically
dominates ϕ if
Z Z
u( x )ϕ(dx ) ≤ u( x )ψ(dx ) for every increasing bounded function u : R → R
Finally, we say that ψ is a mean-preserving spread of ϕ if there exists a pair of random
variables (Y, Z ) such that
E [ Z | Y ] = 0, Y =
d d
ϕ and Y + Z = ψ
In other words ψ is a mean-preserving spread of ϕ if it adds noise without changing
the mean.
Page 2
Second Edition
Additional Exercises
John Stachurski
(Solutions available on request)
Page 1
, P RELIMINARIES
The questions below use the material in the textbook plus the following two additional
results.
Theorem 1 (Neumann Series Lemma). Let A be n × n, let b be n × 1 and let I be the n × n
identity matrix. If r ( A) < 1, then I − A is nonsingular and x = Ax + b has the unique
solution
x ∗ = ( I − A)−1 b = ∑ Ak b.
k ≥0
In the theorem above, r ( A) is the spectral radius of A, defined by
r ( A) := max{|λ| : λ is an eigenvalue of A} (1)
Here |λ| indicates the modulus of the complex number λ.
In the next result, we take S and T to be self-maps on M ⊂ Rn and let ≤ be the point-
wise partial order on Rn . We say that T dominates S on M if Sx ≤ Tx for all x ∈ M.
Theorem 2. If T dominates S on M and, in addition, T is monotone increasing and globally
stable on M, then its unique fixed point dominates any fixed point of S.
Also, for a given pair of distributions ϕ, ψ on R, we say that ψ first order stochastically
dominates ϕ if
Z Z
u( x )ϕ(dx ) ≤ u( x )ψ(dx ) for every increasing bounded function u : R → R
Finally, we say that ψ is a mean-preserving spread of ϕ if there exists a pair of random
variables (Y, Z ) such that
E [ Z | Y ] = 0, Y =
d d
ϕ and Y + Z = ψ
In other words ψ is a mean-preserving spread of ϕ if it adds noise without changing
the mean.
Page 2
