Exercise 1. Given is an input-output system with output y(t) and input u(t) and inputoutput relation
y(t) = α1 y(t − 1) + α2 y(t − 2) + β0 u(t) + β1 u(t − 1).
Construct an input-state-output representation of this system with state
y(t − 1)
x(t) =
.
y(t − 2)
Exercise 2. Consider the system described by equations (1.1), (1.2), (1.4) and (1.5) in
Example 1.1.
(a) Determine the equilibrium values of the variables, i.e. the values D0 , S0 , P0 and P̂0
such that D(t) = D0 , S(t) = S0 , P (t) = P0 and P̂ (t) = P̂0 . Rewrite the model in
terms of variables in deviation from equilibrium: d(t) = D(t) − D0 , s(t) = S(t) − S0 ,
p(t) = P (t) − P0 , and p̂(t) = P̂ (t) − P̂0 .
(b) Derive an input-state-output representation of the system, using p̂ as an unobserved
state variable.
(c) Eliminate the auxiliary variables p and p̂ and derive a first order difference equation
for the quantity traded.
(d) In the previous choose λ = 1 and give a graphical illustration of the development of
S(t) over time. Assume that α1 < 0, β1 > 0, and distinguish the cases −α1 > β1 , and
−α1 < β1 .
Exercise 3. Consider the system described by equations (1.1), (1.2), (1.4) and
P̂ (t) = P (t − 1) + λ P (t − 1) − P (t − 2)
in Example 1.1.
(a) Determine the equilibrium values of the variables, i.e. the values D0 , S0 , P0 and P̂0
such that D(t) = D0 , S(t) = S0 , P (t) = P0 and P̂ (t) = P̂0 . Rewrite the model in
terms of variables in deviation from equilibrium: d(t) = D(t) − D0 , s(t) = S(t) − S0 ,
p(t) = P (t) − P0 , and p̂(t) = P̂ (t) − P̂0 .
(b) Derive an input-state-output representation of the system. Choose an appropriate state
variable that summerizes all the past information that is relevant for the future evolution of the system.
(c) Eliminate the auxiliary variables p and p̂ and derive a difference equation for the
quantity traded. What is the order of the difference equation?