WEALTH, DISPOSABLE INCOME AND CONSUMPTION - Economics

58
E[qˆ t + 1 qˆ t + 2
( qˆ t)
= qˆ( k)]
= φk l
l = 1m
= 1
, φ l m
, qˆ()qˆ l ( k)
(A5)
Terms further into the future can be constructed in a similar fashion. The
solutions (A4) and (A5) bring home the magnitude of the task of computing
the cumulative growth factors. The terms (A4) and (A5) are only the
first two terms of an infinite summation, and this infinite summation has to
be computed in every state of the system.
Fortunately, there is a closed-form solution for Γ based on the
approximated system, which makes the task of computing the cumulative
growth factors much more manageable. Let Γ be an N × 1 vector of all the
cumulative growth factors in the discrete system. In addition, let Ω be an
N × N matrix with all its rows being Q , where is the transpose operator.
The vector of cumulative growth factors then has the following
representation:
τ
τ
∞
(A6)
where • denotes element-by-element multiplication, ι is an N × 1 vector
of ones, and I is the identity matrix. Computing the vector of cumulative
growth factors in every state of the system therefore amounts to
inverting a large matrix. With grids of 25 points, the matrix to be inverted
is 625 ×
625 .
N
∑
N
∑
Γ ( Φ• Ω)
α ∑ ι [ I – Φ •Ω]
1 –
= = ( – I)ι
α = 1