WEALTH, DISPOSABLE INCOME AND CONSUMPTION - Economics

expectation operator. Instead, the expectation in (2.4) must be evaluated
directly.
This is accomplished by approximating the continuous-valued VAR
as a discrete-valued finite-state vector Markov chain. If net income growth,
the real interest rate and their joint distribution are made to be discrete,
expectations may be solved as a probability weighted sum rather than as
an intractable integral. The approximation procedure is due to Tauchen
(1986) and it is implemented by means of 25-point grids for net income
growth and the real interest rate. The finite-state system therefore has
25 states, and the transition matrix describing the dynamics of the
system is . The statistical properties of the discrete-valued system
are summarized in the bottom panel of Table 2. The circumflex on and
distinguishes the series of discrete values from the true series of continuous
values. From a comparison of the upper and lower panels of Table 2, it
is apparent that the finite-state vector Markov chain closely mimics the statistical
properties of the VAR.
2
= 625
625 × 625
xˆ rˆ
With the approximated system, the cumulative growth factors Γ can
be computed directly for every state of the system. The explicit solution for
Γ is given in Appendix 2, and a sample of the results is provided in Table 3.
The values for xˆ and rˆ in the second column and row respectively are the
discrete values that net income growth and the real interest rate (both
measured at quarterly rates) can take on. For example, suppose, xˆ is in
state 3 and rˆ
is in state 24. The net income growth is –2.19 per cent at quarterly
rates, the discount rate is 3.41 per cent at quarterly rates, and the
cumulative growth factor is 85.5. If the level of net income was, say,
$3,000 at quarterly rates, then real per capita human wealth would be
$3,000 x 85.5 = $256,500.
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