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Pb. 2.16 The expression for the National Income is given by:

y(k) = c(k) + i(k) + g(k)
where c is consumer expenditure, i is the induced private investment, g is the
government expenditure, and k is the accounting period, typically corre-
sponding to a particular quarter. Samuelson theory, introduced to many
engineers in Cadzow’s classic Discrete Time Systems (see reference list),
assumes the following properties for the above three components of the
National Income:

1. Consumer expenditure in any period k is proportional to the
National Income at the previous period:

c(k) = ay(k – 1)

2. Induced private investment in any period k is proportional to the
increase in consumer expenditure from the preceding period:

i(k) = b[c(k) – c(k – 1)] = ab[y(k – 1) – y(k – 2)]

3. Government expenditure is the same for all accounting periods:

g(k) = g
Combining the above equations, the National Income obeys the second-
order difference equation:
y(k) = g + a(1 + b) y(k – 1) – aby(k – 2) for k = 1, 2, 3, …

The initial conditions y(–1) and y(0) are to be specified.
Plot the National Income for the first 40 quarters of a new national entity,
assuming that: a = 1/6, b = 1, g = $10,000,000, y(–1) = $20,000,000, y(0) =
$30,000,000.

How would the National Income curve change if the marginal propensity to
consume (i.e., the constant a) is decreased to 1/8?

2.5 Convolution-Summation of a First-Order System with
Constant Coefficients
The amortization problem in Section 2.2 was solved by obtaining the present
output, y(k), as a linear combination of the present and all past inputs, (u(k),

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