7. Household model strategies
|16 October 1999|
Another approach is to assume that the households spend more when they earn more. Assume also that the households expect certain incomes = wages and dividends (CU/year), Uwa and Udiv. The households have desired expenditures (CU/year), Uexp, for employees and capital owners together. If they earn more, they spend the ratio rexp of their surplus incomes. If they earn less, they spend less. The fraction rexp of the surplus income have to be positive but less than one (0 < rexp < 1).
These assumptions give the linear and static model strategy of table 7.1:1 below:
|Spending part of surplus||X(9) = Uexp + rexp*((X(8)-Uwa)+ rexp*(X(13)-Udiv)|
|Spending part of surplus||-rexp*X(8) + X(9) - rexp*X(13) =|
= Uexp - rexp*(Uwa + Udiv)
Table 7.1:1. Household linear and static model strategy.
The households may also relate their consumption to the previous period. They want to consume the same volume of goods and services as the previous period. It is hard to find a realistic model without savings and loans accounts. We will test two models.
First, we assume that a fraction of an income rise is used to increase the consumption. This model has the disadvantage of not taking price changes of goods and services into account. A second model is to assume that the households spend a fraction of the savings of the previous period. The households adjust gradually to both changed incomes and changed prices.
The excess incomes over the previous period would be X(8)(t)
+ X(13)(t) - X(9)(t-Dt). The fraction
rexp of the excess incomes is spent. We have to divide
by the price pr(CU/pmy) to get the increase in volume X(7)(t)
- X(7)(t-Dt). We also have to consider
that the adjustment during the time Dt
is the fraction (Dt/1year) of
a whole years adjustment:
X(7)(t) - X(7)(t-Dt) = (X(8)(t) + X(13)(t) - X(9)(t-Dt))/pr * (Dt/1year) (pmy/year)
The second model would consider the savings of the previous period
X(10)(t-Dt) as the margin for
X(7)(t) - X(7)(t-Dt) = X(10)(t-Dt) / pr * (Dt/1year) (pmy/year)
By rearranging these equations, we get the equations of table 5.3:4.
|Spend fraction of income increase||pr*X(7)(t) - Dt/1year * X(8)(t) - Dt/1year * X(13)(t) = pr*X(7)(t-Dt) - Dt/1year * X(9)(t-Dt)||Eq. 7.2:1
|Spend fraction of last periods savings||pr*X(7)(t) = pr*X(7)(t-Dt) + Dt/1year * X(10)(t-Dt)||Eq. 7.2:2
Table 7.2:1. Household linear and dynamic model strategy.
The units of the equations are payment flows, currency units per year (CU/year). This does not mean that X(9) = pr*X(7) can be substituted into the equation all over. Note that the price pr is valid at time t but not for the previous sample at time (t-Dt). Thus X(9) (t-Dt) is not equal to pr*X(7) (t-Dt). This distinction comes from our assumption that the consumers want to consume the same real goods and services as before (not the same amount of money as before).
The terms X(7)(t-Dt), X(9)(t-Dt),
X(10)(t-Dt) in the right member
of equations 7.2:1 and 7.2:2 introduce new state variables and
turn the strategies into a dynamic strategies.
I will try to add some exercises to each chapter. I do not publish any answers at the moment but readers are welcome with their suggestions to my e-mail. If the reader wishes, I can add the answers to this document. I reserve the right to add my own comments to the answers. If contributors wish, I can also, as far as I have time, return personal comments by e-mail.
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