## 7. Household model strategies |
16 October 1999 |

- Static model strategies
- Dynamic model strategies
- Simulations with linear model strategies
- Simulations with dynamic model strategies
- Simulations with non linear model strategies
- Exercises.
- References.

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:

Strategy | Equation
| Eq. no |

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 | Eq. 7.1:1 |

**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
extra expenditures:

*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.

Strategy | Equation
| Eq. no |

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
(CU/year) |

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
(CU/year) |

**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.

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