### Three Sectors Model of National Income Determination Including Population Structure

The role of demographic structure was not considered in the classic Hansen-Samuelson model. In order to analyze how the consumption intention and investment intention of different age groups influence the multiplier-accelerator model and then play a role in the fluctuation of national income, we have classified the entire population into two types, the young and the aged. Suppose the total young population is NOP, the total aged population is ROP and the total population is

*N*, the proportions of the young and the aged are

$$ \mathrm{NOPOP}=\frac{\mathrm{NOP}}{N},\kern0.875em \mathrm{ROPOP}=\frac{\mathrm{ROP}}{N},\kern0.875em \mathrm{NOPOP}+\mathrm{ROPOP}\equiv 1 $$

(6.7)

Correspondingly, both the marginal consumption intention and investment accelerator of the whole society are weighted averages of the young and the aged

$$ \beta =\mathrm{NOPOP}\times {\beta}_N+\mathrm{ROPOP}\times {\beta}_R $$

(6.8)

$$ v=\mathrm{NOPOP}\times {v}_N+\mathrm{ROPOP}\times {v}_R $$

(6.9)

where

*β*_{N} and

*β*_{R} are the marginal consumption intentions for the young and the aged respectively, and 0 <

*β*_{N} < 1 and 0 <

*β*_{R} < 1 are established; and

*v*_{N} and

*v*_{R} are the investment accelerators for the young and the aged respectively, and

*v*_{N} > 1 and

*v*_{R} > 1 are established.

When total social consumption is constituted by the young and the aged, the total consumption is the weighted average of the two parts, that is,

*C*_{t} = NOPOP ×

*C*_{NOPOP, t} + ROPOP ×

*C*_{ROPOP, t}, and the total investment is also the weighted average of the two parts, that is,

*I*_{t} = NOPOP ×

*I*_{NOPOP, t} + ROPOP ×

*I*_{ROPOP, t}. Therefore, the national income of the three sectors is determined by the following equations:

$$ {\mathrm{GDP}}_t={C}_t+{I}_t+{G}_t $$

(6.10)

$$ {C}_t=\mathrm{NOPOP}\times {C}_{\mathrm{NOPOP},t}+\mathrm{ROPOP}\times {C}_{\mathrm{ROPOP},t} $$

(6.11)

$$ {I}_t=\mathrm{NOPOP}\times {I}_{\mathrm{NOPOP},t}+\mathrm{ROPOP}\times {I}_{\mathrm{ROPOP},t} $$

(6.12)

For the young, total consumption should be equal to their marginal consumption intention multiplied by the previous total output:

$$ {C}_{\mathrm{NOPOP},t}={\beta}_N\times {\mathrm{GDP}}_{\mathrm{NOPOP},t} $$

(6.13)

The total output of the young should also be equal to the proportion of the population multiplied by the total social output:

$$ {\mathrm{GDP}}_{\mathrm{NOPOP},t}=\mathrm{NOPOP}\times {\mathrm{GDP}}_{t-1} $$

(6.14)

According to the relationship between total consumption and total output, the total consumption of the young is:

$$ {C}_{\mathrm{NOPOP},t}={\beta}_N\times \mathrm{NOPOP}\times {\mathrm{GDP}}_{t-1} $$

(6.15)

Correspondingly, the total consumption of the aged should also be equal to their marginal consumption intention multiplied by the proportion of the population and then the total social output in the next period:

$$ {C}_{\mathrm{ROPOP},t}={\beta}_R\times \mathrm{ROPOP}\times {\mathrm{GDP}}_{t-1} $$

(6.16)

Similarly, regarding investment, the total investment of the young is the accelerator multiplied by their previous consumption balance:

$$ {\displaystyle \begin{array}{l}{I}_{\mathrm{NOPOP},t}={v}_N\times \left({C}_{\mathrm{NOPOP},t-1}-{C}_{\mathrm{NOPOP},t-2}\right)\\ {}\kern4.25em ={v}_N\times \left({\beta}_N\times \mathrm{NOPOP}\times {\mathrm{GDP}}_{t-1}-{\beta}_N\times \mathrm{NOPOP}\times {\mathrm{GDP}}_{t-2}\right)\\ {}\kern4.25em ={v}_N\times {\beta}_N\times \mathrm{NOPOP}\times \left({\mathrm{GDP}}_{t-1}-{\mathrm{GDP}}_{t-2}\right)\end{array}} $$

(6.17)

It is the same with the aged:

$$ {\displaystyle \begin{array}{l}{I}_{\mathrm{ROPOP},t}={v}_R\times \left({C}_{\mathrm{ROPOP},t-1}-{C}_{\mathrm{ROPOP},t-2}\right)\\ {}\kern4.25em ={v}_R\times \left({\beta}_R\times \mathrm{ROPOP}\times {\mathrm{GDP}}_{t-1}-{\beta}_R\times \mathrm{ROPOP}\times {\mathrm{GDP}}_{t-2}\right)\\ {}\kern4.25em ={v}_R\times {\beta}_R\times \mathrm{ROPOP}\times \left({\mathrm{GDP}}_{t-1}-{\mathrm{GDP}}_{t-2}\right)\end{array}} $$

(6.18)

To bring (

6.13)–(

6.18) into (

6.10), (

6.11) and (

6.12):

$$ {\mathrm{GDP}}_t={C}_t+{I}_t+{G}_t $$

(6.19)

$$ {C}_t=\left({\mathrm{NOPOP}}^2\times {\beta}_N+{\mathrm{ROPOP}}^2\times {\beta}_R\right)\times {\mathrm{GDP}}_{t-1} $$

(6.20)

$$ {I}_t=\left({\mathrm{NOPOP}}^2\times {v}_N\times {\beta}_N+{\mathrm{ROPOP}}^2\times {v}_R\times {\beta}_R\right)\times \left({\mathrm{GDP}}_{t-1}-{\mathrm{GDP}}_{t-2}\right) $$

(6.21)

Let

*β*_{1} = NOPOP

^{2} ×

*β*_{N} + ROPOP

^{2} ×

*β*_{R} and

*v*_{1} = NOPOP

^{2} ×

*v*_{N} ×

*β*_{N} + ROPOP

^{2} ×

*v*_{R} ×

*β*_{R}, then

$$ {\mathrm{GDP}}_t=\left({\beta}_1+{\beta}_1\times {v}_1\right){\mathrm{GDP}}_{t-1}+{v}_1\times {\mathrm{GDP}}_{t-2}+{G}_t $$

(6.22)

When

*G*_{t} =

*G*_{0} and

*β*_{1} ×

*v*_{1} < 1, the general solution of the above state Eq. (

6.22) is:

$$ {\mathrm{GDP}}_t={r}_1^t\left({A}_1\sin {\omega}_1t+{B}_1\cos {\omega}_1t\right)+\frac{G_0}{1-{\beta}_1} $$

(6.23)

where

*r*_{1} = (

*β*_{1} ×

*v*_{1})

^{1/2}. Obviously, when

*t* → ∞, GDP

_{t} is convergent to (

*G*_{0}/1 −

*β*_{1}), which means the long-term steady-state equilibrium is:

$$ {\mathrm{GDP}}_1^{\ast }=\frac{G_0}{1-{\beta}_1} $$

(6.24)

### Comparison and Verification

First, let’s compare the steady-state output. In the function GDP = *G*_{0}/(1 − *β*), since (dGDP/d*β*) = [(*G*_{0}/(1 − *β*)^{2}) > 0], which is a monotonically increasing function, the smaller the marginal intention, the lower the long-term steady-state output.

Also because *β*_{1} = NOPOP^{2} × *β*_{N} + ROPOP^{2} × *β*_{R} and *β* = NOPOP × *β*_{N} + ROPOP × *β*_{R}, *β* − *β*_{1} = NOPOP × *β*_{N} × (1 − NOPOP) + ROPOP × *β*_{R} × (1 − ROPOP) is established. According to the equation NOPOP + ROPOP ≡ 1 in (6.7), 0 < NOPOP < 1 and 0 < ROPOP < 1 are true, so that *β* − *β*_{1} > 0 holds, which means *β* > *β*_{1}. Then according to the monotonically increasing nature of the function GDP = *G*_{0}/(1 − *β*), \( {\mathrm{GDP}}^{\ast }>{\mathrm{GDP}}_1^{\ast } \) is established, which verifies Conclusion 1: *An uneven population structure is unfavorable for the growth of long-term output of GNP*.

Then, let’s look into the impact of the reduction in the proportion of the young on steady-state output. Because \( \left({\mathrm{dGDP}}_1^{\ast }/\mathrm{dNOPOP}\right)=\left({G}_0/{\left(1-{\beta}_1\right)}^2\right) \)\( \times \left(\mathrm{d}{\beta}_1/\mathrm{dNOPOP}\right) \) and (d*β*_{1}/dNOPOP) = 2 × NOPOP × *β*_{N} + 2 × ROPOP × *β*_{R} × (dROPOP/dNOPOP), when NOPOP + ROPOP ≡ 1, (dROPOP/dNOPOP) = − 1, so (d*β*_{1}/dNOPOP) = 2 × (NOPOP × *β*_{N} − ROPOP × *β*_{R}).

Despite the fact that the consumption intention of the young *β*_{N} may be greater than that of the aged *β*_{R}, NOPOP × *β*_{N} − ROPOP × *β*_{R} < 0 will be established when the young takes a small enough proportion in the population structure, so \( \left({\mathrm{dGDP}}_1^{\ast }/\mathrm{dNOPOP}\right)<0 \), causing even lower steady-state output after the reduction of the proportion of the young. Therefore, Conclusion 2 is reached: *When the population structure is out of balance, if the proportion of the young is reduced sufficiently to offset the addition in consumption by the aged, the long-term output of GNP will decline even more*.

An unbalanced demographic structure will also affect the changes in the consumption intention of the entire society. According to the definitions of Eq. (6.24) and *β*_{1}, \( \left({\mathrm{dGDP}}_1^{\ast }/\mathrm{d}{\beta}_N\right)=\left[\left({G}_0\times {\mathrm{NOPOP}}^2/{\left(1-{\beta}_1\right)}^2\right)\right]>0 \) and \( \left({\mathrm{dGDP}}_1^{\ast }/\mathrm{d}{\beta}_R\right)=\left[{G}_0\times {\mathrm{ROPOP}}^2/{\left(1-{\beta}_1\right)}^2\right]>0 \) are established. Then it can be concluded that it is the young and the aged who can contribute to the growth of a long-term steady-state output by increasing consumption intentions.

With the aging of the population, as the young people decrease, the demand for durable consumer goods and non-durable consumer goods will become saturated, and then the consumption intention will decline when the rigid demand is met. In other words, the decline in *β*_{N} leads to the reduction of GDP_{1}^{∗}. Meanwhile, the demand for products required by the aged will be boosted in a short period of time as the aging population is growing, so that \( {\mathrm{GDP}}_1^{\ast } \) will grow as *β*_{R} becomes greater, but it will decline afterwards with a lower marginal consumption intention when the aged pass away, which means the decline in *β*_{R} leads to a further reduction of \( {\mathrm{GDP}}_1^{\ast } \). Therefore, in the aging process, the total social output will go through a ‘fall-rise-fall’ fluctuation as the marginal demand of the young decreases while the that of the aged increases and then decreases. Hence, Conclusion 3 can be reached: *In an aged society, the marginal consumption intentions of the young and the aged are not synchronized, so GNP may recover in the short term but will definitely decline in the long run*.

Finally, let’s look into the effect of accelerator. Because (d*r*_{1}/d*v*_{1}) = [*β*_{1}(*v*_{1}*β*_{1})^{−1/2}(NOPOP^{2}*β*_{N}d*v*_{N} + ROPOP^{2}*β*_{R}d*v*_{R})]/2, when the proportion of the young significantly reduces, their investment intention d*v*_{N} will fall too. Although the investment intention of the aged d*v*_{R} may rise temporarily, its degree of growth will not be very high (and will also be lower than that of the young) and will even shrink in the long run as the aged successively pass away. Therefore, the long-term investment intention of the entire society after the decrease of the young, that is, d*v*_{1}, will become smaller, resulting in a slower adjustment rate of short-term GNP toward long-term steady-state equilibrium. Even if macro policies are adopted to encourage such a shift, it will still take a longer time to realize due to the weakened investment intention, reducing the efficiency of the stimulus policies. So, Conclusion 4 is reached: With the aging of the demographic structure, the short-term investment intention of the aged will be stimulated, but the long-term investment intention of the entire society will decline because of the population decrease of the young. The fluctuation in investment of the population once again indicates a ‘fall-rise-fall’ trend. The long-term GNP will decline with the reduction in total social investment, while macro stimulus policies will also need longer time to regulate due to the decline in investment intention, accompanied by the reduced effectiveness of monetary and fiscal policy.