Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations

Economists and sociologists have been paying attention to the speed of new products, hoping to establish a mathematical model to describe it and use it to guide production [3].

Suppose a new product is to be launched on the market and the number of new products sold at t is X(t). Assuming that the product is easy to use and has excellent performance, these new products in use actually act as promotional materials, attracting the unpurchased. Assuming that each new product attracts K customers on average per unit time, X(t) satisfies the differential equation: (2) {dXdt=KXX(0)=X0 \left\{{\matrix{{{{dX} \over {dt}} = KX}\cr{X(0) = {X_0}}\cr}} \right.

Solution (2): (3) X(t)=X0eKt X(t) = {X_0}{e^{Kt}}

If t = 0 is taken to represent the moment of the birth of a new product, then X(0) = 0 can be obtained from X(t) ≡ 0 (2). This result is inconsistent with the facts. Because, model (1) only considers the role of physical advertising and ignores the possibility that manufacturers can advertise new products through other means, thereby opening up the market. If the product of X₀ has been put into use through hard work, then the growth of X(t) in the initial stage fits well with the result of Eq. (2). In Eq. (3), if we let t → ∞, then X(t) → ∞, which is also inconsistent with the facts. In fact, X(t) should have an upper bound. Suppose the upper bound of demand is M, the number of households that have not yet used new products is M − X(t). According to the statistical law, dXdt {{dX} \over {dt}} is X(M − X) proportional to and the proportional coefficient is r, then (4) dXdt=rX(M−X) {{dX} \over {dt}} = rX(M - X)

Its solution is: (5) X(t)=M1+ce−Mrt X(t) = {M \over {1 + c{e^{- Mrt}}}}

The curve X(t) becomes a growth curve or logistic curve (see Figure 1).

Fig. 1

According to Eq. (5), its first and second derivatives can be obtained: (6) X'(t)=cM2re−Mrt1+ce−Mrt {X^{'}}(t) = {{c{M^2}r{e^{- Mrt}}} \over {1 + c{e^{- Mrt}}}} (7) X″(t)=cM3r2e−Mrt(ce−Mrt−1)(1+ce−Mrt)2 {X^{''}}(t) = {{c{M^3}{r^2}{e^{- Mrt}}(c{e^{- Mrt}} - 1)} \over {{{(1 + c{e^{- Mrt}})}^2}}}

It is not difficult to see that when X^′(t) > 0, X(t) increases monotonically, and ce^−Mrt = 1 is derived from X^″(t₀) = 0, and X(t0)=M2 X({t_0}) = {M \over 2} at this time. When t < t₀, X^″(t₀) > 0, that is, X^′(t) increases monotonically, which means that when the sales volume is less than half of the maximum demand, the sales speed X^′(t) keeps increasing; when t > t₀, X^″(t₀) < 0, X^′(t) monotonously decreases, which means that when the sales volume reaches half the demand, (t = t₀). The product was the most popular, followed by (t > t₀), and the sales rate began to decline. From this survey by many economic experts and product salespeople, the sales curve of many new products is quite close to the logistic curve. Therefore, it is possible to conduct an in-depth analysis of the sales curve characteristics of new products and conclude that new products should be produced in a small amount in the early stage of sales, and at the same time, publicity and advertising should be strengthened. When the new product consumers reach 20–80%, from time to time, new products can be produced in large quantities; when the consumers of new products exceed 80%, then the company should choose the right time to switch production in order to achieve better interests of the company [4].

Price p is the main factor affecting demand D and supply S. Hypothesis (8) D=c−dpS=−a+bp \matrix{{D = c - dp}\cr{S =- a + bp}\cr}

Among them, p is the price, and a, b, c and d are normal numbers. The rate of price increase is proportional to the excess demand D-S, so there is a mathematical model of price: (9) dpdt=α(D−S)=α(c−dp+a−bp)=α(c+a−(b+d)p) {{dp} \over {dt}} = \alpha (D - S) = \alpha (c - dp + a - bp) = \alpha (c + a - (b + d)p)

Among them, k = α(b + d), h = α(a + c). The general solution of Eq. (9) is (10) p(t)=ce−kt+hkp¯=a+cb+d \matrix{{p(t) = c{e^{- kt}} + {h \over k}}\cr{\bar p = {{a + c} \over {b + d}}}\cr}

And when D = S, that is, c − d p = −a + bp, p is the equilibrium price, so p¯ \bar p is the equilibrium price; so p(t)=ce−kt+p¯ p(t) = c{e^{- kt}} + \bar p .

We see that although p(t) fluctuates, when t → ∞, p(t) tends to the equilibrium price p¯ \bar p , and the market price at this time tends to stabilise. If supply and demand are constant, but D > S, then dpdt=α(D−S) {{dp} \over {dt}} = \alpha (D - S) , (11) p(t)=ceα(D−S)tlimt→+∞p(t)=+∞ \matrix{{p(t) = c{e^{\alpha (D - S)t}}}\cr{\mathop {\lim}\limits_{t \to+ \infty} p(t) =+ \infty}\cr}

This is inflation. It is caused by a short supply. In order to stabilise prices, it is necessary to reduce consumption funds, reduce demand or increase the supply of goods. For example, reducing the number of government officials and redundant employees in enterprises can reduce consumption funds. Or the implementation of measures such as the promotion of commercial housing is to increase the supply of goods, which can play a role in restraining price increases [5].

Under a certain abnormal economic situation, consumers’ shopping psychology is abnormal. The more the price increases, the more they buy, and the faster the price increases, the more they buy, so demand D. The relationship between supply S and the price p(t) is: (12) {D=c1−d1p+c2dpdt+d2d2pdt2S=−a1+b1p+a2dpdt+b2d2pdt2 \left\{{\matrix{{D = {c_1} - {d_1}p + {c_2}{{dp} \over {dt}} + {d_2}{{{d^2}p} \over {d{t^2}}}}\cr{S =- {a_1} + {b_1}p + {a_2}{{dp} \over {dt}} + {b_2}{{{d^2}p} \over {d{t^2}}}}\cr}} \right. (13) dpdt=α(D−S)=α[(d2−b2)d2pdt2+(c2−a2)dpdt−(d1+b1)p+(c1+a1)] {{dp} \over {dt}} = \alpha (D - S) = \alpha \left[ {\matrix{{\left({{d_2} - {b_2}} \right){{{d^2}p} \over {d{t^2}}}}\cr{+ \left({{c_2} - {a_2}} \right){{dp} \over {dt}}}\cr{- \left({{d_1} + {b_1}} \right)p}\cr{+ \left({{c_1} + {a_1}} \right)}\cr}} \right]

Finished up: (14) α(d2−b2)d2pdt2+[α(c2−a2)−1]dpdt−α(d1+b1)p=−α(c1+a1) \alpha \left({{d_2} - {b_2}} \right){{{d^2}p} \over {d{t^2}}} + \left[ {\alpha \left({{c_2} - {a_2}} \right) - 1} \right]{{dp} \over {dt}} - \alpha \left({{d_1} + {b_1}} \right)p =- \alpha \left({{c_1} + {a_1}} \right)

Let dpdt=q {{dp} \over {dt}} = q , then (15) {dpdt=qdqdt=1−α(c2−a2)α(d2−b2)q+d1+b1d2−b2p−a1+c1d2−b2 \left\{{\matrix{{{{dp} \over {dt}} = q} \hfill\cr{{{dq} \over {dt}} = {{1 - \alpha \left({{c_2} - {a_2}} \right)} \over {\alpha \left({{d_2} - {b_2}} \right)}}q + {{{d_1} + {b_1}} \over {{d_2} - {b_2}}}p - {{{a_1} + {c_1}} \over {{d_2} - {b_2}}}\quad} \hfill\cr}} \right.

The singularity of Eq. (15) is M(a1+c1d1+b1,0) M({{{a_1} + {c_1}} \over {{d_1} + {b_1}}},0) , and the characteristic equation is (16) λ2−1−α(c2−a2)α(d2−b2)λ−d1+b1d2−b2=0λ1,2=12[1−α(c2−a2)α(d2−b2)±[1−α(c2−a2)α(d2−b2)]2+4(d1+b1)d2−b2] \matrix{{{\lambda ^2} - {{1 - \alpha \left({{c_2} - {a_2}} \right)} \over {\alpha \left({{d_2} - {b_2}} \right)}}\lambda- {{{d_1} + {b_1}} \over {{d_2} - {b_2}}} = 0} \hfill\cr{{\lambda _{1,2}} = {1 \over 2}\left[ {\matrix{{{{1 - \alpha ({c_2} - {a_2})} \over {\alpha ({d_2} - {b_2})}} \pm}\cr{\sqrt {{{\left[ {{{1 - \alpha ({c_2} - {a_2})} \over {\alpha ({d_2} - {b_2})}}} \right]}^2} + {{4({d_1} + {b_1})} \over {{d_2} - {b_2}}}}}\cr}} \right]} \hfill\cr}

λ₁ and λ₂ are real numbers with different signs, and the singularity is a saddle point, as shown in Figure 2.

Fig. 2

It can be seen from the phase diagram that the right half-plane is divided into two areas by the ray AC, the trajectory starting from above AC, when t → ∞, p(t) → ∞. The trajectory starts from below AC, and when t → ∞, p(t) → 0 due to consumers psychological abnormalities and unstable prices. Taking the ray AC as the dividing line, when the price state is above AC, inflation will be triggered, and the government intervenes in the market and forcibly lowers prices, causing the price state to fall below the AC ray, and prices will continue to fall again the consequences of depression. At this time c₂ < 0, d₂ < 0, and it should be supplemented by publicity and guidance to correct the abnormal shopping psychology of consumers. When consumers return to their normal shopping mentality, they are unwilling to shop when prices are increased or accelerated. On the other hand, when prices are increased or accelerated, manufacturers believe that they are profitable and increase supply. At this time, a₂ > 0, b₂ > 0. So in the characteristic root expression (17) 1−α(c2−a2)α(d2−b2)<04(d1+b1)d2−b2<0 \matrix{{{{1 - \alpha \left({{c_2} - {a_2}} \right)} \over {\alpha \left({{d_2} - {b_2}} \right)}} < 0}\cr{{{4\left({{d_1} + {b_1}} \right)} \over {{d_2} - {b_2}}} < 0}\cr}

λ₁ and λ₂ are two negative real numbers or conjugate complex numbers with negative real parts, M is a stable node or focus, and (18) ∂∂p(q)+∂∂q(1−α(c2−a2)α(d2−b2)q+d1+b1d2−b2p−a1+c1d2−b2)=1−α(c2−a2)α(d2−b2)<0 {\partial\over {\partial p}}(q) + {\partial\over {\partial q}}\left({\matrix{{{{1 - \alpha ({c_2} - {a_2})} \over {\alpha ({d_2} - {b_2})}}q}\cr{+ {{{d_1} + {b_1}} \over {{d_2} - {b_2}}}p - {{{a_1} + {c_1}} \over {{d_2} - {b_2}}}}\cr}} \right) = {{1 - \alpha ({c_2} - {a_2})} \over {\alpha \left({{d_2} - {b_2}} \right)}} < 0

Therefore, it is known from Theorem 1 that there is no closed track, that is, prices will not oscillate periodically and can only tend to a stable value a1+c1d1+b1 {{{a_1} + {c_1}} \over {{d_1} + {b_1}}} as time increases, resulting in a better market situation [6].

If the output value fluctuates too much over time, it will cause instability of economic life and social life, and economic adjustment must be made to make it develop in a relatively healthy manner [7].

Suppose Y (t) is the output value, I(t) is induced investment, A is a spontaneous investment, C(t) is consumption, where C(t) = cY (t) is, and suppose (19) dIdt=−k(I−νdYdt)dYdt=−λ(Y−C−I−A) \matrix{{{{dI} \over {dt}} =- k\left({I - \nu {{dY} \over {dt}}} \right)} \hfill\cr{{{dY} \over {dt}} =- \lambda \left({Y - C - I - A} \right)} \hfill\cr}

Then (20) dYdt=−λY+λ[(1−s)Y+I+A] {{dY} \over {dt}} =- \lambda Y + \lambda \left[ {(1 - s)Y + I + A} \right] where 1 − s = c is obtained from (13) (21) I=1λdYdt+sY−A I = {1 \over \lambda}{{dY} \over {dt}} + sY - A

Derivation from (20) (22) dIdt=1λd2Ydt2+sdYdt {{dI} \over {dt}} = {1 \over \lambda}{{{d^2}Y} \over {d{t^2}}} + s{{dY} \over {dt}}

From (21) (23) 1λd2Ydt2+sdYdt=−k(1λdYdt+sY−A)+kνdYdtd2Ydt2+(λs+k−kλν)dYdt+ksλY=kλA \matrix{{{1 \over \lambda}{{{d^2}Y} \over {d{t^2}}} + s{{dY} \over {dt}} =- k\left({{1 \over \lambda}{{dY} \over {dt}} + sY - A} \right) + k\nu {{dY} \over {dt}}} \hfill\cr{{{{d^2}Y} \over {d{t^2}}} + \left({\lambda s + k - k\lambda \nu} \right){{dY} \over {dt}} + ks\lambda Y = k\lambda A} \hfill\cr}

Converted into an equivalent system of equations (24) {dYdt=ududt=(kλν−k−λs)u−ksλY+kλA \left\{{\matrix{{{{dY} \over {dt}} = u} \hfill\cr{{{du} \over {dt}} = \left({k\lambda \nu- k - \lambda s} \right)u - ks\lambda Y + k\lambda A} \hfill\cr}} \right.

The singularity is P0=(Y0,u0)=(As,0) {P_0} = \left({{Y_0},{u_0}} \right) = \left({{A \over s},0} \right) . The characteristic equation is (25) |0−μ1−ksλ(kλν−k−λs)−μ|=0μ2−(kλν−k−λs)μ+ksλ=0μ1,2=12[(kλν−k−λs)±(kλν−k−λs)2−4ksλ] \matrix{{\left| {\matrix{{0 - \mu} & 1\cr{- ks\lambda} & {\left({k\lambda \nu- k - \lambda s} \right) - \mu}\cr}} \right| = 0}\cr{{\mu ^2} - \left({k\lambda \nu- k - \lambda s} \right)\mu+ ks\lambda= 0}\cr{{\mu _{1,2}} = {1 \over 2}\left[ {\matrix{{\left({k\lambda \nu- k - \lambda s} \right) \pm}\cr{\sqrt {{{\left({k\lambda \nu- k - \lambda s} \right)}^2} - 4ks\lambda}}\cr}} \right]}\cr}

Remember Δ = (kλ ν − k − λs)² − 4ksλ.

Fig. 3

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Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

In order to be economically stable, kλ ν − k − λ s ≤ 0 must be increased, s must be increased and s = 1 − c must be decreased. c, c is the consumption coefficient, so it is necessary to compress consumption, reduce the proportion of consumption funds or decrease ν, ν, which is the investment coefficient, that is, control. The speed of inducing investment does not make the induced investment grow too fast due to the stimulation of increased production [8].