The effect of delay and the economic cycles

The paper considers one of the well-known mathematical models in economics, called "Aggregate Demand - Aggregate Supply model"(AD-AS). Originally this model takes the form of the following ordinary differential equation
\dot{p} = D(p) − S(p),
where $p = p(t)$ – price, $D(p)$ – demand, $S(p)$ – supply. This functions are considered with the natural assumptions for macroeconomics.
(i) Both of these functions are positive-definite and they are differentiable a sufficient number of times at $t \in (0, ∞)$;
(ii) $D'(p) < 0, S'(p) > 0;$
$(iii)$ Let $\lim\limits_{p\to 0} D(p) = D_0 $, $\lim\limits_{p\to ∞}D(p) = D_∞ $, $\lim\limits_{p\to0}S(p) = D_0$, $\lim\limits_{p\to ∞}S(p) = D_∞ $. Suppose that constants $D_0, S_∞$ are sufficiently large (or $D_0 = ∞, S_∞ = ∞$). On the contrary, $D_∞, S_0$ – are sufficiently small constants (it is also possible that they are equal to 0).
We note that this equation has only one equilibrium - (global) attractor $p(t) = p_0 > 0$. In particular, the equation AD-AS does not have periodic solutions, i.e. it can not describe the cycles that are typical for the real economy.
In this paper we propose to consider the delay differential equation D.-D.-S. instead of the traditional equation D.-S.
$$\dot{p} = D(p) − S(p_h), p_h = p(t − h), h > 0.$$
In this functional differential equation the function of demand depends on the price at the previous moment of time. This assumption is quite natural and meaningful from an economic point of view.
Let $p(t) = p_0 + x(t)$. Then the D.-D.-S. equation will take the form
$$\dot{x} = −ax(t) − by + a_2x − b_2y + a_3x − b_3y + . . . ,\;\;\;\;\;\;\;\;\;\;\;\;(\alpha)$$
where $y = x(t − h)$, the point means terms of higher order of smallness.
We will also consider the linear form of the previous equation
$$\dot{x} = −ax − by.$$

Lemma.There exist $b = b_∗(a)$, such that at $b < b_∗(a)$ the zero solution for both equations is asymptotically stable and unstable at $b > b_∗(a)$.
If $b = b_∗(a)$, then the linear equation has the periodic solution
$$x(t) = \xi \: exp (i\sigma t),$$
where $\xi ∈ C$, and $\sigma$ we will find after the analysis of characteristic equation
$$λ = −a − b\;exp(λh).$$

Now let $b = b_∗(1 + ε), where ε ∈ (0, ε_0), 0 < ε_0 << 1$. For such parameter b the following statement is true.
Theorem. There exist $ε_0 > 0$, such that for all $ε ∈ (0, ε_0)$ equation α has the asymptotically orbital stable cycle $x_∗(t, ε)$, for which the amplitude is proportional to ε^1/2.
The proof of the theorem is kept to studying of the normal form, i.e. to ordinary differential equation on the central two-dimensional manifold.
The cycle of the basic equation D.-D.-S. can be find by the formula
$$p_∗(t, ε) = p_0 + x_∗(t, ε).$$

Key words: differentional equations, economical cycle, delay, stability, bifurcation