# Delay two-sector economic growth model with a Cobb–Douglas production function

## Abstract

This study demonstrates the possibility of cyclic capital accumulation in the case in which there are delays in capital implementation and estimation of capital depreciation. For this purpose, a two-sector growth model with Cobb–Douglas production function is built. It is shown that the stability of the balanced growth may change as lengths of delay change. It is also shown that on the stability switching curve the stability is lost and bifurcates to a limit cycle via a Hopf bifurcation.

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1. 1.

We can arrive at the same result to be obtained even if $$\nu <0$$ is assumed.

2. 2.

Parameters $$\alpha$$ and $$\beta$$ denote the capital distribution rates of production sectors 1 and 2. It is natural to assume that both are positive and less than unity. $$\theta$$ is the sum of the depreciation coefficient and the growth rate of labor (i.e., $$\mu +n$$) that could take a small value. These parameter values are selected for illustration, but other values would not change the qualitative properties shown here.

3. 3.

Mathematica, version 12.1 is used for simulations.

4. 4.

Under the specified parameter values, the trajectory becomes negative for the first time at $$t_{a}\simeq 1027.71.\$$Since the dynamic system is delayed, it can generate real solutions until $$t= t_{b}\simeq 1078.71$$ at which a solution becomes complex. Notice that the difference of the critical times, $$t_{b}-t_{a}=51,$$ is equal to the length of the delay.

5. 5.

The functions satisfying those conditions guarantee that (37) is the characteristic equation for a delay system. For more detail, see Appendix A of Matsumoto and Szidarovszky (2018).

6. 6.

It is possible that the same region may be considered on the left with respect to one point of the curve and be considered as on the right on another point of the curve.

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Correspondence to Akio Matsumoto.