Model Description
This is a two-sector growth model that couples the dynamics of human demographics and a renewable resource base. The two sectors are agricultural and manufacturing sectors. To capture both the positive (Malthusian) and negative (modern growth) type relationships between population growth and output, it is important to model the shifting composition of output from agricultural to manufacturing as growth occurs. Thus, the model is a two sector (productions and consumptions in agricultural sector and manufacturing sector), three factor growth model with a renewable resource base.The three factors of production are labor, human-made capital, and natural capital. The human population dynamics is influenced by the per capita consumption of goods from each of the two sectors.
The main purpose of this model is to reveal under what conditions, in terms of investment, demographic, and economic parameters, a developing economy become susceptible to population overshoot and resource collapse. In one set of parameter space, any reasonable initial condition with high biophysical capital and low population will evolve to a stable steady state (perhaps through a series of damped oscillations). In the other set, it will converge to a limit cycle.
Scenarios
Here, we can test three scenarios: (scenario 1) if investment dynamics is fast relative to the regeneration rate of the resource base, investment is a fundamentally destabilizing force making the system more prone to overshoot and collapse; (scenario 2) if the feedback between manufactured goods consumption and birth rate b2 is sufficiently strong, the fertility transition can prevent overshoot and collapse; and (scenario 3) the higher the savings rate 's' and the stronger the feedback between manufactured goods consumption and the death rate d2 , the birth rate b2 must become higher to prevent overshoot and collapse.
To test scenario 1, assume an agricultraul intensive economy (set alpha1=0.7, alpha2=0.3) and set b2 = d2=0. Then vary the parameter for investment level 's' from low (s=0.09) to high (s=0.29) and observe how population and natural capital changes. As 's' is increased from low to high values, a stable equilibrium will turn into a limit cycle.
To test scenario 2, set b2>0 and examine the potential for the fertility transition to prevent overshoot and collapse. The implication of this analysis is that the greater the savings rate s, the greater b2 (the strength of the fertility transition) must be to prevent over-shoot and collapse. In fact, if b2>1, the system does not undergo a Hopf bifurcation (for example, with b2=1, and s=0.2, the equilibrium birth (and death rate) is approximately 1.1 percent versus 3.1 percent with b2=0 and s=0.09).
To test scenario 3, vary d2 from 0 to 1. The larger the effect the growth has on reducing death rates, the greater must be its effect on reducing birth rates to avoid over-shoot and collapse dynamics. In addition, assume that the economic structure is more capital intensive (set alpha1=0.3 and alpha2=0.7). Set the savings rate 's' at 0 .2 and d2=0. For a given value of b2, as the capital intensity in agriculture increases, the system moves from the stable region to the limit cycle region. The lower b2, the higher the value of alpha1 below which the system will exhibit overshoot and collapse behavior.
$\Large Y_{1}=E_{1}k_{r}^{\alpha _{r}}h^{\alpha _{1}}k_{h}^{1-\alpha_{1} } \ \ \ \normalsize \bf \mbox {(Eq. 1)}$ |
| Agricultural production |
$\Large Y_{2}=E_{2}h^{\alpha _{2}}k_{h}^{1-\alpha_{2} } \ \ \ \normalsize \bf \mbox {(Eq. 2)}$ |
| Manufacturing good production |
$\Large Max : U(q_{1}, q_{2})=(q_{1})^{c_{1}}(q_{2})^{1-c_{1}} \ \ \ \normalsize \bf \mbox {(Eq. 3)}$ |
| Consumers maximize their utility. Here, U is utility, q1 and q2 are the per capita consumption rates of agricultural and manufacturing goods,P1and P2 are their respective prices,M is per capita income, s is the savings rate, and c1 is the preference for agricultural goods. |
$\Large Subject \ to : P_{1}*q_{1}+P_{2}*q_{2}\leq (1-s)M \ \ \ \normalsize \bf \mbox {(Eq. 4)}$ |
$\Large q_{1}=\frac{c_{1}M(1-s)}{P_{1}} \ \ and \ \ q_{2}=\frac{(1-c_{1})M(1-s)}{P_{2}} \ \ \ \normalsize \bf \mbox {(Eq. 5)}$ |
| The solution to the optimization problem specified by Eq. 3 and 4. |
$\Large Y_{1}=E_{1}\beta ^{\alpha _{1}}\gamma ^{1-\alpha _{1}}k_{r}^{\alpha _{r}}h^{\alpha _{1}}k_{h}^{1-\alpha_{1} } \ \ \ \normalsize \bf \mbox {(Eq. 6)}$ |
| The equilibrium level of output in agricultural sector |
$\Large Y_{2}=E_{2}(1-\beta) ^{\alpha _{2}}(1-\gamma)^{1-\alpha _{2}}h^{\alpha _{2}}k_{h}^{1-\alpha_{2} } \ \ \ \normalsize \bf \mbox {(Eq. 7)}$ |
| The equilibrium level of output in manufactured good sector |
$\Large \beta=\left (\left ( \frac{1}{c_{1}(1-s)}-1 \right ) \frac{\alpha _{2}}{\alpha _{1}}+1 \right )^{-1} \ \ \ \normalsize \bf \mbox {(Eq. 8)}$ |
| Part of Eq. 6 and 7. |
$\Large \gamma=\left (\left ( \frac{1}{c_{1}(1-s)}-1 \right ) \frac{1-\alpha _{2}}{1-\alpha _{1}}+1 \right )^{-1} \ \ \ \normalsize \bf \mbox {(Eq. 9)}$ |
| Part of Eq. 6 and 7. |
$\Large I=\frac{sY_{2}}{1-c_{1}(1-s)} \ \ \ \normalsize \bf \mbox {(Eq. 10)}$ |
| Investment |
$\Large b=b_{0}(1-e^{-b_{1}q_{1}})e^{-b_{2}q_{2}} \ \ \ \normalsize \bf \mbox {(Eq. 11)}$ |
| Net birth rate: The term b0(1 -exp^b1q1) represents increases in birth rates up to a maximum of b0 as q1 (i.e. nutrition) increases. The parameter b1 measures the sensi-tivity of birth rates to such increases. The term exp^b2q2 represents downward pressure on birth rates as q2 increases (fertility transition). Again, b2measures the sensitivity of birth rates to changes in q2. |
$\Large d=d_{0}e^{-q_{1}(d_{1}+d_{2}q_{2})} \ \ \ \normalsize \bf \mbox {(Eq. 12)}$ |
| Net death rate: Improved nutrition reduces death rates (i.e. through improved immunity, etc.) via the term q1d1 while improved infrastructure reduces death rates via the term q1d2q2 (mortality transition) where as above, d1 and d2 are sen-sitivity parameters. |
$\Large \frac{dh}{dt}=(b-d)h \ \ \ \normalsize \bf \mbox {(Eq. 13)}$ |
| Rate of change in population. |
$\Large \frac{dk_{h}}{dt}=I-\delta k_{h} \ \ \ \normalsize \bf \mbox {(Eq. 14)}$ |
| Rate of change in human-made capital. |
$\Large \frac{dk_{r}}{dt}=n_{r}k_{r}(1-k_{r})-\eta Y_{1} \ \ \ \normalsize \bf \mbox {(Eq. 15)}$ |
| Rate of change in natural capital. Note that natural capital exhibits logistic growth. It is degraded by agricultural production. |
$\mbox {(NOTE)} \ \ \ \ \Large q_{1}=\frac{Y_{1}}{h} \ \ \ and \ \ \ q_{2}=\frac{Y_{2}}{h}$ |
#human ecosystem model with production functions #options econ.ops #=====define parameters #c1= agricultural good preference #s =savings rate #al1, al2=labor productivities #alr=natural capital productivity #dp= depreciation rate #d1,d2,b1,b2=demographic parameters. par al1=0.3,al2=0.7, alr=0.75, c1=0.3,s=0.23, dp=0.05, E1=1,E2=1 par nr=0.1, eta=0.1 par d2=0,b2=0,d1=5,b1=1,d0=0.1, b0=0.05 #=====define some hidden variables===== #al2=al1 be1=1-al1 be2=1-al2 A=(1/(c1*(1-s)))-1 gamma=1/((A*be2)/be1+1) beta=1/((A*al2)/al1+1) #============economic functions BB1=(beta**al1)*(gamma**be1) BB2=((1-beta)**al2)*((1-gamma)**be2) Y1(N,L,K)=E1*BB1*(N**alr)*(L**al1)*(K**be1) Y2(L,K)=E2*BB2*(L**al2)*(K**be2) #==================demographicfunctions d(q1,q2)=d0*exp(-q1*(d1+d2*q2)) b(q1,q2)=b0*exp(-b2*q2)*(1-exp(-b1*q1)) #variation on birth equation #b(q1,q2)=0.5*exp(-b2*q2**20/(q2tran**20+q2**20))*(1-exp(-b1*q1)) agout=Y1(kr,h,kh) manout=Y2(h,kh) brate=b(agout/h,manout/h) drate=d(agout/h,manout/h) hdot=(brate-drate)*h #=====auxiliary quantities================= #aux pagout=agout/h #aux pmanout=manout/h #aux bbrate=brate #aux ddrate=drate #aux eqcaplabrate=(s*BB2/(dp*(1-c1*(1-s))))**(1/al2) #aux abeta=beta #aux agamma=gamma #======right hand sides dh/dt=hdot dkh/dt=s*manout/(1-c1*(1-s))-dp*kh dkr/dt=nr*kr*(1-kr)-eta*agout #=============initial data init kh=0.1,kr=1,h=0.1 @ meth=qualrk @ bound=1000 @ total=2000 @ dt=0.1 @ xp=kr,yp=h @ xlo=0,xhi=1,ylo=0,yhi=2 done
Yu JHD, Arizona State University.
Bozicevic M, Arizona State University.
. 2003. Economic development, demographics, and renewable resources: a dynamical systems approach. Environment and Development Economics. 8:219-246.
