A Renewable Stock Constrained by a Renewable Resource
A renewable resource and the capital used to harvest it form linked stocks. Profits can fund more harvesting capacity even as the resource declines, creating a risk that harvest will exceed regeneration and cause collapse.
THE IDEA
A Renewable Stock Constrained by a Renewable Resource
This model, corresponding to Figures 42–45, describes a two-stock system: a renewable resource and capital. The resource stock changes via regeneration (inflow) and harvest (outflow). The capital stock changes via investment (inflow) and depreciation (outflow).
Equations and Parameters
Resource Stock:
- Stock:
resource(t) = resource(t - dt) + (regeneration - harvest) * dt - Initial value:
resource = 1000 - Inflow:
regeneration = resource * regeneration rate - Outflow:
harvest = capital * yield per unit capital
Capital Stock:
- Stock:
capital(t) = capital(t - dt) + (investment - depreciation) * dt - Initial value:
capital = 5 - Inflow:
investment = minimum of profit or growth goal - Outflow:
depreciation = capital / capital lifetime
Converters:
capital lifetime = 20growth goal = capital * 10%profit = (price * harvest) - capital
Nonlinear Relationships
- Price: Starts at 1.2 when yield per unit capital is high (resource abundant), rises to 10 as yield falls (resource scarce). The relationship is nonlinear; its full curve is not provided in this excerpt.
- Regeneration rate: Zero when resource is fully stocked or completely depleted; peaks near 0.5 at intermediate resource levels; the full curve is not provided here.
- Yield per unit capital: Starts at 1 when resource is fully stocked, declines nonlinearly as resource stock declines. Efficiency increases across Figures 43, 44, and 45, which are referenced but not included here.
Dynamics
The model shows how capital growth (investment) is constrained by profit, which depends on price and harvest. As the resource depletes, yield per unit capital falls, raising price and potentially allowing continued profit, but if harvest exceeds regeneration, the resource collapses, leading to capital depreciation. The three figures explore different levels of harvesting efficiency.
When to use it
Sustainable fisheries management: setting catch limits to keep harvest below regeneration rate.
Groundwater extraction: modeling the balance between pumping and aquifer recharge to avoid depletion.
Grazing on communal pastures: regulating herd size to prevent overgrazing and land degradation.
Biomass harvesting: examining whether investment in harvesting equipment grows faster than the resource can regenerate.
FROM THEORY TO EVERYDAY LIFE
In practice
Fishery Management
A fishery with a fish stock and fishing fleet. As fish are caught, yield per boat decreases, increasing fish price. If profits remain high, investment in new boats continues, potentially leading to overfishing and stock collapse if regeneration is insufficient.
Forestry and Logging
A forest with trees and logging equipment. Harvest depends on forest size and equipment efficiency. If demand (price) for timber stays high, more equipment is bought, accelerating harvest and possibly causing deforestation if regrowth is slow.
A MOMENT TO REFLECT
Put it to work.
How does the model demonstrate the risk of overshoot and collapse when capital growth outpaces resource regeneration?
What real-world assumptions might alter the model's outcome, such as time lags in regeneration or alternative income sources?
How could policy interventions (e.g., quotas or taxes) be represented in this model to prevent resource depletion?
CONTINUE THE CONVERSATION
Source texts
- Thinking in Systems A Primer