The Hotelling rule, named after economist Harold Hotelling, provides a condition for the optimal depletion of exhaustible resources over time.

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It is a principle used in natural resource economics, particularly for non-renewable resources like minerals or fossil fuels. The Hotelling condition suggests that the economic rent—the difference between the market price and extraction cost—should increase at the rate of interest over time to maximize the net present value of resource extraction.

Mathematically, the Hotelling rule can be expressed as follows:

[ r = \frac{p – c}{p} ]

Where:

- ( r ) is the rate of change in the rent over time.
- ( p ) is the market price of the resource.
- ( c ) is the marginal cost of extraction.

The key implication of the Hotelling rule is that the resource rent, or the economic surplus derived from extracting and selling the resource, should increase over time at a rate equal to the prevailing interest rate. This condition reflects an intertemporal optimization strategy, where the decision to extract or conserve a finite resource is based on balancing present and future benefits.

If the rent does not increase at the rate of interest, it implies that either extraction is happening too quickly, leading to premature depletion, or too slowly, resulting in suboptimal resource utilization. The Hotelling rule provides a guideline for policymakers and resource managers to make decisions about the optimal rate of extraction, considering economic, environmental, and intergenerational equity considerations.

It’s worth noting that the Hotelling rule makes certain assumptions, including perfect competition, no technological progress, and constant extraction costs. In reality, these conditions may not always hold, and adaptations may be needed for a more accurate analysis of resource depletion.