Dynamic optimization is often applied to analyze the optimal rate of extraction of exhaustible resources in the context of monopoly.

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In this scenario, a monopolist controls the entire supply of a finite resource and aims to maximize its profit over time, considering both current and future periods.

The key concept in dynamic optimization is the Hotelling’s Rule, which provides insights into the optimal rate of resource extraction. According to Hotelling’s Rule, in a competitive market, the price of an exhaustible resource should increase at the rate of interest. However, when a monopoly is involved, the situation is different.

In the case of a monopoly, the monopolist takes into account not only the scarcity of the resource but also its ability to influence the market price. The monopolist will extract the resource at a rate such that the marginal cost of extraction equals the rate of pure time preference plus the rate of increase of the resource’s scarcity.

Mathematically, the condition for the monopolist’s optimal extraction rate ((E_t)) can be expressed as:

[ \text{Marginal Cost of Extraction (MC)} = \text{Rate of Pure Time Preference (r)} + \text{Rate of Increase of Scarcity} ]

This equation guides the monopolist in determining the optimal intertemporal allocation of the resource extraction to maximize its discounted profits over time.

In summary, in a monopoly setting with exhaustible resources, dynamic optimization involves finding the extraction rate that balances the current profit from extraction with the consideration of future profits, reflecting the interplay between market power and the resource’s scarcity.