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The optimal use of renewable resources is a central concern in the Economics of Social Sector and Environment. Renewable resources, unlike non-renewable ones, have the capacity to regenerate over time, but only up to a sustainable limit. The economic challenge is to determine the rate of use that maximizes social welfare while maintaining the resource base. This involves both biological and economic dynamics, framed in either discrete or continuous time. ### Optimal Use of Renewable Resources in Discrete Time In discrete-time models, time progresses in steps—e.g., year to year. Consider a basic renewable resource like a fish population that regenerates according to a known function. Let: * $x_t$: stock of the resource at time $t$ * $h_t$: harvest at time $t$ * $f(x_t)$: natural regeneration function * $\delta$: discount factor (where $0 < \delta < 1$) The dynamic of the resource can be written as: $$ x_{t+1} = f(x_t) - h_t $$ The goal is to choose $h_t$ in each period to maximize the present value of net benefits: $$ \max_{h_t} \sum_{t=0}^{\infty} \delta^t \cdot \pi(h_t) $$ subject to: $$ x_{t+1} = f(x_t) - h_t \quad \text{and} \quad 0 \leq h_t \leq f(x_t) $$ Here, $\pi(h_t)$ is the benefit function associated with harvesting $h_t$ units. In a typical case, $\pi(h_t) = p h_t$, where $p$ ____ _____ _______ _______ ___ __________ ________ ______ ______ _________.
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