Limited factors and why optimal growth has led to destruction

Carmen Camacho (), Weihua Ruan () and Benteng Zou
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Carmen Camacho: PJSE - Paris Jourdan Sciences Economiques - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École nationale des ponts et chaussées - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École nationale des ponts et chaussées - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Weihua Ruan: Purdue University [West Lafayette]

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Abstract: We revisit the classical Ramsey (1928) model with time discounting and a linear production function, explicitly accounting for the inevitable limitations of tangible production factors, which must remain both finite and positive. By employing Pontryagin's (1962) maximum principle, we transform state constraints into control constraints and provide a complete solution for all impatience rates under the linear production framework. While we classify the levels of impatience as established in the existing literature, we show that the behaviors associated with this threshold fundamentally differ when input limitations are considered -a factor previously overlooked. Our analysis extends beyond the literature's traditional focus on agents with mild impatience, encompassing the entire spectrum of impatience. For highly patient agents, the policymaker prioritizes investment over consumption, ensuring the economy reaches its maximum capital level in finite time. Once this level is attained, consumption stabilizes indefinitely, achieving the golden rule trajectory -an outcome previously deemed unattainable under time discounting. Conversely, beyond the classical impatience threshold, capital and consumption decline over time. For agents with extreme impatience, we identify a second threshold where investment ceases entirely, leading to rapid depletion of capital and output.

Keywords: Economic growth; Optimal Control; Dynamic Programming; Limited resources; Linear Production; Discount (search for similar items in EconPapers)
Date: 2025-01
New Economics Papers: this item is included in nep-gro
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Working Paper: Limited factors and why optimal growth has led to destruction (2025)
Working Paper: Limited factors and why optimal growth has led to destruction (2025)
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