Limited factors and why optimal growth has led to destruction

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.