Carmen Camacho-Perez

The dynamics of capital distribution across space are an important topic in economic geography and, more recently, in growth theory. In particular, the spatial AK model has been intensively studied in the latter stream. It turns out that the positivity of optimal capital stocks over time and space for any initial capital spatial distribution has not been entirely settled even in the simple linear AK case. We use Ekeland’s variational principle together with Pontrya-gin’s maximum principle to solve an optimal spatiotemporal AK model with a state constraint (non-negative capital stock), where the capital law of motion follows a diffusion equation. We derive the necessary optimality conditions to ensure the solution satisfies the state constraints for all times and locations. The maximum principle enables the reduction of the infinite-horizon optimal control problem to a finite-horizon problem, ultimately proving the uniqueness of the optimal solution with positive capital and the non-existence of such a solution when the time discount rate is either too large or too small.

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.

Under the threat of a rapid expanding virus like the 2020 COVID-19, policy-makers need to decide relatively fast whether and under which conditions to invest in a vaccine, and eventually adopt other protective measures like social distancing or lockdowns, or to wait for natural herd immunity. Taking into account that vaccines take time to be fully developed and effective, this paper considers a unified framework at the crossroad between economics and epidemiology to study optimal public spending in medical research to obtain a vaccine against an infectious disease evolving according to a SIR dynamics. We prove that developed economies always invest in the search of a vaccine. The more individuals care about consumption, the more they actually reduce their current consumption and the more they invest in the vaccine research program to recover their consumption potential at the earliest. Our model would only recommend economies with very poor technology to restrain from investment and wait for herd immunity.

We construct a Susceptible–Infected–Vaccinated Economic two‐sector growth model to explore the dynamics of inequality in an economy with distinct groups of workers exposed to a transmissible disease. Our analysis reveals a spectrum of outcomes in the long term, ranging from a disease‐free economic environment to a scenario where only the most susceptible group suffers from the disease. Long‐term outcomes are influenced by the reproduction rates both of the overall economy and those of the two groups of workers. If one group remains infected over time, the other will surely follow, leading to a perpetual disease burden for both. Additionally, because long‐term equilibria may not be unique, there is a possibility of long‐term uncertainty, posing additional challenges for policymakers. Notably, our calibrated model suggests that if the vaccination rate exceeds 24%, the relationship between disease exposure and inequality in capital assets becomes nonmonotonic.