Optimal pattern of technology adoption under embodiment with a finite planning horizon : A multi-stage optimal control approach

By deriving the necessary conditions for a multi-stage discounted optimal control problem where the endogenous switching instants between regimes appear as an argument of the objective function and the state equation, we analyze the optimal pattern of technology adoption under embodiment with a finite planning horizon. The economy is characterized by the existence of an exogenously growing technology frontier and technology specific learning by doing. We obtain time varying durations for the adopted technologies to be in use due to finite planning horizon. We analyze numerically the effects of planing horizon, speed of learning, growth rate of technology and impatience rate on the optimal pattern.Multi-stage optimal control, technology adoption, learning by doing, embodiment

The Development problem under embodiment

We study technology adoption in an optimal growth model with embodied technical change. The economy consists of the final good sector, the capital sector, and the technology sector whose role is the imitation of exogenous innovations. Labor resources are scarce. They are freely allocated to the technology and final good sectors. The final good is freely allocated to consumption and to the capital sector. We analytically characterize the optimal allocation decisions in the long run. Using a calibrated version of the model, we find that an acceleration in the rate of embodied technical change should not be responded by an immediate and strong adoption effort. Instead, adoption labor should decrease in the short run, and the optimal technological gap is shown to increase either in the short or in the long run. The state of the institutions and policies around the technology sector is key in the design of the optimal adoption timing.Embodiment, Technology adoption, Technological gap, Transition dynamics

Quality of Knowledge Technology, Returns to Production Technology and Economic Development

Presenting a discrete time version of the Romer (1986) model, this paper analyzes optimal paths in a one-sector growth model when the technology is not convex. We prove that for a given quality of knowledge technology, the countries could take-off if their initial stock of capital are above a critical level; otherwise they could face a poverty-trap. We show that for an economy which wants to take-off by means of knowledge technology requires three factors : large amount of initial knowledge, small fixed costs and a good quality of knowledge technology.Optimal Growth;optimal path;value fuction;poverty-trap;increasing returns

Technology adoption under embodiment : A two-stage optimal control approach

We use two stage optimal control techniques to solve some adoption problems under embodied technical change. We first solve a benchmark problem without learning behavior. At the date of switching, the consumption level is shown to drop, as the relative price of capital goes down (obsolescence). In such a case, the economy sticks to the initial technology, or immediately switches to a new technology with a higher level of embodiment, depending on how the obsolescence costs compare to the induced growth advantage. In a second step, we introduce learning. The learning curve involves fixed costs and incentives to wait as well. Adoption is shown to depend on the growth advantage of switching net of obsolescence and learning fixed costs. The economy will switch if and only if this indicator is positive. If it is big enough to “compensate” the option of waiting, then the economy switches immediately. Otherwise, the economy waitsOptimal control, adoption, learning, embodiment

Endogenous Time Preference and Strategic Growth

This paper presents a strategic growth model that analyzes the impact of Endogenous preferences on equilibrium dynamics by employing the tools provided by lattice theory and supermodular games. Supermodular game structure of the model let us provide monotonicity results on the greatest and the least equilibrium without making any assumptions regarding the curvature of the production function. We also sharpen these results by showing the differentiability of the value function and the uniqueness of the best response correspondence almost everywhere. We show that, unlike globally monotone capital sequences obtained under corresponding optimal growth models, a non-monotonic capital sequence can be obtained. We conclude that the rich can help the poor avoid poverty trap whereas even under convex technology, the poor may theoretically push the rich to her lower steady state.Lattice programming, Endogenous time preference

Optimal control in infinite horizon problems: a Sobolev spaces approach

In this paper, we make use of the Sobolev space W1,1 (R+,Rn) toderive at once the Pontryagin conditions for the standard optimalgrowth model in continuous time, including a necessary and sufficienttransversality condition. An application to the Ramsey model is given.We use an order ideal argument to solve the problem inherent to thefact that L1 spaces have natural positive cones with no interior points.Optimal control, Sobolev spaces, Transversality conditions, Order ideal

Optimal control in infinite horizon problems : a Sobolev space approach

In this paper, we make use of the Sobolev space W1,1(R+, Rn) to derive at once the Pontryagin conditions for the standard optimal growth model in continuous time, including a necessary and sufficient transversality condition. An application to the Ramsey model is given. We use an order ideal argument to solve the problem inherent to the fact that L1 spaces have natural positive cones with no interior points.Optimal control, Sobolev spaces, transversality conditions, order ideal.

Existence, Optimality and Dynamics of Equilibria with Endogenous Time Preference

To account for the development patterns that differ considerably among economies in the long run, a variety of one-sector models that incorporate some degree of market imperfections based on technological external effects and increasing returns have been presented. This paper studies the dynamic implications of, yet another mechanism, the endogenous rate of time preference depending on the stock of capital, in a one-sector growth model. The planner's problem is presented and the optimal paths are characterized. We show that development or poverty traps can arise even under a strictly convex technology. We also show that even under a convex-concave technology, the optimal path can exhibit global convergence to a unique stationary point. The multipliers system associated with an optimal path is proven to be the supporting price system of a competitive equilibrium under externality and detailed results concerning the properties of optimal (equilibrium) paths are provided. We show that the model exhibits globally monotone capital sequences yielding a richer set of potential dynamics than the classic model with exogenous discounting.Endogenous time preference; Optimal growth; Competitive equilibrium; Multiple steady-states

Existence, Optimality and Dynamics of Equilibria with Endogenous Time Preference

To account for the development patterns that differ considerably among economies in the long run, a variety of one-sector models that incorporate some degree of market imperfections based on technological external effects and increasing returns have been presented. This paper studies the dynamic implications of, yet another mechanism, the endogenous rate of time preference depending on the stock of capital, in a one-sector growth model. The planner’s problem is presented and the optimal paths are characterized. We show that development or poverty traps can arise even under a strictly convex technology. We also show that even under a convex-concave technology, the optimal path can exhibit global convergence to a unique stationary point. The multipliers system associated with an optimal path is proven to be the supporting price system of a competitive equilibrium under externality and detailed results concerning the properties of optimal (equilibrium) paths are provided. We show that the model exhibits globally monotone capital sequences yielding a richer set of potential dynamics than the classic model with exogenous discounting.Endogenous time preference; Optimal growth; Competitive equilibrium; Multiple steady-states

Optimal timing of regime switching in optimal growth models: A Sobolev space approach

This paper analyses the optimal timing of switching between alternative and consecutive regimes in optimal growth models. We derive the appropriate necessary conditions for such problems by means of the standard techniques from calculus of variations and some basic properties of Sobolev spaces.Multi-stage optimal control ; Sobolev spaces ; Optimal growth models