We reconsider the optimal population size problem in a continuous time economy populated by homog... more We reconsider the optimal population size problem in a continuous time economy populated by homogenous cohorts with a fixed life span. Linear production functions in the labor input and standard rearing costs are also considered. First, we study under which conditions the successive cohorts will be given the same consumption per capita. We show that this egalitarian rule is optimal whatever the degree of altruism when life spans are infinite. However, when life spans are finite, this rule can only be optimal in the Benthamite case, i.e. when the degree of altruism is maximal. Second, we prove that under finite life spans the Millian welfare function leads to optimal extinction at finite time whatever the lifetime. In contrast, the Benthamite case is much more involved: for isoelastic utility functions, it gives rise to two threshold lifetime values, say T 0 < T 1 : below T 0 , finite time extinction is optimal; above T 1 , balanced growth paths are optimal. In between, asymptotic extinction is optimal. Third, optimal consumption and population dynamics are given in closed-form.
A family of optimal control problems for economic models, where state variables are driven by del... more A family of optimal control problems for economic models, where state variables are driven by delay differential equations (DDEs) and subject to constraints, is treated by Bellman&#x27;s dynamic programming in infinite dimensional spaces. An existence theorem is provided for the associated Hamilton-Jacobi-Bellman (HJB) equation: the value function of the control problem solves the HJB equation in a suitable sense (although such value function cannot be computed explicitly). An AK model with vintage capital and an advertising model with delay effect are taken as examples.
Asia-Pacific Journal of Accounting & Economics, 2011
In this paper, we devise a social criterion in the spirit of the critical utility level of to stu... more In this paper, we devise a social criterion in the spirit of the critical utility level of to study an optimal population size problem in an endogenously growing economy populated by workers living a fixed amount of time and without capital accumulation. Population growth is endogenous. The problem is analytically solved, yielding closed-form solutions to optimal demographic and economic dynamics. It is shown that provided the economy is not driven to optimal finite time extinction, the optimal solution is egalitarian for appropriate choices of the critical utility levels: all individuals of any cohort are given the same consumption. The results obtained do not require any priori restriction of the values of the elasticity of intertemporal substitution unlike in several related papers.
In this paper the dynamic programming approach is exploited in order to identify the closed loop ... more In this paper the dynamic programming approach is exploited in order to identify the closed loop policy function, and the consumption smoothing mechanisms in an endogenous growth model with time to build, linear technology and irreversibility constraint in investment. Moreover the link among the time to build parameter, the maximum capital reproduction rate, and the magnitude of the smoothing effect is deeply investigated and compared with what happens in a vintage capital model characterized by the same technology and utility function. Finally we have analyzed the effect of time to build on the speed of convergence of the main aggregate variables.
We consider the following economic problem: modelling the vintage capital structure of an economi... more We consider the following economic problem: modelling the vintage capital structure of an economic system. The problem is naturally formulated as an optimal control problem where the state equation is either a first order PDE or a delay equation. The optimal state-control couple describe the behavior of the economic agent. We formulate the problem as the optimal control of an ODE's in a Hilbert space using standard techniques of semigroup theory and we develop the dynamic programming approach for this problem. The HJB equation cannot be studied with the existing techniques due the special features of the problem (the semigroup appearing in the state equation is not analytic, boundary control terms appear, the cost is not smooth). We show some new results on this equation and we apply them to find an optimal feedback for the economic problem
This paper deals with an endogenous growth model with vintage capital and, more precisely, with t... more This paper deals with an endogenous growth model with vintage capital and, more precisely, with the AK model proposed in [R. Boucekkine, O. Licandro, L.A. Puch, F. del Rio, Vintage capital and the dynamics of the AK model, J. Econ. Theory 120 (1) (2005) 39-72]. In endogenous growth models the introduction of vintage capital allows to explain some growth facts but strongly increases the mathematical difficulties. So far, in this approach, the model is studied by the Maximum Principle; here we develop the Dynamic Programming approach to the same problem by obtaining sharper results and we provide more insight about the economic implications of the model. We explicitly find the value function, the closed loop formula that relates capital and investment, the optimal consumption paths and the long run equilibrium. The short run fluctuations of capital and investment and the relations with the standard AK model are analyzed. Finally the applicability to other models is also discussed. ✩ We thank Davide Fiaschi for useful discussions. We also thank an anonymous referee and an anonymous associated editor for deep comments and useful suggestions that led to an improved version of the paper.
A benchmark AK optimal growth model with maintenance expenditures and endogenous utilization of c... more A benchmark AK optimal growth model with maintenance expenditures and endogenous utilization of capital is considered within an explicit vintage capital framework. Scrapping is endogenous, and the model allows for a clean distinction between age and usage dependent capital depreciation and obsolescence. It is also shown that in this set-up past investment profile completely determines the size of current maintenance expenditures. Among other findings, a closed-form solution to optimal dynamics is provided taking advantage of very recent development in optimal control of infinite dimensional systems. More importantly, and in contrast to the pre-existing literature, we study investment and maintenance co-movements without any postulated ad-hoc depreciation function. In particular, we find that optimal investment and maintenance do move together in the short-run in response to neutral technological shocks, which seems to be more consistent with the data.
We present an application of the Dynamic Programming (DP) and of the Maximum Principle (MP) to so... more We present an application of the Dynamic Programming (DP) and of the Maximum Principle (MP) to solve an optimization over time when the production function is linear in the stock of capital (Ak model). Two views of capital are considered. In one, which is embraced by the great majority of macroeconomic models, capital is homogeneous and depreciates at a constant exogenous rate. In the other view each piece of capital has its own finite productive life cycle (vintage capital). The interpretation of the time patterns of macroaggregates is quite different between the two cases. A technological shock generates an oscillatory movement in the time pattern of per capita output when capital has a vintage structure; conversely an instantaneous adjustment with no transitional dynamics occurs when capital is homogeneous. From a methodological point of view it emerges that the DP approach delivers sharper results than the MP approach (for instance it delivers a closed form solution for the optimal investment strategy) under slacker parameter restrictions. Cross-time and cross-country data on investments, income, and consumption drawn from the Penn World Table version 6.2 are used to evaluate the vintage and standard Ak model.
The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to opti... more The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded controlrelated term in the state equation in Hilbert setting. An existence-anduniqueness result is obtained.
Economic and demographic models governed by linear delay differential equations are expressed as ... more Economic and demographic models governed by linear delay differential equations are expressed as optimal control problems in infinite dimensions. A general objective function is considered and the concavity of the Hamiltonian is not required. The value function is a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation and a verification theorem is proved.
The present paper continues the study of infinite dimensional calculus via regularization, starte... more The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô's type formula applied to f (t, X(t)) where f : [0, T ] × H → R is a C 0,1 function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of mild solution for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.
We reconsider the optimal population size problem in a continuous time economy populated by homog... more We reconsider the optimal population size problem in a continuous time economy populated by homogenous cohorts with a fixed life span. Linear production functions in the labor input and standard rearing costs are also considered. First, we study under which conditions the successive cohorts will be given the same consumption per capita. We show that this egalitarian rule is optimal whatever the degree of altruism when life spans are infinite. However, when life spans are finite, this rule can only be optimal in the Benthamite case, i.e. when the degree of altruism is maximal. Second, we prove that under finite life spans the Millian welfare function leads to optimal extinction at finite time whatever the lifetime. In contrast, the Benthamite case is much more involved: for isoelastic utility functions, it gives rise to two threshold lifetime values, say T 0 < T 1 : below T 0 , finite time extinction is optimal; above T 1 , balanced growth paths are optimal. In between, asymptotic extinction is optimal. Third, optimal consumption and population dynamics are given in closed-form.
A family of optimal control problems for economic models, where state variables are driven by del... more A family of optimal control problems for economic models, where state variables are driven by delay differential equations (DDEs) and subject to constraints, is treated by Bellman&#x27;s dynamic programming in infinite dimensional spaces. An existence theorem is provided for the associated Hamilton-Jacobi-Bellman (HJB) equation: the value function of the control problem solves the HJB equation in a suitable sense (although such value function cannot be computed explicitly). An AK model with vintage capital and an advertising model with delay effect are taken as examples.
Asia-Pacific Journal of Accounting & Economics, 2011
In this paper, we devise a social criterion in the spirit of the critical utility level of to stu... more In this paper, we devise a social criterion in the spirit of the critical utility level of to study an optimal population size problem in an endogenously growing economy populated by workers living a fixed amount of time and without capital accumulation. Population growth is endogenous. The problem is analytically solved, yielding closed-form solutions to optimal demographic and economic dynamics. It is shown that provided the economy is not driven to optimal finite time extinction, the optimal solution is egalitarian for appropriate choices of the critical utility levels: all individuals of any cohort are given the same consumption. The results obtained do not require any priori restriction of the values of the elasticity of intertemporal substitution unlike in several related papers.
In this paper the dynamic programming approach is exploited in order to identify the closed loop ... more In this paper the dynamic programming approach is exploited in order to identify the closed loop policy function, and the consumption smoothing mechanisms in an endogenous growth model with time to build, linear technology and irreversibility constraint in investment. Moreover the link among the time to build parameter, the maximum capital reproduction rate, and the magnitude of the smoothing effect is deeply investigated and compared with what happens in a vintage capital model characterized by the same technology and utility function. Finally we have analyzed the effect of time to build on the speed of convergence of the main aggregate variables.
We consider the following economic problem: modelling the vintage capital structure of an economi... more We consider the following economic problem: modelling the vintage capital structure of an economic system. The problem is naturally formulated as an optimal control problem where the state equation is either a first order PDE or a delay equation. The optimal state-control couple describe the behavior of the economic agent. We formulate the problem as the optimal control of an ODE's in a Hilbert space using standard techniques of semigroup theory and we develop the dynamic programming approach for this problem. The HJB equation cannot be studied with the existing techniques due the special features of the problem (the semigroup appearing in the state equation is not analytic, boundary control terms appear, the cost is not smooth). We show some new results on this equation and we apply them to find an optimal feedback for the economic problem
This paper deals with an endogenous growth model with vintage capital and, more precisely, with t... more This paper deals with an endogenous growth model with vintage capital and, more precisely, with the AK model proposed in [R. Boucekkine, O. Licandro, L.A. Puch, F. del Rio, Vintage capital and the dynamics of the AK model, J. Econ. Theory 120 (1) (2005) 39-72]. In endogenous growth models the introduction of vintage capital allows to explain some growth facts but strongly increases the mathematical difficulties. So far, in this approach, the model is studied by the Maximum Principle; here we develop the Dynamic Programming approach to the same problem by obtaining sharper results and we provide more insight about the economic implications of the model. We explicitly find the value function, the closed loop formula that relates capital and investment, the optimal consumption paths and the long run equilibrium. The short run fluctuations of capital and investment and the relations with the standard AK model are analyzed. Finally the applicability to other models is also discussed. ✩ We thank Davide Fiaschi for useful discussions. We also thank an anonymous referee and an anonymous associated editor for deep comments and useful suggestions that led to an improved version of the paper.
A benchmark AK optimal growth model with maintenance expenditures and endogenous utilization of c... more A benchmark AK optimal growth model with maintenance expenditures and endogenous utilization of capital is considered within an explicit vintage capital framework. Scrapping is endogenous, and the model allows for a clean distinction between age and usage dependent capital depreciation and obsolescence. It is also shown that in this set-up past investment profile completely determines the size of current maintenance expenditures. Among other findings, a closed-form solution to optimal dynamics is provided taking advantage of very recent development in optimal control of infinite dimensional systems. More importantly, and in contrast to the pre-existing literature, we study investment and maintenance co-movements without any postulated ad-hoc depreciation function. In particular, we find that optimal investment and maintenance do move together in the short-run in response to neutral technological shocks, which seems to be more consistent with the data.
We present an application of the Dynamic Programming (DP) and of the Maximum Principle (MP) to so... more We present an application of the Dynamic Programming (DP) and of the Maximum Principle (MP) to solve an optimization over time when the production function is linear in the stock of capital (Ak model). Two views of capital are considered. In one, which is embraced by the great majority of macroeconomic models, capital is homogeneous and depreciates at a constant exogenous rate. In the other view each piece of capital has its own finite productive life cycle (vintage capital). The interpretation of the time patterns of macroaggregates is quite different between the two cases. A technological shock generates an oscillatory movement in the time pattern of per capita output when capital has a vintage structure; conversely an instantaneous adjustment with no transitional dynamics occurs when capital is homogeneous. From a methodological point of view it emerges that the DP approach delivers sharper results than the MP approach (for instance it delivers a closed form solution for the optimal investment strategy) under slacker parameter restrictions. Cross-time and cross-country data on investments, income, and consumption drawn from the Penn World Table version 6.2 are used to evaluate the vintage and standard Ak model.
The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to opti... more The paper concerns with the infinite dimensional Hamilton-Jacobi-Bellman equation related to optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded controlrelated term in the state equation in Hilbert setting. An existence-anduniqueness result is obtained.
Economic and demographic models governed by linear delay differential equations are expressed as ... more Economic and demographic models governed by linear delay differential equations are expressed as optimal control problems in infinite dimensions. A general objective function is considered and the concavity of the Hamiltonian is not required. The value function is a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation and a verification theorem is proved.
The present paper continues the study of infinite dimensional calculus via regularization, starte... more The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô's type formula applied to f (t, X(t)) where f : [0, T ] × H → R is a C 0,1 function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of mild solution for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.
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Papers by Giorgio Fabbri