Parameter identification, population and economic growth in an extended Lucas and Uzawa-type two sector model

The aim of this paper is twofold. First of all we re-examine the long-run relationship between population and economic growth. To do this we extend the Lucas-Uzawa model along two different directions: we introduce the growth of the physical capital stock into the human capital supply equation and include in the intertemporal maximization problem of the representative household a preference parameter controlling for the degree of agents\u2019 altruism towards future generations. These two extensions allow us to capture eventual complementarity/substitutability links between physical and human capital in the production of new human capital and to study how such links, along with agents\u2019 altruism, may impact on the interplay between economic and demographic growth along the balanced growth path equilibrium. In the second part of this paper we develop the inverse problem for this extended Lucas-Uzawa model. The method we are going to use is based on fractals and has been developed by two of the authors in recent papers. Through the solution of the inverse problem one can get the estimation of some key-parameters such as the total factor productivity, the productivity of human capital in the production of new skills, the physical capital share in total income, the inverse of the intertemporal elasticity of substitution in consumption, the depreciation rate of (physical and human) capital and the parameter controlling for the degree of altruism towards future generation

Fractal-based methods and inverse problems for differential equations : current state of the art

We illustrate, in this short survey, the current state of the art of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. We review several methods based on the Collage Theorem and its extensions. We also discuss two innovative applications: the first one is related to a vibrating string model while the second one considers a collage-based approach for solving inverse problems for partial differential equations on a perforated domain

Parameter identification, population and economic growth in an extended Lucas and Uzawa-type two sector model

The aim of this paper is twofold. First of all we re-examine the long-run relationship between population and economic growth. To do this we extend the Lucas-Uzawa model along two different directions: we introduce the growth of the physical capital stock into the human capital supply equation and include in the intertemporal maximization problem of the representative household a preference parameter controlling for the degree of agents\u2019 altruism towards future generations. These two extensions allow us to capture eventual complementarity/substitutability links between physical and human capital in the production of new human capital and to study how such links, along with agents\u2019 altruism, may impact on the interplay between economic and demographic growth along the balanced growth path equilibrium. In the second part of this paper we develop the inverse problem for this extended Lucas-Uzawa model. The method we are going to use is based on fractals and has been developed by two of the authors in recent papers. Through the solution of the inverse problem one can get the estimation of some key-parameters such as the total factor productivity, the productivity of human capital in the production of new skills, the physical capital share in total income, the inverse of the intertemporal elasticity of substitution in consumption, the depreciation rate of (physical and human) capital and the parameter controlling for the degree of altruism towards future generations

Solving inverse problems for the Hammerstein integral equation and its random analog using the "collage method" for fixed points

Many inverse problems in applied mathematics can be formulated as the approximation of a target element $u$ in a complete metric space $(X,d_X)$ by the fixed point $\bar x$ of an appropriate contraction mapping $T : X \mapsto X$. The method of collage coding seeks to solve this problem by finding a contraction mapping $T$ that minimizes the so-called collage distance $d(x,Tx)$. In this paper, we develop a collage coding framework for inverse problems involving deterministic or random Hammerstein integral operators. Such operators are used to model image blurring. We illustrate the method with example

Solving inverse problems for biological models using the collage method for differential equations

In the first part of this paper we show how inverse problems for differential equations can be solved using the so-called collage method. Inverse problems can be solved by minimizing the collage distance in an appropriate metric space. We then provide several numerical examples in mathematical biology. We consider applications of this approach to the following areas: population dynamics, mRNA and protein concentration, bacteria and amoeba cells interaction, tumor growt