scholarly journals Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

2021 ◽  
Author(s):  
Yves Achdou ◽  
Jiequn Han ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions ◽  
Benjamin Moll
Keyword(s):  
Continuous Time ◽  
Wealth Distribution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heterogeneous Agent ◽  
Saving Behavior ◽  
Income And Wealth Distribution ◽  
Heterogeneous Agent Models ◽  
Special Case

Abstract We recast the Aiyagari-Bewley-Huggett model of income and wealth distribution in continuous time. This workhorse model – as well as heterogeneous agent models more generally – then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (i) an analytic characterization of the consumption and saving behavior of the poor, particularly their marginal propensities to consume; (ii) a closed-form solution for the wealth distribution in a special case with two income types; (iii) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including – but not limited to – the Aiyagari-Bewley-Huggett model.

A CLOSED-FORM SOLUTION OF THE EVOLUTION OPERATOR IN TWO-LEVEL SYSTEMS

1994 ◽  
Vol 08 (08n09) ◽  
pp. 505-508 ◽  
Author(s):  
XIAN-GENG ZHAO
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Evolution Operator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Two Level Systems ◽  
Special Case

It is demonstrated by using the technique of Lie algebra SU(2) that the problem of two-level systems described by arbitrary time-dependent Hamiltonians can be solved exactly. A closed-form solution of the evolution operator is presented, from which the results for any special case can be deduced.

Effects of Geometry on the Performance of a Downhole Orbital Vibrator

2002 ◽  
Vol 124 (2) ◽  
pp. 77-82
Author(s):  
Robert R. Reynolds ◽  
Jack H. Cole ◽  
Zhen Yuan
Keyword(s):  
Pressure Field ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Confined Water ◽  
Dimensional Model ◽  
Concentric Cylinder ◽  
Fe Method ◽  
Two Dimensional Model ◽  
Special Case

The influence of geometry on the pressure field within the confined, water-filled annulus between a central, vibrating cylinder and an outer, rigid enclosure is determined. A two-dimensional model is constructed using the finite element (FE) method and parameters are identified to characterize the eccentricity of the nominal cylinder position, the size of the annulus relative to the inner cylinder and the degree to which the annulus is not circular (i.e., it is elliptic). The FE solution is verified using a closed-form solution for the special case of a concentric cylinder and annulus. It is shown that the system acts as a force multiplier. Analyses of the asymmetrical geometries indicate that while the pressure field on the surface of the cylinder and enclosure can be highly asymmetric, the system is relatively insensitive to minor variations in annulus shape except when the vibrating cylinder is not centrally located within the fluid region or the annulus size itself is small.

scholarly journals A CLOSED-FORM SOLUTION TO A MODEL OF TWO-SIDED, PARTIAL ALTRUISM

2012 ◽  
Vol 16 (2) ◽  
pp. 230-239
Author(s):  
Ana Fernandes
Keyword(s):  
Closed Form ◽  
Overlapping Generations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Allocation Of Resources ◽  
Overlapping Generations Model ◽  
Parents And Children ◽  
Saving Rates ◽  
Markov Strategies

This paper presents a closed-form characterization of the allocation of resources in an overlapping generations model of two-sided, partial altruism. Three assumptions are made: (i) parents and children play Markov strategies, (ii) utility takes the CRRA form, and (iii) the income of children is stochastic but proportional to the saving of parents. In families where children are rich relative to their parents, saving rates—measured as a function of the family's total resources—are higher than when children are poor relative to their parents. Income redistribution from the old to the young, therefore, leads to an increase in aggregate saving.

Theory for Multilayered Anisotropic Plates With Weakened Interfaces

1996 ◽  
Vol 63 (4) ◽  
pp. 1019-1026 ◽  
Author(s):  
Zhen-qiang Cheng ◽  
A. K. Jemah ◽  
F. W. Williams
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Layer Model ◽  
General Representation ◽  
Anisotropic Plates ◽  
First Order ◽  
Multilayered Plates ◽  
Interface Parameters ◽  
Zigzag Theory ◽  
Special Case

Rigorous kinematical analysis offers a general representation of displacement variation through thickness of multilayered plates, which allows discontinuous distribution of displacements across each interface of adjacent layers so as to provide the possibility of incorporating effects of interfacial imperfection. A spring-layer model, which has recently been used efficiently in the field of micromechanics of composites, is introduced to model imperfectly bonded interfaces of multilayered plates. A linear theory underlying dynamic response of multilayered anisotropic plates with nonuniformly weakened bonding is presented from Hamilton’s principle. This theory has the same advantages as conventional higher-order theories over classical and first-order theories. Moreover, the conditions of imposing traction continuity and displacement jump across each interface are used in modeling interphase properties. In the special case of vanishing interface parameters, this theory reduces to the recently well-developed zigzag theory. As an example, a closed-form solution is presented and some numerical results are plotted to illustrate effects of the interfacial weakness.

scholarly journals Solving discrete time heterogeneous agent models with aggregate risk and many idiosyncratic states by perturbation

2020 ◽  
Vol 11 (4) ◽  
pp. 1253-1288 ◽  
Author(s):  
Christian Bayer ◽  
Ralph Luetticke
Keyword(s):  
Perturbation Method ◽  
Recent Work ◽  
Discrete Time ◽  
Continuous Time ◽  
Dynamic Equilibrium ◽  
Stationary Equilibrium ◽  
Heterogeneous Agent ◽  
Aggregate Risk ◽  
Aggregate Dynamics ◽  
Heterogeneous Agent Models

This paper describes a method for solving heterogeneous agent models with aggregate risk and many idiosyncratic states formulated in discrete time. It extends the method proposed by Reiter (2009) and complements recent work by Ahn, Kaplan, Moll, Winberry, and Wolf (2017) on how to solve such models in continuous time. We suggest first solving for the stationary equilibrium of the model without aggregate risk. We then write the functionals that describe the dynamic equilibrium as sparse expansions around their stationary equilibrium counterparts. Finally, we use the perturbation method of Schmitt‐Grohé and Uribe (2004) to approximate the aggregate dynamics of the model.

The Pressure Field Around a Vibrating Cylinder in a Fluid Annulus

2001 ◽  
Author(s):  
Robert R. Reynolds ◽  
Jack H. Cole ◽  
Zhen Yuan
Keyword(s):  
Pressure Field ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Confined Water ◽  
Dimensional Model ◽  
Concentric Cylinder ◽  
Fe Method ◽  
Two Dimensional Model ◽  
Special Case

Abstract The influence of geometry on the pressure field within the confined, water-filled annulus between a central, vibrating cylinder and an outer, rigid enclosure is determined. A two dimensional model is constructed using the finite element (FE) method and parameters are identified to characterize the eccentricity of the nominal cylinder position, the size of the annulus relative to the inner cylinder and the degree to which the annulus is not circular (i.e. it is elliptic). The FE solution is verified using a closed form solution for the special case of a concentric cylinder and annulus. It is shown that the system acts as a force multiplier. Analyses of the asymmetrical geometries indicate that while the pressure field on the surface of the cylinder and enclosure can be highly asymmetric, the system is relatively insensitive to minor variations in annulus shape except when the vibrating cylinder is not centrally located within the fluid region or the annulus size itself is small.

A Quantitative Evaluation of Two Classical Approximations Used to Predict the Extent of Vertical Hydraulic Fractures

1983 ◽  
Vol 105 (4) ◽  
pp. 512-527 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Critical Parameters ◽  
Hydraulic Fractures ◽  
Fracture Width ◽  
Special Case

An integral equation was developed to predict the critical parameters (fracture width and length) associated with the propagation of a vertical hydraulic fracture and a numerical solution procedure was developed. The effects of the classical approximations of pressure and fracture width were investigated both separately and together. It was found that the effects associated with the pressure approximation were relatively insignificant, whereas those associated with the fracture width approximation were significant, particularly when the formation was only moderately permeable. Finally, an exact closed-form solution of the integral equation was developed for a special case. It was shown that when the formation is only moderately permeable, this solution provides a better approximation of the exact solution than the classical solution of Carter [2].

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