1. 
 Puu, Tönu, 1936, et al.
(författare)

A business cycle model with cubic nonlinearity
 2004

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 19:3, s. 597612

Tidskriftsartikel (refereegranskat)abstract
 This paper deals with a simple multiplieraccelerator model of the business cycle, including a cubic nonlinearity. The corresponding two dimensional iterative map is represented in terms of its bifurcation diagram in parameter space. A number of bifurcation sequences for attractors and their basins are studied.


2. 
 Puu, Tönu, 1936, et al.
(författare)

A Hicksian multiplieraccelerator model with floor determined by capital stock
 2005

Ingår i: Journal of Economic Behavior and Organization.  01672681. ; 56:3, s. 331348

Tidskriftsartikel (refereegranskat)abstract
 This article reconsiders the Hicksian multiplieraccelerator model with the “floor” related to the depreciation on actual capital stock. Through the introduction of the capital variable, a growth trend is created endogenously by the model itself, along with growth rate oscillations around it. The “ceiling” can be dispensed with altogether. As everything is growing in such a model, a variable transformation is introduced to focus relative dynamics of the income growth rate and the actual capital output ratio.


3. 
 Puu, Tönu, 1936, et al.
(författare)

Cournot duopoly when the competitors operate under capacity constraints
 2003

Ingår i: Chaos, Solitons & Fractals.  Elsevier.  09600779. ; 18:3, s. 577592

Tidskriftsartikel (refereegranskat)abstract
 The paper considers Cournot duopoly where the competitors have capacity constraints. An isoelastic demand function, which always results when consumers maximise utility functions of the Cobb–Douglas type, is used. It has been demonstrated that isoelastic demand, combined with constant marginal costs, results in complex dynamics. The purpose of the present paper is to reconsider the case, using in stead cost functions with capacity limits. This is a point on which Edgeworth insisted as important. Comparisons between cases of few large and many small competitors cannot be made when firms have constant returns and hence are all infinitely large in potential.


4. 
 Puu, Tönu, 1936, et al.
(författare)

On the Change of Periodicities in the Hicksian MultiplierAccelerator Model with a Consumption Floor
 2006

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 29:3, s. 681696

Tidskriftsartikel (övrigt vetenskapligt)abstract
 The Hicksian multiplieraccelerator model with “floor” and “ceiling” continues to be the most successful machine generating business cycles. This is, no doubt, due to its capability of explaining both downturn and upswing through one single model. The “ceiling” is due to a full employment constraint, whereas the “floor” is due to a limit to disinvestment when no worn out capital at all is replaced. However, another “floor” to consumption at zero level seems never to have been discussed. Hence, net disinvestments, even if they are bounded downwards, may also give rise to negative consumption, which is absurd. As we will show, the effect of an additional constraint to avoid this is easy to analyze, and results in a change of the periodicities according to a simple rule.


5. 
 Puu, Tönu, 1936
(författare)

On the Stability of Cournot Equilibrium when the Number of Competitors Increases
 2008

Ingår i: Journal of Economic Behavior and Organization.  01672681. ; 66:34, s. 445456

Tidskriftsartikel (refereegranskat)abstract
 This article reconsiders whether the Cournot equilibrium really becomes a perfect competition equilibrium when the number of competitors goes to infinity. It has been questioned whether the equilibrium remains stable with an increasing number of firms. Contraindications were given for linear and for isoelastic demand functions. However, marginal costs were then taken as constant, which means adding more potentially infinitesized firms. As we want to compare cases with few large firms to cases with many small firms, the model is tuned so as to incorporate capacity limits, decreasing with an increasing number of firms. Then destabilization is avoided.


6. 
 Puu, Tönu, 1936, et al.
(författare)

The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints
 2006

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 28:2, s. 403413

Tidskriftsartikel (refereegranskat)abstract
 This article considers Cournot oligopoly with three competitors, given an isoelastic demand function, and cost functions that asymptotically go to infinity when capacity limits are approached. It is shown that for some parameter values (capacity limits) the Cournot equilibrium point loses stability through a NeimarkSacker bifurcation. In addition global bifurcation scenarios are simulated numerically, and some are illustrated through Arnol'd tongues in the parameter plane.


7. 
 Puu, Tönu, 1936
(författare)

The Hicksian Trade Cycle with Floor and Ceiling Dependent on Capital Stock
 2007

Ingår i: Journal of Economic Dynamics and Control.  01651889. ; 31:2, s. 575592

Tidskriftsartikel (refereegranskat)abstract
 This article reconsiders the Hicksian multiplieraccelerator model with 'floor' and 'ceiling'. The new thrust is that these constraints are tied to the actual stock of capital, the floor to the depreciation on this stock, the ceiling to capital as a limiting production factor according to the fixed proportions technology that also underlies the principle of acceleration. For capital formation just the Hicksian investment theory is used. The result is one unified model creating economic growth and growth rate cycles.


8. 
 Sushko, Iryna, et al.
(författare)

Regular and chaotic growth in a Hicksian Floor/ceiling model
 2010

Ingår i: Journal of Economic Behavior and Organization.  Elsevier.  01672681. ; 75:1, s. 7794

Tidskriftsartikel (refereegranskat)abstract
 In some previous papers the present authors reassembled the Hicksian trade cycle model in a new way. The floor was tied to depreciation on capital, itself the cumulative sum of past net investments, for which the principle of acceleration provided an explanation. Hence no alien elements were needed to include capital, and so close the system. The resulting model created a growth trend along with growth rate cycles, which could be periodic or quasiperiodic. In the current paper, the ceiling, using capital stock as a capacity limit for production, is added. It then turns out that pure growth no longer exists, and chaos and multistability become possible, which they were not in the previous model. A variety of bifurcation scenarios is explored, and a full understanding of the working of the fourpiece, originally threedimensional, piecewise smooth map, is attained, using a reduction to a onedimensional return map.


9. 
 Sushko, Irina, et al.
(författare)

The Hicksian floor–roof model for two regions linked by interregional trade
 2003

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 18:3, s. 593612

Tidskriftsartikel (övrigt vetenskapligt)abstract
 The Hicksian multiplier–accelerator model with the original floor–roof limits to investments is studied for the case of two regions linked by interregional trade. The result is a piecewise linear continuous four dimensional map, which is reduced to three dimensions through the choice of an appropriate distributed consumption lag. The attractors, basins, and bifurcations of the map are studied under the assumption of a certain symmetry between the regions. The Neimark–Hopf bifurcation for piecewise linear maps is described in detail which gives rise to the appearance of an attracting closed invariant curve homeomorphic to a circle. The structure of resonance regions in the parameter space are investigated.


10. 
 Sushko, Irina, et al.
(författare)

Tongues of periodicity in a family of twodimensional discontinuous maps of real Mobius type
 2004

Ingår i: Chaos, Solitons & Fractals.  09600779. ; 21:2, s. 403412

Tidskriftsartikel (refereegranskat)abstract
 In this paper we consider a twodimensional piecewisesmooth discontinuous map representing the socalled "relative dynamics" of an Hicksian business cycle model. The main features of the dynamics occur in the parameter region in which no fixed points at finite distance exist, but we may have attracting cycles of any periods. The bifurcations associated with the periodicity tongues of the map are studied making use of the firstreturn map on a suitable segment of the phase plane. The bifurcation curves bounding the periodicity tongues in the parameter plane are related with saddlenode and bordercollision bifurcations of the firstreturn map. Moreover, the particular "sausages structure" of the bifurcation tongues is also explained.

