Politico-economic cycles of regulation and deregulation

European

Journal

of Political

Economy

7 (1991) 469485.

Politico-economic and deregulation Gustav

Feichtinger

North-Holland

cycles of regulation

and Franz Wirl*

Institute of Econometrics, Operations Research and Systems Theory, Technical University oJ Vienna, A-1040 Vienna, Austria Accepted

for publication

November

1990

This paper considers politico-economic cycles that do not depend on the exogenous electoral cycles. More precisely, the paper develops a positive model of intertemporal subsidy strategies for an authoritarian and dynastic government. It will be shown - applying the Hopf bifurcation theorem - that cyclical strategies, i.e. waves of regulation, populism alternating with deregulation, cuts in social programmes, etc., may be optimal.

1. Introduction The bulk of the literature on political business cycles relies extensively on the (exogenous) electoral cycle in order to explain when (and why) political decisions generate cycles in real economic variables. The typical argument runs as follows. The voters’ evaluation depends on inflation and unemployment observed at the election date. The politicians exploit this behaviour strategically to maximize the turnout at the ballots. Usually, the politicians manipulate the inflation-unemployment trade off along a short run Phillips curve. The literature on this line of reasoning is huge since the original contribution of Nordhaus (1974); e.g. see the recent surveys and discussions of Schneider and Frey (1988), in Nordhaus (1989) and in the public choice text book of Mueller (1989). This paper investigates whether and when we should observe cycles in important economic variables that are not due to the electoral cycle. Or in other words, do other kinds of oscillations underly the political evolution that are not due to the exogenous electoral cycle? At the extreme one could think of a dynastic government’ that is not subject to the reelection *We presented a preliminary version of the paper at operations Research 1990 and acknowledge the helpful suggestions from the participants, in particular, the elaborate comments from Professor William A. Brock and Franz X. Hop. We are further grateful for the able research assistance of Andreas Novak and for the suggestions from one of the editors. ‘E.g. in monarchies, see Usher (1989), in democracies with dominant parties like in Japan (The Liberals) and in Mexico (The Revolutionary Party) or underlying the electoral cycles in two party democracies. 0176-2680/91/$03.50

0

1991-Elsevier

Science Publishers

B.V. All rights reserved

470

G. Feichtinger and F. Wit-l, Cycles of regulation and deregulation

constraint. This idea of cycles in regulations and subsidies and laissez faire capitalism is developed in a recent book by Phillips (1990). In particular, Phillips predicts that the U.S. is up to a ‘populist revolt’ following the ‘heyday capitalism’ of the two Reagan administrations. He argues by reference to previous experiences of populism in the U.S., namely by William Jennings Bryan and Franklin Delano Roosevelt, both movements counteracting the capitalistic periods of the 1880s and the Roaring Twenties. This paper attempts to explain such long run cycles in political fashions that reach beyond the electoral cycle. Two recent papers, Usher (1989) and Beenstock (1989) ask related questions; another related work is Olson’s (1982) investigation of the rise and fall of nations. Usher (1989) considers how a dynasty introduces despotism to replace anarchy and vice versa. Beenstock (1989) considers successful rent seeking which will come to a halt when the exploited fraction of the electorate becomes the majority. The following argument is set up in the context where the government hands out subsidies to ‘buy’ support. Of course, subsidies could be extensively understood as transfer payments and regulation. Both instruments generate private revenues but bear social costs, i.e. they increase the public deficit. Many examples fit into that description, e.g. social security payments, farmers’ support programmes, import quotas (e.g. on sugar that raises the income of sugar growing farmers). Once a subsidy is granted, it has to be paid in the future unless deregulation, cutting subsidies, takes place. Deregulation becomes necessary because large deficits finally lower electorate’s support and/or reduce the politicians benefits due to lower discretion. The politicians face only a ‘soft intertemporal budget constraint’ that offers them considerable discretion. The paper is organized as follows. Section 2 introduces the model. Section 3 derives the results using two mathematical theorems that are documented in the appendix. Section 4 presents an example where cyclical strategies are rational. A summary in section 5 completes this investigation. 2. The model The objective of the politician is to maximize an intertemporal aggregate (using the constant time preference rate r>O) of public support (or his reputation) because a higher popularity offers benefits from potential discretion, or on the job ‘shirking’ as Kalt and Zupan (1990) call this behaviour. For example, a popular politician may engage on a crusade to implement his ideological and/or religious goals. The infinite planning horizon stresses that we are interested in regularities that are not caused by the electoral cycles. In the narrow sense, this long run objective applies to an authoritarian government, e.g. for a monarchy. Despite the power inherited by authoritar-

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation

ian governments, popularity and Frey (1988, p. 263).

max 7 e-“!‘(t)

is also

vital

to their

interest,2

471

see Schneider

dt.

(1)

0

The current approval rating or popularity, P(t), depends on subsidies, the total subsidies, S, and the most recently approved ones, 0, and on the budget B (or respectively on the budget deficit if B
P(S,o,B): P,>O, PsssO,P,>O, P,,O, P,,jO,P jointlyconcave.

(2)

The importance of transfers for support seeking politicians as opposed to elusive (from the voter’s point of view) ‘macro-economic’ variables is stressed in Buchanan and Wagner (1977); see also the excellent summary in Mueller *Indeed, we conjecture that in the end authoritarian governments, either left or right wing, are more ‘populistic’ than democracies. This conclusion complies with recent experiences of dictatorships in Latin America, other developing countries and Eastern Europe. Dictators are often unable to lower subsidies for food and energy prices, despite (or because?) exercising considerable oppression. ‘Peltzman (1990) finds that voters perform this task efficiently. This seems paradox, similar to the voting paradox, since it does not pay off to do so.

472

G. Feichtinger

and F. Wirl, Cycles of regulation and deregulation

(1989).4 As a consequence, politicians tend to increase transfer cheques (either social security, pensions, etc.) before elections,’ but withhold price increases for public services. 6 Subsidies proceed by precedence so that S is the result of historically granted (or abolished) ‘rights’, i.e. the evolution of the subsidies S is sticky; compare Alt and Chrystal (1983). The inclusion of the new subsidies G in the approval rating accounts for the likely effect that most recently successful rent seekers are more loyal than formerly satisfied pressure groups. Or the other way round: abolishing an existing privilege, i.e. negative subsidies 0, will sharply alienate some voters and thus lower support. This particular type of benefits (to politicians) and its nonlinearity (P,,
S=cJ, S(O)=&,

B(0) = B,,

(3)

(4) (5)

i.e. he searches for an intertemporal strategy to grant or to abolish subsidy progammes accounting for a ‘soft’ budget constraint. The amount S denotes the subsidies received by the constituents. However, ‘Mueller writes (on p. 295): ‘More generally, Buchanan and Wagner (1977) have argued that transfer and expenditure increases are so much more attractive for vote-seeking politicians than tax increases that budget deficits are a natural consequence’. 5Most prominent is Nixon’s trick to secure re-election - he signed a 20% increase of social security cheques in October 1972 shortly before his reelection; cf. Nordhaus (1989). 6E.g. Hubka and Obermann (1977) document this effect for Austria. ‘The legacy of the Keynesian economics lowered this esteem for balanced budgets.

G. Feichtinger

and F. Wirl, Cycles

ofregulationand deregulation

473

subsidy payments and costs of financing this payment may differ. The convex cost function C(S), C’>O, c” 20, C(0) =O, C’(0) = 1, describes the financial burden and accounts for the fact that a subsidy creates a deadweight loss so that the social costs C(S) exceed the subsidy payment S. Various reasons contribute to this divergence C(S) > S, e.g., the necessity to create (or expand) institutions that manage the distribution of subsidies, or disincentive effects. Becker (1983) calls these additional expenses the deadweight costs of rent seeking. The assumption C’(O)= 1 states that no deadweight costs accrue for (infinitesimally) small subsidies, i.e. C(S)%: for S close to zero. The convexity, C(S) > S for S >O, states that the deadweight costs increase with increased subsidy payments. Two reasons justify this convexity: (a) ‘efficient’ competition among the pressure groups, i.e. proposals with lower deadweight costs tend to win, see Becker (1983); (b) dis-incentive effects associated with transfer payments. In any case, this convexity is not crucial to the following arguments, i.e. C”=O is also compatible with cyclical strategies. The interest 6 is earned on budget surpluses or respectively, has to be paid for deficits. In general, 6 may itself depend on the budget B, in particular if B is negative, i.e. the interest rate 6 may rise for large deficits. We assume a constant 6 that is less than the time preference of politicians, r>6. The budget accounting according to (4) seems unconventional. The reason is that the differential equation (4) considers only the budget allocated to subsidies (or additional taxes), but not the entire budget. In other words, all the taxes and expenses for the provision of public services are balanced and thus they are netted out to simplify our argument. For example, suppose that all ‘conventional’ taxes are ear-marked for the provision of public goods. The analysis of this control problem proceeds in the usual manner, H denotes the Hamiltonian and 2,~ denote the costate variables, all in current values. The following equations summarize the necessary and sufficient [since the control problem is concave; see Feichtinger and Hart1 (1986)] optimality conditions:

H=P+rl(JB-C)+,ua,

(6) (7) (8)

,Li=rp+AC’-P,,

(9)

lim e-“A(t)B(t) r-rm

=O,

(10)

lim e-‘%(t)S(t) f’rn

=O.

(11)

414

G. Feichtinger

and F. Wirl, Cycles of regulation

and deregulation

The stationary solution of states and costates, if existing, is indicated the subscript cc and follows from the (simultaneous) solution of following set of equations:

by the

B, = WJP,

(12)

pm = C(r- 6)Ps- P&‘l/Cr(r - 41,

(14)

cTs=o,

(15)

P,(~,,S,,O)= The stability section.

properties

(16)

-Pa;.

of this equilibrium

are the subject

of the following

3. Stability analysis The following investigation assumes that interior solutions describe the optimal policy for P = P(S, 0, B). Standard boundary conditions ensure this. Therefore, the maximum principle allows to solve H,=O for the control D in dependence of states and costates, c=s(B, S,p), since I does not appear in (7). Substitution into state and costate differential equations yields the so called canonical equations in (B, S, i, p). The central question is to characterize the intertemporal equilibrium strategy, whether it is constant, or cyclical, stable or unstable. The model (3)(5) fits into a class of optimal control models proposed in Wirl (1992) where stable limit cycles may characterize the optimal strategy. The following analysis draws on two important theorems, which are summarized in the Appendix. The first theorem, the Hopf bifurcation theorem, is applied to prove the existence of stable limit cycles. The application of this theorem requires the (analytical) computation of the eigenvalues of the linearized dynamic system that approximates the four dimensional system of the canonical equations. A theorem derived in Dockner (1985) allows the explicit calculation of these eigenvalues, which are crucial for the (local) stability analysis.

G. Feichtinger and F. Wirl, Cycles

ofregulationand deregulation

475

The procedure is now the following. First, calculate the Jacobian of the canonical equation system evaluated at an equilibrium, equations (12)--(16). Second compute the eigenvalues of this Jacobian drawing on Dockner’s formula (see appendix, Theorem 2). This investigation provides necessary and sufficient conditions for stability and necssary conditions for the possibility of cyclical strategies. Moreover, such cyclical strategies do indeed exist as will be proven with the Hopf bifurcation theorem (see Appendix, Theorem 1) for a particular example. First, we compute the Jacobian:

J=

r -

1-

6 P UB P

CT0

pBBpcwP

p:B

pBSpm

0

-- P us P CD

0

-

poBp,S

P 6L7

00

00

of J

to calculate the eigenvalues of J and the coefficient K.

jjJll = {[PBBc’2 + 2P&6 - r(r - UPA

K = [6(r - S)] + +

2c{-

These two coefficients

P bb P LYE P bb

.

we have

to

compute

according

the

+ PBc”6

(18) rPos}/P,,]

)/pml~

determine

(17)

P rSLs --1 P 00

P,&WP,,,

[{PBC"/(r - 6) - P,, -

p,Bc'

1

c

+ Psss2] -P&r-P&+ +

0

r-6

p,,p,,p:, pBc” +--P CT0 r-6

pBSpm pcrSpoB P

In order determinant

-C'

(19) to Theorem

2 the eigenvalues

(ei):

(20) and thus the (local) stability properties of the canonical equations. Dockner and Feichtinger (1991) identify five cases depending on these two coefficients; see table 1. The case (iv) is crucial to apply the Hopf bifurcation theorem. This condition and the exclusion of (i) implies immediately the necessary condition (/J/I > 0 and K > 0. All five cases are conceivable for the problem (3)-(5). Indeed, even a much

476

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation Table Different

1

types of equilibria:

arithmetical

conditions.

Case Inequalities

Properties

i

saddlepoint stability, real roots, two are negative and two are positive;

K
ii \\J((>(/K)*

of the equilbrium

saddlepoint

)/JI\-($K)2-+-2K>0

transient

stability,

complex

roots,

oscillations;

iii JIJ(J
one negative real eigenvalue, all other eigenvalues have positive real parts, instability except for a one dimensional manifold;

iv

/(J((>(~K)* /1+(~K)*-fr2K=0

two purely imaginary

eigenvalues;

”

l\JllN1K12

complex

positive

\lJll-(tK)‘-fr2K<0

parts, locally unstable

eigenvalues,

real

spirals.

Table 2 Different

cases and associated

properties

(r > 6)

Case

Some favourable

i

r-6, IP_,\ sufficiently small, P, and C” large, (P,,I and C’ not too large, le(P,)(>e(C’)/e(c).

ii

(P,,/ and C’ sufficiently

111

C’ and IP,,l small, or P, and C” large Je(PB)J<~(C))/E(C), e.g. P,,= 0 is sufficient.

iv

P,, sufftciently negative, P, and c” small, Is( > a(C)/~(c), r>& (P,,I large.

v

IP,,\ and C’ sufficiently large, Ia( c” small, r > 6 and IP,,J very large

% defines logarithmic CSjC.

properties”

large, Je(P,)I > s(C’)/e(c).

derivatives,

so

that

>E(~‘)/E(c),

i.e. elasticities,

e.g. t(C) =

simplified version of the model, shown below, retains this property simplifies the formulas considerably. Thus, we assume the function additive so that all mixed second order derivatives vanish; moreover, assume linear dependence with respect to S, P,,=O.This yields:

IlJll= cp,Lc2+ p,c"mp,,, K =

qr -6)

+ P&"/[(r-

but

P is we

(21) 6)P,,].

Table 2 outlines some of the economic properties benefits, the slope of marginal benefits and similarly characteristic for the different cases.

(22) - in terms of marginal for the costs - that are

G. Feichtinger and F. Wirl, Cycles

of regulation and deregulation

477

Hence, case (i) - saddlepoint stability, real eigenvalues and thus (local) monotonicity of the paths - is given when (a) interest and discount rate are roughly equal, (b) for highly convex deadweight costs C, (c) the concavity with respect to budgets and the marginal costs C’ are neither too low nor too large, (d) the concavity of the approval index with respect to new subsidies is small (so that K ~0) but not too small. The transition (i) to (ii), complex roots but saddlepoint stability, is facilitated by various means, e.g. enlarging both items listed under (c) above. Instability, case (iii) results for example if P is linear with respect to budgets, P,,=O. The conditions opposite to case (i), i.e., P,,, P,, sufficiently negative and C” small, favour pure imaginary roots, case (iv), and thus facilitate cyclical strategies. These algebraic conditions mean in economic terms that (a) low deadweight costs of subsidy programmes [e.g. because only the ‘efficient’ proposals of pressure groups win according to Becker (1983)], (b) sufficient concavity of the evaluation of budget surpluses, (c) a strong concavity of popularity with respect to most recent subsidy concessions, i.e. the marginal gain of popularity from granting an additional dollar declines rapidly with respect to the concessions; or the other way round: any substantial deregulation bears large costs in terms of popularity, are favourable for those conditions that are necessary for cyclical policies. An example, which will be discussed in the following section, proves then the existence of such cyclical policies. Thus, an efficient process of rent seeking, everything else equal, favours cyclical strategies. Indeed, the case of no deadweight costs, C(S) = S, allows to solve explicitly the equality (iv) in table 1 to calculate the bifurcation point (indicated by a hat) of the parameter P,,: B,, = 4P,,/{ S(r -8) [(r2 - 62) + r(r + S)]}.

(23)

The choice of P,, as the bifurcation parameter is here natural because it rescales the determinant without affecting the equilibrium. Of course, one can also vary the discount rate r to establish bifurcations. Formula (23) suggests that even a small discount rate Y (but greater 6) satisfies the equality (iv) from table 1 provided that P,, is sufficiently large. Hence, highly discounting politicians are not necessary, but helpful, when explaining cyclical policies. Further motions away from the bifurcation point lead to case (v), complex eigenvalues and presumably complete instability. The conditions associated with this case, low esteem for balanced budgets, the impossibility to deregulate (IP,,( large), large subsidies (thus large marginal costs C’), low weights for future generations (I large), could comply with the developments in Eastern Europe. Eastern Europe, despite being ruled by dictatorships, was subject to populism in the sense of subsidization of large parts of everyday economic life. Of course, these subsidies favoured consumption at the

478

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation

expense of investments and increased simultaneously the public deficit, see Schneider and Frey (1989). Thus, governments tried at several occasions to revert these disastrous tendencies, i.e. to cut subsidies, but unsuccessfully or at least insufficiently, due to public opposition and strikes (e.g. in Poland). Ultimately, these oscillations depleted the entire financial and capital stocks of these countries, which contributed to the recent collapse of the central planned economies. All cases, except (iii), are characterized by a positive determinant of the Jacobian. According to formula (21), ((J((>O is equivalent to the following economic characterization: the absolute elasticity of marginal benefits, denoted Is(P must exceed the ratio of the elasticity of marginal costs (of financing subsidies) divided by the elasticity of the costs C. For example, linear costs and strict concavity of P with respect to B are sufficient to meet this requirement. On the other hand, P linear with respect to B and convex costs C are sufficient to violate this condition. In this case, (iii) according to the numbering in the above tables, stability is restricted to a one-dimensional manifold in the four-dimensional space of states and costates. Hence, the equilibrium is unstable, except for particular initial conditions (of zero Lebesgue measure). The above analysis does not carry over to farsighted politicians, r ~6, for two reasons. First, the costate i switches sign 3,, pL, >O thus contradicting P,>O. This conclusion is intuitively plausible. With discounting (r) below the opportunity costs of capital (6) it simply ‘pays off’ to tax today in order to accumulate budget surpluses that allow to hand out even larger subsidies in the future (even from a present value point of view).

4. Cyclical policies - an example The central task of this investigation is to locate stable cyclical policies through the application of the Hopf bifurcation theorem. This theorem requires in layman terms (for details see appendix, Theorem 1): (i) a parameter constellation such that (20) yields two pure imaginary eigenvalues; (ii) the derivative of the real part of the complex eigenvalue must not vanish when crossing the imaginary axis; (iii) stability of the cycle, which requires a negative coefficient on the quadratic term of a third order Taylor approximation of the canonical equations in a proper (nonlinear) coordinate system. For this purpose consider the following specifications:

‘(‘, ~,‘) = a,S + b,(B - B,i”)B+ d

+

~y02,

479

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation

a,>o, C(S)=S+JkS2,

b,>O,

O
y-Co, &j”
(24)

k>O.

(25)

The account (24) of the function P neglects eventual constants and represents a weighted average of the individual contributions to P; w.1.o.g. the weight from the popularity bonus with respect to the factor 0 is set equal to one. This implies at the equilibrium P,= 1 = --pm since go0 =O. Variations of the weights a, and b, determine whether a surplus (and thus subsidies) or a deficit (and thus taxes) describe the stationary solution. The parameter Bmin (presumable negative) shifts the origin how budget surpluses and deficits are appreciated. Additionally, the power function constrains budget deficits to B> Bmin. The first condition for a limit cycle demands that the Jacobian J has two pure imaginary eigenvalues. The equation (22) and the discussion in table 1 reveal that the parameter P,,= y is highly suitable to achieve K > 0 without changing the (positive) sign of the determinant. Moreover, the choice of y does not affect the equilibrium position. Therefore, we choose y as our bifurcation parameter, i.e. we vary y until the bifurcation curve, /(J/l = (+K)’ - $r’K, is crossed. And this is indeed possible. For example consider the parameters a0 = 1, b, = 50, Bmin = - 1, Y= 1.0, 6=0.2; table 2 recommends a sufficiently concave benefit schedule, hence, /I small, e.g. /?= 0.1, and almost linear costs, i.e. k small, e.g. k =O.Ol. This example leads to an equilibrium with a budget surplus, B, =2.57 that facilitates positive subsidies S, = 0.51. However, less appreciation of budgets, e.g., a lower value of the parameter bO, could lead to a deficit and taxation as equilibrium strategy. The canonical equations exhibit saddlepoint stability for (y[ small, but pass the bifurcation curve at 9% -4.793. At this point a Hopf bifurcation occurs and a stable limit cycle is born as numerical calculations contirm.8 Thus, a family of stable cyclical strategies exists in a local left hand neighborhood at the bifurcation point, i.e., for y such that y 5 -4.793 but sufficiently close. The following figures show the cyclical strategies, either projected into phase planes or drawn in the time domain. The parameter y is very close to its critical value and all functions are well behaved along the cycle, in particular, P,>O. More precisely, these figures show the limit cycle but omit the transient behaviour. Fig. 1 shows a counter clockwise strategy in the budget-subsidy plane, while fig. 2 reports a clockwise motion in the subsidy plane, accumulated and newly granted subsidies. This second circle underlines how phases of regulation and populism, o>O, alternate with attempts ‘We are very grateful to Andreas Novak who applied Wan (1981) to check the stability of the closed orbit.

the code of Hassard,

Kazarinoff

and

480

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation

S .60

7

.55

--

.50

--

.45

--

B

II!IIUIIIII!I~~II~~II!~~I~~~~,~!~,~~I,,~!,~,,I,,,,!,,,,I,,,,!,,,~

,40

2.2

Fig. 1. Optimal

2.3

2.4

2.5

2.6

2.7

2.8

23

paths (limit cycle, no transients) in the state phase plane - subsidies - (B, 9.

and budgets

to deregulate and to lower public subsidization programmes, a ~0. Moreover, it is worth mentioning that the cycles have sizable amplitudes, +20x for subsidies and rfr 10% for the budget. Figs. 3 and 4 show the results in the time domain. The granting (eradication) of privileges (a) proceeds the evolution of the subsidies and the budget.

5. Summary This paper attempts to explain politico-economic cycles that do not depend on the exogenous election cycle. This question is further motivated by recent arguments of political scientists that populism (in the United States) will re-appear. This paper considered this question for the concrete example of intertemporal subsidy strategies of an authoritarian government. The supposition of an authoritarian government eliminates the re-election constraint. It was demonstrated that stationary policies but also more complex strategies, e.g. stable limit cycles, could be optimal, which was proven through applying the Hopf bifurcation theorem. Thus, politico-

481

G. Feichtinger and F. Wirl, Cycles of regulationand deregulation

.02

-

.Ol

-

0.

--

701

702

-

_,03

~L~l~I~I~~~I~I~I~I~/~I~I~I~I~~~i~l~l~l,~

40

Fig. 2. Optimal

.45

s

paths (limit cycle, no transients)

economic cycles of this type may underly, tional business cycles.

Appendix- Mathematical

.55

.50

.60

in the control-state

compete

phase plane of (S, CT).

or reinforce

the conven-

theorems

This appendix documents the mathematical requisites that are necessary to derive the results of the paper. The first theorem states the Hopf bifurcation which gives sufficient conditionsion stable limit cycles; theorem, Guckenheimer and Holmes (1983) is a good reference on this topic. Theorem I. Hopf Bifurcation Theorem [Guckenheimer and Holmes (1983)]. Suppose that a family of dynamic systems i = f (z; m), z E R4 and m E R has an equilibrium (2,; rit) at which the following property is satisfied: the Jacobian

has a simple pair of pure imaginary

eigenvalues

and no other eigenvalues

with

G. Feichtinger and F. Wirl, Cycles of regulation and deregulation

482

Is;

I;,;,-_ ,!,!,!

I ~,~,~,~,~1~,~,~,~,~

;

j

[_!

1

5

0

10

15

t

Fig. 3. Time paths

u(t),

20

B(t) along a periodic solution.

real parts zero; A is called the critical value of the bifurcation parameter m, or shortly the bljiurcation point. Then there exists a smooth curve of equilibria depending

on the bifurcation

parameter

The eigenvalues c(m) of J(m), with m. If, moreover,

&(Rei(m))=d#O

m, denoted z,(m), so that zm(rh)=im. which are imaginary at m =rk, vary smoothly

at m=rit,

then there exists a unique three-dimensional

manifold passing through (.?,;tk)

in

G. Feichtinger and F. Wirl, Cycles

of regulation and deregulation

S-

483

-_-a

03c

.60

.OZE .!a

021 .56

01: .54 II11 .52

00:

0

.50

00’ .48 OlC .46 O!E .44 021 .42

32:

03!

.40 0

5

15

10

20

t

Fig. 4. Time paths

a(t), S(t) along a periodic solution,

R4 x R and a smooth system of coordinates for which the Taylor expansion of degree 3 on the center manifold is given by the following normal form: x=[d(m-m)+a(x2+y2)]x-[y+c(m-m)+b(x2+y2)]y, 1; = [y + c(m -m) + b(x2 + y2)]x + [d(m -m) + a(x2 + y2)] y. Zf a # 0, there is a surface of periodic solutions in the center mantfold which has quadratic tangency with the eigenspace of i(m) and i(m) agreeing to second order with the paraboloid m = m-(a/d)(x2 + y’). Moreover, if a <0 the periodic solutions are stable limit cycles, while in the case a>0 the periodic solutions are repelling.

G. Feichtinger

484

and F. Wit-l, Cycles of regulation and deregulation

This theorem is applied to z=(B, S, L,P)~, the equilbirium z, = (B,, S,,I,,,p,)‘, and the bifurcation parameter m=y. A necessary and crucial requirement for the application of Theorem 1 is the determination pure imaginary eigenvalues of the linearized approximation of the non-linear dynamic system. Dockner (1985) derives the following explicit formula for the eigenvalues if the dynamic systems results from the necessary optimality conditions of a two state variable control problem. Theorem 2 [Dockner (1985)]. Consider the canonical equations (B, S, L, p) obtained from the necessary optimality conditions of an optimal control problem with two state variables (here B and S). The eigenualues (e,, i = 1,. . ,4) of the linear dynamic system that approximates the canonical equations around an equilibrium can be explicitly computed:

34-l le2 -2r-

+

(~r)2-~K+)JKZ-4~~J~~ J

where

aB

aS a2

al

al

a& K

aB

aS as

+ ap aA as

+2

ai3 as al as

aB ap al . ap

References Ah, J.E. and A.K. Chrystal, 1983, Political economics (University of California Press, Los Angeles, CA). Beenstock, M., 1989, A democratic model of the ‘rent-sought’ benefit cycle, Public Choice 63, 1-14. Becker, G.S., 1983, A theory of competition among pressure groups for political influence, Quarterly Journal of of Economics 98, 371400. Buchanan, J.M. and R.E. Wagner, 1977, Democracy in deficit (Academic Press, New York). Dockner, E., 1985, Local stability analysis in optimal control problems with two state variables, in: G. Feichtinger, ed., Optimal control theory and economic analysis 2 (North-Holland, Amsterdam) 89-103. Dockner, E. and G. Feichtinger, 1991, On the optimality of limit cycles in dynamic economic systems, Journal of Economics 53, 31-50. Guckenheimer, J. and P. Holmes, 1983, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields (Springer-Verlag. New York). Feichtinger, G. and R.F. Hartl, 1986, Optimale Kontrolle iikono-mischer Prozesse (DeGruyter, Berlin). Hassard, B.D., N.D. Kazarinoff and Y.H. Wan, 1981, Theory and application of Hopf bifurcation, in: London Math. Sot. Lecture notes 41 (Cambridge University Press, Cambridge).

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