IERCU Institute of Economic Research, Chuo University 50th Anniversary Special Issues Discussion Paper No.210 Discrete an Continuous Dynamics in Nonlinear Monopolies Akio Matsumoto Chuo University Ferenc Sziarovszky University of Pécs September 2013 I INSTITUTE OF ECONOMIC RESEARCH Chuo University Tokyo, Japan
Discrete an Continuous Dynamics in Nonlinear Monopolies Akio Matsumoto y Chuo University Ferenc Sziarovszky z University of Pécs Abstract Dynamic monopolies are investigate with iscrete an continuous time scales by assuming general forms of the price an cost functions. The existence of the unique pro t maximizing output level is prove. The iscrete moel is then constructe with graient ajustment. It is shown that the steay state is locally asymptotically stable if the spee of ajustment is small enough an it goes to chaos through perio-oubling cascae as the spee becomes larger. The non-negativity conition that prevents time trajectories from being negative is erive. The iscrete moel is converte into the continuous moel augmente with time elay an inertia. It is then emonstrate that stability can be switche to instability an complex ynamics emerges as the length of the elay increases an that instability can be switche to stability as the inertia coe cient becomes larger. Therefore the elay has the estabilizing e ect while the inertia has the stabilizing e ect. Keywors: Time elay, Inertia, Graient ynamics, Continuous an iscrete ynamics, Boune rationality, Monopoly The authors highly appreciate the nancial supports from the MEXT-Supporte Program for the Strategic Research Founation at Private Universities 2013-2017, the Japan Society for the Promotion of Science (Grant-in-Ai for Scienti c Research (C) 24530201 an 25380238) an Chuo University (Grant for Special Research). The usual isclaimers apply. y Professor, Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan. akiom@tamacc.chuo-u.ac.jp z Professor, Department of Applie Mathematics, University of Pécs, Ifúság, u. 6, H-7624, Pécs, Hungary. sziarka@gmail.com 1
1 Introuction The literature on monopolies an oligopolies plays a central role in mathematical economics. The existence an uniqueness of the equilibrium were the research issues in early stage an then ynamic extensions become the main topic of researchers. Linear moels were rst examine, where local asymptotical stability implies global stability. Each moel is base on a particular output ajustment scheme. In applying best response ynamics, global information is neee about the pro t function while in the case of graient ajustments only local information is neee to assess the marginal pro t. The early results on static an ynamic oligopolies are summarize in Okuguchi (1976) an their multiprouct generalization are iscusse in Okuguchi an Sziarovszky (1999). During the last two ecaes an increasing attention has been given to nonlinear ynamics. Bischi et al. (2010) gives a comprehensive summary of the newer evelopment. As a tractable case many authors have examine ynamic monopolies. Baumol an Quant (1964) have investigate cost-free monopolies, the ynamic extensions of which were examine in both iscrete an continuous time scales. Their ajustment scheme resulte in convergent processes. Puu (1995) assume cubic price an linear cost functions an consiere only iscrete time scales. Naimzaa an Ricchiuti (2008) assume cubic price an linear cost functions, an their moel was generalize by Askar (2013) with a general from of the price function keeping the linearity of the cost function. It is shown in these stuies that chaotic ynamics arises via perio oubling bifurcation. In this paper, we reconsier monopoly ynamics in three i erent points of view. First, we generalize the moel of Askar (2013) by introucing more general types of the cost function an stuy local an global stability with the nonnegativity conition that prevents time-trajectories from being negative. The iscrete-time moel is converte into the continuous-time moel by introucing a time elay an an inertia in the irection of output change. Secon, after examining the stability with continuous time scale, we reveal that the iscrete moel is less stable than the continuous moel. Lastly, we numerically an analytically show that the continuous moel gives rise to complex ynamics involving chaos ue to elay an inertia. The paper is organize as follows. In Section 2, the iscrete-time moel is presente, conitions are erive for local asymptotic stability. In Section 3, the continuous-time moel is constructe from the iscrete-time moel. Stability with respect to time elay an inertia is consiere. In the nal section, concluing remarks are given. 2 Discrete Time Moel In this section we construct a iscrete time ynamic moel of a monopoly, after etermining the pro t maximizing output. Consier a monopoly, where q is its output, p(q) = a bq the price function (a; b > 0; 1) an C(q) = cq (c > 0; 2) its cost function. The pro t of the monopoly is given 2
as By i erentiation an (q) = (a bq )q cq : 0 (q) = a b( + 1)q cq 1 (1) 00 (q) = b( + 1)q 1 c( 1)q 2 < 0: So (q) is strictly concave in q, an since 0 (0) = a > 0 an lim q!1 0 (q) = 1; there is a unique positive pro t maximizing output, q; which is the solution of the rst-orer conition b( + 1)q + cq 1 = a: (2) The left han sie is strictly increasing, its value is 0 at q = 0 an converges to in nity as q! 1, therefore the unique positive solution can be obtaine by simple computer methos (see, for example, Sziarovszky an Yakowitz 1978). In the numerical consierations to be one below, we always use the following speci cation of the parameters, a = 4; b = 3=5 an c = 1=2 which are also use by Naimzaa an Ricchiuti (2008) an Askar (2013). Equation (2) is epicte by the roughly-meshe surface shown in Figure 1 where the black ots on the surface represent the values of q for integer values of an in interval [2; 8]. It can be seen that the pro t maximizing output is ecreasing in an : Figure 1. Determination of the optimal q 3
In applying graient ynamics it is assume that the monopoly ajusts its output level in proportion to the marginal pro t, q(t + 1) q(t) = k 0 (q(t)) where k > 0 is the spee of ajustment. Substituting equation (1) results in the following ajustment equation of output: q(t + 1) = q(t) + k a b( + 1)q(t) cq(t) 1 : (3) If = 1 an = 2; then this equation is linear with uninteresting global ynamics. So will assume that either > 1 or > 2 or both occur. So uner this assumption, equation (3) is a nonlinear i erence equation, the local asymptotic stability of which can be examine by linearization. Equations (2) an (3) imply that the steay state of this system is the pro t maximizing output q. Linearization of the right han sie aroun q results in the linear equation q (t + 1) = 1 k b( + 1)q 1 + c( 1)q 2 q (t) with q (t) = q(t) q. Let then we have A = b( + 1)q 1 an B = c( 1)q 2 ; q (t + 1) = (1 k(a + B)) q (t): The steay state of (3) is locally asymptotically stable if an locally unstable if j1 k(a + B)j < 1 j1 k(a + B)j > 1: Thus we have the following result on local stability: Theorem 1 The steay state of system (3) is locally asymptotically stable if k < k S an locally unstable if k > k S where k S = 2 A + B : If we write i erence equation (3) in function iteration form, then it becomes q(t + 1) = '(q(t)) where '(q) = q + k a b( + 1)q cq 1 4
with an 8 1 >< if > 1 an > 2; ' 0 (0) = 1 kc( 1) if > 1 an = 2; >: 1 kb( + 1) if = 1 an > 2 lim q!1 '0 (q) = 1 an ' 00 (q) < 0: Since the right han sie of i erence equation (3) takes a unimoal pro le, the steay state coul procee to chaotic uctuations via the perio-oubling bifurcation if '(q) has strong nonlinearities an its steay state is locally unstable. Naimzaa an Ricchiuti (2008) numerically con rm the generation of complex ynamics uner = 3 an = 1. Later Askar (2013) extens their analysis to the case with 3: We also conuct numerical simulations on global behavior in the more general case of 1 an 2: Figure 2 is a bifurcation iagram with = 4 an = 2 in which the steay state loses stability at k = k S (' 0:154): It is seen that the trajectories exhibit perioic an aperioic uctuations an are economically feasible (i.e., non-negative) for k < k N (' 0:228): In Appenix we consier the non-negativity conition that guarantees the non-negativity of the trajectories for all t 0 an etermine the value of k N : The main result obtaine there is summarize as follows: Theorem 2 There is a positive k N the non-negativity conition such that R(q m (k N ); k N ) = q M (k N ) an R(q m (k); k) q M (k) hols for k k N where, given k; R(q; k) = '(q); q m (k) maximizes '(q) for q 0 an q M (k) solves '(q) = 0: 5
Figure 2. Bifurcation iagram of (3) with respect to k 3 Continuous Time Moel In this section we consier continuous-time ynamics base on the iscrete-time moel. Notice rst that the iscrete system can be rewritten as [q(t + 1) q(t)] + q(t) = '(q(t)): There are many ways to transform a iscrete-time moel into a continuous-time moel. One of the simplest ways is to replace the i erence, q(t+1) q(t); with the erivative _q(t); _q(t) = q(t) + '(q(t)) or _q(t) = k 0 (q(t)): Since 00 (q) < 0; this continuous-time moel is always locally asymptotically stable. Furthermore, its trajectory cannot be negative. 1 Another simple way in obtaining a corresponing continuous moel is the following. We rst assume elaye ajustment (i.e., '(q(t ))) an secon, an inertia in the irection of output change (i.e., q(t + 1) q(t) = _q(t)) to get the elay i erential equation _q(t) = q(t) + '(q(t )) 1 When it goes to negative from positive, the trajectory has to go through zero where _q(t) = ka > 0: So the trajectory turns back to the positive region. 6
or _q(t) = q(t) + q(t ) + k a b( + 1)q(t ) cq(t ) 1 (4) where > 0 is a constant an > 0 is the elay. 2 It is clearly seen that the elay i erential equation (4) reuces to the i erence equation (3) in iscrete time step if = 0. This suggests that the evolution in iscrete time might escribe well the evolution in continuous time if is small enough. In the following, we pursue a possibility that the elay contiuous-time moel gives rise to complex ynamics when its iscrete version also generates complex ynamics an she light on the conition uner which this is a case. The linearization with respect to q(t ) results in the linear equation _q (t) = q (t) + (1 k(a + B)) q (t ): In orer to get the characteristic equation, assume that q(t) = e t u; then + 1 (1 k(a + B)) e = 0: (5) We rst show that the roots of the characteristic equation (5) are single. On the contrary, suppose that is not single. Then is a root of its erivative + (k(a + B) 1)e ( ) = 0 from which (k(a + B) 1)e = : Substituting this relation into equation (5) gives + 1 + = 0 where the only multiple eigenvalue can be = 1 1 : Consequently all pure complex roots are single. If = 0; then = k (A + B) < 0: Hence the steay state is locally asymptotically stable if there is no elay (i.e., = 0) regarless of the value of the spee of ajustment. If = 0; then the continuous system is reuce to the iscrete system an its stability epens on the value of the spee of ajustment as con rme in Theorem 1. Our main concern in this section is place on two issues: one is whether a positive elay estabilizes the stable steay state an the other is whether a positive inertia 2 For this transformation, see Berezowski (2001) where the logistic map is connecte with some physical process of e nite inertia. 7
stabilizes the unstable steay state. The remaining part of this section is ivie into two parts, the rst issue is consiere in the rst part an so is the secon issue in the secon part. Before proceeing, Figure 3 epicts a bifurcation iagram of the elay i erential equation (4) with respect to k similarly to Figure 2. For each value of k the equation is simulate for 2000 times where = 2 an = 0:1. The ata for 1900 < t < 2000 are plotte vertically above the value of k: There are similarities an issimilarities between this gure an Figure 2. For relatively smaller values of k, the continuous system is stable an thus trajectories converge to the stationary state. For values greater than the threshol value of k1 S ; a limit cycle appears with perio-oubling an then bifurcates to a perio-8 limit cycle with two-times oubling at the secon threshol value of k2 S. For even larger values of k, ynamic behavior of output is very aperioic. So we will look more carefully into how the elay a ects this bifurcation process. Figure 3. Bifurcation iagram of (4) with respect to k 3.1 Delay E ect In orer to n stability switches in the case of > 0; we assume that = i! with! > 0. Substituting it into equation (5) yiels i! + 1 (1 k(a + B)) (cos! i sin!) = 0: By separating the real an imaginary parts, we get two equations for the two unknowns an!: 1 [1 k(a + B)] cos! = 0;! + [1 k(a + B)] sin! = 0; 8
which can be written as cos! = sin! = 1 1 k(a + B) ;! 1 k(a + B) : (6) Aing the squares of the two equations an arranging the terms, we have or [1 k(a + B)] 2 = 1 + (!) 2! 2 = [1 k(a + B)]2 1 2 : (7) If j1 k(a + B)j < 1; then there is no positive solution of!, therefore there is no stability switch, that is, the steay state is locally asymptotically stable with arbitrary length of the elay. This is the case, when k < k S = 2 A + B : Notice that this is the stability conition in the iscrete moel. This follows the general unerstaning that a continuous-time moel is more stable than a iscrete-time moel in the sense that the former stability region in the parameter space is larger than the latter s. Lemma 1 If the iscrete-time moel (3) is locally asymptotically stable, then so is the continuous-time moel (4). Assume next that k > k S or 1 k(a + B) < 1. Then from (7) the solution for! is q [1 k(a + B)] 2 1! = : (8) Since 1 k(a + B) is negative, equations (6) imply that sin! is positive an cos! is negative, so =2 + 2n! < + 2n an n = 1 sin 1! + 2n for n = 0; 1; 2; ::: (9)! k(a + B) 1 We will next examine the irection of the stability switches. By selecting as the bifurcation parameter, we verify how the change in the length of the elay a ects the real parts of the roots of the characteristic equations. Consiering as the function of ; = (); an implicitly i erentiating the characteristic equation (6), we have (1 k(a + B))e = 0: 9
For the sake of analytical convenience, we solve for (=) 1, 1 = ( + 1) (10) where the relation + 1 = (1 k(a + B)) e from (5) is use to obtain the form at the right han sie. Substituting = i! with! > 0 into equation (10) an then taking the real part represent " # 1 (! + i) Re = Re!(! i)(! + i) = 2 (!) 2 + 1 > 0: The last inequality implies that increasing inuces the crossing of the imaginary axis from left to right. So stability is lost at the critical elay with n = 0 in equation (9), 0 = q [k(a + B) 1] 2 1 sin 1 s1!! 1 [k(a + B) 1] 2 : (11) an stability cannot be regaine later. In short, we summarize the results as follows: Lemma 2 In the case of k > k S ; the continuos moel (4) is locally asymptotically stable if < 0 : Lemmas 1 an 2 together imply the following: Theorem 3 Given k > 0; the elay i erential equation (4) is locally asymptotically stable an elay is harmless if either k < k S or k > k S an < 0 : We now numerically examine global ynamics when the stability of the steay state is lost. Figure 4(A) epicts the ownwar-sloping partition curve escribe by equation (11) with = 3 an = 1: 3 The curve ivies the (k; ) plane into two regions. The steay state is locally unstable in the region above the curve an locally asymptotically stable in the yellow region below the curve. This stable region is further ivie into two by the vertical otte line 3 Neeless to say, speci e values of an o not a ect the qualitative aspects of the results to be obtaine below. 10
at k = k S : In the region left to the line, k < k S hols an thus the steay state is locally stable regarless of the length of elay. Such a elay is often calle harmless. In the region right where k > k S ; we select the value k 0 (= 0:29) an epict the vertical otte-real line at k = k 0 crossing the partition curve at = 0 : Lemma 2 implies that the steay state is locally asymptotically stable for < 0 : On the partition curve, the real part of one eigenvalue is zero an its erivative with respect to is positive. If becomes larger than 0 by crossing the otte vertical line at 0 from left to right, then one pair of complex eigenvalues changes the sign of their real part from negative to positive. Thus increasing > 0 estabilizes the moel an the steay state bifurcates to a limit cycle as shown in Figure 4(B). At = 1 ; another pair of eigenvalues oes the same. In consequence, a new limit cycle emerges from the existing limit cycle with oubling the perio of the cycle. 4 So perio-oubling bifurcation occurs. In aition to this, carefully observing Figure 4(B) inicates the emergence of another new limit cycle for 1 < < 2. 5 Thus the perio of the limit cycle is more than ouble. At the next critical value, = 2 ; a new pair of eigenvalues changes real part from negative to positive again an the existing cycle bifurcates to a new limit cycle with oubling perio an another new limit cycles are born. In this way the bifurcation procees. If k 1 < < k ; then exactly k pairs have positive real parts an if! 1; then k! 1; so by increasing the values of ; the ynamics of the system becomes more an more complex generating complex ynamics involving chaos. This is well illustrate in the bifurcation iagram in Figure 4(B) in which we call such bifurcation quasi perio-oubling because the perio is increase to more than ouble. Global behavior with respect to is summarize as follows: Proposition 1 Given k; the steay state loses stability at = 0 an bifurcates to chaos via the quasi perio-oubling cascae as increases. 4 The values of i for i = 1; 2; :::; 5 are etermine by a rule of thumb. 5 The same phenomeon is often observe in a elay i erential equation. However, it is not clear yet why the new cycles arise suenly. 11
(A) Partition curve (B) Bifurcation iagram 3.2 Inertia E ect Figure 4. Delay e ect In the same way as in the previous section, we can etect the e ect cause by a change in on ynamics. We rst eal with the benchmark case of = 0: The characteristic equation is 1 + (k(a + B) 1) e = 0: If = + i with > 0; then it is written as 1 + (k(a + B) 1) e (cos i sin ) = 0: Notice that stability epens on the sign of : Separating real an imaginary parts yiels two equations an 1 + (k(a + B) 1) e cos = 0 (12) (k(a + B) 1) e sin = 0: (13) sin = 0 is e nitely etermine from (13). On the other han, the sign of cos in (12) is not etermine unless the sign of k(a + B) 1 is speci e, if k(a + B) 1 < 0, then cos > 0 implying cos = 1 an if k(a + B) 1 > 0, then cos < 0 implying cos = 1: 12
So from (12), Then 1 + (k(a + B) 1) e (+1) = 0 if k(a + B) 1 < 0; 1 + (k(a + B) 1) e ( 1) = 0 if k(a + B) 1 > 0: e 1 = accoring to sign[k(a + B) 1] = k(a + B) 1 Since ouble-sign is in the same orer, > 0 if an only if either or (i) k(a + B) 1 < 0 an k(a + B) 1 < 1 (ii) k(a + B) 1 > 0 an k(a + B) 1 > 1: Conition (i) cannot occur as A > 0 an B > 0. It is therefore shown that the real part is negative if k < k S an positive if k > k S : Hence, the stability conition in the continuous moel with = 0 is the same as the one in the iscrete moel: Theorem 4 The ynamic equation (4) with = 0 is locally asymptotically stable if k < k S an locally unstable if k:k S : Now we procee to the case of > 0: If is given, then the critical values of can be obtaine from equations (8) an (9), q [k(a + B) 1] 2 1 n = q : (14) (2n + 1) sin 1 1 1 [k(a+b) 1] 2 They are the partition curves in the (; ) space. Largest n occurs at n = 0: n ecreases in n an converges to zero as n! 1: To verify whether stability switch takes place on the partition curve, we suppose that = (): Implicitly i erentiating the characteristic equation yiels implying that + (1 k(a + B))e 1 = ( + 1) = 0 : If = i! with! > 0; then the real part is " # 1 (! 2 i!) Re = Re! 2 = < 0: 13
That is, when the value of crosses a critical value from left to right, then one pair of complex eigenvalues changes sign of real part from positive to negative. Let 0 () an 0 () enote the right han sies of equations (9) an (14) with n = 0: Both functions are strictly increasing. Assume that the value of is xe an assume that > 0 = 0 (): Then = 0 ( 0 ) < 0 (); so Lemma 2 implies the local asymptotic stability of the continuous moel with parameters an : When the value of crosses 0 from right to left, then one pair of eigenvalues changes real part from negative to positive. Then at 1 ; another pair oes the same, at 2 a new pair changes the sign of their real part from negative to positive, an so on. At each n the number of eigenvalues with positive real parts increases by two when the value of crosses n from right to left. So if n < < n 1 ; then exactly n pairs of eigenvalues have positive real parts, that is, at any > 0 there are only nitely many such eigenvalues an as ecreases, their number increases. Furthermore this number tens to 1 as! 0: This is the reason of the "chaotic" behavior for small values as shown in the bifurcation iagrams of Figure 5(B). As increases, the number of eigenvalues with positive real parts ecreases making the ynamics of the moel more simple. This is also illustrate in Figure 5(B). Proposition 2 Given ; ynamics becomes simple from complex via quasiperio halving cascae as increases to 0 where stability is gaine an never lost for > 0 : (A) Partition curve (B) Bifurcation iagram Figure 5. Inertia e ect In summary we have the following result: Theorem 5 The steay state of the continuous moel (4) is locally asymptotically stable when the inertia oes not a ect stability if either k < k S 14
or k > k S an > 0 : 4 Concluing Remarks A iscrete an corresponing continuous ynamics of a monopoly were examine with general price an cost functions. First the existence of the unique pro t maximizing output was prove an then stability conitions were erive for the ynamic extensions. The iscrete system is locally asymptotically stable if the spee of ajustment is su ciently small, in which case the continuous system is also locally asymptotically stable with any length of the elay. It was shown that the continuous system may become stable even in cases when the iscrete system is unstable with selecting su ciently small length of the elay. In examining global behavior we have seen a similarity between the epenence from the elay an the inertia coe cient : The same quasi perio oubling phenomenon occurs with increasing values of an with ecreasing values of : This interesting fact was analytically prove an illustrate with numerical stuies. 15
Appenix In this Appenix, we consier the non-negativity conition for the trajectories. It is assume that 1 an 2 where > 1 or > 2 or both. The ynamic equation is given by q(t + 1) = R(q(t); k) where with Furthermore R(q; k) = q + k(a b( + 1)q cq 1 ) (A-1) R(0; k) = ka an lim R(q; k) = 1: q!1 R 0 (q) = 1 k(b( + 1)q 1 + c( 1)q 2 ): (A-2) Notice that R(q) is strictly concave. Result 1 Given k > 0; there is a unique q M (k) such that R(q M (k)) = 0: Proof. Equation (A-1) implies that q M is the solution of the equation b( + 1)q + cq 1 = a + 1 k q (A-3) where the left han sie is enote by f(q): At q = 0; the left han sie of equation (A-3) is f(0) = 0 an is less than the right han sie. As q goes to in nity, f(q) converges to 1 faster than the right han sie. So there is at least one solution. Since f(q) is strictly convex an the right han sie is linear, the solution is unique. Furthermore R(q; k) > 0 if q 2 (0; q M ) an R(q; k) < 0 if q > q M : an It is veri e from (A-3) that lim q M (k) = 1, lim q M (k) = q (A-4) k!0 k!1 q 0 M (k) = q=k 2 f 0 (q M ) 1=k > 0 where the inequality is ue to the fact that at q = q M ; the left han sie crosses the right han sie from below. In the same way we have the following. Result 2 Given k; there is a unique q m (k) that maximizes R(q; k) for q 0: 16
Proof. At any interior maximum, b( + 1)q 1 + c( 1)q 2 = 1 k : (A-5) Let g(q) enote the left han sie, which strictly increase in q an tens to 1 as q! 1. Notice that 8 b( + 1) if = 1 an > 2; >< g(0) = c( 1) if > 1 an = 2; >: 0 otherwise. If g(0) < 1=k, then there is a unique solution q m (k) of equation (A-5). Furthermore, g(q) < 1=k as q < q m (k) an g(q) > 1=k as q > q m (k), so q m (k) is the unique maximizer. If g(0) 1=k, then g(q) > 1=k for all q > 0 implying that @R(q; k)=@q < 0; therefore q m (k) = 0 is the unique maximizer. Notice that q m (k) satis es the following limit relations: lim q m(k) = 1. k!0 This is clear if g(0) < 1=k from equation (A-5), which becomes the case if k is su ciently small. Furthermore lim q m(k) = 0 k!1 which is also a simple consequence of equation (A-5) for interior maximum, otherwise q m (k) = 0 for large enough values of k. Notice also that q m (k) strictly ecreases in k if it is positive an therefore solution of equation (A-5). Result 3 q M (k) > q m (k) for any k > 0. Proof. Since R(0; k) > 0 an R(q; k) 0 as q q M ; the maximizer q m has to be less than q M ; so we have q M (k) > q m (k) for all k > 0: If g(0) > 0; then there is a k 1 such that g(0) = 1=k 1 an if k < k 1 ; then g(0) < 1=k, an if k k 1 ; then g(0) 1=k. So for k k 1 ; q m (k) = 0 an if k < k 1 ; then q m (k) is interior. If g(0) = 0; then q m (k) is interior for all k > 0; so we may select k 1 = 1 in this case. Result 4 R(q m (k); k) takes a U-shape pro le with respect to k an the q m (k) curve passes through its minimum point. Proof. Assume k < k 1 ; then q m (k) is interior. Di erentiating R(q m (k); k) with respect to k gives R k = @R q m @q k + @R @k q=qm q=qm 17
where the rst term on the right han sie is zero at q = q m : So R k = a b( + 1) (q m (k)) c (q m (k)) 1 = 1 k (R(q m(k)) q m (k)) where equation (A-1) evaluate at q = q m (k) is use at the secon step. The pro t maximizing output q is etermine by f(q) = a an inepenent from k: Since q m (k) is strictly ecreasing in k while positive, lim k!0 q m (k) = 1 an lim k!1 q m (k) = 0; there is a unique threshol value k such that q m ( k) = q: In consequence, since q m (k) > q for k < k, we have R(q m (k); k) < q m (k) that then leas to R=k < 0: In the same way, since q m (k) < q for k > k, we have R(q m (k); k) > q m (k) that then leas to R=k > 0: Therefore the R(q m (k); k) curve takes the U-shape pro le for k < k 1 an the q m (k) curve passes the minimum value. From Results 1-4, we have the following conition for the non-negativity of the trajectory which is given earlier in Theorem 2. Result 5 There is a positive k N such that R(q m (k N ); k N ) = q M (k N ) an the non-negativity conition hols for k k N : Proof. For k < k; R(q m (k); k) q M (k) R(q m (k); k) < q m (k) < q M (k): For k > k; R(q m (k); k) is increasing in k an converges to 1 as k! 1 while q M (k) is ecreasing. Thus there is a unique k N > k where R(q m (k N ); k N ) = q M (k N ) such that R(q m (k); k) q m (k) < q M (k) for k k an R(q m (k); k) < q M (k) for k < k < k N R(q m (k); k) > q M (k) for k > k N : 18
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