Information and uncertainty in a queueing system

Transcription

1 Information and uncertainty in a queueing system Refael Hassin December 7, 7 Abstract This aer deals with the effect of information and uncertainty on rofits in an unobservable single server queueing system. We consider scenarios in which the service rate, or the service quality, or the waiting conditions are random variables that are known to the server but not to the customers. We ask whether the server is motivated to reveal these arameters. We investigate the structure of the rofit function, and its sensitivity to the variance of the random variable. We consider and comare variations of the model according to whether or not the server can modify the service rice after observing the realization of the random variable. Introduction There is a growing interest in queueing models where servers and customers react to each other s action, such as rices and queue disciline set by the server, and arrival rocess generated in a non-cooerative way by the customers see [9] for a survey). Much of the literature on this subject assumes that the system s arameters are fixed and known to the articiating agents. In other cases, customers own rivate information on the individual arameters, and one asks whether an adequate ricing system can be used to motivate them to reveal this information to the queue oerator. In contrast, this aer deals with the common situation where some arameters are random variables whose realization at a articular time is known to the queue oerator but not to the otential customers. For examle, the rate of service may vary due to various reasons, the quality of the service may be subject to changes, and even the conditions of waiting in the queue which may be of imortance to a customer when deciding on whether or not to join, may vary. The subject of this aer is the effect that this uncertainty has on the decisions made by the server and the customers, and in articular on the exected rate of rofit. We ask whether or not the server is motivated to reveal the information, what is the effect of increased uncertainty, and what haens when the server is restricted to a fixed rice indeendent of the realization of the random variable. Several aers have dealt with the value of information in congested systems. Some have roved that information may reduce customers welfare. This henomenon can be broadly interreted as a result of the non-cooerative way in which customers behave. Hassin [6] asked whether forcing a rofit maximizing server to inform customers about the length of the queue is necessarily beneficial, given that with or without) this information, customers behavior is not necessarily otimal. It turns out that, deending on the inut arameters, it may or it may not be desired to reveal the length of the queue in front of a rofit maximizing server to customers who consider whether or not to join it. Guo and Zikin [5] considered customers with time values uniformly distributed on [,], and three levels of delay information: no information, length of queue, and exact waiting time. School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. hassin@ost.tau.ac.il; htt:// hassin/. This research was suorted by the Israel Science Foundation grant no. 37/.

2 They show how to comute the erformance measures in the three systems. Their model has no entry ricing, and welfare increases when more accurate delay information is available. Debo, Parlour, and Rajan [4] consider a firm that knows the quality of the service it rovides but cannot credibly communicate it to its otential customers. There are two tyes of customers, characterized by their rivate signal on the quality of the service. Customers udate their beliefs after observing the length of the queue and decide whether or not to join. The authors show that in general, when waiting in the queue is costly, the equilibrium behavior is not of the threshold tye. Other results on the value of information in queueing systems are discussed in [, 7, 8, ]. Arnott, de Palma, and Lindsey [, 3] analyzed effects of information on articiation and time-of-use decisions in congestible systems, when caacity and demand fluctuate. Other aers dealing with related questions are mentioned in these references. They conclude that with aroriate rice regulation of the system, added information can imrove the efficiency of the system. Otherwise, the effect of information may be negative. In this aer, we consider a structured model of a congested system - that of a Markovian queue managed by a single server who aims to maximize rofits. To simlify the derivations we assume that the random variable in question may only obtain two values. For this model, we are able to obtain exlicit answers to the effect of information about the system s arameters on the servers rofits and system s overall welfare. We consider an M/M/ system with a large otential demand of risk neutral customers, service rate, waiting cost of C er unit time, entry fee T, and value of service R. The queue length is unobservable to the customers while making their decision of whether or not to join it. We consider three tyes of uncertainty, defined by assuming that a random arameter takes on one of two values. In the first, the caacity of the server, measured by is random. In the second, the cost of sending time in the system, as reflected by C, is random. In the third, the quality of the service as reflected by R is a random variable For examle, the service rate and quality may deend on the articular server on duty, and the waiting cost may be affected by local time-variable conditions or quality of accommodation. In each case we consider three sub-models. In the first, the customers are not informed about the realized value of the random arameter. In articular, this means that the server sets a single rice indeendent of the realized value of the arameter, because rice differentiation may serve as a signal to the customers about the value of the arameter. In the other two cases, the realized value is revealed to the customers. In the second case, the server may set a different rice for each realized value, charging a higher fee for better quality or for faster service. In the third case this is not ossible, for examle, because the server cannot observe the realized value, or because it is technically imossible or too costly to modify the rices. Main results For a given set of arameters and information, customers join the system according to a Nash) equilibrium rate. By our assumtion of large otential demand, this rate is such that a customer is indifferent between joining and not joining. This means that both in the case of uninformed customers and in the case of informed customers and two rices, the exected net benefit of a customer, after deduction of the admission fee and waiting cost, is. It is well known see for examle [9]) that since in this case the server extracts all the customer surlus, the server s objective coincides with the social objective, and therefore the server s decisions conform with welfare maximization. Note that with informed customers and a single rice, customers may have ositive surlus and the rofit maximizing solution is in general not socially otimal. Let Π un, Π in, and Π in denote the server s maximum rofit when customers are uninformed or informed, and in the latter case when a single rice or two rices are set by the server, resectively. Since customers arrival rate in equilibrium under the rofit maximizing olicy is socially otimal, additional information to the customers can only increase social welfare. Therefore, Π in Πun. Obviously, Π in Πin. However, when the server is restricted to a single See [] for how rice may serve as a signal for quality.

3 rice, there is no straightforward answer to whether or not informing the customers increases rofits and social welfare). We find that with uncertain service rate or waiting cost, Π in Π un. However, with uncertain value of service this inequality does not hold in general. In articular, when the waiting cost is sufficiently small we get the reverse inequality. For uncertain service rate, we rove Theorem 4.3) a result which is stronger than the obvious Π in Πun : Informing customers is desired even if the rice is exogenously fixed, and is not otimally set by the server. Moreover, the difference between the rofits obtained with informed and uninformed customers increases with the amount of uncertainty. Note that in such a case customers enjoy a ositive consumer surlus and the server s objective differs from the social objective. With uncertain waiting cost, for rices that induce ositive demand in the case with no information, the same level of social welfare is attained no matter whether customers are informed or not, but informing customers may be beneficial when one of the ossible values of the waiting cost is so high that without information no customers will show u. With uncertain quality of service and a single rice, when the difference between the ossible values of service is large, the otimal rice is so high that no customer will choose to arrive if the low value is realized. It follows that, for small values of C, the server is motivated to conceal the realized value of service from the customers. For large values of C the oosite is true. The effect of increasing uncertainty on the three rofit functions is different from case to case: With informed customers and two rices, increasing the variance has a ositive effect on rofits and welfare in all three tyes of uncertainty that we consider. With informed customers and a single rice, the effect is ositive excet for when the variance is small and the random variable is the quality of service. With uniformed customers, rofits are indeendent of the variance when the random variable is the waiting cost or quality of service. However, with random service rate, rofits decrease as the variance increases. The aer is organized as follows. In Section 3 we describe the well-known outcome of the model when all arameters are known with certainty. Sections 4,5, and 6 deal with uncertain service rate, waiting cost, and quality of service, resectively. Each section is divided into five subsections. The first three deal with the cases of uninformed customers, informed customers with two rices, and informed customers with a single rice, resectively. The fourth subsection comares the results and draws conclusions on the effect of the assumtions on the server s exected rofit. The last subsection analyzes the effect of uncertainty, as exressed by the variance of the random variable, in each of the three models. 3

4 3 Background For given values of the arameter R, C,, and T, if R T < C then the equilibrium arrival rate is. Otherwise, it satisfies R T = C λ, or λ = C R T. Suose that R, C, and are given, and that T is a decision variable. The rate of rofit is Π = λt. If R < C then it is not ossible to gain any rofits. Otherwise, Π is otimized by setting RC T = R. ) Thus, the maximum rate of rofit is R ) Π C R = C R < C see, for examle, Table 3. in [9]). In the following we measure uncertainty by the variance of the random arameter. Suose that the uncertain arameter is q and it obtains the values q and q with robabilities and resectively. Denote its exected value by q, then, V arq) = q q) + ) q q ). We consider two ways of controlling the variance. In one we fix q and obtain desired values of V arq) and q by adjusting and q. In the other we fix and adjust q and q. In both tyes we maintain a constant value of q = q + )q. Thus, q = q q and V arq) = 4 Uncertain service rate ) q q). 3) Assume that = with robability, and = with robability, where >. Let and v = R T C, r = R C. Thus, v and r reresent the values of service and net gain from service after deduction of service fee, resectively, normalized with resect to the time value. 4. Uniformed customers Without loss of generality, R T C + ), 4) since otherwise λ = and Π =. The roof of the following lemma is given in the aendix: Lemma 4. The equilibrium value of v is a root of ) [ + ) ) ] v 4 + ) ) [ + ) ) ] v 3 + [ + ) r ) [ + + ) )] ) ) ] v + r ) [ ) + ) ) ) ) ] v + r [ r ) + ) r ) ) ) ) ] =. 4

5 We solve the above olynomial for v and then comute Π un = λt = λr Cv). 4. Informed customers - two rices Suose that customers are informed about the service rate, and the server charges different rices deending on the realization of. In this case we use ) to obtain that the server s rate of rofit is Π in = C r ) + )C r ) r C r ) < r < r. 4.3 Informed customers - single rice Assume that the firm must set a fixed rice, and cannot change it when the value of is revealed. Customers are informed however about the realized value of. Assume first that the rice T is not too large, so that λ > for both values of. The arrival rate, given = i, is λ i = i C R T. The exected arrival rate, given T, is λ = C ) + ) C ) R T R T = + ) C R T. 6) Define = + ). The exected rate of rofit, given T, is ΠT ) = T λ. Its derivative with resect to T is C R T T C R T ). The first-order otimality condition is C R T ) R =, which gives T = R ± RC. C Since T < R is required, only the root associated with the minus is relevant, and λ = R. Therefore, Π = R ) RC 5) ) ) C R = C R C. 7) The firm may choose however a high rice which doesn t attract any customers if =. In this situation, the firm will set the rice to gain maximum rofit when the high rate is realized. CR Thus, T = R, as in ), and ) R Π = C C. 8) The firm will choose the solution giving the higher value between 7) and 8). The two values are equal if ) ) R C R = C, In our numerical solution we obtained at most two real roots that satisfy 4), only one of which corresonds to an equilibrium, see Remark6.. 5

6 or Equivalently, r = η, where r = r ). η = ). Remark 4. By concavity of the square root function, + ) >. This inequality can be used to show that + > or η >, for any, ). Denote the resulting rofit by Π in, then C r ) r η Π in = C r ) r η r. 9) 4.4 Discussion Denote by λ in T ) the equilibrium exected arrival rate when customers are informed and there is a single rice T. Similarly, let λ un T ) be the arrival rate when customers are not informed. Theorem 4.3 Consider fixed inut values of,, and. For every rice T, < T < R C + ), λ in T ) λ un T ). Proof: By 6) and 9), we need to rove that ) v v + + S ), or that S [ v + ) ) ]. It is sufficient to show that [ v + ) + ] ) v ) ) v + ) ). This is equivalent to ), and the claim follows. Note that the inequality in the theorem is strict excet for when {, }, the case with no uncertainty. The difference in the exected arrival rates is concave and it is maximized when =. Thus, we may say that the difference in the exected arrival rates increases with the amount of uncertainty. Also note that the range of rices in the theorem is the allowed range for the uninformed customers case. For the informed customers case, the allowed range is wider, and in the added values we have λ in T ) > whereas λ un T ) =, conforming with the sirit of the theorem. Corollary 4.4 Π in Πin Πun. Therefore, the firm is motivated to reveal the realized value of to its customers. Proof: The first inequality is obvious, and the second follows from Theorem 4.3 The relative behavior of the rofit functions under the three models differs in different regions of r as follows we use here the inequality η > given in Remark 4.): ) If r, then Π un = Π in = Πin =. ) If r, + then = Π un < Π in = Πin. ) +, If r ) If r, then < Π un < Π in < Π in where k = ) ). then < Π un < Π in = Πin.. The difference Πin Πin is equal to kc r 6

7 Figure left) gives the exected rate of rofit for =., C = 3, =.5 and =.5. 3 Remark 4.5 Clearly, when = we have Π un = Π in = Π in. However, lim Π un ) < Π un ), whereas lim Π in j ) = Πin j ), for j =,. The functions when are illustrated in Figure right). The reason is that when = we may have < λ <, whereas for any < we must have λ < to guarantee a finite value for the exected waiting time. 4 Actually, the value λ = c R for =, is given by the root corresonding to the lus sign in 8). 3 5 C=3, =. =.5, = C=3, =.5, = Π in Π in Π in =Πin Π un 5 Π un R 5 5 R Figure : Profits when C = 3 and =.5 left) and when right) 4.5 The effect of uncertainty We consider two tyes of changes in uncertainty. In both tyes we fix. In the first case we also fix > recall that > ), and simultaneously change and so that is reserved at the same level and V ar) varies. In Figure left) we assumed = and =.5, so that by 3) V ar) = 4 ). We note that this is an increasing function of. In the second case we fix and modify and, again reserving. In Figure right) we set =.5 and =, giving V ar) = ). As exected from 9), Π in deends only on and not on the individual values of and, as long as η r. For η > r Π in linearly increases with in Figure left) recall that is fixed). We obtain that Π in = Πin when is large enough so that to maintain = the value of is smaller than C R. 5 Figure right) gives similar results but this time = is fixed,, 3.4) and = 4 C R =.6, ) so that again = is reserved. The two figures are essentially identical excet for a nonlinear change of scale. The function ) is convex. Suose that V ar) is increased while maintaining. Then, Π in in the first two cases of 5) also increases. Therefore, Π in, which is the maximum of these two exressions and ) is a nondecreasing function of the variance and strictly increasing where the rofit is ositive). Finally, Π un decreases when the variance increases due to the convexity of the exected waiting time as a function of λ for any given. 3 In this examle, η, corresonding to R. The R values corresonding to, +, and, are, 5, and 6, resectively. 4 In the other cases that we consider later, where C or R are random, we naturally have Π un = Π in = Π in in the limit, when. 5 Secifically, for the arameters used for this figure, is obtained when, or V ar).7. 9

8 3 3 5 C=3, R=5 =.5, + ) = Π in =Πin 5 C=3, R=5 =.5, + ) = Π in Π in Π in Π in 5 5 Π un Π un Var) Var) Figure : The effect of uncertain : changing, ) left) and, ) right) 5 Uncertain waiting cost Suose that C = C with robability and C = C with the comlementary robability, where C < C. Denote the exected value of C by C = C + )C. 5. Uniformed customers Given a rice T, the equilibrium arrival rate is λ = rofit is Π un = { R ) C R C 5. Informed customers - two rices C R T ) +. By ), the maximum rate of R < C. Suose that customers are informed about the service rate, and the server charges different rices deending on the realization of C. Then, Π in = R C ) + ) R C ) R C R C ) C < R < C R C. 5.3 Informed customers - single rice Given the rice T and the information that C i is realized, the equilibrium arrival rate is λ i = ) +. Ci R T For T < R C we have λ, λ >. The exected rate of rofit is then Π = T [λ + )λ ] { = T C } R T + ) ) C R T = T CT R T. ) ) 8

9 This is the same exression as in the uniformed customers case and the maximum rofit is Π = R C). ) Exression ) is correct if the maximizing rice satisfies T < R C. Otherwise, it underestimates the rofit by assuming a negative λ, whereas the true value is. For T R C, R C ) we have λ > and λ =. In this range, the rofit maximizing rice is R RC and the rofit is Π = R C ). 3) We comare 3) with ) to comute the maximum ossible rofit given R and. Thus, { Π in = max ) ) } R C, R C. The two values are equal if R C = R C ), or R = γ where ) C C γ =. Note that from C > C it follows that γ > C. Therefore, Π in = R ) C R γ R C ) C < R < γ R C. 4) 5.4 Discussion The relative behavior of the rofit functions under the three models differs in different regions of R defined by C C C γ < as follows: If R, C ) then Π un = Π in = Πin =. If R C, C) then = Π un < Π in = Πin. If R C, C ) then < Π un < Π in = Πin. If R C, γ) then < Π un < Π in < Πin. If R γ, ) then < Π un = Π in < Π in. The difference Π in Π in is equal to κ R where κ = C C ) ) C. Figure 3 shows the rofit functions for =., C =, and C = 8. The variable corresonding to the x axis is R which is the value roduced by the server er unit of time of service. 6 Corollary 5. Π in Πin Πun. Therefore, the firm is motivated to reveal the realized value of C to its customers. 6 With this data, C = 68, γ 7.7, and κ.39. 9

10 5 =, =. c =, c =8 5 Π in Π un =Π in 5 Π in Π in =Πin Π un R Figure 3: Random waiting cost examle 5.5 The effect of uncertainty We consider two tyes of changes in uncertainty. In both tyes we fix C. In the first case we also fix C C recall that C C ), and simultaneously change and C so that C is reserved at the same level. In Figure 4 left) we assumed C = 3 and C = giving by 3, V ar) =. In the second case we fix and modify C and C, again reserving C. An increase in C comes with an adequate decrease of C. In Figure 4 right) we assumed =.5 and C = 3 giving, by ), V ar) = C 3) =, R=5 c =, c + )c = =, R=5 =.5, c + )c = Π in Π in 3.5 Π in.5 Π un 3.5 Π in Π un VarC) VarC) Figure 4: The effect of uncertain C: changing, C ) left) and C, C ) right) 7 Π in = Π in when R C, see ) and 4). With the arameters used for Figure 4 right) this means C > 5,, and equivalently V arc) >. In Figure 4 left) the analogous condition is C or V arc) Similarly, from ) and 4), Π in = Π un ), if R ρ. In Figure 4 left) this means 3 5 or aroximately V arc) 6. In Figure 4 right) this means C ) given aroximately V arc) 46.

11 6 Uncertain quality of service Let R = R with robability, and R = R with the comlementary robability. R > R and that and C are fixed. Assume 6. Uninformed customers Given a rice T, the equilibrium arrival rate is λ = rofit is Π un = 6. Informed customers - two rices C R T ) +. By ), he maximum rate of { R C ) R C R < C. 5) Suose that customers are informed about the service quality, and the server charges different rices deending on the realization of R. Then, R R C) + ) C) R C Π in = R C) R < C < R 6) R C. 6.3 Informed customers - single rice ) + For a given rice T, the arrival rate is λ i = C R i T if Ri is realized. To have ositive gain, the rice cannot exceed R C. We distinguish the two cases according to whether or not T < R C. Consider first the case T < R C : The exected rofit is ΠT ) = T [λ + )λ ] = T C ) )C + ) R T R T = T CT R T + ). R T The first-order otimality conditions are Π = C R T + ) CT R T Multilication by R T ) R T ) C gives R T ) + ) R T ) =. C R T ) R T ) R T )R T ) + )R T ) R T ) ) T R T ) )R T ) ) =, or C R T ) R T ) R R T ) )R T ) R =. This gives the following olynomial

12 C T 4 R) C R + R )T 3 + C R + R + 4R R ) T + R R ) C R + R ) T where R = R + )R. + R R C R R R )R ) =, 7) Suose now that T R C. In this case λ = and the otimal rice gives rofit R C). Figure 5 left) shows the rofit as a function of the rice for C =, =., R = 7, =, and R = 3. With these arameters R C =, and we see that the server gains more by raising the rice to a level that induces no arrivals when R is realized. Lowering R to 6 and increasing R to 4 and then R C = ) we obtain a new situation, shown in Figure 5 right), in which it is worth lowering the rice to attract customers also when R is realized C=, =., = R =7, R = C=, =., = R =6, R = Price Price Figure 5: Otimal rice is greater left) or smaller right) than R C 6.4 Discussion When C is close to, Π un = Π in = R > [maxr, R )] = Π in. Thus, for small values of C, if the server is restricted to a single rice, the server is motivated to conceal the realized value of R from the customers. However, for large values of C the oosite is true. Figure 6 gives the rofits as a function of C, when R =,R = 3, =., and =. Note that these are actually functions of C The effect of uncertainty We fix R and thus R = R R. In the first case we also fix R R recall that R > R ), and simultaneously change and R so that R is reserved. In Figure 7 left) we assumed R = 3 and R = 5 giving by 3)) V ar) = 4. In the second case we fix and modify 8 For this instance, Π un > for C < R < 44, and the other curves are ositive for C < R =.

13 35 3 =., = R =, R =3 5 5 Π in Π in 5 Π un Π in = Πin Waiting cost er unit of time Figure 6: Random value of service R and R, again reserving R. In Figure 7 right) we assumed =.5 and R = 3 giving V ar) = R 3). We see that Π un only deends on R and is not affected by the change in uncertainty, as exected from 5). We also see that in the case of informed customers and two rices, rofits increase with uncertainty. However, in the case of a single rice, rofits initially decrease as a function of the variance, and only for larger variance there is an increase c=, = R =5, R + )R =3 c=, = =.5, R + )R = Π in Π in.6.4 Π in..8 Π un. Π in Π un VarR) VarR) Figure 7: The effect of uncertain R: changing, R ) left) and R, R ) right) 9 We have Π in = Π in if R C. With the arameters used to create the figures, this means R. In the left figure this means, and thus V arr). In the right figure it means R 4, and thus V arr). 3 3

14 References [] E. Altman, T. Jiménez, R. Núñez-Queija, and U. Yechiali 4) Otimal routing among /M/ queues with artial information, Stochastic Models 4) [] R. Arnott, A. de Palma, and R. Lindsey, Information and usage of free-access congestible facilities with stochastic caacity and demand, International Economic Review ) 8-3. [3] R. Arnott, A. de Palma, and R. Lindsey, Information and time-of-usage decisions in the bottleneck model with stochastic caacity and demand, Euroean Economic Review ) [4] L. G. Debo, C. A. Parlour, and U. Rajan, The value of Congestion, 5. [5] P. Guo and P. Zikin, Analysis and comarison of queues with different levels of delay information, 4. [6] R. Hassin, Consumer information in markets with random roducts quality: The case of queues and balking, Econometrica ) [7] R. Hassin, On the advantage of being the first server, Management Science 4 996) [8] R. Hassin and M. Haviv Equilibrium strategies and the value of information in a two line queueing system with threshold jockeying, Communications in Statistics - Stochastic Models 994) [9] R. Hassin and M. Haviv To Queue or not to Queue: Equilibrium Behavior in Queueing Systems, Kluwer International Series, 3. [] C. Larsen, Investigating sensitivity and the imact of information on ricing decisions in an M/M// queueing model, International Journal of Production Economics ) [] P. Milgrom and J. Roberts, Price and advertising signals of roduct quality, Journal of Political Economy ) Aendix Proof of Lemma 4.: In equilibrium, or and finally, v = λ + λ, v λ) λ) = + ) λ, vλ + λ v v ) + v ) =, λ = + v + ) v v ) ± 4v[v ) ] { v v = } v + + ) ± v + ) + v ) ). 8) Assuming >, the square root is minimized when =, and then it is equal to v ). Therefore, the root corresonding to the lus sign is greater than. We conclude that the equilibrium is defined by the root corresonding to the minus sign. Hence, λ = { } v + + ) S, 9) 4

15 where S = v + ) + ) ) ) = v v ) + 4 ). v Note that by our assumtions that, ) and >, it follows that S is strictly ositive. The derivative of λ with resect to v is λ v = { v v ) } ), S and with resect to T λ T = dv λ v dt = Cv { v ) } ). S The rofit Π un is equal to λt. The first-order otimality condition is T λ T + λ =, or T v Cv + + ) ) ) + ) S v + + ) S =. Multily by S allowed since S > ): S T Cv ) v + + = T Cv 3 + T ) Cv + v + ) + ). v Substitute T = R vc: S R ) Cv + + = R Cv 3 + R ) Cv + ) + ). ) v Recall that r R C. Multily the left-hand side of ) by v3 and square: LS = v + ) + ) ) r v + v 6 + ) r + )v 4) v = r + + ) v 4 r + )v + )r )v + ) r v + ) + ) v 6 r ) + )v 4 + ) ) + ) v 5 4r ) ) + )v 3. Multily the right-hand side of ) by v 3 and square: RS = r + r ) ) v + ) ) v 4 + ) 4 v 6 +r ) )v + r ) )v + r ) v 3 +r ) ) v 3 + r ) ) 3 v 4 + ) ) 3 v 5. By ), LS RS =. Division by v gives ) [ + ) ) ] v 4 + ) ) [ + ) ) ] v 3 + [ + ) r ) [ + + ) )] ) ) ] v + r ) [ ) + ) ) ) ) ] v + r [ r ) + ) r ) ) ) ) ] =. With = =, the olynomial reduces to ) v = 4r, or v = r. This gives T as in ).) 5

16 Remark 6. Figure 8 shows the roots as a function of R, with =., =.5, and =.5. The figure shows the real roots that satisfy T R C + ). Recall that by 4) T must be bounded this way. ) For large values of R we get two such roots, marked Π un and Π un. For examle, with R = 3, the values of λ, v, T, and Π un are.44,.8, 65.4, 4.) and.,.8, 96., ), resectively. However, the new root, denoted Π in in Figure 8, does not corresond to a new equilibrium. Substituting these values in ) we obtain that the left-hand side is negative, whereas the right-hand side is ositive and equal to the left-hand side in its absolute value). Thus, the root Π un results from the squaring of the two sides of ). C=3, =.4 =.5, =.5 8 Π un 6 4 Π un R Figure 8: Profits corresonding to the two real roots Π = if R < C + ) = 44. 6