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1 OPERATIONS RESEARCH doi 0.87/opre ec e-copaio ONLY AVAILABLE IN ELECTRONIC FORM ifors 009 INFORMS Electroic Copaio Equilibriu Capacity Expasio Uder Stochastic Dead Growth by Alfredo Garcia ad Zhijiag She, Operatios Research, doi 0.87/opre

2 ec e-copaio to Author: Alfredo Garcia ad Zhijiag She Proofs of Stateets EC.. Proof of Theore. (cotiued) EC... Regio S I the paper, we have checked that there is o (strictly) profitable deviatio fro y whe κ > ρ(+g) θg i regio S. The proof follows by checkig this coditio i all reaiig regios. EC... Regio S If the iitial relative capacities (k, 0 k) 0 are i this regio, there will be o ivestet accordig to strategy y. If fir deviates by choosig a target of capacity k > δk, 0 the the derivative of fir s objective fuctio is: D(k ) = κ + βθ(), δk 0 ) + β( θ) (k, δk 0 ) (EC.) If (k 0, k 0 ) {k 0 k δ < k0 i (+g), i =, }, the assuptio δ < guaratees the relative capacities 3δ +g after depreciatio will be regio S. They are either (k, δk 0 ) if o growth o the dead side or, δk0 ) if dead grows. The forer capacity pair is i regio IV ad the latter oe is i regio I. Thus we have: (k, δk k ) 0 = δκ, δk 0 ) = R, δk 0 ) + δ κ Substitutig i EC. we obtai the followig expressio for the derivative: D(k ) = ( βδ)κ + βθ() R, δk 0 ) Sice R k < 0, R < 0 i regio I ad δk 0 k > k, k > δk 0 > k,so it is true that Thus, for all (k 0, k 0 ) {k 0 k δ D(k ) < ( βδ)κ + βθ() R ( k, k ) = 0 < k0 i (+g), i =, } fir ca ot be better off by ivestig. 3δ

3 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec3 If (k, 0 k) 0 {k 0 ki 0 > (+g), i =, }, the relative capacities after depreciatio will reai i 3δ regio IV o atter whether dead growth or ot. We kow the short ter equilibriu reveue R is a costat i regio IV, so fir has o icetive to ivest i this case. Therefore, it is ot profitable for fir to ivest i regio S. EC..3. Regio S 3 If the iitial relative capacities (k, 0 k) 0 are i this regio, strategy y prescribes o ivestet for fir. We defie fir s ivestet target as ϕ(k) 0 which is solely decided by the relative capacity level of fir. If fir ivests to reach a relative capacity level k, we have: The derivative of the objective is: (k, δk k ) 0 = R (k, δk k ) 0 + δκ, δk 0 ) = R, δk 0 ) + δ κ D(k ) = ( βδ)κ + βθ() R, δk 0 ) + β( θ) R (k, δk 0 ) (EC.) We ow show that it is ever optial for fir to choose a capacity level i regio I uder our assuptio > ρ(+g). The derivative for this case is: κ θg D(k ) = ( βδ)κ + β βδk0 β ( θg) ( θg)k which is decreasig i k. Whe k reaches the right boudary of regio I, that is k = δk 0, D(k ) is iiized: ( θg) D(k ) > ( βδ)κ β + Sice δk 0 > k for ay iitial k 0 capacity i regio S 3, we have: ( θg) D(k ) > ( βδ)κ β + 3( θg) βδk 0 So it is ever optial for fir to choose a capacity level i regio I. 3( θg) βk > 0 Based o the relative capacities (k, δk) 0 ad, δk0 ), EC. ay have differet expressios for a give k 0.

4 ec4 e-copaio to Author: Alfredo Garcia ad Zhijiag She Case : If (k, δk 0 ) IV ad +g, δk0 +g ) I: D (k ) = ( βδ)κ + βθ[ (k + δk 0 )] Case : If (k, δk 0 ) II ad +g, δk0 +g ) I: D (k ) = ( βδ)κ + βθ[ (k + δk 0 )] + β( θ) ( k ) Notice that D (k ) > D (k ). That eas for the sae δk 0, if the locally optial poit appears i Regio IV (D (k ) = 0) the D (k ) ust be positive. It is optial for fir to ivest to ake the capacity pair (k, δk 0 ) eter regio IV. Case 3: If (k, δk) 0 II ad, δk0 ) II:, the derivative of the objective is: D 3 (k ) = ( βδ)κ + θ β β( + θ)k Case 4: Ad if (k, δk 0 ) IV ad +g, δk0 +g ) II: D 4 (k ) = θ β ( βδ)κ β( + θ)k Sice there will be o ivestet i regio S, fir s ivestet level is bouded by (+g) 3. For a fixed δk 0, if there is oly oe locally optial k (that is D(k ) = 0), this k ust be the optial target of capacity ivestet for fir ; If for exaple, D 3 (k ) > 0 ad D 4 (k ) < 0, the optial level of ivestet for fir will be lie k = ; If there are ore tha oe locally optial level 3 of k, we ust copare the value of objectives correspodig to each targets levels i order to idetify the real optial level of ivestet level of capacity for fir. EC..4. Regio S 4 If the iitial relative capacities (k, 0 k) 0 are i this regio, fir will ivest to ake its relative capacity i the ext period equal to ϕ(k). 0 We are goig to check if it is profitable for fir to deviate by akig a little ivestet i this regio. Fir will ivest oly if the relative capacities after ivestet, ϕ(k0 ) ) are i regio I, otherwise fir s oe period reveue is oly decided

5 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec5 by fir s relative capacity. (accordig to the clai i the previous part, (k, ϕ(k 0 )) ca ot be i regio I). We have: The derivative of the objective is: (k, ϕ(k k )) 0 = δκ, ϕ(k0 ) ) = R, ϕ(k0 ) ) + δ κ D(k ) = ( βδ)κ + βθ() R, ϕ(k0 ) ) + β( θ) R (k, ϕ(k 0 )) (EC.3) Case : If the relative capacities (k, ϕ(k 0 )) IV ad, ϕ(k0 ) ) I: we kow that for fir Thus, D (ϕ(k 0 )) = ( βδ)κ + βθ[ (δk0 + ϕ(k 0 ))] = 0 ϕ(k 0 ) = () [ ( βδ)κ ] δk0 βθ ad if fir ivests to reach k > δk 0, the D(k ) = ( βδ)κ + βθ[ (k + ϕ(k 0 ))] = [ ( βδ)κ + βθ( 4k δk 0 )] < [ ( βδ)κ + βθ( 3k )] < 3k [ ( βδ)κ + βθ( )] = 0 Hece, it is ot profitable for fir to ivest i this case. Case : If (k, δk) 0 III ad, δk0 ) I: we kow fro the previous part that the ivest- et target for fir is: D (ϕ(k 0 )) = ( βδ)κ + βθ[ (ϕ(k0 ) + δk 0 )] + β( θ) ( ϕ(k0 )) = 0 Thus, ϕ(k 0 ) = ( + ( θ)(+g) θ ) ( [ ) β( + θ) ( βδ)κ]() δk 0 βθ

6 ec6 e-copaio to Author: Alfredo Garcia ad Zhijiag She ad k + ϕ(k 0 ) > δk 0 + ϕ(k 0 ) = [ β( + θ) ( βδ)κ]() [ + βθ > [ β( + θ) ( βδ)κ]() βθ > 3k ( θ)() ]ϕ(k 0 θ ) Hece, D(k ) = ( βδ)κ + βθ[ (k + ϕ(k 0 ))] < 3k [ ( βδ)κ + βθ( )] = 0 ad it is ot profitable for fir to ivest i this case. EC.. Proof of Theore, whe ρ < κ ρ(+g) θg Siilar with the proof i the previous part, based o the iitial relative capacity prior depreciatio, we partitio the etire relative capacity space R + ito 4 regios slightly differet with before as figure EC. idicates: k S 3 S ( + g) 3 S S 4 * k 3δ ( + g) 3δ k Figure EC. The partitio of relative capacity space whe ρ < κ ρ(+g) θg.

7 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec7 S = {k 0 ki 0, i =, } 3δ S = {k 0 ki 0 > (+g), i =, } 3δ S 3 = {k 0 k 0 < k}\(s 0 S ) S 4 = {k 0 k 0 > k}\(s 0 S ) We propose a ivestet strategy cobiatio ŷ whe ρ < ρ(+g), forally described as κ θg follows where ax{k δk ; 0} k i S ŷ (k) = ax{ϕ(k) δk ; 0} k i S 3 0 otherwise k = [β ( βδ)κ]() 3β( θg) ad ϕ(k) is a fuctio to be defied later. Because of syetry, it is eough to check the optiality of strategy ŷ whe fir follows strategy ŷ. We defie BR (k ) = k, BR (k ) = (+g) k whe k k ad defie BR (k ), BR (k ) accordigly whe k > k. EC... Regio S Suppose the iitial relative capacities are such that δk 0 i k ad fir s capacity at the begiig of the ext period after ivestet is k = δk 0 + y. Accordig to strategy ŷ, fir s decisio is to achieve a relative capacity level of k. So fir s objective fuctio is ax{r k (k, 0 k) 0 (k δk)κ 0 + β[θ()v, k ) + ( θ)v (k, k )]} The derivative D(k ) of the objective fuctio above with respect to k is: D(k ) = κ + βθ(), k ) + β( θ) (k, k ) (EC.4) If fir chooses a target capacity ivestet level k k, v (k, k ) = R (k, k ) (k δk )κ + C(k ) v, k ) = R, k ) (k δk )κ + C(k ) where C(k ) = β[θ()v ( k +g, k +g ) + ( θ)v (k, k )] is a costat oly related to the capacity level i equilibriu k. Thus, (k, k ) = R (k, k ) + δκ

8 ec8 e-copaio to Author: Alfredo Garcia ad Zhijiag She, k ) = R, k ) + δ κ Substitutig i EC.4 we obtai the followig expressio for the derivative: Note that D(k ) = ( βδ)κ + βθ() R, k ) + β( θ) R (k, k ) = ( βδ)κ + βθ[ ( k + k )] + β( θ)[ (k + k )] D(k ) = ( βδ)κ + βθ() R, k ) + β( θ) R (k, k ) k =k k = ( βδ)κ + βθ( 3k ) + β( θ)( 3k ) = β ( βδ)κ 3βk ( θg ) = 0 k [β ( βδ)κ]() = 3β( θg) k is i regio I oly if ρ < ρ(+g), where ρ = βδ. κ θg β The differece betwee D(k ) ad D(k ) is D(k ) D(k ) = βσ( θg )(k k ) We have that D(k ) > D(k ) = 0 whe k < k ad D(k ) < D(k ) = 0 for all k < k k δ. If the iitial relative capacities satisfy k 0 i S {k 0 i k < δk 0 k () ad k 0 k 0 }, ad fir chooses a target capacity ivestet level k, fir will ot ivest accordig to strategy ŷ. We have k =k v (k, δk 0 ) = R (k, δk 0 ) (k δk )κ + C(k ) v, δk 0 ) = R, δk 0 ) (k δk )κ + C(k ) Siilar to EC.4, the first order coditio of fir s objective is: D(k ) = κ + βθ()[ R, δk 0 ) + δκ ] + β( θ)[ R (k, δk 0 ) + δκ] = ( βδ)κ + βθ() R, δk 0 ) + β( θ) R (k, δk 0 )

9 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec9 = ( βδ)κ + βθ[ ( k + δk0 )] + β( θ)[ (k + δk 0 )] = 0 We get the ivestet target for fir i this case k (k 0 ) = [β ( βδ)κ]() β( θg) δk0 = 3k δk0 If the iitial relative capacities satisfy k 0 i S {k 0 i k < δk 0 k () ad k 0 < k 0 }, ad fir chooses a target capacity ivestet level k slightly greater tha δk 0, we have v (k, k (k 0 )) = R (k, k (k 0 )) (k δk )κ + C(k ) v, k (k 0 ) ) = R, k (k 0 ) ) (k δk )κ + C(k ) The derivative of fir s objective is: D(k ) = κ + βθ()[ R, k (k 0 ) ) + δκ ] + β( θ)[ R (k, k (k 0 )) + δκ] Sice k (k 0 ) = 3k = ( βδ)κ + βθ[ σ( k + k (k 0 ) )] + β( θ)[ σ(k + k (k 0 ))] δk0, the derivative becoes D(k ) = ( βδ)κ + βθ[ σ( k + k (k 0 ) )] + β( θ)[ σ(k + k (k 0 ))] = β ( βδ)κ βσ( θg )(k + 3k δk0 ) < β ( βδ)κ βσ( θg )( 3δk0 + 3k ) < β ( βδ)κ 3βσk ( θg ) = 0 So it is ot profitable for fir to ivest i this case. The siilar aalysis ca be carried out for iitial relative capacities i regio k 0 i S {k 0 i k (+ g) t < δk 0 k () t+ ad k 0 k 0 } ad k 0 i S {k 0 i k () t < δk 0 k () t+ ad k 0 < k 0 } where t =,, 3,... Therefore, we coclude that for all the iitial relative capacities (k 0, k 0 ) i regio S, k (k 0 ) ad k (k 0 ) describes the optial ivestet target for each fir. If both k 0 ad k 0 are less tha k /δ,

10 ec0 e-copaio to Author: Alfredo Garcia ad Zhijiag She they both ivest to reach k at the begiig of ext period; if both k 0 ad k 0 are greater tha the correspodig level described by k (k) 0 ad k (k), 0 there will be o ivestet at all; for the other cases, oly the fir with less relative capacity will ivest to a level of capacity decided by the iitial capacity of the oppoet fir. EC... Regio S If (k, 0 k) 0 {k 0 ki 0 > (+g), i =, }, the relative capacities after depreciatio will reai at regio 3δ IV o atter whether dead growth or ot. We kow the short ter equilibriu reveue R is a costat i regio IV, so o fir has icetive to ivest for this case. EC..3. Regio S 3 Sice we already kow the optial ivestet strategy ŷ i regio S, we ca idetifiy the states trajectories startig fro ay iitial relative capacities i regio S. Therefore, the expected value v (k) i regio S is kow. We divide regio S 3 ito a series of horizotal stripes I t = {k 0 i k 0 i S 3 ad 3 ()t < δk 0 ( + 3 g)t+ } where t = 0,,,... If (k, 0 k) 0 I 0, we defie fir s ivestet target as ϕ(k) 0 which is solely decided by the iitial relative capacity level of fir. If fir ivests to reach a relative capacity level k, the derivative of the objective is: D(k ) = κ + βθ(), δk 0 ) + β( θ) v (k, δk 0 ) (EC.5) Sice there will be o ivestet i regio S, fir s ivestet level is bouded by (+g) 3. For a fixed δk 0, if there is oly oe locally optial k (that is D(k ) = 0), this k ust be the optial target of capacity ivestet for fir ; If for exaple, D(k ) > 0 whe k < 3σ ad D(k ) < 0 whe k >, the optial level of ivestet for fir will be lie k 3σ = ; If there are ore tha 3σ oe locally optial level of k, we ust copare the value of objectives correspodig to each targets levels i order to idetify the real optial level of ivestet level of capacity for fir. After we get fir s ivestet target as ϕ(k 0 ) i regio I 0, we are able to calculate the expected value fuctio of fir v (k) i this stripe. The we ca idetify fir s ivestet target whe

11 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec (k, 0 k) 0 I. Fir s optial ivestet target ϕ(k) 0 for those stripes with low idex ubers I 0, I,... ay ot be cotiuous alog k, 0 the reaso for that is because the short ter reveue is oly piecewise cotious i the capacity space. Let s exaie ϕ(k) 0 for I t with relatively large idex uber. If fir ivests to reach a relative capacity level k, we have: The derivative of the objective is: (k, δk k ) 0 = R (k, δk k ) 0 + δκ, δk 0 ) = R, δk 0 ) + δ κ D(k ) = ( βδ)κ + βθ() R, δk 0 ) + β( θ) R (k, δk 0 ) Case : If (k, δk) 0 II ad, δk0 ) II:, the derivative of the objective is: Case : If (k, δk 0 ) IV ad +g, δk0 +g ) II: D (k ) = θ β ( βδ)κ β( + θ)k D (k ) = θ β ( βδ)κ β( )k Followig the guidlies we just etioed above, if there is oly oe local axiu, this is the optial target ivestet level; if D (k ) > 0 whe k < 3 ad D (k ) > 0 whe k > 3, ϕ(k0 ) = (+g) ; if D 3 (k ) > 0 whe k < ad D 3 (k ) < 0 whe k >, 3 ϕ(k0 ) = ; ad if there are two 3 local axia levels, we eed to copare the expected value fuctio explicitly. EC..4. Regio S 4 If the iitial relative capacities (k, 0 k) 0 are i this regio, fir will ivest to ake its relative capacity i the ext period equal to ϕ(k). 0 We are goig to check if it is profitable for fir to deviate by akig a little ivestet i this regio. Fir will ivest oly if the relative capacities after depreciatio ad after dead growth ( δk0, ϕ(k0 ) ) are i regio I, otherwise fir s oe period

12 ec e-copaio to Author: Alfredo Garcia ad Zhijiag She reveue is oly decided by fir s relative capacity. Siilar with the aalysis for regio S 3, we divide regio S 4 ito a series of vertical stripes H t = {k 0 i k 0 i S 4 ad 3 (+g)t < δk 0 3 (+g)t+ } where t = 0,,,... For the first stripe H 0, if (k 0, k 0 ) H 0 ad fir chooses a target level of ivestet k 3δ we have: v (k, ϕ(k 0 )) = R (k, ϕ(k 0 )) + β[θ()v, ϕ(k0 ) ) + ( θ)v (k, ϕ(k 0 ))] Thus, (k, ϕ(k 0 k )) = β( θ) [ R (k, ϕ(k k )) 0 + βθ(), ϕ(k0 ) )] The derivative of the objective is: D(k ) = κ + β( θ) [β( θ) R (k, ϕ(k k )) 0 + βθ(), ϕ(k0 ) )] (EC.6) If the relative capacities +g, ϕ(k0 ) +g ) are above the coectio of k (k 0 ) ad k (k 0 ), fir will ot ivest accordig to the optial strategy ŷ i regio S. So we have The derivative i EC.6 becoes, ϕ(k0 ) ) < κ D(k ) < κ + β( θ) [β( θ) R (k, ϕ(k 0 )) + βθκ] (i) If the relative capacities +g, ϕ(k0 ) +g ) are uder the coectio of k (k 0 ) ad k (k 0 ), fir will ivest to k (k 0 ) or k accordig to the optial strategy ŷ i regio S. So we have The derivative i EC.6 becoes, ϕ(k0 ) ) = R, ϕ(k0 ) ) + δ κ D(k ) = ( βδ)κ + βθ() R, ϕ(k0 ) ) + β( θ) R (k, ϕ(k 0 )) (ii) Case : If the relative capacities (k, ϕ(k 0 )) IV ad D(k ) < κ +, ϕ(k0 ) βθκ = ( β) < 0 β( θ) ) I: the derivative i (i) is

13 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec3 Sice it is true for fir that ( βδ)κ + βθ[ (δk0 + ϕ(k 0 ))] = 0 Thus, ϕ(k 0 ) = () [ ( βδ)κ ] δk0 βθ ad the derivative i (ii) becoes D(k ) = ( βδ)κ + βθ[ (k + ϕ(k 0 ))] < [ ( βδ)κ + βθ( 3k )] < 3k [ ( βδ)κ + βθ( )] = 0 Hece, it is ot profitable for fir to ivest i this case. Case : If (k, δk) 0 III ad, δk0 ) I: the derivative i (i) is Sice it is ture for fir that: D(k ) < κ + βθκ = ( β) < 0 β( θ) ( βδ)κ + βθ[ (ϕ(k0 ) + δk 0 )] + β( θ) ( ϕ(k0 )) = 0 Thus, ad ϕ(k 0 ) = ( + ( θ)(+g) θ ) ( [ ) β( + θ) ( βδ)κ]() δk 0 βθ k + ϕ(k 0 ) > δk 0 + ϕ(k 0 ) = [ β( + θ) ( βδ)κ]() [ + βθ > [ β( + θ) ( βδ)κ]() βθ > 3k ( θ)() ]ϕ(k 0 θ ) Hece, the derivative i (ii) is D(k ) = ( βδ)κ + βθ[ (k + ϕ(k 0 ))]

14 ec4 e-copaio to Author: Alfredo Garcia ad Zhijiag She < 3k [ ( βδ)κ + βθ( )] = 0 ad it is ot profitable for fir to ivest i this case. S. Case 3: If (k, δk) 0 I ad, δk0 ) I: the aalysis is the sae as what we did i regio The siilar arguet holds for (k 0, k 0 ) H t, t =,,... I Suary, for ay iitial relative capacities (k 0, k 0 ), there exists a MPE strategy ŷ whe ρ < κ ρ(+g) θg so that ay deviatio fro which is ot profitable. EC.3. Uiqueess of y Cosider the class of ivestet strategies, say ỹ(k), that prescribe ivestets ỹ i (k) = k i δk i wheever δk i k i ad k k. Suppose the iitial capacity stock k 0 is such that δki 0 k i. Fir i s objective fuctio is ax{r i (k 0 ) (k i δk 0 k i i )κ + β[θ()v i ( k i, kj ỹ) + ( θ)v i(k i, k j ỹ)]} The (ecessary) first order coditio for equilibriu for i {, } is: ( βδ)κ = βθ() R i k i ( k i, kj ) + β( θ) R i ( k i, k k j ) i Without loss of geerality, it is eough to cosider the coditio where k i k j. The argial profits are as follows: Regio I: { R i k i ( k i, k j ) = ( k i + k j ) R i k j ( k i, k j ) = ( k i + k j ) Regio II: { R i k i ( k i, k j ) = k i R i k j ( k i, k j ) = 0 Regio IV : { R i k i ( k i, k j ) = 0 R i k j ( k i, k j ) = 0

15 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec5 Note that best reply fuctios for fir i ad j are liear with differet slopes whe ( k i, k j ) ad ( k i +g, k j +g ) {I, IV }. So there ca oly exists a syetric equilibriu i these cases. If ( k i, k j ) II ad ( k i, k j ) I. The best reply of fir i is: k (i) j The best reply of fir j is: = βθ [ θ β( + θ) ( βδ)κ β( + θ) k i ] k (j) j The slope of fir i s best reply equals = ( ρκ θ ) k i ( θ ()( θ) + θ) = θ θ which is steeper tha fir j s best reply. We check for a itersectio of best reply aps whe ( k i, k j ) II ad ( k i +g, k j +g ) I. Whe k i = 3, k i locates at the right boudary of this regio, the correspodig k j fro fir i s best reply is k (i) j The correspodig k j fro fir j s best reply is Hece, the differece is: = ()ρκ ( + θ + 5θg) 6θ θ k (j) j = 6 ( + 3g) ()ρκ θ k (i) (j) j k j = ()ρκ ( θ + θg) 6θ θ Reeber that we require κ > ρ(+g) θg i Theore, we get κ < θg. Plug this relatio ito the ρ(+g) above expressio, this leads to k (i) (j) j k j > θg ( θ + θg) 6θ θ = ()( θ) 0 6θ Therefore, fir i s best reply is always above fir j s best reply, so these two best replies will ot itersect i this regio. If (( k i, k j ) II ad ( k i +g, k j +g ) II)or (( k i, k j ) IV ad ( k i +g, k j +g ) II), we have that the right side of first order coditio for equilibriu for fir j is always equal to 0. This eas fir j has o icetive to ivest to these regios.

16 ec6 e-copaio to Author: Alfredo Garcia ad Zhijiag She EC.4. Algebraic Derivatio i 5. H() + F (r θg) = θ()r ( k (), k EC.5. Etry Deterrece = θ[ k () ]k () + ( θ) ) + ( θ)r (k (), k ) ρκk () ( ) ρκk () + = θ[ + ( ρκ θ ) ρκ θ )]k () + ( θ) = θ ( ρκ ) ( ) k () + ( θ) + θ + ( ρκ ) ( ) θ = θ() + ( θ) + + ( ) + The paraeters i this uerical illustratio are κ =, β = 0.87, δ =, θ = 0.6, g = 0., F = ad let free. With this choice of paraeters, > 5, provided >.5. Assue icubet firs wat to deter the etry by aitaiig a level of capacity k > k (). Note that i order to deter etry, k does ot eed to be greater tha (+g). Figure EC. shows the state trajectory whe etry occurs. Etry is ot profitable if ad oly if: ax x { F κx + β[θ()v( x, ( ) () Etry occurs whe there is dead growth. Sice ( ) (+g) ( ) ) + ( θ)v(x, )]} 0 (EC.7) ( )k (), the firs will retur to syetric capacity levels equal to k () just oe period after the dead growth has bee ( ) k' * ( ) k ( ) ( ) K i * * ( k ( ), ( ) k ( )) x + K i Figure EC. The Capacity Trajectory Whe Etry Occurs.

17 e-copaio to Author: Alfredo Garcia ad Zhijiag She ec7 observed. Sice there are firs havig a capacity stock greater tha k (), the optial scale x for the etrat (i.e. the -th fir) ust be saller tha k (). We have v(x, ad ( ) ) = R(x, ( ) x ) + β[θ()v(, ( ) () ( ) ) + ( θ)v(x, )] (EC.8) x ( ) v(, () ) = R( x ( ), () ) (k () x )κ (EC.9) +β[θ()v( k (), ( )k () ) + ( θ)v(k (), ( )k ())] Fro EC.7-EC.9, we copute the derivative of the left side of EC.7: ( ) R(x, ) β( θ) x + βθ() R( x, ( ) ) +g (+g) ( β)κ 0 x (EC.0). Suppose (x, ( ) ) falls i regio II. I this case, the derivative becoes β( θ)( ( ) x+ x) + βθ[ ] ( β)κ +g = x This derivative equals to 0 whe x = So the optial level is x = i{ , 6 }. Suppose (x, ( ) ) falls i regio IV. I this case, the derivative becoes ( ) x+ βθ[ ] ( β)κ +g = x 0.3 This derivative is ootoe decreasig i x ad equals to 0 whe x = So the optial level of x is: x = ax{ , 6 } I suary, the optial level of x should be <.735 x =.735 < First, let s see if the etrat ca acquire positive profit whe x = The expected value of the etrat is: F κx + β[θ()v( x F κx + β( θ) β( θ), ( ) +g (+g) = R(x, ( ) ) + ( θ)v(x, ( ) )] ) + βθ(+g) β( θ) v( x +g, ( ) (+g) )

18 ec8 e-copaio to Author: Alfredo Garcia ad Zhijiag She Also, x ( ) v(, () ) = R( x ( ), () ) (k () x )κ + β β( + θg) [θ()r i(k ()) + ( θ)r i ( k () ( θ)g ) κk ()] =.0839( 0.935)( ) Therefore, the expected value of the etrat whe.4943 <.735 is: F κx + β( θ) β( θ) which is positive whe > R(x, ( ) ) + βθ(+g) (.0839( 0.935)( )) β( θ) =.0609(.5096)( +.35) I a aalogous way, we ca fid out the rage of which ight deter etry for the other two cases: whe x =,.the expected value of the etrat is:.0566(.5097)( +.64) which is 6 always positive sice.735 < whe x = , the expected value of the etrat is:.0596(.55)(+.37) which is positive sice > i this case. This result idicates that whe >.5096, it is always profitable for a ew fir to eter the arket. The icubets ca ever deter the occurrece of etry by aitaiig a higher level of capacity. Ackowledgets The authors gratefully ackowledge all the coets ad observatios received fro the referees which have iproved the paper sigificatly. Refereces See refereces list i the ai paper.

2.6 Rational Functions and Their Graphs

2.6 Rational Functions and Their Graphs .6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a