Startseite Timing of Adopting a Flexible Manufacturing System and Product Differentiation
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Timing of Adopting a Flexible Manufacturing System and Product Differentiation

  • Chia-Hung Sun EMAIL logo
Veröffentlicht/Copyright: 11. Januar 2020

Abstract

Based on a circular product-space model with continuous time, we investigate a dynamic game in which each firm decides whether and when to adopt a flexible manufacturing system (FMS) at the beginning of the game and then chooses its quantity supplied at each time. We show that the equilibrium outcomes may be either joint adoption at the beginning of the game or sequential adoption, depending on the range of an initial adoption cost. For a given basic product, we find that a larger product differentiation decelerates the adoption of FMS. We also investigate competition behavior when the decisions on product locations are made endogenously and conduct welfare analysis, showing that there is market failure in the adoption timing choices.

JEL Classification: L13; D21; O30; R10

Funding statement: This work was supported by the Ministry of Science and Technology (Funder Id: http://dx.doi.org/10.13039/100007225, Grant Number: 106-2410-H-031-004).

Acknowledgements

Financial support by the Ministry of Science and Technology (106-2410-H-031-004) is deeply appreciated.

Appendix

A

Proof of Lemma 1. We define ri(tj)argmaxti[0,tj]Πi(ti,tj)=L(ti,tj) as firm i’s best response for titj and define ri+(tj)argmaxti[tj,)Πi(ti,tj)=F(ti,tj) as firm i’s best response for titj. We now investigate firm 2’s incentive to deviate to t2<t1. We note F(r2+(t1),t1)L(r2(t1),t1) is a strictly decreasing function of t1. We also note F(r2+(t1),t1)L(r2(t1),t1)0, for t1=max{0,t¯1}, and that F(r2+(t1),t1)L(r2(t1),t1)0, for t1=max{0,t¯2}. We conclude that there is a unique t1=d[max{0,t¯1},max{0,t¯2}], satisfying F(r2+(t1),t1)L(r2(t1),t1)=0. If t1 ≤ d, then F(r2+(t1),t1)L(r2(t1),t1)0 and firm 2’s best response is r2(t1)=r2+(t1)=max{0,t¯2}. If t1 ≥ d, then F(r2+(t1),t1)L(r2(t1),t1)0 and firm 2’s best response is r2(t1)=r2(t1)=max{0,t¯1}. By symmetry of the model, we obtain Lemma 1. Q.E.D.

Proof of Lemma 2. Denote ΠiN(tj) as firm i’s profit under no adoption, given firm j’s timing of adoption tj. If c¯>c¯2(x1)=(πbπf)/(πbπf)(α+r)(α+r), then firm 2’s equilibrium timing of adoption is t2=t¯2>0 and:

(15)F(t¯2,t1)Π2N(t1)=(πbπf)t¯2ersdsc(t¯2)=c¯(α+r)re(r+α)t¯2>0.

If c¯c¯2(x1)=(πbπf)/(πbπf)(α+r)(α+r), then t1=0 and firm 2’s equilibrium timing of adoption is t2=0 and:

(16)F(0,t1)Π2N(t1)=(πbπf)0ersdsc(0)=(πbπf)rc¯>0.

We conclude F(t2,t1)>Π2N(t1) and that adoption is better than no adoption from the follower’s viewpoint. On the other hand, since L(t1,t2)L(t2,t2)=F(t2,t2)>Π1N(t2), adoption is also better than no adoption from the leader’s viewpoint. Q.E.D.

Proof of Proposition 3. We calculate (V2V1)(πbπf) as:

(17)(V2V1)(πbπf)=Δ72(Δ12τx12+12Δx12+16τx1316Δx13),

which is a decreasing function of τ:

(18)[(V2V1)(πbπf)]τ=118Δx12(34x1)0.

We solve (V2V1)(πbπf)=0 to yield:

(19)τ=Δ(1+12x1216x13)4x12(34x1).

From eq. (19), τ is an increasing function of x1:

(20)τx1=3Δ(12x1)2x13(34x1)20.

The function τ thus reaches its maximum at τ(x1=1/2)=0, resulting in (V2V1)(πbπf)<0,x1[0,1/2] and t2<t2w. Subtracting (πlπ0) from (V1V0) yields:

(21)(V1V0)(πlπ0)=Δ72(Δ12τx12+16τx13),

which is decreasing in τ:

(22)[(V1V0)(πlπ0)]τ=118Δx12(34x1)0.

Solving for (V1V0)(πlπ0)=0 yields:

(23)τ¯=Δ4x12(34x1).

We find that τ¯ decreases with x1:

(24)τ¯x1=3Δ(12x1)2x13(34x1)20.

The function thus reaches its minimum at τ¯(x1=1/2)=Δ. It follows that (V1V0)(πlπ0)<0 and t1<t1w for a relatively high original adjustment cost (τ>τ¯). On the contrary, (V1V0)(πlπ0)>0 and t1>t1w for a relatively low original adjustment cost (τ<τ¯).

We note that eq. (21) is an increasing function of Δ: [(V1V0)(πlπ0)]/[(V1V0)(πlπ0)]ΔΔ>0. Solving for (V1V0)(πlπ0)=0 obtains:

(25)Δ¯=12τx1216τx13.

We conclude (V1V0)(πlπ0)<0 and t1<t1w for a relatively small innovation size (Δ<Δ¯) and that (V1V0)(πlπ0)>0 and t1>t1w for a relatively large innovation size (Δ>Δ¯).

We note that eq. (21) is a decreasing function of x1:

(26)[(V1V0)(πlπ0)]x1=τΔx13(12x1)0.

Since (V1V0)(πlπ0)=Δ2/Δ27272>0 for x1 = 0 and (V1V0)(πlπ0)=Δ(τΔ)/Δ(τΔ)7272<0 for x1 = 1/2, there exists x¯1 such that (V1V0)(πlπ0)<0 and t1<t1w when x1>x¯1 and (V1V0)(πlπ0)>0 and t1>t1w when x1<x¯1. Q.E.D.

Proof of Proposition 4. We calculate the first-order derivative of the leader’s discounted sum of profits with respect to x1 as:

(27)Lx1=πox1(1ert1)r+πlx1(ert1ert2)r+πbx1ert2r.

Since πo, πl, and πb are all increasing functions of x1 for a given x2 = 0, firm 1’s best response in product location choice is to choose x1 = 1/2. We now investigate firm 2’s product location choice. Differentiating the follower’s discounted sum of profits with respect to x2 yields:

(28)Fx2=πox2(1ert1)r+πfx2(ert1ert2)r+πbx2ert2r.

By symmetry of the model, πo, πf, and πb are all increasing functions of x2 for a given x1 = 0. It is thus best for the follower to choose x2 = 0, provided x1 = 1/2, and maximal dispersion in firms’ basic products is the unique result.

Substituting (x1=1/2,x2=0) into eq. (5), we yield:

(29)t1=1αln(c¯(α+r)πlπo)>0,forc¯>πlπoα+rc¯1(x1=1/2),t2=1αln(c¯(α+r)πbπf)>0,forc¯>πbπfα+rc¯2(x1=1/2)<c¯1(x1=1/2),

where πlπo=(Δ/54)(6a3τ+2Δ) and πbπf=(Δ/54)(6a3τ+Δ). We differentiate (πlπo) and (πbπf) with respect to τ as:

(30)(πlπo)τ=(πbπf)τ=Δ18<0,

which are both decreasing functions of τ. Differentiating (πlπo) and (πbπf) with respect to Δ yields:

(31)(πlπo)Δ=6a3τ+4Δ54>0,(πbπf)Δ=6a3τ+2Δ54>0,

which are both increasing functions of Δ. Q.E.D.

Proof of Proposition 5. A social planner chooses x1 and x2 to maximize social welfare:

(32)SW=V00t1ertdt+V1t1t2ertdt+V2t2ertdtc(t1)c(t2).

We calculate social welfare’s first-order derivatives with respect to x1 and x2 as:

(33)SWx1=V0x1(1ert1)r+V1x1(ert1ert2)r+V2x1ert2r,SWx2=V0x2(1ert1)r+V1x2(ert1ert2)r+V2x1ert2r.

By symmetry of the model, we conclude that SW/x1>0 provided x2 = 0, and that SW/x1>0 provided x1 = 0. Therefore, the socially optimal product location is also maximum differentiation. Substituting (x1=1/2,x2=0) into eq. (12) yields the socially optimal timing of adoption:

(34)t1w=1αln(c¯(α+r)V1V0)>0,forc¯>V1V0α+rc¯3(x1=1/2),t2w=1αln(c¯(α+r)V2V1)>0,forc¯>V2V1α+rc¯4(x1=1/2)<c¯3(x1=1/2),

where V1V0=(Δ/216)(24a15τ+11Δ) and V2V1=(Δ/216)(24a15τ+4Δ). Differentiating (V1V0) and (V2V1) with respect to τ yields:

(35)(V1V0)τ=(V2V1)τ=5Δ72<0.

Differentiating (V1V0) and (V2V1) with respect to Δ yields:

(36)(V1V0)Δ=24a15τ+22Δ216>0,(V2V1)Δ=24a15τ+8Δ216>0.

Combining the results in Proposition 3, we obtain Proposition 5. Q.E.D.

Proof of Proposition 6. The optimal timing of adoption by firm 2, t2, equalizes its incremental benefits from delaying adoption, c(t2), to the marginal cost of waiting, (πbπf)ert2. Thus, firm 2 adopts FMS at the same time in the preemption game as in the precommitment game (t2=t¯2). Taking the follower’s reaction into account, we specify the difference in the discounted sum of profits to the leader and follower as a function of the leader’s timing of adoption, t1, and define a new function G(t1):

(37)G(t1)=(πlπf)t1t2(x1)ertdtc(t1)+c(t2(x1)).

It can be shown that G(t1) is quasiconcave in t1 with a single peak and maximized at t¯¯1<t¯1, solving G(t¯¯1)/t1=0. In equilibrium both firms obtain the same level of discounted sum of profits. There is a unique timing of first adoption, t1<t¯¯1<t¯1, solving G(t1)=0 in eq. (37). We now define the function, f(t1,t2(x1))=r[c(t1)c(t2(x1))][ert1ert2(x1)]. We thus rewrite the equilibrium condition as:

(38)(πlπf)f(t1,t2(x1))=0.

The implicit function theorem indicates that the function t1(x1) has partial derivatives with respect to x1:

(39)t1x1=(πlπf)x1ft2t2x1ft1.

We differentiate the function f with respect to t1 to obtain:

(40)f(t1,t2)t1=rer(t1+t2)[ert1[c(t1)+rc(t1)](er(t2t1)1)+r[c(t1)ert1c(t2)ert2]][ert1ert2]2.

The conditions of er(t2t1)>1+r(t2t1), [c(t)ert]/t<0, and 2[c(t)ert]/t2>0 guarantee ert1[c(t1)+rc(t1)]<[c(t1)ert1c(t2)ert2]/[c(t1)ert1c(t2)ert2](t2t1)(t2t1) and:

(41)f(t1,t2)t1<r2er(t1+t2)[ert1[c(t1)+rc(t1)](t2t1)+[c(t1)ert1c(t2)ert2]][ert1ert2]2<0.

By similar procedures, we can show that f(t1,t2)/t2<0. From eq. (39) we note that πlπf=(Δ/36)(6a2τ+Δ) is constant in x1 and t2/x1>0, implying t1/x1<0.

If the innovation size is not too drastic, then there is also a joint adoption equilibrium in the preemption game. In this equilibrium, both firms adopt the new technology at the same time t1=t2=tm, and a firm’s discounted sums of profits are:

(42)M(tm,tm)=πo0tmertdt+πbtmertdtc(tm).

Solving the first-order condition for maximizing the profit of a firm yields:

(43)tm=1αln(c¯(α+r)πbπo),

where πbπo=(Δ/108)(6a2τ+Δ48τx12+24Δx12+64τx1332Δx13). Since (πbπo) decreases with x1, the timing of adoption tm is increasing in x1. Q.E.D.

References

Argenziano, R., and P. Schmidt-Dengler. 2012. “Inefficient Entry Order in Preemption Games.” Journal of Mathematical Economics 48: 445–60.10.1016/j.jmateco.2012.08.007Suche in Google Scholar

Argenziano, R., and P. Schmidt-Dengler. 2014. “Clustering in N-Player Preemption Games.” Journal of the European Economic Association 12: 368–96.10.1111/jeea.12054Suche in Google Scholar

Astebro, T. 2002. “Noncapital Investment Costs and the Adoption of CAD and CNC in US Metalworking Industries.” The RAND Journal of Economics 33: 672–88.10.2307/3087480Suche in Google Scholar

Boyer, M., and M. Moreaux. 1997. “Capacity Commitment Versus Flexibility.” Journal of Economics & Management Strategy 6: 347–76.10.1162/105864097567129Suche in Google Scholar

Dixon, H. D. 1994. “Inefficient Diversification in Multi-Market Oligopoly with Diseconomies of Scope.” Economica 61: 213–19.10.2307/2554958Suche in Google Scholar

Dutta, P. K., S. Lach, and A. Rustichini. 1995. “Better Late than Early: Vertical Differentiation in the Adoption of a New Technology.” Journal of Economics and Management Strategy 4: 563–89.10.3386/w4473Suche in Google Scholar

Eaton, B. C., and N. Schmitt. 1994. “Flexible Manufacturing and Market Structure.” American Economic Review 84: 875–88.Suche in Google Scholar

Fudenberg, D., and J. Tirole. 1985. “Preemption and Rent Equalization in the Adoption of a New Technology.” Review of Economic Studies 52: 383–401.10.2307/2297660Suche in Google Scholar

Gil-Molto, M. J., and J. Poyago-Theotoky. 2008. “Flexible Versus Dedicated Technology Adoption in the Presence of a Public Firm.” Southern Economic Journal 74: 997–1016.10.1002/j.2325-8012.2008.tb00877.xSuche in Google Scholar

Gotz, G. 2000. “Strategic Timing of Adoption of New Technologies Under Uncertainty: A Note.” International Journal of Industrial Organization 18: 369–79.10.1016/S0167-7187(98)00016-2Suche in Google Scholar

Goyal, M., and S. Netessine. 2007. “Strategic Technology Choice and Capacity Investment Under Demand Uncertainty.” Management Science 53: 192–207.10.1287/mnsc.1060.0611Suche in Google Scholar

Gupta, B., F. C. Lai, D. Pal, J. Sarkar, and C. M. Yu. 2004. “Where to Locate in a Circular City?” International Journal of Industrial Organization 22: 759–82.10.1016/j.ijindorg.2004.03.002Suche in Google Scholar

Hoppe, H. C., and U. Lehmann-Grube. 2001. “Second-Mover Advantages in Dynamic Quality Competition.” Journal of Economics and Management Strategy 10: 419–33.10.1162/105864001316908008Suche in Google Scholar

Hoppe, H. C., and U. Lehmann-Grube. 2005. “Innovation Timing Games: A General Framework with Applications.” Journal of Economic Theory 121: 30–50.10.1016/j.jet.2004.03.002Suche in Google Scholar

Hotelling, H. 1929. “Stability in Competition.” Economic Journal 39: 41–57.10.1007/978-1-4613-8905-7_4Suche in Google Scholar

Kim, T., L. H. Roller, M.M. Tombak. 1992. “Strategic Choice of Flexible Production Technologies and Welfare Implications: Addendum et Corrigendum.” The Journal of Industrial Economics 40: 233–35.10.2307/2950514Suche in Google Scholar

Lambertini, L. 2002. “Equilibrium Locations in a Spatial Model with Sequential Entry in Real Time.” Regional Science and Urban Economics 32: 47–58.10.1016/S0166-0462(01)00073-4Suche in Google Scholar

Matsumura, T., and D. Shimizu. 2015. “Endogenous Flexibility in the Flexible Manufacturing System.” Bulletin of Economic Research 67: 1–13.10.1111/j.1467-8586.2012.00458.xSuche in Google Scholar

Matsushima, N. 2001. “Cournot Competition and Spatial Agglomeration Revisited.” Economics Letters 73: 175–77.10.1016/S0165-1765(01)00481-5Suche in Google Scholar

Meza, S., and M. Tombak. 2009. “Endogenous Location Leadership.” International Journal of Industrial Organization 27: 687–707.10.1016/j.ijindorg.2009.03.001Suche in Google Scholar

Milliou, C., and E. Petrakis. 2011. “Timing of Technology Adoption and Product Market Competition.” International Journal of Industrial Organization 29: 513–23.10.1016/j.ijindorg.2010.10.003Suche in Google Scholar

Norman, G., and J. F. Thisse. 1999. “Technology Choice and Market Structure: Strategic Aspects of Flexible Manufacturing.” Journal of Industrial Economics 47: 345–72.10.1111/1467-6451.00104Suche in Google Scholar

Reinganum, J. 1981a. “On the Diffusion of New Technology: A Game-Theoretic Approach.” Review of Economic Studies 48: 395–405.10.2307/2297153Suche in Google Scholar

Reinganum, J. 1981b. “Market Structure and the Diffusion of New Technology.” Bell Journal of Economics 12: 618–24.10.2307/3003576Suche in Google Scholar

Riordan, M. H. 1992. “Regulation and Preemptive Technology Adoption.” Rand Journal of Economics 23: 334–49.10.2307/2555866Suche in Google Scholar

Riordan, M. H., and D. J. Salant. 1994. “Preemptive Adoptions of an Emerging Technology.” Journal of Industrial Economics 42: 247–61.10.2307/2950568Suche in Google Scholar

Roller, L. H., and M. M. Tombak. 1990. “Strategic Choice of Flexible Production Technologies and Welfare Implications.” The Journal of Industrial Economics 38: 417–31.10.2307/2098348Suche in Google Scholar

Salop, S. 1979. “Monopolistic Competition with Outside Goods.” Bell Journal of Economics 10: 141–56.10.2307/3003323Suche in Google Scholar

Shimizu, D. 2002. “Product Differentiation in Spatial Cournot Markets.” Economics Letters 76: 317–22.10.1016/S0165-1765(02)00060-5Suche in Google Scholar

Sun, C. H. 2017. “Timing of Entry and Product Location in a Linear Barbell Model: Application to Flexible Manufacturing Systems.” International Review of Economics & Finance 52: 29–38.10.1016/j.iref.2017.09.004Suche in Google Scholar

Ungern-Sternberg, T. 1988. “Monopolistic Competition and General Purpose Products.” The Review of Economic Studies 55: 231–46.10.2307/2297579Suche in Google Scholar

Upton, D. 1995. “What Really Makes Factories Flexible?” Harvard Business Review 73: 74–84.Suche in Google Scholar

Published Online: 2020-01-11

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 22.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/bejeap-2019-0094/html
Button zum nach oben scrollen