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.
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
Proof of Lemma 1. We define
Proof of Lemma 2. Denote
If
We conclude
Proof of Proposition 3. We calculate
which is a decreasing function of τ:
We solve
From eq. (19),
The function
which is decreasing in τ:
Solving for
We find that
The function thus reaches its minimum at
We note that eq. (21) is an increasing function of Δ:
We conclude
We note that eq. (21) is a decreasing function of x1:
Since
Proof of Proposition 4. We calculate the first-order derivative of the leader’s discounted sum of profits with respect to x1 as:
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:
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
where
which are both decreasing functions of τ. Differentiating
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:
We calculate social welfare’s first-order derivatives with respect to x1 and x2 as:
By symmetry of the model, we conclude that
where
Differentiating (
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,
It can be shown that G(t1) is quasiconcave in t1 with a single peak and maximized at
The implicit function theorem indicates that the function
We differentiate the function f with respect to t1 to obtain:
The conditions of
By similar procedures, we can show that
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
Solving the first-order condition for maximizing the profit of a firm yields:
where
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Artikel in diesem Heft
- Research-Articles
- The Wage Response to a Reduction in Income Tax Rates: The Israeli Tax Reform
- Employee Choice and the Demand for Health Insurance Coverage: Evidence from Random Coefficients Models
- Marathon, Hurdling, or Sprint? The Effects of Exam Scheduling on Academic Performance
- Sharing Competition: An Agent-Based Model for the Short-Term Accommodations Market
- Timing of Adopting a Flexible Manufacturing System and Product Differentiation
- Childcare Allowances and Public Pensions: Welfare and Demographic Effects in an Aging Japan
- The Crucial Role of Social Welfare Criteria and Individual Heterogeneity for Optimal Inheritance Taxation
- The Economics of Enlightenment: Time Value of Knowledge and the Net Present Value (NPV) of Knowledge Machines, A Proposed Approach Adapted from Finance
- The Consequences of the 1959–1961 Chinese Famine for Educational Attainment
- Investment in Green Technology and Entry Deterrence
- Analyzing the Effect of Mandatory Water Restrictions on Water Usage
- Acquisitions for Sleep
- Gender Differences in Performance and Risk-taking among Children, Teenagers, and College Students: Evidence from Jeopardy!
- Letters
- Accounting for Differences in Female Labor Force Participation between China and India
- Conflict Intensity and Birth Outcomes – Evidence from the West Bank
- Search and Bidding Costs