Abstract
This research analyzes a firm’s timing of bringing a new product to the market, based on a product-innovation timing game in which the quality of a new product increases over time. We explore the equilibrium outcomes when both firms can precommit to their timing of market entry (the so-called precommitment game), when they cannot credibly precommit to their timing of market entry (the so-called preemption game), and when only one firm can precommit to its timing of market entry (the so-called mixed precommitment game). Sequential entry is the unique equilibrium outcome in all three games. We finally discuss two extended timing games with endogenous commitment choices. The equilibrium involves a precommitment subgame in the endogenous precommitment game with observable delay, while mixed precommitment subgames appear to be the equilibria in undominated strategies in the endogenous precommitment game with action commitment.
Funding source: Ministry of Science and Technology
Award Identifier / Grant number: MOST 107-2410-H-031-014
Proof of Lemma 1.
Firm 1 chooses t 1 to maximize its discounted sums of profits, taking firm 2’s choice as given. The first-order condition for profit maximization is calculated as:
Substituting the first-order condition for profit maximization into the second-order derivative of the leader’s profit yields:
where
Substituting the first-order condition for profit maximization into the second-order derivative of the follower’s profit yields:
where
where
Proof of Lemma 2.
Differentiating L(t 1, t 2) and F(t 2, t 1) with respect to t 2 and t 1 yields:
The cross-partial derivatives of L(t 1, t 2) and F(t 2, t 1) are calculated as:
where
Proof of Lemma 4.
When t 1 ≤ t 2, the slope of the leader’s best response on the plane (t 1 − t 2) is:
The slope of the follower’s best response is:
Therefore,
where
If t
2 > t
1, then h(t
1, t
2) > 0, implying
Proof of Corollary 1.
Suppose that t 1 ≤ t 2. If the R&D cost is s(t) = t, then:
where
Proof of Lemma 5.
We note L(t
1, g
f
(t
1)) is continuous in t
1 ∈ [0, d] and that F(t
1, g
l
(t
1)) is continuous in
Proof of Proposition 2.
(1) If the commitment firm is the leader
(2) If the commitment firm is the follower
Proof of Corollary 2.
Because
Proof of Proposition 5.
We first consider the equilibrium strategy in the precommitment game
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Articles in the same Issue
- Frontmatter
- Research Articles
- FRAND Licensing of Standard-Essential Patents: Comparing Realistic Ex-Ante and Ex-Post Contracts
- To Commit or Not to Commit in Product-Innovation Timing Games
- Coordinated Minimum Wage Policies: Impacts on EU-Level Income Inequality
- Regulatory Contestability and Cost Pass-Through
- Explaining the Economic Characteristics of Different International Peacekeeping Institutions
- Setting Reserve Prices in Repeated Procurement Auctions
- Public and Private School Grade Inflation Patterns in Secondary Education
- Estimating Labor Supply Elasticities in Korea: The Role of Limited Commitment Between Spouses
- Strategic Brand Proliferation: Monopoly versus Duopoly
- Letter
- Parental Investments During Labor Shocks: Evidence from Vietnam’s Marine Disaster
Articles in the same Issue
- Frontmatter
- Research Articles
- FRAND Licensing of Standard-Essential Patents: Comparing Realistic Ex-Ante and Ex-Post Contracts
- To Commit or Not to Commit in Product-Innovation Timing Games
- Coordinated Minimum Wage Policies: Impacts on EU-Level Income Inequality
- Regulatory Contestability and Cost Pass-Through
- Explaining the Economic Characteristics of Different International Peacekeeping Institutions
- Setting Reserve Prices in Repeated Procurement Auctions
- Public and Private School Grade Inflation Patterns in Secondary Education
- Estimating Labor Supply Elasticities in Korea: The Role of Limited Commitment Between Spouses
- Strategic Brand Proliferation: Monopoly versus Duopoly
- Letter
- Parental Investments During Labor Shocks: Evidence from Vietnam’s Marine Disaster