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2021 | OriginalPaper | Buchkapitel

10. Delaying Product Introduction in a Duopoly: A Strategic Dynamic Analysis

verfasst von : Herbert Dawid, Serhat Gezer

Erschienen in: Dynamic Economic Problems with Regime Switches

Verlag: Springer International Publishing

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Abstract

We analyze the optimal timing for the introduction of a new product in a duopoly. Two incumbent firms are active on a homogeneous product market and one of these firms has an option to additionally introduce a new product, thereby incurring costs of product adoption. We assume that the innovator can commit on the time of product introduction and numerically derive the optimal introduction time as well as the associated Markov-perfect equilibria for investment in production capacities. We find that depending on the initial capacities for the established product and the size of the adoption costs, three scenarios are possible for the innovator: innovating immediately, delaying introduction, and abstaining from product introduction. In case of delayed introduction, the innovator strategically reduces capacities on the established market prior to product introduction, whereas the dynamics of the non-innovator’s capacity is ambiguous. Furthermore, in this case, the firm commits to a market introduction time such that at the time of market introduction it has incentives to further delay the product adoption.

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Fußnoten
1
Several studies [1, 2, 28] have found out that a large fraction of product innovations is not brought to the market.
 
2
Since we use a quadratic, rather than a linear quadratic, cost function we have implicitly normalized the price of capital, both when buying and selling, to zero. Having a positive price of capital would not qualitatively change our results, as long as the price is not too large.
 
3
When solving the investment problem in mode \(m_2\) the non-negativity constraint for the capacities is not explicitly included in the optimization problem. Rather, we have verified ex-post that the constraint holds under the optimal investment strategy. A similar statement also applies to our solution of the problem in mode \(m_1\).
 
4
Actually, H(K) is the Hamiltonian where the co-state variable is replaced by the state derivatives of the scrap value (cf. Pontryagin’s maximum principle with finite time horizon e.g. in [13]).
 
5
Note that for higher choices of \(\epsilon \), inequality (10.25) might be satisfied for all T and hence would not yield a large T.
 
6
The idea of considering large values for the terminal time has been employed by several works, e.g. in [17].
 
7
The value functions of immediate and no switching intersect at a point where the slopes of the value functions are very different and hence there is a kink. As usual in endogenous timing problems the option of delaying ’smoothes’ the value function of firm A.
 
8
In the literature, such indifference points or curves, separating different basins of attraction of the dynamics under optimal investments, are called Skiba points (curves), see e.g.  [20, 32]. For a discussion of issues related to the existence of Skiba points under Markov Perfect Equilibria of differential games see [7]. In general, the value functions have a kink at such Skiba points since, depending on whether firm A plans to introduce the new product or not, equilibrium investments are different. This means that equilibrium investment strategies exhibit jumps at this point in the state space and accordingly the value functions exhibit a kink. In Fig. 10.6 below, the Skiba curve can be seen as the line separating the area with \(T = \infty \) from those with \(T=0\) respectively \(T < \infty \).
 
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Metadaten
Titel
Delaying Product Introduction in a Duopoly: A Strategic Dynamic Analysis
verfasst von
Herbert Dawid
Serhat Gezer
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-54576-5_10