Investment with network externality under uncertainty

Lee et al. (2009) investigate the valuation of information technology investments by real options analysis. However, the feature of network-externality is not incorporated.

Many studies applying real option method to analyze the network-externality issues do not take consumer’s waiting-to-buy decision into account (Benaroch and Kauffman 1999; Balasubramanian et al. 2000; Courchane et al. 2002; Hori and Mizuno 2006).

For simplicity and analytical tractability, the jump model, such as the model proposed by Wu (2003), is not considered in this paper.

Because the discount rate is equal to the risk free rate r plus the hazard rate μ ₂ and according to Ito’s lemma, dN ² = 2NdN + (dN)² = (2μ + σ ²)N ² dt + 2σN ² dz, the assumption σ ² + 2μ − μ ₂ − r < 0 can ensure the convergence of the expected present value of the future profit H(N).

By the assumption σ ² + 2μ − μ ₂ − r < 0, it is straightforward to find C > 0. On the other hand, if μ < 0, μ − μ ₂ − r < 0. If μ ≥ 0, μ − μ ₂ − r < 0 since \( \mu - \mu_{2} - r < - (\sigma^{2} + \mu ) { < 0} \) by the assumption σ ² + 2μ − μ ₂ − r < 0. Therefore, D > 0.

It is straightforward to show that \( {\frac{\partial }{{\partial \sigma^{2} }}} ({\frac{{ 2\mu { - }\mu_{ 1} - \mu_{2} + \sigma^{2} - r}}{{ 2\mu - \mu_{2} + \sigma^{2} - r}}} )=\,{\frac{{\mu_{ 1} }}{{ ( 2\mu - \mu_{2} + \sigma^{2} - r )^{ 2} }}} > 0 \) and \( {\frac{\partial }{{\partial \sigma^{2} }}} ({\frac{\beta - 1}{\beta }} ) { = }\,{\frac{1}{{\beta^{ 2} }}}{\frac{\partial \beta }{{\partial \sigma^{2} }}} < 0. \)

For the case without consumer’s waiting effect, \( G(N) = a\,\left( {{\frac{{2\mu - \mu_{1} - \mu_{2} + \sigma^{2} - r}}{{2\,\mu - \mu_{2} + \sigma^{2} - r}}}} \right)\,N^{2}. \)

As the volatility is approaching to \( \sqrt {r - 2\mu + \mu_{2} } \), i.e., \( \beta \to 2. \) Under this situation, G(N) of the case without consumer’s waiting effect is almost twice as large as that of the case with consumer’s waiting effect. Moreover, under this situation, the difference ratio is the largest.

In an unreported result, we show that the model with risk averse is qualitatively similar to that with risk neutral. The conclusion in this paper is not affected by the assumption of consumers risk preference. We would like to thank an anonymous referee for this suggestion.

In this case, consumer and monopolist both use the NPV rule to make their decisions. Therefore, the increase in the network externality raises the consumer’s utilities gained by buying the product with network externality so that monopolist can charge a higher price. The total expected present value of the profit is denoted by GNPV, and GNPV is an increasing function of the number of consumers N and network externality θ. Using GNPV-I = 0, we can find the trigger network size of investment for the NPV rule. It is also found that \( {{\partial N_{npv}^{*} } \mathord{\left/ {\vphantom {{\partial N_{npv}^{*} } {\partial \theta }}} \right. \kern-\nulldelimiterspace} {\partial \theta }} = {{ - {\tfrac{\partial (GNPV)}{\partial \theta }}} \mathord{\left/ {\vphantom {{ - {\tfrac{\partial (GNPV)}{\partial \theta }}} {{\tfrac{\partial (GNPV)}{\partial N}}}}} \right. \kern-\nulldelimiterspace} {{\tfrac{\partial (GNPV)}{\partial N}}}}, \) where \( N_{npv}^{*} \) here is the trigger network size of investment for the NPV rule. Thus we have \( {{\partial N_{npv}^{*} } \mathord{\left/ {\vphantom {{\partial N_{npv}^{*} } {\partial \theta }}} \right. \kern-\nulldelimiterspace} {\partial \theta }} < 0. \) As a result, the relationship between the network externality and trigger network size of investment is always negative when the NPV rule is used.