Timing Advantage: Leader/ Follower Value Functions If the Market Share Follows a Birth and Death Process

For a duopoly environment, we model the leader and follower real value functions assuming that the leader’s “market share” evolves according to an immigration (birth) and death process. We derive analytical solutions for the follower and leader options to invest, and numerical solutions for the leader’s optimal investment timing. Then we calculate the partial derivatives of the leader and follower value functions to market share, birth/death parameters, and market profitability. This model is possibly more realistic than that proposed by some other authors studying the advantages of being first (and also being a follower). We show that over certain ranges of the parameter values, the leader and follower real options to wait to invest, and not to wait to invest, are sometimes surprising, but possibly on refection plausible. The follower’s value function is usually less sensitive (and of opposite sign) than the leader’s value function to market share or the rate of customer arrivals/departures until the expected revenue exceeds the follower’s trigger investment level, but the sensitivity is dependent on the relative parameters, particularly the revenue/trigger. The follower’s trigger increases with market share, the immigration-death ratio and revenue volatility. The leader’s value function “deltas” are highly sensitive and unstable as revenues approach the follower’s trigger, confirming the adage, if you’re ahead, “watch the competition”.