This is a study of the economic behavior of vendors of service in competition. A simple model with two competing exponential servers and Poisson arrivals is considered. Each server is free to choose his own service rate at a cost (per time unit) that is strictly convex and increasing. There is a fixed reward to a server for each customer that he serves. The model is designed to study one specific aspect of competition. Namely, competition in speed of service as a means for capturing a larger market share in order to maximize long run expected profit per time unit. A two person strategic game is formulated an its solutions are characterized. Depending on the revenue per customer served and on the cost of maintaining service rates, the following three situations may arise. (i) a unique symmetric strategic (Nash) equilibrium in which expected waiting time is infinite; (ii) a unique symmetric strategic equilibrium in which expected waiting time is finite; and (iii) several, non symmetric strategic equilibria with infinite expected waiting time. An explicit expression for the market share of each server as a function of the service rates of the two servers is also given.