Reduction of fuel consumption is one of the primary goals of modern automotive engineering. While in the past the focus was on more efficient engine design and control there is an upcoming interest on economic context aware control of the complete vehicle. Technical progress will enable future vehicles to interact with other traffic participants and the surrounding infrastructure, collecting information which allow for reduction of fuel consumption by predictive vehicle control strategies. The principle of Model Predictive Control allows a straightforward integration of e.g. navigation systems, on-board radar sensors, V2V- and V2I-communication whilst regarding constraints and dynamic of the system. This paper presents a Linear Model Predictive Control approach to Cooperative Adaptive Cruise Control, directly minimizing the fuel consumption rather than the acceleration of the vehicle. To this end the nonlinear static fuel consumption map of the internal combustion engine is included into the control design by a piecewise quadratic approximation. Inclusion of a linear spacing policy prevents rear end collisions. Simulation results demonstrate the fuel and road capacity benefits, for a single vehicle and for a string of vehicles, equipped with the proposed control, in comparison to vehicles operated by a non-cooperative adaptive cruise control. Full information on the speed prediction of the predecessor is assumed, hence the purpose of this paper is twofold. On the one hand, best achievable benefits, of the proposed control, due to perfect prediction are demonstrated. On the other hand, the paper studies the behavior of the considered control and the influence of the prediction horizon.