The aim of this work is to demonstrate analytically under what conditions activating elasticity of demand of consumers could be beneficial for the social welfare. It has added to the literature on analyzing the use of price signals in eliciting demand response by an analytical approach. We develop so an analytical Nash model to quantify the effect of implementing demand response, via price signals, on social welfare and energy exchanges. A prior results show that the trade-off between producing locally and exporting energy depends on the opportunity cost of the energy and the global efficiency of the generation technology. Results are moreover impacted by the degree of integration between the countries. The novelty of this research is the demonstration of the existence of an optimal region of price signal for which demand response leads to increase the social welfare. This optimality region is negatively correlated to the degree of competitiveness of the generation technologies and to the market size of the system. We particularly notice that the value of un-served energy or energy reduction the producers could lose from such demand response program would limit the effectiveness of its implementation. This constraint is strengthened when energy exchanges between countries are limited. Finally, we demonstrate that when we only consider the impact in term of consumers' surplus, more aggressive DR could be adopted. The intensity of DR program is however negatively correlated to the degree of the elasticity of demand.