We introduce the idea and possibility of loan in a modified Dragulescu–Yakovenko model in econophysics [1]. This is the implementation of the discrete version of the modified Z-model introduced by Pomeau and Lopez-Ruiz in 2015 [2]. This new model is equipped with a parameter that provides an idea of the level of debt that the agents can reach when they trade in a free market. When this parameter is adequately modified, the total debt in the system increases and a financial bubble is created. Once a certain limit is surpassed, the bubble is oversized and the financial system becomes unstable. Simulations and calculations are displayed for this system and they are put in correlation with real data. It is shown that a critical value in the system is found when the bubble, money created in relation to real money, reaches the factor of 5.18, a value in good agreement with the real data of bubbles formed in Western countries during the 2008 financial crisis.

[1] Dragulescu A and Yakovenko VM. Statistical mechanics of money. The European Physical Journal B, 17:723-729, 2000.

[2] Pomeau Y and López-Ruiz R. Study of a model for the distribution of wealth. In Lopez-Ruiz, Fournier-Prunaret, Nishio and Gracio, editors, Nonlinear Maps and Their Applications, vol. 112, p. 1-12, Springer Int Publishing, 2015.