Abstract: This article proposes a mathematical model to explore the relationship between customer loyalty and customer profitability deeply.By researching the function property of the mathematical model,it is demonstrated that the impact of customer loyalty on profitability is non-monotonic.Only if the cost of cultivating loyalty is fixed,profitability increases monotonically with the size of customer loyalty.Otherwise,there is a threshold which lets profitability increase with the size of customer loyalty when θ∈[0,θ^*) and decrease when θ∈[θ^*,1].In addition,the paper discusses how the cost rate of cultivating loyalty,purchase arrival rate and monetary at each time affect the size of θ^*.Finally,some management suggestions are given according to the findings.

Key words: customer loyalty, profitability, customer relationship management