The profit maximization model formulation Essay

Profit maximized model formulation
We consider a customer's choice of decision to buy a product depends on following factors
1. Current and future price of the product.
2. Valuation for the product.
3. Loyalty reward value.
4. Possibility to rebuy in the next period.
As assumed in extant literature, we consider the operation of loyalty program in two periods. A customer will earn the reward if he/she makes a purchase in both two periods. Similarly to Singh et al and Gandomi et al. we assume that only a certain proportion of first period buyers proceed to the second period. This proportion is modeled as a parameter, . So is the probability that a first buyer become a potential buyer in the second period. As a result. is the probability of buyers in period 1 fails to proceed to period 2.
Let surplus, S1 be the factors that influence the customers' decision to buy a product in the first period. A customer's surplus from purchasing a product is the value he/she will gain by obtaining the product subtracted by the amount he/she has to pay for it. If a customer gets a nonnegative surplus, he/she will make a purchase. So, the equation for customer's surplus from buying a unit of product in period 1 is given by
Where
= valuation of the customer for the product
(i=1, 2) = offer price in period i
= loyalty reward value
is the sum of the first period surplus and the expected surplus from buying the product in period 2 since customer are assumed to be strategic. The expected surplus in period 2 is the surplus value multiplied by the probability of proceeding to the second period.
A customer buys if the firm's offer gives him/her a nonnegative surplus. Thus the probability that a customer makes a purchase in period 1 is
Where F (.) denotes the CDF of the customers' valuation in the first period. Customers who buy in period 1 will proceed to the second period with the probability of . For a given customer in period 2, the utility derived from making a purchase is
Where is the valuation of the customer in the second period. Here and are neither equal nor independent. To model this dependency, we assume for each customer. This implies that the past purchase experience leads to a shift size in the valuation of the customer. For a satisfied customer is nonnegative and for dissatisfied customer is negative. Therefore we infer that and are dependent. Thus, the probability that a first period buyer makes a purchase in period 2 is conditioned on the fact that his/her surplus in period 1 has been nonnegative. The equation is given by,
But
Thus, we get
By applying the conditional probability theorem, eqn above can be expressed as following piecewise function
Now, based on the firms's profit from selling a product and customers' probabilities of buying in each period , we can find the firm's expected total profit function as follows