Prices and quantities

The buyers will wish to buy a quantity from sellers given the price per unit p which is proposed and are able to resell in their own local market where they face a demand function , which determines the relationship between the price they obtain and the quantity q that they bring to the local market. Let us suppose in order to simplify matters that is known by the buyers, is the same for all buyers and that it is a simple function of q such as:

The particular choice of the function is of no importance and is made to facilitate calculations and to provide clear benchmarks. For the model any monotonic decreasing function would suffice. The buyer's profit is then:

We suppose then that the buyer knows the demand curve he faces and is thus able to compute the quantity that will maximise his profit for a given price p proposed by the seller. This quantity is:

We make similar assumptions for the sellers, in particular that they know the behavior of buyers described by the three equations above, and they can therefore maximize their own profit per transaction:

with respect to the price p that they charge to the buyers, where is the price at which the sellers themselves purchase the fish. (Since the price p which maximises equation 4 is the solution of a third degree equation, its expression is rather complicated and not given here).

In order to simplify assumptions as much as possible, let us suppose that:

• All sellers propose a single price p to all their customers.
• Customers choose one shop everyday. As long as the shop has supplies, they are offered goods at price p; customers then purchase q given by equation 3. Whether they do get supplies when they visit the shop depends upon the time they visit the shop, which is randomly chosen in the simulations. It also depends on how much the sellers bring each day to the market.
• Each day sellers bring a fish quantity Q in relation to , the expected number of customers for that day. In the simplest version of the model, this fish quantity is , where is the number of customers who visited yesterday and a constant extra fraction brought to attract for the future extra customers that could show up.