# Hotelling’s law

The previous Market Simulation (MO-121 Hot Beach) explored what would happen to the Profit Maximizing Price of water for Vendors distributed evenly along a hot beach. This previous example is a variation of the Hotelling Law experiments designed to find the ideal location for Vendors to sell their Products.

According to Wikipedia, Hotelling’s Law predicts that a street with two Vendors will also find both Vendors right next to each other at the same halfway point. Each Vendor will serve half the market; one will draw customers from the west (left-hand-side), the other all customers from the east (right-hand-side).

Another example of the law in action or practice is to think of two food pushcarts at a beach. Assume one starts at the south end of the beach and one starts at the north. Again assuming a rational consumer and equal distribution along the beach, each cart will get 50% of the customers, divided along an invisible line equidistant from the carts. But, each cart owner will be tempted to push his cart slightly towards the other, in order to move the invisible line so that it encompasses more than 50% of the beach. Eventually, the pushcart operators end up next to each other in the center of the beach.

Hotelling’s Law has been well explored for the 2-Vendor case. But economists have not yet found a generalized solution for more than 2-Vendors, such as the 3-Vendor case or the 4-Vendor case. Furthermore, the models that do exist assume that each Vendor will set the same fixed Price, and that the Vendors do not attempt to alter Prices in the face of (sometimes fierce) competition.

This Market Simulation is designed to provide a generalized method to predict how Vendors may shift along the beach, while simultaneously changing Price, in order to find a Profit Maximizing Location.

The Market Simulation starts with 4-Vendors. They are spread out, but all start nearer to the beginning of the beach. At each iteration, the Vendors engage in intense Price Competition (a ‘Price War’). The Vendors also test whether conditions would improve if they were to iteratively shift either up or down the beach.

This Case Study provides provides a glimpse into the premium Market Optimization nodes. These premium nodes are not available in the Free Community Edition of Scientific Strategy. But a selection of problems that can be solved with the Free Community Edition nodes can be found in Case Studies and Market Simulation.

# #1 Shift Vendor Location

In the beginning, the Vendors are all located near the beginning of the beach:

• Vendor A is located 40% along the beach
• Vendor B is located 30% from the beginning
• Vendor C is located at 20%
• Vendor D is located nearest the beginning at 10% along the beach

There are two loops in this workflow.

The outer-loop iterates between each of the Vendors. Vendor_A first tries shifting up and down the beach to find a more Profitable location. Then Vendor_B tries, then Vendor_C, and finally Vendor_D. When all Vendors have made a decision as to whether or not to shift, the process repeats, starting again with Vendor_A.

The inner-loop tests how much each Vendor should shift. The beach starts at 0% and extends out to 100%. The Vendors each try shifting:

• -2% back towards the beginning of the beach
• 0% staying where they currently are
• +2% forward towards the far end of the beach

The walking length of the beach is set in the same way as the previous workflow (MO-121 Hot Beach) and can be adjusted so more or less Customers find it convenient to buy from each Vendor.

# #2 Generate Market

The new market conditions are generated in very much the same way as the previous workflow (MO-121 Hot Beach). Each Customer’s Willingness To Pay (WTP) is proportional to their distance from each Vendor. Customers are evenly distributed along the beach, but their locations can be changed in the workflow. Water is a commodity, and the value each Customer places upon the water itself is fixed, but again the workflow provides the flexibility to change the shape of this Customer Distribution.

Once the WTP of each Customer has been calculated with respect to each Product, the Vendors engage in a Price War. Each Vendor’s goal from the Price War is to change Price until they maximize their own Profitability.

The next Vendor then starts experimenting with shifting their location. The inner-loop runs three times to test:

• shifting -2% back towards the beginning of the beach
• 0% staying where they currently are
• shifting +2% further along the beach

Price Wars are run at each test location, and the Vendor selects the location that maximizes their Profitability.

# #3 Round Robin

The outer-loop iterates 120 times, with each Vendor taking their turn in round-robin fashion to experiment with adjusting location. Hence each Vendor has 120 / 4 = 30 iterations in which to try shifting location.

The final results demonstrate that, while Vendors generally shift down the beach, they do not come to rest at an obvious symmetrical balance point (such as 20%, 40%, 60%, and 80%). Instead, the Vendors settle down at these locations:

• Vendor A shifts from 40% and settles at 84%
• Vendor B shifts from 30% to 54%
• Vendor C shifts from 20% to 24%
• Vendor D shifts from 10% to only 16%

The Vendors with plenty of room to move (Vendors A and B) generally enjoy much greater Market Share and Profitability. While the Vendors who start out more constrained (Vendors C and D) don’t shift very far from their starting points. Their markets remain squeezed, and they do not gain large Market Shares nor Profitability.