1. A "coffee shop" serves the entire market of consumers on the main street of Ventura, CA. The shop earns total profits of (N, n) = N[V - (t/2n) - c] - nF, where N is the number of consumers spaced evenly along the (one-mile) street, n is the number of locations, V is the consumers reservation price, t is the transport cost per unit of distance travelled, c is the cost of supplying a single homogenous cup of coffee, and F is the fixed cost of setting up a new location.

(a) Use the profit function to carefully derive the (discrete) condition that shows when it is profitable for the coffee shop to add an extra location.

(b) Use your condition to formally explain how a decrease in the number of consumers (N) impacts the optimal number of locations (n).