Question

the harrisons live in a country where they are the only business allowed to supply concumers with leather shoes, what type of economy is this an example of

Answer 1

This economy would be a rough example of a "controlled economy", since the government seems to be making it illegal for another other business to compete with this existing company. 

Answer 2

In the economy described there is a certain economic authority that governs what is happening in a certain industrial sector, by giving permission to only one producer to supply the whole demand of leather shoes.

It is an example of a command economy, that is governed by centrally-planned schedules enacted by the authorities instead of by the forces of supply and demand and the free interactions of economic agents in the markets (which is the manner in which economic decisions are reached in market economies).