Question

There is a walled city that is circular with radius R. Assume that when the population if this city is N, the benefit per person is ((π(R^2))/N)^2. The cost of defending is given by (b2πR)/N. Walls for defense were already built a long time ago, so people in the city do not have to bear the cost of building walls. Since there is a risk of epidemic, its cost is (C(N/πr)^2). Because this city still has much capacity, the government can control the population of the city by accepting immigrants or increasing birth rates.

a. Describe the net social benefit (NSB) that the government wants to maximize.

b. Find the optimal population.

Answer 1

In Europe the custom of the walled city continued as evidenced by sites such as the Oppidum of Manching (located near modern-day Ingolstadt, Germany) which was a 3rd century BCE Celtic community of the Vindelici tribe. The Roman city of Lugo in Galicia, Spain was surrounded by enormously thick walls considered utterly impregnable. The most famous wall of antiquity in Europe, however, is Hadrian's Wall in Britain.

The Roman Emperor Hadrian (r. 117-138 CE) grew tired of incursions into the Roman provinces in Britain and so, in the year 122 CE, began building a wall across the northern border of Roman Britain to separate it from the invading Caledonian tribes much in the same way that Shulgi built his wall almost two thousand years earlier to keep out the Amorites (as with the Great Wall of China and the Anastasian Wall). It took six years to build, stretched for 80 miles (128 kilometres) across the land, and was, at points, over nine feet wide (2.7 metres) and twenty feet (6 metres) high. It was fortified by towers along the way and served as a symbol of Roman military might and power