Question

7. Consider a nation with a fixed population of 12 million that is initially divided equally between two cities, Circusville and Dullsville. The urban utility curve is u(w) = 15 + 12w − w2. Suppose a dictator in Circusville initiates free circuses, financed by coercive transfer payments from people outside the region. The circus program initially increases the utility of living in Circusville by 3 utils. Illustrate the effects of the circuses on the regional economy, including values for (i) city sizes and (ii) regional utility.

Answer 1

Answer:

We have given information as follows:

A nation has 12 million of population divided equally between two cities Circusville (C) and Dullsville (D)

it means C = 6 mn and D = 6 mn

The urban utility curve is u(w)= 15 + 12w -w²

Circusville initiates free circuses financed by coercive transfer payments from people outside the region

The circus program initially increases the utility of living in Circusville by 3 utils

We have utility function u(w)= 15+12w-w² --------------------(1)

We will solve this equation for i) city sizes and ii) regional utility

Differentiating equation with respect to w to get maximization,

du(w)/dw= 12-2w=0 ----------------------(2)

2w=12

w=6

Taking second order derivative of equation (2) for maximum we get,

d²u(w)/d²w = -2

The circus program initially increases the utility of living in Circusville by 3 utils

Putting this value in original equation we get,

u(w) = 15+12*3-3²

= 51-9

= 42

du(w)/dw= 12-2w

= 12-2*3

= 12-6

= 6

After increase in utility in C by 3 regional utility remains same for both the cities