Question

Production in the cities of Westeros is a function of the number of people in each city. If we denote the number of people in a city as N, the amount of private good that could be producted in that city will be N^1/2/N and the amount of the public good that could be producted in the city will be N^1/2. Citizens of Westeros all have a utility function of the form U=(pri)^1/2(pub)^1/2, where pri is the amount of the private good in the city and pub is the amount of public goods in the city. Find an equation for citizens utility as a function of the number of citizens in the city.

Answer 1

Consider the given problem here “N” be the number of people and numbers of private good produced is given by, “N^1/2/N = N^(-1/2)”. Similarly, numbers of public good produced is given by, “N^1/2”. Now, the utility function is a function of "numbers of private goods produce" and the "numbers of public good produced", is given below.

=> U = (Private)^1/2*(Public)^1/2 = (N^-1/2)^1/2*(N^1/2)^1/2 = N^-1/4*N^1/4 = N^0 = 1.

So, the required utility function is given by, “U(N) = 1”. So, the utility function is constant for values of “N”.