Consider an economy in which there are 50 people, who consume a private good (bread) and a public good (circuses). There are, however, two types of people (type 1 and type 2). Each type 1 person's marginal rate of substitution is
where z is the number of circuses. Each type 2 person's marginal rate of substitution is
There are 20 type 1 people and 30 type 2 people. The economy's marginal rate of transformation is
Find the optimal number of circuses.