1.There are two agents with utility functions
u1(x1,x2)=−(x1−1)2+x2; u2(x1,x2)=−(x2−1)2+x1, where agent i controls variable x(i),i=1,2.
Calculate the symmetric (involving equal values of x’s) Pareto optimum.

2.Firm 1 produces
x units of its output which it sells at price p=12 per unit. This firm’s own production cost is cown(x)=2x2. The firm also imposes external costs cext(x)=x2 on the neighboring firm 2. What is the highest level of the total transaction costs that will still enable the firms to settle the externality through Coasean bargaining?


3.If in the setup of the previous problem the government imposes a Pigouvian tax on firm 1 to control the externality, what should be the tax rate to ensure social optimality?


4.Suppose that there are
n individuals, each with sufficiently high monetary endowment w. Each individual has the same utility function u(xi,G)=xi+2G−−√, where xi=w−gi is her private consumption (wealth net of the contribution gi towards the provision of public good), and G=∑ni=1gi is the public good.
If all agents make identical voluntary contributions towards the provision of the public good, how will these contributions be affected by the growing number of agents. (Select one option)
They will be declining.
They will be increasing.
None of the above.


5In the rent-seeking model from the lecture assume
F(x)=x√ and calculate levels of x (resources contributed towards production) for the following values of θ: 0.5; 0.25. (Enter two numbers separated by space and rounded to the nearest hundredth; use round half up tie-breaking rule)