Consider a 2 person 2 good economy where there is a private good $x$ and a public good $y$. Agent 1 has an endowment of 10 units of the private good and Agent 2 has an endowment of 20 units of the private good. Initially, there is no public good in the economy. In order to produce $y$ units of the public good, $y^2$ units of the private good should be used. That is, the cost function is $c(y)=y^2$.
Utility functions of the agents are as follows;
$$u_1(x_1,y)=x_1+y$$ $$u_2(x_2,y)=x_2y$$
Firstly, I need to find the Pareto efficient allocations where 4 units of public good are produced.
Secondly, I need to find the private consumption of both agents as well as the public good level at the equilibrium.
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In other to find the Pareto efficient allocations, I maximize the sum of payoffs of both agents
$$max\{ u_1(x_1,y)+ u_2(x_2,y)-c(y)\}$$
First order condition with respect to $y$ is $6+x_2-2y=0$
So, I found $x_2=2\times 4-6=2$
But my attempt is not true. I cannot do the correct solution. I will appreciate if you help me to solve the parts of this question.