Imagine there are two students who are deciding how to spend their time between consuming a private good, studying alone (A_i), and a public good, studying together (T). Each student has utility function U_i (A_i, T) = 2log (A_i) + log (T). This utility depends on private consumption of studying alone and consumption of the total amount of studying together, which is the sum of how much time each student invests in studying together: T₁ + T₂ = T. Each student has 100 hours to allocate (i.e. A_i + T_i = 100).

1. What will be the equilibrium levels of A_i and T in this market if the students make private decisions?
2. What is the socially optimal. equilibrium in this market?
3. Will studying together be under-provided, over-provided, or provided at the right level in the equilibrium in which the students make private decisions? Explain.