1. Consider a situation where there are two goods: x and G. There are two individuals: 1 and 2. Each individual privately consumes x₁ and x₂ of good x, respectively. In contrast, they both consume G amount of the second good. The price of good x is p, and each individual makes an income m₁ and m₂ respectively.

a. Suppose each individual contributes a different share to the aggregate good G. The share they contribute is t₁ of total G for individual 1, and t₂ of total G for individual 2. Write their budget constraints.

b. Using the general utility functions u₁(x₁, G) and u₂(x₁, G), where the budget constraints are given above, explain and describe the first order conditions for each individual.

c. Suppose each individual could choose their own optimal level of G, so that individual 1 consumes an amount G₁ and individual 2 consumes an amount G₂. Using the utility functions:

u₁ (x₁,G₁₎ = a₁ ln(G₁) + ln x₁

u₂(x₂, G₂) = a₂ ln(G₂) + ln x₂

Solve for each consumer's optimal choice of G.

d. Suppose we impose the constraint that G must be equal to the sum of the two individuals' contributions to G Solve for the optimal level of G.

e. Suppose in a society you had two individuals who wanted different quantities of a common good G so that G₁ > G₂. How would you determine what value of G you would choose? Explain the logic behind your answer.