Consider a situation where there are two goods:XandG. There are two individuals: 1 and 2. Each individual privately consumes x₁and x₂of goodX, respectively. In contrast, they both consumeGamount of the second good. The price of goodXisp, and each individual makes an incomem₁andm₂respectively.

a. Suppose each individual contributes a different share to the aggregate goodG. The share they contribute ist₁of totalGfor individual 1, andt₂of totalGfor individual 2. Write their budget constraints.

b. Using the general utility functionsu₁(x_1,G) andu₂(x_1,G), where the budget constraints are given above, state and describe the first order conditions for each individual.

c. Suppose each individual could choose their own optimal level ofG, so that individual 1 consumes an amountG₁and individual 2 consumes an amountG₂. Using the utility functions:

u₁(x_1,G₁) = a₁ln(G₁) + ln x₁

u₂(x_2,G₂) = a₂ln(G_{2) +}ln x₂

Solve for each consumer's optimal choice ofG.

d. Suppose we impose the constraint thatGmust be equal to the sum of the two individuals' contributions toG. Solve for the optimal level ofG.

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