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(x₁,G) andu₂(x₂,G), where the budget constraints are given above, what are the first order conditions for each individual.