Elle's preferences over apples (A) and bananas (B) are represented by the Cobb-Douglas utility function; u(A; B) = AB. (a) Write down the optimization problem. What is the objective function? What are the choice variables? What is the constraint? (b) What are the optimality conditions? Explain. (c) Derive her individual demand curves for apples and bananas as functions of the prices and income. (d) What is her optimal bundle if her income is $100, and bananas are $1 a pound and apples are $2 a pound. (e) What is her new optimal bundle if price of apples go down by 1 dollar? What is the substitution effect of this price change? What is the income effect of this price change?