(10 pts) Consider that there are two goods, good X and good Y, which are perfect substitutes. The consumer has the utility function given by: U(X,Y)=X+aY The prices of the goods are given by pX=3 and pY=6, respectively. The income of the consumer is given by I=100. a) Choose a value for parameter " a " by yourself. Graphically illustrate the budget line of the consumer and draw some example indifference curves. Then, find the consumption bundle that maximizes the consumer's utility. b) List the possible outcomes for different values of parameter " a ". (10pts) Suppose that there are three goods, good X, good Y, and good Z, and suppose that the consumer has the utility function denoted by U(X,Y,Z) where X,Y,Z denote the quantities of goods. The prices of the goods are denoted by px,py, and pz, respectively, and the income of the consumer is denoted by I. a. Write down the consumer's optimization problem (formally, as we wrote in the lectures). b. Apply the Lagrangian method. Find the necessary conditions (NC) for the consumer's optimization problem of allocating budget between consumption of three goods (good X, good Y, good Z). c. Provide at least two interpretations of the NC (explain what these conditions mean in words). d. What if there were n goods listed as {X1,X2,,Xn} ? What would be the necessary condition for the optimal quantities of any pair of goods Xi and Xj