DEMAND CURVES and ENGLE CURVES: Consider the following consumer's problem: U(X,Y) = X1/2 Y1/2 . Suppose that she has income (I) and faces prices Px and Py. a) Sketch the budget set. What is the slope of the Budget Line? What are maximal possible consumptions of X and Y? (note: these will depend on what values of I, Px and Py are given). b) Show that the MRSXY = Y/X. c) Show that the consumer will spend of her income on Good X and on Good Y. d) Now, sketch the Demand Curve for X and then for good Y. Show that both are downward sloping. e) Sketch the Engel Curve for X and then for good Y. Show that both are upward sloping. f) Suppose income (I) is $1000, Px = $6 and Py = $4. Find the optimal consumption bundle. g) At the optimum, what is the Demand Elasticity for good x? recall: demand elasticity = [X/Px][Px/X]. Interpret the value you calculate. h) At the optimum, what is the Income Elasticity for good x? recall: Income elasticity = [X/I][I/X]. Interpret the value you calculate. Q2: (20 points) SUBSTITUTION and INCOME EFFECTS: Suppose we are given the following utility function for a consumer: U(X,Y) = X1/2y1/2 : Suppose also that her income (I) is $1000, Px = $6 and Py = $4. a) Find the consumer's optimal choice given the prices and income above. What is the utility she derives from this income? b) Find the new optimum if Py falls to $3. c) Show that the income required to just make the previous utility from (a) attainable with Px = $6 and Py = $3 is $866.03. Show and explain the process you use to get this result. (Eg. you have the answer so just show the steps to get there.) d) Given the "new" income in (c) with Px = $6 and Py = $3, find the new optimum. Confirm that it yields the same utility as in (a). e) What are the Hicks Substitution and Income Effects of the fall in the price of y? eg find X and Y. f) What is the Compensating Variation for the fall in Py? Explain your reasoning. g) Show that the income required to just make the new utility in (b) attainable at the old prices (Px = $6 and Py = $4) is $1154.70. Show and explain the process to get this result. h) What is the Equivalent Variation for the fall in Py? Explain your reasoning