1. Mary spends all of her income on goods X and Y. She has a monthly income of 8, and the prices of the goods are PX = 4/9 and FY = 1. Her utility function is U(X,Y) = XY. a. Calculate Mary's optimal consumption bundle, and the utility level she attains at this bundle. Call this point (X0, Y0). What happens to Mary's demand for good X if the price of X increases to PX = 1? Call the new optimum (X1,Y1). Now decompose the change in demand for X into the substitution and income effects. To do this, assume that the price of X changes to PX = 1, and at the saine time Mary can attain exactly the same utility level as she did in part a). Note: this is the Hicks substitution effect. What are the values of X and Y such that Mary attains exactly the same level of utility as she did in part a), but the slope of the indifference curve is now equal to the new price ratio PX/PY = 1? Call the point that you found in part c) (Xg,YS). Then, the total change in demand for X, X1 X0 can be decomposed as follows: X1_XU=(X1_XS)+(XS_XD)- Which part is the substitution effect? Which part is the income effect? Draw a graph with the budget constraint, indifference curves, and the three points you found in parts a)c). On a graph with X on the horizontal axis and PX on the vertical axis, draw the regular (Marshallian) demand curve, and the (Hicksian) compensated demand curve. [The compensated demand curve is the curve that shows how the demand for X changes in response to changes in PX if there were only a substitution effect] Which curve is more elastic? If X were an inferior good (clearly not possible if the utility function is Cobb-Douglas), which demand curve would be more elastic, the regular demand curve or the compensated demand curve? 2. A representative individual in the country of Magnolia has gLss income equal to 10,000 Magnolian dollars (1V[Ds). Everyone in Magnolia pays a 20% income tax, so that the individual's n_et income is 8,000 MDs. The individual derives utility from consumption of gasoline and all other goods. The price of gasoline is 2N[Ds per gallon, and the price of all other goods is normalized to lMD per unit. At the current level of income and taxes, the individual purchases 1,000 gallons of gasoline per year. As part of a comprehensive reform of the tax system, the government of Magnolia is considering a plan to remove the income tax completely, and to replace it with a tax of ZNID per every gallon of gasoline. (a) In a well labeled graph, show the individual's budget constraint before and after the tax reform. Also, mark the individual's optimal consumption bundle before the reform. (b) A government economist claims that the proposed reform will yield exactly the same amount of tax revenues as the old tax system. Is this claim correct? Explain. 3. Consider the following utility functions: i) U(X, Y) = XZY ii) U(X, Y) = mir{2X, Y] Assume that the consumer maximizes utility subject to the usual budget constraint, PXX + PYY = I . For each utility function, calculate the demand Jnction for X, and the elasticity of demand for X with respect to Px, PY and I. 4. A consumer's utility function is U(X,Y) = iX2 +10X+ Y. The price of Yis PF}. 2 The consumer's monetary income is B=100. a) Calculate the consumer's demand function for X. b) Calculate the elasticity of demand for X with respect to PX. c) For which values of X is demand elastic (i.e., the elasticity is smaller than -1 meaning further from zero)? For which values is demand inelastic (elasticity is greater than -1 meaning closer to zero)? For which value is demand unit elastic? d) Draw a graph with X on the horizontal axis, and the consumer's expenditure on good X (Expenditure = Px-X) on the vertical axis. At what value of X is total expenditure maximized? How is your answer related to your answer to part c