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1. Suppose a consumer has an income of 10. and we have pi = p2 = 1. Further. this consumer has 'standard' preferences (decreasing MRS). and chooses z, = 2 5 on the resulting budget line. a) Show this utility maximizing choice on a graph of the budget set with an indifference curve. Now suppose that the government imposes tax of $1.00 per unit of z1. The con- sumer chooses point r1 = 3. 12 = 4 on the resulting budget line. Show this outcome on your graph, including an indifference curve through the new point. b) The tax in a) raises a revenue of $1.00 x 3 - $3. Suppose the government imposes an income tax of $3.00 rather than using a per unit tax on r1. Show this policy on your graph. Which policy is better from the consumer's point of view? Why? c) Now suppose that instead of the above two policies. the government imposes a per unit tax of of $0.50 on both goods. Show the effect of this policy in a graph. How does this policy rank against the other two from the consumer's point of view? d) Are these preferences consistent with a Cobb Douglas utility function? Explain. 2. A consumer has a utility function given by u1 = 2 22. a) Derive an expression for the two marginal utilities: A/U, (21. 12) and AU2 (11. 12). Since MRS = - MUI - 1707: use these marginal utilities to derive a simple expression for the AIRS (11. 12). b) Optimal choice on the part of the consumer implies AIRS = - 2. Suppose A = 20. p1 = p2 = 1. Show the optimal choice in this case on a well-labelled graph of the budget set. Include an indifference curve consistent with these preferences. c) Now keep income at 20, and p2 = 1, but set p1 = 2. Show the optimal choice in this case on a well-labelled graph of the budget set. Include an indifference curve consistent with these preferences. d) Now consider any set A. p1. pz. We have two equations that must hold: p12 + p242 = M and MRS (21.22) = - P. Use these equations to solve for the consumer's demand function for good 1