1. For the following statements, determine whether each is true or false, and explain your answer:
a) Asim likes sushi (the x-axis good) and the more money he has, the more sushi he will eat. If the price of sushi increases, the Engel curve for sushi shifts to the right.
b) The Hicksian demand for a Giffen good is upward sloping.
c) The income effect for perfect complements is zero.
d) For a given increase in income only one good can be inferior. (Explain your answer graphically).
e) On an inverse demand curve where price is on the y-axis and quantity is on the x-axis, the Marshallian demand curve is steeper than the Hicksian demand curve when the good is normal.
2. Alex's preferences for goods x and y are represented by the following utility function: U(X,Y)=X°(Y - k)1-a. 0 < a < land K>0. Px and Py represent the prices of x and y respectively, and I represents Alex's income.
a) Solve for the optimal bundle using Lagrangian optimiation, and simplify as much as you can. Don't forget to consider both interior and corner solutions.
b) Now consider only interior solutions: Is X a normal or an inferior good? Is X a gross complement or substitute for good y? show with math and explain in words your results (HINT: two goods are gross complements if as the price of one good decreases consumption of the other good increases; two goods are gross substitutes if as the price of one good decreases consumption of the other good decreases).
c) Give your economic interpretation of the parameter K.
3. Say Morgan has $1000 she wants to allocate between spending this year and spending next year. She can buy goods for $1 per unit today, or she can save her money and earn %5 on her savings. That is, if she puts MX) in the bank today, interest will accrue and the bank will give her $105 next year (assume dollars here are in real terms, i.e. they are inflation-adjusted). Then she can spend the $105 on consumption next year, which costs $1 per unit as well. Denote consumption today as Co (on the x-axis) and consumption next year as Ci(on the -axis). We are interested in how government policy might affect savings behavior. Assume that Morgan's utility function is given by U(Co,Ci) = NrCWI. Keep in mind that the price of a unit of consumption today is Po = $1. Also, since Morgan accrues interest on savings the price of saving for a unit of consumption tomorrow is only P1 = th.
a) Write Morgan's budget constraint mathematically. Graph Morgan's budget constraint with C1 on the vertical axis (on the y-axis). Label intercepts and slope, and give interpretations of them in the context of the savings problem.
b) Using the Lagrange or substitution method, find Morgan's optimal consumption and savings. Add this to your graph.
c) Now, the government imposes a tax on savings. This tax is levied as a rate t (where 0 < t < 1) such that Morgan's actual after-tax savings would be (1 + r)(1-t). Let's suppose that the government starts out with a 20 percent tax on savings.
(1) Before you do any calculations, do you expect the new tax to affect Morgan's savings behavior? How?
(2) Modify the budget constraint and optimal consumption bundle in light of the government's new tax. (Ent: All that's changing is Morgan's effective interest rate on savings).
(3) On your graph, draw Morgan's new budget line and compute her new optimal savings bundle to the nearest dollar. Comment on how these differ from the original amounts in light of the new tax.
(4) Compute the substitution effect of the tax (as a change in pre-tax savings in dollars).
(5) Compute the income effect of the tax (as a change in pre-tax savings in dollars).
(6) Compare the SE and IE. Comment.
(7) Based on your answer to (5), do you think that the Cobb-Douglas utility function is appropriate for modeling changes in savings behavior in a real-world context? Explain.