**Problem Set 2**

**Due 2/26/96 in class or by 5 p.m. to Michael Kreutz, E52-251**

__Theory__

1. Nicholson, Problems 3.2, 3.4, 4.4, 4.5, and 5.8.

2. Let *U*(*X*,*Y*)=*X*^(1/4)* * Y*^(3/4).
Suppose that prices are *P _{x}* and

(a) Calculate "indirect utility," or the utility at the optimal
choices, *V*(*P _{x}*,

(b) For a given utility level *U*_{0}, solve the dual expenditure-minimization
problem, and compute the optimal choices of *X* and *Y* (the
"compensated demand functions," *h _{x}*(

(c) Calculate the minimum expenditure function *E*(*P _{x}*,

(d) Consider two famous applications of the envelope theorem in consumer theory, Shephard's Lemma and Roy's Identity ((E5.1) and (E5.2) in Nicholson). State a verbal interpretation of each of these results.

(e) Verify directly that Shephard's Lemma and Roy's Identity hold in
this problem.

__Application ("Irish Potato" article)__

1. Draw a diagram illustrating Giffen's paradox. Your diagram should
illustrate a consumer's 2-good utility maximization problem for two different
budget constraints, where *P _{X}* is
higher under the second constraint.

2. Consider Figure 1 of Dwyer and Lindsay. Explain why the top diagram is consistent with the your diagram from question (1). Why is the lower diagram a better description of the Irish potato famine?

3. Dwyer and Lindsay make two claims about Giffen goods: "For a good to be Giffen, some normal good must be displaced by the inferior good as the price rise lowers real income." "Inferiority is necessary for a good to be Giffen." Formally prove these two statements, and then briefly state a reason why each is unlikely to hold for the case of the Irish potato famine.

4. We are interested in measuring the effects of a price change on the
amount demanded of potatoes. Graph the budget constraint facing a person
with income *I* and the ability to purchase potatoes and bread if
the price of bread is $1.00 and the cost of potatoes is *P*. Draw
a standard set of indifference curves and illustrate the optimal bundle.

5. Show how the budget constraint and optimal bundle change when the
price rises from *P*=*P*_{1} to *P*=*P*_{2},
*P*_{1}<*P*_{2}, for each of the following
assumptions.

a) *I* is independent of the price.

b) *I* = *SP*, where *S* is fixed.

c) *I* = *SP*, where we first consider *P*=*P*_{1}
and *S*=*S*_{1}, and then we consider *P*=*P*_{2}
and *S*=*S*_{2}, where *S*_{2}<*S*_{1}
(price rises and supply falls).

Explain how the trade-off between potatoes and bread changes in (a)-(c). What does this say about the interpretation of the Irish potato famine as an example of the kind of experiment which might be used to determine whether a good is Giffen?

6. What is an alternative example of a Giffen good (in addition to the one suggested in the article)? Why is this a more plausible scenario?