08 Jun Question Problem Set 3 – ECO420K (Fall 2015) H
Question
Problem Set 3 – ECO420K (Fall 2015)
Hand-in at BRB 1.116 by Thursday 24 Sept 2015, 4:45PM
Exercise 1 – Consumer choice problems (duality) I
Preferences over goods 1 and 2 are given by
u(q1 , q2 ) = (q1 − a)(q2 − b)
1. Set up the consumer’s utility maximisation problem when prices are p1 , p2 and available income is y. Assuming there exists an interior solution, set up the Lagrangian
and solve for the Marshallian demands. (You may wish to substitute them back into
the budget constraint to check that your answers are correct).
2. Compute the indirect utility function v(p1 , p2 , y).
3. Now set up the consumer’s expenditure minimisation problem when prices are p1 , p2
and the minimum required utility level is v . Assuming there exists an interior so¯
lution, set up the Lagrangian and solve for the Hicksian demands. (You may wish
to substitute them back into the utility constraint to check that your answers are
correct).
4. Show that the expenditure function is
√
¯
e(p1 , p2 , v ) = p1 a + p2 b + 2 v p1 p2 .
¯
5. Show that inverting the expenditure function with respect to its third argument gives
the indirect utility function, and vice-versa.
6. Show that evaluating the Marshallian demands when y = e(p1 , p2 , v ) gives the Hick¯
sian demands. Then show that evaluating the Hicksian demands when v = v(p1 , p2 , y)
¯
gives the Marshallian demands.
7. State Shephard’s Lemma and apply it to derive the Hicksian demands.
8. State Roy’s identity and apply it to derive the Marshallian demands.
9. Verify that the Slutsky equation holds.
1
Exercise 2 – Consumer choice problems (duality) II
Individuals consume three breakfast goods – cereal q1 , bacon q2 and eggs q3 . Preferences
are modelled by an indirect utility function
v(p1 , p2 , p3 , y) =
y
p1 (p2 + p3 )
,
where y denotes total breakfast spending and (p1 , p2 , p3 ) are the prices of the three goods.
1. What is the expenditure function?
2. What are the Hicksian demands?
3. What are the Marshallian demands?
4. Are preferences homothetic? How can you tell?
5. Assume prices are given by (p1 , p2 , p3 ) = (2, 2, 4) and total budget is given by y = 24.
(a) What are the utility-maximising consumption levels for goods 1,2 and 3, subject
to the budget constraint?
(b) How much would the consumer spend on this utility-maximising bundle?
(c) What would be her utility from consuming the utility-maximising bundle?
6. What would be the answers to question 5) if the consumer’s income were halved, i.e.
if her total budget were now given by y = 12 ? (Assuming prices are unchanged.)
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