Problem Set 1
ECH-32306 Advanced Microeconomics, Part 1, Fall 2014
Instructor: Dusan Drabik, De Leeuwenborch 2105
Email: [email protected]
1. A consumer has a preference relation on R1 which can be represented by the utility function
u(x) = x2 + 4x + 4. Is this function quasi-concave? Briefly explain. Is there a concave utility
function representing the consumer's preferences? If so, display one; if not, why not?
2. A consumer has Lexicographic preferences on R 2 if the relationship
satisfies x1
x2
whenever x11  x12 , or x11  x12 and x12  x22 . Show that lexicographic preferences on R 2 are
rational, i.e., complete and transitive.
3. A consumer with convex, monotonic preferences consumes non-negative amounts of x1 and
x2.
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
a.) If u  x1, x2   x1 x22 represents those preferences, what restrictions must there be on the
value of parameter  ? Explain
b.) Given those restrictions, calculate the Marshallian demand functions.
4. There are two
goods,
x   x1, x2  R2 . A consumer has the utility function
U  x   u1  x1   u2  x2  where each ui is twice continuously differentiable with ui '  xi   0
and ui ''  xi   0 for all xi  R1 . Each ui also satisfies the condition: limxi 0 ui '  xi    .
Assume that prices of both goods are strictly positive, each pi > 0, and wealth is strictly
positive, w > 0. [The conditions in this problem are sufficient to guarantee that the optimal
bundle of goods x* is interior, i.e. x* >> 0. So you can ignore inquality constraints of the
form x  0 .]
a.) Write the consumer's problem as a constrained optimization problem and display the first
order conditions for this optimization problem.
b.) Show that if wealth increases then the demand for good 1 increases.
c.) What is the sign of the effect of a change in the price of good 1 on the consumer’s
demand for good 1? Show your work.
5. In a two-good case, show that if one good is inferior, the other must be normal.
6. Consider the three-good setting in which the consumer has utility function



u  x    x1  b1   x2  b2   x3  b3  .
a.) Write down the first-order conditions for the utility maximization problem, and derive the
consumer’s Marshallian demand and indirect utility functions. This system of demands is
known as the linear expenditure system and is due to Stone (1954).
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b.) Verify that these demand and indirect utility functions satisfy the properties covered in
the class.
7. How would you determine whether the function
X  px , p y , I  
2 px I
px 2  p y 2
could be demand function for commodity x of a utility maximizing consumer with
preferences defined over the various combinations of x and y? Is it a demand function?
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