Problem Set 1
14.04, Fall 2020
Prof: Robert Townsend
TA: Laura Zhang and Michael Wong
1 Problem 1: Preference Relations and Utility
Functions
a) Let X = R2+ and there be two points x = (x1 , x2 ), y = (y1 , y2 ).
Suppose x y if x1 > y1 or if x1 = y1 and x2 ≥ y2 .
Is the preference relation complete? Transitive? Why or why not?
b) John has preferences over consumption bundles (A, B) ∈ R2+ characterized
1 2
by utility function U (A, B) = A 3 B 3 . Show that John’s preferences satisfy
strict monotonicity, local non-satiation, strict convexity, and continuity.
c) Consider the following constrained maximization problem using the utility
function from part b)
1 2
max U (A, B) = A 3 B 3
s.t. p A A + pB B ≤ I
A ≥ 0 and B ≥ 0
where pA , pB , I > 0. Let A∗ , B ∗ denote the solution to the above problem.
i. Can we ever have A∗ = 0 or B ∗ = 0? Why or why not?
ii) Can we ever have pA A∗ + pB B ∗ < I? Why or why not?
iii) Set up the consumer’s Lagrangian and find the first-order conditions.
How do you know that these first-order conditions are sufficient to
characterise the solution to the consumer’s problem? For what values
of pA , pB will the consumer consume twice as much A as B?
2 Problem 2: Income and Substitution Effects
A (potential) worker has utility over consumption c and leisure l given by
cδ lδ
U (c, l) = α +β
δ δ
where δ < 1. She has T hours to allocate between leisure and work. For
each hour she works, she earns a wage w to spend on consumption c, which we
normalize the price of c to one. However, because her wife works, she receives an
1
additional ‘non-labor income’ Y regardless of how much she works. Assume she
takes Y as given (i.e. her own decisions do not affect her wife’s labor supply).
She therefore maximizes utility subject to the following constraints:
c ≤ w(T − l) + Y
c≥0
0≤l≤T
a) Without writing down the Lagrangian or solving the optimization problem,
identify which constraints above will always bind (hold with equality) at
the optimum, and which constraints will always be slack (not hold with
equality). Are there any constraints which fall into neither category?
b) Set up the Lagrangian and write out all the relevant conditions for a solution,
using your answer to a) to help simplify things.
c) Assume now that the solutions are at an interior point. How do c and l
change as non-labor income Y increases? What does this tell us about
whether c, l are normal goods?
d) How do c and l change as the wage w increases? Show that your result
can be interpreted as income and substitution effects. Note: An intuitive
answer will get you most of the points.
3 Problem 3: Production Functions and Feasi-
ble Allocations
Recall the Leontief input-output model from lecture 4. Suppose we have two
commodities and input-output matrix given by
.2 .7
A=
.6 .1
Specifically, producing one unit commodity 1 costs .2 units of commodity 1 and
.7 units of commodity 2, and producing one unit commodity 2 costs .6 units of
commodity 1 and .1 units of commodity 2
a) Suppose John has a demand vector given by D = [3, 1]. Find the production
vector X = [X1 , X2 ]0 that satisfies this demand.
b) Now suppose John has a utility function given by UJ (Y1 , Y2 ) = αY1 + βY2
where α, β > 0. Characterize the set of production vectors X that gives
John a utility of V > 0. (Hint: this will be a linear equation of X1 and
X2 in terms of α, β, and V )
c) Suppose Sally does not like it when X2 is produced in either too much or too
little quantity. Specifically, Sally’s utility is given by US (X2 ) = −γ|X2 −X|
2
where γ > 0. Find the production vector X ∗ that maximizes Sally’s utility
subject to keeping John’s utility constant at V. (Hint: you should not use
any calculus to solve this problem)
4 Problem 4: Giffen Good
See the compressed folder on the Giffen Good exercise. Please recreate the
results and include screenshots of your Stata output in your submission.
3
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14.04: Intermediate Microeconomic Theory
Fall 2020
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