Econ 301 Fall 2009
Study questions for Exam #2
(Some of these study questions were discussed in class. (UMP refers to the Utility
Maximization Problem, EMP refers to the Expenditure Minimization Problem)
Please know the definition of a utility function.
Remember the method we used in class to verify that a complete, transitive, and continuous
preference relation can always be represented by a utility function. What number will that
method assign to bundle x = (3, 6) for the following preferences : x is at least as good as y if
x1 + x2 ≥ y1 +y2?
u(3, 6) = 4.5
Use the Kuhn-Tucker Theorem to solve the UMP for the following utility function :
U(x1, x2, … , xN) = α1lnx1 + α2lnx2 + … + αNlnxN, for α1, α2, … , αN > 0.
Kuhn-Tucker First Order conditions are
α1/x1 − λp1 ≤ 0, x1 ≥ 0 at least one weak inequality must hold as equality
α2/x2 − λp2 ≤ 0, x2 ≥ 0 at least one weak inequality must hold as equality
↓
αN/xN − λpN ≤ 0, xN ≥ 0 at least one weak inequality must hold as equality
p1x1 + p2x2 + … + pNxN − m ≥ 0 , λ ≥ 0 at least one weak inequality must hold as equality
Here, we note that if xi = 0 for some i, then u(x) = −∞ (because ln(0) = −∞), hence the
solution to the UMP must be x* >> 0. Second, the budget constraint must be binding : p1x1 +
p2x2 + … + pNxN = m.
Therefore K-T first order conditions simplify to αi/xi − λpi = 0 for i = 1, 2, … , N
Multiply both sides by xi : αi − λpixi = 0, add up all N first order conditions : ∑αi − λ∑ pixi =
0, solve for λ: λ = (∑αi)/(∑ pixi). Since ∑ pixi = m this is the budget constraint binding part,
we have λ = (∑αi)/m. Substitute for λ in the to αi/xi − λpi = 0 and solve for xi : x*i = (m/pi)(αi
/∑αi).
Use the Kuhn-Tucker Theorem to solve the UMP for the following utility function :
U(x1, x2) = x2 + αlnx1, for α > 0.
We solved this is class. Please read the detailed explanations at the end of these notes.
Solve the UMP for the following utility function : U(x1, x2) = min{x1, x2} + max{x1, x2}.
Kuhn Tucker conditions are not very useful here. We have to go case by case. Let p1 > p2.
Consider x1 = 0, x2 = m/p2 as a candidate solution. u = m/p2.
SMALL Correction to the solution
Use the Kuhn-Tucker Theorem to solve the EMP for the following utility function :
U(x1, x2) = x1x2. What is the minimum amount of income needed to reach the utility level of
bundle (2, 8) at prices p1 = 2, and p2 = 4?
We solved this example in class. Here is a brief reminder:
Kuhn-Tucker First Order conditions are
−p1 + λα1x1(α1−1) x2 ≤ 0, x1 ≥ 0 at least one weak inequality must hold as equality
−p2 + λα2x1x2(α2−1) ≤ 0, x2 ≥ 0 at least one weak inequality must hold as equality
x1x2 − u ≥ 0 , λ ≥ 0 at least one weak inequality must hold as equality
Again x* >> 0 and the “utility constraint” is binding i.e., u(x*) = u, where u is the “target
level of utility”
Take −p1 + λα1x1(α1−1) x2 = 0, multiply both sides by x1, solve for x1, substitute into the utility
constraint x1x2 = u, do the same with −p2 + λα2x1x2(α2−1) = 0, but this time multiply both sides
by x2.
You will see that the u’s in the utility constraint will cancel each other. Solve this λ, your λ
will be in terms of α’s and p’s. Once you have λ go to the −p1 + α1λx1(α1−1) x2 = 0, substitute
for λ and solve for x*1. Repeat with −p2 + λα2x1x2(α2−1) = 0.
The answer is x*1(p1, p2, u) = [u(p2/p1)]0.5, and x*2(p1, p2, u) = [u(p1/p2)]0.5.
Solve the EMP for the following utility function :
U(x1, x2) = min{x2, x1}.
We did this in class. The answer is x*1(p1, p2, u) = u, and x*2(p1, p2, u) = u. The expenditure
function is e(p1, p2, u) = (p1+p2)u
Describe the expenditure function in plain language. Write down the expenditure function for
the previous two questions.
For given prices and a given utility level it tells us the minimum expenditure to reach that
given utility level at these prices.
Use the Kuhn-Tucker Theorem to solve the EMP for the following utility function :
U(x1, x2) = x2 + αlnx1, for α > 0.
This will be done in class on Monday Dec 7.
Can the following functions be the expenditure function for a utility maximizing consumer
with monotone preferences that are represented by a continuous utility function? Explain
your answer.
a.
e(p, u) = (p1p2)u2.
b.
e(p, u) = (p1+p2)u.
c.
e(p, u) = [p1(1−p2)2]u
Please remember the properties of the e function : e(αp, u) = αe(p, u) must hold for all α > 0, e
must be increasing in u, non-decreasing in prices, and concave in p.
Let p1 and p2 be two price vectors, define pt = tp1 + (1−t)p2 for t between 0 and 1. Fix u. e(p,
u) is concave in p, if for any p1 and p2 we have e(pt, u) ≥ te(p1, u) + (1−t)e(p2, u).
Consider a consumer with the following utility function: u(x1, x2) = x1x2.
a.
Let p1 = 1, p2 = 2, m = 4. Find the solution to the UMP.
b.
What is the maximum level of utility the consumer can reach at p1 = 1, p2 = 2, m = 4?
c.
The expenditure function for this consumer is e(p1, p2, u) = 2[u(p1p2)]1/2. What is the
minimum income this consumer needs to have a utility level of at least u = 2?
d.
Comment on your answers to part b and c.
e.
Prices change: Now p1 = 2, p2 = 3. What is the minimum income this consumer needs
to maintain his utility level in part a?
a. The solution of the UMP for general prices and income is x1(p1, p2, m) = m/2p1, and x2(p1,
p2, m) = m/2p2. So for these specific values x1 = 2, and x2 = 1.
b. max utility is u(2, 1) = 2.
c. min income is 4.
d. comment : UMP and EMP are very closely related to each other. If at given prices with
income mo max utility that you can reach is u* then to reach utility level u* min income you
will need is mo.
e. e(2, 3, 2) = 2[12]1/2 = 4[3]1/2 = 6.93
CORRECTION to the demand functions
Consider a world with two goods x1 and x2. Here are the demand functions for x1 and x2 :
x1 = αp1/βp2 is now changed to x1 = αp2/(β+p1)
x2 = m/γp2 − δ
α, β, γ, and δ are parameters. Based on what we have seen in class, what must be the
parameter values so that these demand functions are not inconsistent with utility
maximization?
Demand function must satisfy (at least) the following two properties :
1. p·x(p, m) = m for all p and m
2. x(αp, αm) = x(p, m) for all α > 0.
We must have β = 0, this is by property 1
With β = 0, Property 2 gives us p1[αp2/p1)] + p2[m/γp2 − δ] = m, simplifies to αp2 + m/γ − δp2
= m. This gives us α = δ and γ = 1.
Suppose x = (2, 4) and y = (3, 1) both solve the utility maximization problem for a consumer
with locally non-satiated and convex preferences that can be represented by a utility function.
a.
Compute the person’s income when p1 = 1.
b.
True of false?
1 x is at least as good as y
2 x is strictly preferred to y
3 y is at least as good as x
4 y is strictly preferred to x
5 x is indifferent to y
6 y is indifferent to x
c.
Find at least one more bundle that also solves the utility maximization problem.
Explain your reasoning.
a. m = 10
b since both x and y solve the UMP x must be indifferent to y.
c. any convex combination of x and y must also solve the UMP. Example : (2.5, 2.5)
Asuman claims that she has strictly convex preferences and with her current income level and
at the existing prices, both x = (12, 8), and y = (8, 16) solve her utility maximization problem.
How is this possible?
Asuman claims that she has convex preferences and with her current income level and at the
existing prices, only the following three bundles x = (12, 8), y = (8, 16), and z = (16, 0) solve
her utility maximization problem. How is this possible?
It cannot be true. If Asuman has convex preferences and if there are more than bundle that
solve the UMP then any convex combination must also solve the UMP.
Let m = 10, p1 = 2, and p2 = 4. Solve the “preference maximization problem” for a person
with lexicographic preferences : x is at least as good as y of either x1 > y1 or x1 = y1 and x2 ≥
y2. Define the “preference maximization problem” as follows : Find the bundle x such that x
is affordable and x is at least as good as y for all y that are also affordable.
Solve the utility maximization problem for general m, p1, and p2.
x*1 = m/p1, x*2 = 0. Here x*1 = 10, x*2 = 0.
True or false.
If a preference relation is complete and transitive it can be represented by a utility function.
If a preference relation can be represented by a utility function it must be complete and
transitive.
If x* satisfies the Kuhn-Tucker first order conditions of the UMP, then x* is a solution to the
UMP.
If x* is a solution to the UMP then it must satisfy the Kuhn-Tucker first order conditions of
the UMP.
False
True
False
True
Let N = 2. Suppose x solves the EMP for prices p’ = (1, 4) and utility level uo and y solves
the EMP for prices p” = (2, 1) and utility level uo. (Same utility level, different prices.)
a.
true or false explain. p’ · x ≥ p’ · y
b.
true or false explain. p” · y ≥ p” · x
c.
true or false explain. p’ · y ≥ p’ · x
b.
true or false explain. p” · x ≥ p” · y
Based on your answers, can you identify the sign of this following expression ? Is it minus or
plus? (p” − p’) · (y − x)
False. x and y both give the utility uo. x minimizes expenditure for prices p’, so y must cost
more at prices p’.
False y minimizes expenditure for prices p’’, so x must cost more at prices p’’.
True
True
We have . p” · y ≥ p” · x and p” · y ≥ p” · x
Take (p” − p’) · (y − x) multiply it out : p” · y − p” ·x − p’· y + p’· x. Using the above
inequalities we have p” · y − p” ·x ≤ 0 and − p’· y + p’· x ≤ 0.