EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
September 26, 2011
Problem Set 2
Due Date: October 3, 2011, 11:59pm
Instructions: Solutions should be put in my mailbox, slipped under the door of my office,
or emailed to [email protected].
1. Profit Maximization
Consider a firm that produces output y with capital k and labor l according to the
technology described by
k a lb ≥ y,
(1)
where 0 < a < 1, 0 < b < 1, and a + b < 1, k, l ∈ R+ . The firm sells each unit of
output at the price p, rents each unit of capital at the rate r, and hires each unit of
labor at the wage w. Hence it chooses y, k, and l to maximize profits
π ≡ py − rk − wl
subject to the constraint just shown in (1).
(a) Set up the Lagrangian for this problem, letting λ denote the multiplier on the
constraint.
(b) Next, write down the conditions that, according to the Karush-Kuhn-Tucker
(KKT) theorem, must be satisfied by the values y ∗ , k ∗ , and l∗ that solve the
firm’s problem, together with the associated value λ∗ for the multiplier.
(c) Assume that the constraint binds at the optimum (can you tell under what conditions this will be true?), and use your results from above to solve for y ∗ , k ∗ , l∗ ,
and λ∗ in terms of the model’s parameters: a, b, p, r, and w.
(d) Finally, use your solutions from above to answer the following questions: What
happens to the optimal y ∗ , k ∗ , and l∗ when
i. the output price p rises, holding all other parameters fixed? In each case,
does the optimal choice rise, fall, or stay the same?
ii. the rental rate for capital r rises, holding all other parameters fixed?
iii. the wage rate w rises, holding all other parameters fixed?
iv. p, r, and w all double at the same time?
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2. Cobb-Douglas Utility Maximization
Consider a consumer who uses his or her income I to purchase c1 units of good 1 at
the price of p1 per unit and c2 units of good 2 at the price of p2 per unit, subject to
the budget constraint
I ≥ p 1 c1 + p 2 c2 .
(2)
Suppose that the consumer has preferences over the two goods described by the utility
function
U (c1 , c2 ) = ca1 c1−a
(3)
2 ,
where 0 < a < 1, c1 , c2 ∈ R+ .
(a) Set up the Lagrangian for the consumer’s problem: choose c1 and c2 to maximize
utility given in (3) subject to the budget constraint shown in (2), letting λ denote
the multiplier on the constraint.
(b) Next, write down the conditions that, according to the KKT theorem, must be
satisfied by the values c∗1 and c∗2 that solve the consumer’s problem, together with
the associated value λ∗ for the multiplier.
(c) Assume that the budget constraint binds at the optimum (again, can you tell
under what conditions this will be true?), and use your results from above to
solve for c∗1 , c∗2 , and λ∗ in terms of the model’s parameters: I, p1 , p2 , and a.
(d) Finally, suppose you have data on this consumer’s income as well as how much of
that income the consumer spends on each of the two goods. Assume that in the
data as in the theory the consumer purchases only these two goods. How could
you use those data to estimate the preference parameter a?
Now suppose the budget constraint (2) stays the same but that the utility function of
the consumer is described by
U (c1 , c2 ) = a ln(c1 ) + (1 − a) ln(c2 ),
(4)
where 0 < a < 1 as before.
(e) Solve the constrained maximization problem for a consumer with utility function
(4) and budget constraint (2) using the KKT approach and compare the solutions
of c∗1 , c∗2 , and λ∗ for this problem with that of the previous problem. What do you
notice?
3. Utility Maximization - Second-Order Conditions
The following result specializes Theorem 19.8 from Simon and Blume’s book to provide
first and second-order conditions for a constrained optimization problem with two
choice variables and a single constraint that is assumed to bind at the optimum.
Theorem 1. Let f : R2 → R and g : R2 → R be twice continuously differentiable
functions, and consider the constrained optimization problem
max f (x, y) subject to θ ≥ g(x, y)
x,y
2
with parameter θ ∈ R. Define the Lagrangian
L(x, y, λ) = f (x, y) − λ(g(x, y) − θ).
Suppose there exist values x∗ , y ∗ , and λ∗ that satisfy the conditions
Lx (x∗ , y ∗ , λ∗ ) = fx (x∗ , y∗) − λ∗ gx (x∗ , y ∗ ) = 0,
Ly (x∗ , y ∗ , λ∗ ) = fy (x∗ , y ∗ ) − λ∗ gy (x∗ , y ∗ ) = 0,
Lλ (x∗ , y ∗ , λ∗ ) = g(x∗ , y ∗ ) − θ ≤ 0,
λ∗ ≥ 0,
and
λ∗ (g(x∗ , y ∗ ) − θ) = 0,
and such that the constraint binds. The “bordered Hessian” matrix


0
gx (x∗ , y ∗ )
gy (x∗ , y ∗ )
H = gx (x∗ , y ∗ ) Lxx (x∗ , y ∗ , λ∗ ) Lyx (x∗ , y ∗ , λ∗ )
gy (x∗ , y ∗ ) Lxy (x∗ , y ∗ , λ∗ ) Lyy (x∗ , y ∗ , λ∗ )
satisfies the second-order condition that |H| > 0, i.e. the determinant of H is strictly
positive. Then (x∗ , y ∗ ) is a constrained local maximizer.
Note that this result provides sufficient conditions for a local constrained maximizer:
it says that if the KKT first-order conditions and the above second-order condition is
satisfied, then (x∗ , y ∗ ) is at least a constrained local maximum.
There is a similar, though generally more involved, second-order condition that maximizers necessarily satisfy. For our current problem with two choice variables and one
constraint, the condition is not difficult to state or check. However, the number of conditions needed to be checked grows rapidly as the dimensions of the problem increase.
For an excellent discussion on how to check whether a symmetric matrix is negative
semidefinite on a subspace (which is essentially what we are doing in the following
theorem), see Appendix M.D in Mas-Colell, Whinston, Greene, particularly Theorem
M.D.3.
Theorem 2. Suppose, as in Theorem 1, f : R2 → R and g : R2 → R are twice continuously differentiable functions, and consider the constrained optimization problem
max f (x, y) subject to θ ≥ g(x, y)
x,y
with parameter θ ∈ R. Suppose that (x∗ , y ∗ ) is a solution to the constrained problem
that satisfies the Karush-Kuhn-Tucker thereom (both the constraint qualification and
the first-order conditions), with λ∗ the associated Lagrangian multipliers. Define H as
3
in the previous theorem and Hπ as follows:


0
gy (x∗ , y ∗ )
gx (x∗ , y ∗ )
Hπ = gy (x∗ , y ∗ ) Lyy (x∗ , y ∗ , λ∗ ) Lxy (x∗ , y ∗ , λ∗ )
gx (x∗ , y ∗ ) Lyx (x∗ , y ∗ , λ∗ ) Lxx (x∗ , y ∗ , λ∗ )
Then, (x∗ , y ∗ , λ∗ ) satisfies the second-order (necessary) condition that |H| ≥ 0 and
|Hπ | ≥ 0.
With this result in mind, return to the problem above solved by a consumer with
preferences represented by the utility function in equation (3) and a budget constraint
given by equation (2).
(a) Restate the KKT first-order conditions you derived earlier.
(b) Show that your previous candidate solution(s) satisfies the Second Order Necessary Condition stated in Theorem 2.
(c) Does (any of) your solution(s) satisfy the Second Order Sufficient Condition stated
in Theorem 1?
4. The Constraint Qualification
A consumer chooses consumption c of a single good in order to maximize the utility
function U (c) subject to the budget constraint pc ≤ I, where I > 0 is the consumer’s
income and p > 0 is the price of the good. Suppose that the utility function is strictly
increasing, so that U 0 (c) > 0 for all values of c.
(a) Can you guess the solution c∗ to this problem without working through the mathematical details?
(b) To see how the KKT theorem leads you to this same solution, define the Lagrangian as
L(c, λ) = U (c) − λ(pc − I),
and use the KKT conditions to find c∗ and the associated value λ∗ .
(c) Now, suppose that the consumer’s budget constraint is given as
(pc − I)3 ≤ 0
so that the Lagrangian is
L(c, λ) = U (c) − λ(pc − I)3 .
Notice that the constraint set is unchanged, since given the parameters I and
p, the same set of values for c that satisfy the original constraint satisfy this
alternative constraint and vice-versa. Write down the KKT conditions that follow
from this alternative definition of the Lagrangian. Are these conditions satisfied
by the same values of c∗ and λ∗ that you found before? Why or why not?
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5. Perfect Substitutes
Suppose a consumer likes two goods, good 1 and good 2, which he or she views as
perfect substitutes: perhaps they are just different brands of the same basic product.
A utility function that captures this idea takes the linear form
U (c1 , c2 ) = c1 + c2 .
Let I > 0 denote the consumer’s income and let p1 > 0 and p2 > 0 denote the prices
of the two goods. Then the consumer’s problem is
max c1 + c2 subject to p1 c1 + p2 c2 ≤ I, c1 ≥ 0, and c2 ≥ 0.
c1 ,c2
Define the Lagrangian
L(c1 , c2 , λ, µ1 , µ2 ) = c1 + c2 − λ(p1 c1 + p2 c2 − I) + µ1 c1 + µ2 c2 .
(a) Can you guess how the optimal choices c∗1 and c∗2 will depend on the parameters
I, p1 , and p2 even without working through the mathematical details?
(b) To see how the KKT theorem can lead you to the same solution, write down the
conditions that, according to the theorem, must be satisfied by the values of c∗1
and c∗2 that solve the problem together with the associated values of λ∗ , µ∗1 , and
µ∗2 , and derives solutions for these variables.
6. Elasticities of Demand
Consider a consumer who purchases three goods to maximize utility subject to a budget
constraint. Assuming that the utility function is such that nonnegativity constraints
on the choice variables can be ignored, the consumer’s problem can be written as
max U (c1 , c2 , c3 ) subject to p1 c1 + p2 c2 + p3 c3 ≤ I.
c1 ,c2 ,c3
Suppose that the utility function is also such that is it possible to find functions
(c∗1 , c∗2 , c∗3 ) of (p1 , p2 , p3 , I) that uniquely determine the optimal choices in terms of the
parameters measuring prices and income (note that here, the subscripts refer to the
three goods and not to the derivatives of the functions). Under most circumstances,
these functions, which correspond to “Marshallian demand curves” for each of the three
goods, can be expected to satisfy two basic conditions. First, under the assumption
that the budget constraint binds at the optimum, it must be that
3
X
pi c∗i (p1 , p2 , p3 , I) = I
(5)
i=1
for all values of (p1 , p2 , p3 , I). Second, because increasing or decreasing all three prices
and the consumer’s income by the same proportions has no effect on the consumer’s
5
optimal choices, it must be that
c∗i (rp1 , rp2 , rp3 , rI) = c∗i (p1 , p2 , p3 , I)
(6)
for any value of r > 0 for each i = 1, 2, 3; that is, the Marshallian demands are
“homogeneous of degree zero.”
Now, for each i = 1, 2, 3 and j = 1, 2, 3, let
εi,j =
pj
∂c∗i (p1 , p2 , p3 , I)
c∗i (p1 , p2 , p3 , I)
∂pj
denote the elasticity of demand for good i with respect to the price of good j, and for
each i = 1, 2, 3, let
∂c∗i (p1 , p2 , p3 , I)
I
ηi = ∗
ci (p1 , p2 , p3 , I)
∂I
denote the income elasticity of demand for good i. Finally, for each i = 1, 2, 3, let
si =
pi c∗i (p1 , p2 , p3 , I)
I
denote the share of his or her total income that the consumer spends on good i.
(a) Use equation (5) to show that for each i = 1, 2, 3, the “share-weighted” price
elasticities of demand must be related via
3
X
sj εj,i = −si .
j=1
(b) Use (5) to show that the “share-weighted” income elasticities of demand must be
related via
3
X
si ηi = 1.
i=1
(c) Finally, use equation (6) to show that for each i = 1, 2, 3, the price and income
elasticities of demand must be related via
3
X
j=1
6
εi,j + ηi = 0.