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The problems in this chapter are primarily mathematical. They are intended to give students
some practice with the concepts introduced in Chapter 2, but the problems in themselves offer
few economic insights. Consequently, no commentary is provided. Results from some of the
analytical problems are used in later chapters, however, and in those cases the student will be
directed back to this chapter.
Solutions
2.1
22
( , ) 4 3 .f x y x y=+
b. Constraining
( , ) 16f x y =
creates an implicit function between the variables. The
2.2 a. Profits are given by
2
2 40 100.R C q q
= − = − + −
The maximum value is found
by setting the derivative equal to 0:
CHAPTER 2:
Mathematics for Microeconomics
Chapter 2: Mathematics for Microeconomics
2
2.3 First, use the substitution method. Substituting
1yx=−
yields
2.4 Setting up the Lagrangian,
(0.25 ).x y xy
= + + −L
The first-order conditions are
b. Substituting for
*,t
c. Differentiation of the original function at its optimal value yields
Chapter 2: Mathematics for Microeconomics
3
b. The first-order condition for maximum volume is given by
d. This would require a solution using the Lagrangian method. The optimal solution
2.7 a. Set up the Lagrangian:
1 2 1 2
5ln ( ).x x k x x
= + + − −L
The first-
order conditions are
Chapter 2: Mathematics for Microeconomics
4
2.8 a. Because
MC
is the derivative of
,TC
TC
is an antiderivative of
.MC
By the fundamental theorem of calculus,
b. For profit maximization,
( ) 1,p MC q q= = +
implying
1.qp=−
But
15p=
implies
14.q=
Profit are
Chapter 2: Mathematics for Microeconomics
5
d. Assuming profit maximization, we have
( ) ( )
p pq TC q
=−
e. i. Using the above equation,
ii. The envelope theorem states that
*( ).d dp q p
=
That is, the derivative of
Analytical Problems
2.9 Concave and quasi-concave functions
The proof is most easily accomplished through the use of the matrix algebra of quadratic forms.
See, for example, Mas Colell et al.,1995, pp. 937–939. Intuitively, because concave functions lie
Chapter 2: Mathematics for Microeconomics
6
2.10 The Cobb–Douglas function
a.
1
1 1 2
0,
x
fx
−
=
b. A contour line is found by setting the function equal to a constant:
12
,y c x x
==
2.11 The power function
2.12 Proof of envelope theorem
Chapter 2: Mathematics for Microeconomics
7
a. The Lagrangian for this problem is
b., c. Multiplication of each first-order condition by the appropriate deriviative yields
d. The optimal value of
f
is given by
( )
( ), ( ), .f x a x a a
Differentiation of this
( )
12
( ), ( ), 0g x a x a a =
yields
f. Multiplying the results from part (e) by
and using parts (b) and (c) yields
g. In Example 2.8, we showed that
8.P
=
This shows how much an extra unit of
2.13 Taylor approximations
a. A function in one variable is concave if
( ) 0.fx
Using the quadratic Taylor
formula to approximate this function at point
a
:
Chapter 2: Mathematics for Microeconomics
8
b. A function in two variables is concave if
2
11 22 12 0.f f f−
2.14 More on expected value
a. The tangent to
()gx
at the point
()Ex
will have the form
()c dx g x+
for all
c. Let
d. Use the hint to break up the integral defining expected value:
e. 1. Show that this function integrates to 1:
Chapter 2: Mathematics for Microeconomics
9
2. Calculate the cumulative distribution function:
3. Using the result from part (c):
4. To show Markov’s inequality use
f. 1. Show that the PDF integrates to 1:
2. Calculate the expected value:
3. Calculate
( 1 0Px−
):
4. All we must do is adjust the PDF so that it now sums to 1 over the new,
smaller interval. Since
( ) 8 9,PA=
5. The expected value is again found through integration:
6. Eliminating the lowest values of x increases the expected value of the
remaining values.
2.15 More on variances
a. This is just an application of the definition of variance:
Chapter 2: Mathematics for Microeconomics
10
b. Here, we let
x
yx
=−
and apply Markov’s inequality to
y
and remember that
x
can only take on positive values.
,
x
1, ,in=
be
n
independent random variables each with expected value
d. Let
12
(1 )X kx k x= + −
and
( ) (1 ) .E X k k
= + − =
e. Suppose that
2
1
Var( )x
=
and
2
2
Var( ) .xr
=
Now
Chapter 2: Mathematics for Microeconomics
11
2.16 More on covariances
a. This is a direct result of the definition of covariance:
Cov( , ) ( ( ))( ( ))
x y E x E x y E y
= − −
b.
22
Var( ) [( ) ] [ ( )]
ax by E ax by E ax by
= −
c. The presence of the covariance term in the result of Problem 2.16b suggests that
the results would differ. In the two-variable case, however, this is not necessarily
the situation. For example, suppose that x and y are identically distributed and that
d. If
12
,x kx=
the correlation coefficient will be either
1+
(if
k
is positive) or
1−
(if
,yx
=+
Chapter 2: Mathematics for Microeconomics
12
