ec400 problem sets

LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli

Department of Economics S.478; x7525

EC400 2010/11

Math for Microeconomics

September Course, Part II

Problem Set 1

1. Show that the general quadratic form of

a11x21 + a12x1x2 + a22x

22

can be written as ( x1 x2 )

(a11 a12

0 a22

)(x1

x2

).

2. List all the principal minors of a general (3 × 3) matrix and denote which are the

three leading principal submatrices.

3. Let C =

(0 0

0 c

), and determine the definiteness of C.

4. Determine the definiteness of the following symmetric matrices:

a)

(2 −1

−1 1

)b)

(−3 4

4 −6

)c)

1 2 0

2 4 5

0 5 6

5. Approximate ex at x = 0 with a Taylor polynomial of order three and four. Then

compute the values of these approximation at h = .2 and at h = 1 and compare with

the actual values.



EC400 2010/11



Problem Set 2

1. For each of the following functions, find the critical points and classify these as local

max, local min, or ‘can’t tell’:

a) x4 + x2 − 6xy + 3y2,

b) x2 − 6xy + 2y2 + 10x + 2y − 5

c) xy2 + x3y − xy

2. Let S ⊂ Rn be an open set and f : S → R be a twice differentiable function.

Suppose that Df(x∗) = 0. State the weakest sufficient conditions the relevant points,

corresponding to the Hessian of f must, satisfy for:

(i) x∗ to be a local max.

(ii) x∗ to be a strict local min.

3. Which of the critical points found in Problem 1 are also global maxima or global

minima?

4. Check whether f(x, y) = x4 + x2y2 + y4 − 3x − 8y is concave or convex using its

Hessian.

2



EC400 2010/11



Problem Set 3

1. A commonly used production or utility function is f(x, y) = xy. Check whether it

is concave or convex using its Hessian.

2. Prove that the sum of two concave functions is a concave function as well.

3. Let f be a function defined on a convex set U in Rn. Prove that the following

statements are equivalent:

(i) f is a quasiconcave function on U

(ii) For all x,y ∈ U and t ∈ [0, 1],

f(x) ≥ f(y)⇒ f(tx + (1− t)y) ≥ f(y)

(iii) For all x,y ∈ U and t ∈ [0, 1],

f(tx + (1− t)y) ≥ min{f(x), f(y)}

4. State the corresponding theorem for quasiconvex functions.

5. For each of the following functions on R1, determine whether they are quasiconcave,

quasiconvex, both, or neither:

a) ex; b) ln x; c) x3 − x.

3



EC400 2010/11



Problem Set 4

1. For the following program

minx

f(x) = x

subject to

−(x2) ≥ 0,

find the optimal solution.

2. Solve the following problem:

maxx1,x2

f(x1, x2) = x21x2

subject to

2x21 + x2

2 = 3.


maxx,y

x2 + y2

subject to

ax + y = 1

when a ∈ [12, 32].

4

4. Consider the following problem:

maxx

f(x)

subject to

g(x) ≤ a

x ∈ X

Let X be a convex subset of Rn, f : X → R a concave function, g : X → Rm a convex

function, a is a vector in Rm. What is the Largrangian for this problem? prove it is

a concave function of the choice variable x on X.

5



EC400 2010/11



Problem Set 5

1. Assume that the utility function of the consumer is

u(x, y) = x +√y

The consumer has a positive income I > 0 and faces positive prices px > 0, py > 0.

The consumer cannot buy negative amounts of any of the goods.

a) Use Kuhn-Tucker to solve the consumer’s problem.

b) Show how the optimal value of u∗,depends on I.


max(min{x, y} − x2 − y2)

6



EC400 2010/11



Problem Set 6

1. Consider the problem of maximizing xyz subject to x + y + z ≤ 1, x ≥ 0, y ≥ 0

and z ≥ 0. Obviously, the three latter constraints do not bind, and we can then

concentrate only on the first constraint (x + y + z ≤ 1). Find the solution and the

Lagrange multiplier, and show how the optimal value would change if instead the

constraint is x + y + z ≤ .9.

2. Consider the problem of maximizing xy subject to x2 + ay2 ≤ 1. What happens to

the optimal value when we change a = 1 to a = 1.1?

3. Consider Problem 1 in Problem set 5. Set the first order conditions, and for the

case of an interior solution use comparative statics to find changes in the endogenous

variables when I and px change (one at a time), i.e., find

(i)∂x

∂I,

∂y

∂I,

∂q0∂I

;

(ii)∂x

∂px,

∂y

∂px,

∂q0∂px

.

7

ec400 problem sets

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