EC400 Problem Sets
description
Transcript of 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.
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 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
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 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
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 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.
3. Solve the following problem:
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
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 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.
2. Solve the following problem:
max(min{x, y} − x2 − y2)
6
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 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