5SSPP232 MATHEMATICS FOR ECONOMICS

1. Suppose that a pro
t maximising
rm can sell as many pencils as it likes at a price

p > 0: Its pro
t function is given by:

(q) = qp wq2

where w > 0.

(a) How much output will the
rm produce?

The FOC is p 2wq = 0: Hence q = p2w : The SOC is 2wq which ensures

that q indeed maximizes the
rms pro
t.

(b) How much pro
t will the
rm make?

(q) = qp wq2 = p24w :

(c) Now suppose that w increases. How does this a¤ect the optimal choice of

output and the
rms pro
t?

For the optimal choice of output just compute dq

dw = p2w2 : Hencet, output is

decreasing in w: For the pro
t, use the envelope theorem to obtain,

d (q)

dw

=

@ (q)

@q

dq

dw

+

@ (q)

@w

=

@ (q)

@w

= q2 = p

2

4w2

Alternatively, you could just compute directly d(q

)

dw = p

2

4w2

: Hence, pro
t is

also decreasing in w:

2. Find the stationary points of the following function, and determine whether each

corresponds to a max, a min, or an inection point.

f(x) = (x 1) 3

p

x2

Any stationary point must satisfy

f 0(x) = 3

p

x2 +

2

3

(x 1)x 13

= x

1

3 (x+

2

3

(x 1))

=

x

1

3

3

(5x 2) = 0;

which only has one solution at x = 25 : Note that f

0(0) = 20 = 1; so x = 0 is

not a stationary point. The second derivative is

f 00(x) = x

4

3

9

(5x 2) + 5

3

x

1

3 ;

which evaluated at x = 25 is f

00(25) = 2:262 > 0: Hence, x =

2

5 is a local minimum.

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3. According to the O¢ ce of National Statistics, 100 GBP in 1970 were equivalent to

1,457.38 GBP in 2017.

(a) Using this data,
nd the anual increase in prices (i.e. ination rate). In other

words, at which rate prices had to increase annually so 100 GBP in 1970

would buy the same basket of goods (i.e. have the same value) as 1,457.38

GBP in 2017.

Call the ination rate I: We are thus looking to
nd

100(1 + I)47 = 1457:38

which yields I = 0:0587 or, in other words, a 5.87% annual ination rate.

(b) Find the anual rate of ination assuming now that prices change monthly.

Now we are looking to
nd

100(1 +

I

12

)4712 = 1457:38;

which yields I = 0:05714, or, in other words, a 5.714% annual ination rate.

4. Calculate the following limits. Please show your work.

(a) Compute the limit from both above and below

lim

x!3

x2 4

x2 5x+ 6 =

5

0

= 1

lim

x!3

x2 4

x2 5x+ 6 =

5

0+

= +1

Hence the limit does not exist :

(b) limx!a x

3a3

x2a2 =

3

2a

(c) limx!1

p

x+2px+1p

x

= 0

5. Find the value of the following integrals

(a)

R 4

1

x2+x1p

x

dx = 22615 15:06

(b)

R

x sin(x)dx = sinx x cosx+ C

6. Determine whether the following matrices are invertible:

(a) det

1 2

1 1

= 1 so its invertible

(b) det

[email protected] 1 2 11 3 4

2 4 2

1A = 0 so its not invertible

2

7. Consider the following system of equations

x1 + ax2 = 3

2x1 + 2x2 = 4

(a) Write these equations in matrix form Ax = b

1 a

2 2

x1

x2

=

3

4

(b) Suppose that a = 2. Find the solution to this system of equations by inverting

A.

detA= det

1 2

2 2

= 2

adj(A) =

2 2

2 1

A1 =

1 1

1 0:5

So the solution to the system of equations is

x1

x2

=

1 1

1 0:5

3

4

=

1

1

8. Consider the following consumer problem:

max

x;y

u(x; y) = 60x+ 90y 2x2 3y2

s.t. 2x+ 4y = 68

(a) Use the Lagrange method to
nd the stationary point(s) of this function.

The Lagrangian is

L(x; y; ) = 60x+ 90y 2x2 3y2 (2x+ 4y 68)

The
rst order conditions are

@L(x; y; )

@x

= 60 4x 2 = 0

@L(x; y; )

@y

= 90 6y 4 = 0

@L(x; y; )

@

= 2x 4y + 68 = 0

3

Use the
rst two FOCs to establish that

30 2x

45 3y =

1

2

This together with the third FOC yields (x; y) = (12; 11):

(b) Find the bordered Hessian matrix and show whether this point(s) is a maxi-

mum or a minimum

HB =

[email protected] 0 2 42 4 0

4 0 6

1A

The minors

det(HB1 ) = 4 < 0

det(HB) = 88 > 0

Hence (x; y) = (12; 11) is a maximum.

(c) Find the value of the Lagrange multiplier . What is its economic interpre-

tation?

Substituting (x; y) = (12; 11) in one of the
rst two FOCs yields = 6. This

is how much u(x; y) increases when the income of the consumer moves away

from 68.

9. Consider the function

f(x; y) = x2 + y2 3xy

a. Find the Hessian matrix and
nd the conditions for f(x; y) to be concave/convex

H =

2 3

3 2

det(H1) = 2 > 0

det(H2) = 5

The Hessian is inde
nite. The function is neither concave or convex.

b. Find the maximum, minimum and saddle points (if any) of this function.

The stationary points are given by the FOCs

@f(x; y)

@x

= 2x 3y = 0

@f(x; y)

@y

= 2y 3x = 0;

which has only one solution (x; y) = (0; 0): This must be a saddle point because

the Hessian is inde
nite.

4

10. Use the Kuhn-Tucker method to solve the following maximisation problem

max

x;y

3x y

s.t. y ex

x 1

See solution to mid-term problem 5, which was the same problem.

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