Exercises in Mathematcs for NEGB01, Quantitative Methods in Economics
Problems marked with * are more difficult and optional.
Part 1: Wisniewski Module A and “Logic and Proofs in Mathematics”
1. The following sets of co-ordinates are two points on a line. Draw the lines in a diagram.
i) (2, 3) and (-1, 0)
ii) (0, 0) and (2, -3)
iii) (5000, 0,5) and (2500, -2)
2. A company’s cost for producing x units is given by:
b(x)= 1000+300x+x2.
Determine
i) b(0),
ii) b(100), iii) b(101), iv) b(101)-b(100) and
v) b(x+1)
3- A demand function is given by the following: D(p)= 6,4-0,3*p where p is the price of
the good:
a) Determine
i) D(8),
ii) D(10)
b) If equilibrium demand D(p)= 3,13 what is the equilibrium price?
4. The cost (in millions of SEK) of reducing pollution in a lake by k% is given by:
10 * k
C (k )
105 − k
a) Determine the cost of reducing pollution in the lake by
i) 0 %
ii) 50%
iii) removing ALL pollution from the lake.
b) Explain (in words) the meaning of the expression C(50+h)-C(50).
5. Simplify without using a calculator.
i) 161/4
ii) 51/7*56/7
iii) (48)-3/16
6. Simplify without using a calculator.
i) (32+42)-1/2
ii) a*a1/2*a1/4*a3/4
(((3 * a ) −1 ) −2 * (2 * a −2 ) −1 )
a −3
iii)
7. a) Solve for K.
3*K-1/2*L1/3= 1/5
b) Solve for x
a*x*(a*x+b)-2/3+(a*x+b)1/3=0
1
8.
K(t) is the capital stock a firm has at the end of year t. Assume that K(t) increases by p%
each year. If K(0)=50; what is:
i) K(1),
ii) K(2),
iii) K(q),
vi) K(10)
9. Change “and” to implication and equivalence signs where appropriate.
a) “x2=9” and “x=3”
b) “x+2 > 3” and” x > 1”
c) “Peter is taller than Paul and Mary is taller than Peter” and “Mary is taller than Paul”
d) “It is a warm summer day” and “we are going swimming”
e) “x4=125x” and “x = 5”
10. Are these two statements true or false?
a) “x is any real number” and “x1/2 is the square root of x”
b) “x is any real number” and “x1/3 is the third root of x”
11. Express the following in terms of ln 3
a) ln 9
b) ln ( 3 )
c) ln ( 5 32 ) d) ln
12. Find x when
a) 3x=8
b) ln x=3
x x-1
x+1
f) 4 -4 =3 - 3x
c) ln (x2-4x+5)=0
13. Solve for t.
a) x=eat+b
b) e-at= 0.5
1
81
d) 3x4x+2=8 e) 3ln x +2ln x2= 6
14. Show that:
a) lnx- 2 = ln(x•e-2)
b) ln x-ln y+ln z = ln
xz
y
c) 3+2ln x=ln(x2e3)
15 . True or False ?
a) ln 5-ln 10 = -ln 2
b) (ln A)4= 4 ln A
c) ln B=2 ln B
A+ B
ln A
d) ln
e)
= ln A(BC)-1
= ln A + ln B − ln C
ln B + ln C
C
16. a) Show that if f(x)=100*x2 ; then f(t*x)=t2*f(x) for all t.
b) Show that if P(x)=x0,5, then P(t*x)= t0,5*P(x).
17. a) In Sweden 2014, the average (full-time equivalent) monthly of women was SEK 29 200
and that of men was 33 600. What was the wage difference between men and women in per
cent?
b) In the age group 16-64 years, the male unemployment rate was 8.3 per cent and the female
was 7.7 per cent. What was the gender difference in percentage points? In per cent?
18. The usual excise tax (“moms) on goods and services in Sweden is 25 %. What proportion
of the price paid by the consumer consists of excise tax?
19. How do real wages change if:
a) Prices increase 5% and nominal wages by 7%?
b) Prices increase 300% and nominal wages by 200%?
2
Part 2 Equations and polynomials
Part 2.1 Linear functions and equations, simultaneous equations
1. Which of the following functions are linear? Identify the slope and intercept of those
functions that are linear. Sketch the functions roughly using this information.
i) f(x)=2x+3
ii) h(g)=4-g/12
iii) y=0.5+2x-3x
iv) z=422+0,5x2
v) T(s)=s
vi) L(p)=0.5p+45
2. Find the equations for the straight lines in question 1. Part 1.
3. Solve the following systems of linear equations. Do a) and b) both by substitution and
elimination.
a)
2x + 3y = 1
(1)
5x – y = 11
(2)
b)
23x + 45y =181
(1)
10x + 15y = 65
(2)
c)
5x + 2y + 3z = 6
(1)
2x – y – 3z =3
(2)
7x + 4y + z = -4
(3)
4. Two linear functions, L(x) and M(x) cut each other at point P.
L passes through the co-ordinates (1,1) and has a slope = 3;
M passes through the co-ordinates (-1, 2) and (3, -1).
Determine the equations of L and M as well as the co-ordinates for P. (Give an exact answer
for coordinates of P.)
5. A national income model is Y = C + I + G
C = a +b(Y-T)
a>0
0<b<0.90
I = d+rY
d<0
0<r<0.10
G, r and T are exogenous
r = the rate of interest, other notation is as in Wisniewski
a) Find the public consumption multiplicator. Interpret it.
b) Find the tax multiplicator. Compare with a). Interpret!
c) Find the public consumption multiplicator if T = tY 0<t<1. Compare with a) and
interpret.
6. Do the following equations have any real solutions? How many?
a) x2 + 1= 0
No solution
2
b) x - 1= 0
Two solutions
c) x4 = 0
One solution
2
2
d) (x-1) + (y-3) = 1
Infinitely many solutions
e) (x-1) 2 + (y-3) 2 = 0
One solution
3
7. Solve the systems of equations:
a) 𝐾𝐾 0.75 𝐿𝐿−0.25 = 2𝐾𝐾 −0.25 𝐿𝐿0.75
2K + 3L = 7 000
1
1
3
b) 𝑥𝑥 2 𝑦𝑦 − 2 = 4
x + y = 100
4
3
1
1
5
c) 3𝑥𝑥 − 2 𝑦𝑦 6 = 2𝑥𝑥 2 𝑦𝑦 − 6
x + y = 30
x, y > 0
Part 2.2 Roots of polynomials
1. Find the roots of the following quadratic equations and write the left sides of the equations
as products of linear factorsx 2 10 x
i) x2+5x+6=0
ii)
iii) 2x2-8x+8=0
iv) 9x2+27x+54=0
+
−4=0
6 12
2. a) Find the zeroes (roots) of p(x) = 2x3-2x2-4x
*b) One of the zeroes of the polynomial p(x) = x3+3/2 x2 – 9x +4 is x = 2. Use the factor
theorem to find the other two. (Hint: Calculations are easier if you write numbers as fractions
rather than with decimals.)
Part 3. Derivatives
Part 3.1
1. Differentiate the following expressions:
a) y=5
b) y=x4
c) y=9x10
d) y=π7
2. Differentiate the following expressions. Assume g´(x) is known.
1
a) y=- g(x)+8
6
g ( x) − 5
b) y ( x) =
3
3. Differentiate the following expressions:
x12
a) s ( x) =
12
−2
b) f ( x) = 2
x
−2
c) B( x) =
x⋅ x
1
d) f(A)= 2
A * A
a 2 + f (a )
e) g(a)= 2*a+h+
h is a constant.
h
4. Differentiate the following expressions:
a) f(x)= x+4
b) f(x)=x+x2-0.6
c) f(x)= 3x5+2x4+1
d) f(x)= 8x4+ 2 x +12
𝑥𝑥
e) 𝑓𝑓(𝑥𝑥) = 2 −
3𝑥𝑥 2
2
+ 5𝑥𝑥 3 − 34
5
5. Differentiate the following expressions using the product rule.
a) y(x)= (2x2-1)(x4-1)
1

b) z (t ) =  t 5 +  ⋅ (t 5 + 1))
t

 x2 +1 
 ⋅ x )
c) h( x) = 
 x 
6. Differentiate using the quotient rule. f ( x) =
x −2
x +1
7. Differentiate
S(p)= p*D(p) where D is a differentiable function of p
Part 3.2 Derivatives continued
1. Determine the values of x where
dy
=0
dx
a) f(x)= 3x2-12x+13
b) f(x)= 0,25(x4-6x2)
x 2 − x3
c) f ( x) =
2x + 2
2. Determine the equations of the tangents to the following functions at the given points.
a) y= 3-x-x2 at x=1
x2 −1
b) y ( x) = 2
at x=1
x +1
If you found a) and b) difficult, do c) and d) too, otherwise skip them.

 1
c) y ( x) =  2 + 1 * (x 2 − 1) at x=2

x
4
x +1
d) y ( x) = 2
at x=0
x + 1 * ( x + 3)
3. Differentiate:
at + b
a) y (t ) =
ct + d
1
b) y (t ) = 2
at + bt + c
y2 + 2
c) L( y ) =
y8
df ( x )
*4. Show that if f ( x ) = x-n then
= -nx-n-1 using the quotient rule.
dx
5. Differentiate:
a) g(x) = (2x+1)3
b) h(x)= (1-x)5
1
c) R( x) =
2
(2 x + 3 x − 4) 2
(
)
6
d) F ( x) = ( x 3 + 1)
1/ 2
 2x +1 
*e) B( x) = 

 x −1 
f) y(x)= (1-x2)33
g) d(t)= (at2+1)-3
h) y(t)= (at+b)n
6. Determine
dy
dx
when y=
b) (f(x))4
a) x+f(x)
c) x*f(x)
d)
dy
using only x in the expression. If y= 5u4 and u= 1+x2
dx
dy
8. Determine
as a function of t if y= -3*(v+1)5 and v=t3/3
dt
dK
as a function of t if K=ALa and L=bt+c.
9. Determine
dt
dy
10. Determine
when y=
dx
1
2
5
a) ( x + x + 1)
7. Determine
*b) x + x + x
c) xa(px+q)b
11. If a(t) and b(t) are both differentiable functions and A, L and B are constants then
determine
x´(t )
x(t )
a) x= (a(t))2*b(t)
b) x=A(a(t))L*(b(t))B
12. Differentiate the following expressions:
a) C(x)=20q -4q(25-0,5x)0.5
b)* F(x)= f(xng(x))
13. Find the second order derivatives of the following functions:
a) y= x5-3x4+2
b) y = x
c) y=(1+x2)10
7
f ( x)
14
a) z= 120t-
t3
3
b) f(x)= g(u(x))
d 3z
dt 3
determine f´´(x)
determine
15. a) f(z)=z-4
determine f(4)(1).
t2
b) g (t ) =
determine g´´(2)
t −1
Part 3.3 Derivatives of exponential functions
In the following, note that ef(x) can be written as exp(f(x))
1. Differentiate the following expressions where f(x) =
3
a) e-3x
b) 2 e x
c) e1/x
d) 5exp(2x2-3x+1)
2. Find the derivatives of the following functions:
a) f(x) = exp(ex)
b) f(t) = et/2+e-t/2
d) g(z)=(exp(z3)-1)1/3
e) h(x) = xex
c) f ( x) =
3. Differentiate the following functions:
a) y= ln(x+1)
b) y= lnx+1
d) y =
x
ln x
1
e + e −x
x
c) y= xlnx
e) y= ln(x2+3x-1)
4. A function is given by y=ln x. Find the equation of the tangents to the graph at the points:
a) x=1
b) x=e
*5. Some derivatives f’(x) are easiest to calculate if instead of f(x) you differentiate
𝑓𝑓′(𝑥𝑥)
g(x) = ln f(x) and use the fact that 𝑔𝑔′ (𝑥𝑥) = 𝑓𝑓(𝑥𝑥) . Use this method (logarithmic
differentiation) to find the derivatives of the following functions:
1/ 3
 x + 1
a) y= 
b) y=xx
c) y= (x-2)1/2(x2+1)(x4+6)

 x − 1
Part 3.4 Elasticities
1. Determine the general elasticity for the following functions where f(x)=
a) 3x-3
b) -100x100
c) x0.5
h)A*g(x)
i) x+x2
j) a+xb
A
x x
k) a+b/x
d)
8
e) 3
f) x+1
g) (1-x2)10
2.* A firm’s total cost function is given by C(Q) where Q is the quantity produced. Average
cost is given by AC(Q) =
C (Q)
Q
Show that AC´(Q)=0 if and only if the elasticity of C with respect to Q, ε =1.
Part 4: Optimization of functions of one variable.
If you are short of time, leave 5) and 7) for later revision.
1.Find the stationary points of the following functions. Determine the nature of these points
(max, min, inflection?) Sketch the functions roughly using this information.
a) f(x)= x3+3x2+2
x
b) f ( x) =
1 + x2
5
c) h(y)=y -5y3
d) z(x)= (x-2)5(2x+1)4
2. A function is given by: F(x)=
1
3x − 5
x−2
where x ≠ 2,
a) Show that 𝐹𝐹(𝑥𝑥) = 3 + (𝑥𝑥−2)
b) Show that F(x) is decreasing for all values of x ≠ 2.
3. Determine the stationary points of the function G(x)= x1/3*(x-7)2 and determine the nature
of these points.
4. A Cobb-Douglas function has the form.
Q( K, L )= AKαLβ
a) Define L as a function of K for an arbitrary fixed value of Q ; Give this function an
economic interpretation.
b) Show that L is a decreasing function of K for all K.
5. Determine the stationary points of the following functions. Determine the nature of these
points, as well.
a) y= -2.05+1.06x-0.04x2
8x
b) h( x) = 2
3x + 4
6. A square piece of cardboard has the dimensions 18 x 18 cm. If we cut out 4 identical
squares from each corner of the cardboard , it can be folded to form an open box. The side of
each cut corner is denoted x.
x
a) Determine an expression V(X) for the volume of the box
18 cm
b) For which value of x is the volume of the box maximized ?
c) For what values is V(x) increasing, decreasing ?
7. a) Show that if f(x)= 3-(x-2)2 then f(x) ≤ 3 for all x.
b) Find the stationary point of the function T(x)= 2-(1-x)1/2
9
18 cm
8. A company in a perfectly competitive market has total costs
TC=Q2+10Q+900.
where Q is quantity of the good produced.
The price of the product is constant and =80 kr.
a) Determine the profit function π for the company.
b) At what Q is profit π =0?
c) At what Q is profit π maximized?
9). Find the max. and min points of:
a) p(x)=a+k(1-e-cx)
where a, k and c>0
b) y= x2e-x
Part 5: Functions of more than one variable.
1. a) z( x,y)= xy2 determine
i) z(0,1)
ii) z(-1,2)
iii) z(a,a)
b) f(x,y)= 3x2-2xy+y3 determine
i) f(1,1)
ii) f(-2,3)
2. If F(K,L)= 10K1/2L1/3 where K and L ≥ 0 then determine
a) F( 2K,2L)
*b) Determine α so that F(tK,tL)= tαF(K,L).
Does this production function have diminishing, constant or increasing returns to scale?
3. Differentiate the following functions with respect to x and y.
a) z(x,y)= x2+3y2
c) P(x,y)= 5x4y2-2xy5
b) K(x,y)=xy
4. Find the partial derivatives of the following functions.
a) F(x,y,z)= 3xyz+x2y-xz3 b) Y ( K , L)
K 2 L2
aL3 + bK 3
Hint: in c) you get less work if you use that
2
5. Let z(x, y) = x -8xy – y
3
𝑑𝑑[𝑙𝑙𝑙𝑙𝑙𝑙(𝑥𝑥)]
𝑑𝑑𝑑𝑑
c) T ( x, y ) =
=
( x 2 y − 4 y)2
(2 y 3 + x 4 )3
𝑑𝑑𝑑𝑑(𝑥𝑥)
𝑑𝑑𝑑𝑑
𝑇𝑇(𝑥𝑥)
a) Find the differential of z
b) if x(t) = 3t and y(t) = 1-t, what are the total differential and total derivative of z with
respect to t?
c) Use substitution to write z as a function of t, differentiate and check that you get the same
answer as in b).
6. A firm has the production function Q = Q(L, K) = 10K1/4L1/2
10
a) How much does the firm produce if K = 625 and L = 64? (Try to calculate without using a
calculator.)
b) Use a) and the total differential to find approximate values of Q(64.05; 625.1) and Q(100,
725)
c) Use a calculator or excel to find Q(64.05;125.10) and Q(100, 725) rounded to 5 decimals.
Comments?
Part 6 Unconstrained and constrained optimisation of functions of several variables.
Part 6.1 Unconstrained optimisation
1. Determine and solve the First Order Conditions (FOC) for the following functions:
a) f(x,y)= -2x2-y2+4x+4y-3
b) P(x,y)= -x2-y2+22x+18y-102.
2. f(x,y)= x2+y2-6x+8y+35 .
a) Solve the FOC of the function.
b) Show that f(x,y) can be written as f(x,y)=10+(x-3)2+(y+4)2.
c) Show that f(x,y) ≥ 10 for all x and y. What does this imply with regard to your answer in a)
3. A firm has the production function F(K,L)= 80-(K-3)2-2(L-6)2-(K-3)(L-6) where K
represents capital and L labour. The price of the product is p=1, the cost of capital is r =0.65
and the wage rate is w= 1.2.
Find the values of K and L that satisfy:
∂F r
=
∂K p
and
∂F w
=
∂L p
. Give an economic interpretation
4. Find the max. , min or saddle points for the following functions:
a) f(x,y) = 2x3+2y3-6xy.
b) h(x,y)= x2y3( 6-x-y ) +15, where x, y >0.
c) f(x,y)= x4+2y2-2xy
d) g(x,y) = ln( 1+x2y)
*5. Find the values of a, b and c such that F(x,y) = ax2y+bxy+2xy2+c has a local minimum at
the point (2/3, 1/3) and F(2/3, 1/3)= -1/9.
11
Part 6.2 Constrained optimization
1. Maximize f(x,y)= x+y given that g(x,y)=x2+y=1
a) Using substitution.
b) Using the relation
f ′ x g′ x
=
f ′ y g′ y
c) Using the Lagrange method. Interpret the Lagrange multiplier.
2. a) Use the Lagrange method to find the points that satisfy the first-order conditions for a
maximum or minimum of the function f(x,y)= 3xy under the constraint that g(x,y)=x2+y2=8.
b) Try to illustrate the problem graphically. Can you determine which of these points
maximizes / minimizes the function.
3. Use the substitution method to determine the minimum of x2+y2 given that x+2y=a.
4. Henrik´s utility function is given by U(x,y)= 10x1/2y1/3.
The price of good X is 3 kr while the price of good Y is 1 kr. Assume that Henrik will spend
150 kr on goods X and Y. Assuming that Henrik is rational, what quantities of X and Y will
he consume?
*5. Bengt´s utility function is given by U(x,y)= 100-e-x-e-y.
The price of good X is p kr, while the price of good Y is q kr.
Bengt will spend R kr on goods X and Y.
Assuming that he is rational, determine an expression for Bengt´s demand of the two goods.
6. A utility function is given by U(x,y)= x3+y3.
The price of good X is 9 kr, while the price of good Y is 4 kr. Determine the optimal
combination of x and y in order to obtain utility U= 35000.
7. A Cobb Douglas production function is Q(K,L)= 80K0,75L0,25; the price of capital is 3,
while the price of labour is 2. Minimize the cost of producing 6400 units.
8. Maximize F(x,y)= ln(xy2) given that 2x+3y=18.
12