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Demand and Supply - Mathematics - Exam, Exams of Mathematics

This is the Past Exam of Mathematics which includes Determine, Represents, Angle, Measured, Formula, Measured in Degrees, Value, Exact Value, Formula, Numerically etc. Key important points are: Demand and Supply, Unit Price, Quantity Demanded, Quantity Supplied, Equilibrium, Surplus, Calculate, After Tax, Coordinates, Evaluate

Typology: Exams

2012/2013

Uploaded on 02/26/2013

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SECTION A
1. The demand and supply functions of a good are given by
Demand : P=3
2QD+ 100,Supply : P= 2QS+ 16,
where P, QDand QSdenote unit price, quantity demanded and quantity supplied
respectively.
(i) Find the equilibrium price of the good, assuming that the market clears.
[4 marks]
(ii) Sketch the demand and supply functions, indicating the equilibrium
values of Pand Q. [4 marks]
(iii) Indicate on your sketch the levels of consumer and producer surplus at
equilibrium. Calculate their values. [4 marks]
(iv) Suppose that a tax of 14 per unit is imposed. Calculate the new
equilibium values of Pand Qand the tax revenue. [4 marks]
(v) Calculate the price elasticity of demand at equilibrium, after tax is
imposed. [4 marks]
2. Sketch graphs of the following quadratic functions, indicating the coordi-
nates of the points (if any) where the graphs cut the axes and the coordinates of
the highest or lowest point on the graph:
(i) y=x24,(ii) y=x2+x+ 2,(iii) y=x22x+ 1.
[9 marks]
3. State what is meant by the term ‘homogeneous function’ in the context of
the production function f(K, L) where Kdenotes capital and Ldenotes labour.
Show that each of the following functions is homogeneous and determine the
nature of its returns to scale.
(i) f(K, L) = 2K2/5L7/10
(ii) f(K, L) = K3+L3
KL
[8 marks]
4. (i) Show that 2 ln(x2y)3 ln(xy3) = ln(xy7). [4 marks]
(ii) Show that log10 100x3/2y5/2
xy != 2 + log10 x+ 2 log10 y. [4 marks]
Paper Code ECON111 Page 2 of 4 CONTINUED
pf3
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SECTION A

  1. The demand and supply functions of a good are given by

Demand : P = −^32 QD + 100, Supply : P = 2QS + 16,

where P, QD and QS denote unit price, quantity demanded and quantity supplied respectively.

(i) Find the equilibrium price of the good, assuming that the market clears. [4 marks] (ii) Sketch the demand and supply functions, indicating the equilibrium values of P and Q. [4 marks]

(iii) Indicate on your sketch the levels of consumer and producer surplus at equilibrium. Calculate their values. [4 marks]

(iv) Suppose that a tax of 14 per unit is imposed. Calculate the new equilibium values of P and Q and the tax revenue. [4 marks]

(v) Calculate the price elasticity of demand at equilibrium, after tax is imposed. [4 marks]

  1. Sketch graphs of the following quadratic functions, indicating the coordi- nates of the points (if any) where the graphs cut the axes and the coordinates of the highest or lowest point on the graph:

(i) y = x^2 − 4 , (ii) y = −x^2 + x + 2, (iii) y = x^2 − 2 x + 1.

[9 marks]

  1. State what is meant by the term ‘homogeneous function’ in the context of the production function f (K, L) where K denotes capital and L denotes labour.

Show that each of the following functions is homogeneous and determine the nature of its returns to scale.

(i) f (K, L) = 2K^2 /^5 L^7 /^10

(ii) f (K, L) =

K^3 + L^3

KL

[8 marks]

  1. (i) Show that 2 ln(x^2 y) − 3 ln(xy^3 ) = ln(xy−^7 ). [4 marks]

(ii) Show that log 10

( 100 x^3 /^2 y^5 /^2 √ xy

) = 2 + log 10 x + 2 log 10 y. [4 marks]

Paper Code ECON111 Page 2 of 4 CONTINUED

  1. Solve the equation 2000 = 4500e−^0.^7 t^ for t, giving the answer correct to two decimal places. [4 marks]
  2. Differentiate the following functions with respect to x. (i) y = 3x^2 + 5x + 1 [2 marks] (ii) e^3 x^ + e−^3 x^ [4 marks]

(iii) y =

2 x^2 x + 1

[4 marks]

  1. A person invests a lump sum of £15,000 for 15 years at an annual interest rate of 5%, compounded monthly.

(i) Find the final value of the investment. [4 marks] (ii) After how many months will the value exceed £20,000? [4 marks] Find also the number of months for the value to exceed £20,000 if the same annual interest were compounded continuously. [3 marks]

  1. Two projects will produce income as shown in the table, in each of the first four years. Calculate the present values of these projects, assuming a discount rate of 5%. Give your answers in £K to 2 decimal places. On these figures, which is the better investment now?

End of Year Project A (£K) Project B (£K) 1 3 2 2 4 2 3 2 4 4 2 3

[10 marks]

Paper Code ECON111 Page 3 of 4 CONTINUED

ECON111 Formulae handbook 2008-

Introduction

This handbook is designed for examination purposes, to be used in the Semester Examination. A copy will be provided with the examination paper.

The information contained below is not exhaustive of all the formulae and relationships derived and/or given in the lectures.

1 Solutions to the Quadratic Equation ax^2 + bx + c = 0 are

x =

−b ±

b^2 − 4 ac 2 a

2 Rules of Indices/Exponents (Power Laws)

(a) b^0 = 1 (b) b^1 = b (c) b−n^ =

bn^

(d) b

1 n (^) = n

b (e) bmbn^ = bm+n

(f)

bm bn^

= bmb−n^ = bm−n (g) (bm)n^ = bmn^ = (bn)m (h) (ab)n^ = anbn

(i)

( a b

)n

an bn^

= anb−n

3 Rules of Logarithms (a) M = bx^ ⇐⇒ x = logb M (b) logb(xy) = logb x + logb y (c) logb

x y

= logb x − logb y

(d) logb(xn) = n × logb x (e) logb( n

x) = logb(x n^1 ) = (^1) n logb x (f) logb(xyn) = logb x + n logb y Note that logb(x ± y) 6 = logb x ± logb y

4 Rules of Differentiation

(a) If y = axn^ where a and n are constants then

dy dx

= naxn−^1.

ECON111 Formulae handbook 2008-

(b) If y = uv, where u and v are functions of x, then

dy dx

= u

dv dx

  • v

du dx

(The Product Rule)

(c) If y =

u v

, where u and v are functions of x, then

dy dx

v du dx − u dv dx v^2

(The Quotient Rule)

(d) If y is a function of u and u is a function of x then

dy dx

dy du

×

du dx

(The Chain Rule)

(e)

Function Derivative

kun^ (k constant, u a function of x) knun−^1 du dx

ln x (^1) x

ln f (x) (or ln(u)) f^

′(x) f (x) (or^

1 u

du dx ) ex^ ex

ef^ (x)^ (or eu) f ′(x)ef^ (x)^ (or eu du dx )

5 Compound Interest (Single Payment) (a) The future value, S, of a principal, P , invested for t years at an annual compound rate of interest of r% is given by

S = P (1 + R)t, where R =

r 100

(b) If the interest is compounded k times per year, then the future value of the investment is given by

S = P

( 1 +

R

k

)kt , where R =

r 100

ECON111 Formulae handbook 2008-

(b) If the discount rate is compounded m times per year, then the present value of S is given by

P = S

( 1 +

R

m

)−mt , where R =

r 100

(c) If the discount rate is compounded continuously, then the present value of S is given by

P = Se−Rt, where R =

r 100

9 Present Value (Annuities and Perpetuities) (a) The present value, P , of an annuity worth A each year for t years at an annual compound rate of interest of r% is given by

P = A ×

1 − (1 + R)−t R

, where R =

r 100

(b) If the annuity worth A each year is paid in perpetuity (t → ∞) then its present value P is given by

P =

A

R

, where R =

r 100

10 Present Value (Irregular Income Streams) The present value, P , of a stream of future payments (B 1 at the end of the first year, B 2 at the end of the second year,... , Bn at the end of the last year) at an annual compound rate of r% is given by

P = B 1 × (1 + R)−^1 + B 2 × (1 + R)−^2 + · · ·+ Bn × (1 + R)−n, where R =

r 100