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SECTION A
- 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]
- 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]
- 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]
- (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
- Solve the equation 2000 = 4500e−^0.^7 t^ for t, giving the answer correct to two decimal places. [4 marks]
- 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]
- 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]
- 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
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
