B.Sc. (Hons) Degree in Social Sciences
B.Sc. (Hons) Degree in Mathematical Science with Finance and Economics
B.Sc. (Hons) Degree in Banking & International Finance
Part II Examination
SS2.06 Economic Theory Treated Mathematically
Friday 2 June 2000
5.00 pm – 8.00 pm
Answer FOUR of the ten questions
INTERNAL EXAMINER:
Professor D. Glycopantis
EXTERNAL EXAMINER:
Professor S. Thompson
Page 1 of 6
1.
The utility function of an individual is given by
U  [ x11  x 21 ] 1
where xi denotes the quantity of Good i. The individual is a price taker and has a fixed
income, M. The price of Good 1 is denoted by p1 and the price of Good 2 by p2 .
2.
(i)
Give first and second order conditions for utility maximisation subject to the
budget constraint.
(ii)
Calculate the demand functions and show that they are homogeneous of degree
zero in prices and income.
(iii)
Obtain the form of the indirect utility function.
homogeneous of any degree in prices and income.
(iv)
Suppose now that the individual has to pay a tax t1 per unit of Good 1 bought and
receives a subsidy s2 per unit of Good 2 bought, where s2  p2 . Obtain the
indirect utility function and explain for which combination of t1 and s1 the
individual remains as well off when the tax and subsidy are introduced as before.
Explain whether it is
The minimum cost function of a firm is given by
1
1
1
C*  2 y 4 p1 2 p2 2  p1  p2
where y denotes the quantity of output and pi the price of Input i.
(i)
Obtain the production function of the firm and show the shape of the isoquants on
an appropriate plane. Explain also whether the production function exhibits
constant, increasing or decreasing returns to scale.
(ii)
Calculate the expressions for
 x1
 x2
, and
where xi is the quantity demanded of
 p2
 p1
Input i and give an intuitive explanation of their signs.
(iii)
Verify the usual relation between the average variable cost and the marginal cost
functions of the firm.
Page 2 of 6
3.
A firm produces two outputs using two factors of production. The production functions
are given by Y1  min( L1 , K1 ) and Y2  min( L2 ,  K2 ) where Yi denotes the quantity of
Output i , Ki the quantity of capital in the production of Output i and Li the quantity of
labour in the production of Output i, and  is a positive constant. The total quantity of
capital at the firm's disposal is 1 unit and the total quantity of labour is 1 unit. The firm
can sell its products at fixed positive prices p1 and p2 where pi denotes the price of Good
i.
The firm's objective is to maximise its revenue.
4.
(i)
Derive the transformation frontier of the firm and explain how its shape varies
with .
(ii)
For =2, explain whether it is possible that the firm specialises in the production
of one good only.
(iii)
Suppose that technological change takes place in the production of Good 1 so that
any combination of labour and capital produce now 2 times as much as before.
Derive the new transformation frontier of the firm.
Consider the following neoclassical growth model:
1
Y  3K 2 L
1
2
 2L
dK  sY  K
dt
0 s 1
L  Loent
where Y denotes the quantity of output, K the quantity of capital and L the quantity of
labour. s, , n and Lo are positive constants. The initial quantity of capital, K(0)  Ko is
also given.
(i)
Interpret the relationships above and explain whether the marginal productivity
theory of distribution holds.
(ii)
Given the initial condition K(0)  Ko  0, discuss the questions of existence,
uniqueness and stability of the steady-state growth paths.
(iii)
Calculate the golden rule capital-labour ratio and, without engaging in any
calculations, give a condition which must be satisfied so that the steady state
capital-labour ratio and the golden rule capital-labour ratio will be equal.
Page 3 of 6
5.
6.
There are two duopolists in a market for a particular product. Their profit functions are
 1  (16  2(q1  q 2 ))q1 and  2  (20  (2q1  3q 2 ))q 2 where qi is the quantity of the
good produced by Firm i.
(i)
Calculate the collusion solution.
(ii)
Calculate the duopolists' reaction functions and the Cournot-Nash equilibrium.
Explain whether it is stable.
(iii)
Prove that the Cournot solution is inefficient and that the collusion solution is
efficient. Give an intuitive explanation of these results.
A particular economy consists of two traders and two goods. The utility function of
2
1
x
Trader 1 is given by U  (1  x11 )e 12 and of Trader 2 by U  log x 21  log x 22 , where
xij denotes the quantity of Good j consumed by Trader i. The initial endowments of
Trader 1 are ( x11, x12 )  (1, 0) and those of Trader 2 are ( x21, x22 )  (0,1).
(i)
Prove that the contract curve of this economy is given by x12 
2 x11
and show
1  x11
it on an Edgeworth-box diagram.
(ii)
Calculate the competitive equilibrium and show it on in your diagram. Check that
the competitive allocation is Pareto efficient.
(iii)
Explain why in calculating the competitive prices only relative prices can be
determined.
Page 4 of 6
7.
(a)
The indirect (maximum) profit function of a firm is given by
* 
Py2
2( PL   PK )
where Py is the price of output, PL the price of labour and PK the price of capital.
Obtain the form of the production function and show the isoquants on an
(L, K) plane. Explain whether the production function exhibits constant,
increasing or decreasing returns to scale.
(b)
A monopolist's total cost function is given by TC  8q  q 2 and his demand
function by p  10  2q where p denotes price and q quantity.
The objective of the firm is to maximise its profit.
8.
(i)
Calculate the solution of the monopolist's profit maximisation problem.
(ii)
Suppose the government taxes the monopolist according to the formula
2
T  To  tq where T is total taxes and To , t are fixed constants.
Calculate the effect of a change in t on the level of profits.
The indirect utility function of a consumer is given by
1 M

U*  
 logp1  logp2 
 3 p1

3
where M is income, pi the price of Good i,  is a positive constant and M  3 p1 .
(i)
Check that U* is homogeneous of degree zero in prices and income.
(ii)
Obtain the demand functions of Goods 1 and 2 and show that they are
homogeneous of degree zero in prices and income.
(iii)
Calculate
(iv)
Suppose an extra unit of income becomes available. Explain how it will be
allocated between the two goods.
(iv)
Obtain the consumer's utility function and show on a graph what the indifference
curves map will look like.
 x1
and give an explanation of its sign.
 p2 U  U
Page 5 of 6
9.
Consider the following linear programming problem:
Minimise C  3 p1  2(2   ) p2  6 p3
Subject to
2 p1  2 p2  0 p1  4
0 p1  2 p2  4 p3  6
p1 , p2 , p3  0
10.
(i)
For 0    1 , obtain the solution of the above problem and the solution of its
dual.
(ii)
Give a possible economic interpretation to the primal problem and the dual
problem.
(iii)
Explain what happens to the solution of the dual when  takes values greater or
equal to 1.
Consider the following macroeconomic system:
(1)
(2)
(3)
(4)
Y  C I G
C  Co  0. 75Y d
Y d  Y  To  tY
(5)
M  kPY 
(6)
YN
(7)
Y ' ( N )  W and
P
W
N
P
(8)
I  Io   r
1
r 1
where Y is real output, Y d is disposable real output, C is consumption, I is Investment, r
is the rate of interest, P is the price level, and M,G, are, respectively, the exogenously
determined supply of money and government expenditure.
Co , To , t , Io , k , ,  and  are positive constants with t<1.
(i)
Interpret the relations above and explain which is the liquidity trap rate of interest.
(ii)
Obtain from the relations (1) to (5) above the IS-LM curves and calculate the
effect on real output of an increase in G. Calculate also the effect on real output
of an increase in M and compare the two effects.
(iii)
Consider now the complete system. Explain briefly whether the system
decomposes into a real subsystem and a money subsystem.
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