UNIVERSITY OF EAST ANGLIA
School of Economics
Main Series UG Examination 2012-13
MATHEMATICAL ECONOMICS
ECO-2A03
Time allowed: 2 hours
ANSWER THREE QUESTIONS
Notes are not permitted in this examination.
Do not turn over until you are told to do so by the Invigilator.
ECO-2A03
Copyright of the University of East Anglia
Module Contact: Mr David Bailey, ECO
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1.
A firm can produce three different goods using three inputs. To produce a unit
of the first good requires two units of the first input, three units of the second input
and one unit of the third input. To produce a unit of the second good requires three
units of the first input, two units of the second input and one unit of the third input,
whilst to produce a unit of the third good requires two units of the first input, one unit
of the second input and three units of the third input. The firm has available 58 units
of the first input, 52 units of the second input and 35 units of the third input.
Its objective is to maximise the value of its sales revenue. The price of good one is
34, the price of good two is 36 and the price of good three is 20.
(a) Use the simplex method to find the amounts of the three goods that the firm
should produce to meet its objective. (18)
(b) What are the values of the dual variables? (3)
(c) By modifying the optimal table in (a), obtain the new optimal solution if a technical
change enables a unit of the third good to be produced with one unit of the first input,
one unit of the second input and three units of the third input. (12)
2.(a) Consider the following two functions:
F ( x1, x2 , y1, y 2 )  x12  2x1x2  x2  y12 y 2  0
G( x1, x2 , y1, y 2 )  x1x2  y10.5  6y 2  12.
(i) Demonstrate that in the vicinity of the point x1  2, x2  4, y1  4, y 2  1 the following
two implicit functions exist:
y1  h( x1, x2 ) and y 2  j ( x1, x2 ).
(9)
(ii) Obtain and evaluate at the above point the following two partial derivatives:
y1
y 2
 h1 and
 j2.
x1
x2
(8)
(b) Consider the following IS/LM model:
Y  C(Y  T (Y ))  I (r )  G  0
L(Y , r )  M  0
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where Y is income, C is consumption, T is income tax, I is investment, r is the rate of
interest, G is government spending, L(Y, r) is the demand for money function and M
is the money supply.
(i) Using your knowledge of macroeconomics, obtain and sign the following two
partial derivatives:
Y
r
and
M
G
(ii) Explain what happens to
Y
as I / (r )  0?
M
(iii) Explain what happens to
r
as Lr  0?
G
(10)
(3)
(3)
3. A firm’s physical output, Q, is given by
Q  256E 0.5N 0.5
where E is the amount of effort each worker puts into the job and N is the number of
workers. The amount of effort is itself a function of the wage, w, paid by the firm and
is given by the following function:
E  w 0.5  4 for w  16.
The firm’s costs are just the labour costs wN. At any wage greater than 16, the firm
will always have a long queue of applicants for jobs. The firm can sell as much as it
likes of the good at the exogenously given price p.
(a) Write down the firm’s profits as a function of p, w and N. (2)
(b) Differentiate the profits function partially with respect to w and N to obtain the
first-order conditions for a maximum. (4)
(c) Show that the profit-maximising value of w is independent of the price of the
product. What is its value? (10)
(d) Derive the elasticity of effort with respect to the wage rate at the profit-maximising
value of the wage rate. (4)
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(e) If the price of the product is 10, how large will employment be? (6)
(f) A serious epidemic reduces the size of the potential labour force and the supply
of labour to the firm, Ns, is now given by
Ns  400  25w
The price of the product is still 10. Explain why the previous equilibrium is no longer
feasible and discuss qualitatively how the firm should respond. (7)
4. A consumer has the following utility function:
U ( x1, x2 ) 
2
x  x21
1
1
She spends all her income on the two goods: x1 and x2; her money income is M and
the prices of the two goods are p1 and p2.
(a) Derive the utility maximising quantities of the two goods as functions of M, p1 and
p2. (16)
(b) Show that these two demand functions are both homogeneous of degree zero. (4)
(c) Prove that the maximum value function is given by:
V  V (M, p1, p2 )  U ( x1* , x 2* ) 
2M
p1  2 p1p2  p2
where x1* and x2* are the utility maximising quantities of the two goods. (6)
(d) Differentiate the maximum value function partially with respect to M and provide
an economic interpretation of this derivative. What is the value of the Lagrangean
multiplier if M  400, p1  4 and p2  4? (7)
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5.(a) On the island of Autarkia, there exists an exhaustible resource that can be
costlessly extracted and then sold. At time t  0 the remaining stock of the resource
is equal to 20A. The price of the resource and demand for the resource at time t are
given respectively by:
p(t )  p(0)exp(rt ) and x(t )  Ap(t )2.
where A is a positive constant, r is the rate of interest and p(0)  1.
(i) Calculate how much consumer surplus the consumers receive at time zero. (4)
(ii) Provide some economic intuition as to why the price should rise through time at a
rate equal to the rate of interest. (4)
(iii) If r  0.02 when will the resource be finally exploited? (4)
(b) Use integration by parts to find the following integrals:
 100t exp(rt )dt and  t
2
exp(rt )dt.
(12)
(c) A firm’s profits function is given by
(t )  100t  t 2
where t is time. The firm enters the industry at time t  0 and leaves at time t  100.
(i) How much profit does the firm earn in the industry? (3)
(ii) If the rate of interest is 0.1, find the present value of the stream of profits from
t  0 until the time the firm leaves the industry. Use the results you derived in part
(b). (6)
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6. (a) The rate of change in the number of Norfolk farmers who have adopted GM
seeds is given by the following differential equation:
dN(t )
 0.002N(t )(500  N(t ))
dt
where N(t ) is the number of farmers who have adopted GM seeds by time t.
(i) If N(0)  1, find the particular solution to the differential equation. How many
farmers finally adopt such seeds? (12)
(ii) Find the value of t at which half the long-run equilibrium number of farmers have
adopted the seeds. (5)
(b) The dynamics of a country’s output are given by the following second-order linear
difference equation:
y t  1.5y t 1  y t 2  150.
(i) Find the general solution to the difference equation. (10)
(ii) Derive the necessary relationship between y (0) and y (1) for output to converge
on the value of the particular integral you found in answering part (i). (6)
END OF PAPER
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