Mathematical Methods: AE04/EE04

First Assessment - 25th September 20204

I
Instructions: Maximum time allowed - 1.5 hours; Maximum marks - 25
QI. State whether the following statements are 'true' or 'false' and provide brief explanation
to support your answer. (2x5 = IO marks)
a) Suppose that a firm produces two outputs y, and y2 using two inputs z, and z2. Let the
input requirement matrix A, output vector y, and input price (in Rupees) vector w be:

Then, wTAy represents total costs incurred by the firm and are equal to Rs. 650.
b) An investor is considering investing in a bond that gives interest rate at continuous
compounding. At 5% interest rate Rs. 300 would double in ten years.
0
c) Ram's utility function depends on number of coconuts (C) he consumes, U(C) = 100-
(C-9)2. Additional coconuts always increase Ram's utility.
d) Suppose that the government has been taxing each person's income at a marginal rate
of 0.25 for every rupee in excess of Rs. 20,000. The government decides to generate
extra tax revenue but wishes to avoid increasing the tax burden on low- or middle-
income earners. Therefore, the government decides to impose a lump-sum surtax of Rs.
''
1,000 on every person who earns Rs. 60,000 or more. The income after tax, written as
a function of income before tax is discontinuous at Rs. 60,000.
e) Consider the following simplified national-income model:
Y = C + I + G; C = a + b Y
where Y and C are endogenous variables and / and G are exogenous variables. The
parameters, a and h, represent the autonomous consumption expenditure and lhe
marginal propensity to consumer, respectively. The solution for Y, written in matrix
1/+G -11
notation using Cramer's rule would be: Y = If -ll
-b 1
Q2. (a) A firm wants to bid for the monopoly franchise to sell samosas at a cricket match. It
estimates the inverse demand function for samosas as: p = 5 - 0.5x, where p is the price in
Rupees and x is samosas in hundreds. It also estimates that it can supply samosas at a constant
unit cost of Rs. 0.50 per samosa. (i) What is the largest bid it would make for the franchise?
(ii) Supposing that the stadium owners levy a royalty of Rs. 0.25 per samosa sold, what will be
the largest bid the firm will make to get the franchise? (4 marks)
(b) The value of a parcel of land bought for speculation is increasing according to V(t) =
2oooet 114 _ If the interest rate is IO percent, how long should the parcel be hold to maximize
the present value? (3 marks)
Q3. (a) In a two industry economy, industry I uses IO paise of its own product and 60 paise of
commodity 11 to produce a Rupee worth of commodity I; industry II does not use any of its
own product, but uses 50 paise of commodity I in producing a Rupee worth of commodity II.
The final demands are, Rs. I 000 billion of commodity I and Rs. 2000 billion of commodity II.
Write down the input-output matrix equation of this economy, and find the equilibrium level
of output using Cramer's rule. (4 marks)
1 3
(b) Show that the matrix, A= (~ ~) is diagonizable as D = (~ ~)- (4 marks)

\
 Mathematical Methods: AE04/EE04
Second Assessment-13 th November 2024

Instructions: Maximum time allowed - l hr 30 mins; Maximum marks - 20

QI. Consider the utility maximization problem of a consumer: u(x, y) "-' x + y' 12, subject to
budget constraint: PiX + pyy ~ I, where Px and py are prices of goods x and y, and / is the income.
The first order conditions (for the case of an interior solution) are:
f-pxX 't- pyy = O; 1-Apx = O; 0.5y 1'i - A.py = 0
where, ,l, is the Lagrange multiplier associated with the constraint. Using comparative statics
show that the impact of change in px on x (i.e., ox~ ) is negative.
/opx
(5 marks)

Q2. An electric company is setting up a power plant and it has to plan its capacity. The peak
period demand for power is given by p, = 400 - q, and the off-peak period demand is given by
pi = 380 - qi, where p & q are prices and quantities in the respective periods. It may be noted
that the capacity, K, is chosen in such a way to meet the peak period demand, and the off-peak
period demand would then automatically met by the chosen capacity. The variable cost to the
company is 20 per unit (in both peak and off-peak periods) and the capacity costs are 10 per
unit (paid only once at installation). The company's optimization problem can be stated as:
Max (400-q,)q, + (380-qi)qi - 20q1 - 20q2 - IOK
Subject to: q, ~ K & q2 < K

Note that q, > 0 & q, > 0. Also, q2 (off-peak demand) will be strictly less than the capacity K.
(i) Write Kuhn-Tucker conditions for this problem;
(ii) Find the optimal outputs (q1 & q2) and capacity (K) for this problem; and
(iii) What is the value of1,, the Lagrange multiplier associated with the first constraint?
(5 marks)
Q3. Solve a competitive firm's profit maximizing use of labour and capital when output, y =
Loi5K°·25, price of output, p = 64, wage rate of labour, w = 2, and price of capital, r = 4. Show
that the solution is a true maximum (i.e., check the relevant sufficiency condition).
(5 marks)
Q4. (i) A company must fill an order for 200 units of its product. It wishes to distribute the
production between its two plants, plant A and plant B. The total cost function is c = 300qA +
200qn + qi + 8000, where qA and qs are the number of units produced at plant A and plant B,
respectively. How many units should be produced at plant A and plant B to minimize costs?
(No need to check sufficiency condition).
(ii) Show that the quadratic form, q(x,, xi, x3) = 4xt2 + 9xl + 2x/ + 8xix3 + 6x1x3 + 6 x1xi is
indefinite.
(3+2 marks)
,
MADRAS SCHOOL OF ECONOMICS

POSTGRADUATE PROGRAMMES IN
ACTUARIAL ECONOMICS / APPLIED QUANTITATIVE FINANCE I
ENVIRONMENTAL ECONOMICS / FINANCIAL ECONOMICS / GENERAL ECONOMICS
[2024-26]

SEMESTER ONE [AUGUST- DECEMEBER, 2024)

~

I REGULAR E~D TERM EXAMINATION, DECEMBER 2024

I Course Name: MATHEMATICAL METHODS Course Code: AE04/EE04

Duration: 2 Hours Maximum Marks: 60

1 Part - A (IO x 1 = 10 Marks)

I
Answer ALL Questions
State whether the following statements are 'true' or 'false' and provide brief explanation to support
your answer.
1. Total tax revenue (R, expressed in rupees) is a function of tax rate (t, expressed in
percentage): R = 50 +25t -75t2. The marginal tax revenue from one-percent increase in tax
rate (when the current tax rate is 10%) is equal to 15.
ij
2. Let the supply and demand for petrol are given by: q = (1/4) p 4; q = 8 p-1, where pis the
; price and q is the quantity. Expressing supply and demand as relations between logarithms
of the price and quantity and solving the resulting pair of simultaneous linear equations
yields p = 2 and q = 4.
3. Consider profit maximization problem of a firm facing constraint on input x. The Lagrange
multiplier reflects the maximum amount the firm is willing to pay to be allowed to use one
more unit of input x.
4. The marginal rate of substitution (between x2 and x1) for a utility function U(x1, x2) = x1+x2

I is equal to 1.
5. One solution to the system x3y- z = 1, x+y2+z3 = 6 is x=l, y=2, z=l. It is possible to
estimate x and y values when z changes to 1.1.
6. Suppose that a government taxes each person's income at a marginal rate of 0.4 for every
rupee in excess of Rs. 25000 (i.e., the first 25000 rupees earned are not taxed). In addition,

I assume that the government imposes a lump-sum tax of Rs. 2000 on every person who
earns Rs. 100000 or more. The income after tax (say, y) as a function of income before tax
(say, x) is discontinuous atx = 100000.
7. Consider the following specific CES production function defined on x1 > 0 and x2 > 0: Y
= f(x1, x2) = [0.3x1-2+0. 7xi2l 112. This production function is homogenous of degree 1.
8. In a linear programming problem involving 3 decision variables and 4 constraints, there
will be 3 positive values in the seven variable solution space.
9. If a resou rce is not fully utilized by a profit maxim izing
resource must be great er than zero.
10. Utility function U=4x1x2 is strictly quasiconcave (for
firm, then the shado w price of the
\
x,>O and x2>O).

Part - B (5 x 10 = 50 Mark s)
Answ er Any Five Questions
11. (i) Sruth i pays incom e tax according to:
T(X) = 0 if X < E; t(X - E) if X ~ E
where , Xis her pre-ta x income; E and t are positive const
ants; t < 1.
Sruth i is also eligib le for an income related transfer:
B(X) = s(P - X) if X < P; 0 if X ~ P
where , P ands are constants; P > 0; t<s<l.
Find the relationship betwe en Sruth i's disposable incom
e, Y and X for (i) E > P, and (ii) E
< P; s+t < 1.
(ii) A wine deale r owns a case of fine wine that can be
sold for k.exp(t 112), t years from
now. There is no storag e cost and interest rate is r. Find
the optim al sellin g time of the wine
case?
(iii) Cons ider a Cobb-Douglas production function, Q = 4K314L114• Let,
1012
K(t,r ) = and L(t,r ) = 6t + 250r. Find the rate of chang e of outpu t (Q)
2
r with respe ct
to chang e in r when t= 10 and r=O. l.

(4+3+3 marks)
12. (i) In welfa re econo mics we consider the marginal
benefits to socie ty from consu ming a
good (MB) and the marginal cost to society from producing
the good (MC). Let MB & MC
be: MB = a + /JQ, + yE; MC = 0 + AQ, where, 13 < 0, y
> 0 and A> 0. Q is level of outpu t
and E is consu mptio n externality from consu ming the
good (E>0 indicates a positive
externality; E <0 indicates a negative externality). The socia
lly effici ent level of outpu t will
be wher e MB= MC= M. (i) Suppose that there are no extern
alitie s (i.e., E=0). What would
be the socially effici ent level of outpu t (Q1)? Assum ing
positi ve consu mptio n externalities
(i.e., E>0), calcu late the new socially effici ent level of
outpu t (Q2). (ii) Based on these
calcu lation s, does a positi ve consu mptio n externality cause
the optim al level of outpu t to
be highe r or lower than the outpu t level with no externality?
(Use matri x notation and Cram er's rule to solve Qi and
Q2).

6) is diagonizable as D = (~
(ii) Show that the matri x, A = (~ ~ 7
~).

(S+S marks)
13. The mark et for a single comm odity is described by
the follow ing set of equations:
Qd = D(P,G ); Qs = S(P,N)
Alon g with the mark et cleari ng condition (viz., Qd = Qs).
Here, G is the price of substitutes,
and N is the price of inputs, and both G and N are
exoge nousl y given . The following
assum ption s are imposed:

j
 an < 0. an > 0. as 0. as 0
aP ' ac •aP > •aN <
· . aP'·aQ'
ap' ·- aQ'
-·and- .
Fmd the expression and sign of the derivatives: -
ac; ' aN ' ac; ' aN
14. (i) A monopoly firm produces two outputs, x 1 and xi, with the linear demand functions:
XJ = ]00-2p1 + pi

xi= 120 + 3p,-5pi

Let the cost function of the firm be: C = 50 + I Ox, + 20x2. Show that the profit maximizing
levels of output are x, • = 41.35 and xi•= 45.53, and find maximum profit.
(Hint: note that to write the profit function you need to find the inverse demand functions
using the given demand functions for the two outputs. Also, you may need calculator).
(ii) Consider the quadratic form Q() = x2+ 2axy+ 2xz+z2. Show that Q(.) is positive
semidefinite if a=0.
(7+3 marks)
15. (i) A firm manufactures a commodity at two different factories. The total cost of
manufacturing depends on the quantities, q l and q2, supplied by each factory, and is
expressed by the joint cost function,
C= f(q1,q2 ) =2qt +q1q2 +q; +500
The company's objective is to produce 200 units, while minimizing production costs.
How many units should be supplied by each factory? Check the second order conditions
also. What is the interpretation of the Lagrange multiplier?
(ii) Show that the function y =xtx~ 12
defined on 9?!+is strictly concave.
(6+4 marks)
16. The consumption of goods by a consumer takes time as well as money. So it is interesting
to consider the utility maximization problem with both a budget and a time constraint.
Use the Kuhn-Tucker conditions to solve the problem:
Max U(x,y) = xy
s.t. x+2y ~ 40;x+ y ~ 24
Consider x and y as strictly greater than zero. Analyze four remaining cases to find the
optimal solution. Which constraint is binding at optimum?
17. Consider the following Linear Programming problem:
Maximize 2x,+4x2+3x3+x4
Subject to: 3x1+x2+x3+4x4 S 12; x,-3xi +2x3+ 3x4 S 7; 2x,+xi+3x3-x4 S 1O
XJ, Xi, XJ, X4 ~ 0
a. Write the dual of this problem.
b. Show that primal optimal solution will be (0, 10.4, 0, 0.4) if the dual optimal solution
is given as (1, 0, 3).
c. Find the primal and dual objective values.
(3+5+2 marks)