Download Question Bank Economics (Hons.) Third year Paper VIIB

Question Bank
Economics (Hons.)
Third year
Paper VIIB
Mathematical Economics
Section A
Each Question carries 4 marks
1) Construct the ordinary demand functions for X1 and X2 for the utility
function and budget constraint
U=1\4logX1 + 3\4logX2
M=P1X1 + P2X2
2) Show that λ=δv\δM, where λ=lagrange multiplier.
3) Given the production function q=k½L½ and the prices of inputs are Pk=Rs
2and Pl=Rs 4, derive the total cost function.
4) If we assume that the production function of a firm is
q=9x1½ +9x2½
If the prices of the inputs are equal then show that the firm will use equal
quantities of x1 and x2.
5) Let u and v be two utility functions with v (w) =f [u (w)], where f is
concave and w denotes wealth. Prove that the co-efficient of Absolute risk
aversion for v is more than that for u.
6) Derive the fundamental differential equation of the Solow growth model,
clearly stating the assumptions.
7) Find the effect of the imposition of a per unit tax on price for a
monopolist if his marginal cost is constant.
8) Mr X intends to invest Rs 5000(his only wealth) is a risky project that
might yield Rs 25000 if the project succeeds and nothing otherwise. If he is
indifferent between these two options and his utility function over wealth is
given by u=w-2w2, find out the probability that the project fails.
9) A publisher would be interested in maximizing profit while the author
will be interested in maximizing sales since he gets a fixed royalty of 20%
on sales proceeds. The demand function is P=20-0.0002Q. The cost
function is given by C=0.00168Q. Find the optimum number of books to be
printed for the author and publisher.
10) suppose the utility function is given by V (ω) =α ω -β ω 2 (α, β>0)
Find how the inclination towards risk changes with increase in wealth.
11) In a game of matching coins with 2 players, A wins 1 unit of value
when there are two heads, wins nothing when there are two tails and loses
1\2 unit of value when there are one head and one tail. Determine the payoff matrix, the best strategy for each player and the value of the game to A.
12) Given a continuous income stream at the constant rate of Rs 1000 per
year, what will be the present value of profit if the income stream lasts for
two years and the continuous discount rate is 0.05 per year?
13) Calculate the value of expected pay-off for the row player in the
following two person zero sum game
1
6
5
4
Where row player plays the first row strategy with probability 2\5 and the
column player plays the first column strategy with probability 1\5.
14) Check whether the following input-output matrix satisfies the HawkinsSimon condition:
0.1 0.3 0.2
A= 0.4 0.1 0.3
0.3 0.5 0.2
15) Construct an indirect utility function that corresponds to the direct
function
U=αlog (Q1 + Q2)
16) Contrast risk aversion with risk loving behaviour in terms of property of
utility function of the individual.
17) What is dominance rule in game? Explain by constructing a suitable
game matrix.
18) What are the assumptions that are required to „close „the Leontief static
open model? Give a description of such a closed model.
19) Prove that the isoquants of a “CES” production function are negatively
sloped.
20) In an n – period utility maximization problem, show that an increase in
the interest rate raises present wealth and leads to greater present
consumption.
21) Let H=eQ where Q=AKαLβ.Show that the function H is homothetic but
not homogenous.
22) How can you determine the intertemporal equilibrium nature of the time
path from a phase diagram? Is the differential equation dy\dt +ay=b
dynamically stable?
23) If the demand function for a profit maximising monopolist is P=274-Q2
and MC=4+3Q, find consumer surplus.
24) Consider the following 2×2 game
Player B
Player A
2
7
5
3
Determine the optimal mixed strategy for the players and also find the value
of the game.
25) In a two sector economy, the input co-efficient matrix and the final
demand vector are given by
0.2
0.2
A=
20
&C=
0.3
0.3
30
Find out the sectoral output X1 and X2. If labour requirements to produce
one unit of X1 is 0.5 and one unit of X2 ,then what will be the minimum
amount of labour supply that will be required to make the output system
productive.
Section B
Each question carries 6 marks
1) Set up the Samuelson model of the multiplier accelerator interaction .Use the
solution of the second order difference to interpret the time path of output Y for
v<(1-s½)2
2) Determine the long run cost function of a firm where short run cost function is given
by C(Q,K)=0.04Q3-0.9Q2+(10-logK)Q+8K2, K>1
3) Set up a complete Keynesian model with commodity market, money market and
labour market. Using Cramer‟s rule find out the effects of change in money supply.
4) A revenue maximizing monopolist requires profit of at least Rs 1500.Her demand
and cost function are P=304-2q and C=500+4q+8q2. Determine his output level and
price.
5) Given two isolated market supplied by a single monopolist, the two corresponding
demand functions are given as:
P1=12-Q1
P2=20-3Q2
The two monopolists total cost function is
C=3+2(Q1+Q2)
What prices, sales and profit will the monopolist earn in a regime of price
discrimination? Find the corresponding values if he cannot discriminate.
6) Given the demand and supply for the cobweb model
Qd,t=18-3Pt, Qs,t=-3+4Pt-1
Find the intertemporal equilibrium price and determine whether the equilibrium is
stable.
7) Consider the following macro model of inflation and unemployment;
P=1\6-3u+π
δπ\δt=3\4(p-π) and du\dt=-1\2(m-p)
Determine the time path of π, p and u.
8) Let the demand and supply be:
Qd=9-P+P‟+3P”
Qs=-1+4P-P‟+5P”
With P(0)=4 and P‟(0)=4. Find out the price path assuming market clearance at
every point of time. Discuss the nature of the time path.
9) Given C= 102+ 0.7y
I= 150 – 100r
Ms= 300
MT= 0.25y and
Mp= 124-200r
Find (i) equilibrium level of income and rate of interest, (ii) the level of C, I, MT &
Mp when the economy is in equilibrium.
10) Derive the supply curve of a firm under conditions of perfect competition if the
equation of the total cost curve is :
C= 0.1q3-2q2+15q+10
11) (i) Check whether the utility function u = (q1 ¼ + q2 ½) ½ is weakly or strongly
separable.
(ii) If u(x1, x2) is homogeneous of degree r in (X1, X2) then U* (P1, P2) is
homogenous of degree –r in (P1, P2), where U* is the indirect utility function. Prove
this result assuming u=x1 x2
12) Determine which of the following utility function exhibits decreasing risk aversion:
a) u(w)= (w+α ) β , where α>0 & 0< β<1
b) u(w)=log (w+α ), where α>0
c) u(w)= w3
13) Find out the optimal mixed strategies & the value of the game for the following payoff matrix of a two person zero sum game:
4
1
3
2
3
1
14) Suppose a person‟s utility function is u (w)= log w. The person takes the following
gamble. A fair coin is flipped until a tail appears. The person wins Rs. 2 n- 1 if it
appears on the nth toss. What is the expected utility of such a gamble?
15) Determine the consumption possibility locus is a two sector static Leontief open
input-output model.
16) In a two industry economy it is know that industry I uses 10 paise worth of its own
product and 60 paise worth of commodity II to produce a rupee worth of commodity
I. Industry II uses none of its own product but uses 50 paise worth of commodity I in
producing a rupee worth of commodity II & the open sector demands Rs 1000 of
commodity I and Rs 2000 of commodity II.
(i)
Check whether the data satisfies H-S condition
(ii) Find out the output levels by Cramer‟s Rule.
17) Suppose a monopolist can sell electricity in two markets. The demand curves are
P1D = 3-1/2 X1
P2D= 2-1/2 X2
The cost function is given by C = 2 1/3 + (X1+X2)
What are the profit maximising prices in each market & what are the corresponding
elasticities of demand.
18) Prove that the elasticity of substitution for a Cobb-Douglas production function is
unity.
19) In a model of intertemporal choice of consumption derive the relevant Slusky
Equation & find out the effect of a change in the interest rate on the intertemporal
choice of consumption.
20) Calculate the elasticity of substitution between K&L for
Q= L α + K α, O<A<1
21) Given the cubic cost function , C= α + βq+ rq2+ δq3
Find out the restrictions on the values of the parameters so that it gives usual average
& marginal cost curves.
22) Find out the optimum level of research (R) and advertising (A) expenditures when
the profit function of the firm is :
λ = -R2-A2+22R+18A-102
23) Show that (1/2, 1/2) is the unique mixed strategy adopted by both the players at
equilibrium in the following 2- person Zero –sum game
1
-1
-1
1
24) Suppose the demand & supply functions are given as follows :St= 10/β (Pt e-α)
Dt= γ-δ Pt
(i)
If Pte=Pt-1, set up the difference equation & solve for the time path.
25) If in a macroeconomic model we have,
C(t)=ay(t) + b
I(t)=K dc(t)\dt
And y(t)=C(t)+I(t), where b,k>0, 0<a<1
Derive the differential equation for y(t)
Solve for the equation if y(0)= b\1-a +ε, ε>0 and find the expression for I(t).
-------------------------------------------------------------------------------------