Topic 1 :
Budget Constraint
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Numerical Problems
Suppose that the price of a commodity Y is Re. 1 per unit and the price of
commodity X is Rs. 2 per unit. Suppose further that an individual’s
money income is Rs. 16 per time period and all is spent on X and Y.
(i)
Draw the budget line for the consumer.
(ii) Find the slope of the budget line.
(iii) Write the equation of the Budget Line.
Draw the budget line for the consumer, find the slope of the budget line
and write the equation of the Budget Line in each of the following
cases :
(i)
p1 = 1, p2 = 1, m = 20
(ii) p1 = 1, p2 = 1, m = 30
(iii) p1 = 1, p2 = 2, m = 20
(iv) p1 = 2, p2 = 1, m = 20
(v) p1 = 0, p2 = 1, m = 20
(vi) p1 = 1, p2 = 0, m = 20
(vii) p1 = p2, m = 20p1
(iv) p1 = p2, m = 20p2
Ranjan spends his entire income of Rs. 800 on clothes and food. If P c =
Rs. 40 and Pf = Rs. 16. Then draw budget line (clothes on y axis) and
calculate its slope.
Sanjiv’s budget line relating to good X and good Y has intercepts of 40
units of good X and 20 units of good Y. If the price of the good X is Rs.
8; what is Sanjiv’s income? Calculate the price of good Y, Sanjiv’s
income and slope of the budget line.
Write the budget equation for the following problem : Anand has 82 units
of local currency and he has spent all of the money to buy 5 plates of
patties (x) and 6 pastries (y). A patty costs 8 units of local currency.
On spending the entire income a consumer can buy either of the two
consumption bundles (4, 6) or (12, 2). Find income of the consumer,
write the equation of the budget line, find the slope of the budget line and
draw the budget line for the consumer.
A person consumes two goods X and Y. On spending the entire income
he can buy either of the two consumption bundles (8, 8) or (10, 4). The
price of good X is Rs. 0.5 per unit. Find the price of good Y and the
1|Page
PMG
Q8.
Q9.
Q10.
Q11.
Q12.
Q13.
income of the consumer. Also, write the equation of the Budget Line, find
the slope of the budget line and draw the budget line for the consumer.
Graph the budget line for apples and oranges, with prices of $2 and $3
respectively and $60 to spend. Now increase the price of apples from $2
to $4 and draw the budget line.
Graph the budget line for apples and oranges, with prices of $2 and $3
respectively and $60 to spend. Now increase expenditure to $90 and draw
the budget line.
Denote the budget set of a household with an income of 200MU (money
units) consuming the goods 1 and 2 with prices of p1 = 20 and p2 = 40,
respectively. Draw the respective picture with consumption of good 1 on
the horizontal and
of good 2 on the vertical axis. What happens if p1 drops to p1 = 10? What
happens if the income is doubled (p1 = 20)? What happens if the income
falls to its half (p1 = 20)?
Originally the consumer faces the budget line p1x1 + p2x2 = m. Then the
price of good 1 doubles, the price of good 2 becomes 8 times larger, and
income becomes 4 times larger. Write down an equation for the new
budget line in terms of the original prices and income.
Suppose that the prices of good x and good y both double, and income
triples. On a graph where the budget line is drawn with x on the
horizontal axis and y on the vertical axis. Which of the propositions is
correct :
(a) The budget line becomes steeper and shifts inward.
(b) The budget line becomes flatter and shifts outward.
(c) The budget line becomes flatter and shifts inward.
(d) The new budget line is parallel to the old budget line and lies
below it.
(e) None of the above.
Parul is preparing for exams in two subjects. She has to read 40 pages of
subject A and 30 pages of subject B. In the same amount of time she
could also read 30 pages of subject A and 60 pages of subject B.
(i)
Assuming that the number of pages per hour that she can read of
either subject does not depend on how she allocates her time, how
2|Page
PMG
Q14.
Q15.
Q16.
Q17.
Q18.
many pages of subject B can she read if she decides to devote all
her time for subject B only.
(ii) Also, determine how many pages of subject A can she read if she
decides to devote all her time for subject A only.
A person uses two goods X and Y. Good X pays him Rs. 3 per unit while
the price of good Y is Rs. 6 per unit.
(i)
If he has 0 units of good X, how much of good Y will he buy?
(ii) If he has 10 units of good X, how much of good Y will he buy?
(iii) Write the equation of budget line and draw the budget set for him.
(i)
Suppose we have three goods X, Y and Z and their prices are Rs. 2,
Rs. 4 and Rs. 6 respectively. The money income of the consumer is
Rs. 600. Write the equation of the budget line.
(ii) Now if a new currency ‘`’ is introduced and the price of good X in
terms of this currency is ` 10 Per unit. If the relative prices remain
unchanged write the prices of Y and Z in terms of this new
currency also write the new equation of budget line.
Rohit consumes three goods Pizza, Burger and Cold drink. He has an
income of Rs. 1000. The price of Pizza is Rs. 100, of Burger Rs. 50 and
of Cold drink is Rs. 20.
(i)
Write and draw his budget line.
(ii) If he buys only one Burger then write and draw his budget line in
terms of other two goods.
A person is hired to help fill empty chairs during speeches of politicians
and administrators. He is paid ` 1 per hour for listening to politicians, ` 2
per hour for listening to administrators and he consumes a good X which
costs him ` 10 per unit. Apart from this he has an income of ` 100 from
other sources.
(i)
Write his budget equation.
(ii) If he decides to consume 10 units of good X, now write his budget
equation in terms of two kinds of speeches and also present this
equation on the graph.
Suppose the price of a good rise by 10% while the income rises by 5%.
Show that a man who spends half his income on this good would become
better off. Is there any exception?
3|Page
PMG
Tax and Subsidy
Q19. Suppose that the government puts a tax of 15 paise a gallon on gasoline
and then latter decides to put a subsidy on gasoline at a rate of 7 paise a
gallon. What net tax is this combination equivalent to?
Q20. Suppose that a budget equation is given by p1.x1 + p2.x2 = m. The
government decides to impose a lump sum tax of u, a quantity tax on
good 1 of t, and a quantity subsidy on good 2 of s. What is the formula
for the new budget line?
Q21. Joseph lives on healthy food. Consequently, he generally buys only two kinds
of
consumption goods : ‘Beer’ and ‘Leberkäs’. The quantity of his beer
consumption, measured in gallons, is x1. The price of beer is p1 = 1 Bavarian
Mark (BAM) per gallon. The quantity of his consumption of Leberkäs,
measured in ounces, is x2. The price of Leberkäs is p2 = 2 BAM per ounce.
Joseph has a budget of m = 100 BAM.
(a) What is the exact form of Joseph’s budget constraint? Draw the
constraint and give an economic interpretation.
(b) Government introduces a quantity tax on the consumption of
Leberkäs. Tax per ounce is t2 = 2 BAM. How does this affect
Joseph’s budget constraint?
(c) How would a general consumption tax in form of a value tax at rate
‘ ’ affect Joseph’s budget constraint? Show in general terms that
this value tax is exactly equivalent to a proportional income tax at
PMG
rate tm, if it is true that t m
1
.
Kinky Budget Line
Q22. You are given the following market information :
(i)
Price of X is Rs. 2 for first 200 units and Rs. 0.50 for all units
purchased in excess of 200 units.
(ii) Price of Y is constant at Re. 1.
Sketch the budget line when the consumer’s income is Rs. 500.
4|Page
x1 / 2 y 1 / 2 , income of the consumer M = 500,
2
x 50
prices of the commodities py = 10 y and p x
. Draw
5
x 50
the budget line.
Q24. What is the shape of the budget line if the Power Regulatory Authority
proposes that a consumer can consume power each month at a subsidy of
‘s’ per unit upto 100 units, and then pays the normal price ‘P’ upto 300
units, but thereafter pays a per unit tax ‘t’ on all consumption beyond 300
units.
Q25. Ashish has Rs. 5,000 to spend on advertisement campaign through
newspapers for a new product launched by his firm. There are two
newspapers X and Y in which ads can be placed.
I: One ad of standard size costs Rs. 500 in newspaper X and Rs. 250 in
newspaper Y.
II : There are two kinds of potential clients for the product A and B. Each
ad in newspaper X is read by 1000 clients of type A and 300 clients of
type B.
III : Each ad in newspaper Y is read by 300 clients of type A and 250
clients of type B.
IV : No one reads more than one ad and nobody reads both the
newspapers.
(i)
If Ashish spends his entire budget on newspaper X then how many
clients of both the types will read the ad.
(ii) If Ashish spends his entire budget on newspaper Y then how many
clients of both the types will read the ad.
(iii) If Ashish spends his money equally between the two newspapers
then how many clients of both the types will read the ad.
(iv) Draw a budget line showing all the combinations of number of
readings by both kind of clients. Does the budget line extends all
the way to the axes?
Q23. For the utility function U
PMG
5|Page
Answer Sheet of Topic 1 :
1.
2.
3.
6.
7.
11.
13.
14.
15.
16.
17.
19.
21.
25.
Numerical Problems
(ii) Slope = -2, (iii) 2x + y = 16,
(i)
x1 + x2 = 20, slope = -1,
(ii) x1 + x2 = 30, slope = -1,
(iii) x1 + 2x2 = 20, slope = -1/2,
(iv) 2x1 + x2 = 20, slope = -2,
(v) x2 = 20, slope = 0,
(vi) x1 = 20, slope = ,
(vii) x1 + x2 = 20, slope = -1,
(viii) x1 + x2 = 20, slope = -1
-2/5,
4.
py = 16, m = 320, Slope = -1/2,
5.
8x + 7y =
82,
m = 16, x + 2y = 16, Slope = -1/2,
py = 0.25, m = Rs. 6, Eq. of B. Line is 0.5x + 0.25y = 6, Slope of B. Line
is -2,
2p1x1 + 8p2x2 = 4m,
12. (e),
Budget line is 3A + B = 150, (i) 150 pages, (ii) 50 pages,
(i)
0 units,
(ii) 5 units,
(iii) y = x/3
(i)
2x + 4y + 6z = 600,
(ii) py = ` 20 and pz = ` 30
(i)
100x1 + 50x2 + 20x3 = 1000,
(ii) 100x1 + 20x3 = 950
(i)
10X – P – 2A = 100,
(ii) P + 2A = 0
Tax of 8 paise per gallon ,
20. (p1 + t).x1 + (p2 – s).x2 = m –
u
(a) x1 + 2x2 = 100,
(b) x1 + 4x2 = 100
(i)
10,000 type A and 3,000 type B,
(ii) 6,000 type A and
5,000 type B,
(iii) 8,000 type A and 4,000 type B
6|Page
PMG
Topic 2 :
Preferences
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Theory Questions
Mr. Smith likes cashews better than almonds and likes almonds better
than walnuts. He likes pecans equally well as macadamia nuts and prefers
macadamia nuts to almonds. Assuming his preferences are transitive,
which does he prefer?
Suppose the chief selector of a football team says that given two players
A and B he would weakly prefer a richer player. Check if this preference
is reflexive, transitive and complete or not.
Consider the following binary relations defined over X where X is the set
of human beings :
(i)
“At least as tall as”
(ii) “Taller than”
(iii) “Is sister of”
Check if each of these relations satisfy reflexivity, completeness and
transitivity.
College football coach says that between two players A and B, he would
strictly prefer the taller and the faster player. Is this preference reflexive,
complete and/or transitive?
Miss X says that given any two drinks she always prefers the one that is
sweeter and colder. Is this preference relation complete? Is it transitive?
Vishnu likes strong coffee; the stronger the better but he can’t distinguish
differences smaller than one teaspoon per six cup pot. He is offered Cup
A using 14 teaspoon of coffee per pot, cup B using 14.75 teaspoon and
cup C using 15.5 teaspoon. For each of the following expressions,
determine whether it is true or false :
(i)
A B,
(ii) B C,
(iii) C ≻ A.
A Basketball coach likes his players to be big, fast and obedient. If player
A is better than player B in two of these three characteristics, then the
coach prefers A to B, but if B is better than A in two of these three
characteristics, then the coach prefers B to A, otherwise the coach is
indifferent between them.
A player P weighs 340 pounds, runs very slowly and is fairly obedient.
A player Q weighs 240 pounds, runs very fast and is very disobedient.
7|Page
PMG
A player R weighs 150 pounds, runs at average speed and is extremely
obedient.
(a) Does coach prefer P to Q.
(b) Does coach prefer Q to R.
(c) Does coach prefer P to R.
(d) Are coach’s preferences
transitive.
(e) Are coach’s preferences complete.
Now, coach has changed his preferences. If player A is better than player
B in all the three characteristics, then the coach prefers A to B, but if B is
better than A in all the three characteristics, then the coach prefers B to A,
otherwise the coach says that A and B are not comparable.
(f)
Are coach’s new preferences transitive.
(g) Are coach’s new preferences complete.
Q8. A consumer is indifferent between X and Y and between Z and Y but
strictly prefers X to Z. which property of the indifference curves is being
violated if the consumers preferences are represented by the indifference
curves?
Q9. A consumer who is unable to detect small differences in the amount of
water in her beer could have a transitive strict preference relation but is
unlikely to have a transitive indifference relation. True or False?
Q10. Sita likes chocolates and ice cream but after 10 slices of cakes, she gets
tired of chocolates and eating more makes her less happy. She always
prefers more ice cream to less.
(i)
If however she is made to eat everything put on her plate, what will
her indifference curves look like?
(ii) If she is allowed to leave anything she doesn’t want on her plate,
what would her indifference curve look like?
Q11. If all the individuals in a society have complete, reflexive and transitive
preferences and the social preference in the society is derived through the
simple majority rule, then do you think that the social preferences will
also necessarily fulfill these properties?
Q12. Suppose coffee and sushi have the same quality : the more you consume,
the more you want. What kind of preferences are these? For a given
budget, should you diversify if you have these kind of preferences?
[Hint : Concave preferences. No]
PMG
8|Page
Q14. Assume, aside from yourself, that there are only two types of students in
the world, Commerce and BBE students. Also assume that you as an
economics student, dislike both.
(i)
If both commerce and BBE students are “Bads” will the
indifference curves have a positive or negative slope? Draw some
smooth indifference curves and indicate the direction of preference.
Label the number of commerce students (C) on the X-axis and the
number of BBE students (B) on the Y axis.
(ii) Now suppose your dislike for students can be written as D =
max(3C, B), where D is the level of dislike associated with a
bundle of commerce and BBE students. In effect, your dislike is
determined by the maximum of a particular type, with 3 BBE
students as tolerable 1 commerce student. Draw some indifference
curves characterized by these preferences.
(iii) Assume now that you do not care one way or the other about being
around BBE students. Draw some indifference curves associated
with these preferences.
Q15. What is the marginal rate of substitution between two goods that are (a)
perfect substitute and (b) perfect complements?
Q16. If good 1 is a “neutral”, what is its MRS for good 2?
Q17. Mr. A likes burgers, while ketchup is a neutral good for him. What is his
MRS of burgers for ketchup?
Q18. Ajay likes oats (x) and fruit juice (y) and has concave preferences
between them. Price per kilogram of oats is Px and price per litre of fruit
juice is Py. His monthly budget for the two commodities is M.
Draw his indifference map and comment on the behavior of the marginal
rate of substitution between oats and fruit juice (MRSxy), as he increase
his consumption of oats. Indicate the possible optimum choices for Ajay
in a representative diagram.
Q19. Can you draw an indifference curve that does not have diminishing MRS,
but that is still allowed?
PMG
9|Page
Numerical Problems
Q1. A consumer consumes two goods x1 and x2. His preference relation
between x1 and x2 can be represented as x 2
k
, where k is some
x1
constant.
(i)
Sketch the IC which passes through the consumption bundle (20, 5)
and name it IC1 and shade the set of consumption bundles which A
weakly prefers to (20, 5).
(ii) Sketch the IC which passes through the consumption bundle (10,
15) and name it IC2 and shade the set of consumption bundles
which A weakly prefers to (10, 15).
(iii) Is the set of bundles which A weakly prefers to (20, 5) a convex
set?
(iv) Is the set of bundles that A considers inferior to (20, 5) a convex
set?
(v) For each of the following statements about A’s preferences write
True or False.
(a) (30, 5) ~ (10, 15),
(b) (10, 15) ≻ (20, 5),
(c) (20, 5) (10, 10),
(d) (24, 4) (11, 9),
(e) (30, 5) ≻ (25, 6),
(f)
(10, 15) ≻ (20, 10)
(vi) Calculate the MRS x x at (5, 20), (10, 10) and (20, 5).
PMG
1 2
Q2.
(vii) Do the indifference curves of A have diminishing MRS.
A consumer consumes two goods x1 and x2. His preference relation
between x1 and x2 can be represented as x2 k 4 x1 , where k is some
constant.
(i)
Sketch the IC which passes through the consumption bundle (1, 16)
and name it IC1 and shade the set of consumption bundles which A
weakly prefers to (1, 16).
(ii) Sketch the IC which passes through the consumption bundle (36, 0)
and name it IC2 and shade the set of consumption bundles which A
weakly prefers to (36, 0).
(iii) Is the set of bundles which A weakly prefers to (1, 16) a convex
set?
(iv) Find the slope of the indifference curve at the following points :
10 | P a g e
(a) (4, 12),
(b) (4, 16),
(c) (9, 8),
(d) (9, 12)
(v) Do the indifference curves of A have diminishing MRS.
Q3. Mohit consumes soft drink which is available in ½ litre and 1 litre bottles.
He is not concerned about the size of the bottle, he is concerned only
about how much soft drink he has to consume. On the graph, draw some
of his indifference curves taking ½ litre bottles on vertical axis.
[Hint : Perfect substitutes, slope -2]
Q4. Rohit consumes soft drink which is available in ½ litre and 1 litre bottles.
He allows himself only half litre of soft drink at a time, so if he has a
bottle of 1 litre he will throw ½ litre in the sink. Draw some of his IC’s
taking ½ litre bottles on vertical axis.
[Hint : Perfect substitutes, slope -1]
Q5.(a)Sakshi has to buy a soft drink from a machine which delivers 1 can of soft drink for 2
horizontal axis and Rs. 5 coin on vertical axis.
(b) How many cans of soft drink she can purchase if she has 4 Rs. 10 coins
and 2 Rs. 5 coins. Also determine the numbers of cans which she can buy
if she has 4 coins each of Rs. 10 and Rs. 5.
(c) Shade the area that Sakshi is indifferent to when she has 4 Rs. 10 coins
and 2 Rs. 5 coins.
(d) Does Sakshi has convex preferences between Rs. 10 and Rs. 5 coins.
(e) Does Sakshi has a bliss point.
[Hint : Perfect complements]
Q6. Darshan works in a factory where he has to face foul smell and smoke.
He hates both foul smell (x1) and smoke (x2) but he has strictly convex
preferences. Sketch the indifference curves for Darshan. What will be the
slope of indifference curves and what happens to the slope when we
increase (x1).
Q7. Mr. Paul consumes two commodities (x1) and (x2). His indifference
curves are concentric circles around his bliss point is (20, 15). Suppose
his current consumption bundle is (25, 3), will he prefer (30, 8) over his
present consumption bundle. Also, draw some of his indifference curves.
[Hint : Use distance formula]
Q8. A teacher conducts two mid-term exams and uses the maximum of the
two scores to determine the course grade for the student.
PMG
11 | P a g e
(a)
A student Bhanu wants to maximize his grade in the course. x1
represents his score in the first mid-term and x2 represents his score
in the second mid-term. Which combination of the scores would
Bhanu prefer x1 = 30, x2 = 80 or x1 = 70, x2 = 70.
(b) Draw an IC representing all combinations of scores that Bhanu
likes exactly as much as x1 = 30, x2 = 80.
(c) Draw an IC representing all combinations of scores that Bhanu
likes exactly as much as x1 = 70, x2 = 70.
(d) Does the student have convex preferences over these
combinations?
(e) What is the preference direction of Bhanu?
Q9. A teacher conducts two mid-term exams and uses the minimum of the
two scores to determine the course grade for the student.
(a) A student Bhanu wants to maximize his grade in the course. x1
represents his score in the first mid-term and x2 represents his score
in the second mid-term. Which combination of the scores would
Bhanu prefer x1 = 30, x2 = 80 or x1 = 70, x2 = 50.
(b) Draw an IC representing all combinations of scores that Bhanu
likes exactly as much as x1 = 30, x2 = 80.
(c) Draw an IC representing all combinations of scores that Bhanu
likes exactly as much as x1 = 70, x2 = 50.
(d) Does the student have convex preferences over these
combinations?
(e) What is the preference direction of Bhanu?
Q10. A person likes to eat lunch at 12 noon. The lunch at 12 noon costs Rs. 5.
However if he eats lunch ‘t’ hours before or ‘t’ hours after 12 noon then
the cost of lunch reduces to Rs. (5 – t). He has Rs. 15 to spend on lunch
and other goods.
(a) If he eats lunch at 12 noon then how much money is left for other
goods.
(b) If he eats lunch at 2 pm then how much money is left for other
goods.
(c) Draw some of his IC’s.
PMG
12 | P a g e
Answer Sheet of Topic 2 :
1.
7.
9.
1.
Theory Questions
C ≻ A ≻ W and P ~ M ≻ A. Therefore P ~ M ≻ A ≻ W
a.
P≻W
b.
M ? C -- not enough information to determine.
(a) Yes, (b) Yes, (c) No, he prefers R to P,
(d)
Yes,
(f)
Yes, (g) No
False,
16. Zero,
PMG
No,
(e)
Numerical Problems
(iii) Yes, (iv)
No, (v)(a) True, (b) True, (c) True, (d)
False,
(e) False, (f)
False, (vi) MRS x x = -4, -1 & -1/4 respectively, (vi)
1 2
2.
(iii)
5.
8.
(d)
(a)
9.
10.
(a)
(a)
13 | P a g e
Yes,
Yes, (iv)(a)
Yes,
Yes, (e) No,
x1 = 30, x2 = 80,
North-east,
x1 = 70, x2 = 50,
Rs. 10,
(b)
-1,
(b)
-1,
(d)
7.
Yes,
No, Concave preferences,
(d) Yes, (e)
Rs. 12,
(c)
-2/3, (d)
North-east,
-2/3, (v)
(e)
Topic 3 :
Utility
Q1.
Theory Questions
What kind of preferences are represented by the following utility
functions, where x1, x2 denote the amount of good 1 and good 2
respectively :
(i)
Q2.
Q3.
Q4.
Q5.
Q6.
Q1.
U ( x1 , x2 )
x1
x2 ,
(ii)
U ( x1 , x2 ) ( x1 x2 ) 2
U ( x1 , x2 ) x1 ln x2
PMG
(iii) U = min (x1, x2)
(iv)
What kind of preferences are represented by a utility function of the form
U ( x1 , x2 )
x1 x2 ?
What
about
the
utility
function
V ( x1 , x2 ) 13x1 13x2 ?
What kind of preferences are represented by a utility function of the form
U ( x1 , x2 ) x1
x2 ? Is the utility function V ( x1 , x2 ) x12 2 x1 x2 x2 a
monotonic transformation of U(x1, x2)?
What kind of preferences are represented by a utility function of the form
U ( x1 , x2 )
x1 x2 ? Is the utility function V ( x1 , x2 ) x12 .x2 a monotonic
2
2
transformation of U(x1, x2)? Is the utility function W ( x1 , x2 ) x1 .x2 a
monotonic transformation of U(x1, x2)?
What kind of preferences are represented by the utility function
U ( x1 , x2 ) x11 / 3 x22 / 3 . Are the following monotonic transformation of the
above stated utility function :
2 / 3 1/ 3
U ( x1 , x2 ) x12 / 3 x22 / 3 ,
(i)
(ii) U ( x1 , x2 ) x1 x2
2/3 4/3
(iii) U ( x1 , x2 ) x1 x2
Depict for the following utility function that by taking its monotonic
transformation the MRS does not change : U = x1/3y2/3.
Numerical Problems
What do you mean by monotonic transformation? Which of the following
are monotonic transformations :
(a) u = v2 for v > 0, (b) u = v2 for v < 0,
(c) u = v2,
14 | P a g e
(d)
u = 3v – 4, (e)
u = 1/v2,
v
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
(f)
u = –1/v2,
(g)
u = ln(v),
–v
(h) u = e ,
(i)
u = –e ,
In a two commodity world (x1, x2), specify utility functions where
(a) x1 and x2 are perfect substitutes with one unit of x1 equivalent to
three units of x2.
(b) x1 and x2 are perfect complements and one unit of x1 is always used
with four units of x2.
A consumer is willing to trade 3 units of X for 1 unit of Y when she has 6
units of X and 5 units of Y. She is also willing to trade 6 units of X for 2
units of Y when she has 12 units of X and 3 units of Y. She is indifferent
between bundle (6, 5) and (12, 3). What is the utility function for goods X
and Y?
A consumer is willing to trade 4 units of X for 1 unit of Y when she is
consuming bundle (8, 1). She is also willing to trade in 1 unit of X for 2
units of Y when she is consuming bundle (4, 4). She is indifferent
between these two bundles. Assuming that the utility function is CobbDouglas of the form U ( x, y ) x y , where
and
are positive
constants. What is the utility function for this consumer?
Nikhil consumes two goods x1 and x2. He has the utility function
u( x1 , x2 ) x1 .x2 . Determine the utility for the consumption bundle (40,
5). Also, draw some of his indifference curves. Mona offers to Nikhil to
give 15 units of x2 if Nikhil gives her 25 units of x1, should Nikhil make
this trade?
Use separate graphs to draw indifference curves for each of the following
utility functions and determine whether they have convex indifference
curves :
(i)
U(x, y) = min(2x + y, 2y + x) (ii) U(x, y) = max(2x + y, 2y + x)
PMG
(iii)
U(x, y) = 3x + y
(v)
U ( x, y )
x
y
(iv)
U ( x, y)
x. y
(vi)
U ( x, y)
x2
y2
2
y2
(vii) U ( x, y ) min(2 x y, 2 y x)
(viii) U ( x, y ) x
Ram consumes two commodities X and Y and his utility function is
Min(x + 2y, y + 2x).
15 | P a g e
(a)
If Ram has the consumption bundle (8, 2), draw his indifference
curve which pass through the point (8, 2).
(b) Draw indifference curve along which the utility of Ram is 6?
(c) At the point where Ram has 5 units of X and 2 units of Y, how
many units of X would he be willing to trade for one unit of Y?
Q8. Reeta has strong preference for having equal number of apples and
oranges. Her utility function is of the form u(x, y) = min{2x – y, 2y – x},
where x represents the number of apples and y represents the number of
oranges.
(a) Draw indifference curve along which the utility of Reeta is 10.
(b) When min{2x – y, 2y – x} = 2y – x, does she has more apples or
more oranges?
(c) When min{2x – y, 2y – x} = 2x – y, does she has more apples or
more oranges?
(d) If Reeta has 10 apples and 9 oranges and she gets 5 more oranges,
will her utility increase or decrease?
(e) If Reeta has 16 apples and the number of oranges is more than the
number of apples. She thinks that her utility is exactly as good as
having 10 apples and 10 oranges, then determine the number of
oranges which she has?
(f)
If Reeta has 16 apples and the number of oranges is less than the
number of apples. She thinks that her utility is exactly as good as
having 10 apples and 10 oranges, then determine the number of
oranges which she has?
Q9. Use separate graphs to draw indifference curves for each of the following
utility functions :
(i)
U(x, y) = min(2x + y, 2y + x) (ii) U(x, y) = max(2x + y, 2y + x)
Are these preferences convex? Why?
Q10. Ruchika’s preferences over bundles that contain non-negative amounts of
2
2
x1 and x2 are represented by the utility function U ( x1 , x 2 ) x1 x 2 .
Draw a few of her indifference curves. What kind of geometric figures
are they? Does ruchika have convex preferences?
Q11. Picabo, an aggressive skier, spends her entire income on skis and
bindings. (Binding are the mechanism by which skiers attach their boots
to the skis.)
PMG
16 | P a g e
(i)
If Picabo wears out one pair of bindings for every one pair of skis,
graph her indifference curves for skis and bindings, illustrating
bindings on the horizontal axis and skis on the vertical axis.
(ii) If Picabo wears out two pairs of bindings for every one pair of skis,
graph her indifference curves for skis and bindings, illustrating
bindings on the horizontal axis and skis on the vertical axis.
[Hint : Perfect Complements]
Q12. Paula, a former actress, spends all her income attending plays and
movies. She likes plays exactly three times as much as she likes movies.
Graph Paula’s indifference curves, illustrating plays on the horizontal
axis and movies on the vertical axis.
[Hint : Perfect Substitutes]
Q13. Mohit consumes soft drink which is available in ½ litre and 1 litre bottles.
He is not concerned about the size of the bottle, he is concerned only
about how much soft drink he has to consume.
(a) Write Mohit’s utility function taking ½ litre bottle as x and 1 litre
bottle as y.
(b) On the graph, draw some of his indifference curves.
(c) Would the utility function u = 100x + 200y represent Mohit’s
preferences?
(d) Would the utility function u = {5x + 10y}2 represent Mohit’s
preferences?
(e) Would the utility function u = x + 3y represent Mohit’s
preferences?
[Hint : Perfect substitutes, slope -1/2]
Q14. Rohit consumes soft drink which is available in ½ litre and 1 litre bottles.
He allows himself only half litre of soft drink at a time, so if he has a
bottle of 1 litre he will throw ½ litre in the sink.
(a) Write Rohit’s utility function taking ½ litre bottle as x and 1 litre
bottle as y.
(b) On the graph, draw some of his indifference curves.
(c) Would the utility function u = {x + y}2 represent Rohit’s
preferences?
[Hint : Perfect substitutes, slope -1]
PMG
17 | P a g e
Q15. U ( x1 , x2 ) 4 x1 x2 . If x1 = 9, x2 = 10 find total utility. If initially 81
units of x1 and 14 units of x2 were being consumed, how much x2 an
individual is willing to give up to consume 40 more x1.
[Hint : Quasi-linear Preferences]
Q16. The utility function of Rohan is u( x1 , x2 ) ( x1 2)(x2 6) .
(a) Determine the slope of his indifference curve at the point where his
consumption bundle is (4, 6). Also, draw the IC which passes
through the point (4, 6).
(b) Find at least one more point which lies on the indifference curve
passing through the point (4, 6). Determine the equation of the
indifference curve passing through the point (4, 6).
(c) Rohan has the bundle (4, 6) and Rosy offers him 9 units of x2 in
exchange of 3 units of x1. Should Rohan make this trade?
(d) Rohan has the bundle (4, 6) and Rosy offers him 3 units of x2 in
exchange of 1 units of x1. Should Rohan make this trade? If yes
what will be his consumption bundle after this trade?
(e) Rohan has the bundle (4, 6) and Rosy offers him 6 units of x2 in
exchange of 2 units of x1. Should Rohan make this trade? If yes
what will be his consumption bundle after this trade?
Q17. Charu has the utility function of the form u(x, y) = max {x, 2y}.
(a) Charu has the consumption bundle (10, 5). Draw the indifference
curve passing through this point. Can you determine the MRS of
such IC?
(b) Does Charu has convex preferences?
Q18. Consider the utility function U = x2y2. What is the marginal utility of
good X? what is the marginal utility of good Y? Also find the marginal
rate of substitution between goods X and Y?
Q19. An individual’s marginal utilities for commodity X and Y are given as :
MU x 40 5 x , MU y 20 3 y
What is the marginal rate of substitution if the consumer is consuming 3
units of X and 5 units of Y?
Q20. A consumer’s utility function is x 2 y and money income is ` 10.
Find MRSxy.
PMG
18 | P a g e
Answer Sheet of Topic 3 :
1.
2.
4.
Theory Questions
(i)
Perfect Substitutes,
(ii)
(iii) Perfect Compliments,
(iv)
Perfect Substitutes,
3.
Yes.
Cobb-Douglas Preferences, No, Yes.
13.
14.
18.
Numerical Problems
(a) Yes, (b) No, (c) No, (d)
No,
(g) Yes, (h) Yes, (i)
Yes.
U(x, y) = x + 3 y, MRS = -1/3,
4.
(i)
Yes, (ii) No, (iii) No, (iv)
No,
(vii) Yes, (viii) No.
Quarter circles centred at origin. No,
preferences.
(a) U(x, y) = x + 2y, (c) Yes, (d)
(a) U(x, y) = x + y,
(c) Yes. 15.
2
2
MUx = 2xy , MUy = 2x y, MRSxy = y/x,
19.
MRS = 5,
1.
3.
6.
10.
Perfect Substitutes,
Quasi Linear.
Quasi-linear
Preferences,
Yes, (e)
No,
PMG
19 | P a g e
20.
U ( x, y)
(f)
x 2 / 3 y1 / 3 ,
Yes, (v)
Yes, (vi)
she doesn’t have convex
Yes, (e) No.
U = 22, x2 = -80/9,
MRS
1
2
y
x
Topic 4 :
Choice
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Theory Questions
A consumer suddenly realizes that MUx/px is less than MUy/py, with the
current commodity bundle. Is he maximizing his utility? If not which
good should be consumed more of, to improve utility without increasing
the expenditure? What happens to these ratios as he begins adjusting his
consumption ? why ?
4
If a consumer has a utility function u(x1, x2) = x1 .x 2 , what fraction of her
income will she spend on good 2?
2
4
If a consumer has a utility function u(x1, x2) = x1 .x2 , what fraction of her
income will she spend on good 1?
Suppose that indifference curves are described by straight lines with a
slope of –b. Given arbitrary prices and money income p1, p2 and m, what
will the consumer’s optimal choices look like?
The slope of indifference curve and the budget line are same at all levels
of consumption. What can you say about the equilibrium of the
consumer?
A consumer has a utility function with the form :
PMG
u ( x1 , x2 )
Q7.
Q1.
( x1 ) a
( x2 ) a
Where it is assumed that a 0 and b 0. For each of the following
additional properties, what additional restrictions are required on the two
parameters a and b ?
(i)
Preferences are quasi-linear.
(ii) Two goods are perfect substitutes.
If a consumer's indifference curves are concave, will he consume both the
goods in equilibrium? Why or why not?
Numerical Problems
An individual’s marginal utilities for commodity X and Y are given as :
MU x 40 5 x , MU y 20 3 y
20 | P a g e
What is the marginal rate of substitution if the consumer is consuming 3
units of X and 5 units of Y? If Px = 5 and Py = 1, will the consumer be in
equilibrium?
Q2. An agent’s utility function is written as U = 4X 4Y 4, and his budget
constraint is
4X + Y = 88. What are the optimal amounts of X and Y ?
Q3. Given the utility function U = (x + 2)(y + 1), and the budget constraint 2x
+ 5y = 51 find the optimum levels of x and y purchased by the consumer.
Q4. Given the utility function U = (x + 2)(y + 1), where x represents
consumption of good 1 and y represents consumption of good 2. The
price of each good is Re. 1 and the income of the consumer is Rs. 11.
(i)
Write the equation of an indifference curve which passes through
the point (2, 8). Also sketch this indifference curve on the graph.
(ii) Determine the MRS.
(iii) Write down the equation of budget line and also find its slope.
(iv) Can the consumer achieve the utility of 11 with this budget.
(v) Determine the optimum consumption bundle of the consumer.
Q5.
Q6.
PMG
xy is the utility function, Px = Py = Rs. 10 and Income = Rs. 100,
If u
find the demands that maximize utility. Check the second order condition.
A consumer’s utility function is given as :
u
x1 x2
where x1, x2 denote the quantities of two products consumed by the
consumer and the prices per unit of the goods are Rs. 20 and Rs. 10
respectively. Determine the optimum level of commodities to maximize
his utility and spend his total income of Rs. 640 on the two goods.
Q7(i) Anjali, a sports woman, enjoys playing two games X and Y each week
and derives utility according to the utility function U
XY . If she has
`24 a week to spend on the two games and the price of playing X and Y is
`4, how will she maximize utility while playing the two games?
(ii) Anjali is also a businesswoman with a very busy schedule. She has only
16 hours time available to devote to these activities each week. If time
taken for game X is 4 hours and that for game Y is 2 hours, how many
hours would she pursue her games under the circumstances of part (i)
y for < 1. Show that :
Q8. Suppose U ( x, y ) x
21 | P a g e
M
x
px 1
Q9.
Q10.
Q11.
Q12.
Q13.
M
y
Py
and
py 1
Px
Px
Py
.
A consumer spends an amount m to buy x units of one good at the price
6/unit and y units of a different good at price 10/unit. m is positive. The
consumer utility function is
U(x, y) = xy + y2 + 2x + 2y
Find the optimal quantities of x* and y* as function of m. what are the
solutions for x* and y* if m ≤ 8?
A consumer’s utility is a function of two goods X and Y and is given by :
U = 100logX + 50logY
If the consumer’s budget is Rs. 10 and the price of X is Rs. 3 and the
price of Y is Rs. 1, find the quantities of X and Y that the consumer
should purchase to maximize utility. What is his marginal utility of
Money?
A consumer has utility function U ( x, y ) ( x a) ( y b) , where x and
y are the quantities of the two goods that he can consume. He buys the
goods at fixed prices px and py, subject to a ceiling M on total
expenditure. If he maximizes U, obtain the demand functions for each
good in terms of income M and prices px and py. show that the total
expenditure on each good is a linear function of M.
Show that two utility maximizing consumers with utility functions :
U ( x1 , x 2 ) x11 / 2 x21 / 2 and V ( x1 , x2 ) x12 x22
respectively have same demand functions
A consumer has the following utility function ;
PMG
U ( x1 , x2 ) ( x1 2)1 / 2 ( x2 )1 / 2 , x1 , x2
0.
Find the optimal consumption of x1 and x2 given prices P1 = Rs. 6, p2 =
Re. 1 and income = Rs. 10.
Q14. While you are at college hostel, your parents give you 300 rupees as
pocket money. You spend the entire amount on videogames (v) and
comic books (c). Videogames cost Rs. 2 per unit and comic books cost
Rs. 10 and your utility function is given by U = c2v. [DSE MA Ent. Eco. 1999]
22 | P a g e
(a)
(b)
Calculate your marginal rate of substitution.
Find the amounts of videogames and comic books consumed in
month. Sketch this in a diagram.
(c) Suppose your father restricts you to play at most 30 videogames in
a month. How does your optimum consumption change? Are you
better off or worse off than before?
Q15. SV College has Rs. 60,000 to spend on computers (C) and other stuff (X).
The UGC wants to encourage computer literacy in colleges and the
following two plans were proposed
(a) Give a grant of Rs. 10,000 to each college, that the college is free
to use as it wishes.
(b) Give a matching grant, for every rupee spent on computers, the
UGC gives the college Re. 0.50.
(i)
Write the budget equation and draw the budget line in each case.
(ii) If SV College has preferences that can be represented by the utility
function U(C, X) = C.X, what will be the amount spent on
computers under each plan.
PMG
Quasi-linear Preferences
Q16. An individual purchases quantities x1 and x2 of two goods whose prices
are p1 and p2 respectively. His utility function is :
U(x1, x2) = x1 + logx2.
Assuming his income is M, find the optimal quantities x1 and x2. Also
find the marginal utility of income.
Q17. An individual purchases quantities x1 and x2 of two goods whose prices
are p1 = 1 and p2 = 2 respectively. His utility function is :
U ( x1 , x2 )
(i)
(ii)
(iii)
(iv)
(v)
23 | P a g e
4 x1
x2
Write the equation of an indifference curve which passes through
the point (25, 0). Also sketch this indifference curve on the graph.
Determine the MRS.
if the income of the consumer is Rs. 24, write down the equation of
budget line and also find its slope.
Determine the optimum consumption bundle of the consumer.
If the income of the consumer increases to Rs. 34, now find his
optimum consumption bundle.
(vi)
If the income of the consumer falls to Rs. 9. Draw his budget line
and sketch the indifference curve which passes through the point
(9, 0) and find the optimum consumption bundle. Is budget line
tangent to the IC at optimum point, if not then which of them is
more steeper.
Concave Preferences
Q18. Ajay likes oats (x) and fruit juice (y) and has concave preferences
between them. Price per kilogram of oats is Px and price per litre of fruit
juice is Py. His monthly budget for the two commodities is M.
(i)
Draw his indifference map and comment on the behavior of the
marginal rate of substitution between oats and fruit juice (MRSxy),
as he increase his consumption of oats. Indicate the possible
optimum choices for Ajay in a representative diagram.
(ii) Specifically, if Ajay’s utility function is U = f(x, y) = x2 + xy + y2,
Px = Rs. 100, Py = Rs. 80 and M = Rs. 1,000, find his optimum
consumption choice.
PMG
Kinky Preferences
Q19. For the utility function U
x1 / 2 y 1 / 2 , income of the consumer M = 500,
2
x 50
p
prices of the commodities py = 10 y and x
. Find
5
x 50
the optimum point.
Q20. Consider a two commodity world : Sugar and Kerosene. Kerosene is sold
through a fair price shop only, at Rs. 10 per litre; the consumer can buy
any quantity she desires. Sugar sells at Rs. 10/kg in the fair price shop
and at Rs. 15/kg in the open market, but the consumer can buy up to 6 kg
of Sugar from the fair price shop. Suppose the consumer’s income is Rs.
150.
[DSE MA Ent. Eco. 2001]
(a) Draw her budget line.
(b) We know the consumer has convex preferences and that she
consumes 8 kg of Sugar in equilibrium. Find out her optimum
consumption of Kerosene.
24 | P a g e
Q21.
Q22.
Q23.
Q24.
Q25.
Q26.
Perfect Complements
A consumer’s utility function is given by U(x, y) = min {2x, y}. Suppose
that the price of good X is ` 1 per unit, price of good Y is ` 0.75 per unit
and income is
` 20. How many units of X and Y will the consumer
demand in this situation?
Use separate graphs to draw indifference curves for each of the following
utility functions :
(i)
U(x, y) = min(2x + y, 2y + x)
(ii) U(x, y) = max(2x + y, 2y + x)
Find the equilibrium bundles for each kind of preferences given Px = Py =
1 and M = 10.
Ram consumes two commodities X and Y and his utility function is
Min(x + 2y, y + 2x). He chooses to buy 8 units of X and 16 units of Y. the
price of commodity Y is ` 0.50. What is the price of commodity X and
income of the consumer?
Anita spends all her pocket money on chocolates and ice cream. The
money spent on chocolates is x and money spent on ice creams is y. Her
utility function is U ( x, y ) min{4 x,2 x y} . Anita consumes 15
chocolates and 10 ice creams. The price of a chocolate is Rs. 10. Find the
price of an ice cream and her pocket money.
Reena has the utility function U(x, y) = min{x, y2}
(i)
Write an equation for the line on which all the optimum solutions
lie.
(ii) Determine her utility if her consumption bundle is (1, 1). Also
draw the indifference curve which passes through (1, 1).
(iii) Determine her utility if her consumption bundle is (4, 2). Also
draw the indifference curve which passes through (4, 2).
(iv) Determine her utility if her consumption bundle is (16, 5). Also
draw the indifference curve which passes through (16, 5).
(v) If px = 1, py = 2 and m = 2, determine the optimum bundle.
(vi) If px = 10, py = 15 and Reena demands 100 units of x, find her
income.
A teacher conducts two mid-term exams and uses the minimum of the
two scores to determine the course grade for the student. A student
25 | P a g e
PMG
Ashish wants to maximize his grade in the course. x1 represents score of a
student in the first mid-term and x2 represents score in the second midterm exam. He has a total of 1200 minutes to study for these exams. If he
doesn’t study at all he will get zero in both the exams, whereas every 10
minutes spent on first test increases the score in first test by one point and
every 20 minutes spent on second test increases the score in second test
by one point.
(a) Write the utility function of Ashish.
(b) Write an equation for the line on which all the optimum solutions
lie.
(c) Write Ashish’s budget line.
(d) Determine how much time he should spend on first test and how
much on second and what is his best score.
Q27. A teacher conducts two mid-term exams and uses the maximum of the
two scores to determine the course grade for the student. A student
Ashish wants to maximize his grade in the course. x1 represents score of a
student in the first mid-term and x2 represents score in the second midterm exam. He has a total of 400 minutes to study for these exams. If he
doesn’t study at all he will get zero in both the exams, whereas if he
spends m1 minutes for first test then his score in first test will be m1 / 5
points and if he spends m2 minutes for second test then his score in
second test will be m2 / 10 points.
(a) Write the utility function of Ashish.
(b) Write an equation for the line on which all the optimum solutions
lie.
(c) Write Ashish’s budget line.
(d) Determine how much time he should spend on first test and how
much on second and what is his best score.
Q28. Mrs. Pathak is very particular about her consumption of tea. She always
takes 50 grams of sugar with 20 grams of ground tea. She has allocated
Rs. 55 for her spending on tea and sugar per month. (Assume that she
doesn’t offer tea to her guests or anybody else and she doesn’t consume
sugar for any other purpose). Sugar and tea are sold at 2 paisa per 10
grams and 50 paisa per 10 grams respectively. Determine how much of
tea and sugar she demands per month.
PMG
26 | P a g e
Q29. Picabo, an aggressive skier, spends her entire income on skis and
bindings. (Binding are the mechanism by which skiers attach their boots
to the skis.)
(i)
If Picabo wears out one pair of bindings for every one pair of skis,
graph her indifference curves for skis and bindings, illustrating
bindings on the horizontal axis and skis on the vertical axis.
(ii) If Picabo wears out two pairs of bindings for every one pair of skis,
graph her indifference curves for skis and bindings, illustrating
bindings on the horizontal axis and skis on the vertical axis.
Now assume that Picabo has $5,760 in income to spend on binding and
skis each year. Skis cost `480 per pair, and bindings cost `240 per pair.
(a) Graph Picabo’s optimal consumption bundle for skis and bindings
under the assumptions in part (i).
(b) Graph Picabo’s optimal consumption bundle for skis and bindings
under the assumptions in part (ii).
Q30.
Q31.
Q32.
Q33.
PMG
Perfect Substitutes
Vivek spends all his pocket money on Pizzas (P) and Burgers (B). His
utility function is given by U = 3P + B. A burger costs Rs. 20 and a pizza
costs Rs. 30, and his pocket money is Rs. 300. What is vivek’s optimum
consumption bundle?
[DSE MA Ent. Eco. 2000]
Suppose that a consumer always consumes 2 spoons of sugar with each
cup of coffee. If the price of sugar is p1 per spoonful and the price of
coffee is p2 per cup and the consumer has m dollars to spend on coffee
and sugar, how much will he or she want to purchase?
Sara a consumer who views Chocolate and Vanilla Ice cream as perfect
substitutes. She likes both and is always willing to trade one scoop of
chocolate ice cream for two scoops of vanilla ice cream. If the price of a
scoop of a chocolate ice cream is three times the price of vanilla ice
cream, will Sara buy both types of ice cream? What happen if the price of
chocolate ice cream is twice the price of vanilla ice cream? Draw
diagrams to illustrate your answer.
Paula, a former actress, spends all her income attending plays and
movies. She likes plays exactly three times as much as she likes movies.
27 | P a g e
(i)
(ii)
Graph Paula’s indifference curves, illustrating plays on the
horizontal axis and movies on the vertical axis.
Paula earns $120 per week. If tickets to plays cost $12 each and
tickets to movies cost $5 each, graph her optimal consumption
bundle, illustrating plays on the horizontal axis and movies on the
vertical axis.
Income Tax versus Quantity Tax
Q34. An agent’s utility function is written as U = 12X 2Y 2, and his budget
constraint is
X + 3Y = 120. What are the optimal amounts of X and Y ? The income tax
is definitely superior to the quantity tax in the sense that you can raise the
same amount of revenue from a consumer and still leave him or her better
off under the income tax than under the quantity tax. Verify this result for
a quantity tax t = 2 on X.
1/ 2 1/ 2
Q35. The utility function of Ayush is U ( x, y) x y , px = 2 and py = 1 and
income of Ayush is 120.
(i)
Find his optimum consumption bundle and level of utility.
(ii) If govt. imposes a quantity tax of Rs. 1 per unit on good X, then
find his new optimum consumption bundle and level of utility.
(iii) Show that by imposing income tax rather than quantity tax the
govt. can collect same revenue but with greater utility for Ayush.
PMG
28 | P a g e
Answer Sheet of Topic 4 :
2.
Theory Questions
4/5,
x1
m
, x2
p1
x1
0, x 2
4.
0
m
p1
3.
p
if 1
p2
b
p1
p2
b
if
1/3,
Any combination of x1 & x2 on the budget line if
6.(i) a
1.
4.
5.
9.
b
PMG
1 and b = 1,
(ii)
a = 1 and b = 1,
Numerical Problems
MRS = 5, Yes,
2.
X = 11; Y = 44,
13, y = 5,
34 x
y
(i)
, (ii) MRS = y 1 ,
x 2
x 2
Yes,
(v) x = 5, y = 6,
Max. U = 5 at x = 5, y = 5,
6.
x2 = 32,
x
p1
p2
40 m
,y
24
m 8
,
8
3.
Max. U = 90 at x =
(iii)
x + y = 11, (iv)
Max. U = 22.63 at x1 = 16,
10. x = 20/9, y = 10/3 & Marginal utility of money
= 15,
(M
11.
bp y ) ap x
x
px
(M
, y
1
ap x )
py
bp y
1
15.(i) C + X = 70000 and 0.5C + X = 60,000,
17.(i) x2
-½,
29 | P a g e
20 4 x1 , (ii)
MRS =
,
2
x1
,
m
p1
1 , x2
p1
,
p2
16.
x1
(iii)
x1 + 2x2 = 24, slope =
24.
25.
26.
27.
29.
31.
(iv) (16, 4),
(v) (16, 9),
indifference curve.
Price = Rs. 30, Pocket money = Rs. 450,
(i)
x = y2 ,
(ii) 1,
(iii) 4,
(vi) 1150,
(a) U = min(x1, x2), (b) x1 = x2,
35.
(9,
0),
(iv)
16,
(c)
10x1 + 20x2 = 1200,
(v)
No,
(4,
2),
(d) 400 minutes, 800 minutes and 40 marks (40, 40).
(a) U = max(x1, x2), (b) x1 = x2,
(c) 5x1 + 10x2 = 400,
(d) 400 minutes, 0 minutes and 80 marks (80, 0).
(a) S = 8, B = 8,
(b) S = 6, B = 12
Let number of cups of coffee is x and number of teaspoons of sugar is 2x,
then
x
33.
34.
(vi)
PMG
m
2 p1
p2
.
P = 10, M = 0
Without tax X = 60, Y = 20 & U = 1,72,80,000; With quantity tax X =
20, Y = 20 & U = 19,20,000; with income tax X = 40, Y = 40/3 & U =
34,13,333.33
(i)
x = 30, y = 60 and U = 1800 , (ii) x = 20, y = 60, U = 1200 &
R* = 20
(iii) x = 25, y = 50, U = 1250 & R* = 20
30 | P a g e
Topic 5 :
Demand
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Theory Questions
Can both goods in a two commodity world be inferior? Explain.
Income Offer Curve
Show that perfect substitutes are examples of homothetic preferences.
Show that Cobb-Douglas preferences are homothetic preferences.
“In a two commodity world of X and Y, if the income consumption curve
approaches the X-axis as income rises, then X (represented on X-axis) is
an inferior good”. True or False. Explain.
PMG
Engel Curve
What does an Engel curve for an inferior good look like?
If Engel curve for good x is a vertical line (good X is measured on Xaxis), can x be a Giffen good. Give reasons.
Price Offer Curve
How does the price consumption curve look like if the demand for the
product is :
(i)
Perfectly inelastic, (ii) unit elastic, (iii) elastic
Demand Curve
Q8. If the income consumption curve for commodity X is a vertical straight
line, can the demand curve for X be downward sloping? Explain?
Q9. At least one of the goods that a consumer consumes must have a
downward sloping demand curve. Do you agree? Why or why not ?
Q10. A consumer spends all his money on coffee and sugar. He only drinks his
coffee with two spoons of sugar and only consumes sugar if he drinks
coffee.
(a) Write his utility function.
(b) Graph his indifference curves.
(c) What is his demand function for coffee?
31 | P a g e
(d)
Write the equation for his Engel curve.
Numerical Problems
Q1.
Q2.
Income Offer Curve
A consumer has the utility function U(x, y) = lnx + y. What is his income
expansion path?
PMG
Engel Curve
What kind of preferences are represented by the following utility
functions, where x1, x2 denote the amount of good 1 and good
respectively :
U ( x1 , x2 ) x1 x2 ,
(iii) U ( x1 , x2 ) x1 ln x2
(i)
Q3.
Q4.
Q5.
(ii)
U ( x1 , x2 ) ( x1
x2 ) 2
Draw Income consumption curve and Engel Curve for x1 , x2.
Suppose Ram’s utility function is Ur = x + y2, while that of Ishank is U1 =
x + log y. Assume price of both commodities to be Re.1. Derive the
income offer curves and Engel curves for commodity x for both of them.
Demand Curve
An individual purchases quantities x1 and x2 of two goods X1 and X2. His
U ( x1 , x2 ) 4 x1 x2
utility function is :
(i)
Determine his demand function for good X1.
(ii) Determine his demand function for good X2.
Arun consumes only two goods x1 and x2. His preferences are represented
by the function u(x1, x2) = x1 + ln x2. The price of two goods are p1 and p2
respectively and Arun’s income is m.
(i)
Determine his demand function for good x2.
(ii) Determine his demand function for good x1.
(iii) Show that if Arun buys both the goods then the amount of money
he spends on x2 depends only on p1.
(iv) If Arun gets extra income for which good will he use that income.
32 | P a g e
Q6.
A consumer has the utility function given by U = y + logx. What can you
say about demand for good x when money income changes?
Q7. Consider a consumer with utility function U(x, y) = y + ex. Derive the
demand curve for commodity X.
Q8. Keerti’s utility function for the two goods Gasoline (G) and Wireless
minutes (W) is U 20 G W . Her income is Rs. 125. Price of Gasoline
is Rs. 1 per gallon and Wireless Rs. 0.50 per minute. Calculate
equilibrium amounts of G and W consumed. Suppose Keerti loses her cell
phone, how much G is required to keep utility constant. How much
should be the income compensation.
Q9. Mohit consumes soft drink which is available in ½ litre and 1 litre bottles.
He is not concerned about the size of the bottle, he is concerned only
about how much soft drink he has to consume. His income is Rs. 30.
Suppose that the price of ½ litre bottle is Rs 0.75 each and price of 1 litre
bottle is Rs. 1 each.
(a) Write demand functions of both types at these prices
(b) Suppose that the price of ½ litre bottle falls to Rs. 0.55 and price of
1 litre bottle remains same. Now write new demand functions.
(c) Suppose that the price of ½ litre bottle falls to Rs. 0.40 and price of
1 litre bottle remains same. Now write new demand functions.
Q10. Casper consumes cocoa and cheese. He has an income of Rs. 16. Cocoa
ia sold in an unusual way. There is only one supplier and the more cocoa
one buys from him, the higher the price one has to pay per unit. In fact x
units of cocoa will cost Casper a total of x2 dollars. Cheese is sold in the
usual way at a price of Rs. 2 per unit. Casper’s budget equation,
therefore, is x2 + 2y = 16 where x is his consumption of cocoa and y is his
consumption of cheese. Casper’s utility function is U = 3x + y.
(a) Write an equation such that the slope of his indifference curves
equals the slope of his budget line.
(b) Find his optimum consumption bundle. Also, show it graphically.
Q11. What kind of preferences are represented by the following utility
functions, where x1, x2 denote the amount of good 1 and good
respectively :
PMG
(i)
33 | P a g e
U ( x1 , x2 )
x1
x2 ,
(ii)
U ( x1 , x2 ) ( x1
x2 ) 2
Drive the price offer curve and the demand curve for good 1 for these
functions.
Q12. Consider the two consumers A and B, each income W. they spend their
entire budget over the two commodities, X and Y. Compare the demand
curves of the two consumers under the assumption that their utility
2
2
function U A ( x, y) x y and U B ( x, y) x y respectively.
Q13. For two consumers with utility functions given by u = log x + 2log y and
u = x1/2 + y1/2, respectively, find their demand function for x and y.
Q14. Monica consumes cakes and pastries. Her demand function for cakes is
Q
m 30 p
20 p , where m is her income, pc is the price of cake
c
c
p
PMG
and pp is the price of a pastry, Qc is her consumption of cakes.
(i)
Is pastry a substitute for cakes or a complement. Explain
(ii) Determine Monica’s demand for cakes if her income is Rs. 100 and
price of a pastry is Re. 1 per piece.
(iii) Write an equation for inverse demand function for cakes using her
income as Rs. 100 and price of a pastry as Re. 1 per piece.
Q15. Ms Shagun consumes 2 units of x2 for each unit of x1.
(i)
Write the demand functions of both the goods.
(ii) If her income is Rs. 20, p1 = Re. 1 and p2 = Rs. 0.75. Find the
demands of both the goods.
Q16. A consumer’s utility function is given by U(x, y) = min {2x, y}. Suppose
that the price of good X is ` 1 per unit, price of good Y is ` 0.75 per unit
and income is
` 20. How many units of X and Y will the consumer
demand in this situation? Derive the demand function for good Y.
Q17. Urmilla’s utility function is U = r + 100l – l2, where r is the number rose
plants and l is the number of lily plants in her garden. She has 500 square
feet in her garden to allocate between roses and lily plants. Each rose
plant needs 1 square foot and each lily plant needs 4 square feet area.
There are no other costs involved.
(i)
Determine the optimum number of rose and lily plants.
(ii) If she acquires extra 100 square feet for her garden, find her new
optimum bundle.
(iii) If her garden size is only 144 square feet, find her optimum bundle.
34 | P a g e
(iv)
What is her minimum garden size such that she grows both roses
and lilies.
Answer Sheet of Topic 5 :
8.
Theory Questions
Yes, for quasilinear preferences demand curve is downward sloping and
income consumption curve is a vertical straight line, eg., U(x, y) = V(x) +
y.
PMG
Numerical Problems
3.
Equation of Ram’s Engel curve is x m r
curve is
x
mI
1,
2
5.
6.
9.
10.
13.
First Consumer x
M
p2
x1 = (m/p1) – 1,
(ii)
(b)
(b)
m
, y
3 px
14.
(i)
Substitute, (ii)
17.
(i)
r = 308, l = 48,
192 sq feet.
35 | P a g e
Equation of Ishank’s Engel
where mr is income of Ram and mI is income of Ishank
2 p2
x1
(i)
,
p1
(i)
x2 = p1/p2, (ii)
x = px/py,
(a) x1 = 0, x2 = 30,
(a) x = 3,
4.
1
,
2
x2
4 p2
p1
(iv)
x1 = 0, x2 = 30,
x = 3, y = 3.5,
x1 ,
(c)
2m
, First Consumer x
3py
Q 120 30 p , (iii)
c
c
(ii) (408, 48), (iii)
x1 = 75, x2 = 0,
m
, y
2 px
m
,
2 py
Q
c
P 5
.
c
30
(0, 36),
(iv)
Topic 6 :
Revealed Preferences
Q1.
Q2.
Q3.
Theory Questions
When prices are (4, 6) Rohit chooses the bundle (6, 6). When the prices
become (6, 3) he chooses the bundle (10, 0).
(i)
Draw the first budget line of Rohit and mark his choice.
(ii) Draw the Second budget line of Rohit and mark his choice.
(iii) Is Rohit’s behavior consistent with weak axiom of revealed
preferences?
Mr. A’s yearly budget for his car is Rs. 1,00,000, which he spends
completely on Petrol(P) and on all other expenses (M). M is measures in
Rs. so price of M is Re. 1. When price of petrol is Rs. 40 per litre, Mr. A
buys 1,000 litres per year. The petrol price rises to Rs. 50 per litre, and to
offset the harm to Mr. A, the govt. gives him a cash transfer of Rs. 10,000
per year.
(i)
Write down Mr. A yearly budget equation under the price rise plus
transfer situation.
(ii) What will happen to his petrol consumption – increase, decrease or
remain the same?
(iii) Will he be better off or worse off after the price rise plus transfer
than he was before?
If the income of the consumer increases, and one of the prices decreases
at the same time, will the consumer be necessarily at least as well off ?
PMG
Numerical Problems
Q1.
WARP
Suppose the consumer’s choices reveal the following bundles and prices
matrix. Do they satisfy WARP?
Bundles
1
2
3
Prices
10
5
11
1
11
10
8
2
36 | P a g e
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
6
8
5
3
A consumer’s behavior was observed in two different price income
relations :
Situation
p1
p2
x1
x2
A
6
3
6
6
B
5
5
10
0
Which consumption bundle has he revealed preferred to other? [Eco. (H) 2002]
Do the following choices satisfy WARP
Prices
Quantities
Situations
Px
Py
X
Y
1
2
1
2
A
2
1
2
1
B
1
1
2
2
C
Define Weak Axiom of Revealed Preference Theory. Do these choices satisfy
the weak axiom of revealed preference. Explain your answer
Situation Price of X Price of Y Quantity of X
Quantity of Y
1
4
6
6
6
2
6
3
10
0
Define WARP. Do the following choices satisfy WARP
Situation
Price
Quantities
A
1
1
4
6
B
2
0.5
8
3
State the weak Axiom of revealed preference? Check the following to
determine whether the choices satisfy WARP :
With Money Income M = 20, P1 = 1, P2 = 1; The choice is x = 15, y = 5
with M = 20, P1 = 1/2, P2 = 2; The choice is x = 1, y = 7.5
A consumer is observed to buy q1 = 20 and q2 = 20 at price p1 = 2 and p2
= 6. He is also seen to buy q1 = 18 and q2 = 4 when p1 = 3 and p2 = 5. Is
this behavior consistent with WARP?
State the Weak Axiom of Revealed Preference. When prices are (p1, p2) =
(1, 2) a consumer demands (x1, x2) = (1, 2) and when prices are (q1, q2) =
(2, 1) the consumer demands (y1, y2) = (2, 1). Does this behavior satisfy
WARP?
37 | P a g e
PMG
( p1 , p 2 ) A (2, 1) ,
( x1 , x2 ) A (1, 2) ,
( p1 , p2 ) B (1, 2) ,
Given
( x1 , x2 ) B (2, 1) . Is this behavior consistent with the model of
maximizing behavior? Which bundle is preferred by the consumer
( x1 , x2 ) A or ( x1 , x2 ) B . Give diagrammatic proof also.
Q10. State Samuelson’s Weak Axiom of Revealed Preference (WARP).
Consider two commodities, apples and oranges. Initially when both
apples and oranges cost Re. 1 per unit each, a consumer buys 4 apples
and 6 oranges. When the price of apple rises to Rs. 2 per unit and that of
orange falls to Rs. 0.50 per unit. She buys 8 apples and 3 oranges. Does
she violate WARP?
Q11. Sheela and her family consume wheat flour and sugar. Six years back,
price per kilogram (kg) of wheat flour was Rs. 14, while price per kg of
sugar was Rs. 20. Sheela purchased 10 kg of wheat flour and 7.5 kg of
sugar. Current prices of the wheat flour and sugar have risen to Rs. 20
and Rs. 35 per kg. Sheela’s preferences have not changed though her
budget has increased to Rs. 500. Is she better off or worse off now?
Explain.
Q12. Here is a table of prices and demands of a consumer whose behavior was
observed in 5 different price income situations :
Situation
P1
P2
X1
X2
1
1
5
35
A
1
2
35
10
B
1
1
10
15
C
3
1
5
15
D
1
2
10
10
E
(i)
Is this behavior consistent with WARP?
(ii) Represent all the budget lines on the graph, mark the choices of the
consumer and determine graphically if his behavior is consistent
with WARP.
Q9.
PMG
38 | P a g e
SARP
Q13. Suppose the consumer’s choices reveal the following bundles and prices
matrix. Do they satisfy SARP?
Bundles
1
2
3
Prices
10
5
11
1
11
10
8
2
6
8
5
3
Q14. Here is a table that illustrates some observed prices and choices for three
different goods at three different prices in three different situations :
p1
p2
p3
x1
x2
x3
Situation
1
2
8
2
1
3
A
4
1
8
3
4
2
B
3
1
2
2
6
2
C
Do these observations violate WARP? Do these observations violate
SARP?
Q15. An individual consumes three goods x1, x2 and x3 at respective prices p1,
p2 and p3. His month-wise consumption amounts of xi at prices pi in three
different months are given in each rows of the table below :
x1
x2
x3
p1
p2
p3
3
2
4
2
3
6
Month 1
4
2
3
4
1
7
Month 2
3
7
2
3
2
1
Month 3
Check if this price and consumption data is consistent with :
(a) weak axiom of revealed preference, and
(b) strong axiom of revealed preference?
PMG
Index Numbers
Q16. Given below are the prices and quantities consumed of 4 staple foods,
which accounted for 2/3 of food budget of a nation.
Price
Quantities
Year
1910
1940
1990
2013
1940
1990
Goods
A
0.14
0.14
0.16
0.19
165
220
B
0.28
0.34
0.66
0.85
22
42
39 | P a g e
C
0.07
0.08
0.10
0.13
120
180
D
0.032
0.044
0.051
0.064
200
200
(i)
Calculate the annual cost of 1940 and 1990 bundles at prices of
1940 and 1990.
(ii) Is 1990 bundle revealed preferred to 1940 bundle?
(iii) Calculate Laspeyres and Paasches quantity index for 1990 with
base year 1940.
(iv) Calculate Laspeyres price index for 1990 with base year 1940.
Ans. (i)
C1940 at P1940 = 49, C1940 at P1990 = 63.12, C1990 at P1940 = 68.28,
C1990 at P1990 = 91.12,
(ii) Yes, (iii) Lq = 1.39, Pq = 1.44,
(iv) Lp = 1.29
PMG
40 | P a g e
Topic 7 :
Slutsky Equation
Q1.
Q2.
Q3.
Theory Questions
Suppose there are only two goods x1 and x2. You know that x2 is an
inferior good. what can you conclude from the Slutsky equation about the
slope of the demand curve for good x1?
Suppose there are only two goods x1 and x2. You know that x1 is a normal
good. what are the signs of the terms in the Slutsky equation for x1. What
are the signs of the terms if x1 is a Giffen good?
“If an individual consumes two goods X and Y that are perfect
substitutes, the change in demand for X as its price changes, is due
entirely to the substitution effect”. Is this always the case ? Substantiate
with suitable exposition.
PMG
Numerical Problems
Q1. Neeraj likes is Mangoes. When the prices of all other goods are fixed at
current levels, Neeraj’s demand function for high quality mangoes is q =
.02m - 2p, where m is his income, p is the price of mangoes, and q is the
number of Kg’s of mangoes that he demands. Neeraj’s income is ` 7,500,
and the price of a Kg of mangoes is `30.
(a) How many Kg’s of Mangoes will Neeraj buy?
(b) If the price of a kg of mangoes rose to `40, how much income
would Neeraj have to have in order to be exactly able to afford the
quantity of mangoes and the amount of other goods that he bought
before the price change? At this income, and a price of 40 pounds,
how many Kg’s would Neeraj buy?
(c) At his original income of 7,500 and a price of 40, how much
mangoes would he demand?
(d) When the price of mangoes rose from `30 to `40, determine change
in demand due to price effect, substitution effect and income effect.
Ans. (a) 90, (b) `8,400, 88 Kg.’s, (c) 70 Kg.’s,
(d) PE = -20, SE = -2, IE = -18
Q2. Colin has demand function for apples given by q = 0.02m – 2p, where m
is income and p is price. Colin’s income is Rs. 6,000 and initial price of
41 | P a g e
Q3.
Q4.
Q5.
Q6.
effect.
apples was Rs. 30 per kg. Price of apples rose to Rs. 40 per kg. What is
the substitution effect and income effect?
Shyam’s demand function for good x is X(px, py, m) =2m/5px. His income
is Rs. 1,000, px = Rs. 5 and py = Rs. 20.
(i)
If px falls to Rs. 4 by how much does his demand change?
(ii) If his income changes at the same time so that he could exactly
afford his old consumption bundle, what was his new income?
What would be the equilibrium bundle at this set of prices?
(iii) What is substitution effect and income effect?
Suppose that demand function for milk is of the form :
m. p1
x1
p12 p 22
Initially, m = 200, p1 = 3 and p2 = 5. If p1 falls to 2, calculate the
substitution effect and income effect through Slutsky equation.
PMG
You have a demand function of the form x1 2
M
, where x1 = quantity
2 P1
of commodity 1, P1 = price of commodity 1 and M = Income. Originally
your income was Rs. 200 and P1 = 5. Suppose the price of commodity 1
falls to 4. What is the total change in demand? how much of it is due to
Slutsky substitution effect?
[DSE MA Ent. Eco. 2003]
Mr. Mohit consumes two goods x1 and x2. His utility function is U(x1, x2)
= x1x2. The price of x1 is `1, the price of x2 is `2, and Mohit’s income is
`40 a day.
(a) Find his optimum consumption bundle. Draw Mohit’s budget line
and Mark his optimum choice with A.
(b) The price of x2 suddenly falls to `1. If, after the price change,
Mohit’s income had changed so that he could exactly afford his old
consumption bundle, then what would be his new income.
(c) Find his optimum consumption bundle with this income and the
new prices. Draw Mohit’s New budget line and Mark his new
optimum choice with B.
(d) Determine the change in demand of x2 due to the substitution
42 | P a g e
(e)
Find his optimum consumption bundle with original income and
the new prices. Draw Mohit’s New budget line and Mark his new
optimum choice with C.
(f)
Determine the change in demand of x2 due to the income effect.
(g) Mark the substitution effect, income effect and price effect on the
vertical axis.
(h) The income effect of the fall in the price of x2 on Mohit’s demand
for bananas is the same as the effect of an (increase, decrease)
_________ in his income of ____ per day.
Ans. (a) (20, 10),
(b) `30, (c) (15, 15),
(d) Increase by 5
units, (e) (20, 20),
(f)
Increase by 5 units,
(h) Increase, ` 10
Q7. A consumer has a utility function U(x, y) = xy. Suppose the initial prices
were Px = 1, Py = 2 and the price of x falls to 0.5. The consumer’s income
is unchanged at Rs. 100. What is the change in demand for x? How much
of this is due to the substitution effect and how much due to income
effect?
[DSE MA Ent. Eco. 2000]
Q8. The utility function of a consumer is : U(x, y) = xy. Suppose income of
the consumer (M) is 100 and the initial prices are Px = 5, Py = 10. Now
suppose that Px goes up to 10, Py and M remaining unchanged. Assuming
slutsky compensation scheme, estimate price effect, income effect and
substitution effect.
Q9. The utility function of a consumer is : U(x, y) = x2y. Suppose income of
the consumer (M) is 100 and the initial prices are Px = 10, Py = 5. Now
suppose that Px goes down to 5, Py and M remaining unchanged.
Assuming slutsky compensation scheme, estimate price effect, income
effect and substitution effect.
Q10. Rahul spends all his income on two goods x1 and x2. According to him x1
and x2 are perfect substitutes. One x1 is just as good as 1 x2. It is known
that p1 = `4 and
p2 = `5.
(a) If Rahul’s income is `120 find his optimum choice. Draw his
budget line and mark optimum choice by A.
(b) If the price of x1 decreases to `3 a unit, will Rahul buy more of it?
What part of the change in consumption is due to the income effect
and what part is due to the substitution effect?
PMG
43 | P a g e
If the price of x2 decreases to `3 a unit, while p1 = `4 and m = 120.
Find his new optimum choice. Draw his new budget line and mark
optimum choice by B. What part of the change in consumption
from A to B is due to the income effect and what part is due to the
substitution effect?
(a) (30, 0),
(b) Yes, all due to income effect, (c) (0, 40),
all due to substitution effect,
The utility function of a consumer is : U(x, y) = 2x + 4y. Suppose income
of the consumer (M) is 100 and the initial prices are Px = 10, Py = 5. Now
suppose that Px goes down to 4, Py and M remaining unchanged.
Assuming slutsky compensation scheme, estimate price effect, income
effect and substitution effect.
The utility function of a consumer is : U(x, y) = min(2x, 4y). Suppose
income of the consumer (M) is 100 and the initial prices are Px = 10, Py =
5. Now suppose that Px goes down to 5, Py and M remaining unchanged.
Assuming slutsky compensation scheme, estimate price effect, income
effect and substitution effect.
The utility function of a consumer is : U(x, y) = x y . Suppose income
of the consumer (M) is 100 and the initial prices are Px = 10, Py = 5. Now
suppose that Px goes down to 5, Py and M remaining unchanged.
Assuming slutsky compensation scheme, estimate price effect, income
effect and substitution effect.
Consider a consumer with utility function U(x, y) = y + ex. Derive the
demand curve for commodity x. Determine the Substitution and Income
effects of a decrease in price of x from Px = e3 to Px = e2, when money
income = 100 and
Py = e2.
A certain utility function takes the form u(x1, x2) = v(x1) + x2. What will
be the income effect on the demand for good 1 if price of good 1 falls.
Illustrate with the help of a diagram.
(c)
Ans.
Q11.
Q12.
Q13.
Q14.
Q15.
44 | P a g e
PMG
Topic 8 :
Buying and Selling
Q1.
Q2.
Q3.
Q4.
Q1.
Q2.
Q3.
Q4.
Q5.
Theory Questions
Using Slutsky equation show that the effect of wage increase on supply of
labour. How can one ensure that labor works the longer hours.
Discuss the shape of labour supply curve when labour is an inferior good.
Aditi has strictly convex preferences between leisure and a consumption
commodity. Let her endowment of time be ‘T’ hours which she can
devote to work or leisure. Her only source of income is the work she
performs. What is the shape of her labour supply curve when :
(i)
Leisure is a normal good.
(ii) Leisure is an inferior good.
With the help of indifference curves, show how an increase in the wage
rate could induce a worker to supply more or less labour, depending on
the strength of income and substitution effects. Also, briefly discuss the
impact of extra non-wage income on labour supply. [DSE MA Ent. Eco. 1999]
PMG
Numerical Problems
If a consumer’s net demand is (5, 3) and her endowment is (4, 4). What
are her gross demand.
The prices are (p1, p2) = (2, 3) and the consumer is currently consuming
(x1, x2) = (4, 4). There is a perfect market for the two goods in which they
can be bought and sold costlessly. Will the consumer necessarily prefer
the bundle (y1, y2) = (3, 5)? Will she prefer having the bundle (y1, y2)?
The prices are (p1, p2) = (2, 3) and the consumer is currently consuming
(x1, x2) = (4, 4). Now the prices change to (q1, q2) = (2, 4). Could the
consumer be better off under these new prices?
A country is importing about half of the oil that it uses. The rest of its
needs are met by domestic production. Could the price of oil rise so much
that the country would be made better off?
Abhishek owns 20 kg mangos and 5 kg apples. He has no income from
any other source, but he can buy or sell either mangos or apples at their
market prices. The price of apples is four times the price of mangos.
There are no other commodities of interest.
45 | P a g e
Q6.
(a) How many mangos could he have if he was willing to do without
apples?
(b) How many apples could he have if he was willing to do without
mangos?
(c) Draw the budget line of Abhishek and label the endowment by the
letter E.
(d) If the price of mango is ` 1 per kg and price of apples is ` 4 per kg,
then write Abhiskek’s budget Equation.
(e) Now, If the price of mango becomes ` 2 per kg and price of apples
becomes ` 8 per kg, what effect does the doubling both prices have
on the set of commodity bundles that Abhishek can afford?
(f)
Suppose Abhishek Decides to sell 10 kg mangos and buy apples
from the money so obtained. Label his new consumption bundle as
A on the graph.
(g) Now, after this trade the price of apples falls to ` 1 per kg, draw his
new budget line.
(h) Using weak axiom of revealed preferences mark that portion of
new budget line which Abhishek will surely not Choose.
Manish has a small garden where he raises onions and tomatoes. He
consumes some of these vegetables and he sells some in the market.
Onions and tomatoes are perfect complements for Manish, since the only
recipes he knows use them together in the ratio 1 : 1. During one week his
garden yielded 30 kg of onions and 10 kg of tomatoes. At that time the
price of each vegetable was ` 5 per kg.
(a) What is the monetary value of Manish’s endowment of vegetables?
(b) Draw the budget line of Manish and label the endowment by letter
E.
(c) Determine the final consumption bundle of Manish and draw an
Indifference curve through his final choice.
(d) Suppose that before Manish makes any trade the price of tomatoes
rise to `15 per kg while the price of onions stays at ` 5 per kg. What
is the value of Manish’s endowment now.
(e) Draw his new budget line and determine the consumption bundle
that he will choose now.
46 | P a g e
PMG
(f)
Q7.
Q8.
Suppose that before Manish makes any trade the price of tomatoes
rise to `15 per kg while the price of onions stays at ` 5 per kg, what
is the change in demand for tomatoes due to substitution effect?
What is the change in demand for tomatoes due to ordinary income
effect? And, what is the change in demand for tomatoes due to
endowment income effect? Also determine the total change in
demand for tomatoes.
Ruchi consumes x and y and derives utility according to u(x, y) = xy. She
gets a gift of 220 units of x and 140 units y from Gaurav when the market
price of x is Re. 1 and of y is Rs. 2. She is allowed to buy and sell in the
market.
(i)
What are the gross and net demands of x and y for Ruchi?
(ii) If the price of y reduces to Re. 1, work out the resulting equilibrium
quantities of x and y.
(iii) Break up the change in quantity demanded of y into the appropriate
substitution effect, ordinary and endowment income effect.
The owner of an egg farm produces 300 eggs a weak. Initially the price of
eggs is Rs. 4 per egg. Her demand function for eggs, for her own
consumption is:
PMG
x = 180 + m
10P
Q9.
Where x is the number of eggs she consumes per week, P is the price of
an egg and m is her income. The price of eggs then falls to Rs. 3 per egg.
How many eggs does she consume before and after the fall in price of
eggs? How much is the endowment income effect on her consumption of
eggs?
Sumit works in a factory. Let C be the number of rupees he spends on
consumer goods and let R be the number of hours of leisure that he
chooses.
(i)
Suppose that Sumit has a wage rate of `w per hour, a non labour
income of `m and that he has 18 hours a day to divide between
labour and leisure. Determine his budget line.
(ii) Sumit’s utility function is U(R, C) = C.R, determine his demand of
leisure as a function of w and m. Also determine his supply
function of labour.
47 | P a g e
(iii)
Sumit has 18 hours in a day to devote to labour or leisure and he
earns `8 per hour of work. If he has a non labour income of `16 per
day, write his budget equation between leisure and consumption
and draw it on the graph.
(iv) Sumit’s utility function is U(R, C) = C.R, determine the number of
hours of leisure and work per day which he will choose?
(v) If Sumit’s wage rate increases to `12 per hour, write the equation of
his new budget line and draw it on the graph.
(vi) If he continued to work exactly as many hours as he did before the
wage increase how much more money would he have each day to
spend on consumption?
(vii) Find his optimal consumption bundle with new budget line.
Q10. Rohan is from Agra and he has just joined a college in Delhi university.
He is from a poor family and his parents can only send `50 per week to
him. He has 50 hours per week as leisure time left after all necessary
activities. He has an option to work in a nearby shop at `5 per hour. His
utility function for leisure (L) and consumption of all other goods (C) is
U(L, C) = LC.
(i)
Write the equation of Rohan’s budget line, assuming that the price
of all other goods is `1.
(ii) Determine his optimal consumption bundle of L and C.
(iii) Determine his optimal labour supply.
Q11. Vipul has 80 hours a week to allocate between work and leisure and he
does not have any other source of income. He gets `4 an hour for the first
40 hours and `6 an hour for every hour beyond 40 hours a week. His
utility function for leisure (R) and consumption (C) is U = C.R.
(i)
Determine the point where Vipul’s budget line has a kink.
(ii) Write the equation of that part of budget line where he does not
work overtime.
(iii) Write the equation of that part of budget line where he chooses to
work overtime.
(iv) Will Vipul choose to work overtime?
(v) What is the optimal bundle for Vipul?
(vi) If Vipul is being paid `5 per hour uniformly, find his optimal
choice.
PMG
48 | P a g e
Q12.
Q13.
Q14.
Q15.
(vii) If given a choice which wage plan Vipul will choose?
A self employed plumber is endowed with 168 hours per week and no
non labour income. Current wage rate is Rs. 10 per hour.
(i)
Write down and draw his budget equation.
(ii) His optimal choice of work is 40 hours per week. He is offered Rs.
20 per hour by an MNC to work for longer hours. But he still
chooses to work only for 40 hours. Do you think that the plumber’s
behavior is rational? Explain graphically.
(iii) What alternative remuneration scheme would you suggest to the
MNC to induce the plumber to work for longer hours? Explain
graphically.
Suppose the labour supply (l) of a household is governed by
maximization of its utility u c 2 / 3 h1 / 3 , where c is the households
consumption and h is leisure enjoyed by the household (with h + l = 24).
Real wage rate (w) is given and the household consumes the entire labour
income (wl). What is the households labour supply? Does it depend on w?
Dudley’s utility function is U(C, R) = C – (12 – R)2, where R = hrs of
leisure per day C is consumption level. He has 16 hrs a day to allocate
between leisure and work. He has an income of Rs. 20 a day from nonlabour sources. Price of consumption goods is Re. 1.
(i)
How many hours will Dudley work if the hourly wage rate is Rs.
10?
(ii) What is the non-labour income drops to Rs. 5 a day?
(iii) Suppose he has to pay an income tax of 20% on all his income and
suppose non labour income remains Rs. 20 a day. How many hours
will he then choose to work?
A worker’s utility function for leisure (R) and consumption (Y) is
PMG
u ( R, Y )
R
Y . If the price of consumption is 1 and wage rate is
‘w’, can his labour supply curve be backward-bending?
49 | P a g e
Answer Sheet of Topic 8 :
1.
2.
3.
4.
5.
6.
7.
8.
9.
Numerical Problems
(9, 1),
No, as (y1, y2) is more expensive. Yes, as it increases the value of her
endowment,
It depends whether the consumer is net buyer or net seller of the good that
has become more expensive.
Yes, If the country turns to being a net exporter of oil.
(a) 40kg, (b) 10 kg,
(d) M + 4A = 40,
(e) No
effect,
(a) `200, (c) (20kg, 20kg),
(d) ` 300,
(e) (15kg,
15kg),
(f)
0kg, -10kg, 5kg, -5kg,
(i)
Gross Demand = (250, 125) & Net Demand = (30, -15), (ii)
(180, 180),
-10,
(i)
wR + C = m + 18R,(ii) D : R = 9 + m/2w, S :18 – R = 9 – m/2w,
PMG
(iii)
10.
11.
(i)
(i)
8R + C = 160,
(vi) `32, (vii)
5L + C = 300,
R = 40, C = 160,
(iv) Yes, (v)
per hour,
50 | P a g e
(iv) 10hours, 8 hours, (v)
R = 25/3 hours, `116,
(ii) L = 30, C = 150, (iii)
(ii) 4R + C = 320,
(iii)
R = 100/3, C = 200,
(vi)
12R + C = 232,
20 hours,
6R + C = 400,
R = 40, C = 200, (vii) `5
Topic 9 :
Intertemporal Choice
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Theory Questions
Write the budget constraints for the consumer in terms of present value
and future value. What would be their geometric interpretation?
Is the following statement true or false : “If both current and future
consumption are normal goods, an increase in the interest rate will
necessarily make a saver save more.” Explain.
A consumer who is initially a lender, remains a lender even after decline
in interest rates. Is this consumer better off or worse off after the change
in interest rates? If the consumer becomes a borrower after the change is
he better off or worse off?
In the context of inter-temporal choice, diagrammatically analyze the
impact of a rise in interest rate for a person who is initially a lender of
capital.
An individual has an endowment of 2 periods (m1, m2). He has option of
lending and borrowing money at given rate of interest r. Show that he
may be borrower or lender depending upon his Intertemporal preferences
[Assume no inflation]. As interest rate declines, a consumer who is
initially a lender, remains a lender even after a decline in interest rates. Is
this consumer better off or worse off after the change in interest rates? If
the consumer becomes a borrower after the change, is he better off or
worse off?
What is the slope of a consumer’s inter-temporal budget line? What is its
shape if the consumer can borrow at a higher interest rate than the one at
which he can lend?
Decide whether each of the following statements is true or false. Then
explain why your answer is correct based on the Slutsky decomposition
into income and substitution effects.
(i)
If both the current and future consumptions are normal goods then
an increase in interest rate will necessary make a saver save more.
(ii) If both the current and future consumptions are normal goods then
an increase in interest rate will necessary make a saver choose
more consumption in second period
51 | P a g e
PMG
Ans. (i)
False,
(ii)
True
Numerical Questions
Present Value
Q1. What is the present value of Rs. 100 one year from now if the interest rate
is 10%? What will be the change in present value if the rate of interest
comes down to 5%?
Q2. Find the present value of Rs. 500 due 10 years hence when interest of
10% is compounded (i) half yearly (ii) continuously.
Ans.(i)
Rs. 188.40, (ii) 183.94,
Q3. Anand is trying to decide whether or not to go to a technical institute. If
he spends 2 years in the technical institute, paying Rs. 15,000 tuition each
year, he will get a job of Rs. 60,000 per year for the rest of his working
life. If he does not go to the institute, he will take up a job immediately.
He will then earn Rs. 30,000 yearly for 3 years, then Rs. 45,000 per year
for the next 3 years and thereafter Rs. 60,000 annually. If the interest rate
is 10%, is the institute training a good financial investment?
Q4. Given the following three projects, which project has the maximum net
discounted present value assuming the market rate of interest as 8% :
(i)
Project A which yields an income of Rs. 200 per year, indefinitely
into the future.
(ii) Project B which yields Rs. 400 per year for 2 years.
(iii) Project C which yields Rs. 100 today and Rs. 200 after 2 years.
Ans. (i)
N. P. V. = 2500, (ii) N. P. V. = 713.31, (iii) N. P. V. =
271.47
Hence the project A has the maximum net discounted present value.
Q5. Suppose that the residents of Vegopolis spend all of their income on
cauliflower,
broccoli, and carrots. In 2001 they buy 100 heads of cauliflower for 200€,
50 bunches of broccoli for 75€, and 500 carrots for 50€. In 2002 they buy
75 heads of cauliflower for 225€, 80 bunches of broccoli for 120€, and
500 carrots for 100€. If the base year is 2001, what is the CPI in both
years? What is the inflation rate in 2002?
PMG
Intertemporal Choices
52 | P a g e
Q6.
Kriti consumes (c1, c2) and earns (m1, m2) in periods 1 and 2 respectively.
Suppose the interest rate is r.
(i)
Write down her Intertemporal budget constraint in present value
terms.
(ii) If c1 = 0 then what is c2? Is this the present value or future value of
endowment of her income?
(iii) What is her maximum possible consumption in period 1? Is this the
present value or future value of endowment of her income?
value,
(iii)
Q7.
c2
(1 r )
c1
Ans.(i)
c1
m1
m2
(1 r ) ,
(ii)
c2
m1 (1 r ) m2 , future
PMG
m1
m2
(1 r ) , Present value
a 1 a
Mona has a Cobb-Douglas utility function U (C1 , C 2 ) C1 C 2 where 0
< a < 1 and C1 & C2 are her consumptions in periods 1 and 2 respectively.
Suppose that Mona’s income is m1 in period 1 and m2 in period 2.
(i)
Write down her budget constraint in terms of present values.
(ii) If we compare this budget constraint with the standard form p1x1 +
p2x2 = m then what is p1, p2 and m here.
(iii) If a = 0.2, solve for Mona’s demand functions for consumption in
each period as a function of m1, m2 and r.
(iv) If interest rate increases than what will be its impact on
consumption of period1 and period 2?
Ans.(i)
m
c2
(1 r )
c1
m1
m1
m2
(1 r ) ,
(ii)
p1 = 1, p2 = 1/(1 + r) &
m2
(1 r )
(iii)
(iv)
Q8.
c1 will decrease and c2 will increase.
A consumer has an income of Rs. 2,000 this year and he expects an
income of Rs. 1,100 next year. He can borrow and lend at an interest rate
53 | P a g e
0.2m1
0.2m2
c
(1 r ) , 2
c1
0.8(1 r )m1 0.8m2 ,
of 10%. Consumption goods cost Rs. 1 per unit this year and there is no
inflation. He has the utility function U = C1.C2. Calculate the following :
(i)
The present value and future value endowment,
(ii) Optimal choice of present and future consumption,
(iii) Borrowing/Saving in first period.
(iv) Now if the interest rate rises to 20% find his new consumption
bundle. Also determine his borrowing/Saving in first period.
Ans.(i)
Rs. 3000 and Rs. 3300, (ii) c1 = 1500, c2 = 1650,
(iii)
savings of Rs. 500, (iv) c1 = 1458.3, c2 = 1750, savings of Rs.
541.7
Q9. Mr. N has an income of Rs. 4,000 this year and Rs. 2,000 next year. He
can borrow and lend at 20% per annum. Assuming no inflation, p1 = p2 =
C1C2
1. Given that his utility function is U (C1 , C2 )
(i)
Draw the budget line.
(ii) Write equilibrium conditions.
(iii) Equilibrium levels of C1, C2.
(iv) Effect on equilibrium levels of C1, C2 if rate of interest changes to
10% and then 30% (only diagrammatic proof)
Q10. Let Mr. N have an income of Rs. 4,000 this year and Rs. 2000 next year.
Rate of interest is 20%. His utility function is U = C1.C2, where C1 is the
consumption in current year and C2 is the consumption in next year. Find
the Marginal Rate of Substitution between C1 and C2 and his equilibrium
C1 and C2, assuming no inflation and prices of consumer goods be Re.1
per unit in both time periods. Will he borrow or lend.
Q11. A consumer has the utility function u(c1, c2) = c1.c2, where c1 and c2 are
the consumptions in period 1 and 2, respectively. He earns an income of
Rs. 1,00,000 in period 1 and Rs. 1,29,600 in period 2. If the objective is
to optimize the consumption choice over time, work out the required
consumption in each period and determine whether he would need to
borrow or lend? Assume that the rate of interest is 8% per annum and
there is no inflation.
Q12. Suppose there are two goods of which your consumption in the year I is
C1 and consumption in year II is C2. Endowments in the two years are m1
= 100 and m2 = 100. The interest rate is given as r = 10%. If your utility
PMG
54 | P a g e
0.75
Q13.
Q14.
Q15.
Q16.
0.25
function is U (C1 , C 2 ) C1 C 2 , determine how much you would like
to borrow or lend in the first year.
Suppose there are two goods of which your consumption in the year I is
C1 and consumption in year II is C2. Endowments in the two years are m1
= 75 and m2 = 120. If the interest rate is given as r = 10%. If your utility
0.8 0.2
function is U (C1 , C2 ) C1 C2 , determine how much you plan to spend
in each year and how much you would like to borrow or lend in the first
year.
Suppose that the utility function of a consumer is given as
U ( x1 , x2 ) x11 / 2 x12 / 2 , where x1 and x2 are consumption in the two
periods. In period I her income is Rs. 5000 whereas she earns nothing in
period II. The market rate of interest is given as 20%. Find the optimum
consumption in each period.
The Intertemporal utility function is given to be homogeneous cobbdouglas of degree 1 and is described as such that is twice that of . The
inter temporal endowment is given as Rs. 100 and Rs. 110 and the market
rate of interest is 10%. Find the optimum inter temporal consumption
function and whether the individual is a lender, borrower or Polonius.
Abhimanyu will earn Rs. 50,000 in first period. In the second period he
will retire and will live on his savings. His utility function is U = C 1.C2,
where C1 is the consumption in current year and C2 is the consumption in
next year. He can borrow and lend at the rate of 10%.
(i)
If the interest rate rises, will his period 1 consumption increase,
decrease or stay the same?
(ii) Would an increase in the interest rate make him consume more or
less in the second period?
(iii) If his income is zero in period 1 and Rs. 55,000 in period 2 would
an increase in the interest rate make him consume more, less or the
same amount in period 1?
PMG
Intertemporal Choices with Inflation
Q17. A consumer has the utility function U(C1, C2) = C1.C2. The consumer has
income Rs. 100 in period I and Rs. 121 in period II.
55 | P a g e
(a)
There is no inflation and the interest rate is 10%. Find the
equilibrium values of C1 and C2.
(b) Suppose the inflation rate is 6% and interest rate is 10%. Suppose
price of period I goods is Re. 1.
Find the exact value of real interest rate and the new budget constraint.
Q18. A consumer has the utility function U(C1, C2) = min(C1, C2). The
consumer has income Rs. 2,000 in period I and Rs. 1,100 in period II. If
the rate of interest is 21% find his optimum consumption for two periods.
Now suppose the money income in the two periods is same as above and
interest rate is also 21% but there is a 10% inflation rate then find his new
optimum choice.
Different Interest Rate for Borrowing and Lending
Q19. Mr. B earns Rs. 500 today and Rs. 500 tomorrow. He can save for future
by investing today in bonds that return tomorrow the principal plus
interest. He can also borrow from his bank paying an interest. When the
interest rates on both bank loans and bonds are 15% Mr. B chooses
neither to save nor to borrow.
(i)
Suppose the interest loan on bank loan goes up to 30% and interest
rate on bonds fall to 5%. Write down the equation of new budget
constraint and draw his budget line.
(ii) Will he lend or borrow? By how much?
Q20. Varun lives in a small city with only one bank that provides the facilities
of borrowing and lending at an interest rate of 6% per annum. Varun has
a job that ensures an income of Rs. 1,04,000 this year and Rs. 1,06,000
the next year. His utility function over the two years is U = f(C0, C1) =
C0C1, where C0 and C1 represent the value of consumption in the current
and next year respectively. Assume that prices do not change over the
two year period.
(i)
Draw the inter-temporal budget line that he faces and determine his
*
*
optimum consumption choice C 0 , C1 . Does he lend or borrow in
the current period?
(ii) After the Government announced a deregulation of the savings
bank deposit interest rate, the bank now provides the facilities of
borrowing and saving at a higher interest rate. Depict and explain
with the help of a representative diagram how this increase in the
PMG
56 | P a g e
(iii)
interest rate affects Varun’s choice of being a borrower or a lender,
and how does his welfare change.
Suppose the bank now decides to give an interest rate of 8% per
annum on deposits and charge an interest of 10% per annum from
borrowers. Draw Varun’s inter-temporal budget line. Find his
optimum consumption mix Cˆ 0 , Cˆ1 . Compare
C 0* , C1* in terms of welfare implications.
Cˆ 0 , Cˆ1
and
Bond
Q21. A person purchased a bond of face value Rs. 100 on which interest is
payable yearly at 4% per annum. He received in all three interest
payments, the first one falling due one year after purchase of the bond.
At the end of three years the bond has matured for payment at par. If the
person has realized an interest yield of 5% per annum in the transaction,
what was the purchase price of the bond?
Q22. An individual has to choose between a perpetual bond giving a return of
Rs. 1,000 every year starting from the year two and one that gives Rs.
50,000 in year 1, Rs. 50,000 in year 2 and a return of Rs. 10,000 and a
scrap value of Rs. 1200 in year 3? Assume a rate of interest of 10%.
Show your calculations.
PMG
57 | P a g e
Topic 10 :
Production
Theory Questions
Q1. If the average product is rising, marginal product is also rising. Do you
agree?
Q2. Show the relation between average product of labour and marginal
product of labour.
Q3. If Diminishing marginal returns set in from the very start what do the
total product, average product and marginal product curves of the
variable input look in this case?
[Hint : AP, MP always falling, TP concave]
Q4. Why marginal rate of technical substitution is always diminishing?
Q5. What is MRTSLK? Show that MRTSLK is equal to the ratio of Marginal
Product of Labour and Marginal Product of Capital.
Q6. Explain how the curvature of an isoquant relates to marginal rate of
technical substitution.
Q7. Suppose the marginal rate of technical substitution between two factors of
production increases, as factor labour is substituted for factor capital.
What would the tangency condition between iso-cost line and an isoquant
indicate? What would be the most preferred production choice? Can
similar choices occur even when the MRTS between labour and capital is
declining?
[Hint : Concave isoquant, corner optimum]
Q8. What are the characteristics of a linear homogenous production function?
Q9. Distinguish between return to scale and economies of scale. Is it possible
to have constant return to scale and economies of scale together?
Q10. Can one have decreasing marginal product of one factor along with
increasing returns to scale? Give an example.
Q11. In a production process is it possible to have decreasing marginal product
in an input and yet increasing returns to scale?
Q12. Is it possible to have diminishing returns to a single factor of production
and constant returns to scale at the same time? Discuss.
Q13. Explain the main properties of technology.
Q14. Give an example of each of the following
PMG
58 | P a g e
(i)
A production function that exhibits diminishing marginal returns,
but increasing returns to scale.
(ii) A production function that exhibits increasing marginal returns, but
decreasing MRTS.
(iii) A production function that exhibits increasing returns to scale, but
decreasing MRTS
Q15. What do we mean by ‘elasticity of substitution’ and ‘marginal rate of
technical substitution’ as used in theory of production. Which is a better
measure of the degree of substitutability of factors?
PMG
Q2.
Numerical Problems
The production function of a firm is given by Q = 20L1/3K2/3. Find the
average and marginal products and evaluate them when L = 8 and K = 27.
The production function of a firm is given by :
Q 8LK L2 K 2 , where L > 0 and K > 0. Find marginal productivity of
labour(L) and capital (K).
Q3.
The production function of a firm is given as Q
Q4.
(i)
Average product of labour and capital,
(ii) Marginal product of labour and capital, when L = 10, K = 5.
3 / 4 1/ 4
The production function of a firm is given by Q 4 L K , L > 0, K >
0. Find the marginal productivities of labour(L) and capital(K). Also
show that :
Q1.
L
Q5.
Q6.
Q7.
Q
L
K
Q
K
L2 K 2
L K
, Find
Q.
If the production function be Q 2L3 / 4 K 1 / 3 , where L and K are inputs
labour and capital.
(i)
Find marginal products of capital (K) and labour (L).
(ii) If the capital is fixed, show that APL curve slopes downward.
For the production function x A.L .K , show that marginal product of L
is times its average product.
For the production function Q = f(L,K) = AKα Lβ ea(K/L)
Where A, α, β and a are positive constants, find the marginal products of
labour and capital. Also verify Euler’s theorem.
59 | P a g e
Q8.
Q9.
Q10.
Q11.
Q12.
Q13.
Given the production function x
2 HLK AL2 BK 2
CL DK
, where H, A, B, C,
D are positive constants, x denotes output and L, K denote labour and
capital inputs respectively.
(i)
Find the average and marginal products of factors of production.
(ii) Verify Euler’s theorem for the production function.
For the production function Q = f(L, K) given implicitly, in the form of
F(L, K, Q) = 0, by the equation 6Q 3 Q 2 6 L 24K L2 4 K 2 50 0 , find
the marginal productivities of labour and capital.
Let Q A.L .K be the production function of a firm. Show that the ratio
of MPL to APL is constant.
The production function for a commodity is :
Q 10L 0.1L2 15K 0.2 K 2 2 KL , where L is labour and K is capital and
Q is production.
(i)
Calculate the marginal products of two inputs when 10 units each
of labour and capital are used.
(ii) 10 units of capital are used. What is the upper limit for use of
labour which a rational producer will never exceed?
[Hint : Find L such that Q/ L 0, when K = 10]
A production function is given by Q = 3L3K2 – 2L2K3, when L and K are
inputs, labour and capital.
(i)
Find average product and marginal product of labour.
(ii) If input K is fixed, what is the value of input L for which AP is
maximum?
(iii) Does the maximum of MPL reach at lower level of employment
than that of APL.
Given X L3 / 4 K 1 / 4 . Does law of DMPL hold if labour increases. Also, if
labour increases will marginal product of capital increase or decrease.
PMG
Q14. Show that
APl
l
l.MPl q
and hence deduce that at maximum APl we
l2
have APl = MPl.
1/ 4 1/ 8
Q15. Suppose the production function is q l k :
(i)
What is the marginal product of labor?
(ii) What is the marginal product of capital?
60 | P a g e
(iii)
(iv)
Q16.
Q17.
Q18.
Q19.
Q20.
What is the firms MRTS?
Does this production technology exhibit increasing, decreasing, or
constant returns to scale?
Consider a cobb douglas production function Y = AL K where K and L
are respectively the capital and labour to produce output Y. Show that if
all the factors are paid according to their marginal products, the total
product will be exhausted if + = 1.
Explain. Which of the following production functions depict diminishing
marginal rate of technical substitution between the two inputs labour (l)
and capital (k) ?
q 100k 0.2 l 0.8
(i)
(ii) q 20k 5l
Given the production function q = f(k, l) = kl(800 – kl), with fl > 0, fk > 0
for all
l > 0 and k > 0, determine the range of positive values of k and
l over which (i) marginal productivities of l and k are diminishing and (ii)
cross productivity effect is positive.
A firm is in the business of assembling personal computers (PCs) and it
1/ 3 1/ 2
has the following production function : q f (l , k ) l k , where l
and k are measures of labour and capital input used to produce q number
of PCs that are sold in the market at a fixed price = p per PC. Cost of one
unit of l and k are ‘w’ and ‘v’ respectively. Does this production function
exhibit increasing, decreasing, or constant returns to scale?
Do the following production functions exhibit increasing, constant, or
decreasing returns to scale? In each case, find if returns to labour (l) are
increasing, decreasing or constant, as capital (k) is held constant.
PMG
(a)
q
3l
q
l 0.8 k
3 k
(b)
q
l2
k2
1/ 2
(c)
Q21. (a) Define elasticity of substitution.
(b) Calculate the elasticity of substitution of the following production
functions:
q(k , l ) k a l b
(i)
(ii) q(k , l ) min(ak , bl)
Where, 0 < (a, b) < 1, l and k are inputs used to produce q.
61 | P a g e
(c)
What complication arises when one generalizes the elasticity of
substitution to the many-input case?
Answer Sheet of Topic 10 :
1.
2.
Numerical Problems
APL = 45, APK = 40/3, MPL = 15, MPK = 80/9,
MPL = 8K – 2L, MPK = 8L – 2K, Degree = 2,
3.(i)
APL
L2
L2
4.
MPL
3L 1 / 4 K 1 / 4 , MPK
MPL
3 1/ 4 1/ 3
L K
2
7.
MPL
8.
APL
K2
, APK
LK
L2 K 2
LK K 2
PMG
AK L
1 aK / L
e
L3 / 4 K
, (ii)
3/ 4
ACL2
MPL
,
aK
, MPK
L
2 HLK AL2 BK 2
, APK
CL2 DLK
MPL
5.
1
AK
7
1
,
, MPK
9
9
2 3/ 4
MPK
L K
3
L e aK / L
2/3
,
aK
L
2 HLK AL2 BK 2
,
CLK DK 2
2 DHK 2 2 ADKL
(CL DK ) 2
BCK 2
, MPK
2 HCL2
2 BCKL BDK 2
(CL DK ) 2
AL2 D
,
9.
3 L
, MPK
9Q 2 Q
MPL
12 4 K
, 11.(i) MPL = 18, MPK = 31,
9Q 2 Q
(ii)
150,
12.(i) APL
3L2 K 2
Yes
62 | P a g e
2 LK 3 , MPL
9 L2 K 2
4 LK 3 ,
(ii)
L
K
3
,
(iii)
L
Topic 11 :
Cost
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q1.
Theory Questions
Can the long run total cost curve of a firm be positively sloped straight
line through the origin? What does it imply? What shapes will the long
run average cost and long run marginal cost take in this case? Can the
short run average cost be ‘U’ shaped?
Is a firm minimizing costs if the Marginal Product of Labour is 6 and
Marginal Product of Capital is 5, the wage rate is Rs. 2, the interest on
capital is Re. 1? If not what must the firm do minimize the costs?
Draw and briefly explain the shape of the total, average and marginal cost
curves for a firm facing constant input prices and has a production
function depicting
(i) Constant Returns to Scale, and (ii) Increasing
Returns to Scale.
Why does the minimum point of AVC curve lie to the left of the
minimum point of the ATC curve?
What do you mean by the cost function of a firm? What is the nature of
relationship between
(i)
The average cost and marginal cost curves
(ii) The long run and short run average cost curves
(iii) The long run and short run marginal cost curves
If the production function exhibits increasing returns to scale everywhere,
a firm’s long run average cost curve must be declining. Do you agree?
PMG
Numerical Problems
The cost function for a good X is given as
C 30 8 x 4 x 2 x 3
Write down TFC, TVC, AVC, AFC, and MC. what will be the shut down
point.
Q2. A firm’s long run cost function is : C = f(Q) = Q3 – 100 Q2 + 2750Q,
where Q is output produced by the firm per period. Find the firm’s
minimum efficient scale (MES) of production. Find the average cost and
marginal cost of production at the MES of production.
63 | P a g e
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Ans.
Q10.
Q11.
Q12.
Find the equilibrium condition for maximizing of output Q = f(L, K)
subject to a budget constraint C = wL + rK with the help of lagrangian
multiplier.
The firm is employing 100 hours of labour (L) and 50 bags of cement (K)
to produce 500 blocks. Labour costs Rs. 40 per hour and cement Rs. 120
per bags for the quantities employed MPL = 3, MPK = 2. Can the firm
produce the same output at a lower total cost. Explain using a diagram.
Derive the cost function with the help of Cobb-Douglas production
function.
Given output function as Q K 1 / 2 L1 / 2 . Price of Labour (W) = 5 and Price
of capital (r) = 4. Write down the long run cost function in the form of C
= C(Q).
Derive the cost function when the production function is represented by
Q = min (L, K).
Let output be given by the following production function.
Q = min (5K, 10L)
Where K is capital and L is labour. Let price of capital be Re. 1 per unit
and price of labor be Rs. 3 per unit. If only 10 units of capital are
employed, find the short run total cost.
If a firm’s production function is given by Q = 5LK, where Q is output, L
is labour and K is capital (all per unit of time). If the price of labour is Re.
1 per unit and that of capital is Rs. 2 per unit, what combination of inputs
should be used to produce 20 units of output?
L = 2 2 , K = 2 , TC = 4 2
A firm aims to produce 4096 units of output using labour (L) and capital
(K). The technology is given as q = KL2. IF the price of L is 15 and K is
10 what are the optimal input levels.
A firm’s production function is Q 2 KL . In the short run capital
equipment is fixed, i.e., K = 100 and Price of capital is Re. 1, Price of
labour is Rs. 4. Determine the short run total cost curve.
A firm’s production technology is given by q = min(l, 2k), where l and k
are measures of labour and capital input. Price of one unit of l and k are
denoted by ‘w’ and ‘v’ respectively.
(i)
If w = v = 1, find the equation of the firm’s long run expansion
path, the conditional input demand functions and the cost function.
64 | P a g e
PMG
(ii)
If the price of l becomes 1.5 times that of k, find the equation of the
firm’s long run expansion path, the conditional input demand
functions and the cost function.
0.25
0.25
Q13. A firm has a production technology given by Q L .K . Initial input
prices are given by w = 1 and r = 1. Suppose for the moment that the
amount of capital is fixed at K = 4.
(a) Is the marginal product of labour diminishing? Why?
(b) Find the short run total cost function, and then the short run
marginal and average cost curves
(c) Suppose now that the period is long enough that both inputs can be
adjusted. Find the capital-labour ratio.
(d) Find the returns to scale of this production technology.
(e) If the rental price of capital is doubled, what happens to the profit
maximizing output level?
Q14. Vijay gives up his small time business to undertake professional training
for one year. He would have a net profit of Rupees five lacs from the
business by the year end. The training program costs Rupees two lacs,
which is paid in the beginning of the program. If the market rate of
interest is constant 10% per annum, and Vijay finishes the training
program, what is the economic cost of undertaking the training program?
Q15. A firm’s production technology is given by q = lk, where l and k are
labour and capital input. Price of one unit of l and k are denoted by ‘w’
and ‘v’ respectively.
(a) Find the equation of the firm’s long run expansion path, the
contingent input demand functions and the long run total cost
function.
(b) Show that the long run total cost function derived in part (a) above
is
(i) homogeneous in input prices, and (ii) concave in input
prices.
PMG
65 | P a g e