Questions on Utility and Preferences
Abhishek
February 20, 2023
Q1)1)Seema has pocket money of Rs 120. She consume 5 units of pizzas
and 10 bottles of coke from pizza hut.The price of coke is half that of Pizza.
The Pizza hut has just announced a special deal. All pizza’s after first Pizza
will be priced at half the original price.
a)Write down her original and new Budget constraint. Draw a neatly labelled
diagram for the same.What are the slope of the two budget constraints.
b)Assume that Seema consumes coke and pizza in fixed proportion as given
above, calculate the new bundle consumed by Seema when faced with the
new budget constraint.Explain with diagram.
2)Naina’s utility function is given by U(x,y)= max(x,2y).Draw representa-
tive indifference curve and label axis properly. Does Naina has convex pref-
erences? Give reason for your answer.
3) Naidu’s utility function is given by U(x,y)=xy. Annu’s utility function is
−1
given by U(x,y)= 100xy. Ena’s utilty function is U(x,y) = xy+1 . Which of
these persons have the same preferences as Naidu’s? Justify your answer.
Q2)1) A person’s utility function is of the form U(x,y)=5xy. The prices of
good x and good y are Px = 2 and Py = 4 respectively.The person’s income
is 1200
a)Work out the optimum consumption bundle.
b)Determine the persons income offer curve(IOC). Draw a representative
IOC.
c) Explain whether each of the two good is normal or inferior.
d) Derive and draw the engel curve for x
Q3)1) Vinita likes both tennis racket and tennis shoes and would like to
consume both.At the moment she has many of both and her MRS of rackets
1
for shoes is 3.Unused racket and shoes may be returned to the stores for
refund.The current price of racket is Rs 200 and the price of shoes is Rs 100.
Is her consumption optimum? Suggest a way to make Vinita better off.
2)Mala is very flexible. She consumes x and y. She says” Give me x or give
me y, i don’t care , i cannot differentiate between the two good.”She is cur-
rently endowed with 8 units of x and 17 units of y. The price of x is 3 times
the price of y. Mala can trade x and y at the going prices but does not have
any other sources of income. How many units of y will Mala consume.
√
3)For a consumer the utility function is given by U(x,y)=2x + 2y + xy. He
has an Income of Rs1000 whilePx = 50 and Py = 50,Find his optimal choice
of consumption.
Q4)1)Which of the following utility function satisfies strict convexity.
a)U(x,y) =3x2 + 31 y 2
b)U(x,y)=min(2x+3,y).
Also draw indifference map of the two.
2)Draw the IOC and Engel curve for commodity x for the following utility
function,assumingPx = Py = 1
a)U(x,y)=max(2x,3y)
b)U(x,y)=x-y.
Q5)1) Anil’s father believes that Anil must devote 9 hours everyday, out
of which 5 hours for phy and 4 hours for Maths in order to crack IIT. If Anil
chosses some other bundle his father gets angry and measures his deviation
from the optimal by
D = |5 − P | + |4 − M |
THe larger the D is worse is Anil in terms of achieving his goal. Draw Indif-
ference curve representing the above preferences.
2)Mr Malik resides in a state where the sex ratio is very skewed.The state
government wants to improve the condition of the girls in the state and is
offering four different schemes:A,B,C and D.Two eligibility criterion is spec-
ified by the government to take advantage of the scheme.
a) Girls only household and
b) Girls above 18 years.
Scheme A :Each qualifying household gets a grant of Rs10000 that it can
spend as it wants.
Scheme B: The government gives a grant of Rs10000 only as long as the
qualifying households spend Rs10000 more than its current spending on ed-
2
ucation of their daughters.
Scheme C: The government offers a matching subsidy of 50 paise on evry
rupee spend on daughters education by the qualifying family.
Scheme D: Same as scheme C with the condition that the matching grant
will not exceed Rs10000
Mr Malik has twin daughters who have recently turned 18 and therfore qual-
ifies for the scheme.He currently earns Rs60000, out of which he spends
Rs20000 on the education of the daughters(D) and Rs40000 on other goods(O).Show
graphically and numerically how Mr Malik’s budget set will look like.
3) Consider the Utility function
U(x,y)=(x1 − 2)(x2 − 3). With the underlying budget set.
P1 X 1 + P 2 X 2 = M .
a) Draw the indifference curve for the above utility function.
b) Find the demand function for good 1. Comment if the two goods are
complements or substitutes.
Q6)1)John enjoys commodity x and y according to the following utility
function:
U(x,y)=x2 + y 2 ; x ≥ 0, and, y ≥ 0
a) Maximise John’s utility if Px = 3, Py = 4andm = 50 and find the demand
bundle.
b)Graph John’s indifference curve and its tanjency point with the budget
set.
c)Write your observation for John’s behaviour and the optimal bundle
d)Give an example of monotonic transformation of John’s utility function.
2) Suppose cofee and sushi have the same quality : the more you consume
the more you want. What kind of preference are these? For a given budget,
should you diversify if you have these kinds of preferences?
3) There is a government subsidy programme that lower the price of food
consumers from Rs3 to Rs2. The price of all other goods(AOG) is Rs 1.
Both food and AOG are normal goods and there is diminishing MRS be-
tween them,
a) Graphicaly show the change in food consumption from the programme.
Q7)1) Let U(x,y)=max[2x,y] and assume that the price of both the com-
modities is Rs 1. Derive the Income offer curve and the engel curve for
commodity x.
Consider the following binary relation defined over X, where X is the set
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of human beings:
a)”At least as tall as ”
b)” Taller than ”
c)”Is the sister of ”
Check if each of theses relations satisfy reflexivity, completeness and transi-
tivity.
3) Ajay likes oats(x) and fruit juice(y) and has concave preferences between
them.
Q8)1) What are the assumptions of ’monotonicity’ and ’averages are pre-
ferred to extremes’ with respect to consumer’s preferences?. What do these
imply for the shape of Indifference curves.
2) Suppose a consumer spends his entire income of Rs 5000 on electricity
and consumption of other goods. The electric company charges Rs 5 per
unit of electric consumption.How does the consumer’s budget line changes if
the government gives full subsidy on electricity consumption up to 100 units,
and a subsidy of Rs4 per unit on the consumption of electricity more than
100 units and up to 500 units and a subsidy Rs 2 per unit subsidy on the
consumption of electricity more than 500 units and up to 700 units and no
subsidy on electric consumption of more than 700 units. Assume that the
price of other goods is Rs 1 per unit.
3) Suppose a consumer consumes two goods X and Y and his preferences are
described by the utility function U (x, y) = lnx + Y . The consumer’s income
is Rs 120 per month and the price of Y is Rs 12 per unit.
a) Find the optimal consumption bundle when the price of X is Rs 3 per
unit.
b) Find his optimal consumption bundle if the price of X increases to Rs 6
per unit.
