ECON7030 MICROECONOMIC ANALYSIS
TUTORIAL EXERCISE 4
Question 1 This is the Featured Example from Lecture 4.
Residents of Cavity City cares only about oral hygiene (good x) and money for all
other goods (good y). Mr. Whiteteeth, a resident of the city, has a preference that can
be represented by the utility function
U(x, y) = 4 lnx + ln y.
The price of 1 unit of oral hygiene is $5. The price of y is $1. Mr. Whiteteeth has an
income of $200. You may also assume that, if Mr. Whiteteeth spends no money on oral
hygiene, he gets 0 unit of oral hygiene.
(a) (Level A) Find the consumption bundle that maximises Mr. Whiteteeth’s utility.
You may use either the tangency method or the Lagrangian method. (Hint: the
derivative of lnx with respect to x is 1/x.)
(b) (Level A) On a clearly labeled diagram with oral hygiene on the horizontal axis,
draw Mr. Whiteteeth’s budget line and indicate his optimal consumption bundle.
Sketch the indifference curve passing through his optimal consumption bundle.
(There is no need to draw it to scale.)
(c) In order to reduce tooth decay, the Council of Cavity City decides to fluoridate
the council water. As a result, all residents of the city receives 20 units of oral
hygiene for free.
(i) (Level A) In the diagram you have drawn for part (b), draw Mr. Whiteteeth’s
new budget line.
(ii) (Level B) Find the new utility-maximsing bundle for Mr. Whiteteeth. Label
the new bundle in the diagram you have drawn, and sketch the indifference
curve passing through the new bundle. (Hint: Pretend that there is no kink
to Mr. Whiteteeth’s new budget line and find the optimal bundle along the
“pretended budget line”. Then verify that what you have found is indeed
on the actual budget line.)
(iii) (Level B) Find the increase in the amount of oral hygiene due to water
fluoridation. Is the difference smaller than, equal to, or bigger than 20 (the
amount given)? Why is this the case?
(d) Mr. Whiteteeth’s neighbour, Mr. Badbreath, has the same income (i.e., $200) and
faces the same prices and the same Council policy as Mr. Whiteteeth. However,
1
Mr. Badbreath’s utility function is
U(x, y) = ln x + 4 ln y.
(i) (Level A) In a separate diagram, first draw Mr. Badbreath’s budget line
after water fluoridation.
(ii) (Level A) For the sake of argument, suppose, instead of receiving 20 units
of oral hygiene for free, the Council hands out $100 (the value of 20 units of
oral hygiene) to each resident. Find Mr. Badbreath’s optimal consumption
bundle under the hypothetical situation above. Label it in your diagram
and sketch the indifference curve passing through it.
(iii) (Level B) Is the bundle you have just found in (d)(ii) affordable under the
water fluoridation scheme?
(iv) (Level B) Now find Mr. Badbreath’s optimal consumption bundle under the
water fluoridation scheme. (Hint: The tangency condition will not hold.
The Lagrangian would not be useful either. However, the diagram you have
drawn in (d)(iii) should tell you where this bundle is.)
Question 2 Professor Egghead consumes only two goods, large (600g) eggs (x) and
jumbo (800g) eggs (y) . He cares only about the total weight of the eggs he consumes,
so his utility function is
U(x, y) = 3x + 4y.
Being a professor, he has an income of $120.
(a) (Level A) Draw an indifference curve of Professor Egghead.
(b) (Level B) Suppose large eggs costs $5 a dozen and jumbo eggs costs $8 a dozen.
With the aid of a diagram, find Professor Egghead’s utility maximising consump-
tion bundle.
(c) (Level B) Now jumbo eggs are on sale and each dozen costs only $6. The price
of large eggs remains at $5 per dozen. Using a separate diagram, find Profes-
sor Egghead’s new utility maximising consumption bundle.
(d) (Level B) Later, the price of jumbo eggs goes back to $8 a dozen. Meanwhile, the
price of large eggs increases to $6 per dozen. Using a separate diagram, describe
the utility maximising consumption bundle(s) for Professor Egghead.
Question 3 (Level B) Ichiro consumes only scoops of ice-cream (x) and cones (y).
Moreover, he insists on consuming these two goods in the combination of 1 scoop of ice-
cream and 1 cone. If there are more scoops of ice-cream than cones, he throws the extra
ice-cream away. If there are more cones than scoops of ice-cream, he throws the extra
2
cones away. His utility function of ice-cream and cones is given by
U(x, y) = min {x, y} .
(The function min {x, y} returns the smaller number between x and y. In other words, if
x < y, min {x, y} = x; if x > y, min {x, y} = y; and if x = y, min {x, y} = x = y.)
(a) Draw a couple indifference curves for Ichiro. Put ice-cream on the horizontal axis.
(Hint: Draw the 45◦ line. Choose a point on the line. If you increase the amount
of x but not y from that point, how would Ichiro’s utility change? If you increase
the amount of y but not x from that point, how would Ichiro’s utility change?)
(b) Suppose each scoop of ice-cream costs $2, and each cone costs $1. Ichiro has an
income of $6. Draw his budget line in the diagram you have drawn in part (a).
(c) Using the diagram you have drawn, find Ichiro’s utility maximising consumption
bundle.
3