Download Exercise for intermediate microeconomics and more Exercises Microeconomics in PDF only on Docsity!
INTERMEDIATE MICROECONOMICS
Tutorial 1
Exercise 1. Charlie’s utility function is U(x A
, x B
) = x A
x B
, where x A
and x B
are the quantities of
apples and bananas consumed. Suppose that the price of apples is 1, the price of bananas is 2, and
Charlie’s income is 40. Find Charlie’s optimal consumption bundle.
Exercise 2. Anne, the nut and berry consumer, has a utility function U(x 1
, x 2
1
, where
x 1
is her consumption of nuts and x 2
is her consumption of berries. Suppose that the price of a unit
of nuts is 1, the price of a unit of berries is 2, and Anne’s income is 24. Find Anne’s optimal
consumption bundle.
Exercise 3. Nancy is trying to decide how to allocate her time in studying for her economics
course. There are two examinations in this course. Her overall score for the course will be the
minimum of her scores on the two examinations. She has decided to devote a total of 1,200 minutes
to studying for these two exams, and she wants to get as high an overall score as possible. She
knows that on the first examination if she doesn’t study at all, she will get a score of zero on it.
For every 10 minutes that she spends studying for the first examination, she will increase her score
by one point. If she doesn’t study at all for the second examination she will get a zero on it. For
every 20 minutes she spends studying for the second examination, she will increase her score by
one point.
a. Write a “budget constraint” showing the various combinations of scores on the two exams
that Nancy can achieve with a total of 1,200 minutes of studying.
b. Find Nancy’s time allocation to maximize her overall score.
Exercise 4. Lily has the utility function U(x, y) = x + 3y.
a. What bundle would she choose if the price of x is 1, the price of y is 2 and her income is
b. What bundle would she choose if the price of x is 1, the price of y is 4, and her income is
Exercise 5. Nick has an income of $2,000 this year, and he expects an income of $1,100 next
year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost $1 per
unit this year and there is no inflation. Suppose that Nick has the utility function U(C 1
, C
2
) = C
1
C
2
a. What is the present value of Nick’s endowment? What is the future value of his
endowment?
b. Find Nick’s optimal consumption path. Will he save or borrow in the first period? How
much?
Exercise 6. Willy owns a small chocolate factory, located close to a river that occasionally floods
in the spring. Next summer, Willy plans to sell the factory and retire. The only income he will
have is the proceeds of the sale of his factory. If there is no flood, the factory will be worth
$500,000. If there is a flood, then what is left of the factory will be worth only $50,000. Willy can
buy flood insurance at a cost of $0.1 for each $1 worth of coverage. Willy thinks that the
probability that there will be a flood this spring is 1/10. Let c F
denote the contingent commodity
dollars if there is a flood and c NF
denote dollars if there is no flood. Willy’s utility function is
U(C
F
, C
NF
𝐹
𝑁𝐹
a. If he buys no insurance, what will be Willy’s contingent commodity bundle?
b. If Willy insures for $x, what will be his contingent commodity bundle?
c. Find Willy’s optimal bundle of contingent commodities.
Exercise 7. The demand curve for ski lessons is given by D(p D
) = 100− 2 p D
and the supply curve
is given by S(p S
) = 3p S
a. What is the equilibrium price? What is the equilibrium quantity?
b. A tax of $10 per ski lesson is imposed on consumers. Find the equilibrium price received
by sellers, equilibrium price paid by consumers and the total number of lessons given.
Exercise 8. The short-run production function of a competitive firm is given by f(L) = 6L
2 / 3
, where
L is the amount of labor it uses. The cost per unit of labor is w = 6 and the price per unit of output
is p = 3. How many units of labor will the firm hire? How much output will it produce? If the firm
has no other costs, how much will its total profits be?
Exercise 9. Nadine sells user-friendly software. Her firm’s production function is f(x 1 , x 2 ) = x 1 +
2 x 2 , where x 1 is the amount of unskilled labor and x 2 is the amount of skilled labor that she
employs.
a. If Nadine faces factor prices (1, 1), what is the cheapest way for her to produce 20 units of
output?
b. If Nadine faces factor prices (1, 3), what is the cheapest way for her to produce 20 units of
output?
Exercise 10. Joe, an avid indoor gardener, has found that the number of happy plants, h, depends
on the amount of light, l, and water, w. In fact, Joe noticed that plants require twice as much light
as water, and any more or less is wasted. Thus, Joe’s production function is h = min{l, 2 w}. Each
unit of light costs w 1
and each unit of water costs w 2
. Write Joe’s conditional factor demand
function for light, l(w 1
, w 2
, h), his conditional factor demand function for water w(w 1
, w 2
, h), and
his cost function c(w 1
, w 2
, h).
Exercise 11. Earl sells lemonade in a competitive market on a busy street corner. His production
function is f(x 1
, x 2
) = x 1
1 / 3
x 2
1 / 3
, where x 1
is the quantity of lemons he uses, and x 2
is the time spent
squeezing them.
a. Where w 1
is the cost of a unit of lemons and w 2
is the wage rate for lemon-squeezers, what
is the cheapest way for Earl to produce lemonade?
b. If he is going to produce y units in the cheapest way possible, then what input quantities
will he use? What is the cost to Earl of producing y units at factor prices w 1
and w 2
c. If one unit of lemons costs $1, the wage rate is $1, and the price of lemonade is p, what is
Earl’s marginal cost function and and his supply function S(p)?
Exercise 12. Consider a competitive industry with a large number of firms, all of which have
identical cost functions c(y) = y
2
- 1 for y > 0 and c(0) = 0.
a. What is the supply curve of an individual firm? If there are n firms in the industry, what
will be the industry supply curve?
b. What is the smallest price at which the product can be sold?
c. Suppose the demand curve for this industry is given by D(p) = 52 − p. What will be the
equilibrium number of firms in the industry? (Hint: Take a guess at what the industry price
will be and see if it works.) What will be the equilibrium output of each firm? What will
be the equilibrium output of the industry?
5 a. Use the present and future value formulas
b. The optimal consumption path satisfies:
(1) budget constraint, and
(2) utility maximization condition:
𝑀𝑈 (𝐶
1
)
𝑀𝑈 (𝐶
2
)
𝑃
1
𝑃
2
a. PV = $3000, FV =
b. C 1
= $1500, C
2
$1650. He saves $
from the first period.
6 a. C F
= m – L, C NF
= m
b. C F
= m – L + (1 – γ)x, C NF
= m – γx
c. The optimal choice satisfies:
(1) budget constraint (written in present or future value form)
(2) utility maximization condition:
𝛾
1−𝛾
𝜋
𝐹
𝑀𝑈 (𝐶
𝐹
)
𝜋
𝑁𝐹
𝑀𝑈 (𝐶
𝑁𝐹
)
a. (C F
, C
NF
b. (C F
, C
NF
0.9x; 500000 – 0.1x)
c. C F
= C
NF
7 a. At equilibrium: D(p D
) = S(p S
) and p D
= p S
b. At equilibrium: D(p D
) = S(p S
) and p D
= t
a. p* = 20, q* = 60
b. p D
= 26, p S
= 16, q* =
8 Profit maximization condition: pMP L
= w
L* = 8, y* = 24, 𝜋 = 24
9 Perfect substitutes: f(x 1
, x 2
) = a 1
x 1
x 2
The optimal input bundle satisfies:
(1) f(x 1
, x 2
) = y, and
(2) cost minimization condition
If
𝑀𝑃
1
𝑤
1
𝑀𝑃
2
𝑤
2
then the firm has multiple optimal input bundles
or ( x
1
*, x
2
*) = ( x,
𝑚−𝑤
1
𝑥
𝑤
2
) ∀ 0 ≤ x ≤
𝑚
𝑝
1
If
𝑀𝑃
1
𝑤
1
𝑀𝑃
2
𝑤
2
then the firm uses only x
1
or ( x
1
*, x
2
𝑚
𝑤
1
If
𝑀𝑃
1
𝑤
1
𝑀𝑃
2
𝑤
2
then the firm uses only x
2
or ( x
1
*, x
2
𝑚
𝑤
2
a. (x 1
*, x 2
b. (x 1
*, x 2
10 Perfect complements: f(x 1
, x 2
) = min{a 1
x 1
, a 2
x 2
The optimal input bundle satisfies:
(1) f(x 1
, x 2
) = y, and
(2) cost minimization condition a 1
x 1
= a 2
x 2
l(w 1
, w 2
, h) = h
h(w 1
, w 2
, h) = h/
c(w 1
, w 2
, h) = w 1
h +
w 2
h/
a. Cost minimization condition:
𝑀𝑃
1
𝑀𝑃
2
𝑤
1
𝑤
2
b. The optimal input bundle satisfies:
(1) f(x 1
, x 2
) = y, and
(2) cost minimization condition:
𝑀𝑃
1
𝑀𝑃
2
𝑤
1
𝑤
2
c. MC = TC’(y)
Competitive firm’s profit maximization condition (supply
decision): P = MC
a. Earl uses the two
inputs lemons and labor
in the ratio of 1:w 1
/w 2
(or
w 1
/w 2
unit of labor per
unit of lemons).
b. x 1
-1/
w 2
1/
y
3/
x 2
1/
w 2
-1/
y
3/
c = 2 w 1
1/
w 2
1/
y
3/
c. MC = 3w 1
1/
w 2
1/
y
1/
If 𝑤
1
= 1 and 𝑤
2
then MC = 3y
1/
S(p) = p
2
12 a. Competitive firm’s profit maximization condition (supply
decision): P = MC
Industry supply: S(p) = ∑ 𝑆
𝑖
𝑛
𝑖=
b. Competitive firm supply in the long run: P ≥ AC min
c. Competitive industry supply in the long run: P = AC min
a. s(p) = P/2, S(p) =
nP/
b. P min
c. n* = 50, s* = 1 , S* =
