Microeconomics I (Fall 2014)
Review Questions
1.
A utility function is given by u(x1 , x2 , x3 ) = a ln x1 + b ln x2 + c ln x3
a) Find demand functions for x1 , x2 , x3 .
b) What is the slope of Engel curve for
c) Find the income elasticity for
path?
x1
x1
?
. What does your finding imply about the shape of the income expansion
a b c
d) Consider v( x1 , x2 , x3 )= x1 x2 x3 . Do these two utility functions (u and v) represent the same preferences?
Prove it.
2.
Raymond’s preferences are represented by the utility function U(x, y) = x/y if y > 0, U(x, y) = 0 if y = 0.
Draw his indifference curves.
3.
Zeynep’s utility function is min{x, 5y+z}. The price of x is 1, the price of y is 10, and the price of z is 3.
Zeynep’s income is 27. How many units of x does Zeynep demand?
4.
Max has the utility function U(x, y) = x(y + 1). The price of x is $2 and the price of y is $1. Income is
$10. How much x does Max demand? How much y? If his income doubles and prices stay unchanged, will
Max’s demand for both goods double?
5.
Use separate graphs to sketch two indifference curves for people with each of the following utility
functions:
a. U(x, y) = x + 2y
b. U(x, y) = min{x, 2y}
c. U(x, y) = max{x, 2y}
6.
Atacan’s preferences over goods x and y are represented with the utility function u(x, y) = x.y + x. The
price of goods x and y are respectively Px and Py. Atacan has mY TL. to spend on these two goods.
Draw Atacan’s indifference curve that passes from points x = 4, y =2.
Calculate his MRS (marginal rate of substitution) at that point.
Write the Lagrange function for Atacan’s choice problem.
Derive the demand function for each good. Draw the demand curve for good x at when Py = 2,m =
100.
(e) Calculate the income elasticity of demand for good x when Py = 2, Px = 1,m = 100.
(a)
(b)
(c)
(d)
7.
Arda consumes strawberries and cream but only in the fixed ratio of three boxes of strawberries to two
cartons of cream. At any other ratio, the excess goods are totally useless to him. The cost of a box of strawberries
is $10 and the cost of a carton of cream is $10. His income is $200. Find his demand for strawberries and for
cream.
8.
Clara's utility function is U(X,Y) = (X + 2)(Y + 1). If her marginal rate of substitution is -3 and she s
consuming 12 units of good X, how many units of good Y must she be consuming?
9. The prices of goods x and y are each $1. Gizem has $20 to spend. She has strictly monotonic and strictly
convex preferences. At the bundle (x,y)=(10,10), her marginal rate
of substitution is -2. Is this an optimal choice for Gizem? If not, which good should she consume more of, and
which good should she consume less of?
10. Berna likes to consume books and candy bars. She has 500$ to spend every week.
The price of each book is pb = 10$ and the price of each candy bar is pc = 1$.
(Assume divisibility of goods)
(a) Suppose that book store gives a 10 percent reduction on book purchases above 10 units (only applies to extra
units). Draw Berna’s budget set.
(b) Berna realizes that if she chooses to purchase candy bars in packages their per unit price is cheaper. Suppose
that each candy bar package contains 50 units and costs 30$. Draw Berna’s budget set.
(c) Suppose both part (a) and (b) applies at the same time. Draw Berna’s
budget set.
11. Hayriye’s utility function is U(x,y)=x+2y, where x is her consumption of good x and y is her consumption of
good y. Her income is 2. The price of y is 2. The cost per unit of x depends on how many units she buys. The
total cost of x units of x is the square root of x. What is (x,y) that maximizes her utility?
12. Cagatay has a utility function U(x; y) = x  1.5 xy  30 y . The prices are px = 1, and py = 1. For incomes
between 20 and 60, draw the Engel curve for good 2.
2
13. Ayse's utility function is U(x; y) = (x + 2)(y + 1). If Ayse's marginal rate of substitution is -2 and she is
consuming 10 units of good x, how many units of good y she is consuming.
14. Justin consumes goodsX and Y and has a utility function U(x; y) = x  y . The price per unit of X is px
and the price per unit of Y is py . He has enough money so that he can afford at least 1 unit of either good. When
he
chooses his best affordable bundle,
(a) his budget line must be tangent to the indifference urve passing through this bundle.
(b) he must consume only x.
2
(c) he must consume only if
p 2x
py
his income.
(d) he must consume some of each good if px = py .
(e) he must consume some of each good if py = px=2.
15.
A consumer has a utility function u(x1, x2) = x1.x2
a) What is the marginal rate of substitution of x1 and x2 when x1 = 5 and x2 = 2.
b) Will consumer ever consume only x1 or only x2?
c) Taking prices as p1 and p2, and income as M, find what fraction of the income of
the consumer will be spent on good 2.
d) Derive the demand for both goods using the Lagrangian multiplier method.
e) What is  ? What does it mean?
f) Derive the income expansion path.
g) Derive the Engel curve for both goods.