ECON 5113 Microeconomic Theory

Winter 2016

Test 1 January 29, 2016

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobile

devices or calculators are allowed. Please write your an-

swers on the answer book provided. Use the right-side

pages for formal answers and the left-side pages for your

rough work. Answers should be provided in complete

and readable essay form, not just in mathematical for-

mulae and notations. Remember to put your name on

the front page. You can keep the question sheet after

the test.

1. Suppose that a consumer’s preference relation % on a

consumption set X ✓ Rn+ is complete and transitive.

(a) Define the following induced relations on X:

• “is strictly preferred to”, �,

• “is indi↵erent to”, ⇠.

(b) Show that ⇠ is an equivalence relation.

(c) Prove that � is transitive.

2. A consumer has lexicographic preferences on X ✓ R2+

if the relation satisfies x % y whenever

• x1 > y1, or

• x1 = y1 and x2 > y2.

(a) Explain if (R2+,%) satisfies the axioms of a ra-

tional consumer (see the appendix).

(b) Can (R2+,%) be represented by a utility func-

tion? Explain.

3. Suppose that a consumer’s preference relation on a

consumption set satisfies the axioms of a rational con-

sumer listed in the appendix.

(a) Define the consumer’s expenditure function

E(p, u).

(b) Show that E is concave in p.

4. A consumer’s utility function is given by

U(x1, x2) = (x

⇢1 + x

⇢2)

1/⇢, 0 6= ⇢ < 1.

(a) Set up the utility maximization problem and find

the ordinary demand function.

(b) Find the indirection utility function.

(c) Find the expenditure function.

5. Consider the Slutsky equation

@di(p, y)

@pj=

@hi(p, u)

@pj� dj(p, y)

@di(p, y)

@y

.

(a) Define the Slutsky matrix S(p, y).

(b) Explain why S is symmetric and negative semi-

definite.

Appendix: Axioms of Consumer Choice

For all a,b, c in the consumption set X, the relation %satisfies the following axioms:

A1 Completeness: Either a % b or b % a.

A2 Transitivity: If a % b and b % c, then a % c.

A3 Continuity: The upper contour set % (a) and the

lower contour set -(a) are closed.

A4 Strict Monotonicity: If a � b, then a % b. If

a � b, then a � b.

A5 Strict Convexity: If a 6= b and a % b, then ta +

(1� t)b � b for all t 2 (0, 1).

ECON 5113 Microeconomic Theory

Winter 2016

Test 2 March 1, 2016

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobile

devices or calculators are allowed. Please write your an-

swers on the answer book provided. Use the right-side

pages for formal answers and the left-side pages for your

rough work. Answers should be provided in complete

and readable essay form, not just in mathematical for-

mulae and notations. Remember to put your name on

the front page. You can keep the question sheet after

the test.

1. Suppose that a consumer’s indirect utility function is

given by

V (p, y) = p

↵1 p

�2y,

where ↵ and � are negative numbers. Find the con-

sumer’s utility function U(x1, x2).

2. Suppose that two distinct bundles x

0and x

1are cho-

sen by a consumer when prices are p

0and p

1respec-

tively.

(a) Express the weak axiom of revealed preference

(WARP) in terms of the Laspeyres quantity in-

dex QL and the Paasche qauntity index QP .

(b) If p

0= (1, 2), x

0= (3, 1), p

1= (2, 2), and

x

1= (1, 2), determined if the consumer’s choices

satisfy WARP or not.

3. Suppose that a firm’s production function f : Rn+ !

R+ satisfies the following assumptions:

(a) continuous,

(b) strictly increasing,

(c) strictly quasi-concave,

(d) f(0) = 0.

Suppose in addition that f is linearly homogeneous.

Prove that f is a concave function.

4. Suppose that a production function with one output

and n inputs is given by

y = A

nY

i=1

x

↵ii , ↵i > 0,

nX

i=1

↵i = 1.

(a) Find the marginal product of input i.

(b) Define the marginal rate of technical substitu-

tion of input j for input i at an input bundle

x 2 Rn++.

(c) Find �ij , the elasticity of substitution of input j

for input i at x.

5. Suppose that a firm produces a single output with n

inputs with a production function f(x).

(a) Define µi, the output elasticity of input i at a

bundle x.

(b) Define µ, the elasticity of scale at x.

(c) Suppose that f exhibits global constant returns

to scale. Show that µ = 1.

“I hear they’re desperate for new users.”

ECON 5113 Microeconomic Theory

Winter 2016

Test 3 March 18, 2016

Answer ALL Questions Time Allowed: 1 hour 20 minutes

1. A firm in a competitive industry has total cost func-tion

c(q) = 2q � 2q2 + q

3.

(a) What is the firm’s minimum e�ciency scale?

(b) Suppose that all firms in the market have thesame cost structure. What is the long-run equi-librium market price?

(c) The market demand is given by the function p =100 � 0.5q. How many firms are there in themarket?

2. Suppose that market demand for a good is Q = ↵��p

where ↵ > 0 and � > 0. Initially there are two firmsin the market, each with cost function Ci(q) = Fi +ciq, where i = 1, 2. Assume that the two firms engagein price competition.

(a) Now suppose that F1 = 0, F2 > 0, and c1 = c2 =c. Do both firms stay in the market? Explain.

(b) If F1 = F2 = 0, and c1 < c2, what is the marketoutcome? What are the prices and outputs ofthe two firms in equilibrium?

3. A consumer has preferences over the single good x

and all other goods m represented by the utility func-tion, U(x,m) = log x + m. Let the price of x be p,the price of m be unity, and let income be y.

(a) Derive the Marshallian demands for x and m.

(b) Derive the indirect utility function, v(p, y).

(c) Suppose that a government project reduces theprice of the first good from p

0 to p

1 without af-fecting the other prices. Derive the compensat-ing variation (CV) of the price change.

4. In a two-person exchange economy the utility func-tions of Adele and Beyonce for coconuts (c) and dates(d) are give by

U

a(ca, da) = min{ca, da},U

b(cb, db) = c

b + d

b,

respectively. Adele has 10 coconuts and Beyonce has10 dates.

(a) Find the contract curve of this economy.

(b) Find the core of this endowment, C(e).

(c) Draw an Edgeworth box showing the endowmente, the contract curve, and C(e).

5. The diagram below depicts the e↵ect of a governmentproject that reduces the price of a good (q) from p

0

to p

1 without a↵ecting the other prices.

(a) Draw a similar graph that shows the equivalentvariation (EV) of the price change.

(b) Explain why the consumer surplus is a good ap-proximation of the CV and EV.

PARTIAL EQUILIBRIUM 181

v(p1, y0) u(B)

q

m($)

q

y0

p0

p1

y0 CV!

"

"

A BC

v0 ! v(p0, y0) u(A) u(C)""

#p1 #p1#p0

y0/p0 y0/p1

Price

q(p, y0)

qh(p, v0)

q(p0, y0) qh(p0, v0)

Figure 4.5. Prices, welfare, and consumer demand.

Note that in this example, CV is non-positive because v is non-increasing in p, increasingin y, and p1 < p0. CV would be non-negative for a price increase (p1 > p0). In either case,(4.23) remains valid. This change in income, CV , required to keep a consumer’s utilityconstant as a result of a price change, is called the compensating variation, and it wasoriginally suggested by Hicks.

The idea is easily illustrated in the upper portion of Fig. 4.5, where the indifferencecurves are those of u(q,m). The consumer is initially at A, enjoying utility v(p0, y0). Whenprice falls to p1, the consumer’s demand moves to point B and utility rises to v(p1, y0).Facing the new price p1, this consumer’s income must be reduced to y0 + CV (recallCV < 0 here) to return to the original utility level v(p0, y0) at point C.

Equation (4.23) and Fig. 4.5 suggest another way to look at CV . Using the familiaridentity relating indirect utility and expenditure functions, and substituting from (4.23),

ECON 5113 Microeconomic Theory

Winter 2016

Final Exam April 21, 2016

Answer ALL Questions Time: 9:00 am – 12:00 pm

Instruction: This is a closed-book exam. No mobile

phones are allowed. Please write your answers on the

answer book provided. Use the right-side pages for for-

mal answers and the left-side pages for your rough work.

Answers should be provided in complete and readable

essay form, not just in mathematical formulae and no-

tations. Remember to put your name on the front page.

You can keep the question sheets after the exam.

1. Suppose that a consumer’s preference relation % on a

consumption set X ✓ Rn

+ is complete and transitive.

(a) Given a bundle a 2 X, define the sets % (a),

�(a), and ⇠(a).

(b) Show that �(a) = %(a) \ ⇠(a).

2. Suppose that a consumer with a homothetic utility

function U(x) faces market prices p. Assuming that

the consumer is rational, show, by invoking any nec-

essary results in consumer theory we covered in class,

that

p

p

Tx

=

rU(x)

U(x)

, and

x

p

Tx

=

re(p)

e(p)

,

where e(p) is the unit expenditure function.

3. Suppose that the production function of a competi-

tive firm is y = f(x), with input vector x 2 Rn

+ and

market prices w. The market price of the output is

p.

(a) Define the profit function ⇡(p,w).

(b) Show that the substitution matrix

S(p,w) =

0

BBBB@

@y

⇤

@p

@y

⇤

@w1· · · @y

⇤

@wn

�@x

⇤1

@p

� @x

⇤1

@w1· · · � @x

⇤1

@wn

.

.

.

�@x

⇤n

@p

� @x

⇤n

@w1· · · � @x

⇤n

@wn

1

CCCCA

is symmetric and positive semidefinite.

4. Suppose that a monopoly faces a market demand

curve p = a � bq. The total production cost is

C = cq + F . All the parameters are positive, with

a > c and (a� c)

2> 4bF .

(a) Set up the monopoly’s profit maximization prob-

lem.

(b) What are the price-quantity combination

(p

m

, q

m

) and the profit?

(c) What is the price elasticity of demand at

(p

m

, q

m

)? Is it in the elastic or inelastic range?

5. List the characteristics of a contestable market. Can

the market outcome achieve the e�ciency of a com-

petitive market?

6. Consider an exchange economy with two consumers.

Consumer 1 has utility function U

1(x1, x2) = x2 and

consumer 2 has utility function U

2(x1, x2) = x1+x2.

(a) Construct an Edgeworth box showing the indif-

ference maps of the two consumers.

(b) Does the contract curve exist? If yes, explain

what is it?

(c) Suppose that the endowments are e

1= (1, 1)

and e

2= (1, 0). Does a Walrasian equilibrium

exist?

7. Consider a competitive production economy with J

firms and n goods. Each firm’s technology is repre-

sented by a production set Y

j ✓ Rn

with the follow-

ing properties:

(a) 0 2 Y

j

.

(b) Y

j

is a compact set.

(c) Y

j

is strongly convex: For all y

1 6= y

2 2 Y

j

and ↵ 2 (0, 1), there exists a y 2 Y

j

such that

y > ↵y

1+ (1� ↵)y

2.

Let Y =

PJ

j=1 Yj

. Show that Y also satisfies the

above three properties.

8. Consider a competitive production economy with n

goods:

E =

�(U

i

, e

i

, ✓

ij

, Y

j

) : i 2 I, j 2 J .

(a) Define the excess demand function z(p) for a

market price p 2 Rn

++.

(b) Show that z(p) is homogeneous of degree zero.

9. Tzipi has a risk averse von Neumann-Morgenstern

utility function U(w), where w is her annual income.

She works for the government with income w1. She

is considering leaving the government and opening a

small business. If the business is successful, she will

make w2 profit per year, where w2 > w1. There is,

however, a 50% chance that the business will fail with

zero profit.

(a) What is the expected value of profit if Tzipi de-

cides to enter the business?

(b) What is Tzipi’s expected utility of entering the

business? What is the necessary condition that

she will leave the government job to set up her

business?

(c) If Tzipi can hire a consultant to make sure that

her business is successful, what is the maximum

amount per year she is will to pay the consul-

tant?

10. Suppose that the von Neumann-Morgenstern utility

function of a consumer is given by U(w) = w

2.

(a) Find the absolute risk aversion R

a

(w) and rela-

tive risk aversion R

r

(w).

(b) Is the consumer risk averse, risk neutral, or risk

loving? Explain.

(c) Consider the game of Lotto 6/49. Probability

theory dictates that any six-number combina-

tion has equal chance of hitting the jackpot.

But this consumer prefers to choose the num-

bers from 32 to 49. Explain if the consumer is

rational or not.

“I still say it’s only a theory.”

2