ECON 5113 Microeconomic Theory

Winter 2018

Test 1 February 2, 2018Answer 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. Let % be a preference relation a consumption set X ✓Rn

+.

(i) Define the relations � and ⇠ induced by %.

(ii) Prove or disprove: � and ⇠ are complete.

(iii) Given any x 2 X, show that -(x) \ %(x) = ⇠(x).

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

a consumption set X ✓ Rn+ is continuous. Let e =

(1, . . . , 1) and for every x 2 X, define B = {t 2 R+ :

te - x}. Prove that B is a closed set in R+.

3. Let U : Rn+ ! R+ be a continuous, increasing, and

strictly quasiconcave utility function.

(i) Define the Hicksian (compensated) demand

function.

(ii) Define the expenditure function association with

U .

(iii) Prove that the expenditure function is homoge-

neous of degree one in prices.

(iv) State and prove Shephard’s lemma.

4. Consider the Slutsky equation

@di(p, y)

@pj=

@hi(p, u)

@pj� dj(p, y)

@di(p, y)

@y,

where di and hi are the ordinary and compensated

demand functions respectively for good i.

(i) State the contrapositive of the law of demand.

(ii) Prove the statement you answered in part (i).

5. A consumer’s utility function is given by

U(x1, x2) = Ax↵1 x

1�↵2 , A > 0, 0 ↵ 1.

(i) Derive the ordinary demand function.

(ii) Find the indirect utility function.

(iii) Derive the expenditure function from the indi-

rect utility function.

6. (Bonus question, one mark) Which of the axioms on

preference relation of the following consumer is not

satisfied? Explain.

“I thought getting bigger rocks would make us happier, but Iguess I was wrong.”

ECON 5113 Microeconomic Theory

Winter 2018

Test 2 March 2, 2018Answer ALL Questions Time Allowed: 1 hour 20 minutes

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

digital devices are allowed except a non-programmable

calculator. Please write your answers on the answer

book provided. Use the right-side pages for formal an-

swers and the left-side pages for your rough work. An-

swers should be provided in complete and readable essay

form, not just in mathematical formulae and notations.

Remember to put your name on the front page. You can

keep the question sheet after the test.

1. Suppose that we are given the indirect utility function

V (p, y) of a consumer.

(a) State the minimization problem that recover the

utility function U(x) from V .

(b) Suppose that V (p, y) = pa1pb2y, where a, b < 0.

Derive the consumer’s utility function.

2. Suppose that at market prices p0and p1

, a consumer

buys bundles x0and x1

respectively.

(a) Determine if the consumer satisfies the weak ax-

iom of revealed preference with the following ob-

servations:

p0= (3, 5), x0

= (18, 4),

p1= (2, 6), x1

= (20, 10).

(b) Do the behaviours of this consumer conform

with a well-defined utility function? Explain.

3. Suppose that a perfectly competitive firm produces

a single product y with market price p. Technology

can be represented by a di↵erentiable, increasing, and

strictly quasi-concave production function y = f(x),where x 2 Rn

+ is an input bundle with market prices

w 2 Rn++.

(a) State the objective of the firm and set up the

mathematical problem to achieve that objective.

(b) Define the profit function of the firm.

(c) State and prove Hotelling’s lemma.

4. Suppose that the production function of a firm is

given by

f(x) = x1 + x2.

(a) Calculate the marginal rate of technical substi-

tution of input 2 for input 1 at the input bundle

x = (1, 1).

(b) Sketch the isoquant map of the firm.

(c) Define the elasticity of substitution �12 at any

input bundle x of this firm.

(d) Suggest a value for �12 at the input bundle x =

(1, 1).

5. Consider again the firm described in question 3.

(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 for all x in the feasible

set.

ECON 5113 Microeconomic Theory

Winter 2018

Test 3 March 23, 2018Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobiledigital devices are allowed except a non-programmablecalculator. Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. An-swers should be provided in complete and readable essayform, not just in mathematical formulae and notations.Remember to put your name on the front page. You cankeep the question sheet after the test.

1. There are J identical firms in the market of a singlegood. Each firm has a cost function

C(qj) = cqj , c > 0, j = 1, . . . , J.

Market demand is given by the function

p = a� bq,

where b > 0, a > c, and q =PJ

j=1 qj is the totaloutput quantity. Let W (q) be the sum of consumersurplus and producer surplus for any q.

(a) Show that, when each firm produces the sameoutput q/J , the total surplus is

W (q) = aq � (b/2)q2 � cq.

(b) Find the output level that maximize total sur-plus.

(c) Explain why when the firms are price takers andthere is no externality, social welfare is maxi-mized.

2. Suppose that Emma’s income is y and she buys asingle good q with market price p.

(a) What is Emma’s demand function?

(b) In period 0, y0 = 10 and p0 = 2. In period 1a government regulation increases the marketprice of the good to p1 = 4. Calculate Emma’sequivalent variation.

(c) Find the change in consumer surplus due to theprice change. Is it a good approximation toEmma’s equivalent variation?

3. The monopoly of a single good faces a cost function

C(q) = 504q � 36q2 + q3,

where q is the output quantity per period. There isno sunk cost in the production process. The marketdemand function for the good is

p = 270� 0.01q.

(a) Are there any reasons that the firm may behaveas a price taker?

(b) Suppose that the firm does behave as a pricetaker. What is the supply function?

(c) The firm has successfully lobbied the govern-ment to grant it an exclusive franchise to sell theproduct without any price restriction. What isthe supply function of the product?

(d) Find the price and quantity of the product underthe franchise.

4. Consider a software company which has considerablemarket power over one of its products. The companysells its products online so the marginal cost is prac-tically zero.

(a) Show that the profit maximizing price-quantitycombination is at the point on the demand curvethat the price elasticity of demand is equal to 1.

(b) Suppose that the inverse demand function isgiven by p = a � bq. Show that unitary elas-ticity is at the mid-point of the curve.

5. Suppose that an industry is supplied by a duopoly,each has a cost function

C(qj) = 0.1qj , j = 1, 2.

Market demand is an a�ne function

p = 100� 0.5(q1 + q2).

(a) Suppose that the firms compete on quantity.Find the output for each firm.

(b) What is the market equilibrium price?

(c) If the firms change their strategies to competeon price. Do you expect the prices to go up ordown?

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ECON 5113 Microeconomic Theory

Winter 2018

Final Examination April 23, 2018Answer ALL Questions Time: 1:00 pm – 4:00 pm

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

digital devices are allowed except a non-programmable

calculator. Please write your answers on the answer

book provided. Use the right-side pages for formal an-

swers and the left-side pages for your rough work. An-

swers should be provided in complete and readable essay

form, not just in mathematical formulae 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 ngoods and services can be represented by a di↵eren-

tiable, strictly increasing, and strictly quasi-concave

utility function.

(a) Set up the consumer’s problem as an equality

constrained optimization problem. Define all the

variables clearly.

(b) What is the Lagrange multiplier theorem for the

above optimization problem.

(c) Give an economic interpretation for the La-

grange multiplier � relating to the welfare of the

consumer.

2. A consumer’s utility function is given by

U(x1, x2) = min

nx1

2,x2

3

o.

(a) Derive the ordinary demand function d(p, y).

(b) Find the indirect utility function V (p, y).

(c) Using V above, derive the expenditure function

E(p, u).

(d) Using d and E, derive the Hicksian demand func-

tion.

3. Consider the Slutsky equation

@di(p, y)

@pj=

@hi(p, u)

@pj� dj(p, y)

@di(p, y)

@y,

where di and hi are the ordinary and Hicksian de-

mand functions respectively for good i.

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

(b) Explain why S(p, y) is symmetric and negative

semi-definite.

4. Leonard has a utility function on wealth given by

UL(w) =pw, while Penny’s is UP (w) = logw.

(a) Suggest two measures of risk attitude on wealth.

(b) Calculate Leonard and Penny’s risk attitudes

with your suggestions.

(c) Without knowing their wealth levels, can you tell

whether Leonard or Penny is more risk averse?

5. Consider the problem of insuring a house against

flooding. The value of the house is $D, the insurance

cost is $I per year, and the probability of flooding is

p.

(a) List the four outcomes in the set A associated

with this risky situation.

(b) Characterize the choice between insurance and

no insurance as a choice between two gambles,

g1 and g2, each involving all four outcomes in A,

where the gambles di↵er only in the probabilities

assigned to each outcome.

6. Kiwi Motor Company produces a truck called Xango

and a sedan called Yangmei. The monthly cost of

making x units of Xango and y units of Yangmei are,

in thousands of dollars, given by

C(x, y) = 0.04x2+ 0.01xy + 0.01y2 + 4x+ 2y + 300.

Market price for the truck in its class is $15 thousand

and the sedan $9 thousand. The production manager

is responsible for the monthly production to maximize

the company’s profit.

(a) Suppose that you are an operation analyst in the

production division, what production levels will

you recommend to the manager?

(b) How much profit the company is making per

month under your plan?

(c) How will you demonstrate to the production

manager that your plan is indeed profit maxi-

mizing.

7. Consider an exchange economy

E =�(%i, ei) : i 2 I

.

Define the following items with detailed descriptions

of all the variables you introduce.

(a) The set of feasible allocations F (e).

(b) A Pareto e�cient allocation x.

(c) The core of the economy C(e).

8. Consider a two-person, two-good exchange economy

with utility functions and endowments as follows:

U1(x1, x2) = x1/2

1 x1/22 , e1 = (3, 1),

U2(x1, x2) = min{x1, x2}, e2 = (1, 3).

(a) Find the contract curve.

(b) Find the Walrasian equilibrium.

(c) Find the Walrasian equilibrium allocations.

9. Suppose that a competitive production economy is

given by

E =�(U i, ei, ✓ij , Y j

) : i 2 I, j 2 J .

(a) What is the optimization problem faced by a

typical consumer?

(b) What is the optimization problem faced by a

typical firm?

(c) Show that the net supply function of a typical

firm, sj(p), is homogeneous of degree zero.

(d) Show that the profit function of a typical firm,

⇡j(p), is linearly homogeneous.

10. Consider again the production economy in question 9.

Each firm’s production set Y j ✓ Rnsatisfies the fol-

lowing properties:

(a) 0 2 Y j.

(b) Y jis a compact set.

(c) Y jis strongly convex: For all y1 6= y2 2 Y j

and ↵ 2 (0, 1), there exists a y 2 Y jsuch that

y > ↵y1+ (1� ↵)y2

.

Suppose that for each j, yjis a desirable Pareto-

e�cient allocation for firm j. Define Y j= Y j�{yj}.

Show that Y jalso satisfies the above three properties.

“Who are you kidding? You’re all about small governmentuntil you get stuck in a tree.”

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