ECON 5113 Microeconomic Theory

Winter 2015

Test 1 January 30, 2015

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobilephones or calculators are allowed. Please write your an-swers on the answer book provided. Use the right-sidepages for formal answers and the left-side pages for yourrough work. Answers should be provided in completeand readable essay form, not just in mathematical for-mulae and notations. Remember to put your name onthe front page. You can keep the question sheet afterthe test.

1. Suppose that a consumer’s preference relation % on aconsumption set X ✓ Rn

+ is complete and transitive.

(a) Define the following induced relations on X:

(a) “is strictly preferred to”, �,

(b) “is indi↵erent to”, ⇠.

(b) Explain if � is complete, reflexive, transitive,circular, symmetric, asymmetric, or antisym-metric.

(c) Show that ⇠ is an equivalence relation.

2. Suppose that a consumer’s preference relation satis-fies A1, A2, A3, A4, and A5’ (but not A5, see ap-pendix A).

(a) Explain if the solution to the utility maximiza-tion problem still exists.

(b) If yes, is the choice for the optimal bundleunique?

(c) Illustrate the answers above with a diagram witha two-good case.

3. Aminata buys two goods, xigua and yam every week.Her utility function is given by

U(x, y) = 2x+ log(y + 4).

The market prices for xigua and yam are $2 and $1per kg respectively. Aminata’s weekly budget for foodis $5.

(a) Set up Aminata’s grocery shopping as a Kuhn-Tucker optimization problem.

(b) Find the optimal bundle.

(c) State the condition of the marginal rate of sub-stitution at Aminata’s optimal bundle in rela-tion with the market prices.

4. A consumer’s utility function is given by

U(x1, x2) = Ax

↵1 x

1�↵2 , A > 0, 0 ↵ 1.

(a) Set up the expenditure minimization problemand find the expenditure function E(p, u).

(b) A cost of living index is defined as

I(p0,p

1, u

0) =E(p1

, u

0)

E(p0, u

0),

where p

0 and p

1 are prices in period 0 and 1respectively, and u

0 is the utility level in period0. Show that the index for our consumer is in-dependent of the utility level.

5. Consider the Slutsky equation

@di(p, y)

@pj=

@hi(p, u)

@pj� dj(p, y)

@di(p, y)

@y

.

State and prove the law of demand. You can assumethe properties of the expenditure function listed inappendix B.

Appendix A: 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 thelower contour set -(a) are closed.

A4’ Local Non-Satiation: For any ✏ > 0, there existsan x 2 B✏(a) \X such that x � a.

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

A5’ Convexity: If a % b, then ta + (1 � t)b % b forall t 2 [0, 1].

A5 Strict Convexity: If a 6= b and a % b, then ta +(1� t)b � b for all t 2 (0, 1).

Appendix B: Properties of Expenditure

Function

Suppose that the consumer’s preference relation satis-fies Axioms A1 to A4. Then the expenditure functionE(p, u) has the following properties:

1. E(p, um) = 0, where um is the minimum value inits domain, that is, um = U(0).

2. E is continuous on its domain.

3. For all p � 0, E is strictly increasing and un-bounded above in u.

4. Shephard’s lemma: If E is di↵erentiable in p, then

rpE(p, u) = h(p, u) = x

⇤.

5. E is an increasing function of p.

6. E is linearly homogeneous in p.

7. E is concave in p.

Local Non-Satiation?

2

ECON 5113 Microeconomic Theory

Winter 2015

Test 2 February 27, 2015

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobilephones are allowed. Please write your answers on theanswer 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 readableessay form, not just in mathematical formulae and no-tations. Remember to put your name on the front page.You can keep the question sheet after the test.

1. Suppose that in period t = 0, 1, a consumer buysbundle x

t when the observed price vector is pt.

(a) State the weak axiom of revealed preference.

(b) Determine if the following observation satisfiesWARP:

p

0 = (2, 6), x

0 = (19, 9),

p

1 = (3, 5), x

1 = (17, 4).

2. Suppose that a consumer’s expenditure function is

E(p1, p2, u) =up1p2p1 + p2

.

(a) Find the indirect utility function.

(b) Use the indirect utility function to find the directutility function.

3. An analyst has collected cost data on firms to studythe cost structure of an industry. She uses a log-linearregression model as follows:

log c = �0 +

nX

i=1

�i logwi + � log y,

where c is the observed total cost, y is the out-put quantity, and wi is the market price of inputi. The parameters estimated in the regression are�0,�1, . . . ,�n and �.

(a) What is the functional form of the cost functionthe analyst is using?

(b) If the firms are competitive, what restrictionsshould the analyst impose on the parameters?

(c) What extension do you suggest to make thefunctional form more flexible?

4. A competitive firm produces a single output y with avector x of n inputs with technology represented bythe production function y = f(x).

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

(b) State and prove Hotelling’s lemma.

(c) What additional properties can you infer fromthe Hotelling’s lemma?

5. Consider a competitive firm with a production func-tion y = f(x).

(a) Define global decreasing returns to scale in pro-duction.

(b) Explain why we assume that technology exhibitsdecreasing returns to scale when we study theprofit function of a firm.

“We’re still the same great company we’ve always

been, only we’ve ceased to exist.”

ECON 5113 Microeconomic Theory

Winter 2015

Test 3 March 20, 2015

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobilephones are allowed. Please write your answers on theanswer 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 readableessay form, not just in mathematical formulae and no-tations. Remember to put your name on the front page.You can keep the question sheet after the test.

1. Consider an economy with I consumers.

(a) Define a homothetic preference relation.

(b) How does homotheticity characterize a utilityfunction?

(c) If all consumers in the economy have identicalhomothetic preferences. Discuss if the aggregatedemand function depends on income distribu-tion.

2. Suppose that a monopoly faces a market demandfunction p = a� bq.

(a) Derive the marginal revenue function of the firm.

(b) The cost function of the firm is given by C(q) =cq. Derive the profit maximizing output qm.

(c) What will be the price pm the firm charges?

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

C(qj) = cqj , c > 0, j = 1, 2.

Market demand is an a�ne function

p = a� b(q1 + q2), a, b > 0, a > c.

(a) Set up the profit maximization problem of thefirms and find the necessary condition.

(b) Find the output for each firm.

(c) Find the market equilibrium price.

4. A competitive industry consists of 100 identical firms.The short-run cost function of each firm is given by

C(q) = 200q + 15q2.

(a) What is the supply function for each firm?

(b) What is the market supply function?

(c) Suppose that the market demand function is

p = 400� 0.1q.

Calculate the market equilibrium price andquantity.

(d) Calculate the market consumer surplus and theproducer surplus.

5. The utility functions and endowments of Aicheng andBailin on two goods are given by

U

a(x1, x2) = x1x2, e

a = (2, 2),

U

b(x1, x2) = x

1/21 x

1/22 , e

b = (1, 1).

(a) Characterize the contract curve of this exchangeeconomy.

(b) Find C(e), the core of the economy.

(c) Draw an Edgeworth box of this exchange econ-omy and show the contract curve and the coreas accurate as possible.

“I’ll stop when it isn’t fun anymore.”

ECON 5113 Microeconomic Theory

Winter 2015

Final Exam April 21, 2015

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

Instruction: This is a closed-book exam. No mobilephones are allowed. Please write your answers on theanswer 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 readableessay 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. A consumer’s preference relation is represented by adi↵erentiable and strictly increasing utility functionU(x) on Rn

+. Suppose that the consumer’s income isy > 0 and the market prices of x are p � 0.

(a) Define the indirect utility function V .

(b) State Roy’s identity.

(c) Prove Roy’s identity.

2. A consumer’s preference relation is represented by adi↵erentiable, strictly quasi-concave, and strictly in-creasing utility function U(x) on Rn

+.

(a) Define the consumer’s expenditure function E.

(b) Show that E is concave in the market prices p.

3. Suppose that E is the expenditure function in ques-tion 2.

(a) Define the compensated demand function h.

(b) State Shephard’s lemma (you do not need toprove it).

(c) Prove Hick’s Third Law:

nX

j=1

@hi(p, u)

@pjpj = 0, 1 = 1, . . . , n.

4. Suppose that the ordinary demand function of a con-sumer is d(p1, p2, y) = (↵y/p1, (1 � ↵)y/p2)T, where0 < ↵ < 1.

(a) Does d exhibit the budget balancedness prop-erty?

(b) Find the Slutsky matrix S (see Appendix).

(c) Normalize p2 to 1. Write down the di↵erentialequation you would need to solve to find the ex-penditure function.

5. Let the set of all possible outcome be

A = {w1, w2, . . . , wn},

where wi � 0 are wealth level of a consumer. The con-sumer has a von Neumann-Morgenstein utility func-tion U on Gs, the set of simple gambles on A. Let

g = (p1 � w1, p2 � w2, . . . , pn � wn).

Define the following terms:

(a) Expected value of g.

(b) Expected utility of g.

(c) Certainty equivalent value of g.

(d) Risk premium of g.

6. Faisal’s von Neumann-Morgenstein utility function isgiven by

U(w) = a+ bw + cw2.

(a) What restrictions if any must be placed on pa-rameters a, b, and c if Faisal is risk averse.

(b) Over what domain of wealth can this utilityfunction be defined?

(c) Find Faisal’s Arrow-Pratt measure of absoluterisk aversion, Ra(w).

7. A propose government project, if carried out, willchange the market price of a product from p0 to p1.

(a) Define a consumer’s equivalent variation (EV)with income y0.

(b) Show that

EV =

Z p0

p1

h(p, u1) dp,

where h is the compensated demand function.

(c) Let the Paasche-Konus cost of living index be

PK(p0, p1, u1) =E(p1, u1)

E(p0, u1),

where E is the expenditure function. Show that

PK(p0, p1, u1) =y0

y0 + EV.

8. Consider an exchange economy with a set of con-sumers I = {1, 2, . . . , I} and n goods:

E =�(%i, ei) : i 2 I

.

Let p 2 Rn++ be a price vector.

(a) Define aggregate excess demand z(p) function ofthe economy.

(b) Show that z is homogenous of degree zero.

9. Consider the exchange economy in question 8. Stateand prove Walras’ law.

10. Suppose that a production economy has a set of firmsJ = {1, 2, . . . , J}, each with a net output vector yj .Each firm’s technology is represented by a productionset Y j ✓ Rn with the following properties:

(a) 0 2 Y j .

(b) Y j is a compact set.

(c) Y j is strongly convex: For all y1 6= y

2 2 Y j

and ↵ 2 (0, 1), there exists a y 2 Y j such thaty > ↵y1 + (1� ↵)y2.

Suppose that for all j 2 J , yj is the profit maximizingoutput vector. Let Y j = Y j � {yj}. Show that

(a) 0 2 Y j ,

(b) Y j is compact,

(c) Y j is strongly convex.

Appendix

Slutsky equation:

@di(p, y)

@pj=

@hi(p, u)

@pj� dj(p, y)

@di(p, y)

@y.

“It’s my mobile wallet.”

2