ECON 5113 Microeconomic Theory

Winter 2017

Test 1 February 3, 2017

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobiledevices 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. Let % be a consumer’s preference relation on a con-sumption set X ✓ Rn

+. Suppose that x 2 X.

(a) Define the sets �(x) and �(x).

(b) Show that �(x) \ �(x) = ?.

2. Suppose that a consumer’s preference relation % ona consumption set X ✓ Rn

+ is complete, transitiveand strictly monotonic. Let e = (1, . . . , 1) and defineU : X ! R+ such that U(x)e ⇠ x.

(a) Show that U is a well-defined function.

(b) Show that the function U represents the con-sumer’s preference relation % on X.

3. Suppose that a consumer’s preference relation on aconsumption set satisfies the axioms of a rational con-sumer listed in the appendix.

(a) Define the consumer’s expenditure functionE(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

1�↵2 .

(a) Set up the utility maximization problem and findthe ordinary demand function.

(b) Find the indirect utility function.

(c) Derive the expenditure function from the indi-rect utility function.

5. Suppose that a consumer’s utility function is di↵er-entiable, increasing, and quasi-concave.

(a) Define, ✏hij , the elasticity of Hicksian demand forgood i with respect to the price of good j.

(b) Show that

nX

j=1

✏

hij = 0, i = 1, . . . n.

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 thelower contour set -(a) are closed.

A4 Strict Monotonicity: If a � b, then a % b. Ifa � 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 2017

Test 2 March 10, 2017

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 when the market prices of three goods

in period 1 are p

1= (2, 3, 3). Khawla buys quanti-

ties x

1= (3, 1, 7). In period 2, prices and quantities

are p

2= (3, 2, 3) and x

2= (7, 3, 1). Does Khawla’s

behaviour satisfy the weak axiom of revealed prefer-

ence? Explain.

2. Suppose that a function E : Rn++⇥R+ ! R+ satisfies

the seven properties of an expenditure function (see

the appendix).

(a) Define the optimization problem that can re-

cover the utility function directly for any given

consumption bundle x 2 Rn+.

(b) Show that the resulting utility function U(x) is

unbounded above.

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

given by

V (p, y) =

y

p

1/21 p

1/22

.

Derive the consumer’s utility function U(x).

4. Let y = f(x) be the production function of a com-

petitive firm that produces one output with n inputs

with a constant returns to scale technology.

(a) Define the average product APi and marginal

product MPi of input i.

(b) Show that

f(x) =

nX

i=1

(MPi)xi.

(c) Suppose that n = 2. Show that if the average

product of input 2 is rising, then the marginal

product of input 1 is negative.

5. Suppose that a competitive firm produces one output

with n inputs with a production function y = f(x).

Let p be the market price of the output and w be the

vector of input prices.

(a) Define the profit function of the firm.

(b) Show that the profit function is increasing in p

and decreasing in w.

Appendix: Properties of an Expenditure

Function

Suppose that a consumer’s utility function is continuous,

increasing, strictly quasi-concave. Then the expenditure

function has the following properties:

1. E(p, um) = 0, where um is the minimum value in

its 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.

ECON 5113 Microeconomic Theory

Winter 2017

Test 3 March 31, 2017

Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: This is a closed-book exam. No mobiledevices 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. Consider a market with two firms. Each firm hasidentical cost function

C(qj) = 1 + q

2j , j = 1, 2.

The inverse market demand function is

p = 10� q1 � q2,

(i) What are the Cournot equilibrium outputs ofthe two firms?

(ii) What is the total profit of the duopoly?

(iii) If the government allows the duopoly to merge,what will be the profit of the monopoly?

2. A market with a Stackelberg structure has one leaderand five followers. The market demand function isgiven by

p = 100� q.

The cost function of each of the followers is

C(qj) = 10 + 2.5q2j , j = 1, . . . , 5.

(i) Find the total supply function of the followers.

(ii) Find the demand function facing the leader.

(iii) If the cost function of the leader is given byC(q) = 10 + 20q, find the market equilibriumprice.

3. A government project changes the market price of agood from p

0 to p

1.

(i) Define the equivalent variation (EV) of the pricechange for a consumer by her indirect utilityfunction.

(ii) Express EV in terms of the Hicksian demandfunction.

(iii) Suggest a method to measure EV in practice.

4. Consider a two-person, two-good exchange economywith utility functions and endowments as follows:

U

1(x1, x2) = x

1/21 x

1/22 , e

1 = (2, 0),

U

2(x1, x2) = x1x2, e

2 = (0, 2).

(i) Given market prices p = (p1, p2), derive the ex-cess demand function for each consumer. (Youcan write down the ordinary demand functionsdirectly if you recognize the utility functionalforms.)

(ii) Find the aggregate excess demand function,z(p).

(iii) Is p⇤ = (1, 1) a Walrasian equilibrium? Explain.

5. Consider a set of consumers I = {1, 2, . . . , I} thatforms an exchange economy

E =�(%i

, e

i) : i 2 I ,

in which each consumer has a rational preference re-lation %i on n goods.

(i) Carefully state the second welfare theorem.

(ii) Explain the economic meaning of the theorem.

ECON 5113 Microeconomic Theory

Winter 2017

Final Examination April 21, 2017

Answer ALL Questions Time: 1:00 pm – 4:00 pm

Instruction: This is a closed-book exam. No mobiledevices 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 pageof every answer book. You can keep the question sheetsafter the exam.

1. Let % be the preference relation of a rational con-sumer on a consumption set X ✓ Rn

+.

(i) Define the following induced relations on X:

(a) “is strictly preferred to”, �,

(b) “is indi↵erent to”, ⇠.

(ii) Show that � is a transitive relation.

2. Suppose that a consumer’s utility function U(x) ona consumption set X ✓ Rn

+ is continuous, increasing,quasi-concave, and linearly homogeneous.

(i) Show that the ordinary demand function is sep-arable in market price p and income y, that is,

x

⇤ = d(p)y.

(ii) Suppose that consumers in an economy have lin-early homogeneous but not identical utility func-tions. Show that market demands depend onincome distribution.

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.

(i) Explain why @hi(p, u)/@pi 0.

(ii) Define a normal good.

(iii) State the law of demand.

4. Consider the von Neumann-Morgenstern utility func-tion

U(w) = ↵+ � logw.

(i) What restrictions if any must be placed on theparameters ↵ and � to display risk aversion? Ex-plain.

(ii) Find the absolute risk aversion.

(iii) Find the relative risk aversion.

5. Nikki’s only asset is her $500,000 house in FortWilliam. Each year there is a 0.003 probability thata fire will destroy the house completely.

(i) Find the expected value of Nikki’s asset.

(ii) Suppose Nikki’s utility function on wealth isU(w) =

pw. Find her expected utility.

(iii) What is the certainty equivalence of Nikki’s ex-pected utility.

(iv) An insurance company o↵ers to fully insureNikki’s house with an annual premium of $500.Will she accept the o↵er?

6. 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.

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

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

7. The diagram below shows the sum of the marginalcosts of all the competitive firms and the demandcurve in a market.

(i) Define market consumer surplus in integral form.

(ii) Define market producer surplus in integral form.

(iii) Show that a competitive market achieves maxi-mum e�ciency.

PARTIAL EQUILIBRIUM 187

qq

p

mc(q)

p(q)

Price

CS

PS

Figure 4.7. Consumer plus producer surplus ismaximised at the competitive marketequilibrium.

Choosing q to maximise this expression leads to the first-order condition

p(q) = mc(q),

which occurs precisely at the perfectly competitive equilibrium quantity when demand isdownward-sloping and marginal costs rise, as we have depicted in Fig. 4.7.

In fact, it is this relation between price and marginal cost that is responsible for theconnection between our analysis in the previous section and the present one. Wheneverprice and marginal cost differ, a Pareto improvement like the one employed in the previoussection can be implemented. And, as we have just seen, whenever price and marginal costdiffer, the total surplus can be increased.

Once again, a warning: although Pareto efficiency requires that the total surplus bemaximised, a Pareto improvement need not result simply because the total surplus hasincreased. Unless those who gain compensate those who lose as a result of the change, thechange is not Pareto improving.

We have seen that when markets are imperfectly competitive, the market equilibriumgenerally involves prices that exceed marginal cost. However, ‘price equals marginal cost’is a necessary condition for a maximum of consumer and producer surplus. It should there-fore come as no surprise that the equilibrium outcomes in most imperfectly competitivemarkets are not Pareto efficient.

EXAMPLE 4.4 Let us consider the performance of the Cournot oligopoly in Section4.2.1. There, market demand is p = a− bq for total market output q. Firms are identical,with marginal cost c ≥ 0. When each firm produces the same output q/J, total surplus,W ≡ cs+ ps, as a function of total output, will be

W(q) =! q

0(a− bξ)dξ − J

! q/J

0cdξ,

8. Consider a set of consumers I = {1, 2, . . . , I} and aset of firms J = {1, 2, . . . , J} that forms a productioneconomy with n goods:

E =�(U i, ei, ✓ij , Y j) : i 2 I, j 2 J

. (1)

Each consumer has a di↵erentiable, increasing, andquasi-concave utility function U i on the n goods.Each firm’s compact and strongly convex productionset Y j ⇢ Rn contains the zero net output vector.

(i) Carefully state the second welfare theorem of theabove production economy.

(ii) Explain the economic meaning of the theorem.

9. Consider the production economy in equation (1).Assume that each of the n goods is produced by onecompetitive firm only.

(i) State the utility maximization problem of a typ-ical consumer i.

(ii) State the profit maximization problem of a typ-ical firm j.

(iii) Pick any two goods k and l, with market equi-librium prices pk and pl respectively. Show thatthe MRSkl of all consumers and the MRTSkl ofall firms are the same.

(iv) What is the economic meaning of the above re-sult?

10. The production economy described in question 8above provide an analytical framework for the aggre-gate economy. Many economists employ this Wal-rasian approach over time to study the business cy-cles of the macroeconomy. Many so-called “dynamicstochastic general equilibrium models” reduce I =J = 1 for simplicity. Write a short critique of theDSGE approach in macroeconomics.

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