ECON 5113 Advanced Microeconomics

Winter 2011

Test 1 February 4, 2011Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: 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. Re-

member to put your name on the front page.

1. Let � be a preference relation on Rn+.

(a) Define the relations � and ∼ induced by �.

(b) Show that � ∩ ∼ = ∅.

(c) Show that for every x ∈ Rn+,

∼(x) ∪ �(x) ∪ ≺(x) = Rn+.

2. Let U : Rn+ → R+ be an increasing, twice differen-

tiable, and quasiconcave utility function.

(a) Define the indirect utility function. Make sure

you explain every variable you use.

(b) State and prove Roy’s identity.

3. Suppose a consumer’s preference structure is repre-

sented by the utility function

U(x) = xα1 x

1−α2 , 0 < α < 1. (1)

(a) Set up the expenditure minimization problem.

(b) Derive the Hicksian demand function h(x, u).

(c) Find the expenditure function.

(d) Find the indirect utility function.

4. Let U : Rn+ → R+ be an increasing, twice differen-

tiable, and quasiconcave utility function.

(a) Define price elasticity �ij .

(b) Define income elasticity ηi.

(c) Prove that for i = 1, . . . , n,

n�

j=1

�ij + ηi = 0.

Hint: The ordinary demand function for good

i is homogeneous of degree zero in (p, y).

5. Suppose a consumer’s utility function is given by

Equation (1) with α = 1/2. Define the Laspeyres-

Konus cost of living index as

PK(p0,p1, u0) =

E(p1, u0)

E(p0, u0),

where E is the expenditure function with the super-

scripts 0 and 1 indicating the time periods.

(a) The consumer’s income changes from y0 to y1.Show that the consumer will be worse off in

period 1 if y1/y0 is less than PK(p0,p1, u0).

(b) In period 0 the prices are p0= (1, 1) and

income is y0 = 12. Suppose in period 1

p1= (2, 1). Calculate the price index PK .

Happy New Year!

ECON 5113 Advanced Microeconomics

Winter 2011

Test 2 March 4, 2011Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: 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. Re-member to put your name on the front page.

1. Suppose that a consumer’s indirect utility functionis given by

V (p, y) =y

pα1 pβ2

, α,β > 0.

Find the direct utility function.

2. Suppose that the function E : Rn++×R+ → R+ sat-

isfies the seven properties of an expenditure func-tion. Define

U(x) = maxu

�u : pTx ≥ E(p, u) ∀p ∈ Rn

++

�.

Show that U is increasing in x.

3. A consumer chooses consumption bundle x0 whenthe market prices are p0 and x1 when the prices arep1.

(a) State the weak axiom of revealed preference(WARP).

(b) Suppose that

p0 = (2, 2),

x0 = (3, 2),

p1 = (2, 1),

x1 = (3, 2).

Decide whether the observed choices satisfyWARP.

4. Suppose that f : Rn+ → R+ is a strictly increasing,

strictly quasiconcave, and differentiable productionfunction.

(a) Define the elasticity of substitution σij be-tween inputs i and j.

(b) If f is homothetic, show that σij can be ob-tained from the linear homogeneous part of falone.

5. Suppose the production function of a firm is givenby

y = xα1 x

β2 .

(a) Find the profit function given output price pand input prices w.

(b) What restrictions on α and β are required toensure that the profit function is well-defined?Explain.

“Next time we’re shopping at Wal-Mart.”

ECON 5113 Advanced Microeconomics

Winter 2011

Test 3 March 25, 2011Answer ALL Questions Time Allowed: 1 hour 20 min

Attention: 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. Re-member to put your name on the front page.

1. Suppose that in an economy every consumer’s pref-erences can be represented by a linearly homoge-neous utility function. Prove or disprove: the mar-ket demand for a good is independent of incomedistribution.

2. A monopoly producing a single good faces an inversedemand function

p = a− bq, a, b > 0.

The cost function is given by

c(q) = d+ eq2, d, e > 0.

(a) Derive the marginal revenue.

(b) Derive the marginal cost.

(c) Find the quantity produced by the firm andthe market price.

(d) Find the price elasticity of demand � at themarket price.

3. An industry consists of J identical firms each withcost function c(q) = 10q. Each firm faces an iden-tical inverse market demand p = 110 − 2Q, whereQ =

�Jj=1 qj is the total output produced all the

firms.

(a) Find the Cournot equilibrium output of a typ-ical firm.

(b) Find the profit of each firm.

4. A consumer’s demand for a single good is given byx = d(p, y) = y/p. Suppose income is y = $20 andthe price p increases from $1 to $3. Find

(a) the compensating variation,

(b) the equivalent variation, and

(c) the change in consumer surplus

of the price change.

5. In a two-person exchange economy the utility func-tions of Adam and Betty for fish and rice are giveby

Ua(xaf , x

ar) = min{xa

f , xar},

U b(xbf , x

br) = min{xb

f , xbr},

respectively. Adam has 100 fish and Betty has 100unit of rice.

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

(b) Find the set of Walrasian equilibrium alloca-tions, W (e).

(c) Draw an Edgeworth box showing the endow-ment e, C(e), and W (e).

“Back off. I don’t believe in the Second Welfare Theorem.”

ECON 5113 Advanced Microeconomics

Winter 2011

Final Examination April 18, 2011Answer ALL Questions Time Allowed: 3 hours

Attention: 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. Re-

member to put your name on the front page.

1. A consumer’s indirect utility function is given by

V (p, y) = f(p)y.

(a) What is the form of this consumer’s expendi-

ture function?

(b) What kind of preferences does this consumer

have?

2. Suppose that a consumer’s preferences can be repre-

sented by the utility function U(x1, x2) = f(x1)+x2

where f is a strictly increasing and strictly concave

function.

(a) Show that the marginal rate of substitution is

independent of x2.

(b) Prove that the income elasticity of demand for

good 1, η1 = 0.

(c) U defined above is called a quasi-linear utility

function. What are the economic meanings of

the above results?

3. Suppose that the Slutsky matrix of a consumer with

a regular utility function of three goods is

S =

? ? ?

? −96 ?

3 ? −6

.

The market prices are p = (3, 1, 5)T.

(a) Supply the missing numbers.

(b) Does the resulting matrix possess all the prop-

erties of a Stutsky matrix?

4. Suppose that a consumer has a von Neumann-

Morgenstern utility function u on a set of gambles

G based on wealth level w ∈ R+.

(a) Define the Arrow-Pratt measure of absolute

risk aversion, Ra(w).

(b) Show that Ra(w) is invariant to an affine trans-

formation

v(w) = α+ βu(w), β > 0.

(c) Let

u(w) = a+ b logw, b > 0.

Show that u displays decreasing absolute risk

aversion.

5. Let U : Rn+ → R+ be an increasing, twice differen-

tiable, and quasiconcave utility function.

(a) Define price elasticity of demand for good iwith respect to price of good j, �ij .

(b) Let si be the expenditure share of good i.Prove the Cournot aggregation,

n�

i=1

si�ij = −sj , j = 1, . . . , n.

6. Suppose that the production function f : Rn+ → R+

is differentiable, strictly increasing, strictly quasi-

concave, and f(0) = 0. Let (p,w) be the output

and input prices.

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

(b) Show that π is linearly homogeneous in (p,w).

(c) State and prove Hotelling’s lemma.

7. Let Y ⊆ Rnbe the production set of a competitive

firm which faces market price vector p.

(a) Define a strongly convex Y .

(b) Suppose that p ∈ Rn++. Show that there is at

most one profit-maximizing production plan.

8. Given a production economy

E = {(�i, ei, θij , Y j)|i ∈ I, j ∈ J }

where ei ∈ Rn+. Assume that each firm produces

only one good. Consumers’ utility functions and

firms’ production functions are differentiable and

the Walrasian equilibrium p ∈ Rn++.

(a) Show that the MRS between two consumption

goods is the same for each consumer.

(b) Show that the MRTS between to inputs is the

same for every firm.

(c) Comment on the information content of Wal-

rasian equilibrium prices.

9. Consider a two-good, two-consumer exchange econ-

omy, with utility functions and endowments

U1(x1, x2) = x1/2

1 x1/22 , e1 = (1, 2),

U2(x1, x2) = x1/2

1 x1/22 , e2 = (2, 1).

(a) Find the set of Pareto-efficient allocations.

(b) Find the core of the economy.

(c) Find a Walrasian equilibrium for this economy

and the associated WEA.

10. Given a production economy

E = {(U i, ei, θij , Y j)|i ∈ I, j ∈ J }

where ei ∈ Rn+.

(a) State the Second Welfare Theorem for a pro-

duction economy.

(b) Let Y ⊆ Rnbe the production set of a compet-

itive firm with 0 ∈ Y . Let y ∈ Y and define a

production set Y = Y −{y}. Show that 0 ∈ Y .

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