ECON 5113 Advanced Microeconomics

Winter 2008

Test 1 February 1, 2008Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: 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. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are askedonly. Do not spend time on what you are not asked todo. Remember 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 for any x ∈ Rn+, ≺ (x) and ! (x)

form a partition of Rn+.

2. Suppose that U : Rn+ → R+ is a continuous, in-

creasing, and quasi-concave utility function whichrepresent a preference relation !. Show that ! is

(a) transitive,

(b) continuous,

(c) convex.

3. State and prove Roy’s identity.

4. Let U be an utility function on a consumption setRn

+.

(a) Define the marginal rate of substitution ofgood i for good j.

(b) For the two-good case (n = 2), if x!1 = 0 andx!2 > 0, what is the relation between MRS 12

and the price ratio of good 1 and good 2?

(c) Illustrate your answer in Part 4b with a dia-gram.

5. A consumer’s utility function is given by

u(x1, x2) = x1.

(a) Find the Marshallian demand function.

(b) Find the indirect utility function.

(c) Find the expenditure function.(d) Find the Hicksian demand function.

6. Suppose that U(x) is a linearly homogeneous func-tion.

(a) Show that the indirect utility function can beexpressed as

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

(b) Use the above result to show that the expen-diture function can be expressed as

E(p, u) = e(p)u.

Local non-satiation

ECON 5113 Advanced Microeconomics

Winter 2008

Test 2 February 29, 2008Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: 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. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are asked only.Do not spend time on what you are not asked to do. Re-member to put your name on the front page.

1. Prove 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.Hint: hi(p, u) = di(p, E(p, u)).

2. (a) Define price elasticity of demand for good iwith respect to good j, "ij , and income elastic-ity of good i, #i.

(b) Prove thatn!

j=1

"ij + #i = 0, i = 1, . . . , n.

3. Let U(x) be a continuous, increasing, and quasi-concave utility function.

(a) With a consumer’s income normalized to y =1, show that

p ="U(x)"U(x)Tx

.

(b) LetU(x) = Ax!

1 x1!!2

for 0 < $ < 1, find the consumer’s inversedemand functions, p1(x1, x2) and p2(x1, x2).

4. The Slutsky matrix below belongs to a consumerwith a regular utility function of three goods, withmarket price p = (1, 2, 6)T:

"

#!10 ? ?

? !4 ?3 ? ?

$

% .

Supply the missing numbers. Does the resulting ma-trix possess all the properties of a Stutsky matrix?

5. Let the observed consumption bundles of a con-sumer in period 0 and period 1 be x0 and x1, giventhe price vectors p0 and p1 respectively.

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

(b) Naomi claims that she prefers Pepsi to Coke.Last week she bought five bottles of Pepsi whenboth drinks were $1 per bottle. This weekCoke is on sale for 90 cents per bottle whilethe price of Pepsi remains the same. Naomibuys six bottles of Coke instead. Is Naomi’sbehaviour consistent with WARP? Explain.

ECON 5113 Advanced Microeconomics

Winter 2008

Test 3 March 28, 2008Answer ALL Questions Time Allowed: 1 hour 20 minutes

Instruction: 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. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are asked only.Do not spend time on what you are not asked to do. Re-member to put your name on the front page.

1. Let f(x1, x2) be a constant returns-to-scale produc-tion function.

(a) Define average products and marginal prod-ucts.

(b) Suppose that the average product of input fac-tor 1 is rising. Show that the marginal productof input factor 2 is negative.

2. Let !(p,w) be the profit function of a competitivefirm that produces a single product with productionfunction f : Rn

+ ! R+ that is continuous, strictlyincreasing, and strictly quasiconcave on Rn

+, andf(0) = 0.

(a) Find the supply function.

(b) Prove that "#w!(p,w) = d(p,w), where d isthe input demand function.

(c) Prove that ! is increasing in p.

3. A new highway project changes the rental rate ofapartment in a suburb from p0 to p1. A consumerwith a well-defined preference structure has incomey0.

(a) Define the equivalent variation (EV) of theconsumer due to the price change using theindirect utility function.

(b) Express EV explicitly using the expenditurefunction.

(c) Express EV as a function of the Hicksian de-mand function.

(d) The cost of living index between period 0 andperiod 1 can be defined as

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

where u1 is the utility level in period 1. Findan expression for I(p0, p1, u1) in terms of EVand y0.

4. A firm j in a competitive industry has constantreturns-to-scale technology. Let w be the vectorof factor prices and p be the output price.

(a) What form does the conditional cost functionhave?

(b) Write down the objective function of the firmusing the conditional cost function above.

(c) Can you find the supply function? Why or whynot?

(d) Suppose all the firms in the industry havethe same technology and face the same factorprices. Explain why in the long-run the num-ber of firms in the industry is indeterminate.

5. An industry consists of J identical firms each withcost function c(q) = q2 + 1. Each firm faces anidentical inverse market demand

p = 10" 15q " (J " 1)q

where q is the output produced by each of the otherfirms.

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

(b) Find the profit of each firm.(c) What would happen if entry to the industry is

possible?

ECON 5113 Advanced Microeconomics

Winter 2008

Final Examination April 14, 2008CB4056 9:00 AM – 12:00 PMInstructor: Kam Yu Answer ALL Questions

Instruction: Please write your answers on the answerbooks provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are asked only.Do not spend time on what you are not asked to do. Re-member to put your name on the front page of everyanswer book.

1. Suppose that U : Rn+ ! R+ is a continuous, in-

creasing, and quasi-concave utility function whichrepresents a preference relation !. Show that ! iscontinuous.

2. A consumer has an utility function given by

U(x1, x2) = min!

x1

a1,x2

a2

",

where a1, a2 > 0.

(a) Find the demand functions for both goods.(b) Find the indirect utility function.(c) Find the expenditure function.

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

U(x1, x2) = f(x1) + x2

where f is a di!erentiable, strictly increasing andstrictly concave function.

(a) Show that the marginal rate of substitution isindependent of x2.

(b) Prove that the income elasticity of demand forgood 1, !1 = 0.

4. Suppose that a consumer has wealth W . There isa probability p that she will lose the amount L ina year. An insurance company o!ers the consumera contract that will fully compensate her loss for acost of I per year.

(a) List the four outcomes in the set A of this riskysituation.

(b) If the consumer is risk-averse and I = pL, willthe consumer accept the insurance o!er? Ex-plain.

5. Suppose a firm produces a single product with pro-duction function f : Rn

+ ! R+ that is continuous,strictly increasing, and strictly quasiconcave on Rn

+,and f(0) = 0.

(a) Define the conditional input demand functionh(w, y).

(b) Show that h is homogeneous of degree zero inw.

6. The market demand for a product is Q = " " #p.The product is provided by two firms with identicalmarginal cost C !(q) = c > 0 and no fixed cost.

(a) What is the profit function of each firm if theyengage in Bertrand competition.

(b) Find the market price of the product.

(c) Find the Bertrand equilibrium if firm 1 hashigher marginal cost than firm 2, that is, c1 >c2 > 0.

7. Consider a two-good, two-consumer exchange econ-omy, with utility functions and endowments

U1(x1, x2) = x1/21 x1/2

2 , e1 = (1, 0),

U2(x1, x2) = x1/21 x1/2

2 , e2 = (0, 1).

(a) Find the set of Pareto-e"cient allocations.

(b) Find the core of the economy.

(c) Find a Walrasian equilibrium for this economyand the associated WEA.

8. Suppose that a competitive firm has productiontechnology represented by a production set Y # Rn.

(a) Define a strongly convex set.(b) Suppose Y is strongly convex and market

prices p $ Rn++. Show that the optimal pro-

duction plan is unique.

9. Given a production economy

E = {(U i, ei, $ij , Y j)|i $ I, j $ J }

where ei $ Rn+.

(a) Define the aggregate excess demand functionfor good k, zk(p).

(b) Prove Walrus’s Law in a production economy,that is, for all p $ Rn

++,

z(p)Tp = 0

10. Given a production economy as in Question 9.

(a) State the Second Welfare Theorem for a pro-duction economy.

(b) Let Y # Rn be the production set of a compet-itive firm with 0 $ Y . Let y $ Y and define aproduction set Y = Y "{y}. Show that 0 $ Y .

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