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Economics 401

Assignment 2Must be handed in, in class, Thursday 22 May 2014

Answer all questions in the space provided.

Please write clearly and concisely.

Name:

Student Number:

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1. Let A be an n byn matrix, possibly not symmetric. LetB A+A. ProveB issymmetric and that B is negative semi-definite if and only ifA is negative semi-definite.

ANSWER

Write A as [aij

]. ThenA = [aji

]. Thus

B= [aij ] + [aji ] = [aij+aji] = [aji +aij] = [aji] + [aij] =B.

B is negative semi-definite if and only if for allz n

zBz 0 iff

z

A+A

z 0 iff

zAz+zAz 0 iff

z

Az+

z

Az

0 iff2zAz 0 iff

zAz 0,

which meansA is negative semi-definite. Note that we can move from line 4 to line 5 in theabove series because zAzis a scalar and for any scalar, its transpose is the same number.

2. MWG, 2.F.16

ANSWER

(a) We have

x1(p1, p2, p3, w) = p2p3

x2(p1, p2, p3, w) = p1p3

x3(p1, p2, p3, w) = w

p3

Its clear that for all >0 and for i = 1, 2, 3

xi(p1, p2, p3, w) =xi(p1, p2, p3, w) ,so the demand system is homogeneous of degree zero in prices and wealth. It is also truethat

p1x1(p1, p2, p3, w) +p2x2(p1, p2, p3, w) +p3x3(p1, p2, p3, w) =p1p2p3

+p2

p1p3

+p3

w

p3=w,

so Walrass Law is satisfied.

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(b) Suppose wealth and prices at time 1 are (p1, p2, p3, w) = (1, 1, 1, 1) and wealth andprices at time 2 are (p1, p2, p3, w) = (2, 2, 1, 1). Then, according to the demand system, thebundle of goods purchased at time 1 will be (1, 1, 1) and the bundle of goods purchasedat time 2 will be (2, 2, 1). Now observe that when the person bought bundle 1, bundle 2was affordable this implies that bundle 1 is revealed preferred to bundle 2. Next observe

that when bundle 2 was purchased that bundle 1 was affordable. This tells us that bundle2 is revealed preferred to bundle 1. Well bundle 1 being revealed preferred to bundle 2 andbundle 2 being revealed preferred to bundle 1 contradicts WARP. So this demand systemviolates WARP.

(c) The Slutsky matrixSis the matrix of substitution effects.

S [sij] =

xipj

+xjxiw

By calculation

S=

0 1/p3 p2/p231/p3 0 p1/p23

p2/p23 p1/p

23 0

Using the logic of problem 1, for v 3, v

S+S

v = 2vSv. Inspection ofS showsS+S is a matrix of zeros. Thus vSv = 0.

3. Definestrong monotonicity,monotonicity, and local nonsatiation. Provestrong mono-tonicityimplies monotonicity, and monotonicityimplies local nonsatiation.

ANSWER

Definitions

LetX= n+, a X, b X: d (a, b) j=n

j=1(aj bj)21/2

, where we use the positive root.

Leta X: N(a) {x X :d (a, x)< , >0}

Strong monotonicity: suppose a X andb X andb aand b =a; then b a.

Monotonicity: suppose a X andb X and b >> a; then b a.

Local non-satiation: supposea X: then every -neighbourhood ofa, N(a) contains ab X such that b a.

Let y (y1, y2,...,yn) X and z(z1, z2,...,zn) X. Then y zmeans yj zj, j =1,...,nandy >> zmeansyj > zj, j = 1,...,n.

How to structure the proofs

Suppose we have the following statements.

Statement 1: A1 B1

and

Statement 2: A2 B2

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and we want to prove that

Statement 1 Statement 2.

Then you start by assuming A2 is true, and argue persuasively that A2 A1. SinceStatement 1 is assumed to be true A1 B1. The final step is to argue persuasively thatB1 B2.

Proof that strong monotonicity implies monotonicity: Suppose a Xand b Xand b >> a,thena Xandb Xandb aand b =a, that is, the assumptions of strong monotonicityare met. Since we are assuming strong monotonicity is true we know its conclusion holds,that is,b awhich is what we needed to prove.

Proof that monotonicity implies local non-satiation: Every epsilon-neighbourhood ofa con-tains some b Xsuch that b >> a. Since we are assuming monotonicity is true we knowits conclusion holds, that is, b awhich is what needed to prove.

4. You are given the following information about a consumers purchases. Goods 1 and2 are the only goods consumed.

Year 1 Year 2Quantity Price Quantity Price

Good 1 100 100 120 100Good 2 100 100 ? 80

For what values of good 2 consumed in year 2 would you conclude:

(a) that the consumers consumption bundle in year 1 is revealed preferred to that inyear 2?

(b) that the consumers consumption bundle in year 2 is revealed preferred to that inyear 1?

(c) that her/his behaviour contradicts the weak axiom?

(d) that good 1 is an inferior good somewherefor this consumer (assume WARP holds)?

(e) that good 2 is an inferior good somewherefor this consumer (assume WARP holds)?

ANSWER

(a) Denote the consumption of good 2 in year 2 by x22.If the consumers consumption bundle in year 1 is revealed preferred to that in year 2

then the year 2 bundle must be affordable with year 1 prices and income. So

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(100) (100) + (100) (100) (100) (120) + (100) x22 or

x22 80.

(b) If the consumers consumption bundle in year 2 is revealed preferred to that in year1 then the year 1 bundle must be affordable with year 2 prices and income. So

(100) (120) + 80x22 (100) (100) + (80) (100) or

x22 75.

(c) This requires that the inequalities in (a) and (b) both hold, so 75 x22 80.

(d) Let bundle C,

xC1, x

C2

, be bought with year 2 prices and just enough wealth to buythe year 1 bundle. Since we are assuming WARP holds xC1 < 100. Now consider year 2bundles withx22 < 75. The cost of these bundles at year 2 prices must be less than the costofC, and here x21 = 120, so wealth must increase going from the year 2 bundle to the Cbundle. The effect of this increase in wealth is to decrease good 1 consumption from 120 tosomething less than 100. Thus somewhere in this range good 1 must be inferior.

(e) Again, work with theCbundle but now suppose 80< x22 100; in this range the costof the year 2 bundle at year 2 prices exceeds the cost of the year 1 bundle at year 2 prices.And if WARP holds,xC2 >100. As we move from the Cbundle to the year 2 bundle wealthincreases but the consumption of good 2 falls, so somewhere in this range good 2 must be

inferior.

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