Am Exercises1

8/12/2019 Am Exercises1

1/8

Answers to some selected exercises to Consumer

Theory

R Finger

Advanced Microeconomics ECH-32306

September 2012

Please report mistakes in typesetting. The current version is based on inputsfrom Thomas Herzfeld.

1.6 Cite a credible example were the preferences of an ordinary consumerwould be unlikely to satisfy the axiom of convexity.

Answer: Indifference curves representing satiated preferences dontsatisfy the axiom of convexity. That is, reducing consumption wouldresult in a higher utility level. Negative utility from consumption ofbads (too much alcohol, drugs, unhealthy food etc.) would ratherresult in concave preferences.

Adapted from J and R Sketch a few level sets for the following utilityfunctions: y = x1 + x2 and y = min[x1, x2]. Discuss the propertiesof the presented preferences. Derive Marshallian demand functions,indirect utility functions and expenditure functions.

Answer

x1

x2

(a) y = x1

+ x2

x1

x2

(b) y= min(x1, x

2

)

Figure 1: Sets to Exercise

See lecture notes and handout on utility functions for other solutions.

Adapted from J and R Suppose f(x1, x2) = (x1x2)2 and g(x1, x2) =

(x21x2)3 to be utility functions.

1


2/8

(a) f(x1, x2) is homogeneous. What is its degree?

f(tx1, tx2) =t4(x1x2)2 k= 4

(b)g (x1, x2) is homogeneous. What is its degree?g(tx1, tx2) =t

9(x21x2)3 k= 9

(c)h(x1, x2) =f(x1, x2)g(x1, x2) is homogeneous. What is its degree?h(x1, x2) = x

81x

52 h(tx1, tx2) = t

13(x81x52 k = 13 Obviously, when-

ever two functions are homogeneous of degree m and n, their productmust be homogeneous of degree m +n.

(d)k(x1, x2) =g (f(x1, x2), f(x1, x2)) is homogeneous. What is its degree?k(tx1, tx2) =t

36(x1x2)18 k= 36

(e) Prove that wheneverf(x1, x2) is homogeneous of degreemandg(x1, x2)is homogeneous of degree n, thenk(x1, x2) =g (f(x1, x2), f(x1, x2)) ishomogeneous of degree mn.k(tx1, tx2) = [t

m (f(x1, x2), f(x1, x2))]n k= mn

1.12 Suppose u(x1

, x2

) and v(x1

, x2

) are utility functions.

(a) Prove that if u(x1, x2) and v(x1, x2) are both homogeneous ofdegreer, thens(x1, x2) u(x1, x2) + v(x1, x2) is homogeneous ofdegree r .

Answer: Whenever it holds that tru(x1, x2) = u(tx1, tx2) andtrv(x1, x2) = v(tx1, tx2) for all r > 0, it must also hold thattrs(x1, x2) u(tx1, tx2) +v(tx1, tx2) =t

ru(x1, x2) +trv(x1, x2).

(b) Prove that ifu(x1, x2) andv(x1, x2) are quasiconcave, thenm(x1, x2)min{u(x1, x2), v(x1, x2)} is also quasiconcave.Answer: Forming a convex combination of the two functions uandv and comparing withm(xt) satisfies the definition of quasi-concavity:

When u(xt) min

u(x1), u(x2)

and

v(xt) min

v(x1), v(x2)

so

m(xt) min

u(xt), v(xt)

.

1.20 Suppose preferences are represented by the Cobb-Douglas utility func-tion, u(x1, x2) = Ax

1 x

12 , 0 < < 1, and A > 0. Assuming an

interior solution, solve for the Marshallian demand functions.

2


3/8

Answer: Use either the Lagrangian or the equality of Marginal Rate

of Substitution and price ratio. The Lagrangian isL= Ax1x

12 + (y p1x1 p2x2). The first-order conditions (FOC)

are

L

x1=Ax11 x

12 p1= 0

L

x2= (1 )Ax1 x

2 p2= 0

L

=y p1x1p2x2= 0

By dividing first and second FOC and some rearrangement, we get

either x1 = x2p2(1)p1 or x2 =

(1)p1x1p2 . Substituting one of these ex-

pressions into the budget constraint, results in the Marshallian demandfunctions: x1=

yp1

and x2= (1)y

p2.

1.21 Weve noted thatu(x) is invariant to positive monotonic transforms.One common transformation is the logarithmic transform, ln(u(x)).Take the logarithmic transform of the utility function in 1.20; then,using that as the utility function, derive the Marshallian demand func-tions and verify that they are identical to those derived in the precedingexercise (1.20).

Answer: Either the Lagrangian is used or the equality of Marginal

Rate of Substitution with the price ratio. The Lagrangian isL= ln(A) + ln(x1) + (1 )ln(x2) +(y p1x1 p2x2). The FOCare

L

x1=

x1 p1 = 0

L

x2=

(1 )

x2 p2= 0

L

=y p1x1p2x2= 0

The Marshallian demand functions are: x1 = yp1

and x2 = (1)y

p2.

They are exactly identical to the demand functions derived in thepreceding exercise.

1.22 We can generalise further the result of the preceding exercise. Supposethat preferences are represented by the utility function u(x). Assum-ing an interior solution, the consumers demand functions, x(p, y), aredetermined implicitly by the conditions in (1.10). Now consider theutility function f(u(x)), where f > 0, and show that the firstorderconditions characterising the solution to the consumers problem inboth cases can be reduced to the same set of equations. Conclude

3


4/8

from this that the consumers demand behavior is invariant to posi-

tive monotonic transforms of the utility function.Answer: The set of firstorder conditions for a utility function, as-suming a linear budget constraint, are as follows:

L

xi=

u

xi pi

L

xj=

u

xj pj

u/xiu/xj

= pipj

The set of firstorder conditions for a positive monotone transform ofa utility function, assuming a linear budget constraint, are as follows:

L

xi=

f(u)

u

u(x)

xi pi

L

xj=

f(u)

u

u(x)

xj pj

f(u)/u

f(u)/u

u/xiu/xj

= pipj

After forming the Marginal Rate of Substitution, the outer derivativecancels out. Therefore, the demand function should be unaffected bythe positive monotonic transformation of the utility function.

1.24 Letu(x) represent some consumers monotonic preferences overx Rn+. For each of the functions F(x) that follow, state whetheror not f also represents the preferences of this consumer. In eachcase, be sure to justify your answer with either an argument or acounterexample.

Answer:

(a) f(x) = u(x) + (u(x))3 Yes, all arguments of the function u are trans-formed equally by the third power. Checking the first- and second-order partial derivatives reveals that, although the second-order partialis not zero, the sign of the derivatives is always invariant and positive.

2f

x2i=

2u

x2i+ 6(u(x))

u

xi+ 3(u(x)2

2u

x2i

Thus, frepresents a monotonic transformation ofu.

4


5/8

(b) f(x) = u(x) (u(x))2 No, function f is decreasing with increasing

consumption for any u(x) < (u(x))2

. Therefore, it can not representthe preferences of the consumer. It could do so if the minus sign isreplaced by a plus sign.

(c) f(x) = u(x) +n

i=1xi Yes, the transformation is a linear one, as thefirst partial is a positive constant, here one, and the second partialof the transforming function is zero. Checking the partial derivatives

proves this statement: fxi

= uxi

+ 1 and 2f

x2i

= 2u

x2i

.

1.38 Complete the proof of Theorem 1.9 by showing thatxh(p, u) =x (p, e(p, u)).

Answer: We know that at the solution of the utility maximisation orexpenditure minimisation problem e(p, u) =y and u = v(p, y). Sub-stitute the indirect utility function v into the Hicksian demand func-tion gives xh(p, v(p, y)). As the new function is a function of pricesand income only, it is identical to the Marshallian demand function.Furthermore, by replacing income by the expenditure function we getthe expressionx (p, e(p, u)). Carefully check the proof on page 45 andtry to follow the argumentation also in this case.

1.40 Prove that Hicksian demands are homogeneous of degree zero inprices.

Answer: We know that the expenditure function must be homoge-neous of degree one in prices. Because any Hicksian demand functionequals, due to Shephards lemma, the first partial derivative of theexpenditure function and, additionally, we know that the derivativesdegree of homogeneity is k1 (Theorem A2.6). The Hicksian demandfunctions must be homogeneous of degree 1 1 = 0 in prices.

1.42 For expositional purposes, we derived Theorems 1.14 and 1.15 sep-arately, but really the second one implies the first. Show that when thesubstitution matrix(p, u) is negative semidefinite, all ownsubstitutionterms will be nonpositive.

Answer: Theorem 1.15 implies Theorem 1.14 because all elements ofthe substitution matrix represent secondorder partial derivatives ofthe expenditure function. The diagonal elements of this matrix mustbe negative because the expenditure function is required to be con-cave in prices. Subsequently, any compensated demand function isrequired to be nonincreasing in its own price. With the substitutionmatrix being negative semidefinite, all principal minors (not only theleading ones) will be non-positive. Removing all but one columns androws (with the same number) will leave us diagonal elements, i.e. allown-substitution terms.

5


6/8

Giffen behavior Read the article of Jensen and Miller. Giffen Behavior

and Subsistence Consumption. Am Econ Rev. 2008 September 1;98(4), pp. 1553 to 1577. Recall and discuss the role of substitutionand income effects.

1.43 In a two-good case, show that if one good is inferior, the other goodmust be normal.Answer: The Engel-aggregation in a two-good case is the product ofthe income elasticity and the repsective expenditure share s11+s22 =1. An inferior good is characterised by a negative income elasticity,thus, one of the two summands will be less than zero. Therefore, to

secure this aggregation, the other summand must be positive (evenlarger one) and the other commodity must be a normal good (even aluxury item).

1.60 Show that the Slutsky relation can be expressed in elasticity formas ij =

hij sji, where

hij is the elasticity of the Hicksian demand

for xi with respect to price pj, and all other terms are as defined inDefinition 1.6.

Answer: The Slutsky relation is given by

xi

pj

=xhi

pj

xjxi

y

.

Multiplying the total expression with y /y and pj gives

xipj

pj =xhipj

pjpjxj

y

xiy

y.

By assuming that xhi = xi before the price change occurs, we candivide all three terms by xi. The result of this operation is

xipj

pjxi

=xhipj

pjxi

sjxiy

y

xi

This is equivalent to: ij =hij sji

4.19 A consumer has preferences over the single good xand all other goodsm represented by the utility function, u(x, m) = ln(x) + m. Let theprice ofx be p, the price ofm be unity, and let income be y.

(a) Derive the Marshallian demands for x and m.

Answer The equality of marginal rate of substitution and priceratio gives 1/x = p. Thus, the Marshallian demand for x is

6


7/8

x= 1/p. The uncompensated demand form separates into two

cases depending on the amount of income available:

m=

0 when y 1

y 1 when y >1.

(b) Derive the indirect utility function,v(p, y).

Answer Again, depending on the amount of income availablethere will be two indirect utility functions:

v(p, y) =

ln1p

when y 1

y 1 + ln1p when y >1.Note that +ln(1/p) can be written as ln(1)-ln(p), i.e. -ln(p)

(c) Use the Slutsky equation to decompose the effect of an own-pricechange on the demand for x into an income and substitution ef-fect. Interpret your result briefly.

AnswerA well-known property of any demand function derivedfrom a quasi-linear utility function is the absence of the incomeeffect. Which can be easily seen in the application of the Slutskyequation:

x

p

= xh

p

+xx

y

We see (using our Marshallian demand derived above) that theincome effect is zero. Thus:xp

= xh

p

Therefore, the effect of an own-price change on the demand forxconsists of the substitution effect only, the partial derivative ofthe compensated demand function with respect to price.

(d) Suppose that the price ofx rises from p0 to p1 > p0. Show thatthe consumer surplus area betweenp0 andp1 gives an exactmea-

sure of the effect of the price change on consumer welfare.AnswerThe consumer surplus area can be calculated by integrat-ing over the inverse uncompensated demand function ofx(we seefrom our above derived demand function that this is p = 1/x):

CS=

p0p1

1

xdx= lnp0 lnp1.

Calculating the change in utility induced by a price change gives:

v= v1(p1, y0)v0(p0, y0) =y1lnp1(y1lnp0) = lnp1+lnp0.

7


8/8

We here assumed y > 1. However, analyzing the case for y

Am Exercises1

Documents

Transcript of Am Exercises1