ECON 121: Intermediate MicroeconomicsSolutions to Problem Set 2

Niccolo Lomys

Spring 2016

Problem 1

Consider an economy in which there are two goods, 1 and 2, whose prices are p1 > 0 and p2 > 0,respectively. The two goods can only be consumed in non-negative amounts x1 and x2, respectively. Aconsumer has preferences over R2C which are represented by the utility function

uWR2C ! R; .x1; x2/ 7! u.x1; x2/ WD .x1 C 2/x2:

The consumer’s income is I > 0.

(a) Formulate the consumer’s utility maximization problem, find the first-order conditions for utilitymaximization, and find the Marshallian demand functions1 x1.p1; p2; I / and x2.p1; p2; I / for goods1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraintholds with equality and that the solution is interior (i.e. x1 > 0 and x2 > 0), thus disregarding thenon-negativity constraints on x1 and x2.) Check that the second order conditions are satisfied.

Solution

Let prices .p1; p2/ be fixed. When solving the consumer’s utility maximization problem (UMP),we fix income and some level I > 0 and we address this question: What is the maximum level ofutility the consumer can achieve when facing a given set of prices with income I?

There are three equivalent ways to formulate the consumer’s utility maximization problem.2 (i)In class, you have seen that the problem can be stated as

max.x1;x2/2R2

C

.x1 C 2/x2

subject to p1x1 C p2x2 � I:

(ii) Note that .x1; x2/ must be an element of R2C. Hence, the restrictions on .x1x2/ are equivalentto the following three inequality constraints: x1 � 0, x2 � 0 and p1x1 C p2x2 � I . Therefore,the following is and equivalent, and probably the most transparent statement of the consumer’sproblem:

max.x1;x2/2R2

.x1 C 2/x2

subject to x1 � 0

1The Marshallian demand function is also known as Walrasian or uncompensated demand function.2There are many more, but here we only discuss three of them.

1

x2 � 0

p1x1 C p2x2 � I:

This formulation helps you not to lose sight of the non-negativity constraints on x1 and x2. Theconstraints x1 � 0 and x2 � 0 will not play a major role in this course, but it is better to keepthem in mind.3

(iii) A more compact notation to formulate the consumer’s utility maximization problem is thefollowing. Given prices p WD .p1; p2/ and income I , let

B .p; I / WD˚.x1; x2/ 2 R2C W p1x1 C p2x2 � I

be the consumer’s budget set. The consumer’s utility maximization problem is

max.x1;x2/2B.p;I/

.x1 C 2/x2:

You can use this last formulation provided it is clear to you that .x1; x2/ 2 B .p; I / is equivalentto the constraints x1 � 0, x2 � 0 and p1x1 C p2x2 � I .

Let’s now solve this problem. Assuming that the budget constraint holds with equality anddisregarding the non-negativity constraints on x1 and x2, we can set up the Lagrangian for theutility maximization problem as4

L .x1; x2; �/ D .x1 C 2/x2 C � .p1x1 C p2x2 � I / ;

where � is the Lagrange multiplier.

The first order (necessary) conditions for an interior solution to this problem are:

@L.x1; x2; �/

@x1D 0” x2 C �p1 D 0;

@L.x1; x2; �/

@x2D 0” x1 C 2C �p2 D 0;

@L.x1; x2; �/

@�D 0” p1x1 C p2x2 � I D 0:

The optimal choices of x1 and x2 solve these three equations simultaneously (also note that thereare three unknowns since the value of � is unknown).

The first two equations can be rewritten in such a way that � is separated on the right hand side;that is, as

x2

p1D ��

andx1 C 2

p2D ��:

3See the following discussion of non-negativity constraints for this utility maximization problem. Moreover, look atProblem 2, where you are actually required to check for corner solutions.

4The budget constraint holds with equality because the utility function is strictly increasing in both arguments (Quiz:Why?). However, no argument can be provided to exclude corner solutions to this utility maximization problem. (Quiz:Why?) More on this in a while.

2

Equating the left hand sides of the two previous equations gives

x2

p1Dx1 C 2

p2

or, equivalently,

x2 Dp1

p2.x1 C 2/ (1)

When dividing by p1 and p2, we use our assumption that prices are strictly positive. This isalways so in this problem set, and so we omit to notice this detail in the remaining part of thesolution sheet.

Replacing (1) into the first order condition for � (equivalent to the budget constraint holding withequality) we obtain

p1x1 C p2p1

p2.x1 C 2/ D I:

Solving the last equation for x1 returns the candidate5 Marshallian demand function for good 1:

x1 .p1; p2; I / DI � 2p1

2p1: (2)

Replacing (2) for x1 into (1) gives the candidate Marshallian demand function for good 2:

x2 .p1; p2; I / DI C 2p1

2p2:

Note that x2 .p1; p2; I / is always strictly positive. However, x1 .p1; p2; I / > 0 if and only ifp1 < I=2. Therefore, the solution is interior only for p1 < I=2. Henceforth, we will assume thatthis condition is satisfied.

Let’s now check whether the second order conditions are satisfied. The bordered Hessian for thisproblem is

Hb WD

24L�� L�x1L�x2

Lx1� Lx1x1Lx1x2

Lx2� Lx2x1Lx2x2

35 D 24 0 p1 p2p1 0 1

p2 1 0

35 :Note that the bordered Hessian does not depend on x1 and x2. This is so because of the functionalform of the utility function in this problem, but this is not usually the case. When the borderedHessian depends on x1 and x2, you have to replace them with x1 .p1; p2; I / and x2 .p1; p2; I /,respectively, before computing the relevant leading principal minors.6 Since

det

�L�� L�x1

Lx1� Lx1x1

�D det

�0 p1p1 0

�D �p1

2 < 0

5Recall that first order conditions are necessary, but not sufficient, for an interior maximum. We haven’t yet verifiedthe second order conditions.

6The leading principal minor of order p, for p D 1; 2; 3, of a 3 � 3 matrix A is the determinant of the p � p matrixobtained by considering the first p rows and columns of A. The relevant leading principal minors for second orderconditions for this kind of problems are those of order p D 2 and p D 3.

3

and

det

24L�� L�x1L�x2

Lx1� Lx1x1Lx1x2

Lx2� Lx2x1Lx2x2

35 D det

24 0 p1 p2p1 0 1

p2 1 0

35D 0 � det

�0 1

1 0

�� p1 � det

�p1 1

p2 0

�C p2 � det

�p1 0

p2 1

�D 0 � p1.�p2/C p2p1 D 2p1p2 > 0;

the relevant leading principal minors of Hb have the required pattern. Therefore, the second orderconditions are satisfied and we conclude that x1 .p1; p2; I / and x2 .p1; p2; I / are interior solutionsto the constrained utility maximization problem. �

(b) Now formulate the corresponding expenditure minimization problem and find the compensateddemand functions7 xc1.p1; p2; NU/ and xc2.p1; p2;

NU/ for the two goods, where NU > 0 is somearbitrary level of utility. (Note again: Use the Lagrangian method. Assume that the solution isinterior and that the other relevant constraint holds with equality. Do not check the second orderconditions for this problem.)

Solution

Let prices .p1; p2/ be fixed. When solving the consumer’s expenditure minimization problem(EMP), we fix some arbitrary level of utility NU and we address this question: What is the minimumlevel of money expenditure the consumer must make facing a given set of prices to achieve a givenlevel of utility NU ? In this construction, we ignore any limitations imposed by the consumer’sincome and simply ask what the consumer would have to spend to achieve some particular levelof utility.8

Again, we can formulate the expenditure minimization problem in three equivalent ways. (i) Thefirst statement, which you have seen in class, goes as follows:

min.x1;x2/2R2

C

p1x1 C p2x2

subject to .x1 C 2/x2 � NU :

(ii) If you want to be explicit about the non-negativity constraints on x1 and x2, you may wantto use the following formulation:

min.x1;x2/2R2

p1x1 C p2x2

subject to x1 � 0

x2 � 0

.x1 C 2/x2 � NU :

Finally, by defining the set of .x1; x2/ feasible pairs that give utility at least as high as NU as

G�NU�WD˚.x1; x2/ 2 R2C W .x1 C 2/x2 � NU

;

7The compensated demand function is also known as Hicksian demand function.8When the range of the utility function u is contained in RC, as it is the case for this problem, we require NU > 0.

Otherwise, the problem becomes trivial.

4

you can state the expenditure minimization problem compactly as

min.x1;x2/2G. NU/

p1x1 C p2x2:

Disregarding the non-negativity constraints on x1 and x2 and assuming that the other constraintholds with equality9, we can set up the Lagrangian for the expenditure minimization problem as

L.x1; x2; �/ D p1x1 C p2x2 C �Œ.x1 C 2/x2 � NU �;

where � is the Lagrange multiplier.

The first order (necessary) conditions for an interior solution to this problem are:

@L .x1; x2; �/

@x1D 0” p1 C �x2 D 0;

@L .x1; x2; �/

@x2D 0” p2 C �.x1 C 2/ D 0;

@L .x1; x2; �/

@�D 0” .x1 C 2/x2 � NU D 0:

The optimal choices of x1 and x2 solve these three equations simultaneously (also note that thereare three unknowns since the value of � is unknown).

Following similar steps as in part (a), the first two equations yield

x2 Dp1

p2.x1 C 2/: (3)

Replacing (3) into the first order condition for � and solving for x1 yields the compensated demandfunction for good 1:

x1c�p1; p2; NU

�D

p2 NU

p1

! 12

� 2: (4)

Replacing (4) for x1 into (3) yields the compensated demand function for good 2:

x2c�p1; p2; NU

�D

p1 NU

p2

! 12

:

We observe that the non-negativity constraint on x1 is binding for some values of the exogenousvariables p1, p2 and NU . Henceforth, we assume NU > 4p1=p2, which ensures an interior minimizer.Moreover we omit checking second order conditions for this problem (which are satisfied).10 �

9It is easy to see that the constraint .x1C 2/x2 � NU has to hold with equality at the optimal choices of x1 and x2. Tryto understand why this is so and, if interested, feel free to ask us.

10Here we would need to check second order conditions for a minimum, which are different that those for a maximum.

5

(c) Compute the indirect utility function V.p1; p2; I / and the expenditure function E.p1; p2; NU/.

Solution

These are simple calculations. Recall that the the indirect utility function V.p1; p2; I / is definedas the maximum value function corresponding to the consumer’s utility maximization problem:the most utility the consumer can get at prices .p1; p2/ with income I . It can be computed byevaluating utility at the Marshallian demand. Thus, we have

V.p1; p2; I / WD max.x1;x2/2B.p;I/

u.x1; x2/ D u .x1 .p1; p1; I / ; x2 .p1; p1; I //

D .x1 .p1; p1; I /C 2/ x2 .p1; p1; I /

D

�I � 2p1

2p1C 2

�I C 2p1

2p2

D

�I � 2p1 C 4p1

2p1

�I C 2p1

2p2

DI C 2p1

2p1

I C 2p1

2p2

D.I C 2p1/

2

4p1p2:

(5)

The expenditure function E�p1; p2; NU

�is defined as the minimum value function corresponding to

the consumer’s expenditure maximization problem: the minimum expenditure required to achieveutility NU at prices .p1; p2/. It can be computed by evaluating expenditure at the compensateddemand. Hence, we have

E�p1; p2; NU

�WD min

.x1:x2/2G. NU/

p1x1 C p2x2 D p1xc1

�p1; p2 NU

�C p2x

c2

�p1; p2 NU

�D p1

24 p2 NUp1

! 12

� 2

35C p2 p1 NUp2

! 12

D�p1p2 NU

�1=2� 2p1 C

�p1p2 NU

�1=2D 2

��p1p2 NU

�1=2� p1

�: �

(d) Is@xc

1.p1;p2; NU/@p1

positive or negative? Moreover, verify that

�

@V .p1;p2;I /

@p1

@V .p1;p2;I /

@I

D x1 .p1; p2; I / :

This is called Roy’s identity.

Finally, verify that@E

�p1; p2; NU

�@p1

D xc1�p1; p2; NU

�:

This is called Shephard’s lemma.

6

Solution

From part (b) we know that

x1c�p1; p2; NU

�D

p2 NU

p1

! 12

� 2 D�p2 NU

� 12p� 1

2

1 � 2:

Therefore,

@xc1�p1; p2; NU

�@p1

D �1

2

�p2 NU

� 12p� 3

2

1 D �1

2

p2 NU

p31

! 12

;

which is negative because p1; p2; NU > 0.

From part (c) we know that

V.p1; p2; I / D.I C 2p1/

2

4p1p2:

Thus,

@V .p1; p2; I /

@p1D16.I C 2p1/p1p2 � 4.I C 2p1/

2p2

.4p1p2/2

D4p2.I C 2p1/.2p1 � I /

.4p1p2/2

and@V .p1; p2; I /

@ID2.I C 2p1/

4p1p2:

It follows that

�

@V .p1;p2;I /

@p1

@V .p1;p2;I /

@I

D4p2.I C 2p1/.I � 2p1/

.4p1p2/2

4p1p2

2.I C 2p1/DI � 2p1

p1D x1 .p1; p2; I / ;

where the last equality follows from our derivation of the Marshallian demand function for good1 in part (a). This establishes Roy’s identity for good 1 in our example, as desired.

From part (c) we also know that

E�p1; p2; NU

�D 2

��p1p2 NU

�1=2� p1

�D 2

�p1p2 NU

�1=2� 2p1:

Therefore,

@E�p1; p2; NU

�@p1

D1

22�p2 NU

� 12p� 1

2

1 � 2 D

p2 NU

p1

! 12

� 2 D x1c�p1; p2; NU

�;

where the last equality follows from our derivation of the compensated demand function for good1 in part (b). This establishes Shephard’s lemma for good 1 in our example, as desired. �

(e) Verify that

V�p1; p2; E.p1; p2; NU/

�D NU and that E .p1; p2; V .p1; p2; I // D I:

7

Moreover, verify that

xi .p1; p2; I / D xci .p1; p2; V .p1; p2; I // and that xci

�p1; p2; NU

�D xi

�p1; p2; E.p1; p2; NU/

�for i D 1; 2. Explain why these are equal.

Solution

Recall again that in part (c) we found

V.p1; p2; I / D.I C 2p1/

2

4p1p2and E

�p1; p2; NU

�D 2

��p1p2 NU

�1=2� p1

�:

Therefore,

V�p1; p2; E

�p1; p2; NU

��D

�E�p1; p2; NU

�C 2p1

�24p1p2

D

�2�p1p2 NU

�1=2� 2p1 C 2p1

�24p1p2�

2�p1p2 NU

�1=2�24p1p2

D4p1p2 NU

4p1p2

D NU ;

and

E .p1; p2; V .p1; p2; I // D 2h.p1p2V.p1; p2; I //

1=2� p1

iD 2

24 p1p2 .I C 2p1/24p1p2

!1=2� p1

35D 2

24 .I C 2p1/24

!1=2� p1

35D 2

�I C 2p1

2� p1

�D I C 2p1 � 2p2

D I;

as desired.

This result is quite simple and intuitive. It says that if E.p1; p2; NU/ is the amount of incomerequired to achieve utility NU , then the most utility a consumer can get with wealth E.p1; p2; NU/is exactly NU . Similarly, if V.p1; p2; I / is the most utility that a consumer can achieve with wealthI at prices .p1; p2/, then to achieve utility V.p1; p2; I / will take wealth at least I .

8

Now let’s show the equalities for demand functions. From part (a) we know that

x1 .p1; p2; I / DI � 2p1

2p1and x2 .p1; p2; I / D

I C 2p1

2p2

and from part (b) we know that

x1c�p1; p2; NU

�D

p2 NU

p1

! 12

� 2 and x2c�p1; p2; NU

�D

p1 NU

p2

! 12

:

Therefore,

xc1 .p1; p2; V .p1; p2; I // D

�p2V.p1; p2; I /

p1

� 12

� 2

D

0@p2 .IC2p1/2

4p1p2

p1

1A 12

� 2

D

.I C 2p1/

2

4p21

! 12

� 2

DI C 2p1

2p1� 2

DI � 2p1

2p1

D x1 .p1; p2; I / ;

and

xc2 .p1; p2; V .p1; p2; I // D

�p1V.p1; p2; I /

p2

� 12

D

0@p1 .IC2p1/2

4p1p2

p2

1A 12

D

.I C 2p1/

2

4p22

! 12

DI C 2p1

2p2

D x2 .p1; p2; I / ;

as desired.

Finally,

x1�p1; p2; E.p1; p2; NU/

�DE.p1; p2; NU/ � 2p1

2p1

D2�p1p2 NU

�1=2� 2p1 � 2p1

2p1

9

D

�p1p2 NU

�1=2� 4p1

p1

D

�p1p2 NU

�1=2p1

� 2

D

p2 NU

p1

! 12

� 2

D xc1�p1; p2; NU

�;

and

x2�p1; p2; E.p1; p2; NU/

�DE.p1; p2; NU/C 2p1

2p2

D2�p1p2 NU

�1=2� 2p1 C 2p1

2p2

D

�p1p2 NU

�1=2p2

D

�p1p2 NU

�1=2p2

D

p1 NU

p2

! 12

D xc2�p1; p2; NU

�;

as desired.

The compensated demand function keeps the consumer’s utility level fixed as expenditure (requiredincome) changes, in contrast to the Marshallian demand function which keeps the income fixed butallows utility to vary. The first two relations say that the Marshallian demand at prices .p1; p2/and income I is equal to the compensated demand at prices .p1; p2/ and the utility level that is themaximum that can be achieved at prices .p1; p2/ and income I . The second two relations say thatthe compensated demand at any prices .p1; p2/ and utility level NU is the same as the Marshalliandemand at those prices and an income level equal to the minimum expenditure necessary at thoseprices to achieve that utility level. �

(f) Verify that the Slutsky equation for good 1 with respect to its own price holds in this case. Todo this, first write the the Slutsky equation for a general utility function uWR2C ! R, .x1; x2/ 7!u.x1; x2/. (Note: Assume that the associated Marshallian demand function is differentiable withrespect to prices and income and that the compensated demand function is differentiable withrespect to prices.) Then use your derivations in (a), (b) and (c) to substitute for the various termsin this expression and show that this equality holds for the current utility function.

Solution

Fix prices p1; p2 > 0 and income I > 0. Consider a general a general utility function. In class,

10

we have seen that the Slutsky equation for good 1 with respect to its own price is of the form

@x1 .p1; p2; I /

@p1D@x1

c�p1; p2; NU

�@p1

ˇˇNUDV.p1;p2;I /

� x1 .p1; p2; I /@x1 .p1; p2; I /

@I: (6)

The first term on the right hand side of the previous equation is the substitution effect, whilethe second one is the income effect. Recall that the Slutsky equation holds when utility is fixedat a level NU given by maximizing utility at the original prices and income; that is, at NU DV.p1; p2; I /. This is way we evaluate the partial derivative of the compensated demand functionat that point.11 Observe that the assumption that the Marshallian and compensated demandfunctions are differentiable allows us to take the partial derivatives in the previous expression.

Now consider the utility function whose functional form is defined by u.x1; x2/ WD .x1 C 2/x2.Take the partial derivative with respect to p1 of the associated Marshallian demand for good 1 wederived in part (a) (see (2)). This way we obtain an expression for left hand side of the Slutskyequation (6):

@x1 .p1; p2; I /

@p1D �

I

2p12:

Then, take the partial derivative with respect to p1 of the compensated demand function for good1 we derived in (b) (see (4)). This way we obtain an expression for the first term of the right handside of the Slutsky equation (6):

@x1c�p1; p2; NU

�@p1

D �1

2

.p2 NU/12

p132

:

Then, we replace NU in the previous equation with the indirect utility function V .p1; p2; I / wederived in part (c) (see (5)) to obtain

@x1c�p1; p2; NU

�@p1

ˇˇNUDV.p1;p2;I /

D �1

4

I C 2p1

p12D �

I C 2p1

4p12:

Finally we multiply the Marshallian demand for good 1 in (2) by its partial derivative with respectto income to obtain an expression for second term of the right hand side of the Slutsky equation(6)

x1 .p1; p2; I /@x1 .p1; p2; I /

@ID

�I � 2p1

2p1

�1

2p1DI � 2p1

4p12:

Combing these expressions we have

@x1c�p1; p2; NU

�@p1

ˇˇNUDV.p1;p2;I /

� x1 .p1; p2; I /@x1 .p1; p2; I /

@ID �

I C 2p1

4p12�I � 2p1

4p12

D �I

2p12

D@x1 .p1; p2; I /

@p1;

which verifies the Slutsky equation for good 1 with respect to its own price holds.

11This is the only term in which NU appears.

11

For the next two questions of this problem, consider a general utility function uWR2C ! R, .x1; x2/ 7!u.x1; x2/ satisfying the usual assumptions economists make. Assume that the associated Marshalliandemand function is differentiable with respect to prices and income and that the compensated demandfunction differentiable is differentiable with respect to prices.12 �

(g) The motivation that is commonly given for studying income and substitution effects is that it helpsunderstand the possibility of a Giffen good. Using the Slutsky equation, explain the connectionbetween Giffen goods and inferior goods. In particular, which of the following two statements isnecessarily true and which one is not always so? Why?

(i) A Giffen good is an inferior good;

(ii) An inferior good is a Giffen good.

Solution

Note again that the Slutsky equation for good i with respect to its own price takes the form

@xi

@piD@xi

c

@pi� xi

@xi

@I; (7)

where we economize on notation because it is now clear what we are talking about (see solutionto part (f)).

Recall also that good i is called a Giffen good if @xi

@pi> 0, and that good i is said to be an inferior

good if @xi

@I< 0.

Suppose that good i is a Giffen good; that is, @xi

@pi> 0. Since @xi

c

@piis negative13, the Slutsky equation

in (7) is satisfied only if xi@xi

@Iis negative, i.e. only if @xi

@I< 0. That is, it must be that good i is

inferior. This shows that the first statement is true: any Giffen good is an inferior good.

Now suppose that good i is an inferior good; that is, @xi

@I< 0, so that the income effect xi

@xi

@Iis

negative. From the Slutsky equation we see that, for the good to be Giffen (i.e., for @xi

@pi> 0), we

require not only @xi

@Ito be negative, but also to be large enough in magnitude to outweigh @xi

c

@pi,

the substitution effect, which we know to be always negative. In other words, for the good to beGiffen, we need the income effect to be negative (good i has to be inferior) and large enough tooutweigh the substitution effect. This shows that an inferior good is not necessarily a Giffen good,which implies that the second statement is false. In particular, the Slutsky equation shows thatif the income effect does not dominate the substitution effect, then, even if the good is inferior, itwill not be Giffen. �

(h) Recall that the compensated and Marshallian demand functions for good i , i D 1; 2, are relatedthrough the equality

xci�p1; p2; NU

�D xi

�p1; p2; E.p1; p2; NU/

�;

12For the mathematically inclined reader: This is so when the utility function is twice continuously differentiable on theinterior of R2C, strictly increasing and quasi-concave.

13We have shown in class that, “under the usual assumptions on utility economists make, compensated demand curvesslope downward”: when the price of some good increases, the compensated demand for that good decreases. This isan immediate consequence of Shephard’s lemma. A simple way to see this graphically is to note that the change incompensated demand given a change in price is a shift along an indifference curve.

12

that is the point of departure for deriving the Slutsky equation. Now differentiated both sides ofthe previous equality for good 1 (i.e., i D 1) with respect to p2. What can you say about incomeeffects and whether goods 1 and 2 are substitutes? (Hint: Use Shephard’s lemma and the factthat @x1=@E D @x1=@I .)

Solution

The compensated and Marshallian demand functions for good 1 are related through the equality

xc1�p1; p2; NU

�D x1

�p1; p2; E.p1; p2; NU/

�: (8)

Differentiating both sides of equation (8) with respect to p2 we get

@x1c

@p2D@x1

@p2C@x1

@E

@E

@p2;

which is equivalent to@x1

@p2D@x1

c

@p2�@x1

@E

@E

@p2: (9)

By Shepard’s lemma, we know that @E=@p2 D xc2. Moreover, observe that we can write @x1=@Eas @x1=@I (this is just notation). Using these facts, (9) is equivalent to

@x1

@p2D@x1

c

@p2� x2

c @x1

@I: (10)

Since the compensated and Marshallian demand functions for good 2 are related through theequality

xc2�p1; p2; NU

�D x2

�p1; p2; E.p1; p2; NU/

�;

it follows that (10) is equivalent to

@x1

@p2D@x1

c

@p2� x2

@x1

@I:

The previous expression is the Slutsky equation for good 1 with respect to the price of good 2.Now recall that goods 1 and 2 are said to be substitutes if @x1

@p2> 0. We know that @x1

c

@p2is positive.

Hence, if good 1 is inferior (that is, if @x1

@I< 0), then the goods are necessarily substitutes. If

good 1 is normal (that is, if @x1

@I> 0), then the goods are substitutes when the substitution effect

dominates the income effect, and they are complements when the income effect dominates thesubstitution effect. �

Problem 2

In most of the utility maximization problems we encounter in this course, the solution is interior. Thismeans that a strictly positive quantity of each good is demanded at the utility maximizing bundle.x�1 ; : : : ; x

�n/ (i.e. x�i > 0 for i D 1; : : : ; n.) However, this is not always the case, as the following

problem shows.

13

Consider again an economy in which there are two goods, 1 and 2, whose prices are p1 > 0 and p2 > 0,respectively. The two goods can only be consumed in non-negative amounts x1 and x2, respectively. Aconsumer has preferences over R2C which are represented by the utility function

uWR2C ! R; .x1; x2/ 7! u.x1; x2/ WDpx1 C x2:

The consumer’s income is I > 0.

(a) Formulate the consumer’s utility maximization problem, writing down explicitly all the constraints(i.e., the budget constraint and the non-negativity constraints on x1 and x2).

Solution

The consumer’s utility maximization problem formulated in the required form is

max.x1;x2/2R2

px1 C x2

subject to x1 � 0

x2 � 0

p1x1 C p2x2 � I: �

(b) Show that the consumer’s utility function is strictly increasing is x1 and x2 (hence, the consumer’spreferences are monotone).

Solution

For this utility function we have

MU1.x1; x2/ WD@u.x1; x2/

@x1D1

2x� 1

2

1 D1

2px1> 0

for all .x1; x2/ 2 Int�R2C�

and, similarly,

MU2.x1; x2/ WD@u.x1; x2/

@x2D 1 > 0:

Since both marginal utilities are strictly positive in the interior of R2C, it immediately followsthat the consumer’s utility function in strictly increasing in x1 and x2. Hence, the consumer’spreferences it represents are monotone. �

(c) Now that you have shown that the utility function is strictly increasing, you know that the budgetconstraint holds with equality at the utility maximizing bundle. Ignore for a moment the thenon-negativity constraints on x1 and x2 and solve the utility maximization problem using thesubstitution method. That is: (a) using the budget constraint, express one choice variable as afunction of the other choice variables and the parameters (exogenous variables) p1, p2 and I ofthe model; (b) plug your expression into the objective function; (c) take the first order conditionand solve for x1.p1; p2; I / and x2.p1; p2; I /. Skip the check of second order conditions.

Solution

14

Since the budget constraint holds with equality, we can rearrange it and express x2 as a functionof x1 and the exogenous variables as

x2 DI

p2�p1

p2x1:

Putting this into the objective function gives

maxx12R

px1 C

I

p2�p1

p2x1:

Taking the first order condition and solving it for x1, we find x1 .p1; p2:I /:

1

2

1px1�p1

p2D 0”

1p2D 2

p1

p2

”px1 D

p2

2p1

” x1 .p1; p2:I / D

�p2

2p1

�2:

Putting x1 .p1; p2:I / back for x1 into the above expression for x2 yields x2 .p1; p2:I /:

x2 DI

p2�p1

p2

�p2

2p1

�2” x2 .p1; p2:I / D

I

p2�p2

4p1: �

(d) Now it’s time to reconsider our non-negativity constraints. Show that x1.p1; p2; I / is alwaysstrictly positive. Provide a condition on p2 under which x2.p1; p2; I / is strictly positive.

Solution

Since prices are strictly positive and the square of a strictly positive real number is strictly positive,it immediately follows that x1.p1; p2; I / is strictly positive.

For x2.p1; p2; I / we have

x2.p1; p2; I / > 0”I

p2�p2

4p1> 0

” 4Ip1 > p22

” 2pIp1 > p2:

That is, x2.p1; p2; I / is strictly positive if and only if p2 < 2pIp1. �

(e) The condition you derived in (d) suggests that the constraint x2 � 0 is a potential problem, thatis, it is binding for a subset of the parameters. This is a case in which the utility maximizationproblem has a corner solution. This type of solutions come about when the parameters of theproblem .p1; p2; I / are such that the slope of the budget constraint is never equal to that of anindifference curve. Now assume that the condition you derived in (d) does NOT hold and derivethe Marshallian demand functions x1.p1; p2; I / and x2.p1; p2; I / for this case.

15

Solution

Now suppose that p2 � 2pIp1. In this case we have a corner solution, in which all income is

spent on a good. The two possible corner solutions are�Ip1; 0�

, in which all income is spent on

good 1, and�0; Ip2

�, in which all income is spent on good 2.

Observe that

u

�I

p1; 0

�> u

�0;I

p2

�”

sI

p1C 0 >

p0C

I

p2

”

sI

p1>I

p2

”I

p1>I 2

p22

” p22 > Ip1

” p2 >pIp1:

Since the last inequality in the previous chain of equivalences (p2 >pIp1) always holds under

the assumption p2 � 2pIp1, we have that in this case u

�Ip1; 0�> u

�0; Ip2

�. Therefore, the

Marshallian demand functions are given by

x1.p1; p2; I / DI

p1and x2.p1; p2; I / D 0:

for p2 � 2pIp1. �

(f) Let’s now put things together. Let p1 and I be fixed. Express and draw the Marshallian demandfor good 1 and the Marshallian demand for good 2 as functions of price p2. (Hint: The graph youwill obtain should show that the demand for either of the goods, as a function of p2 when I andp1 are kept constant, has a point of non-differentiabibility; but despite this, it is continuous.)

Solution

Fix p1 > 0 and I > 0. The Marshallian demands for good 1 and good 2 as functions of price p2are given by

x1.p1; p2; I / D

8<:�p2

2p1

�2if 0 < p2 < 2

pIp1

Ip1

if p2 � 2pIp1

and

x2.p1; p2; I / D

�Ip2�

p2

4p1if 0 < p2 < 2

pIp1

0 if p2 � 2pIp1

;

respectively. �

16

Problem 3

Consider once again an economy in which there are two goods, 1 and 2, whose prices are p1 > 0

and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x1 and x2,respectively. A consumer has preferences over R2C which are represented by the utility function

uWR2C ! R; .x1; x2/ 7! u.x1; x2/ WD�˛x�21 C ˇx2

�2��1=2

;

where ˛ and ˇ are strictly positive real numbers. The consumer’s income is I > 0. (Note: This is anexample of CES utility function)

(a) Formulate the consumer’s utility maximization problem, find the first-order conditions for utilitymaximization, and find the Marshallian demand functions x1.p1; p2; I / and x2.p1; p2; I / for goods1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraintholds with equality and that the solution is interior (i.e. x1 > 0 and x2 > 0), thus disregardingthe non-negativity constraints on x1 and x2. Do not check second order conditions.)

Solution

The agent’s utility maximization problem is:

max.x1;x2/2R2

C

�˛x�21 C ˇx

�22

�� 12

subject to p1x1 C p2x2 � I:

Assuming that the budget constraint holds with equality and disregarding the non-negativityconstraints on x1 and x2, we can set up the Lagrangian for the utility maximization problem as

L .x1; x2; �/ D�˛x�21 C ˇx

�22

�� 12 C � .p1x1 C p2x2 � I / ;

where � is the Lagrange multiplier.

The first order (necessary) conditions for an interior solution to this problem are:

@L.x1; x2; �/

@x1D 0”

�˛x�31

� �˛x�21 C ˇx

�22

�� 32 C �p1 D 0

@L.x1; x2; �/

@x2D 0”

�ˇx�32

� �˛x�21 C ˇx

�22

�� 32 C �p2 D 0

@L.x1; x2; �/

@�D 0” p1x1 C p2x2 � I D 0

The optimal choices of x1 and x2 solve these three equations simultaneously (also note that thereare three unknowns since the value of � is unknown).

From the first two first order conditions we get

x1

x2D

�ˇp1

˛p2

�� 13

17

which we can solve for x1 to obtain

x1 D x2

�ˇp1

˛p2

�� 13

: (11)

Replacing (11) into the first order condition for � we obtain

p1x2

�ˇp1

˛p2

�� 13

C p2x2 D I:

Solving the previous expression for x2 returns the Marshallian demand for good 2:

x2 .p1; p2; I / DI

p1

�ˇp1

˛p2

�� 13

C p2

DI�

ˇ

˛

�� 13

p23

1 p13

2 C p2

: (12)

Replacing (12) into (11) gives the Marshallian demand function for good 1:

x1 .p1; p2; I / D

�ˇp1

˛p2

�� 13

I

p1

�ˇp1

˛p2

�� 13

C p2

DI

p1 C p2

�ˇp1

˛p2

� 13

DI

p1 C�p1

ˇ

˛

� 13

p23

2

: �

(b) Find the price elasticity of demand for good 1. Is demand for this good elastic or inelastic?

Solution

To find the price elasticity of demand for good 1, we can work with the expression for the totalexpenditure on good 1, i.e. with

p1x1 .p1; p2; I / DI

1C�ˇ

˛

� 13

p23

2 p� 2

3

1

: (13)

Differentiating this expression with respect to p1, we get:

@p1x1 .p1; p2; I /

@p1D �

I

D2

�ˇ

˛

� 13

p23

2

��2

3

�p� 5

3

1

D2I

3D2

�ˇ

˛

� 13

p23

2 p� 5

3

1 ;

where D is the denominator of the expression in (13). The sign of this derivative is positive(by assumption, p1; p2; I; ˛; ˇ > 0), indicating that demand is inelastic. To see why a positivederivative for total expenditure indicates an elasticity of less than one and therefore inelasticdemand, note that

@p1x1

@p1D x1 C p1

@x1

@p1;

18

which is equivalent to@p1x1

@p1D x1

�1C

p1

x1

@x1

@p1

�D x1 .1 � e/ ;

where e WD �p1

x1

@x1

@p1is the (own) price elasticity of demand.

If demand is inelastic (e < 1), the derivative of total expenditure is positive, and viceversa.Therefore, in this case we established that demand is inelastic. �

(c) Find the derivative of the quantity of good 2 demanded with respect to the price of good 1. Usethis to indicate whether goods 1 and 2 are substitutes or complements.

Solution

Recall that the Marshallian demand for good 2 is

x2 .p1; p2; I / DI�

ˇ

˛

�� 13

p23

1 p13

2 C p2

:

The derivative of the quantity of good 2 demanded with respect to the price of good 1 is thus

@x2 .p1; p2; I /

@p1D �

I

D2

2

3

�ˇ

˛

�� 13

p� 1

3

1 p13

2 < 0:

Therefore the goods are complements.

Now consider a consumer whose preferences over R2C are represented by the CES utility function

uWR2C ! R; .x1; x2/ 7! u.x1; x2/ WD�˛x

�1 C ˇx2

��1=�

; (14)

where �, ˛ and ˇ are real numbers such that 0 ¤ � < 1, ˛ > 0 and ˇ > 0 The consumer’s income isI > 0. �

(d) Why does the utility function

vWR2C ! R; .x1; x2/ 7! v.x1; x2/ WD ˛x�1

�C ˇ

x�2

�

represent the same preferences over R2C as the function u in (14)?

Solution

Utility is an ordinal concept. In class, we argued that utility functions are defined up strictlyincreasing transformations. This means that if the utility function uWX ! R, x 7! u.x/ representspreferences over a nonempty set X , and the function f Wu.X/ ! R, where u.X/ is the range ofu, is strictly increasing on u.X/, then vWX ! R, v.x/ WD f .u.x// also represents those samepreferences over X .14

14The range u.X/ of the function uWX ! R, x 7! u.x/ is defined as u.X/ WD ft 2 R W t D u.x/ for some x 2 Xg. Observethat u.X/ � R. If the function f is differentiable on u.X/, an easy way to check that it is strictly increasing on u.X/ isto show that f 0.t/ > 0 for all t in u.X/.

19

Since ˛; ˇ > 0, the range u�R2C�

of u is RC. Define f WRC ! R as t 7! f .t/ WD 1�t� and observe

that

f .u.x1; x2// D1

�

��˛x

�1 C ˇx2

��1=���

D1

�

�˛x

�1 C ˇx2

��

D ˛x�1

�C ˇ

x�2

�

D v.x1; x2/;

which shows that v is a transformation of u. To show that v represents the same preferences as u,we only need to show that the transformation f is strictly increasing on RC. This is easily shownby observing that

f 0.t/ D1

��t��1 D t��1 > 0

for all real numbers t > 0.

Now forget about the previous utility functions. For the next question, consider generic Marshalliandemand functions for goods 1 and 2, x1.p1; p2; I / and x2.p1; p2; I /, resulting from a consumer’s utilitymaximization problem. Assume that Marshallian demands are differentiable functions of prices. �

(e) Write the budget constraint as

p1x1.p1; p2; I /C p2x2.p1; p2; I / D I:

Now take a derivative of this identity with respect to p1. (Note: Your derivative should have threeterms, two of which are related to elasticity and one of which tells you whether these goods aresubstitutes or complements.) Use this derivative to argue that, for any demand function, if thedemand for good 1 is inelastic, then the goods are complements. (Note: This tight relationshipdoes not generalize beyond the case of two goods. This is an unusual case in which a result dependson there being only two goods in our model.)

Solution

Writing the budget constraint using the Marshallian demand function we get

p1x1.p1; p2; I /C p2x2.p1; p2; I / D I:

Differentiating the previous identity with respect to p1 yields

x1 C p1@x1

@p1C p2

@x2

@p1D 0

Dividing both sides of the previous identity by x1, we obtain that

1Cp1

x1

@x1

@p1Cp2

x1

@x2

@p1D 0

or, equivalently,

1 � e Cp2

x1

@x2

@p1D 0;

20

where e WD �p1

x1

@x1

@p1is the (own) price elasticity of demand.

Now suppose that the demand for good 1 is inelastic, i.e. e < 1. Therefore, 1 � e > 0, and so thefirst term on the left hand side is positive. Since the sum of the two terms equals zero and p2

x1> 0,

the term @x2

@p1must be negative, which means the two goods are complements. �

Problem 4

As you started to see in this course, concave functions and convex sets play an important role in economicanalysis. This problem will help you to gain a better grasp of these concepts.

In what follows, whenever convenient for notation, we denote a generic element .x1; : : : ; xn/ of Rnsimply as x. For any � 2 R and any x 2 Rn, the scalar multiplication between � and x, denoted by�x, is the element of Rn defined as �x.�x1; : : : ; �xn/. For example, if � D 2 and x D .3; 5/ 2 R2, then�x D .6; 10/.

A subset S of Rn is said to be convex if for all x; y 2 S and all � 2 Œ0; 1�, we have �xC .1��/y 2 S .In words, the definition says that a subset S of Rn is convex if for any two elements x and y belongingto S there are no elements of the straight line segment between x and y that are not elements of S . Anexpression of the form �x C .1� �/y, where x; y 2 Rn and � 2 Œ0; 1�, is called a convex combination ofx and y.

(a) Suppose that A and B are arbitrary convex subsets of Rn. Show that A\B is convex, while A[Bneed not be. (Hint: You may want to use a graphical representation of the two set operations toguide your reasoning.)

Solution

Let’s start by showing that A \ B is a convex set. Fix two arbitrary elements x and y of A \ Band an arbitrary � in Œ0; 1�. To show that A \ B is convex, we need to show that �x C .1 � �/yis an element of A \ B.

Since x and y are elements of A \ B, by definition of intersection we know that x and y areelements of A. Since A is convex, �x C .1 � �/y is an element of A.

Analogously, since x and y are elements of A\B, they are also elements of B. Since B is convex,�x C .1 � �/y is an element of B.

Therefore, �xC .1��/y is both an element of A and an element of B. Therefore, we conclude bydefinition of intersection that �x C .1 � �/y is an element of A \ B, as we wanted to show.

Now we show that the statement ”h A convex and B convex i ) A [ B convex” is false.

In general, when you have to show that a statement is false, you just need to provide a coun-terexample (i.e., an exception to the proposed statement). Sometimes counterexamples are easyto find, but sometimes they are really hard. In our case, finding a counterexample is trivial.

Let x and y be two real numbers such that x ¤ y. The sets fxg and fyg are obviously convexsubsets of R. However, their union fxg [ fyg D fx; yg is not. Indeed, for any � 2 .0; 1/ we have�x C .1 � �/y 62 fx; yg.

For n � 2, we observe that fxg �Rn�1 and fyg �Rn�1 are convex subsets of Rn, while their union�fxg � Rn�1

�[�fyg � Rn�1

�D fx; yg � Rn�1 is not for an analogous argument as the one given

above. �

21

(b) Consider an economy in which there are two goods, 1 and 2, whose prices are p1 > 0 and p2 > 0,respectively. The two goods can only be consumed in non-negative amounts x1 and x2. Thebudget set of a consumer with income I > 0 is defined as

B.p1; p2; I / WD˚.x1; x2/ 2 R2C W p1x1 C p2x2 � I

:

Show that B.p1; p2; I / is a convex subset of R2C.

Solution

Fix two arbitrary elements x WD .x1; x2/ and y WD .y1; y2/ of B.p1; p2; I / and an arbitrary � inŒ0; 1�. To show that B.p1; p2; I / is convex, we need to show that �.x1; x2/ C .1 � �/.y1; y2/ D.�x1C.1��/y1; �x2C.1��/y2/ is an element of B.p1; p2; I /. Given the definition of B.p1; p2; I /,this amounts to prove that the following three inequalities are satisfied

�x1 C .1 � �/y1 � 0;

�x2 C .1 � �/y2 � 0;

p1Œ�x1 C .1 � �/y1�C p2Œ�x2 C .1 � �/y2� � I:

For the first inequality, observe that since .x1; x2/ and .y1; y2/ are elements of B.p1; p2; I /, x1 � 0and y1 � 0. Since � 2 Œ0; 1�, so that also .1 � �/ 2 Œ0; 1�, it immediately follows that �x1 C .1 ��/y1 � 0. The second inequality is established analogously. For the third inequality, observe that

p1Œ�x1 C .1 � �/y1�C p2Œ�x2 C .1 � �/y2� D p1�x1 C p1.1 � �/y1 C p2�x2 C p2.1 � �/y2

D �.p1x1 C p2x2/C .1 � �/.p1y1 C p2y2/

� �I C .1 � �/I

D I;

where the inequality holds because .x1; x2/ and .y1; y2/ are elements of B.p1; p2; I /, and so p1x1Cp2x2 � I and p1y1 C p2y2 � I . This completes the proof.

Let S be a convex subset of Rn. A function f WS ! R, x 7! f .x/ is said to be concave if for allx; y 2 S and all � 2 Œ0; 1�, we have

f .�x C .1 � �/y/ � �f .x/C .1 � �/f .y/:

Let S be a convex subset of Rn. A function f WS ! R, x 7! f .x/ is said to be quasi-concave if forall x; y 2 S and all � 2 Œ0; 1�, we have15

f .�x C .1 � �/y/ � min ff .x/; f .y/g : �

(c) Give a graphical example of a concave function f WS ! R, x 7! f .x/, where S is a convex subsetof R. Clearly illustrate the role of the convex combination �xC.1��/y of x and y for all � 2 Œ0; 1�.Is this function also quasi-concave?

(d) Now draw a function f WS ! R, x 7! f .x/, where S � R and convex, which is quasi-concave, butfails to be concave. What is the difference between concave and quasi-concave functions?

15Quiz (OPTIONAL): Do you see why, when defining concave or quasi-concave functions, we require their domain (i.e.,the set S in the current definition) to be a convex set?

22

(e) Argue formally that every concave function is also quasi-concave. (Note: The example you pro-vided in (d) shows that the converse statement is false.)

Solution

Let S be a convex subset of Rn and f WS ! R, x 7! f .x/ be a concave function on S . Fix twoarbitrary elements x and y of S and an arbitrary � 2 Œ0; 1�. Then, we have

f .�x C .1 � �/y/ � �f .x/C .1 � �/f .y/

� �min ff .x/; f .y/g C .1 � �/min ff .x/; f .y/g

D min ff .x/; f .y/g ;

where the first inequality holds because f is a concave function, and the second inequality holdsbecause f .x/ � min ff .x/; f .y/g, f .y/ � min ff .x/; f .y/g and �; 1 � � � 0. Therefore, f isquasi-concave. �

23

ECON 121: Intermediate MicroeconomicsSolutions to Problem Set 6

Niccolo Lomys

Spring 2016

Problem 1

Consider an exchange economy in which there are only two agents, Robinson and Friday, and twogoods, 1 and 2. Denote with

�wR1 ; w

R2

�2 R2C and

�wF1 ; w

F2

�2 R2C the initial endowments of Robinson

and Friday, respectively. The preferences of the two agents are described by the real-valued utilityfunctions

uR�xR1 ; x

R2

�WD 2

qxR1 C x

R2 ;

and

uF�xF1 ; x

F2

�WD xF1 C 2

qxF2 ;

defined on R2C.

Q.1 Compute the demand functions of Robinson and Friday. [Note: Assume that the budget con-straint holds with equality, that the solution is interior and that the second order conditionsare satisfied. Moreover, assume that prices are strictly positive.]

Solution

Robinson’s utility maximization problem is

max.xR

1 ;xR2 /2R

2C

2

qxR1 C x

R2

s.t. p1xR1 C p2x

R2 D p1w

R1 C p2w

R2 :

The Lagrangian for this constrained maximization problem is

L�xR1 ; x

R2 ; �

�D 2

qxR1 C x

R2 C �

�p1x

R1 C p2x

R2 � p1w

R1 � p2w

R2

�;

where � is the Lagrange multiplier. The first order (necessary) conditions for an interior solution tothe problem are

@L

@xR1D 0”

1qxR1

C �p1 D 0; (1)

1

@L

@xR2D 0” 1C �p2 D 0; (2)

and@L

@�D 0” p1x

R1 C p2x

R2 � p1w

R1 � p2w

R2 D 0: (3)

Equation (2) implies that � D � 1p2

(provided p2 > 0). Plugging this into Equation (1) gives us thethe demand function for good 1:

xR1�p1; p2; w

R1 ; w

R2

�D

�p2

p1

�2:

Replacing xR1�p1; p2; w

R1 ; w

R2

�for xR1 in Equation (3), we obtain the demand function for good 2:

xR2�p1; p2; w

R1 ; w

R2

�DwR1 p1

p2C wR2 �

p2

p1:

Friday’s utility maximization problem is

max.xF

1 ;xF2 /2R

2C

xF1 C 2

qxF2

s.t. p1xF1 C p2x

F2 D p1w

F1 C p2w

F2 :

The Lagrangian associated to this constrained maximization problem is

L�xF1 ; x

F2 ; �

�D xF1 C 2

qxF2 C �

�p1x

F1 C p2x

F2 � p1w

F1 � p2w

F2

�;

where � is the Lagrange multiplier. The first order (necessary) conditions for an interior solution tothe problem are:

@L

@xF1D 0” 1C �p1 D 0; (4)

@L

@xF2D 0”

1qxF2

C �p2 D 0; (5)

and@L

@�D 0” p1x

F1 C p2x

F2 � p1w

F1 � p2w

F2 D 0: (6)

Equation (4) implies that � D � 1p1

(provided p1 > 0). Plugging this into Equation (5) gives us thedemand function for good 2:

xF2�p1; p2; w

F1 ; w

F2

�D

�p1

p2

�2:

Replacing xF1�p1; p2; w

F1 ; w

F2

�for xF1 in Equation (6), we obtain the demand function for good 1:

xF1�p1; p2; w

F1 ; w

F2

�DwF2 p2

p1C wF1 �

p1

p2: �

2

Q.2 Assume that Robinson’s endowment is�wR1 ; w

R2

�D .2; 0/, and Friday’s endowment is

�wF1 ; w

F2

�D

.0; 2/. Compute the competitive equilibrium price and allocation. To do so, formulate themarket-clearing conditions. These may look complicated. To proceed, remember that: (a) byWalras’ law, it is enough to restrict attention to one market clearing condition: (b) you canchoose one good (either one) to be the numeraire and set its price equal to 1.

Solution

The market clearing condition for good 1 requires the aggregate demand for good 1 to equal theaggregate endowment of good 1. That is,

xR1 C xF1 D w

R1 C w

F1 :

Similarly, for good 2xR2 C x

F2 D w

R2 C w

F2 :

By Walras’ law, in an economy with n goods, it suffices to consider only n � 1 market clearingconditions (as the condition for the n-th market would be redundant). Consider good 1. Given thedemand functions derived above, the market clearing condition for good 1 in this economy is�

p2

p1

�2CwF2 p2

p1C wF1 �

p1

p2D wR1 C w

F1 :

Given endowments�wR1 ; w

R2

�D .2; 0/ and

�wF1 ; w

F2

�D .0; 2/, the market clearing condition becomes�

p2

p1

�2C2p2

p1�p1

p2D 2:

Since we are always free to choose the price of one good (the numeraire), let’s set p1 D 1. Theprevious equation thus becomes

p32 C 2p22 � 2p2 D 1;

which is satisfied for p2 D 1. We conclude that a competitive equilibrium price vector is .p1; p2/ D.1; 1/. Notice that p1 > 0 and p2 > 0, as we assumed.1

Substituting the values for prices and endowments in the demand functions, we obtain the com-petitive equilibrium allocation: �

xR1 ; xR2

�D .1; 1/�

xF1 ; xF2

�D .1; 1/: �

Q.3 Compare the utility of each Robinson and Friday at the competitive equilibrium to that whichthey would receive if they did not trade with one another. Comment your finding (one line).

Solution

Notice thatUR.1; 1/ D 3 > 2

p2 D UR.2; 0/:

1Any price vector .p1; p2/ such that p1 D p2 is a competitive equilibrium price vector for this economy.

3

Similarly,U F .1; 1/ D 3 > 2

p2 D U F .0; 2/:

Hence, both Robinson and Friday achieve a higher utility than that they would attain if they did nottrade and instead consumed their endowments. All gains from trade exhausted. �

Q.4 Let us now consider an application. Suppose that we think of Robinson and Friday as twocountries. Let their utility functions be the same as before, but let their initial endowments be�wR1 ; w

R2

�D .3; 9/ and

�wF1 ; w

F2

�D .1; 3/. Hence, we can think of Robinson as a relatively rich

country and Friday as a relatively poor country, in the sense that Robinson has more of bothgoods. Find the competitive equilibrium. [Note: You don’t have to start over from scratch todo this. You can simply repeat your derivations from parts Q.1 and Q.2, once again makingthe guess that p1 D p2 and inserting the new values for the endowments.] If the rich countryhas more of everything, why are there gains from trade? One sometimes hears calls for the U.S.to curtail its foreign trade. Thinking of the U.S. as Robinson, what effect would this have inthis model? Given your answer, why would people suggest such curtailment or, equivalently,what would such advocates think of as missing from our model?

Solution

Robinson’s and Friday’s endowments are now�wR1 ; w

R2

�D .3; 9/ and

�wF1 ; w

F2

�D .1; 3/. Substituting

the new values for the endowments in the market clearing condition for good 1, we obtain�p2

p1

�2C3p2

p1�p1

p2D 3;

which is again satisfied at any price vector .p1; p2/ such that p1 D p2. Therefore, any price vectorsatisfying p1 D p2 is a competitive equilibrium price vector.

Plugging the values for prices and the new endowments in the demand functions, we obtain thecompetitive equilibrium allocation: �

xR1 ; xR2

�D .1; 11/;

and �xF1 ; x

F2

�D .3; 1/:

Notice thatUR.1; 11/ D 13 > 2

p3C 9 D UR.3; 9/;

andU F .3; 1/ D 5 > 1C 2

p3 D U F .1; 3/:

That is, both the rich country and the poor country are better off with trade (gains from trade).Trade makes the rich country better off, even though such country already has more of both goods,because it allows the country to obtain the optimal mix of goods (in this case, consuming more ofgood 2 than the endowment would allow).

If we think of the U.S. as the rich country in this model, then a policy that curtailed foreign tradewould make the U.S. worse off. Note that our framework models an exchange economy and abstractsaway from production. Advocates of such curtailment policy likely have in mind a framework where

4

the U.S. and poor countries produce and trade goods, which involves additional implications notcaptured by our model. �

Problem 2

Solve the following questions.

Q.1 Consider the following Edgeworth box economy. There are two consumers, A, and B, and twogoods, 1 and 2. The two consumers have identical preferences represented by the real-valuedutility functions

uj�xj1 ; x

j2

�WD min

nxj1 ; x

j2

o;

j D A;B, defined on R2C. There are 20 units of good 1 and 10 units of good 2. Characterizethe set of Pareto efficient allocations.

Solution

Pareto efficient allocations��xA1 ; x

A2

�;�xB1 ; x

B2

��are characterized by the following conditions:

xA1 � xA2 ; xB1 � x

B2 ; xA1 C x

B1 � 20; xA2 C x

B2 D 10: �

Q.2 Consider an exchange economy with two consumers, A and B, and two goods, 1 and 2. Thetwo consumers have identical preferences represented by the utility functions

uj�xj1 ; x

j2

�WD x

j1x

;2

j D A;B, defined on R2C. for j D A;B. Their initial endowments are�wA1 ; w

A2

�D .1; 1/ and�

wB1 ; wB2

�D .1; 3/. Compute the competitive equilibrium price and allocation.

Solution

As only relative prices matter, we normalize p2 D 1. Consumer A’s utility maximization problem is

max.xA

1 ;xA2 /2R

2C

xA1 xA2

s.t. p1xA1 C x

A2 D p1 C 1:

Hence, consumer A’s demands are

xA1 Dp1 C 1

2p1;

and

xA2 Dp1 C 1

2:

Similarly, we find consumer B’s demands:

xB1 Dp1 C 3

2p1;

5

and

xB2 Dp1 C 3

2:

Market clearing conditions require that

xA1 C xB1 D w

A1 D w

B1 :

Given A’s and B’s demand functions and initial endowments for good 1, the market clearing conditionfor good 1 becomes

p1 C 1

2p1Cp1 C 3

2p1D 2;

i.e.2p1 C 4

2p1D 2;

i.e. p1 D 2.

We conclude that any competitive equilibrium price vector .p1; p2/ satisfies p1

p2D 2, and that the

competitive equilibrium allocation is��xA1 ; x

A2

�;�xB1 ; x

B2

��D

��3

4;3

2

�;

�5

4;5

2

��: �

Q.3 An exchange economy has three consumers (A, B and C ), and three goods (1, 2 and 3).Consumers’ utility functions and initial endowments are as follows:

uA�xA1 ; x

A2 ; x

A3

�WD min

˚xA1 ; x

A2

;

�wA1 ; w

A2 ; w

A3

�D .1; 0; 0/I

uB�xB1 ; x

B2 ; x

B3

�WD min

˚xB2 ; x

B3

;

�wB1 ; w

B2 ; w

B3

�D .0; 1; 0/I

uC�xC1 ; x

C2 ; x

C3

�WD min

˚xC1 ; x

C3

;

�wC1 ; w

C2 ; w

C3

�D .0; 0; 1/:

Find the competitive equilibrium price and allocation.

Solution

Since only relative prices matter, let’s normalize p1 D 1. Consumer A’s demand satisfies

xA1 D xA2 and xA1 C p2x

A2 D 1:

Hence,

xA1 D xA2 D

1

1C p2:

Similarly, for consumer B and consumer C we obtain˝xB2 D x

B3 and p2x

B2 C p3x

B3 D p2

˛H) xB2 D x

B3 D

p2

p2 C p3;

and ˝xC1 D x

C3 and xC1 C p3x

C3 D p3

˛H) xC1 D x

C3 D

p3

1C p3:

6

Together with the market clearing conditions, we have(1

1Cp2C

p3

1Cp3D 1

11Cp2

Cp2

p2Cp3D 1

;

which implies that p2 D p3 D 1.

Hence, a competitive equilibrium price vector is .1; 1; 1/, and the competitive equilibrium alloca-tion is��

xA1 ; xA2 ; x

A3

�;�xB1 ; x

B2 ; x

B3

�;�xC1 ; x

C2 ; x

C3

��D

��1

2;1

2; 0

�;

�0;1

2;1

2

�;

�1

2; 0;

1

2

��: �

Problem 3

Consider an exchange economy in which there are two agents, A and B, and two goods, 1 and 2.The two agents have preferences described by the real-valued utility functions

uA�xA1 ; x

A2

�WD ˛ ln xA1 C .1 � ˛/ ln xA2 ;

anduB�xB1 ; x

B2

�WD .1 � ˛/ ln xB1 C ˛ ln xB2 ;

where ˛ 2 .1=2; 1/. Suppose that consumers start with initial endowments�wA1 ; w

A2

�D�wB1 ; w

B2

�D

.1; 1/. The agents have agreed to share their resources. They have also agreed that the weight thatA receives in the economy is A 2 .0; 1/, and the weight that B receives is B WD 1 � A. [Note:For this problem, feel free to be loose and ignore the fact that the natural logarithm function is notdefined at zero.]

Q.1 For every weight A 2 .0; 1/, find the allocation which would maximize the social surplusgiven the weights; that is, we are interested in finding the allocation

��xA1 ; x

A2

�;�xB1 ; x

B2

��which

maximizes the sum Au

A�xA1 ; x

A2

�C�1 � A

�uB�xB1 ; x

B2

�subject to the resource constraints of the economy.

Solution

The relevant constrained maximization problem is given by

max.xA

1 ;xA2 ;x

B1 ;x

B2 /2R

4C

A�˛ ln xA1 C .1 � ˛/ ln xA2

�C�1 � A

��.1 � ˛/ ln xB1 C ˛ ln xB2

�s.t. xA1 C x

B1 D 2

xA2 C xB2 D 2:

The first order conditions (after eliminating the Lagrange multipliers) are

A˛xB1 D .1 � A/ .1 � ˛/ x

A1 ;

A .1 � ˛/ xB2 D .1 � A/ ˛x

A2 ;

7

xA1 C xB1 D 2;

xA2 C xB2 D 2:

Solve this system of equations to obtain

xA1 D2 A˛

A˛ C .1 � A/ .1 � ˛/; (7)

xA2 D2 A .1 � ˛/

A.1 � ˛/C .1 � A/˛; (8)

xB1 D2.1 � A/.1 � ˛/

A˛ C .1 � A/.1 � ˛/; (9)

xB2 D2.1 � A/˛

A.1 � ˛/C .1 � A/˛: � (10)

Q.2 For every weight A 2 .0; 1/, find initial endowments of goods 1 and 2 among agents A and Band a pair of prices such that the Pareto efficient allocation actually constitutes an equilibriumof the market. [Here we decentralize the Pareto efficient allocation via a market equilibrium.]

Solution

For every weight A 2 .0; 1/, let the initial endowment be given by the system of Pareto efficientallocations in (7)-(10). Normalize the price of good 1 to p1 D 1. We need to find the price of good2 that together with endowments given by (7)-(10) will constitute a competitive equilibrium.

Since (7)-(10) is a Pareto efficient allocation, the marginal rate of substitution between good 1and good 2 are the same for agent A and agent B (verify by yourself that this is actually true). Next,recall that, at a competitive equilibrium, the marginal rate of substitution equals the price ratio. Wewill use this property to recover the price for good 2:

MRSB2;1�xB1 ; x

B2

�D

p2

p1

”˛xB1

.1 � ˛/xB2D p2

”˛ 2.1� A/.1�˛/

A˛C.1� A/.1�˛/

.1 � ˛/ 2.1� A/˛

A.1�˛/C.1� A/˛

D p2

” A.1 � ˛/C .1 � A/˛

A˛ C .1 � A/.1 � ˛/D p2:

Thus, the price vector

.p1; p2/ D

�1; A.1 � ˛/C .1 � A/˛

A˛ C .1 � A/.1 � ˛/

�and the allocation given by (7)-(10) constitute a competitive equilibrium in an economy where initialendowments are given by (7)-(10). It will be just a simple exchange economy where everybodyconsumes his own endowment at the equilibrium prices. �

8