Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10

Dirk Bergemann

Department of Economics Yale University

Solutions by Olga Timoshenko

Economics 121b: Intermediate Microeconomics Problem Set 2

1/20/10

This problem set is due on Wednesday, 1/27/10.

Preliminary remarks

The problem set is graded out of 10 points: Q1 - 2pts, Q2 - 1 pt, Q3 - 2pts, Q4 - 1 pt, Q5 - 3 pts, Q6 - 1 pt.

A typical mistake in Q5 and Q6 is failure to check second order conditions (SOC) for local maxima (the second derivative is negative) or minima (the second derivative is positive). Next, please pay attention to the graph of the indifference curves of the utility function u 2 in Q3: the indifference curves never touch the y-axes (otherwise you are saying that x ₁ can take zero value).

1. A consumer’s preferences are represented by the utility function u(x, y) = lnx + 2lny.

(a) Which of the two bundles (x _A , y _A ) = (1, 4) or (x _B , y _B ) = (4, 1) does the consumer prefer?

Solution

Recall that for any two bundles Z and Z ^′ the following equiva- lence holds

Z ≽ Z ^′ ⇔ u(Z) ≥ u(Z ^′ ) Calculate utility from each bundle.

u(A) = u(1, 4) = ln 1 + 2 ln 4 = 0 + 2 ∗ 1.4 = 2.8

u(B) = u(4, 1) = ln 4 + 2 ln 1 = 1.4 + 2 ∗ 0 = 1.4

Thus, since u(A) > u(B) the consumer prefers bundle A to bundle B.

(b) Take as given for now that this utility function represents a con- sumer with convex preferences. Use this information and your an- swer to part (a) to determine which of the two bundles (x _C , y _C ) = (2.5, 2.5) or (x _B , y _B ) = (4, 1) the consumer prefers. Verify your answer.

Solution

Convex preferences imply that for any three bundles X, Y, and Z, if Y ≽ X and Z ≽ X then αY + (1 − α)Z ≽ X.

From the example above take Y to be bundle A, and X to be bundle B. Since preferences are reflexive B ≽ B, thus, we can take Z to be bundle B. Thus, convexity of preferences implies

αA + (1 − α)B ≽ B

Notice that C = 1/2A + 1/2B. Thus C ≽ B. To verify the answer, calculate utility from C

u(C) = u(2.5, 2.5) = ln(2.5) + 2 ln(2.5) = 2.7 > 1.4 = u(B)

(c) Derive an equation for the indifference through the bundle (x B , y B ) = (4, 1).

Solution

A bundle Z with consumptions given by (x Z , y Z ) is indifferent to the given bundle B when it yields the same utility as bundle B.

Utility that bundle Z yields

u(Z) = u(x _Z , y _Z ) = ln x _Z + 2 ln y _Z

should be equal to u(B) = 1.4. Thus, an equation for the indif-

ference through the bundle (x _B , y _B ) = (4, 1) is

ln x _Z + 2 ln y _Z = ln 4 ln x Z y _Z ² = ln 4

x Z y _Z ² = exp ln 4 y _Z ² = 4

x Z

y Z = 2

x ^1/2 _Z

(d) Derive an equation for the marginal rate of substitution between x and y for an individual with these preferences. Interpret the M RS. What is the M RS at the point (x _C , y _C ) = (2.5, 2.5)?

M RS(x, y) = − M U _x M U y

= − 1/x 2/y = − y

2x M RS(2.5, 2.5) = − 2.5

2 ^. 2.5 = − 1 2

MRS is the rate at which two goods should be exchanges to keep the overall utility constant. For example, at a point (2.5 ; 2.5) 1 unit of good x should be exchanged for 1/2 units of good y.

2. John thinks that margarine is just as good as butter. Margarine is sold in 8oz packages while butter is sold in 16 oz packages.

(a) Draw John’s indifference curves for packages of margarine and butter.

Margarine and butter are perfect substituted for John. Since margarine is sold in 8oz packages and butter is sold in 16 oz packages, one package of butter delivers twice as much utility as one package of margarine. Thus, John’s utility is given by

u(b, m) = m + 2b

where m is the number of packages of margarine, and b is the

number of packages of butter. The following figure depicts indif-

ference curves in the (Margarine, Butter) plane

Butter

Margarine

5 10 15 20 25 30 35 40 45 50

5 10 15 20 25 30 35 40 45 50

(b) A package of margarine costs $1. A package of butter costs $1.50.

How will John spend his money if he has $6 dollars to spend on margarine or butter?

With the utility function specified above, the marginal rate of substitution between margarine and butter is 2, while the price ratio _p ^p

m

= 3/2 < 2. Since the market exchanges butter for mar- garine at a rate that is smaller than John is willing to exchange the two goods, John will maximize his utility by spending all of his money on butter. He will buy _$1.50 ^$6 = 4 packages of butter.

(Some more intuition: If John gives up one package of butter, he will have $1.50 to spend on margarine, and will get 1.5 packages of margarine. But to be as well of as before the exchange John will have to receive 2 packages of margarine. Thus, such exchange will make him worse off, and John will not give up any number of packages of butter to exchange for margarine: He will consume butter only.)

3. Consider the following utility functions u ₁ (x ₁ , x ₂ ) = 3x ² ₁ x ² ₂ , u 2 (x 1 , x 2 ) = ln x 1 + x 2

u 3 (x 1 , x 2 ) = ln x 1 + ln x 2

u ₄ (x ₁ , x ₂ ) = min {2x 1 , x ₂ } . (a) For each of these utility functions:

i. find the marginal utility of each good M U _i . Are the prefer- ences monotone?

ii. find the marginal rate of substitution M RS.

iii. Define an indifference curve. Show that each indifference curve (for some positive level of utility) is decreasing and convex.

iv. Plot a few indifference curves.

Solution

• u 1 (x 1 , x 2 ) = 3x ² ₁ x ² ₂

M U ₁ = ∂u 1

∂x ₁ = 6x ₁ x ² ₂ M U ₂ = ∂u ₁

∂x ₂ = 6x ² ₁ x ₂ M RS = − M U 1

M U ₂ = − x 2

x ₁

Preferences described by this utility function are mono- tone since the utility function is increasing in both of its arguments, x 1 and x 2 . An increase in the number of each good in a bundle will increase utility, and therefore a bundle with more of both goods will be preferred to a bundle with less of each good.

An indifference curve is a set of points (x ₁ , x ₂ ) that yield the same level of utility u. Thus, an indifference curve is described by the following equation

3x ² ₁ x ² ₂ = u Equivalently

x 2 = C x ₁ where C = ( _u

3

)

2

To show that an indifference curve is decreasing we need to compute its first derivative ^dx _dx

1

and show that it is negative.

dx ₂ dx 1

= − C

(x 1 ) ² < 0

since C > 0 and (x 1 ) ² > 0. Thus, an indifference curve is decreasing.

To show that an indifference curve is convex we need to compute its second derivative _d(x ^d

^x

1

)

and show that it is positive.

d ² x 2

d(x 1 ) ² = C 2(x 1 ) ³ > 0

for x ₁ > 0. Thus, the indifference curve is convex.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• u 2 (x 1 , x 2 ) = ln x 1 + x 2

M U ₁ = 1 x 1

M U ₂ = 1

M RS = 1

x ₁

These preferences are monotone by the same argument as above.

An indifference curve is described by

ln x 1 + x 2 = u

Equivalently

x 2 = u − ln x 1

dx 2

dx 1

= − 1 x 1

< 0 for x 1 > 0

Thus, the indifference curve is decreasing.

d ² x 2

d(x 1 ) ² = 1 (x 1 ) ² > 0 Thus, the indifference curve is convex.

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

• u 3 (x ₁ , x ₂ ) = ln x ₁ + ln x ₂

M U ₁ = 1 x 1

M U 2 = 1 x 2

M RS = x ₂ x 1

These preferences are monotone by the same argument

as above.

An indifference curve is described by

ln x ₁ + ln x ₂ = u

ln x 2 = u − ln x 1

exp(ln x 2 ) = exp(u − ln x 1 ) x ₂ = exp(u)

exp(ln x ₁ ) x 2 = C

x 1

where C = exp u

The indifference curve is x 2 = _x ^C

1

. Observe the similarity with the first utility function.

dx ₂ dx 1

= − C

(x 1 ) ² < 0

since C > 0 and (x 1 ) ² > 0. Thus, an indifference curve is decreasing.

d ² x ₂

d(x ₁ ) ² = C 2(x ₁ ) ³ > 0

for x ₁ > 0. Thus, the indifference curve is convex.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• u 4 (x 1 , x 2 ) = min {2x 1 , x 2 }

First, we observe that function u ₄ is not differentiable at a point (t, 2t) for any t > 0.

M U ₁ = 0 if 2x ₁ > x ₂ 2 if 2x ₁ < x ₂ undefined if 2x 1 = x 2

M U 2 = 1 if 2x 1 > x 2

0 if 2x ₁ < x ₂ undefined if 2x ₁ = x ₂ M RS = 0 if 2x ₁ > x ₂

−∞ if 2x 1 < x 2

undefined if 2x 1 = x 2

These preferences are monotone by the same argument as above.

An indifference curve is described by

min {2x 1 , x 2 } = u Equivalently

x 2 = u if 2x 1 > x 2

x ₁ = u

2 if 2x ₁ ≤ x 2

In the case of Leontief preferences we need to adhere to the graph of the indifference curve to determine whether an indifference curve is decreasing. On the segment 2x ₁ >

x 2 the value of x 2 is constant and equals to u. On the segment 2x ₁ < x ₂ , x ₂ can take any value that is greater or equal to u. Thus, an indifference curve is decreasing.

An indifference curve is convex since its upper contour set

(a set of points that lie above the graph of an indifference

curve) is a convex set.

0 1 2 3 4 5 6 7 8 9 10 0

1 2 3 4 5 6 7 8 9 10

(b) For the utility function u 3 (x 1 , x 2 ) can you find a utility function that represent the same preferences (but as function of x ₁ and x ₂ simply represents a product of x ₁ and x ₂ )? Find the relevant monotone transformation f (u).

Solution

A monotone transformation of a given utility function will yield a new utility function that represents the same preferences.

Confider the following monotone transformation f (u) = exp(u)

We obtain

u ^new (x 1 , x 2 ) = exp(u 3 ) =

= exp(ln(x 1 ) + ln(x 2 )) =

= exp(ln(x ₁ x ₂ )) =

= x ₁ x ₂

Thus, utility u ^new (x 1 , x 2 ) = x 1 x 2 represents the same preferences as utility function u 3 (x 1 , x 2 ) = ln(x 1 ) + ln(x 2 ). The relevant monotone transformation is f (u) = exp(u).

4. Suppose a consumer always consumes 3 teaspoons of jam with each

bagel. Write a utility function that would represent her preferences,

define 1 teaspoon as the unit of jam and 1 bagel as the unit of bread.

If the price of jam is p ₁ per teaspoonful and the price of bread is p ₂ per bagel and the consumer has m dollars to spend on bread and jam, how much will he or she want to purchase?

Solution

Since the consumer likes to consume jam and bagel in fixed propor- tions, those two goods are perfect complements to him. The utility function that represents his preferences is given by

u(x _b , x _j ) = min {3x b , x _j }

where x b is the consumption of bagels and x j is the consumption of jam.

Since the goods are perfect complements, at the optimum they will be consumed in the fixed proportion given by the utility function. In this example 3x _b = x j . If the equality is not satisfied, for example the consumer chooses a bundle where 3x _b < x _j , he can give up some of the consumption of jam without effecting the utility. He can then use that money to buy some more bagels and strictly improve his utility.

Thus at the original point where 3x _b < x _j he was not maximizing his utility.

Now, that we have established that at the optimum 3x _b = x _j , we will use that equality and the budget line to find the optimal consumption allocation. The two equations are

3x _b = x j

x b p b + x j p j = m

⇒ x _b p _b + 3x _b p _j = m

⇒

x _b = m

p _b + 3p _j

x j = 3m

p _b + 3p j

5. Consider the real-valued function f defined over the interval [ −3, 3] by f (x) =

{ 4 − (x + 2) ² , if x ≤ −1 x ² (2 − x) , if x ≥ −1 (a) Sketch a rough graph of the function.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3

−10

−8.5

−7

−5.5

−4

−2.5

−1 0.5 2 3.5 5

(b) Find all critical points (i.e., points with zero first derivative) of the function. Identify the interior local maxima and minima.

Solution

All critical points satisfy the following first order conditions { −2(x + 2) = 0 if x ≤ −1

4x − 3x ² = 0 if x ≥ −1 { ⇔

−2x = 4 if x ≤ −1 x(4 − 3x) = 0 if x ≥ −1

 ⇔





x = −2 if x ≤ −1 x = 0 if x ≥ −1 x = 4/3 if x ≥ −1

Thus, the set of interior critical points of the function is {−2, 0, 4/3}.

To determine local maxima and minima we will calculate the sign

of the second derivative at each of those points.

d ² x

dx ² (x = −2) = −2 < 0 d ² x

dx ² (x = 0) = 4 − 6x = 4 > 0 d ² x

dx ² (x = 4/3) = 4 − 6x = −4 < 0

Thus, the set of local interior maxima is {−2, 4/3}; the set of local interior minima is {0}.

(c) Is the function discontinuous anywhere? Is the function non- differentiable anywhere? If your answer to either of these ques- tions is yes, is there a local maximum or minimum at these points?

Solution

The function is continuous on its domain since both pieces are continuous functions, and at the point x = −1 they take the same value of 3.

The function is not differential at the point x = −1 since the left derivative (f ₋ ^′ ( −1) = −2) is not equal to the right derivative (f ₊ ^′ (−1) = −7) at that point. There is no local maximum or minimum at that point.

(d) Does the function have any local maxima or minima at its end- points?

Solution

The function does not have any local maxima or minima at its end-points by the definition of a local extremum point. (x ^∗ is a local maxima (minima) if ∃ϵ > 0 s.t ∀x s.t. |x − x ^∗ | < ϵ f(x) <

f (x ^∗ ) (f (x) > f (x ^∗ )). Since ϵ-ball is not defined around boundary points, those are not considered local extremum points.)

(e) Find its global maximum and minimum.

Solution

Candidate points for the global maximum are local interior max-

ima and boundary points. Check the value of the function at

those points.

f (−2) = 4 f (4/3) = 1.2

f (3) = −9 f ( −3) = 3

Thus the global maximum is x = −2.

Candidate points for the global minimum are local interior min- ima and boundary points. Check the value of the function at those points.

f (0) = 0 f (3) = −9 f (−3) = 3

Thus the global minimum is x = 3.

6. In each of the following questions, x and y must be non-negative num- bers:

(a) Maximize xy subject to x + 2y = 12 by substitution;

Solution

max x,y xy

s.t. x + 2y = 12

From the constraint we find x = 12 −2y. Substitute this equation for x into the objective function to obtain the following uncon- strained maximization problem

max y y(12 − 2y)

⇔

max y 12y − 2y ²

The first order condition implies

12 − 4y = 0 4y = 12

y = 3

Verify that y = 3 is the maximizer by checking the sign of the sec- ond derivative being negative. Next, use previously found equa- tion for x to determine its optimal value

x = 12 − 2y = 12 − 2 ^. 3 = 6 Answer: (x, y) = (6, 3).

(b) Minimize x + 2y subject to xy = 18 by substitution.

Solution

x>0,y>0 min x + 2y s.t. xy = 18

From the constraint we find x = ¹⁸ _y . Substitute this equation for x into the objective function to obtain the following unconstrained maximization problem

min y>0 f (y) = 18 y + 2y

The first order condition implies

− 18

y ² + 2 = 0 9

y ² = 1 y ² = 9

y = 3

Verify that you have found a minimizer by showing that the sec- ond derivative at that point is negative. Next, use previously found equation for x to determine its optimal value

x = 18 y = 18

3 = 6 Answer: (x, y) = (6, 3).

Reading Assignments:

NS: Chapters 2,3,4.