Econ 301 Intermediate Microeconomicsmweretka/301_17f/Exams/M1.pdf · Econ 301 Intermediate Microeconomics Prof. Marek Weretka Midterm 1 (A) You have 70 minutes to complete the exam

Econ 301Intermediate MicroeconomicsProf. Marek Weretka

Midterm 1 (A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (40+15+20+25=100points) + bonus (just for fun). Make sure you answer the �rst four questions before working on the bonusone!

Problem 1 (40p) (Well-behaved preferences)Lila spends her income on two goods: food, x1; and clothing, x2.a) The price food is p1= 2 and one piece of clothing costs p2= 4 . Show geometrically Lila�s budget set

if her income is m = 30. Find the relative price food in terms of clothing (one number)? Where can therelative price be seen in the graph of a budget set? (one sentence)

b) Lila�s preferences are represented by utility function

U(x1; x2) = (x1)2(x2)

1:

We know that her preferences can be alternatively represented by function

V (x1; x2) = 2 lnx1+ lnx2:

Explain the idea behind �monotonic transformation�(one sentence) and derive function V from U .c) Assuming utility function V (x1; x2) = 2 lnx1+ lnx2-Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (8; 8) �nd the

value of MRS (one number). Give economic interpretation of MRS (one sentence). Which of the goods ismore valuable given consumption (8; 8)?

-Write down two secrets of happiness that determine optimal choice given parameters p1; p2 and m.Explain economic intuition behind the two conditions (two sentences for each).

-Using secrets of happiness derive optimal consumption x1,x2; given values of p1= 2; p2= 4 andm = 30:Is the solution corner or interior (chose one)

d) (Harder) Using magic formulas for Cobb-Douglass preferences argue that the two commodities are1) ordinary; 2) normal; and 3) neither gross complements nor gross substitutes (one sentence for eachproperty).

Problem 2 (15p) (Perfect substitutes)You are planing a budget for the state of Wisconsin. The two major budget positions include education,

x1 and health care, x2: Your preferences over the two are represented by function

U(x1; x2) = 3x1+x2:

a) Find marginal rate of substitution (give number).b) Find optimal consumption of x1 and x2, given prices p1= 2 and p2= 1 and available funds m = 50

(two numbers).c) Is solution interior? (yes-no answer). Is marginal utility of a dollar equalized? (give two numbers and

yes-no answer)

Problem 3 (20p) (Intertemporal choice)Zoe is a professional Olympic skier. Her income when �young�is high (m1= 30) but her future (period

two) is not so bright (m2= 20)a) Depict Zoe�s budget set, assuming that she can borrow and save at the interest rate r = 100%.

Partition her budget set into three regions: the area that involves saving, borrowing and none of the two.

1

b) Find PV and FV of Zoe�s lifetime income (two numbers) and show the two values in the graph.Interpret economically PV.

c) Zoe�s utility function is U(C1; C2) = lnC1+11+� lnC2 where discount rate is � = 2. Using magic

formulas, �nd optimal consumption plan (C1; C2) (two numbers) and the corresponding level of sav-ings/borrowing, S.

d) Is Zoe tilting her consumption over time? (yes-no answer).

Problem 4 (25p) (Short questions)a) Given utility function U (C;R)=min (2C;R); daily endowment of time 24h, price and wage pc= w = 2; �nd

optimal choice of C; relaxation time R and labor supply L. (three numbers, use secrets of happiness forperfect complements).

b) Find optimal choice given quasilinear preferences U (x1; x2)= x1+100 lnx2; prices p1= $8; p2= $1and income m = $60. Is your solution corner or interior?

c) Assume Interest rate r = 10%. Choose one of the two: a consol (a type of British government bond)that pays annually $1000 forever, starting next year or cash $12; 000 now (compare PV).

d) Your annual income when working (age 21-60) is $60; 000 and then you are are going to live for thenext 30 years. Write down equation that determines constant (maximal) level of consumption during yourlifetime. Assume annual interest rate r = 2%:

Bonus question (Just for fun)a) Derive magic formulas for perfect complements U (x1; x2)=min (ax1; bx2) that give optimal choices

x1; x2 as a function of a; b; p1; p2;m:b) Provide economic intuition for the magic formula for perfect complements (economic interpretation

for the numerator and denominator).

2


Midterm 1 (B)





U(x1; x2) = (x1)2(x2)

1:


V (x1; x2) = 2 lnx1+ lnx2:

Explain the idea behind �monotonic transformation�(one sentence) and derive function V from U .c) Assuming utility function V (x1; x2) = 2 lnx1+ lnx2-Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2; 2) �nd the







U(x1; x2) = x1+x2:



yes-no answer)




3





Problem 4 (25p) (Short questions)a) Given utility function U (C;R)=min (2C;R); daily endowment of time 24h, price and wage pc= w = 1; �nd



c) Assume Interest rate r = 10%. Choose one of the two: a consol (a type of British government bond)that pays annually $1000; starting next year or cash $8; 000 now (compare PV).





4


Midterm 1 (C)





U(x1; x2) = (x1)1(x2)

2: (1)


V (x1; x2) = lnx1+2 lnx2: (2)

Explain the idea behind �monotonic transformation�(one sentence) and derive function V from U .c) Assuming utility function V (x1; x2) = lnx1+2 lnx2-Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2; 2) �nd the







U(x1; x2) = 2x1+2x2:



yes-no answer)




5





Problem 4 (25p) (Short questions)a) Given utility function U (C;R)=min (C; 2R); daily endowment of time 24h, price and wage pc= w = 1; �nd



c) Assume Interest rate r = 10%. Choose one of the two: a consol (a type of British government bond)that pays annually $500 starting next year or cash $4; 000 now (compare PV).





6


Midterm 1 (D)





U(x1; x2) = (x1)1(x2)

2: (3)


V (x1; x2) = lnx1+2 lnx2: (4)

Explain the idea behind �monotonic transformation�(one sentence) and derive function V from U .c) Assuming utility function V (x1; x2) = lnx1+2 lnx2-Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2; 2) �nd the







U(x1; x2) = 3x1+3x2:



yes-no answer)




7





Problem 4 (25p) (Short questions)a) Given utility function U (C;R)=min (C; 2R); daily endowment of time 24h, price and wage pc= w = 2; �nd



c) Assume Interest rate r = 10%. Choose one of the two: Choose one of the two: a consol (a type ofBritish government bond) that pays annually $100; starting next year or $1; 200 now (compare PV).





8

Midterm 1 Solution for Version A

[Econ 301] Spring 2014

Problem 1 (40p) (Well-behaved preferences)

Lila spends her income on two goods: food, x1, and clothing, x2.

a) 6pts The price food is p1 = 2 and one piece of clothing costs p2 = 4. Showgeometrically Lila’s budget set if her income is m = 30. Find the relativeprice food in terms of clothing (one number)? Where can the relative pricebe seen in the graph of a budget set? (one sentence)

Ans) The relative price food in terms of clothing is p1p2

= 24 = 1

2 . Absolute valueof slope of budget line is the relative price.

b) 7pts Lila’s preferences are represented by utility function

U(x1, x2) = (x1)2(x2)1


V (x1, x2) = 2 lnx1 + lnx2

Explain the idea behind ”monotonic transformation”(one sentence) andderive function V from U.

Ans) Taking logarithm of U becomes V . Logarithm is a positive monotonetransformation, and monotonic transformation of utility function does notdistort original preference order. That means, whenever U(a, b) > U(c, d),it is true that V (a, b) > V (c, d) and vice versa.

V (x1, x2) = 2 lnx1 + lnx2 = lnx21 + lnx2 = ln (x1)2(x2)1 = lnU(x1, x2)

1

c) 15pts Assuming utility function V (x1, x2) = 2 lnx1 + lnx2

- Find marginal rate of substitution (MRS) for all bundles (derive for-mula). For bundle (8, 8) find the value of MRS (one number). Giveeconomic interpretation of MRS (one sentence). Which of the goodsis more valuable given consumption (8, 8)?

Ans) MRS12 = −MU1

MU2= − 2x2

x1. At (8, 8), MRS = − 16

8 = −2. MRS12

means the amount of x2 which consumer is willing to give up inorder to get additional unit of x1 while holding same utility level. x1is more (twice) valuable than x2.

- Write down two secrets of happiness that determine optimal choicegiven parameters p1,p2 and m. Explain economic intuition behindthe two conditions (two sentences for each).

Ans)

x1p1 + x2p2 = m

MU1

P1=MU2

P2or MRS12 = −P1

P2

The first equation is a budget constraint which requires consumersto spend all of their income on the consumption of two goods, x1and x2. The second condition is about efficient combination of twogoods. Mathematically, the slope of the indifference curve passingthrough the optimal consumption bundle to be tangent to the priceratio. Economically, marginal utility per dollar of each good shouldbecome equal at the optimal consumption bundle, or consumer’s rel-ative evaluation about two goods in terms of marginal utility shouldcoincide market’s relative evaluation in terms of price.

- Using secrets of happiness derive optimal consumption x1,x2, givenvalues of p1 = 2, p2 = 4 and m = 30. Is the solution corner or interior(chose one)

Ans) Since utility function is Cobb-Douglas, optimal consumption is fol-lowing.

x1 =a

a+ b

m

p1=

2

3

30

2= 10

x2 =b

a+ b

m

p2=

1

3

30

4= 2.5

Both consumptions are positive, so it is interior solution.

2

d) 12pts (Harder) Using magic formulas for Cobb-Douglass preferences argue thatthe two commodities are 1) ordinary, 2) normal, and 3) neither grosscomplements nor gross substitutes (one sentence for each property).

Ans) When p1 increases x1 decreases, and when p2 increases x2 decreases:∂x1

∂p1= − a

a+bmp21< 0, ∂x2

∂p2= − b

a+bmp22< 0, so both are ordinary. When m

increases, both x1 and x2 increases: ∂x1

∂m = a(a+b)p1

> 0, ∂x2

∂m = b(a+b)p2

> 0.

Therefore, both are normal. When p1 increases x2 doesn’t change, andwhen p2 increases x1 doesn’t change: ∂x1

∂p2= 0, ∂x2

∂p2= 0. Therefore, x1 and

x2 are gross neutral (neither gross complements nor gross substitutes).

Problem 2 (15p) (Perfect substitutes)

You are planing a budget for the state of Wisconsin. The two major budgetpositions include education,x1 and health care, x2. Your preferences over thetwo are represented by function

U(x1, x2) = 3x1 + x2

a) 3pts Find marginal rate of substitution (give number).

Ans) MRS12 = −MU1

MU2= − 3

1 = −3.

b) 4pts Find optimal consumption of x1 and x2, given prices p1 = 2 and p2 = 1and available funds m = 50 (two numbers).

Ans) Compare marginal utility per dollars. Since MU1

P1= 3

2 = 1.5 is larger thanMU2

P2= 1

1 = 1, optimal choice is consuming only x1. x1 = mp1

= 502 = 25

and x2 = 0.

c) 8pts Is solution interior? (yes-no answer). Is marginal utility of a dollar equal-ized? (give two numbers and yes-no answer)

Ans) No, it is a corner solution since I consume zero x2. No, MU1

P1= 3

2 = 1.5 6=MU2

P2= 1

1 = 1.

Problem 3 (20p) (Intertemporal choice)

Zoe is a professional Olympic skier. Her income when ”young”is high (m1 = 30)but her future (period two) is not so bright (m2 = 20).

3

a) 4pts Depict Zoe’s budget set, assuming that she can borrow and save at theinterest rate r = 100%. Partition her budget set into three regions: thearea that involves saving, borrowing and none of the two.

Ans) For c1 ≤ 30, c2 ≤ 20, neither borrowing nor saving is required. c1 <30, c2 > 20 means saving and c1 > 30, c2 < 20 means borrowing.

b) 6pts Find PV and FV of Zoe’s lifetime income (two numbers) and show thetwo values in the graph. Interpret economically PV.

Ans) With r = 1, PV = m1 + m2

1+r = 30 + 20/2 = 40, FV = (1 + r)m1 +m2 =2 ∗ 30 + 20 = 80. PV is x-intercept of budget line and FV is y-interceptof budget line. PV means Zoe’s life time income evaluated at time 1 andit represents the maximum amount of c1 Zoe can afford by borrowingm2

1+r = 10.

c) 6pts Zoe’s utility function is U(C1, C2) = lnC1 + 11+δ lnC2 where discount rate

is δ = 2. Using magic formulas, find optimal consumption plan (C1, C2)(two numbers) and the corresponding level of savings/borrowing, S.

Ans) Given utility function represents same preference of V (C1, C2) = C(1+δ)1 C2 =

C31C2. Define P1 = 1, P2 = 1/(1 + r) = 1/2, and m = PV = 40. Magic

formulas of Cobb-Douglas tell us to consume C1 = 30 and C2 = 20. Thereis no needs for borrowing or saving, so S = 0.

Alternative approach PV (m) = PV (c) and MU1 = (1 + r)MU2 are

4

two conditions for optimal choice.

m1 +m2

1 + r= C1 +

C2

1 + r1

C1=

1 + r

(1 + δ)C2⇔ C1 =

(1 + δ)C2

1 + r

Plug in numbers, then we get C1 = 32C2 and 40 = C1 + C2/2. Therefore

C1 = 30 and C2 = 20. S = m1 − C1 = 0

d) 4pts Is Zoe tilting her consumption over time? (yes-no answer).

Ans) Yes, since time preference parameter δ = 2 whereas interest rate r = 1,i.e. since δ > r, it is optimal to consume more today (C1 = 30) and toconsume less tomorrow (C2 = 20). Therefore, C1 6= C2, and it means shetilts her consumption over time.

Problem 4 (25p) (Short questions)

a) 6pts Given utility function U(C,R) = min(2C,R), daily endowment of time24h, price and wage pc = w = 2, find optimal choice of C, relaxation timeR and labor supply L. (three numbers, use secrets of happiness for perfectcomplements).

Ans) Two formulas are ax1 = bx2 and p1x1 + p2x2 = m. Here, 2C = R and2C+2R = 48 are optimal conditions. C = 8, R = 16 and L = 24−R = 8.

b) 7pts Find optimal choice given quasilinear preferences U(x1, x2) = x1+100 lnx2,prices p1 = $8, p2 = $1 and income m = $60. Is your solution corner orinterior?

Ans) MU1

P1= MU2

P2gives me threshold of non-linear good (x2), and budget con-

straint p1x1 + p2x2 = m need to be satisfied.

MU1

P1=MU2

P2⇔ 1

8=

100

x2⇔ x2 = 800

I want to buy 800 units of x2 first, and then want to buy x1 as much as Ican afford. However, I have only $60, so I cannot afford both goods (800units of x2 costs $800). Optimal choice is buying only x2 good. x1 = 0,x2 = m

p2= 60

1 = 60. My solution is corner because I don’t buy any x1good.

5

c) 6pts Assume Interest rate r = 10%. Choose one of the two: a consol (a typeof British government bond)that pays annually $1, 000, starting next yearor cash $12, 000 now (compare PV).

Ans) Consol bond pays coupon $1, 000 forever. r = 0.1. The cash flow of consolbond is same with perpetuity. Recall that the present value of perpetuityis

∞∑n=1

x

(1 + r)n=

x1+r

1− 11+r

=x

r.

PV (consol) =x

r=

1, 000

0.1= 10, 000

PV (cash) = 12, 000

Current cash doesn’t require discount. Taking cash is better: 12, 000 >10, 000.

d) 6pts Your annual income when working (age 21-60) is $60, 000 and then you aregoing to live for the next 30 years. Write down equation that determinesconstant (maximal) level of consumption during your lifetime. Assumeannual interest rate r = 2%.

Ans) For first 40 years, I will earn $60, 000 annually, and I will consume constantlevel c annually for 70 years. Recall that the present value of annuity cashflow is

T∑n=1

x

(1 + r)n=

x1+r

[1− ( 1

1+r )T]

1− 11+r

=x

r

(1− 1

(1 + r)T

),

and r = 0.02.

PV (income) =60, 000

0.02

(1−

[1

1.02

]40)

PV (consumption) =c

0.02

(1−

[1

1.02

]70)

∴60, 000

0.02

(1−

[1

1.02

]40)=

c

0.02

(1−

[1

1.02

]70)

6

Midterm 1 Solution for Version B






= 24 = 1



U(x1, x2) = (x1)2(x2)1


V (x1, x2) = 2 lnx1 + lnx2



V (x1, x2) = 2 lnx1 + lnx2 = lnx21 + lnx2 = ln (x1)2(x2)1 = lnU(x1, x2)

1

c) 15pts Assuming utility function V (x1, x2) = 2 lnx1 + lnx2


Ans) MRS12 = −MU1

MU2= − 2x2

x1. At (2, 2), MRS = − 4

2 = −2. MRS12



Ans)

x1p1 + x2p2 = m

MU1

P1=MU2

P2or MRS12 = −P1

P2




x1 =a

a+ b

m

p1=

2

3

60

2= 20

x2 =b

a+ b

m

p2=

1

3

60

4= 5


2



∂p1= − a

a+bmp21< 0, ∂x2

∂p2= − b



∂m = a(a+b)p1

> 0, ∂x2

∂m = b(a+b)p2

> 0.


∂p2= 0, ∂x2





U(x1, x2) = x1 + x2


Ans) MRS12 = −MU1

MU2= − 1

1 = −1.



P1= 1

2 = 0.5 is smaller

than MU2

P2= 1

1 = 1, optimal choice is consuming only x2. x1 = 0 and

x2 = mp2

= 501 = 50.


Ans) No, it is corner solution since I consume zero x1. No, MU1

P1= 1

2 = 0.5 6=MU2

P2= 1

1 = 1.



3






1+r = 10.







4


m1 +m2

1 + r= C1 +

C2

1 + r1

C1=

1 + r

(1 + δ)C2⇔ C1 =

(1 + δ)C2

1 + r

Plug in numbers, then we get C1 = 42C2 and 50 = C1 + C2/2. Therefore

C1 = 40 and C2 = 20. S = m1 − C1 = 0.




a) 6pts Given utility function U(C,R) = min(2C,R), daily endowment of time24h, price and wage pc = w = 1, find optimal choice of C, relaxation timeR and labor supply L. (three numbers, use secrets of happiness for perfectcomplements).

Ans) Two formulas are ax1 = bx2 and p1x1 + p2x2 = m. Here, 2C = R andC +R = 24 are optimal conditions. C = 8, R = 16 and L = 24−R = 8.


Ans) MU1

P1= MU2



MU1

P1=MU2

P2⇔ 1

8=

100

x2⇔ x2 = 800


p2= 80


5

c) 6pts Assume Interest rate r = 10%. Choose one of the two: a consol (a typeof British government bond)that pays annually $1, 000, starting next yearor cash $8, 000 now (compare PV).

Ans) A consol bond pays coupon $1, 000 forever. r = 0.1. The cash flow ofa consol bond is same with perpetuity. Recall that the present value ofperpetuity is

∞∑n=1

x

(1 + r)n=

x1+r

1− 11+r

=x

r.

PV (consol) =x

r=

1, 000

0.1= 10, 000

PV (cash) = 8, 000

Current cash doesn’t require discount. Taking a consol is better: 10, 000 >8, 000.



T∑n=1

x

(1 + r)n=

x1+r

[1− ( 1

1+r )T]

1− 11+r

=x

r

(1− 1

(1 + r)T

),

and r = 0.1.


0.1

(1−

[1

1.1

]40)

PV (consumption) =c

0.1

(1−

[1

1.1

]75)

∴70, 000

0.1

(1−

[1

1.1

]40)=

c

0.1

(1−

[1

1.1

]75)

6

Midterm 1 Solution for Version C






= 24 = 1



U(x1, x2) = (x1)1(x2)2


V (x1, x2) = lnx1 + 2 lnx2



V (x1, x2) = lnx1 + 2 lnx2 = lnx1 + lnx22 = ln (x1)1(x2)2 = lnU(x1, x2)

1

c) 15pts Assuming utility function V (x1, x2) = lnx1 + 2 lnx2


Ans) MRS12 = −MU1

MU2= − x2

2x1. At (2, 2), MRS = − 2

4 = −0.5. MRS12



Ans)

x1p1 + x2p2 = m

MU1

P1=MU2

P2or MRS12 = −P1

P2




x1 =a

a+ b

m

p1=

1

3

60

2= 10

x2 =b

a+ b

m

p2=

2

3

60

4= 10


2



∂p1= − a

a+bmp21< 0, ∂x2

∂p2= − b



∂m = a(a+b)p1

> 0, ∂x2

∂m = b(a+b)p2

> 0.


∂p2= 0, ∂x2





U(x1, x2) = 2x1 + 2x2


Ans) MRS12 = −MU1

MU2= − 2

2 = −1.



P1= 2

2 = 1 is smaller

than MU2

P2= 2


x2 = mp2

= 1001 = 100.



P1= 2

2 = 1 6=MU2

P2= 2

1 = 2.



3






1+r = 20.







4


m1 +m2

1 + r= C1 +

C2

1 + r1

C1=

1 + r

(1 + δ)C2⇔ C1 =

(1 + δ)C2

1 + r

Plug in numbers, then we get C1 = 42C2 and 100 = C1 +C2/2. Therefore

C1 = 80 and C2 = 40. S = m1 − C1 = 0.




a) 6pts Given utility function U(C,R) = min(C, 2R), daily endowment of time24h, price and wage pc = w = 1, find optimal choice of C, relaxation timeR and labor supply L. (three numbers, use secrets of happiness for perfectcomplements).

Ans) Two formulas are ax1 = bx2 and p1x1 + p2x2 = m. Here, C = 2R andC +R = 24 are optimal conditions. C = 16, R = 8 and L = 24−R = 16.


Ans) MU1

P1= MU2



MU1

P1=MU2

P2⇔ 1

8=

50

x2⇔ x2 = 400


p2= 40


5

c) 6pts Assume Interest rate r = 10%. Choose one of the two: a consol (a type ofBritish government bond)that pays annually $500, starting next year orcash $4, 000 now (compare PV).

Ans) A consol bond pays coupon $500 forever and r = 0.1. The cash flow ofa consol bond is same with perpetuity. Recall that the present value ofperpetuity is

∞∑n=1

x

(1 + r)n=

x1+r

1− 11+r

=x

r.

PV (consol) =x

r=

500

0.1= 5, 000

PV (cash) = 4, 000

Current cash doesn’t require discount. Taking a consol is better: 5, 000 >4, 000.



T∑n=1

x

(1 + r)n=

x1+r

[1− ( 1

1+r )T]

1− 11+r

=x

r

(1− 1

(1 + r)T

),

and r = 0.1.


0.1

(1−

[1

1.1

]40)

PV (consumption) =c

0.1

(1−

[1

1.1

]75)

∴70, 000

0.1

(1−

[1

1.1

]40)=

c

0.1

(1−

[1

1.1

]75)

6

Midterm 1 Solution for Version D






= 48 = 1



U(x1, x2) = (x1)1(x2)2


V (x1, x2) = lnx1 + 2 lnx2



V (x1, x2) = lnx1 + 2 lnx2 = lnx1 + lnx22 = ln (x1)1(x2)2 = lnU(x1, x2)

1

c) 15pts Assuming utility function V (x1, x2) = lnx1 + 2 lnx2


Ans) MRS12 = −MU1

MU2= − x2

2x1. At (2, 2), MRS = − 2

4 = −0.5. MRS12



Ans)

x1p1 + x2p2 = m

MU1

P1=MU2

P2or MRS12 = −P1

P2


- Using secrets of happiness derive optimal consumption x1,x2, givenvalues of p1 = 4, p2 = 8 and m = 120. Is the solution corner orinterior (chose one)


x1 =a

a+ b

m

p1=

1

3

120

4= 10

x2 =b

a+ b

m

p2=

2

3

120

8= 10


2



∂p1= − a

a+bmp21< 0, ∂x2

∂p2= − b



∂m = a(a+b)p1

> 0, ∂x2

∂m = b(a+b)p2

> 0.


∂p2= 0, ∂x2





U(x1, x2) = 3x1 + 3x2


Ans) MRS12 = −MU1

MU2= − 3

3 = −1.



P1= 3

2 = 1.5 is smaller

than MU2

P2= 3


x2 = mp2

= 1001 = 100.



P1= 3

2 = 1.5 6=MU2

P2= 3

1 = 3.



3






1+r = 20.





formulas of Cobb-Douglas tell us to consume C1 = 5006 = 250

3 ≈ 83.33 andC2 = 100

3 ≈ 33.33. She needs to borrow B = 10/3 (because c1 > m1), orher saving is S = − 10

3 .


4


m1 +m2

1 + r= C1 +

C2

1 + r1

C1=

1 + r

(1 + δ)C2⇔ C1 =

(1 + δ)C2

1 + r

Plug in numbers, then we get C1 = 52C2 and 100 = C1 +C2/2. Therefore

C1 = 2503 and C2 = 100

3 . Saving is S = m1 − C1 = − 103 (or borrowing

B = C1 −m1 = 103 ).


Ans) Yes, since time preference parameter δ = 4 whereas interest rate r = 1,i.e. since δ > r, it is optimal to consume more today (C1 ≈ 83.33) and toconsume less tomorrow (C2 ≈ 33.33). Therefore, C1 6= C2, and it meansshe tilts her consumption over time.


a) 6pts Given utility function U(C,R) = min(C, 2R), daily endowment of time24h, price and wage pc = w = 2, find optimal choice of C, relaxation timeR and labor supply L. (three numbers, use secrets of happiness for perfectcomplements).

Ans) Two formulas are ax1 = bx2 and p1x1 + p2x2 = m. Here, C = 2R and2C+2R = 48 are optimal conditions. C = 16, R = 8 and L = 24−R = 16.


Ans) MU1

P1= MU2



MU1

P1=MU2

P2⇔ 1

8=

50

x2⇔ x2 = 400


p2= 60


5

c) 6pts Assume Interest rate r = 10%. Choose one of the two: a consol (a type ofBritish government bond)that pays annually $100, starting next year orcash $1, 200 now (compare PV).

Ans) A consol bond pays coupon $100 forever and r = 0.1. The cash flow ofa consol bond is same with perpetuity. Recall that the present value ofperpetuity is

∞∑n=1

x

(1 + r)n=

x1+r

1− 11+r

=x

r.

PV (consol) =x

r=

100

0.1= 1, 000

PV (cash) = 1, 200

Current cash doesn’t require discount. Taking cash is better since 1, 200 >1, 000.



T∑n=1

x

(1 + r)n=

x1+r

[1− ( 1

1+r )T]

1− 11+r

=x

r

(1− 1

(1 + r)T

),

and r = 0.1.


0.1

(1−

[1

1.1

]40)

PV (consumption) =c

0.1

(1−

[1

1.1

]75)

∴70, 000

0.1

(1−

[1

1.1

]40)=

c

0.1

(1−

[1

1.1

]75)

6


Midterm 1 (A)


Problem 1 (45p) (Well-behaved preferences)Freddie Frolic consumes only two types of commodities: cheese curds, x1; and diet coke, x2.a) The price of one portion of cheese curds is p1= 5 an one diet coke is p2= 2. Freddie�s income spent

entirely on the two commodities is m = 60. Show geometrically Freddie�s budget set. Find relative price ofcheese curds in terms of coke (number). Give economic interpretation of the relative price (one sentence).Where can the relative price be seen in the graph of a budget set? (one sentence)

b) Suppose that due to shortages in cheese supply, cheese curds are rationed, i.e., each consumer canbuy at most �ve portions. Show the new budget set on the graph.

c) Freddie� s utility isU(x1; x2) = a lnx1+b lnx2:

-Find Marginal Rate of Substitution (MRS) as a function of parameters a; b and x1; x2 (derive formula).For parameters a = 4; b = 2 and bundle (7; 7) �nd value of MRS (one number). Give economic interpretationof MRS (one sentence). Which of the goods is more valuable at the bundle (7; 7)? What is the geometricinterpretation of MRS? (one sentence)

-Write down two secrets of happiness that determine optimal choice. Explain economic intuition behindthe two conditions (two sentences for each).

-Derive optimal choice of x1 and x2 as a function of a; b; p1; p2 and m (show the derivation of magicformulas). Is your solution corner or interior (chose one and provide mathematical argument)

d) Assume a = 4; b = 2 and p2= 2;m = 60: Find geometrically and determine analytically price o¤ercurve and demand curve (give two functions). Is x1 and ordinary or Gi¤en good and why (one sentence)?

e) Show that utility functions V (x1; x2) = xa1x

b2 and U(x1; x2) = a lnx1+b lnx2 represent the same

preferences (derive one function from the other).

Problem 2 (15p) (Quasilinear Preferences)You are asked to plan a budget of University of Wisconsin, Madison for the next year. The two major

expenses involve computers, x1 and classroom furniture, x2: The university�utility function is given by

U(x1; x2) = 2 lnx1+x2:

a) Find marginal rate of substitution as a function of (x1; x2) (give formula).b) Using two secrets of happiness �nd optimal �consumption�of computers and furniture if corresponding

prices are p1 = 2 and p2 = 4 and the available funds are m = 40 (give two numbers).c) Suppose the price of a computer goes down to p1= 1. Find optimal choice after the price change (two

numbers). Decompose the change in x1 into a substitution and income e¤ect (two numbers).d) Find optimal consumption for p1= 2 , p2= 4 andm = 4 (give two numbers). Is your solution interior?

(yes-no answer). Is marginal utility of a dollar equalized? (give two numbers and yes-no answer)

Problem 3 (15p) (Perfect complements, Intertemporal choice)Casper is a manager in a small startup �rm. His income today is relatively small (m1= 50) but in the

future (period two) he expects to become very rich (m2= 200)a) Depict Casper�s budget set assuming that he can borrow and save at the interest rate r = 100%.

Mark consumption plans on the budget line that involve savings and the plans that require borrowing. FindPresent and Future Value of Casper�s income (two numbers) and show the two in the graph.

b) Casper�s utility function is U(C1; C2) =min (C1; C2). In the commodity space plot Casper�s�indif-ference curves.

1

c) Find optimal consumption plan (C1; C2) (give two numbers: use two secrets of happiness for per-fect complements and the fact that p1= 1 and p2= 1=(1 + r)). Find the level of savings/borrowing inequilibrium (one number). Is Casper smoothing his consumption over time? (yes-no answer)

Problem 4 (25p) (Short questions)a) Assume utility function U (C;R)= C �R and the daily endowment of time equal to 24h: Find

optimal choice of consumption C; relaxation time R and labor supply L as a function of real wage ratew=pc:(three nubmers) use magic formula). Is labor supply elastic or inelastic (one sentence)?

b) Find optimal choice given utility function U (x1; x2)= 3x1+x2; prices p1= $8; p2= $2 and incomem = 100: Is your solution corner or interior?

c) You are going to save $10; 000 when working (age 21-70) and then you are going to live for the next30 years. Write down equation that determines constant (maximal) level of consumption during retirementage given your savings. Assume annual interest rate r = 3%:

d) Derive Present Value formula for perpetuity.

Bonus question (Just for fun)a) Prove that for perfect complements U (x1; x2)=min (ax1; bx2) ; MRS is equal to zero for all bundles

below the optimal proportion line and equal to �1 for bundles above it.b) Explain in words why the solution to a linear optimization problem such as with perfect substitutes

is called a bang bang solution.

2


Midterm 1 (B)



entirely on the two commodities is m = 120. Show geometrically Freddie�s budget set. Find relative priceof cheese curds in terms of coke (number). Give economic interpretation of the relative price (one sentence).Where can the relative price be seen in the graph of a budget set? (one sentence)


c) Freddie�s utility isU(x1; x2) = a lnx1+b lnx2:

-Find Marginal Rate of Substitution (MRS) as a function of parameters a; b and x1; x2 (derive formula).For parameters a = 4; b = 2 and bundle (7; 14) �nd value of MRS (one number). Give economic interpre-tation of MRS (one sentence). Which of the goods is more valuable at the bundle (7; 7)? What is thegeometric interpretation of MRS? (one sentence)









U(x1; x2) = 2 lnx1+x2:


prices are p1 = 4 and p2 = 8 and the available funds are m = 80 (give two numbers).c) Suppose the price of a computer goes down to p1= 2. Find optimal choice after the price change (two

numbers). Decompose the change in x1 into a substitution and income e¤ect (two numbers).d) Find optimal consumption for p1= 4, p2= 8 andm = 8 (give two numbers). Is your solution interior?






3










4


Midterm 1 (C)















U(x1; x2) = 4 lnx1+x2:


prices are p1= 4 and p2= 2 and the available funds are m = 20 (give two numbers).c) Suppose the price of a computer goes down to p1= 2. Find optimal choice after the price change (two







5










6


Midterm 1 (D)















U(x1; x2) = 4 lnx1+x2:









7










8


Makeup exam








-Derive optimal choice of x1 and x2 as a function of a; b; p1; p2 and m (show the derivation of magicformulas). Are two goods gross substitutes, gross complements or none?

d) Assume a = 4; b = 2 and p1= 3; p2= 6: Find geometrically and determine analytically income o¤ercurve and Engel curve (give two functions). Is x1 and normal or inferior, and why (one sentence)?

e) Show that utility functions V (x1; x2) = x3a1 x

3b2 and U(x1; x2) = a lnx1+b lnx2 represent the same




U(x1; x2) = lnx1+1

2x2:








9

b) Casper�s utility function is U(C1; C2) = 2min (C1; C2). In the commodity space plot Casper�s�indi¤erence curves.




b) Find optimal choice given utility function U (x1; x2)= 10x1+5x2; prices p1= $8; p2= $2 and incomem = 40: Is your solution corner or interior?






10


Midterm 1 (A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (45+15+15+25=100 points)+ bonus (just for fun). Make sure you answer the first four questions before working on the bonus one!

Problem 1 (45p) (Well-behaved preferences)a) Show geometrically Freddie’s budget set. (5 points)

Find relative price of cheese curds in terms of coke. (2.5 points) 1 unit of cheese curds = 2.5 units of diet coke

Give economic interpretation of the relative price. (2.5 points)

This is the REAL PRICE of the two goods, meaning that in an exchange economy this would be the rateof exchange between the two goods

Where can the relative price be seen in the graph? (2.5 points)The relative price can be seen (geometrically) as the slope of the budget constraint.

b) Cheese curds are rationed, show the new budget set on the graph.(5 points)The area shaded red is the new budget set.

c) Find (MRS) as a function of parameters (2.5 points)

MRS = −ax2

bx1

For parameters a = 4, b = 2 and bundle (7, 7) find value of MRS. (2.5 points)MRS(7,7) = -2Give economic interpretation of MRS. (1 point)The marginal rate of substitution measure the rate at which the consumer is willing to substitute lessconsumption of good x1 for more consumption of good x2 while remaining at the same level of utility.Which of the goods is locally more valuable? ( 1

2 point)x1 is locally more valuable at the bundle (7,7)What is the geometric interpretation of MRS? (1 point)The MRS is the slope of the indifference curve

1

Write down two secrets of happiness that determine optimal choice. Explain economic intuition behind thetwo conditions. (5 points)

M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2

The first equation is a budget constraint which requires consumers to spend all of their income on the con-sumption of two goods, x1 and x2. The second condition is a utility maximization condition that requiresthe slope of the indifference curve passing through the optimal consumption bundle to be tangent to theprice ratio. The second condition will effectively require the marginal utility per dollar of each good toequate at the optimal consumption bundle.

Derive optimal choice of x1 and x2 as a function of a, b, p1, p2 and m (show the derivation of magic formulas).Is your solution corner or interior (5 points)

M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2

Solve for P1x1 in the second equation and substitute that into the budget constraint and solve for X1. To getx2 simply solve for P2x2 in the second equation and substitute that into the budget constraint.

ax2

bx1=

Px1

Px2

(1)

Px1 x1 =ab

x2P2 (2)

M =ab

x2P2 + x2P2 (3)

M = (a + b

bP2)x2 (4)

x2 =b

a + b×

mP2

(5)

and

ax2

bx1=

Px1

Px2

(6)

ba

Px1 x1 = x2P2 (7)

M =ba

x1P1 + x1P1 (8)

M = (a + b

aP1)x1 (9)

x1 =a

a + b×

mP1

x1 =a

a + b×

mP1

=⇒ x1 =2m3P1

x2 =b

a + b×

mP2

=⇒ x2 =m

3P2

d) Assume a = 4, b = 2 and p2= 2,m = 60. Find geometrically and determine analytically price offer curve(2.5 points) and demand curve (2.5 points). The price offer curve can be plotted by taking the partial deriva-tive of the demands for x1 and x2 with respect to P1. This will give us the slope of the price offer curve,with the partial derivative of the demand for x2 being the “rise” and the partial derivative of the demandfor x1 being the “run”. Since the demand for x2 does not depend on P1 there is no vertical change in the

2

price offer curve, making it a horizontal line plotted at 10 units of x2

The demand curve for x1 can be plotted by taking the partial derivative of the demand for x1 withrespect to P1. The slope of this curve is ∂x1(M,P1

∂P1) = − 2M3P2

1

Is x1 and ordinary or Giffen good and why? (2 points)

This is an ordinary good because there is a negative relationship between price and the quantity of thegood demanded. This can be seen by taking the partial derivative of the demand function with respect toits own price:

∂x1(M,P1)∂P1

= −M

2P21

e) Show that utility functions V(x1, x2) = xa1xb

2 and U(x1, x2) = a ln x1+b ln x2 represent the same preferences(3 points).

The function U() appears to be the result of applying the natural logarithm to the function V(). Thatmeans that we can define the function F(x) = ln(x) so that U(x1, x2) = F(V(x1, x2)). The natural logarithmis a concave function (strictly increasing), meaning that applying the natural logarithm to a function willpreserve the ordinality of that function. Thus, the function U() is merely a monotonic transformation ofthe function V(), meaning they represent the same preference order.

Problem 2 (15p) (Quasilinear Preferences)a) Find marginal rate of substitution as a function of (x1, x2) (1 point).

MRS =MUx1

MUx2

=

2x1

1=

2x1

b) Using two secrets of happiness find optimal consumption choices (4 points)

1)40 = 2x1 + 4x2

2) 2x1

= 24 =⇒ x1 = 4

Plug x1 = 4 into the first secret (budget constraint) to reveal: x1 = 4, x2 = 8

c) Suppose the price of a computer goes down to p1= 1. Find optimal choice after the price change (1 point).

To find the new optimal choice repeat the above steps with P1 = 1:

3

1)40 = x1 + 4x2

2) 2x1

= 14 =⇒ x1 = 8


Decompose the change in x1 into a substitution effect (2 points) and income effect (2 points).To calculate the substitution and income effects we need to find an ‘intermediate point’. This is the optimal

consumption choice given the purchasing power necessary to buy the original optimal consumption bundle(4,8) given the new prices P1 = 1, P2 = 4:

M’ = (1)4 + (4)8 =⇒ M’ = 36

To find the ‘intermediate’ optimal choice repeat the above steps with P1 = 1 and M = 36:1) 36 = x1 + 4x2

2) 2x1

= 14 =⇒ x1 = 8


The substitution effect is the change in consumption of x1 due only to the price change (holding pur-chasing power constant). That means the substitution effect is the difference between the amount of x1

originally consumed (4 units) and how much is consumed after the price change (8 units). The substitutioneffect is then 4 units of x1, while the income effect is zero (when we increase purchasing power from M =36 to M = 40 we do not change our level of x1 consumption, only x2.

d) Find optimal consumption for p1= 2 , p2= 4 and m = 4 (1 point). 1) 4 = 2x1 + 4x2

2) 2x1

= 24 =⇒ x1 = 4

Plug x1 = 4 into the first secret (budget constraint) reveals that: x1 = 4, x2 = −1. We cannot consumernegative amounts of a consumption good, so this indicates we would prefer to give up 1 unit of x2 in orderto consume more x1. This is not an option, however, as the lower bound on x2 is zero.

Is your solution interior? (1 point).

Our solution is x1 = 2, x2 = 0, which is not interior (corner solution). Interior solutions are those atwhich consumption of each good is strictly greater than zero, which is not the case here.

Is marginal utility of a dollar equalized? (3 points)

No, we will very rarely see marginal utility per dollar of each consumption good equalized at a cornersolution. In this case we have:

MUx1

P1=

2x1

2=

12,

14

=MUx2

P2

Problem 3 (15p) (Perfect complements, Intertemporal choice)Casper is a manager in a small startup firm. His income today is relatively small (m1= 50) but in the future

(period two) he expects to become very rich (m2= 200)a) Depicts Casper’s budget set assuming that he can borrow and save at the interest rate r = 100%(1 point).

Mark consumptions plans on the budget line that involve savings and the plans that require borrowing.(1point) Find Present and Future Value of Casper’s income and show the two in the graph.(2 points)

4

b) In the commodity space plot Casper’s’ indifference curves.(2 points)

c) Find optimal consumption plan (C1,C2) (5 points). Find the level of savings/borrowing in equilibrium(2 points). Is Casper smoothing his consumption over time? (2 points)

1) 150 = C1 + 11+r C2

2) C1 = C2

Plug C2 into the budget constraint for C1 gives Casper the equation: 150 = 1+r1+r C2 + 1

1+r C2. Plug in r = 1 andwe see that this determines C2 = 100 and looking back at the second secret of happiness, it also determinesC1 = 100. We find that C1 = C2, which means that Casper is consumption smoothing. In order to do this,he borrows $50 from his second period income in the first period (reducing his second period consumptionby $100 and increasing his first period consumption by $50 in the process.

Problem 4 (25p) (Short questions)a) Assume utility function U (C,R) = C × R and the daily endowment of time equal to 24h. Find optimal

choice of consumption C, relaxation time R and labor supply L as a function of real wage rate w/pc.(threenubmers) use magic formula). Is labor supply elastic or inelastic (one sentence)?

b) Find optimal choice given utility function U(x1, x2 = 3x1 + x2), prices p1=$8 and p2=$2 as well as incomem=100. Is your solution corner or interior?

MUx1

P1=

38<

12

=MUx2

P2

The marginal utility per dollar is constant for both goods, but higher for good two, so in this case allincome is spent on x2. This makes the solution (0,50).

c) You are going to save $10,000 when working (age 21-70) and then you are going to live for the next 30years. Write down equation that determines constant (maximal) level of consumption during retirement agegiven your savings. Assume annual interest rate r=3%.

$10, 000r

(1 − (1

1 + r)50) =

Cr

(1 − (1

1 + r)30(

11 + r

)50


5

The formula of a stream of payments that never ends is:

PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + ....

=1

(1 + r)[x +

x(1 + r)

+x

(1 + r)2 + ....]

=1

(1 + r)[x + PV]

so we can solve for PV to get a more concise solution:

(1 −1

1 + r)PV =

11 + r

x

(1 + r1 + r

−1

1 + r)PV =

11 + r

x

(r

1 + r)PV =

11 + r

x

PV =xr

Bonus question (Just for fun)a) Prove that for perfect complements U (x1, x2) = min (ax1, bx2) ,MRS is equal to zero for all bundles below

the optimal proportion line and equal to −∞ for bundles above it.b) Explain in words why the solution to a linear optimization problem such as with perfect substitutes is

called a bang bang solution.

6

Midterm 1 (B)



Find relative price of cheese curds in terms of coke. (2.5 points) 1 unit of cheese curds = 2.5 units of diet coke






MRS = −ax2

bx1


2 point)x1 is locally more valuable at the bundle (7,7)What is the geometric interpretation of MRS? (1 point)The MRS is the slope of the indifference curveWrite down two secrets of happiness that determine optimal choice. Explain economic intuition behind thetwo conditions. (5 points)

M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2

7



M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2


ax2

bx1=

Px1

Px2

(10)

Px1 x1 =ab

x2P2 (11)

M =ab

x2P2 + x2P2 (12)

M = (a + b

bP2)x2 (13)

x2 =b

a + b×

mP2

(14)

and

ax2

bx1=

Px1

Px2

(15)

ba

Px1 x1 = x2P2 (16)

M =ba

x1P1 + x1P1 (17)

M = (a + b

aP1)x1 (18)

x1 =a

a + b×

mP1

x1 =a

a + b×

mP1

=⇒ x1 =2m3P1

x2 =b

a + b×

mP2

=⇒ x2 =m

3P2

d) Assume a = 4, b = 2 and p2= 2,m = 60. Find geometrically and determine analytically price offer curve(2.5 points) and demand curve (2.5 points). The price offer curve can be plotted by taking the partial deriva-tive of the demands for x1 and x2 with respect to P1. This will give us the slope of the price offer curve,with the partial derivative of the demand for x2 being the “rise” and the partial derivative of the demandfor x1 being the “run”. Since the demand for x2 does not depend on P1 there is no vertical change in theprice offer curve, making it a horizontal line plotted at 10 units of x2

8


∂P1) = − 2M3P2

1



∂x1(M,P1)∂P1

= −M

2P21





MRS =MUx1

MUx2

=

2x1

1=

2x1


1)40 = 2x1 + 4x2

2) 2x1

= 24 =⇒ x1 = 4



To find the new optimal choice repeat the above steps with P1 = 1:1)40 = x1 + 4x2

9

2) 2x1

= 14 =⇒ x1 = 8




M’ = (1)4 + (4)8 =⇒ M’ = 36

To find the ‘intermediate’ optimal choice repeat the above steps with P1 = 1 and M = 36:1) 36 = x1 + 4x2

2) 2x1

= 14 =⇒ x1 = 8





2) 2x1

= 24 =⇒ x1 = 4






MUx1

P1=

2x1

2=

12,

14

=MUx2

P2




10



1) 150 = C1 + 11+r C2

2) C1 = C2


1+r C2. Plug in r = 1 andwe see that this determines C2 = 100 and looking back at the second secret of happiness, it also determinesC1 = 100. We find that C1 = C2, which means that Casper is consumption smoothing. In order to do this,he borrows $50 from his second period income in the first period (reducing his second period consumptionby $100 and increasing his first period consumption by $50 in the process.



The two secrets to happiness are:

1) (24 − R)W = PcC

2)MUx1

MUx2

=P1

P2

=⇒ PcC = WR=⇒ 24W = PcC + PcC

=⇒ C =12W

PcR = 12 L = 12

We find that labor supply is perfectly inelastic because it does not vary with the real wage rate.b) Find optimal choice given utility function U(x1, x2 = 5x1 + x2), prices p1=$10 and p2=$1 as well as income

m=100. Is your solution corner or interior?

MUx1

P1=

510<

11

=MUx2

P2

11

The marginal utility per dollar is constant for both goods, but higher for good two, so in this case allincome is spent on x2. This makes the solution (0,100).

c) You are going to save $10,000 when working (age 21-70) and then you are going to live for the next 30years. Write down equation that determines constant (maximal) level of consumption during retirement agegiven your savings. Assume annual interest rate r=3%.

$10, 000r

(1 − (1

1 + r)50) =

Cr

(1 − (1

1 + r)30(

11 + r

)50

d) Derive Present Value formula for perpetuity.The formula of a stream of payments that never ends is:

PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + ....

=1

(1 + r)[x +

x(1 + r)

+x

(1 + r)2 + ....]

=1

(1 + r)[x + PV]


(1 −1

1 + r)PV =

11 + r

x

(1 + r1 + r

−1

1 + r)PV =

11 + r

x

(r

1 + r)PV =

11 + r

x

PV =xr




12

Midterm 1 (C)



Find relative price of cheese curds in terms of coke. (2.5 points) 1 unit of cheese curds = 3 units of diet coke






MRS = −ax2

bx1



M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2

13



M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2


ax2

bx1=

Px1

Px2

(19)

Px1 x1 =ab

x2P2 (20)

M =ab

x2P2 + x2P2 (21)

M = (a + b

bP2)x2 (22)

x2 =b

a + b×

mP2

(23)

and

ax2

bx1=

Px1

Px2

(24)

ba

Px1 x1 = x2P2 (25)

M =ba

x1P1 + x1P1 (26)

M = (a + b

aP1)x1 (27)

x1 =a

a + b×

mP1

x1 =a

a + b×

mP1

=⇒ x1 =2m3P1

x2 =b

a + b×

mP2

=⇒ x2 =m

3P2


14


∂P1) = − 2M3P2

1=⇒ x1 = −40

P1



∂x1(M,P1)∂P1

= −M

2P21





MRS =MUx1

MUx2

=

4x1

1=

4x1


1)20 = 4x1 + 2x2

2) 4x1

= 42 =⇒ x1 = 2



To find the new optimal choice repeat the above steps with P1 = 1:1)20 = 2x1 + 2x2

15

2) 4x1

= 22 =⇒ x1 = 4




M’ = (2)2 + (2)6 =⇒ M’ = 116

To find the ‘intermediate’ optimal choice repeat the above steps with P1 = 2 and M = 16:1) 16 = 2x1 + 2x2

2) 4x1

= 22 =⇒ x1 = 4





2) 4x1

= 42 =⇒ x1 = 2






MUx1

P1=

4x1

4=

44,

12

=MUx2

P2




16



1) 125 = C1 + 11+r C2

2) C1 = C2


1+r C2. Plug in r = 1 andwe see that this determines C2 = 250

3 and looking back at the second secret of happiness, it also determinesC1 = 250

3 . We find that C1 = C2, which means that Casper is consumption smoothing. In order to do this, heborrows 100

3 from his second period income in the first period (reducing his second period consumption by003 and increasing his first period consumption by 33.33 dollars in the process.




MUx1

P1=

58>

12

=MUx2

P2

The marginal utility per dollar is constant for both goods, but higher for good one, so in this case allincome is spent on x1. This makes the solution (12.5,0).

c) You are going to save $50,000when working (age 21-80) and then you are going to live for the next 20years. Write down equation that determines constant (maximal) level of consumption during retirement agegiven your savings. Assume annual interest rate r=2%.

$50, 000r

(1 − (1

1 + r)60) =

Cr

(1 − (1

1 + r)20(

11 + r

)60


17


PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + ....

=1

(1 + r)[x +

x(1 + r)

+x

(1 + r)2 + ....]

=1

(1 + r)[x + PV]


(1 −1

1 + r)PV =

11 + r

x

(1 + r1 + r

−1

1 + r)PV =

11 + r

x

(r

1 + r)PV =

11 + r

x

PV =xr




18

Midterm 1 (D)



Find relative price of cheese curds in terms of coke. (2.5 points) 1 unit of cheese curds = 1 unit of diet coke






MRS = −ax2

bx1



M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2

19



M = x1p1 + x2p2

MUx1

MUx2

=Px1

Px2


ax2

bx1=

Px1

Px2

(28)

Px1 x1 =ab

x2P2 (29)

M =ab

x2P2 + x2P2 (30)

M = (a + b

bP2)x2 (31)

x2 =b

a + b×

mP2

(32)

and

ax2

bx1=

Px1

Px2

(33)

ba

Px1 x1 = x2P2 (34)

M =ba

x1P1 + x1P1 (35)

M = (a + b

aP1)x1 (36)

x1 =a

a + b×

mP1

x1 =a

a + b×

mP1

=⇒ x1 =2m3P1

x2 =b

a + b×

mP2

=⇒ x2 =m

3P2


20


∂P1) = − 2M3P2

1=⇒ x1 = −60

P1



∂x1(M,P1)∂P1

= −M

2P21





MRS =MUx1

MUx2

=

4x1

1=

4x1


1)80 = 4x1 + 4x2

2) 4x1

= 44 =⇒ x1 = 4



To find the new optimal choice repeat the above steps with P1 = 2:1)80 = 2x1 + 4x2

21

2) 4x1

= 24 =⇒ x1 = 8




M’ = (2)4 + (4)16 =⇒ M’ = 72

To find the ‘intermediate’ optimal choice repeat the above steps with P1 = 2 and M = 72:1) 72 = 2x1 + 4x2

2) 4x1

= 24 =⇒ x1 = 8





2) 4x1

= 44 =⇒ x1 = 4






MUx1

P1=

4x1

4=

44,

14

=MUx2

P2




22



1) 300 = C1 + 11+r C2

2) C1 = C2


1+r C2. Plug in r = 1 andwe see that this determines C2 = 200 and looking back at the second secret of happiness, it also determinesC1 = 200. We find that C1 = C2, which means that Casper is consumption smoothing. In order to do this,he borrows from his second period income in the first period (reducing his second period consumption by$200 and increasing his first period consumption by $100 in the process.




MUx1

P1=

58>

12

=MUx2

P2

The marginal utility per dollar is constant for both goods, but higher for good one, so in this case allincome is spent on x1. This makes the solution (5,0).

c) You are going to save $6,000when working (age 21-80) and then you are going to live for the next 20years. Write down equation that determines constant (maximal) level of consumption during retirement agegiven your savings. Assume annual interest rate r=4%.

$6, 000r

(1 − (1

1 + r)60) =

Cr

(1 − (1

1 + r)20(

11 + r

)60


23


PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + ....

=1

(1 + r)[x +

x(1 + r)

+x

(1 + r)2 + ....]

=1

(1 + r)[x + PV]


(1 −1

1 + r)PV =

11 + r

x

(1 + r1 + r

−1

1 + r)PV =

11 + r

x

(r

1 + r)PV =

11 + r

x

PV =xr




24


Midterm 1 (Group A)


Problem 1 (50p) (Standard choice)Jim has two pleasures in his life, drinking Port wine (x1) and reading books (x2).a) Port wine x1 costs p1 = $20 a bottle and each book x2, p2 = $20; and his daily budget is m = $600:

Show geometrically the budget constraint in the commodity space. Mark the two extreme consumption bun-dles (mark concrete values). On the same graph, show how the budget set would be a¤ected by introductionof ad valorem tax on imported wine (but not books) assuming tax rate � = 100%:

b) Jim seems to be a fairly sophisticated fellow with quite a complicated utility function

U (x1; x2) = 123� loghexp

h(6 lnx1 + 3 lnx2)

2i+ 10

i322Argue that in fact Jim is not really that sophisticated as his preferences can represented by a signi�cantly

simpler utility function. (one sentence + simpler utility function). If you are unable to answer point b) inthe remaining part of Problem 1, you can assume U (x1; x2) = x21x2.

c) Find Jim�s marginal rate of substitution (MRS) for any bundle (x1; x2) (give a formula for MRS).What is the value of MRS at consumption bundle (1; 2) (give a number)? Which of the two commodities ismore precious to Jim?

d) Write down mathematically two secrets of happiness, assuming that p1; p2;m are parameters (andnot concrete values).

- Provide some economic intuition behind the two conditions (ca. two sentences for each).- Describe how the two "secrets of happiness" can be seen in the graph.(graph + 2 sentences)- Derive the optimal consumption of x1 and x2 as a function of p1; p2;m (show the derivation).e) Using your formula from d), �nd the optimal consumption levels of both types of commodities (x1; x2)

for:- p1 = $40; p2 = $20 and m = $600 (give two numbers).and after the price of Port wine decreased:- for p1 = $20; p2 = $20 and m = $600 (give two numbers).What is the total change in consumption of Port wine? (give a number). Illustrate the change on the

graph. Is Port wine an ordinary of a Gi¤en good? (Chose one + one sentence explaining why.)f) Decompose the total change in consumption of x1 from e) into a substitution and income e¤ect.

(Calculate the two numbers and show how can you �nd the e¤ect on the graph.)

Problem 2 (20p) (Intertemporal choice)Marky Mark is a rap singer who earns m1 = $100 when young and m2 = $300 when old.a) What is the present value (in terms of $ from period one) of Marky Mark�s income (give one number).b) Write down Marky Mark�s budget constraint (one inequality) and plot budget set given interest rate

r = 200% in the graph. Mark all consumption plans on the budget line that require borrowing and the onesthat require saving.

c) Marky Mark�s intertemporal utility is given by U (C1; C2) = ln (c1) +11+� ln (c1) and discount rate

is � = 2: Provide an economic interpretation of parameter � (one sentence). Using magic formulas, �ndthe optimal consumption plan and the optimal saving strategy. (give three numbers C1; C2; S). Does Samsmooth his consumption? (yes no + one sentence) Is Sam tilting his consumption? (yes no + one sentence)

d) (Perpetuity) Your sister has just promised to send you pocket money of $500 each month startingnext month and she will keep doing it forever. What is the present value of "having such a sister" if monthlyinterest rate is equal to 1% (one number).

1

e) (Annuity) You are going to work for 45 years (from 20 to 65) earning income of $1000,000 a year.Then you are going to retire and going to live for another 35 years (till you are 100). Assume that the annualinterest is given by r = 5%: Write down equation that would allow you to determine the maximal constantlevel of consumption C throughout your whole life (80 years). (write down equation that determines PV ofincome and consumption, but you need not calculate C).

Problem 3 (15p) (Perfect complements)The old recipe for Pierogies (Polish dumplings) requires that sauerkraut x1 is mixed with portabella

mushrooms x2 in a �xed (and sacred) proportion of 4:1.a) Propose a utility function over sauerkraut and mushrooms (function U (x1; x2)) assuming that piero-

gies is a good.b) In the commodity space, carefully depict indi¤erence curves (marking the optimal proportion line).c) Assume that p1 = 2 and p2 = 2 and income is m = 40: Write down two secrets of happiness (give

two equations) that determine the optimal choice (two numbers). Explain the economic intuition behindthe conditions (one sentence for each secret). Is your solution interior (yes or no)?

d) Without any calculations, in two separate graphs plot the price o¤er curve and income o¤er curve(just plot two curves).

Problem 4 (15p) (Perfect substitutes)Sam does not have any money but is endowed with !1 = 10 apples and !1 = 30 oranges and prices of

the two are p1 = p2 = 1. His utility function is given by U (x1; x2) = 2x1 + x2a) Find Sam�s budget line, marking the endowment point.b) Find the optimal choice of the two commodities if his utility is given by U (x1; x2) = 2x1 + x2 (give

two numbers). Is your solution interior?c) Find net demands in optimum (give two numbers). Is Sam a net buyer or a net seller of apples. How

about oranges? (chose one for each commodity)d) In mathematics, the solution to a problem with perfect substitutes is called "bang, bang" solution.

Provide intuition why (one sentence).

Bonus question (Just for fun)Derive PV formula for annuity and perpetuity.

2


Midterm 1 (Group B)






h(4 lnx1 + 2 lnx2)

2i+ 10


simpler utility function. (one sentence + simpler utility function). If you are unable to answer point b) inthe remaining part of Problem 1, you can assume U (x1; x2) = x21x

12.












3

e) (Annuity) You are going to work for 45 years (from 20 to 65) earning income of $500,000 a year. Thenyou are going to retire and going to live for another 35 years (till you are 100). Assume that the annualinterest is given by r = 1%: Write down equation that would allow you to determine the maximal constantlevel of consumption C throughout your whole life (80 years). (write down equation that determines PV ofincome and consumption, but you need not calculate C).






Problem 4 (15p) (Perfect substitutes)Sam does not have any money but is endowed with !1 = 100 apples and !1 = 200 oranges and prices

of the two are p1 = p2 = 1. His utility function is given by U (x1; x2) = x1 + 3x2a) Find Sam�s budget line, marking the endowment point.b) Find the optimal choice of the two commodities if his utility is given by U (x1; x2) = x1 +3x2 (give





4

Intermediate MicroeconomicsProf. Marek Weretka

Midterm 1 (Group C)


Problem 1 (50p) (Standard choice)Jim has two pleasures in his life, drinking Port wine (x1) and reading books (x2).a) Port wine x1 costs p1 = $5 a bottle and each book x2, p2 = $5; and his daily budget ism = $60: Show

geometrically the budget constraint in the commodity space. Mark the two extreme consumption bundles(mark concrete values). On the same graph, show how the budget set would be a¤ected by introduction ofad valorem tax on imported wine (but not books) assuming tax rate � = 100%:



h(10 lnx1 + 5 lnx2)

2i+ 10



12.












e) (Annuity) You are going to work for 40 years (from 20 to 60) earning income of $100,000 a year. Thenyou are going to retire and going to live for another 30 years (till you are 90). Assume that the annual

5

interest is given by r = 1%: Write down equation that would allow you to determine the maximal constantlevel of consumption C throughout your whole adult life (70 years). (write down equation that determinesPV of income and consumption, but you need not calculate C).







of the two are p1 = p2 = 1. His utility function is given by U (x1; x2) = x1 + 4x2a) Find Sam�s budget line, marking the endowment point.b) Find the optimal choice of the two commodities if his utility is given by U (x1; x2) = x1 +4x2 (give





6

Intermediate MicroeconomicsProf. Marek Weretka

Midterm 1 (Group D)





U (x1; x2) = 12�rhlnh(20 lnx1 + 10 lnx2)

2i+ 10



12.












7

e) (Annuity) You are going to work for 40 years (from 20 to 60) earning income of $200,000 a year. Thenyou are going to retire and going to live for another 30 years (till you are 90). Assume that the annualinterest is given by r = 1%: Write down equation that would allow you to determine the maximal constantlevel of consumption C throughout your whole adult life (70 years). (write down equation that determinesPV of income and consumption, but you need not calculate C).







of the two are p1 = p2 = 1. His utility function is given by U (x1; x2) = 2x1 + x2a) Find Sam�s budget line, marking the endowment point.b) Find the optimal choice of the two commodities if his utility is given by U (x1; x2) = 2x1 + x2 (give





8

ECON 301 - ANSWER KEY TO MIDTERM 1(GROUP A) TA : CHANG-KOO(CK) CHIMARCH 4, 2011

Problem 1 (50pts)

(a) (5pts) The budget constraint is given by 20x1 + 20x2 = 600. With the tax τ = 100%, it becomes20(1 + τ)x1 + 20x2 = 40x1 + 20x2 = 600. Since it plays a role in increasing p1 by 100%, the budgetline must shift in as displayed in Figure 1.

0 5 10 15 20 25 300

5

10

15

20

25

30

Demand for Port Wine(x1)

Dem

and

for

book

s(x 2)

After Tax is levied

Figure 1: The Budget Set


Dem

and

for

Boo

ks(x

2)

The Optimal Bundle

Figure 2: Two Secrets of Happiness

(b) (5pts) His sophisticated utility function is simply a monotone transformation of 2 ln x1 + ln x2.

(c) (5pts) From the previous part, we shall write his utility function U(x1, x2) = 2 ln x1 + ln x2 withoutloss of generality. Differentiating with respect to x1 and x2 gives us

MU1 =2x1

and MU2 =1x2

Hence the marginal rate of substitution, defined as the (negative) ratio of two marginal utilities, is

given by MRS = −2/x1

1/x2= −2x2

x1. At the bundle (1, 2), it takes −2 · 2

1= −4. Since its absolute value

is greater than 1, the port wine is more valuable from his perspective.

(d) (10pts) Two secrets of happiness state

(i) p1x1 + p2x2 = m

(ii) MRS(= −2x2

x1

)= − p1

p2

The first condition indicates Jim’s budget line; he has to utilize all his monetary resource to maximizeutility. And the second condition requires the MRS coincide with the relative price. It implies thatthe consumer’s subjective exchange ratio(MRS) must be the same as the market’s objective exchangeratio at the optimal bundle. Figure 2 illustrates it in the graph. For a bundle to be optimal, it must beon the budget line and the indifference curve through it must be tangent to the budget line.

Note that solving (ii) for x1 gives x1 = 2p2

p1x2. Plugging it in (i) for x1 gives

p1

(2

p2

p1x2

)+ p2x2 = m → x2 =

m3p2

1

Consequently, x1 = 2p2

p1

m3p2

=2m3p1

.

(e) (15pts) When p1 = 40, p2 = 20, m = 600,

x1 =2 · 6003 · 40

= 10 and x2 =600

3 · 20= 10.

When p1 falls down to p1 = 20,

x1 =2 · 6003 · 20

= 20 and x2 =600

3 · 20= 10.

The total change in x1 is 20 − 10 = 10. Note that the demand for Port wine is increasing as p1decreases. It is an ordinary good. Figure 3 illustrate the total change in x1.

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Demand for Port Wines (x1)

Dem

and

for

Boo

ks (

x 2)

(10,10) (20,10)

Total Change in x1

Figure 3: Total Change in x1

0 7.5 15 22.5 300

7.5

15

22.5

30

Demand for Port Wines(x1)

Dem

and

for

Boo

ks(x

2)

Original B/LFinal B/LA CPivot @ AB

AC

B

SE

IE

Figure 4: Substitution and Income Effect

(f) (10pts) In order to decompose it, we have to adjust money income m′, which represents the amountof money income that is necessary to make the original bundle(i.e. before p1 changes it was (10,10))affordable at the new price.

m′ = 20 · 10 + 20 · 10 = 400.

After p1 changes, Jim needs m′ = 400 to purchase the original commodity bundle. Then we needfigure out what is the optimal choice for Port wine at the new price and this m′. Plugging p1 = 20 and

m = 400 into the formula x1 =2m3p1

gives us x1 =403

. The substitution effect is given by the difference

between the new demand 403 and the original one 10

SE =403

− 10 =103

The associated income effect immediately follows from Slutsky’s equation, which tells us that the totalchange can be decomposed into the substitution and income effect.

IE = Total Change − SE = 10 − 103

=203

In figure 4, I denote by A,B and C the original optimal choice, the final optimal choice, and theintermediate optimal choice we obtained from m′, respectively. The substitution effect is the gapbetween A and C on the horizontal axis while the income effect is between C and B.

2

Problem 2 (20pts)

(a) (2pts) PV = 100 +300

1 + r= 100 +

3003

= 200

(b) (3pts) The budget constraint is written by

C1 +13

C2 = 200

To wit briefly, the left-hand side is the present value of consumption stream (C1, C2). Observe thatI discounted C2 by 1

1+r = 13 to express it in terms of present value. The right-hand side simply

represents the present value of his income we got in part (a). The budget line tells us that they shouldbe balanced. See figure 5.

0 100 2000

300

600

Consumption Today (C1)

Con

sum

ptio

n To

mor

row

(C2)

Endowment

Saving

Borrowing

Future Value=(1+r)m

1+m

2

Present Value=m

1 + m

2/(1+r)

Figure 5: Intertemporal Choice Model

(c) (7pts) For δ = 2, observe that the utility function comes down to

U(C1, C2) = ln C1 +13

ln C2

which is a Cobb-Douglas function. Using its magic formula, we obtain

C1 =a

a + b· m

p1=

11 + 1

3· 200

1= 150

C2 =b

a + b· m

p2=

13

1 + 13· 200

13

= 150.

To consume 150 today, he must save 100 − 150 = −50, that is, he must borrow 50. He does smoothhis consumption plan because C1 = C2 = 150.

(d) (3pts) PV =xr=

5000.01

= 50, 000.

(e) (5pts) The characterization equation is given by

1, 000, 0000.05

[1 − 1

(1 + 0.05)45

]=

C0.05

[1 − 1

(1 + 0.05)80

]The left-hand side and the right-hand side represents the present value of earnings for next 45 yearsand the present value of consumptions for next 80 years, respectively. And the constant level ofconsumption C comes from the equation in which these two values are balanced.

3

Problem 3 (15pts)

(a) (2pts) U(x1, x2) = min{x1, 4x2}

(b) (3pts) See figure 6.

0 5 10 15 200

2

4

6

8

10

12

14

16

18

20

Demand for Sauerkraut(x1)

Dem

and

for M

ushr

oom

s(x 2)

Optimal Proportion Line=Price Offer Curve

=Income Offer Curve

The Optimal Bundle (16,4)

Figure 6: Perfect Complements

(c) (6pts) The associated ”two secrets of happiness” with perfect complements is

(i) 2x1 + 2x2 = 40(ii) x1 = 4x2.

The first condition indicates the consumer’s balanced budget, and its economic intuition is the sameas before: All income must be exhausted. The second condition represents the optimal proportion(4 : 1) between quantities demanded for sauerkraut and mushrooms.

To figure out the optimal choice, we plug in 4x2 for x1 in the budget line and obtain

2(4x2) + 2x2 = 10x2 = 40 ⇒ x2 = 4.

Substituting into x1 = 4x2 we get x1 = 16. The solution is interior as x1 and x2 are strictly positive.

(d) (4pts) Since the two goods are perfect complements, both the price offer curve and the income offercurve would coincide with the optimal proportion line. Refer Figure 6.

Problem 4 (15pts)

(a) (4pts) x1 + x2 = 40. The value of endowment, the total resources available for consumption, isp1ω1 + p2ω2 = 10 + 30 = 40. See Figure 7.

(b) (4pts) Since p1 = p2 = 1 and the two goods are perfect substitutes, Sam is willing to consume the goodwith more marginal utilities. In this example, it follows from the utility function that apples(good 1)yield more marginal utilities. Therefore, x1 = 40 and x2 = 0. Since x2 = 0, the solution is not interiorbut it is at the corner.

(c) (4pts) The net demand for apples is x1 − ω1 = 40 − 10 = 30, which is positive. It means that Sam is anet buyer of apples. The net demand for oranges, on the other hand, is x2 − ω2 = 0 − 30 = −30 < 0.Hence he is a net seller for oranges.

(d) (3pts) It is because apples and oranges, in this example, are perfectly substitutable. Hence the con-sumer would choose either one good only, depending on the marginal utility from a dollar.

4

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Demand for Apples (x1)

Dem

and

for

Ora

nges

(x 2) Endowment

Figure 7: Budget Line and Endowment Point

Bonus Problem

Perpetuity

A perpetuity in an annuity that has no definite end, or a stream of cash payments, say x, that continuesindefinitely. With the (risk-free) interest rate r, we can write its present value as

PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 +

x(1 + r)4 + · · ·

=x

(1 + r)+

[x

(1 + r)2 +x

(1 + r)3 +x

(1 + r)4 + · · ·]

=x

(1 + r)+

1(1 + r)

[x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + · · ·

]︸︷︷︸

=PV

In the second step, I just partitioned the sum into the first term and the remainder using the parenthesis.The key step is the next one, factoring 1

1+r out of the remainder. Then we come up with the same sequenceas the original PV. It leads to the following simple equation

PV =x

(1 + r)+

1(1 + r)

· PV

Solving for PV gives us the desired formula, PV =xr

.

Annuity

The annuity is an asset that promises a terminating stream of fixed payments over a prespecified period oftime. Its value is closely linked to ”No Arbitrage Principle”, the most fundamental principle in finance. Itstates that ”two assets with identical cash flows must trade at the same price.” With this principle in hand,consider one annuity which guarantees to annually give us payment x over T years. Now it is the key ideathat with the two perpetuities we can replicate the identical cash flows as the annuity.

• Perpetuity 1 - It promises x payment forever from now on.

• Perpetuity 2 - It promises x payment forever but T + 1 years from now.

5

Observe that the cash flow of the annuity is the same as the cash flow of ”Perpetuity 1 − Perpetuity 2”. Byno arbitrage condition, then, it must be the case that

PV(Annuity) = PV(Perpetuity 1) − PV(Perpetuity 2)

=xr− x

r· 1(1 + r)T

=xr

[1 − 1

(1 + r)T

].

6

ECON 301 - ANSWER KEY TO MIDTERM 1(GROUP B) TA : CHANG-KOO(CK) CHIMARCH 4, 2011

Problem 1 (50pts)


0 3 6 9 120

3

6

9

12


Dem

and

for

book

s(x 2)

After Tax is levied



Dem

and

for

Boo

ks(x

2)

The Optimal Bundle




MU1 =2x1

and MU2 =1x2



1/x2= −2x2





(i) p1x1 + p2x2 = m

(ii) MRS(= −2x2

x1

)= − p1

p2




p1

(2

p2

p1x2

)+ p2x2 = m → x2 =

m3p2

1


p1

m3p2

=2m3p1

.

(e) (15pts) When p1 = 20, p2 = 10, m = 120,

x1 =2 · 1203 · 20

= 4 and x2 =120

3 · 10= 4.


x1 =2 · 1203 · 10

= 8 and x2 =120

3 · 10= 4.

The total change in x1 is 8 − 4 = 4. Note that the demand for Port wine is increasing as p1 decreases.It is an ordinary good. Figure 3 illustrate the total change in x1.

0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)

Total Change inx

1

(4,4)(8,4)


0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)


AC

B

SE

IE



m′ = 4 · 10 + 4 · 10 = 80.



gives us x1 =163



SE =163

− 4 =43


IE = Total Change − SE = 4 − 43=

83


2

Problem 2 (20pts)

(a) (2pts) PV = 20 +60

1 + r= 20 +

603

= 40


C1 +13

C2 = 40




0 20 400

60

120


Con

sum

ptio

n To

mor

row

(C2)

Endowment

Saving

Borrowing

Future Value=(1+r)m

1+m

2

Present Value=m

1 + m

2/(1+r)



U(C1, C2) = ln C1 +13

ln C2


C1 =a

a + b· m

p1=

11 + 1

3· 40

1= 30

C2 =b

a + b· m

p2=

13

1 + 13· 40

13

= 30.

To consume 30 today, he must save 20 − 30 = −10, that is, he must borrow 10. He does smooth hisconsumption plan because C1 = C2 = 30.

(d) (3pts) PV =xr=

5000.1

= 5, 000.


500, 0000.01

[1 − 1

(1 + 0.01)45

]=

C0.01

[1 − 1

(1 + 0.01)80


3

Problem 3 (15pts)

(a) (2pts) U(x1, x2) = min{4x1, x2}


0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20


Dem

and

for M

ushr

oom

s(x 2)

Optimal Proportion LIne= Price Offer Curve

= Income Offer Curve




(i) 2x1 + 2x2 = 40(ii) 4x1 = x2.



2x1 + 2(4x1) = 10x1 = 40. ⇒ x1 = 4.



Problem 4 (15pts)


(b) (4pts) Since p1 = p2 = 1 and the two goods are perfect substitutes, Sam is willing to consume the goodwith more marginal utilities. In this example, it follows from the utility function that oranges(good 2)yield more marginal utilities. Therefore, x2 = 300 and x1 = 0. Since x1 = 0, the solution is not interiorbut it is at the corner.

(c) (4pts) The net demand for apples is x1 − ω1 = 0 − 100 = −100, which is negative. It means that Samis a net seller of apples. The net demand for oranges, on the other hand, is x2 − ω2 = 300 − 200 =100 > 0. Hence he is a net buyer for oranges.


4

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

x1 − Demand for apples

x 2 − D

eman

d fo

r ora

nges Endowment (100,200)


Bonus Problem

Perpetuity


PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 +

x(1 + r)4 + · · ·

=x

(1 + r)+

[x

(1 + r)2 +x

(1 + r)3 +x

(1 + r)4 + · · ·]

=x

(1 + r)+

1(1 + r)

[x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + · · ·

]︸︷︷︸

=PV



PV =x

(1 + r)+

1(1 + r)

· PV


.

Annuity




5



=xr− x

r· 1(1 + r)T

=xr

[1 − 1

(1 + r)T

].

6

ECON 301 - ANSWER KEY TO MIDTERM 1(GROUP C) TA : CHANG-KOO(CK) CHIMARCH 4, 2011

Problem 1 (50pts)

(a) (5pts) The budget constraint is given by 5x1 + 5x2 = 60. With the tax τ = 100%, it becomes 5(1 +τ)x1 + 5x2 = 10x1 + 5x2 = 60. Since it plays a role in increasing p1 by 100%, the budget line mustshift in as displayed in Figure 1.

0 3 6 9 120

3

6

9

12


Dem

and

for

book

s(x 2)

After Tax is levied



Dem

and

for

Boo

ks(x

2)

The Optimal Bundle




MU1 =2x1

and MU2 =1x2



1/x2= −2x2





(i) p1x1 + p2x2 = m

(ii) MRS(= −2x2

x1

)= − p1

p2




p1

(2

p2

p1x2

)+ p2x2 = m → x2 =

m3p2

1


p1

m3p2

=2m3p1

.

(e) (15pts) When p1 = 10, p2 = 5, m = 60,

x1 =2 · 603 · 10

= 4 and x2 =60

3 · 5= 4.


x1 =2 · 603 · 5

= 8 and x2 =60

3 · 5= 4.


0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)

Total Change inx

1

(4,4)(8,4)


0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)


AC

B

SE

IE



m′ = 4 · 5 + 4 · 5 = 40.

After p1 changes, Jim needs m′ = 40 to purchase the original commodity bundle. Then we need figureout what is the optimal choice for Port wine at the new price and this m′. Plugging p1 = 5 and m = 40

into the formula x1 =2m3p1

gives us x1 =163

. The substitution effect is given by the difference between

the new demand 163 and the original one 4

SE =163

− 4 =43



83


2

Problem 2 (20pts)

(a) (2pts) PV = 40 +120

1 + r= 40 +

1203

= 80


C1 +13

C2 = 80




0 40 800

120

240


Con

sum

ptio

n To

mor

row

(C2)

Endowment

Saving

Borrowing

Future Value=(1+r)m

1+m

2

Present Value=m

1 + m

2/(1+r)



U(C1, C2) = ln C1 +13

ln C2


C1 =a

a + b· m

p1=

11 + 1

3· 80

1= 60

C2 =b

a + b· m

p2=

13

1 + 13· 80

13

= 60.


(d) (3pts) PV =xr=

10000.02

= 50, 000.


100, 0000.01

[1 − 1

(1 + 0.01)40

]=

C0.01

[1 − 1

(1 + 0.01)70


3

Problem 3 (15pts)

(a) (2pts) U(x1, x2) = min{3x1, x2}


0 5 10 15 200

5

10

15

20

25

30

35

40

45


Dem

and

for M

ushr

oom

s(x 2)


=Income Offer Curve




(i) 2x1 + 2x2 = 80(ii) 3x1 = x2.



2x1 + 2(3x1) = 8x1 = 80 ⇒ x1 = 8.



Problem 4 (15pts)


(b) (4pts) Since p1 = p2 = 1 and the two goods are perfect substitutes, Sam is willing to consume the goodwith more marginal utilities. In this example, it follows from the utility function that oranges(good 2)yield more marginal utilities. Therefore, x2 = 300 and x1 = 0. Since x1 = 0, the solution is not interiorbut it is at the corner.

(c) (4pts) The net demand for apples is x1 − ω1 = 0 − 100 = −100, which is negative. It means that Samis a net seller of apples. The net demand for oranges, on the other hand, is x2 − ω2 = 300 − 200 =100 > 0. Hence he is a net buyer for oranges.


4

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350


Dem

and

for

Ora

nges

(x 2)

Endowment


Bonus Problem

Perpetuity


PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 +

x(1 + r)4 + · · ·

=x

(1 + r)+

[x

(1 + r)2 +x

(1 + r)3 +x

(1 + r)4 + · · ·]

=x

(1 + r)+

1(1 + r)

[x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + · · ·

]︸︷︷︸

=PV



PV =x

(1 + r)+

1(1 + r)

· PV


.

Annuity




5



=xr− x

r· 1(1 + r)T

=xr

[1 − 1

(1 + r)T

].

6

ECON 301 - ANSWER KEY TO MIDTERM 1(GROUP D) TA : CHANG-KOO(CK) CHIMARCH 4, 2011

Problem 1 (50pts)


0 3 6 9 120

3

6

9

12


Dem

and

for

book

s(x 2)

After Tax is levied



Dem

and

for

Boo

ks(x

2)

The Optimal Bundle




MU1 =2x1

and MU2 =1x2



1/x2= −2x2





(i) p1x1 + p2x2 = m

(ii) MRS(= −2x2

x1

)= − p1

p2




p1

(2

p2

p1x2

)+ p2x2 = m → x2 =

m3p2

1


p1

m3p2

=2m3p1

.

(e) (15pts) When p1 = 40, p2 = 20, m = 240,

x1 =2 · 2403 · 40

= 4 and x2 =240

3 · 20= 4.


x1 =2 · 2403 · 20

= 8 and x2 =240

3 · 20= 4.


0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)

Total Change inx

1

(4,4)(8,4)


0 3 6 9 120

3

6

9

12


Dem

and

for

Boo

ks(x

2)


AC

B

SE

IE



m′ = 4 · 20 + 4 · 20 = 160.



gives us x1 =163



SE =163

− 4 =43



83


2

Problem 2 (20pts)

(a) (2pts) PV = 80 +240

1 + r= 80 +

2403

= 160


C1 +13

C2 = 160




0 80 1600

240

480


Con

sum

ptio

n To

mor

row

(C2)

Endowment

Saving

Borrowing

Future Value=(1+r)m

1+m

2

Present Value=m

1 + m

2/(1+r)



U(C1, C2) = ln C1 +13

ln C2


C1 =a

a + b· m

p1=

11 + 1

3· 160

1= 120

C2 =b

a + b· m

p2=

13

1 + 13· 160

13

= 120.


(d) (3pts) PV =xr=

2000.1

= 2, 000.


200, 0000.01

[1 − 1

(1 + 0.01)40

]=

C0.01

[1 − 1

(1 + 0.01)70


3

Problem 3 (15pts)

(a) (2pts) U(x1, x2) = min{x1, 3x2}


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80


Dem

and

for M

ushr

oom

s(x 2)


=Income Offer Curve




(i) 2x1 + 2x2 = 160(ii) x1 = 3x2.



2(3x2) + 2x2 = 8x2 = 160 ⇒ x2 = 20.



Problem 4 (15pts)


(b) (4pts) Since p1 = p2 = 1 and the two goods are perfect substitutes, Sam is willing to consume the goodwith more marginal utilities. In this example, it follows from the utility function that apples(good 1)yield more marginal utilities. Therefore, x1 = 300 and x2 = 0. Since x2 = 0, the solution is not interiorbut it is at the corner.

(c) (4pts) The net demand for apples is x1 − ω1 = 300 − 100 = 200, which is positive. It means that Samis a net buyer of apples. The net demand for oranges, on the other hand, is x2 − ω2 = 0 − 200 =−200 < 0. Hence he is a net seller for oranges.


4

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350


Dem

and

for

Ora

nges

(x 2)

Endowment


Bonus Problem

Perpetuity


PV =x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 +

x(1 + r)4 + · · ·

=x

(1 + r)+

[x

(1 + r)2 +x

(1 + r)3 +x

(1 + r)4 + · · ·]

=x

(1 + r)+

1(1 + r)

[x

(1 + r)+

x(1 + r)2 +

x(1 + r)3 + · · ·

]︸︷︷︸

=PV



PV =x

(1 + r)+

1(1 + r)

· PV


.

Annuity




5



=xr− x

r· 1(1 + r)T

=xr

[1 − 1

(1 + r)T

].

6


Midterm 1 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (50+20+15+15=100points) + bonus (10 extra "e" points). Make sure you answer the �rst four questions before working on thebonus one!

Problem 1. (50 points) To reenergize for Econ 301 class, in the morning, Tony always drinks MountainDew (x1) and eats Burritos (x2).

a) Suppose Mountain Dew costs p1 = $2; burrito costs p2 = $10; and his daily budget is m = $40:Show graphically the budget constraint in the commodity space. Mark the two extreme consumption bundles(mark concrete values). On the same graph, show how the budget set is a¤ected by in�ation of 100% thata¤ects prices of both commodities but does not a¤ect income (so his income stays the same m = $40)?

b) Tony�s preferences are given by the following utility function

U (x1; x2) = x71x72:

Find Tony�s marginal rate of substitution (MRS) as a function of x1; x2 (give a formula for MRS).- What is the value of MRS at consumption bundle (2; 1) (give a number)?- Which of the two goods is more valuable, soda or burrito, if Tony drinks two Mountain Dews and

consumes one burrito?- Depict his indi¤erence curve map in a commodity space. Mark the slope of the indi¤erence curve at

the bundle (2; 1).c) In the commodity space (x1; x2) ; �nd (geometrically) Tony�s optimal choice,.assuming pre-in�ation

prices p1 = $2; and p2 = $10. Describe how the two properties of the optimal bundle, known as two"secrets of happiness" (two short sentences) can be seen in the graph.


- Provide some economic intuition behind the two conditions (ca. two sentences for each).- Derive the optimal consumption of x1 and x2 as a function of p1; p2;m (show the derivation).- What fraction of income is spent on burritos (give the percentage)?- Find analytically and geometrically the demand curve for Mountain Dew (given p2 = $10 and m =

$40) and Engel curve (given p1 = $2; and p2 = $10)- Are they Gi¤en goods? Why? (yes/no answer + one sentence).e) Using your formula from d) �nd the optimal consumption levels of both types of commodities (x1; x2)

for:- p1 = $2; p2 = $10 and m = $40 (give two numbers).and after the price of Mountain Dew decreased:- for p1 = $1; p2 = $10 and m = $40 (give two numbers).What is the total change in consumption of Mountain Dew? (give a number). Illustrate the change on

the graph.f) decompose the total change in consumption of x1 from e) into a substitution and income e¤ect.


Problem 2. (20 points) Bill is a wild-animal lover. From his recent trip to Galapagos Islands hebrought a small Iguana. His new pet has only three legs: one left and two right. (Iguanas use magma heatedsoil to warm their eggs and his favorite pet lost one left leg during the last volcano eruption). To survivethe famous Madisonian winter, the iguana has to wear shoes,.left (x1) and right ones (x2).

a) Write down Bill�s utility function representing his preferences over right and left shoe (functionU (x1; x2)).

b) In the commodity space (x1; x2), carefully depict Bill�s indi¤erence curves.

1

c) Find analytically Bill�s demand for shoes if p1 = $6 and p2 = $2 and Bill�s budget for iguana shoesis m = $40: Is the solution interior? (give two numbers and a yes/no answer).

d) Illustrate Bill�s optimal choice on the graph including the indi¤erence curves and the budget set.e) Suppose the price of left shoe goes down to p1 = $1: Find Bill�s new demand for shoes. What can

you say about the substitution e¤ect? How about the income e¤ect? (Answer the latter question withoutany calculations, using only a graph).

Problem 3. (15 points) Ramon decides about his new collections of postage stamps. He is interested intwo themes: "the birds of the world", x1, (measures the number of stamps in the subcollection with birds)and "the famous mathematicians", x2. The utility derived from the collection is given by

U (x1; x2) = x1 + 100� lnx2:

a) What is the optimal collection of stamps if the prices are p1 = 1 and p2 = 1 and m = 200. (�nd twonumbers x1 and x2). Is your solution interior, or corner?

b) Find the optimal collection if the prices are still p1 = 1 and p2 = 1, but the income is only m = 50.Depict Ramon�s optimal choice in the commodity space.

Problem 4. (15 points) Jacob can use his 24h for leisure R; or work. The hourly wage rate is w = 10:Jacob spends all his money on cheese curds C.

a) Draw Jacob�s budget set, given the price of cheese is pc = $5 (mark the endowment point).Let the utility function be given by

U (x1; x2) = R+ C:

b) Are leisure and cheese curds perfect complements, perfect substitutes or none of them?c) What is the optimal choice of leisure, cheese curds and labor supply? (Find geometrically and give

three numbers).d) Harder: Plot a graph with labor supply (horizontal axis) and wage rate (vertical one) assuming

pc = $5 (hint: for what value of w do we go from one "bang" to the other "bang" solution?)

Bonus Problem. (extra 10 points) Let U (x1; x2) = x31x2; and V (x1; x2) = 3 lnx1 + lnx2 be twoutility functions.

a) Show that U (�) is a monotone transformation of V (�), and hence they de�ne the same preferences.b) Derive MRS for each of the two functions. Using the two formulas for MRS, argue that the functions

de�ne the same indi¤erence curve maps.

2


Solutions to midterm 1 (Group A)

Problem 1. (50 points)a) With p1 = $2; p2 = $10 and m = $40 the budget set is (two extreme consumption bundles are 20

and 4). In�ation that a¤ects only prices shifts budget line inwards.

b) Tony�s marginal rate of substitution (MRS)

MRS = �MU1MU2

= �x2x1

- The value of MRS at consumption bundle (2; 1) is

jMRSj = j � 12j = 1

2

- Burrito (x2) is more valuable than Mountain Dew (x1)- Tony�s indi¤erence curve map is. (the slope of her indi¤erent curve that passes through bundle (2; 1)

is � 12 .

c) Tony�s optimal choice is

- the two geometric properties of the optimal bundle, known as two "secrets of happiness" are:1. At the optimal bundle, the indi¤erence curve is tangent to a budget set2. The optimal bundle is located on budget lined) mathematically the two secrets of happiness, are�

MRS = �p1p2

p1x1 + p2x2 = m

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- the economic intuition behind the two conditions is:The individual value of x1 in terms of x2 coincides with the market valueThe income of a consumer is exhausted- the optimal consumption of x1 and x2 as a function of p1; p2;m can be found as followsFrom the MRS condition

MRS = �x2x1= �p1

p2

hencex2 =

p1p2x1

plugging in budget constraint

p1x1 + p2

�p1p2x1

�= m

Solving for x1 gives

x1 =1

2

m

p1

Plugging in

x2 =p1p2

�1

2

m

p1

�=1

2

m

p2

- the fraction of income spent on burritos is

p1x1m

=1

2= 50%

- and the demand curve for burritos book (given p2 = $10; and m = $40) and Engel curve (givenp1 = $2; and p2 = $10)

Demand curve

x1 =1

2

m

p1=1

2

40

p1=20

p1

and hence inverse demand is

p1 (x1) =20

x1

Geometrically

Engel curve: Since

x1 =1

2

m

p1

at p1 = $2

x1 =1

2

m

2=1

4m

hencem (x1) = 4x1

Geometrically

2

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- are they Gi¤en goods? Why? (yes/no answer + one sentence).No, because the demand curve is downwardslopping on the whole domain.e) The optimal consumption levels for (x1; x2) :- at p1 = $2; p2 = $10 and m = $40

x1 =1

2

m

p1=1

2

40

2= 10

and

x2 =1

2

m

p2=1

2

40

10= 2

and after the price of science-�ction book decreased, for p1 = $1; p2 = $10 and m = $40

x1 =1

2

m

p1= 20

and

x2 =1

2

m

p2=1

2

40

10= 2

Hence the total change in consumption of x1 is

�x1 = 20� 10 = 10

Geometrically

f) Substitution e¤ect: auxiliary budget

m0 = 10� 1 + 10� 2 = 30

and hence

x1 =1

2

30

1= 15

so SE is equal toSE = 15� 10 = 5

and income e¤ect isIE = 10� 5 = 5

Problem 2. .a) Bill�s utility function is

U (x1; x2) = min (2x1; x2)

b) indi¤erence curves.in the commodity space (x1; x2) are

c) Bill�s demand for shoes is

x2 = 2x1

6x1 + 2x2 = 40

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6x1 + 2 (2x1) = 40

x1 =40

10= 4

x2= 8

d) geometrically Bills�s optimal choice is

e) when the price of a left shoe goes down to p1 = $1: the new demand is given by the system ofequations

x2 = 2x1

x1 + 2x2 = 40

and hence demand is

x1 = 8

x2 = 16

The substitution e¤ect is zero (perfect complements) and the income e¤ect is 4.

Problem 3.a) the two secrets of happiness are

� x2100

= �1x1 + x2 = 200

and hence x2 = 100 and x1 = 100: Since both are positive, this is interior solution.b) the two secrets of happiness are

� x2100

= �1x1 + x2 = 50

and hence secrets of happiness give x2 = 100 and x1 = �50: Since consumption must be non-negative theoptimal consumption is x1 = 0 and x2 = 50, which is a cornet solution.

Problem 4.a) Jacob�s budget set, with w = $10 and pc = $5 isIncome is m = 10� 24 = $240

b) They are perfect substitutes

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c) jMRSj = 1 < wpc= 2 which implies that Jacob cares less about leisure than consumption,. therefore

he will spend the whole day at work

R = 0; LS = 24 and C = 2410

5= 48

d)

Bonus Problem. (extra 10 points)a) Monotone transformation ln () : Take a log of U

lnU () = lnx31x2 = lnx31 + lnx2 = 3 lnx1 + lnx2 = V ()

where we used two properties of ln function.b) For U () ; marginal rate of substitution is

MRS = �MU1MU2

= �3x21x2x31

= �3x2x1

and for V ()

MRS = �MU1MU2

= �3=x11=x2

= �3x2x1

and hence MRS coincides for all (x1; x2).It follows that the slopes of indi¤erence curves are the same at anypoint and hence they must be the same.

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Solutions to midterm 1 (Group A)

Problem 1. (60 points)a) Maggie�s marginal rate of substitution (MRS)

MRS = �MU1MU2

= �12

x2x1

- the value of MRS at consumption bundle (2; 2) is

MRS = �12

- "The Marginal Rate of Substitution is a (marginal) value of a science �nction books in terms of romancenovels"

- after taking away 0.00001 of a science-�ction book, to keep her indi¤erent one must compensate Maggiein terms of romance books

0:00001�12= 0:000005

- her indi¤erence curve map is. (the slope of her indi¤erent curve that passes through bundle (2; 2) is� 12 .

b) With p1 = $10; p2 = $5 and m = $300 the budget set is (two extreme consumption bundles are 30and 60). The budget set with ad valorem tax shifts inwards.

c) In the commodity space (x1; x2) ; �nd (geometrically) Maggie�s optimal choice.

- the two geometric properties of the optimal bundle, known as two "secrets of happiness" are:1. At the optimal bundle, the indi¤erence curve is tangent to a budget set2. The oprimal bundle is locaded on budget lined) mathematically the two secrets of happiness, are�

MRS = �p1p2

p1x1 + p2x2 = m

- the economic intuition behind the two conditions is:

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The individual value of x1 in temrms of x2 coincides with the market valueThe income of a consumer is exchausted- the optimal consumption of x1 and x2 as a function of p1; p2;m can be found as followsFrom the MRS condition

MRS = �12

x2x1= �p1

p2

hencex2 = 2

p1p2x1


p1x1 + p2

�2p1p2x1

�= m


x1 =1

3

m

p1

Pluging in

x2 = 2p1p2

�1

3

m

p1

�=2

3

m

p2

- the fraction of income spent on science - �ction novels is

p1x1m

=1

3= 33%

- and the demand curve for science �nction book (given p2 = $5; and m = $300) and Engel curve(given p1 = $10; and p2 = $5)

Demand curve

x1 =1

3

m

p1=100

p1

and hence invese demand is

p1 (x1) =100

x1

Geometrically

Engel curve: Since

x1 =1

3

m

p1

at p1 = $10

x1 =1

3

m

10=1

30m

hencem (x1) = 30x1

Geomettrically

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- are science-�ction books normal goods? Why? (yes/no answer + one sentence).Yes, because their demand increases in income- are they Gi¤en goods? Why? (yes/no answer + one sentence).No, becase the demand curve is downwardslopping on the whole domain.e) The optimal consumption levels for both types of books (x1; x2) :- for p1 = $10; p2 = $5 and m = $300

x1 =1

3

m

p1=1

3

300

10= 10

and

x2 =2

3

m

p2=2

3

300

5= 40


x1 =1

3

m

p1=1

3

300

5= 20

and

x2 =2

3

m

p2=2

3

300

5= 40


�x1 = 20� 10 = 10

Geometrically


m0 = 10� 5 + 5� 40 = 250

and hence

x1 =1

3

250

5=50

3= 16

2

3so SE is equal to

SE = 162

3� 10 = 62

3and income e¤ect is

IE = 10� 623= 3

1

3"The substitution e¤ect is attributed to the pure change in relative price induced by the decrease of

nominal price p1""The income e¤ect can be attribute to the pure change in real income induced by the decrease of nominal

price p1"

g) From the functions

V (x1; x2) = 30�x101 � x202

�+ 3

V (x1; x2) = 10x1 � 20x2V (x1; x2) = 10 lnx1 � 20 lnx2 + 2V (x1; x2) = 10 lnx1 + 20 lnx2 + 7

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the �rst and last represent Maggie�s preferences - they are monotone transformations of U () : The �rsttransformation is

f (U) = 30U + 3

and the second isf (U) = lnU + 7

Problem 2. .a) Jimmy�s utility function is

U (x1; x2) = min (x1; 4x2)

b) in the commodity space (x1; x2), carefully depict Jimmy�s indi¤erence curves.

c) Jimmy�s demand for parts is interior and is given by

x2 =1

4x1

50x1 + 100x2 = 600

50x1 + 1001

4x1 = 600

x1 =600

75= 8

x2=1

4�8 = 2

d) geometrically Jimmy�s optimal choice is

e) when the price of a wheel goes down to p1 = $25: the new demand is

x2 =1

4x1

25x1 + 100x2 = 600

x1 = 12

x2 = 3

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Thte substitution e¤ec is zero (perfect complements) and the income e¤ect is 4 wheels.

Problem 3.a) Ramon�s budget set, with p1 = $5 and p2 = $1 isBudget set: m = 25 + 50 = 75

b) The optimal demand for packs and single cigars can be found as follows. Since

�MRS = 10 > p1p2= 5

therefore the total income will be invested in packs

x1 =75

5= 15 and x2 = 0

c) When prices are p1 = 20 and p2 = 1

�MRS = 10 < p1p2= 20

and hence the total income will be invested in x2: At such prices demands are

x1 = 0 and x2 =5� 20 + 50

1= 150

d) The demands areIf p1p2 < 10 then

x1 =p1!1 + p2!2

p1x2 = 0

if p1p2 > 10 then

x1 = 0

x2 =p1!1 + p2!2

p2

and if p1p2 = 10

x1 = �p1!1 + p2!2

p

x2 = (1� �) p1!1 + p2!2p2

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for � 2 (0; 1)

The demand curve for x1 is (p2 = 1; !1 = 5 and !2 = 50).

Bonus Problem. (extra 10 points) Depict a map of indi¤erence curves that is consistent witha) inferior goods

b) Gi¤en goods

(Make sure you explain why these graphs represent the respective preferences.)

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Midterm 1 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (50+25+15+10=100 points).

Problem 1. (50 points)Patrick spends his income on books (x1) and CD (x2).a) Suppose the price of a book is p1= $10, the price of a CD is p2= $5; and Patrick�s daily budget is m = $40: Show

graphically Patrick�s budget constraint, marking his real incomes in terms of books and CDs. On the same graph, showhow his budget set is a¤ected by a gift of 2 CDs (assume that he can always dispose the gift).

b) Patrick�s preferences are given by the following utility function

U (x1; x2)= x1+2 lnx2:

Find Patrick�s marginal rate of substitution (MRS) for any bundle (x1; x2) (give the formula for MRS).- What is the value of MRS at consumption bundle (5; 8) (give a number)?- Suppose Patrick �consumes� 5 books and 8 CDs and one takes away 0:0001 of a book. What is compensation in

terms of CDs is su¢ cient to make Patrick indi¤erent?- Depict the indi¤erence curve map in a commodity space. Mark the slope of the indi¤erence curve at bundle (5; 8).c) From now on assume no gift. In the commodity space (x1; x2) ; �nd (geometrically) Patrick�s optimal choice.

Describe how the two "secrets of happiness" can be seen geometrically in the graph (two short sentences).d) Write down mathematically two secrets of happiness, assuming that p1; p2;m are parameters (and not concrete

values). Provide economic intuition behind the two conditions (ca. two sentences for each).e) Using the two conditions from d) �nd the optimal consumption levels of both types of commodities (x1; x2) for:- p1= $10; p2= $5 and m = $40 (give two numbers).and after the price of a book increases:- for p1= $30; p2= $5 and m = $40 (give two numbers).Is each of the solutions interior? Illustrate the change on the graph.f) Is the marginal utility of a dollar invested in books and CD equal? (Find two numbers for the parameters before

and after the change.) In case they are not, explain why not equalizing the marginal utility of a dollar is consistent withoptimum.

Problem 2. (25 points) Michael always consumes three hamburgers x1 along with one Coke x2 (this is the onlyhealthy combination of the two products!) .

a) Propose Michael�s utility function that represents his preferences over hamburgers and Coke (function U (x1; x2)).b) In the commodity space (x1; x2), carefully depict Michael�s indi¤erence curves (and mark the optimal proportion

line).c) Write down two secrets of happiness (give two equations) that determine his optimal choice (for parameters p1 and

p2 and m). Explain economic intuition behind the conditions (one sentence for each secret).d) Find Michaels�s optimal choice of x1 and x2 as a function of (p1; p2 and m): Is the choice (solution) interior for

any price and income? (Give formulas x1(p1; p2;m) and a yes-no answer.)e) Using x1(p1; p2;m) derived in d), determine whether goods are 1) ordinary or Gi¤en, 2) normal or inferior and

3) gross substitutes or gross complements (for points 1-3 points chose one option and give one sentence explaining yourchoice).

f) Compare the substitution and income e¤ects relative to a total change of consumption TCH? (You do not have togive any number. Just relate two e¤ects to TCH:)

Problem 3. (15 points) Adam spends all his income on food (x1) and clothing x2):He is a fairly sophisticated fellowand his utility function is quite complicated

U (x1; x2)=

"700�

rlnh(2 lnx1+ lnx2)

2i+10

#800:

1

a) Argue that Adam is not really that sophisticated, as his preferences can be represented by a signi�cantly simplerutility function. (one sentence + simpler utility function)

b) What is his optimal choice of x1 and x2if the prices are p1= 4 and p2= 4 and m = 1200 (�nd two numbers x1 andx2). Is your solution interior, or corner?

c) Assume p2= 4 and m = 1200: Find analitically and geometrically the demand curve and the price o¤er curve.Hint: In b) and c) you can use the magic formula.

Problem 4. (10 points) Frank can use his 24h for leisure R or work. The hourly wage rate is w = 20: Frank is acommitted skier and uses all his income on ski passes in Devil�s Head Resort C.

a) Draw Frank�s budget set, given that the price of one ski pass is pc= $10 (mark the endowment point). What is theslope of his budget line? Interpret this slope economically.

Let the utility function be given byU (x1; x2)= RC

5:

b) What is the real wage? (formula + nubmer) How can the real wage be seen in the graph of a budget set?c) What is the optimal choice of leisure, ski passes and labor supply? (Find the optimal choice geometrically and give

three numbers).

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Midterm 1 (Group B)





U (x1; x2)= 2x1+6 lnx2:

Find Patrick�s marginal rate of substitution (MRS) for any bundle (x1; x2) (give the formula for MRS).- What is the value of MRS at consumption bundle (3; 6) (give a number)?- Suppose Patrick �consumes�3 books and 6 CDs and one takes away 0:0001 of a CD. What is compensation in terms

of CDs is su¢ cient to make Patrick indi¤erent?- Depict the indi¤erence curve map in a commodity space. Mark the slope of the indi¤erence curve at bundle (3; 6).c) From now on assume no gift. In the commodity space (x1; x2) ; �nd (geometrically) Patrick�s optimal choice.


values). Provide economic intuition behind the two conditions (ca. two sentences for each).e) Using the two conditions from d) �nd the optimal consumption levels of both types of commodities (x1; x2) for:- for p1= $10; p2= $10 and m = $50 (give two numbers).and after the price of a book increases:- for p1= $20; p2= $10 and m = $50 (give two numbers).Is each of the solutions interior? Illustrate the change on the graph.f) Is the marginal utility of a dollar invested in books and CD equal? (Find two numbers for the parameters before


Problem 2. (25 points) Michael always consumes �ve hamburgers x1 along with one Coke x2 (this is the only healthycombination of the two products!) .

a) Propose Michael�s utility function that represents his preferences over hamburgers and Coke (function U(x1; x2).b) In the commodity space (x1; x2), carefully depict Michael�s indi¤erence curves (and mark the optimal proportion






Problem 3. (15 points) Adam spends all his income on food (x1) and clothing (x2) :He is a fairly sophisticated fellowand his utility function is quite complicated

U (x1; x2)=

"12�

rlnh(6 lnx1+2 lnx2)

2i+ 3

#300:

3



c) Assume p2= 10 and m = 80: Find analytically and geometrically the demand curve and the price o¤er curve.Hint: In b) and c) you can use the magic formula.

Problem 4. (10 points) Frank can use his 24h for leisure R or work. The hourly wage rate is w = 20: Frank is acommitted skier and uses all his income on ski passes in Devil�s Head Resort C.


Let the utility function be given byU (x1; x2)= R

17C34:

b) What is the real wage? (formula + number) How can the real wage be seen in the graph of a budget set?c) What is the optimal choice of leisure, ski passes and labor supply? (Find the optimal choice geometrically and give

three numbers).

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Midterm 1 (Group C)





U (x1; x2)= 4x1+4 lnx2:






Problem 2. (25 points) Michael always consumes one hamburger x1 along with two Cokes x2 (this is the only healthycombination of the two products!) .








U (x1; x2)= ln

"0:5�

rh(10 lnx1+2 lnx2)

2i+ 3

#300:

5




Problem 4. (10 points) Frank can use his 24h for leisure R or work. The hourly wage rate is w = $120: Frank is acommitted skier and uses all his income on ski passes in Devil�s Head Resort C.



3C:


three numbers).

6


Midterm 1 (Group D)





U (x1; x2)= 8x1+8 lnx2:






Problem 2. (25 points) Michael always consumes one hamburger x1 along with four Cokes x2 (this is the onlyhealthy combination of the two products!) .








U (x1; x2)= ln

"0:5�

rh(12 lnx1+6 lnx2)

2i+ 17 � 21� 7

#300:

7




Problem 4. (10 points) Frank can use his 24h for leisure R or work. The hourly wage rate is w = $120: Frank is acommitted skier and uses all his income on ski passes in Devil�s Head Resort C.



3C:


three numbers).

8

ECON 301: MIDTERM ANSWER KEY - GROUP A TA: CHANG-KOO(CK) CHI

FEBRUARY 25, 2010

1 Problem 1

(a) His budget line without the CD gifts is straightforward: 10x1 + 5x2 = 40 and x1-intercept4010 = 4 and x2-intercept 40

5 = 8 represent his real incomes in terms of books and CDs,respectively. The red line in the figure below exhibits his budget line after those CD gifts.

(b) Since MU1 = ∂U∂x1

= 1 and MU2 = ∂U∂x2

= 2x2

, we can write his MRS as

MRS1,2 = − 12/x2

= − x2

2. (1.1)

At (5, 8), its value is −4(or just its absolute value 4). Note that it does not rely upon x1, thequantity demanded for books. And he has to be compensated 0.0001× 4 CDs in order topreserve his current utility level. The indifference is described below, and the slope at (5,8)should be the same as the value of MRS, 4.

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

Books

CD

s

Figure 1: Budget Line

0 1 2 3 4 50

10

20

30

40

50

60

70

80

90

100

Books

CD

s

Figure 2: Indifference Curve

(c) Two things are required to be an optimal; first, the tangency condition: the budget lineshould be a tangent line to the indifference curve. Second, the optimal point should belocated on the budget line.

(d) The first Secret of Happiness(SOH) says that the MRS should coincide with (negative) therelative price at equilibrium: MRS = − p1

p2. It means that rate of exchange at which the

consumer is willing to stay put(MRS) must be equal to the price ratio. The second SOH isp1x1 + p2x2 = m, which simply means that all the money should be exhausted by consumingboth of the goods.

(e) Since MRS= − x22 , x2 = 4 immediately follows from the first SOH. Note that he can afford

x2 = 4. The extra money $40− $5× 4 = $20 must be spent on good 1 by the second SOH

1

and thus x1 = 2. Now if p1 increased to $30, the first SOH says x22 = 30

5 = 6, that is, x2 = 12but it is not affordable since he needs $60 for 12 CDs. Plugging x2 = 12 into the budget line

30x1 + 5× 12 = 40→ x1 = −23

< 0. (1.2)

Hence x1 = 0 and x2 = mp2

= 405 = 8. While the first bundle (x1, x2) = (2, 4) is interior, the

second bundle (x1, x2) = (0, 8) is at the corner.

(f) As depicted below, (2, 4) is in the interior. Hence the marginal utilities from a dollar forbooks and for CDs at (2, 4) must be the same. You can easily verify that

when (p1, p2) = ($10, $5),MU1

p1=

110

=2/4

5=

2/x2

5=

MU2

p2. (1.3)

At (x1, x2) = (0, 8), on the other hand, their marginal utilities from a dollar would not be thesame. Observe that

when (p1, p2) = ($30, $5),MU1

p1=

130

butMU2

p2=

2/x2

p2=

2/85

=1

20(1.4)

2 Problem 2

(a) Since hamburgers and cokes are perfect complements for Michael, his utility function overtwo goods must take a form of

U(x1, x2) = min{x1, 3x2}. (2.1)

(b) Recall that the indifference curve has a ”L” shape in case of perfect complements. And theoptimal preference line should be x1 = 3x2.

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

Hamburgers(X1)

Cok

es(X

2)

Figure 3: Indifference Curve and Optimal Proportion Line

2

(c) Two secrets of Happiness in case of perfect complements are as follows:

(i) x1 = 3x2 (2.2)(ii) p1x1 + p2x2 = m. (2.3)

The condition (i) provides us with the economic intuition that he is willing to consume twogoods in the same proportion 2 : 1. The second condition says that his income must beexhausted.

(d) Plugging (2.2) into (2.3) gives us the equation of x1 only;

p1(3x2) + p2x2 = x2(3p1 + p2) = m. (2.4)

Hence x2 = m3p1+p2

and x1 = 3m3p1+p2

. Since x1, x2 > 0(without loss of generality, his incomem > 0.) they are interior.

(e) They are ordinary since as x1 and x2 are downward sloping. They are normal since as m goesup, so do x1 and x2. Finally, they are gross complements since as the price of one good goes upthe quantity of the other good will decrease.

(f) By Slutsky equation we can decompose the total change in demand into two effects; substi-tution and income effects. However, its substitution effects would be zero because they areperfect complements. Its main intuition follows from the fact that he is willing to consumein the same proportion so his optimal choice will be determined by the set of kinks in hisindifference curve.

3 Problem 3

0 100 200 300 400 500 600 700 8001

2

3

4

5

6

7

8

9

10

Demand for X1

Pric

e of

X1(

P1)

Figure 4: Demand Curve for x1

0 2 4 6 8 100

50

100

150

Demand for X1

Dem

and

for

X2

Figure 5: Price(p1)-Offer Curve

(a) Let a function V(x1, x2) = 2 log x1 + 1 log x2. You can easily see that Adam’s complicatedutility function is just a monotone transformation of V. In fact, we shall rewrite his originalutility function in terms of V as

U(x1, x2) =[

700×√

log V2 + 10]800

. (3.1)

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(b) Part (a) enables us to analyze his consumption behavior using the simple function V(x1, x2).Since it is a Cobb-Douglas utility, the magic formulae lead us to his demand function:

x1 =a

a + bmp1

=2

2 + 11200

4= 200

x2 =b

a + bmp2

=1

2 + 11200

4= 100.

(c) Assume that p2 = 4 and m = 1200. Now we can think of x1’s demand function whichdisplays how the quantity of x1 changes as p1 varies, as well as the p1-offer curve whichdisplays how the set of optimal bundles (x1, x2) changes as p1 varies. When p2 = 4 andm = 1200, the above demand function comes down to

x1 =a

a + bmp1

=2

2 + 11200

p1=

800p1

(3.2)

x2 =b

a + bmp2

=1

2 + 11200

4= 100. (3.3)

(3.2) addresses his demand function which is depicted below. With both of them (3.2) and(3.3), you can draw p1-offer curve which must be flat.

4 Problem 4

0 5 10 15 20 250

5

10

15

20

25

30

35

40

45

50

Leisure(R)

Ski

Pas

ses(

C)

A=(24,0)

Figure 6: Frank’s Budget Line and His Endowments

(a) If Frank enjoys leisure during R hours, his labor income would be w(24− R) = 20× (24−R). Those money will be totally spent on ski passes(C). Denoting its price by pC, we canwrite his budget line as

pCC = w(24− R). (4.1)

Hence you end up with 10C + 20R = 20× 24 when pC = 10 and w = 20. Note that he isjust endowed with his daily hours, 24 hours. Point A = (24, 0) in the figure indicates hisendowment.

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(b) The real wage is simply wpC

= 2. It is the slope of his budget line.

(c) Since U(R, C) = RC5 is Cobb-Douglas, his demand for R and C are

R =a

a + bmw

=1

1 + 548020

= 4

andC =b

a + bmpC

=5

1 + 548010

= 40,

respectively. From R = 4, his labor supply is immediate; 24− 4 = 20 hours.

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Midterm 1 (Group B)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (50+20+15+15=100points) + bonus (10 extra "e" points). Make sure you answer the �rst four questions before working on thebonus one!

Problem 1. (50 points) To reenergize for Econ 301 class, in the morning, Tony always drinks MountainDew (x1) and eats Burritos (x2).

a) Suppose Mountain Dew costs p1 = $2; burrito costs p2 = $10; and his daily budget is m = $40:Show graphically the budget constraint in the commodity space. Mark the two extreme consumption bundles(mark concrete values). On the same graph, show how the budget set is a¤ected by in�ation of 100% thata¤ects prices of both commodities but does not a¤ect income (so his income stays the same m = $40)?

b) Tony�s preferences are given by the following utility function

U (x1; x2) = x31x32:

Find Tony�s marginal rate of substitution (MRS) as a function of x1; x2 (give a formula for MRS).- What is the value of MRS at consumption bundle (1; 2) (give a number)?- Which of the two goods is more valuable, soda or burrito, if Tony drinks one Mountain Dew and

consumes two burritos?- Depict his indi¤erence curve map in a commodity space. Mark the slope of the indi¤erence curve at

the bundle (1; 2).c) In the commodity space (x1; x2) ; �nd (geometrically) Tony�s optimal choice,.assuming pre-in�ation

prices p1 = $2; and p2 = $10. Describe how the two properties of the optimal bundle, known as two"secrets of happiness" (two short sentences) can be seen in the graph.


- Provide some economic intuition behind the two conditions (ca. two sentences for each).- Derive the optimal consumption of x1 and x2 as a function of p1; p2;m (show the derivation).- What fraction of income is spent on burritos (give the percentage)?- Find analytically and geometrically the demand curve for Mountain Dew (given p2 = $10 and m =

$40) and Engel curve (given p1 = $2; and p2 = $10)- Are they Gi¤en goods? Why? (yes/no answer + one sentence).e) Using your formula from d) �nd the optimal consumption levels of both types of commodities (x1; x2)

for:- p1 = $2; p2 = $10 and m = $40 (give two numbers).and after the price of Mountain Dew decreased:- for p1 = $1; p2 = $10 and m = $40 (give two numbers).What is the total change in consumption of Mountain Dew? (give a number). Illustrate the change on

the graph.f) decompose the total change in consumption of x1 from e) into a substitution and income e¤ect.


Problem 2. (20 points) Bill is a wild-animal lover. From his recent trip to Galapagos Islands hebrought a small Iguana. His new pet has only three legs: two left and one right. (Iguanas use magma heatedsoil to warm their eggs and his favorite pet lost one right leg during the last volcano eruption). To survivethe famous Madisonian winter, the iguana has to wear shoes, left (x1) and right ones (x2).

a) Write down Bill�s utility function representing his preferences over right and left shoe (functionU (x1; x2)).

b) In the commodity space (x1; x2), carefully depict Bill�s indi¤erence curves.

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c) Find analytically Bill�s demand for shoes if p1 = $2 and p2 = $1 and Bill�s budget for iguana shoesis m = $15: Is the solution interior? (give two numbers and a yes/no answer).

d) Illustrate Bill�s optimal choice on the graph including the indi¤erence curves and the budget set.e) Suppose the price of left shoe goes down to p1 = $1: Find Bill�s new demand for shoes. What can

you say about the substitution e¤ect? How about the income e¤ect? (Answer the latter question withoutany calculations, using only a graph).

Problem 3. (15 points) Ramon decides about his new collections of postage stamps. He is interested intwo themes: "the birds of the world", x1, (measures the number of stamps in the subcollection with birds)and "the famous mathematicians", x2. The utility derived from the collection is given by

U (x1; x2) = x1 + 10� lnx2:

a) What is the optimal collection of stamps if the prices are p1 = 1 and p2 = 2 and m = 15. (�nd twonumbers x1 and x2). Is your solution interior, or corner?

b) Find the optimal collection if the prices are still p1 = 1 and p2 = 2, but the income is only m = 8.Depict Ramon�s optimal choice in the commodity space.

Problem 4. (15 points) Jacob can use his 24h for leisure R; or work. The hourly wage rate is w = 5:Jacob spends all his money on cheese curds C.

a) Draw Jacob�s budget set, given the price of cheese is pc = $10 (mark the endowment point).Let the utility function be given by

U (x1; x2) = R+ C:

b) Are leisure and cheese curds perfect complements, perfect substitutes or none of them?c) What is the optimal choice of leisure, cheese curds and labor supply? (Find geometrically and give

three numbers).d) Harder: Plot a graph with labor supply (horizontal axis) and wage rate (vertical one) assuming

pc = $10 (hint: for what value of w do we go from one "bang" to the other "bang" solution?)

Bonus Problem. (extra 10 points) Let U (x1; x2) = x1x22; and V (x1; x2) = lnx1 + 2 lnx2 be twoutility functions.

a) Show that U (�) is a monotone transformation of V (�), and hence they de�ne the same preferences.b) Derive MRS for each of the two functions. Using the two formulas for MRS, argue that the functions

de�ne the same indi¤erence curve maps.

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Solutions to midterm 1 (Group B)

Problem 1.a) With p1 = $2; p2 = $10 and m = $40 the budget set is (two extreme consumption bundles are 20

and 4). In�ation that a¤ects only prices shifts budget line inwards.

b) Tony�s marginal rate of substitution (MRS)

MRS = �MU1MU2

= �x2x1

- The value of MRS at consumption bundle (1; 2) is

jMRSj = j � 21j = 2

- Burrito (x2) is less valuable than Mountain Dew (x1)- Tony�s indi¤erence curve map is. (the slope of her indi¤erent curve that passes through bundle (1; 2)

is �2.

c) Tony�s optimal choice is

- the two geometric properties of the optimal bundle, known as two "secrets of happiness" are:1. At the optimal bundle, the indi¤erence curve is tangent to a budget set2. The optimal bundle is located on budget lined) mathematically the two secrets of happiness, are�

MRS = �p1p2

p1x1 + p2x2 = m

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- the economic intuition behind the two conditions is:The individual value of x1 in terms of x2 coincides with the market valueThe income of a consumer is exhausted- the optimal consumption of x1 and x2 as a function of p1; p2;m can be found as followsFrom the MRS condition

MRS = �x2x1= �p1

p2

hencex2 =

p1p2x1


p1x1 + p2

�p1p2x1

�= m


x1 =1

2

m

p1

Plugging in

x2 =p1p2

�1

2

m

p1

�=1

2

m

p2

- the fraction of income spent on burritos is

p1x1m

=1

2= 50%

- and the demand curve for burritos book (given p2 = $10; and m = $40) and Engel curve (givenp1 = $2; and p2 = $10)

Demand curve

x1 =1

2

m

p1=1

2

40

p1=20

p1

and hence inverse demand is

p1 (x1) =20

x1

Geometrically

Engel curve: Since

x1 =1

2

m

p1

at p1 = $2

x1 =1

2

m

2=1

4m

hencem (x1) = 4x1

Geometrically

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- are they Gi¤en goods? Why? (yes/no answer + one sentence).No, because the demand curve is downwardslopping on the whole domain.e) The optimal consumption levels for (x1; x2) :- at p1 = $2; p2 = $10 and m = $40

x1 =1

2

m

p1=1

2

40

2= 10

and

x2 =1

2

m

p2=1

2

40

10= 2


x1 =1

2

m

p1= 20

and

x2 =1

2

m

p2=1

2

40

10= 2


�x1 = 20� 10 = 10

Geometrically


m0 = 10� 1 + 10� 2 = 30

and hence

x1 =1

2

30

1= 15

so SE is equal toSE = 15� 10 = 5

and income e¤ect isIE = 10� 5 = 5

Problem 2. .a) Bill�s utility function is

U (x1; x2) = min (x1; 2x2)

b) indi¤erence curves.in the commodity space (x1; x2) are

c) Bill�s demand for shoes is

2x2 = x1

2x1 + x2 = 40

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4x2 + x2 = 40

x2 =40

5= 8

x1= 16

d) geometrically Bills�s optimal choice is

e) when the price of a left shoe goes down to p1 = $1: the new demand is given by the system ofequations

2x2 = x1

x1 + x2 = 40

and hence demand is

x1 =80

3= 26

2

3

x2 =40

3= 13

1

3

The substitution e¤ect is zero (perfect complements) and the income e¤ect is 4.10 23

Problem 3.a) the two secrets of happiness are

�x210

= �12

x1 + 2x2 = 15

and hence x2 = 5 and x1 = 5: Since both are positive, this is interior solution.b) the two secrets of happiness are

�x210

= �12

x1 + 2x2 = 8

and hence secrets of happiness give x2 = 5 and x1 = �5: Since consumption must be non-negative theoptimal consumption is x1 = 0 and x2 = 4, which is a cornet solution.

Problem 4.a) Jacob�s budget set, with w = $5 and pc = $10 isIncome is m = 5� 24 = $120

b) They are perfect substitutes

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c) jMRSj = 1 > wpc= 1

2 which implies that Jacob cares more about leisure than consumption,.thereforehe will spend the whole day at home

R = 24; LS = 0 and C = 0

d)

Bonus Problem. (extra 10 points)a) Monotone transformation ln () : Take a log of U

lnU () = lnx1x22 = lnx1 + lnx

22 = lnx1 + 2 lnx2 = V ()

where we used two properties of ln function.b) For U () ; marginal rate of substitution is

MRS = �MU1MU2

= � x222x1x2

= � x22x1

and for V ()

MRS = �MU1MU2

= �1=x12=x2

= � x22x1

and hence MRS coincides for all (x1; x2).It follows that the slopes of indi¤erence curves are the same atany point and hence they must be the same.

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Midterm 1 (Group B)

You have 70 minutes to complete the exam. The midterm consists of 3 questions (60+25+15=100 points)+ bonus (10 points). Make sure you answer the �rst three questions before working on the bonus one!

Problem 1. (60 points) Maggie likes to read science �ction (x1) and romance (x2) novels.Her preferences over the two types of books are represented by a utility function

U (x1; x2) = (x1)6(x2)

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a) Find Maggie�s marginal rate of substitution (MRS) as a function of x1; x2 (give a formula).- what is the value of MRS at consumption bundle (2; 2) (give a number).- complete the sentence: "The Marginal Rate of Substitution is a (marginal) value of a ... in terms of

..."- how much one must compensate Maggie in terms of romance books, after taking away 0.00001 of a

science-�ction book, in order to keep her indi¤erent? (give a number, assume she consumes bundle (2; 2)).- depict her indi¤erence curve map in a commodity space. Mark the slope of her indi¤erent curve that

passes through bundle (2; 2).b) Suppose the price of a science-�ction book is p1 = $4; a romance book costs p2 = $10 and her total

monthly spending on books is m = $600: Show graphically her budget constraint in the commodity space.Mark the two extreme consumption bundles (give values). On the same graph, show how the budget setwould be a¤ected by the introduction of an ad valorem tax on science �ction books (x2) at rate 100%?

c) In the commodity space (x1; x2) ; �nd (geometrically) Maggie�s optimal choice.- describe the two properties of the optimal bundle, known as two "secrets of happiness" (two short

sentences).d) Write down mathematically two secrets of happiness, assuming that p1; p2;m are parameters (and

not concrete values).- provide some economic intuition behind the two conditions (ca. two sentences for each).- derive the optimal consumption of x1 and x2 as a function of p1; p2;m (show the derivation).- what fraction of income is spent on science - �ction novels (give the percentage).- �nd analytically and geometrically the demand curve for science �nction book (given p2 = $10; and

m = $600) and Engel curve (given p1 = $4; and p2 = $10 )- are science-�ction books normal goods? Why? (yes/no answer + one sentence).- are they Gi¤en goods? Why? (yes/no answer + one sentence).e) Using your formula from d) �nd the optimal consumption levels for both types of books (x1; x2) :- for p1 = $4; p2 = $10 and m = $600 (give two numbers).and after the price of science-�ction book decreased:- for p1 = $2; p2 = $10 and m = $600 (give two numbers).What is the total change in consumption of x1? (give one number). Illustrate this change on the graph.f) decompose the total change in x1 from f) into a substitution and income e¤ect. (Calculate the two

numbers, show how you found them on the graph.) Complete the two sentences:"The substitution e¤ect is attributed to the pure change in .... induced by the decrease of nominal price

p1""The income e¤ect can be attribute to the pure change in .... induced by the decrease of nominal price

p1"

g) which of the following alternative utility functions represents Maggie�s preferences (there are two such

1

functions)?

V (x1; x2) = 6x1 � 3x2 + 3V (x1; x2) = 30

�x61 � x32

�+ 3

V (x1; x2) = 6 lnx1 � 3 lnx2V (x1; x2) = 6 lnx1 + 3 lnx2 + 7

Explain why the utility functions you have selected represent Maggie�s preferences (one sentence). Suggestthe transformation of U () function that makes the two V () functions equivalent.

Problem 2. (20 points) Jimmy�s favorite hobby is slot car racing. He assembles slot trucks from parts,by adding ten wheels (x1) to an engine (x2) (these are supertrucks, with �ve wheels on each side). Hepurchases the parts on the market.

a) write down Jimmy�s utility function representing his preferences over wheels and engines (functionU (x1; x2)).

b) in the commodity space (x1; x2), carefully depict Jimmy�s indi¤erence curves.c) �nd analytically Jimmy�s demand for parts if one wheel costs p1 = $2; an engine is p2 = $10 and

Jimmy�s budget for slot cars is m = $120: Is the solution interior (give two numbers and yes/no answer).d) illustrate Jimmy�s optimal choice on the graph including the indi¤erence curves and the budget set.e) suppose the price of one wheel goes down to p1 = $1: Find Jimmy�s new demand for the parts. What

can you say about the substitution e¤ect? How about the income e¤ect? (you can answer the last questionwithout any calculations, using only a graph).

Problem 3. (20 points) Ramon Gonzales M. Panetelas is a specialist in Habanos cigars (famous Cubancigars). Cuban cigars are sold either in 5 cigar packs (x1) ; or in singles (x2). Ramon has no income. Insteadhe is initially endowed with !1 = 10 packs and !2 = 50 cigars.

a) draw Ramon�s budget set, given the price of a pack is equal to p1 = $5 and a single cigar is p2 = $1(mark the endowment point).

b) Illustrate geometrically Ramon�s optimal demand for packs and single cigars, given his utility function

U (x1; x2) = 5x1 + x2

(Give two numbers (x1; x2), and mark them on the graph, including budget set and the indi¤erence curves.)c) What is your answer to b) when prices are p1 = $10 and p2 = $1. (Give two numbers (x1; x2) ; and

plot the graph.)d) Harder: Give the formula for the demands x1; x2 as a function of p1; p2 and endowments !1 and !2.

Show the demand on the graph, assuming p2 = $1; !1 = 10 and !2 = 50.

Bonus Problem. (extra 10 points) Depict a map of indi¤erence curves that is consistent witha) inferior goodsb) Gi¤en goods(Make sure you explain why these graphs represent the respective preferences.)

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Solutions to midterm 1 (Group B)

Problem 1. (60 points)a) Maggie’s marginal rate of substitution (MRS)

MRS = −MU1MU2

= −21

x2x1

- the value of MRS at consumption bundle (2, 2) is

MRS = −2

- "The Marginal Rate of Substitution is a (marginal) value of a science finction books in terms of romancenovels"

- after taking away 0.00001 of a science-fiction book, to keep her indifferent one must compensate Maggiein terms of romance books

0.00001×2 = 0.000002- her indifference curve map is. (the slope of her indifferent curve that passes through bundle (2, 2) is

−2.

b) With p1 = $4, p2 = $10 and m = $600 the budget set is (two extreme consumption bundles are(150, 0) and (0, 60). The budget set with ad valorem tax shifts inwards.

c) In the commodity space (x1, x2) , find (geometrically) Maggie’s optimal choice.

- the two geometric properties of the optimal bundle, known as two "secrets of happiness" are:1. At the optimal bundle, the indifference curve is tangent to a budget set2. The oprimal bundle is locaded on budget lined) mathematically the two secrets of happiness, are½

MRS = −p1p2

p1x1 + p2x2 = m

- the economic intuition behind the two conditions is:The individual value of x1 in temrms of x2 coincides with the market value

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The income of a consumer is exchausted- the optimal consumption of x1 and x2 as a function of p1, p2,m can be found as followsFrom the MRS condition

MRS = −2x2x1= −p1

p2

hence

x2 =1

2

p1p2x1


p1x1 + p2

µ1

2

p1p2x1

¶= m


x1 =2

3

m

p1

Pluging in

x2 =1

2

p1p2

µ2

3

m

p1

¶=1

3

m

p2

- the fraction of income spent on science - fiction novels is

p1x1m

=2

3= 66%

- and the demand curve for science finction book (given p2 = $10, and m = $600) and Engel curve(given p1 = $4, and p2 = $10)

Demand curve

x1 =2

3

m

p1=400

p1

and hence invese demand is

p1 (x1) =400

x1Geometrically

Engel curve: Since

x1 =2

3

m

p1

at p1 = $4

x1 =2

3

m

4=1

6m

hencem (x1) = 6x1

Geomettrically

- are science-fiction books normal goods? Why? (yes/no answer + one sentence).

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Yes, because their demand increases in income- are they Giffen goods? Why? (yes/no answer + one sentence).No, becase the demand curve is downwardslopping on the whole domain.e) The optimal consumption levels for both types of books (x1, x2) .- for p1 = $4, p2 = $10 and m = $600

x1 =2

3

m

p1=2

3

600

4= 100

and

x2 =2

3

m

p2=1

3

600

10= 20

and after the price of science-fiction book decreased, for p1 = $2, p2 = $10 and m = $600

x1 =2

3

m

p1=2

3

600

2= 200

and

x2 =2

3

m

p2=1

3

600

10= 20


∆x1 = 200− 100 = 100

Geometrically

f) Substitution effect: auxiliary budget

m0 = 100× 2 + 20× 10 = 400

and hence

x1 =2

3

400

2= 133

1

3

so SE is equal to

SE = 1331

3− 100 = 331

3and income effect is

IE = 100− 3313= 66

2

3"The substitution effect is attributed to the pure change in relative price induced by the decrease of

nominal price p1""The income effect can be attribute to the pure change in real income induced by the decrease of nominal

price p1"

g) From the functions

V (x1, x2) = 6x1 × 3x2 + 3V (x1, x2) = 30

¡x61 × x32

¢+ 3

V (x1, x2) = 6 lnx1 × 3 lnx2V (x1, x2) = 6 lnx1 + 3 lnx2 + 7

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the second and last represent Maggie’s preferences - they are monotone transformations of U () . Thefirst transformation is

f (U) = 30U + 3

and the second isf (U) = lnU + 7

Problem 2. .a) Jimmy’s utility function is

U (x1, x2) = min (x1, 10x2)

b) in the commodity space (x1, x2), carefully depict Jimmy’s indifference curves.

c) Jimmy’s demand for parts is interior and is given by

x2 =1

10x1

2x1 + 10x2 = 120

2x1 + 101

10x1 = 120

x1 =120

3= 40

x2=1

10×40 = 4

d) geometrically Jimmy’s optimal choice is

e) when the price of a wheel goes down to p1 = $25. the new demand is

x2 =1

10x1

x1 + 10x2 = 120

x1 = 60

x2 = 6

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Thte substitution effec is zero (perfect complements) and the income effect is 20 wheels.

Problem 3.a) Ramon’s budget set, with p1 = $5 and p2 = $1 isBudget set: m = 10× 5 + 50× 1 = 100

b) The optimal demand for packs and single cigars can be found as follows. Since

−MRS = 5 =p1p2= 5

therefore it does not matter how Ramon allocates his income. For example his endowment point is optimal

c) When prices are p1 = 10 and p2 = 1

−MRS = 5 <p1p2= 10

and hence the total income will be invested in x2. At such prices demands are

x1 = 0 and x2 =10× 10 + 50

1= 150

d) The demands areIf p1p2 < 5 then

x1 =p1ω1 + p2ω2

p1x2 = 0

if p1p2 > 5 then

x1 = 0

x2 =p1ω1 + p2ω2

p2

and if p1p2 = 5

x1 = αp1ω1 + p2ω2

p

x2 = (1− α)p1ω1 + p2ω2

p2

for α ∈ (0, 1)

The demand curve for x1 is (p2 = 1, ω1 = 5 and ω2 = 50).

5

Marek Weretka

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Bonus Problem. (extra 10 points) Depict a map of indifference curves that is consistent witha) inferior goods

b) Giffen goods

(Make sure you explain why these graphs represent the respective preferences.)

6

Marek Weretka

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Midterm 1 (A)


Problem 1 (55p) (Well-behaved preferences)Ava is a big fun of delicious gourmet steaks: their robust and hearty beef �avors is what she likes the

most. Her two most favorite steaks are marinated ribeye (x1) and top sirloin (x2).a) The price of a marinated ribeye steak is p1 = $50, the price of top sirloin is p2 = $25, and her income

(spent entirely on steaks) is m = $500. Show geometrically Ava�s budget constraint. Mark the two extremeconsumption bundles (give concrete values). Show on the graph how her budget set would be a¤ected byin�ation that increased the prices of both steaks by 100%, leaving m unchanged (draw a new budget line).

b) Ava�s utility function over both types of steaks is U (x1; x2) = x51x52. Find her marginal rate of

substitution (MRS) for any bundle (x1; x2) (give a formula for MRS). What is the value of MRS at bundle(2; 4) (one number)? Which of the two steak types is more valuable to Ava?

d) Write down two secrets of happiness, assuming that p1; p2;m are parameters (two equations).- Provide economic intuition behind the two conditions (two sentences for each).- Derive optimal consumption of x1 and x2 as a function of p1; p2;m (show the derivation of magic

formulas).- Using magic formulas determine whether the solution is interior for all values of p1; p2;m (one sen-

tence).- What fraction of total income is spent on top sirloin?e) Find the optimal consumption levels of two types of steak (x1; x2) for:- p1 = $50; p2 = $25 and m = $500 (give two numbers).and after the price of ribeye decreased:- for p1 = $25; p2 = $25 and m = $500 (give two numbers).What is the total change in consumption of marinated ribeye? (give a number). Illustrate the change on

the graph. Is marinated ribeye an ordinary of a Gi¤en good? (Chose one + one sentence explaining why.)f) Decompose the total change in consumption of x1 from e) into a substitution and income e¤ect.


Problem 2 (15p) (Perfect complements)Alex is a hotdog lover. In each hotdog bun (x1) Alex always inserts two hotdogs (x2): he hates hotdogs

with ratios of buns to hotdogs di¤erent from 1:2.a) Propose a utility function that captures Alex�s preferences over (x1) and (x2) (function U (x1; x2)).b) In the commodity space, carefully depict Alex�s indi¤erence curves (marking the optimal proportion

line).c) Assume prices p1 = 4 and p2 = 3 and income m = 40. Write down two secrets of happiness (two

equations) and determine the optimal choice (two numbers). Is your solution interior (yes or no)?d) Suppose price of a hotdog bun goes down to p1 = 2; while price p2 = 3 incomem = 40 are unchanged.

Find optimal choice (two numbers). Without calculations, how much of this change can be attributed to asubstitution e¤ect (number + one sentence explaining the number).

Problem 3 (15p) (Perfect substitutes)Pepsi x1 and Coke x2 both are goods that are indistinguishable.a) Propose a utility function (the �craziest� function you can imagine), which captures preferences for

such perfect substitutes.b) Plot the budget line, assuming p1 = $2; p2 = $3 and m = $12: In the same graph plot indi¤erence

curves.

1

c) Find the optimal bundle for U (x1; x2) = x1 + x2. (give two numbers). Is your solution interior orcorner? (one sentence)

d) Plot income o¤er curve and Engel curve given �xed prices p1 = $2; p2 = $3: (two graphs). Is Pepsinormal or inferior (choose one + one sentence)

Problem 4 (15p) (Quasilinear Preferences, Labor Supply)Your best friend Aiden asked you to help him to determine how much time he should spend at work.

His total available time (each day) is 24h and his only source of income is labor income given wage ratew = $10. His spending on consumption good C is equal to what he earns. The price of such good is pc = 2

a) What is his real wage rate? (number+interpretation, one sentence )b) Plot his budget set on the graph.c) Aiden has quasilinear utility function U (R;C)= lnR+ C where R is leisure (or relaxation time)

and C is consumption of other goods. Write down two secrets of happiness that determine optimal choiceof R;C for arbitrary w; pc. Find optimal R as a function w; pc (you can assume interior solution).

d) What is Aiden�s labor supply L if w = $1 and pc= $10 (one number). How about w = $2 andpc= $10 (one number). Is labor supply increasing in real wage rate, when preferences are quasilinear? (yesno answer)

Bonus question (Just for fun)Let MRSV and MRSU be marginal rates of substitution of utility functions V (x1; x2) and U(x1; x2)

respectively. Moreover, assume that V (x1; x2) = f [U(x1; x2)] where f [�] is a strictly monotone transfor-mation. Show that MRSV=MRSU . (Hint: Use formula for the derivative of a composite function)

2


Midterm 1 (B)





b) Ava�s utility function over both types of steaks is U (x1; x2) = x31x32. Find her marginal rate of







Problem 2 (15p) (Perfect complements)Alex is a hotdog lover. In each hotdog bun (x1) Alex always inserts three hotdogs (x2): he hates hotdogs







curves.

3










4


Midterm 1 (C)





b) Ava�s utility function over both types of steaks is U (x1; x2) = x121 x

122 . Find her marginal rate of







Problem 2 (15p) (Perfect complements)Alex is a hotdog lover. In each hotdog bun (x1) Alex always inserts four hotdogs (x2): he hates hotdogs







curves.

5










6


Midterm 1 (D)





b) Ava�s utility function over both types of steaks is U (x1; x2) = x131 x

132 . Find her marginal rate of







Problem 2 (15p) (Perfect complements)Alex is a hotdog lover. In each hotdog bun (x1) Alex always inserts two hotdogs (x2): he hates hotdogs







curves.

7










8

Midterm Exam 1 Solutions(A) The Pink

Q1) (55 points)

a) Check �gure 1 for the budget set (3 p)If there is 100% in�ation, the prices double.The new budget set is also shown on Figure 1. (3 p)

b) U(X1; X2) = X51X

52

MRS12 = ((MU1)=(MU2)) = (@U=@X1)=(@U=@X2) = ((5X41X

52 )=(5X

51X

42 )) =

X2=X1 =MRS12

(4 p)at the point x1 = 2 & x2 = 4: MRS12 = (4=2) = 2 (2 p)

which means:MU1 > MU2 So 1st good, ribeye, is more valuable than top sirleon to Ava

at the consumption level ofx1 = 2 & x2 = 4 (4 p)

d)

secret 1 : p1x1 + p2x2 = m (3p)so 50x1 + 25x2 = 500This means "spend all of your money". If the consumer does not spend all

of his/her money s/he is wasting his/her opportunity to increase his/her utilitysince money does not have an e¤ect on utility itself. (2p)

secret 2 : MRS12 = (MU1)=(MU2) = P1=P2 (3p)! X2=X1 = P1=P2

"the last spent on each good should give the same utility" OR "marginalutility of a $ spent on each good should be equal " . If this condition does nothold, lets say last $ spent on good 1 brings more utility than the last $ spent ongood 2 , then the consumer should buy less of the second good and buy moreof the �rst good to increase his/her utility. (2p)

start with secret 2 : X2=X1 = P1=P2 ! X2 = ((X1P1)=(P2)) (1p)now plug this in secret 1 :p1x1 + p2x2 = m = P1x1 + P2(X1P1)=(P2) = m = 2P1x1 (2p)! Then X1 = (m=(2p1)) = (1=2)(m=p1)Since X2 = ((X1P1)=(P2)) = ((mP1)=(2P1P2)) = (m=(2p2))

= (1=2)(m=(p2)) = X2

1

X1 = (m=(2p1))

X2 = (m=(2p2)) (2p)

The solution is interior since none of good�s optimal consumption level iszero. Furthermore in a Cobb-Douglas case the solution has to be interior aslong as income level (m) is larger than zero ! (we also assume positive prices)(1p)

Using the magic formula X2 = (m=(2p2)) ! X2p2 = m=2note that the left hand side is the total money spent on good 2 (sirleon).

So half of the income is spent on sirleon. (2p)

e) X1 = (m=(2p1)) = ((500)=(2 � 50)) = 5 (1p)

X2 = (m=(2p2)) = ((500)=(2 � 25)) = 10 (1p)

X0

1 = (m=(2p0

1)) = ((500)=(2 � 25)) = 10 (1p)

X0

2 = (m=(2p2)) = ((500)=(2 � 25)) = 10 (1p)

Total change in consumption of ribeye is X0

1 �X1 = 10� 5 = 5 (1p)

Check �gure 2 for the illustration of the change. (3p)

Ribeye is an ordinary good since it�s consumption increases as it�s pricedecrease. We can also see this fact from the magic formula . Since the price ofribeye is in the denominator in optimal consumption formula of ribeye, as theprice goes down the consumption of it will increase ! (2p)

f) How much money does Ava need to consume the original bundle withthe new prices ?

p0

1x1 + p2x2 = m0= ($25 � 5) + ($25 � 10) = $375 is enough (3p)

Now calculating the optimal bundle with this imaginary income :X

s

1 = (m0=(2p

0

1)) = ((375)=(2 � 25)) = 15=2 (2p)

Xs

2 = (m0=(2p2)) = ((375)=(2 � 25)) = 15=2

So Substitution E¤ect (S.E.) is : 15=2� 5 = 5=2 (2p)

T:E = S:E:+ I:E: then : I:E: = 5� (5=2) = 5=2 (1p)

Check �gure 2 for the illustration. (3p)

2

Q 2) (15 points)a) the bundle : (1x1 + 2x2)A suitable utility function would be : min(2x1; 1x2) (1p)

b) Check �gure 3 (2p)c) Two secrets are : (1.5p + 1.5p)

2x1 = x2

p1x1 + p2x2 = m

hence

4x1 + 3(2x1) = 40

x�1 =m

p1 + 2p2and x�2 = 2

m

p1 + 2p2

x�1 =40

4 + (2 � 3) = 4

and x�2 = 240

4 + (2 � 3) = 8

(1.5p + 1.5p)The solution is interior since none of the optimal consumption levels are

zero. (1p)

d) Using the magic formulas :

x�1 =40

2 + (2 � 3) = 5

and x�2 = 240

2 + (2 � 3) = 10

(1.5p + 1.5p)S.E. is zero. In a perfect complements case there is no substitution e¤ect

related to a price change. (2p)

Q.3) (15 points)a) U(X1; X2) = 2X1 + 2X2 can be a utility function for this kind of

preferences. (2p)b) Check the �gure 4 (2p)c) MRS12 = ((MU1)=(MU2)) = 1=1 = 1 given the utility function. (2p)comparing MRS to the price ratio : 1 > 2

3 =P1P2

3

So only good 1 is consumed : X�1 = m=P1 = 12=2 = 6 and X�

2 = 0 (3p)

The solution is a corner one. (1p)d) check �gure 5 for the income-o¤er curve and the engel curve. (2p+2p)Pepsi is normal as the consumption of it increases as the income increases.

(1p)

Q.4)a) 10

2 = 5 is her real wage. (1p)This number is her purchasing power in terms of consumption good. (1p)b) check �gure 6 (3p)c)secret 1: w �R+ pcC = 24 � w (2p)or 10R+ 2C = 24 � 10 = 240secret 2 : MRSRC = ((MUR)=(MUC)) = w=Pc ! 1=R = w=Pc (2p)

! R = Pcw (2p)

d) If w = $1 and Pc = $10 the labor supply : R = Pcw = 10hr (1p)

If w = $2 and Pc = $10 the labor supply : R = Pcw = 5hr (1p)

The labor supply (24-R) is increasing in real wage ! (2p)

Bonus QuestionV (x1; x2) = f [U(x1; x2)]

MRSU = ((MU1)=(MU2)) = (@U=@X1)=(@U=@X2)MRSV = ((MV1)=(MV2)) = (@V=@X1)=(@V=@X2)

= @f [U(x1;x2)]=@X1

@f [U(x1;x2)]=@X2by chain rule :

= f0(:)�@U(x1;x2)=@X1

f 0 (:)�@U(x1;x2)=@X2= @U(x1;x2)=@X1

@U(x1;x2)=@X2=MRSU

4

S

Budget line after inflation

Budget Line 1 20

5 R

10

10

Figure 1

S

Budget Line 2

Budget Line 1

20

5 R

10

10

Figure 2

20

Initial consumption bundle

Last consumption bundle

7.5

7.5

I.E. S.E

Hotdogs

Indifference Curves

Optimal Proportion Line : y=2x

6

2

Buns

4

3

Figure 3

1

2

C

Budget Line

120

R

24

Figure 7

4

Coke

Optimal Consumption Point

Budget Line 5

Pepsi

4

6 Figure 4

6

3

2

1

5 3 2 1

4

Coke

Income offer curve

Budget Lines 5

Pepsi

4

6 Figure 5

6

3

2

1

5 3 2 1

4

income

Engel curve

5

Pepsi

4

6 Figure 6

6

3

2

1

5 3 2 1

Midterm Exam 1 Solutions(B) The Green

Q1) (55 points)


b) U(X1; X2) = X31X

32

MRS12 = ((MU1)=(MU2)) = �(@U=@X1)=(@U=@X2) = (�(3X21X

32 )=(3X

31X

22 )) =

�X2=X1 =MRS12(4 p)at the point x1 = 3 & x2 = 6: MRS12 = (�6=3) = �2 (2 p)



d)






= (1=2)(m=(p2)) = X2

1

X1 = (m=(2p1))

X2 = (m=(2p2)) (2p)




e) X1 = (m=(2p1)) = ((100)=(2 � 10)) = 5 (1p)

X2 = (m=(2p2)) = ((100)=(2 � 5)) = 10 (1p)

X0

1 = (m=(2p0

1)) = ((100)=(2 � 5)) = 10 (1p)

X0

2 = (m=(2p2)) = ((100)=(2 � 5)) = 10 (1p)


1 �X1 = 10� 5 = 5 (1p)




p0

1x1 + p2x2 = m0= ($5 � 5) + ($5 � 10) = $75 is enough (3p)


s

1 = (m0=(2p

0

1)) = ((75)=(2 � 5)) = 15=2 (2p)

Xs

2 = (m0=(2p2)) = ((75)=(2 � 5)) = 15=2

So Substitution E¤ect (S.E.) is : 15=2� 5 = 5=2 (2p)

T:E = S:E:+ I:E: then : I:E: = 5� (5=2) = 5=2 (1p)


2



3x1 = x2

p1x1 + p2x2 = m

hence

4x1 + 2(3x1) = 40

x�1 =m

p1 + 3p2and x�2 = 3

m

p1 + 3p2

x�1 =40

4 + (2 � 3) = 4

and x�2 = 340

4 + (2 � 3) = 12


zero. (1p)


x�1 =40

2 + (2 � 3) = 5

and x�2 = 340

2 + (2 � 3) = 15




preferences. (2p)b) Check the �gure 4 (2p)c) MRS12 = ((MU1)=(MU2)) = 1=1 = 1 given the utility function. (2p)comparing MRS to the price ratio : 1 < 4

2 =P1P2

3


1 = 0 (3p)

The solution is a corner one. (1p)d) check �gure 5 & 6 for the income-o¤er curve and the engel curve. (2p+2p)Pepsi is normal as the consumption of it does not decrease as the income

increases. (1p)(Answers of "inferior" and "ambigous" are also accepted IF NECESSARY

EXPLANATIONS ARE DONE!)

Q.4)a) 10


! R = Pcw (2p)






= @f [U(x1;x2)]=@X1


= f0(:)�@U(x1;x2)=@X1

f 0 (:)�@U(x1;x2)=@X2= @U(x1;x2)=@X1

@U(x1;x2)=@X2=MRSU

4

S


Budget Line 1 20

3.3 R

6.6

10

Figure 1

S

Budget Line 2

Budget Line 1

20

5 R

10

10

Figure 2

20



7.5

7.5

I.E. S.E

Hotdogs

Indifference Curves


9

2

Buns

6

3

Figure 3

1

3

C

Budget Line

120

R

24

Figure 7

4

Coke


Budget Line

5

Pepsi

4

6 Figure 4

6

3

2

1

5 3 2 1

4

Coke

Income offer curve

Budget Lines

5

Pepsi

4

6 Figure 5

6

3

2

1

5 3 2 1

4

income

Engel curve 5

Pepsi

4

6 Figure 6

6

3

2

1

5 3 2 1

Midterm Exam 1 Solutions(C) The Yellow

Q1) (55 points)


b) U(X1; X2) = X1=21 X

1=22

MRS12 = ((MU1)=(MU2)) = �(@U=@X1)=(@U=@X2) = (�( 12X�1=21 X

1=22 )=( 12X

1=21 X

�1=22 )) =




d)






= (1=2)(m=(p2)) = X2

1

X1 = (m=(2p1))

X2 = (m=(2p2)) (2p)




e) X1 = (m=(2p1)) = ((200)=(2 � 10)) = 10 (1p)

X2 = (m=(2p2)) = ((200)=(2 � 5)) = 20 (1p)

X0

1 = (m=(2p0

1)) = ((200)=(2 � 5)) = 20 (1p)

X0

2 = (m=(2p2)) = ((200)=(2 � 5)) = 20 (1p)


1 �X1 = 20� 10 = 10 (1p)




p0

1x1 + p2x2 = m0= ($5 � 10) + ($5 � 20) = $150 is enough (3p)


s

1 = (m0=(2p

0

1)) = ((150)=(2 � 5)) = 15 (2p)

Xs

2 = (m0=(2p2)) = ((150)=(2 � 5)) = 15

So Substitution E¤ect (S.E.) is : 15� 10 = 5 (2p)

T:E = S:E:+ I:E: then : I:E: = 10� 5 = 5 (1p)


2



4x1 = x2

p1x1 + p2x2 = m

hence

6x1 + 1(4x1) = 40

x�1 =m

p1 + 4p2and x�2 = 4

m

p1 + 4p2

x�1 =40

6 + (1 � 4) = 4

and x�2 = 440

6 + (1 � 4) = 16


zero. (1p)


x�1 =40

4 + (1 � 4) = 5

and x�2 = 440

4 + (1 � 4) = 20




preferences. (2p)b) Check the �gure 4 (2p)c) MRS12 = ((MU1)=(MU2)) = 1=1 = 1 given the utility function. (2p)comparing MRS to the price ratio : 1 < 6

2 =P1P2

3


1 = 0 (3p)

The solution is a corner one. (1p)d) check �gure 5 & 6 for the income-o¤er curve and the engel curve. (2p+2p)Pepsi is normal as the consumption of it does not decrease as the income

increases. (1p)(Answers of "inferior" and "ambigous" are also accepted IF NECESSARY

EXPLANATIONS ARE DONE!)

Q.4)a) 10


! R = Pcw (2p)






= @f [U(x1;x2)]=@X1


= f0(:)�@U(x1;x2)=@X1

f 0 (:)�@U(x1;x2)=@X2= @U(x1;x2)=@X1

@U(x1;x2)=@X2=MRSU

4

S


Budget Line 1 40

10 R

20

20

Figure 1

S

Budget Line 2

Budget Line 1

40

10 R

20

20

Figure 2

40



15

15

I.E. S.E

Hotdogs

Indifference Curves


12

2

Buns

8

3

Figure 3

1

4

C

Budget Line

120

R

24

Figure 7

4

Coke


Budget Line

5

Pepsi

4

6 Figure 4

6

3

2

1

5 3 2 1

4

Coke

Income offer curve

Budget Lines

5

Pepsi

4

6 Figure 5

6

3

2

1

5 3 2 1

4

income

Engel curve 5

Pepsi

4

6 Figure 6

6

3

2

1

5 3 2 1

Midterm Exam 1 Solutions(D) The Blue

Q1) (55 points)


b) U(X1; X2) = X1=31 X

1=32

MRS12 = ((MU1)=(MU2)) = �(@U=@X1)=(@U=@X2) = (�( 13X�1=31 X

1=32 )=( 13X

1=31 X

�1=32 )) =




d)






= (1=2)(m=(p2)) = X2

1

X1 = (m=(2p1))

X2 = (m=(2p2)) (2p)




e) X1 = (m=(2p1)) = ((80)=(2 � 4)) = 10 (1p)

X2 = (m=(2p2)) = ((80)=(2 � 2)) = 20 (1p)

X0

1 = (m=(2p0

1)) = ((80)=(2 � 2)) = 20 (1p)

X0

2 = (m=(2p2)) = ((80)=(2 � 2)) = 20 (1p)


1 �X1 = 20� 10 = 10 (1p)




p0

1x1 + p2x2 = m0= ($2 � 10) + ($2 � 20) = $60 is enough (3p)


s

1 = (m0=(2p

0

1)) = ((60)=(2 � 2)) = 15 (2p)

Xs

2 = (m0=(2p2)) = ((60)=(2 � 2)) = 15

So Substitution E¤ect (S.E.) is : 15� 10 = 5 (2p)

T:E = S:E:+ I:E: then : I:E: = 10� 5 = 5 (1p)


2



2x1 = x2

p1x1 + p2x2 = m

hence

2x1 + 2(2x1) = 30

x�1 =m

p1 + 2p2and x�2 = 2

m

p1 + 2p2

x�1 =30

2 + (2 � 2) = 5

and x�2 = 230

2 + (2 � 2) = 10


zero. (1p)


x�1 =30

1 + (2 � 2) = 6

and x�2 = 230

1 + (2 � 2) = 12




preferences. (2p)b) Check the �gure 4 (2p)c) MRS12 = ((MU1)=(MU2)) = 1=1 = 1 given the utility function. (2p)comparing MRS to the price ratio : 1 > 2

6 =P1P2

3


2 = 0 (3p)

The solution is a corner one. (1p)d) check �gure 5 & 6 for the income-o¤er curve and the engel curve. (2p+2p)Pepsi is normal as the consumption of it increases as the income increases.

(1p)

Q.4)a) 10


! R = Pcw (2p)






= @f [U(x1;x2)]=@X1


= f0(:)�@U(x1;x2)=@X1

f 0 (:)�@U(x1;x2)=@X2= @U(x1;x2)=@X1

@U(x1;x2)=@X2=MRSU

4

S


Budget Line 1 40

10 R

20

20

Figure 1

S

Budget Line 2

Budget Line 1

40

10 R

20

20

Figure 2

40



15

15

I.E. S.E

Hotdogs

Indifference Curves


6

2

Buns

4

3

Figure 3

1

2

C

Budget Line

120

R

24

Figure 7

4

Coke


Budget Line

5

Pepsi

4

6 Figure 4

6

3

2

1

5 3 2 1

4

Coke

Income offer curve

Budget Lines

5

Pepsi

4

6 Figure 5

6

3

2

1

5 3 2 1

4

income

Engel curve : x1=m/2

5

Pepsi

4

6 Figure 6

6

3

2

1

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Econ 301 Intermediate Microeconomicsmweretka/301_17f/Exams/M1.pdf · Econ 301 Intermediate Microeconomics Prof. Marek Weretka Midterm 1 (A) You have 70 minutes to complete the exam