Econ 301 Intermediate Microeconomics Prof. Marek Weretka ...

Econ 301

Intermediate Microeconomics

Prof. Marek Weretka

Midterm 2 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (25,30,25 and 20 points)+Just

For Fun question.

Problem 1. (25p) (Uncertainty)

Oscar is an owner of Lamborghini Veneno, one of the most expensive cars ever made. Its market value is estimated

at 8 million dollars. In case of a car collision the value of the car drops to 4 million dollars. The probability of a

collision is �c= 0:5. In short, Lamborghini is a lottery (8; 4):

a) Oscar�s Bernoulli utility function is given by u(c) = 10 ln c. Write down his Von Neumann-Morgenstern

(expected) utility function over lotteries U(Cc; Cnc) (give a formula). Is Oscar risk averse, risk neutral or risk loving

(choose one)? Plot Oscar�s indi¤erence curves in the commodity space (Cc; Cnc).

b) In Madison area insurance for ridiculously expensive cars is provided by State Farm. Derive Oscar�s budget

constraint if State Farm insurance premium is = 0:5 (give equation for Cc and Cnc in terms of coverage x; and

then reduce the two equations to one budget constraint.) Plot the corresponding budget set in the commodity space.

c) Find the optimal level of wealth (Cc; Cnc) and the coverage x. (three numbers) Is Oscar fully insured (yes-no

answer).

d) Demonstrate that if the premium is greater than the probability of �ood, > �c Oscar will not purchase full

insurance. (Use "MRS" secret of happiness to show that in optimum Cc< Cnc).

Problem 2. (30p) (Edgeworth box and equilibrium)

Consider an economy with two goods (apples and oranges) and two agents, Elisa and Bob. Elisa is initially

endowed with !E = (5; 30) of apples and oranges respectively and Ben�s endowment is !B = (20; 20). Elisa and Ben

have the same utility given by

U (x1; x2) = 2 lnx1 + 2 lnx2

a) Plot an Edgeworth box and mark the initial endowments.

b) Give a de�nition of Pareto e¢ cient allocation (one sentence). Using graph argue that the necessary and

su¢ cient condition for Pareto e¢ ciency of a (interior) feasible allocation is MRSE=MRSB .

c) Derive the contract curve (give formula) and depict it in the Edgeworth box.

d) Find the competitive equilibrium (give six numbers) and show it in the Edgeworth box.

e) Verify that the allocation in the competitive equilibrium is Pareto e¢ cient.

f) Give two other prices that are consistent with a competitive equilibrium? (give two numbers without any

calculations)

1

Problem 3. (25p) (Producers)

A producer has the following technology

y =pK + L

a) Show that the production function exhibits decreasing returns to scale (provide formal argument).

b) Find analytically (the variable) cost function given prices of inputs wK = 1 and wL = 2 (formula). Plot the

cost function in the graph.

c) Assuming �xed cost F = 1 �nd analytically yMES and ATCMES (give two numbers) and plot a supply of the

�rm, marking a threshold for non-zero production.1

d) Determine the number of �rms operating in the industry if demand is D(p) = 40� p, �rms are competitiveand there is free entry in the market. (one number)

Problem 4 (20p). (Short questions)

a) Robert�s Bernoulli utility function is given by u(c) = 10c. A lottery ticket pays 10 with probability 12 and

zero otherwise and thus the lottery is C = (10; 0). Find certainty equivalent CE of lottery C (one number) and the

expected value E(C) of the lottery (one number). Is the certainty equivalent bigger or smaller than the expected

value? Why? (one sentence)

b) Consider economy with two agents, Andy and Bob. They have identical utility functions U(x1; x2) = x1+x2

but di¤erent endowments !A= (10; 30) and !B= (10; 20), for Andy and Bob, respectively. Without any calculations,

plot the contract curve in an Edgeworth box and suggest (one) competitive equilibrium (six numbers).

c) Given production function y = 4 �K7L

12 and short-run level of capital �K = 1 derive labor demand (formula)

of a competitive �rm. Find equilibrium real wage rate if labor supply is given by Ls= 16 (one number). Find the

unemployment rate if the minimal real wage is wmin=p = 2 (one number+graph).

Just For Fun

Give a formal argument that from the perspective of the whole society free (competitive) markets e¢ ciently

allocate resources. (i.e., show that an allocation in competitive equilibrium is Pareto e¢ cient).

1 If you do not know how to answer b) to get partial credit you can assume c(y) = y2

2

Econ 301


Prof. Marek Weretka

Midterm 2 (Group B)


For Fun question.












answer).







U (x1; x2) = 3 lnx1 + 3 lnx2








calculations)

3



y =pK + L



















Just For Fun




4

Econ 301


Prof. Marek Weretka

Midterm 2 (Group C)


For Fun question.












answer).





endowed with !E = (10; 60) of apples and oranges respectively and Ben�s endowment is !B = (40; 40). Elisa and

Ben have the same utility given by

U (x1; x2) = 8 lnx1 + 8 lnx2








calculations)

5



y =p2K + 2L



















Just For Fun




6

Econ 301


Prof. Marek Weretka

Midterm 2 (Group D)


For Fun question.












answer).







U (x1; x2) = 2 lnx1 + 2 lnx2








calculations)

7



y =p2K + 2L












b) Consider economy with two agents, Andy and Bob. They have identical utility functions U(x1; x2) = 3x1+3x2







Just For Fun




8

Econ 301 Intermediate Microeconomics Spring 2013Prof.Marek Weretka

Solution to Midterm 2 (Group A)

Problem 1. (a) expected utility function over lotteries

U(Cc, Cnc) =1

2× 10 lnCc +

1

2× 10 lnCnc

= 5 lnCc + 5 lnCnc (3′)

Oscar is risk averse since U(c) = 10 lnC is concave. (1’)

Let U(Cc, Cnc) = u Then Cnc = eu/5

Cc. So we can obtain the indiffer-

ence curve as below. (1’)

Cc

Cnc

Figure 1: Problem 1(a)

(b) The Cc and Cnc can be represented as

Cc = 4− γx+ x = 4 + 0.5x

Cnc = 8− γx = 8− 0.5x (2′)

Then,

Cnc +γ

1− γCc = 8 + 4

γ

1− γ=⇒ Cnc + Cc = 12 (3′)

The budget line is shown as below: (2’)

(c) By the utility function in part (a) and budget constraint in part (b),optimal choice of wealth levels are given by the short-cut formula:

Cc =5

5 + 5× 12

1= 6 (2′)

Cnc =5

5 + 5× 12

1= 6 (2′)

1

Cc

Cnc

12

12

Figure 2: Problem 1(b)

Then Cc = Cnc So Oscar is fully insured. (1’)By the formula in part(b), x = 4. (2’)

(d) First calculate

MRS =5/Cnc5/Cc

=CcCnc

(2′)

then by the ‘secret of happiness’,

MRS =pncpc

=⇒ CcCnc

=1− γγ

(2′)

If γ > πc = 0.5, then 1−γγ < 1. Therefor, Cnc > Cc. That is, Oscar

is not fully insured. (2’)

Problem 2. (a) Total endowment ω = ωE + ωB = (5, 30) + (20, 20) = (25, 50).(1’)The Edgeworth box and initial endowment are in the following figure.

(b) An allocation is Pareto Efficient if there is no other allocation whichwould make one of them strictly better off without hurting any ofthe others. (2’)=⇒) Suppose that MRSE 6= MRSB when an allocation is ParetoEfficient. Then the two indifference curves passing through this pointwill intersect as in figure 4. It’s obvious that the points in the shad-owed area will make both of them not worse off and at least one ofthem strictly better off. That’s a contradiction with the assumption.Then the necessity follows immediately. (3’)⇐=) Now assume the condition MRSE = MRSB is satisfied at an

allocation a. Then the two indifference curves passing through thispoint will be tangent with each other as in figure 5. Then check all

2

E

E

B

30

5

20

20


the points in the areas A,B,C,D except point a in the Edgeworthbox and all of them will make at least of Elisa and Ben strictly worseoff. So allocation a is Pareto efficient by definition. (3’)

(c)

MRSE = MRSB =⇒ xE2xE1

=xE2xE1

(2′)

AlsoxE1 + xB1 = 25, xE2 + xB2 = 50 (2′)

Substitute the last two equation to the first one.

xE2 = 2xE1 , xB2 = 2xE1 (2′)

the contract curve is plotted in figure 6. (2’)

(d) Normalize p2 = 1. the budget constraints are:

2xE1 + xE2 = 5p1 + 30

2xB1 + xB2 = 20P1 + 20 (1′)

Then

xE1 =2

2 + 2

5p1 + 30

p1

xB1 =2

2 + 2

20P1 + 20

p1(1′)

Thus

2

2 + 2

5p1 + 30

p1+

2

2 + 2

20P1 + 20

p1= 25 =⇒ p1 = 2 (2′)

3

A

E

B

ICE

ICB


ICE

ICB a

E

B

AB

C

D


Solve the equations and the result follows:

xE1 = 10, xE2 = 20 (2′)

xB1 = 15, xB2 = 30 (2′)

The competitive equilibrium is indicated in the Edgeworth box inFigure 7.

(e) Calculate and compare the MRS’s.

MRSE =xE2xE1

=20

10= 2 (1′)

MRSB =xB2xB1

=30

15= 2 (1′)

So obviously MRSE = MRSB .

(f) any numbers satisfying p1

p2= 2, for example, p1 = 4, p2 = 2. (2’)

4

E

B

contract

curve

Figure 6: Problem 2(c)

E

B

equilibrium

Figure 7: Problem 2(d)

Problem 3. (a) The production function y = f(K,L) =√K + L

f(rK, rL) =√rK + rL =

√r(K + L) =

√r√K + L = r

12 f(K,L) < rf(K,L) (3′)

So production function is decreasing returns to scale(DRS).

(b)

TRS =∂y/∂L

∂y/∂K=

12√K+L1

2√K+L

= 1 (2′)

Notice ωL

ωK= 2 so TRS < ωL

ωK. (2’)

Hence L = 0 and y =√K =⇒ K = y2 (2’)

Then c(y) = 2× 0 + 1× y2 = y2 (1’)The cost function is plotted by Figure 8. (1’)

(c) TC(y) = FC + V C(y) = 1 + y2 (1’)

ATC(y) =TC(y)

y=

1 + y2

y(1′)

MC(y) = 2y (1′)

To find minimum efficient scale(MES), let MC(y) = ATC(y) =⇒2y = 1+y2

y =⇒ yMES = 1 (2’)

5

y

C(y)

C(y) = y2


Then pMES = MC(yMES) = 2× 1 = 2 (1’)When the firm supply positive quantity, p = MC(y) =⇒ y = p

2 .(1’)Supply curve is given by (2’)

y =

{p/2 if p ≥ 2,

0 if p < 2.

The supply curve is provided by Figure 9. (1’)

p

S(P)

2

1


(d) When there is free entry, the firm will production quantity yMSE = 1and the market price is pMSE = 2. (2’)Total demand D(pMSE) = 40− pMSE = 38 (1’)

Total number of firms N = D(pMSE)yMSE = 38 (1’)

Problem 4. (a) E(C) = 12 × 10 + 1

2 × 0 = 5 (2’)E(u(C)) = 1

2 × 10× 10 + 12 × 10× 0 = 50 Then u(CE) = Eu(C) =⇒

CE = 5 (2’)Thus E(C) = CE (1’)That’s because Robert’s utility function is linear thus he is risk neu-tral. (1’)

6

(b) Notice both Andy and Bob’s utility function is U(x1, x2) = x1 + x2and thus MRSE = MRSB ≡ 1 for all allocations. Therefore, anypoints in the Edgeworth box is Pareto efficient. (2’)Then p1

p2= MRSE = 1. Say p1 = 1 = p2. (2’)

The competitive equilibrium is any points on the budget line withinthe Edgeworth box: (4’)

xE1 + xE2 = 10 + 30 = 40

xB1 + xB2 = 10 + 20 = 30

xE1 + xB1 = 10 + 10 = 20

xE2 + xB2 = 30 + 20 = 50

for example xE1 = 5, xE2 = 35;xB1 = 15, xB2 = 25.

(c) When K = 1, the production function is y = 4L12 . So MPL = ∂y

∂L =2√L

= ωp Thus Ld = ( p

2ω )2 (2’)

If Ls = 16, real wage ωp = 2√

16= 1

2 (2’)

Similarly, if ωmin

p = 2, Ld = (22 )2 = 1. So unemployment rate =

(16-1)/16 = 15/16. (1’)The graph is given by Figure 10. (1’)

L

ω

12

2

1 16

unemployment


7


Solution to Midterm 2 (Group B)


U(Cc, Cnc) =1

2× 10 lnCc +

1

2× 10 lnCnc

= 5 lnCc + 5 lnCnc (3′)





Cc

Cnc



Cc = 8− γx+ x = 8 + 0.5x

Cnc = 16− γx = 16− 0.5x (2′)

Then,

Cnc +γ

1− γCc = 16 + 8

γ

1− γ=⇒ Cnc + Cc = 24 (3′)



Cc =5

5 + 5× 24

1= 12 (2′)

Cnc =5

5 + 5× 24

1= 12 (2′)

1

Cc

Cnc

24

24



(d) First calculate

MRS =5/Cnc5/Cc

=CcCnc

(2′)


MRS =pncpc

=⇒ CcCnc

=1− γγ

(2′)



Problem 2. (a) Total endowment ω = ωE + ωB = (1, 6) + (4, 4) = (5, 10). (1’)The Edgeworth box and initial endowment are in the following Figure1. (1’)



2

E

E

B

6

1

4

4



(c)


=xE2xE1

(2′)

AlsoxE1 + xB1 = 10, xE2 + xB2 = 5 (2′)


xE2 = 2xE1 , xB2 = 2xE1 (2′)



2xE1 + xE2 = p1 + 6

2xB1 + xB2 = 4P1 + 4 (1′)

Then

xE1 =3

3 + 3

p1 + 6

p1

xB1 =3

3 + 3

4P1 + 4

p1(1′)

Thus

3

3 + 3

p1 + 6

p1+

3

3 + 3

4P1 + 4

p1= 25 =⇒ p1 = 2 (2′)

3

A

E

B

ICE

ICB


ICE

ICB a

E

B

AB

C

D



xE1 = 2, xE2 = 4 (2′)

xB1 = 3, xB2 = 6 (2′)



MRSE =xE2xE1

=4

2= 2 (1′)

MRSB =xB2xB1

=6

3= 2 (1′)



p2= 2, for example, p1 = 4, p2 = 2. (2’)

4

E

B

contract

curve


E

B

equilibrium


Problem 3. (a) The production function y = f(K,L) =√K + L

f(rK, rL) =√rK + rL =

√r(K + L) =

√r√K + L = r

12 f(K,L) < rf(K,L) (3′)


(b)

TRS =∂y/∂L

∂y/∂K=

12√K+L1

2√K+L

= 1 (2′)

Notice ωL

ωK= 1

2 so TRS > ωL

ωK. (2’)

Hence K = 0 and y =√L =⇒ L = y2 (2’)

Then c(y) = 2× 0 + 1× y2 = y2 (1’)The cost function is plotted by Figure 8. (1’)

(c) TC(y) = FC + V C(y) = 1 + y2 (1’)

ATC(y) =TC(y)

y=

1 + y2

y(1′)

MC(y) = 2y (1′)


y =⇒ yMES = 1 (2’)

5

y

C(y)

C(y) = y2




y =

{p/2 if p ≥ 2,

0 if p < 2.


p

S(P)

2

1




Problem 4. (a) E(C) = 12 × 12 + 1

2 × 0 = 6 (2’)E(u(C)) = 1

2 × 2 × 12 + 12 × 2 × 0 = 12 Then u(CE) = Eu(C) =⇒


6


p2= MRSE = 1. Say p1 = 1 = p2. (2’)


xE1 + xE2 = 10 + 40 = 50

xB1 + xB2 = 20 + 20 = 40

xE1 + xB1 = 10 + 20 = 30

xE2 + xB2 = 40 + 20 = 60


(c) When K = 1, the production function is y = 12L12 . So MPL =

∂y∂L = 6√

L= ω

p Thus Ld = ( p6ω )2 (2’)


9= 2 (2’)

Similarly, if ωmin



L

ω

2

6

1 9

unemployment


7


Solution to Midterm 2 (Group C)


U(Cc, Cnc) =1

2× 10 lnCc +

1

2× 10 lnCnc

= 5 lnCc + 5 lnCnc (3′)





Cc

Cnc



Cc = 1− γx+ x = 1 + 0.5x

Cnc = 2− γx = 2− 0.5x (2′)

Then,

Cnc +γ

1− γCc = 2 + 1

γ

1− γ=⇒ Cnc + Cc = 3 (3′)



Cc =5

5 + 5× 3

1= 1.5 (2′)

Cnc =5

5 + 5× 3

1= 1.5 (2′)

1

Cc

Cnc

3

3



(d) First calculate

MRS =5/Cnc5/Cc

=CcCnc

(2′)


MRS =pncpc

=⇒ CcCnc

=1− γγ

(2′)






2

E

E

B

60

10

40

40



(c)


=xE2xE1

(2′)

AlsoxE1 + xB1 = 50, xE2 + xB2 = 100 (2′)


xE2 = 2xE1 , xB2 = 2xE1 (2′)



2xE1 + xE2 = 10p1 + 60

2xB1 + xB2 = 40P1 + 40 (1′)

Then

xE1 =8

8 + 8

10p1 + 60

p1

xB1 =8

8 + 8

40P1 + 40

p1(1′)

Thus

8

8 + 8

10p1 + 60

p1+

8

8 + 8

40P1 + 40

p1= 50 =⇒ p1 = 2 (2′)

3

A

E

B

ICE

ICB


ICE

ICB a

E

B

AB

C

D



xE1 = 20, xE2 = 40 (2′)

xB1 = 30, xB2 = 60 (2′)



MRSE =xE2xE1

=40

20= 2 (1′)

MRSB =xB2xB1

=60

30= 2 (1′)



p2= 2, for example, p1 = 4, p2 = 2. (2’)

4

E

B

contract

curve


E

B

equilibrium


Problem 3. (a) The production function y = f(K,L) =√

2K + 2L

f(rK, rL) =√

2rK + 2rL =√r(2K + 2L) =

√r√

2K + 2L = r12 f(K,L) < rf(K,L) (3′)


(b)

TRS =∂y/∂L

∂y/∂K=

1√K+L1√K+L

= 1 (2′)

Notice ωL

ωK= 2 so TRS < ωL

ωK. (2’)

Hence L = 0 and y =√

2K =⇒ K = 12y

2 (2’)Then c(y) = 4× 0 + 2× 1

2y2 = y2 (1’)

The cost function is plotted by Figure 8. (1’)

(c) TC(y) = FC + V C(y) = 1 + y2 (1’)

ATC(y) =TC(y)

y=

1 + y2

y(1′)

MC(y) = 2y (1′)


y =⇒ yMES = 1 (2’)

5

y

C(y)

C(y) = y2




y =

{p/2 if p ≥ 2,

0 if p < 2.


p

S(P)

2

1




Problem 4. (a) E(C) = 12 × 16 + 1

2 × 0 = 8 (2’)E(u(C)) = 1

2 × 10× 16 + 12 × 10× 0 = 80 Then u(CE) = Eu(C) =⇒


6


p2= MRSE = 1. Say p1 = 1 = p2. (2’)


xE1 + xE2 = 10 + 40 = 50

xB1 + xB2 = 20 + 20 = 40

xE1 + xB1 = 10 + 20 = 30

xE2 + xB2 = 40 + 20 = 60



∂y∂L = 8√

L= ω

p Thus Ld = ( p8ω )2 (2’)


16= 2 (2’)

Similarly, if ωmin



L

ω

2

8

1 16

unemployment


7


Solution to Midterm 2 (Group D)


U(Cc, Cnc) =1

2× 10 lnCc +

1

2× 10 lnCnc

= 5 lnCc + 5 lnCnc (3′)





Cc

Cnc



Cc = 4− γx+ x = 4 + 0.5x

Cnc = 8− γx = 8− 0.5x (2′)

Then,

Cnc +γ

1− γCc = 8 + 4

γ

1− γ=⇒ Cnc + Cc = 12 (3′)



Cc =5

5 + 5× 12

1= 6 (2′)

Cnc =5

5 + 5× 12

1= 6 (2′)

1

Cc

Cnc

12

12



(d) First calculate

MRS =5/Cnc5/Cc

=CcCnc

(2′)


MRS =pncpc

=⇒ CcCnc

=1− γγ

(2′)






2

E

E

B

30

5

20

20



(c)


=xE2xE1

(2′)

AlsoxE1 + xB1 = 25, xE2 + xB2 = 50 (2′)


xE2 = 2xE1 , xB2 = 2xE1 (2′)



2xE1 + xE2 = 5p1 + 30

2xB1 + xB2 = 20P1 + 20 (1′)

Then

xE1 =2

2 + 2

5p1 + 30

p1

xB1 =2

2 + 2

20P1 + 20

p1(1′)

Thus

2

2 + 2

5p1 + 30

p1+

2

2 + 2

20P1 + 20

p1= 25 =⇒ p1 = 2 (2′)

3

A

E

B

ICE

ICB


ICE

ICB a

E

B

AB

C

D



xE1 = 10, xE2 = 20 (2′)

xB1 = 15, xB2 = 30 (2′)



MRSE =xE2xE1

=20

10= 2 (1′)

MRSB =xB2xB1

=30

15= 2 (1′)



p2= 2, for example, p1 = 4, p2 = 2. (2’)

4

E

B

contract

curve


E

B

equilibrium


Problem 3. (a) The production function y = f(K,L) =√

2K + 2L

f(rK, rL) =√

2rK + 2rL =√r(2K + 2L) =

√r√

2K + 2L = r12 f(K,L) < rf(K,L) (3′)


(b)

TRS =∂y/∂L

∂y/∂K=

12√K+L1

2√K+L

= 1 (2′)

Notice ωL

ωK= 1

2 so TRS > ωL

ωK. (2’)

Hence K = 0 and y =√

2L =⇒ L = 12y

2 (2’)Then c(y) = 4× 0 + 2× 1

2y2 = y2 (1’)

The cost function is plotted by Figure 8. (1’)

(c) TC(y) = FC + V C(y) = 1 + y2 (1’)

ATC(y) =TC(y)

y=

1 + y2

y(1′)

MC(y) = 2y (1′)


y =⇒ yMES = 1 (2’)

5

y

C(y)

C(y) = y2




y =

{p/2 if p ≥ 2,

0 if p < 2.


p

S(P)

2

1




Problem 4. (a) E(C) = 12 × 10 + 1

2 × 0 = 5 (2’)E(u(C)) = 1

2 × 10× 10 + 12 × 10× 0 = 50 Then u(CE) = Eu(C) =⇒


6


p2= MRSE = 1. Say p1 = 1 = p2. (2’)


xE1 + xE2 = 10 + 30 = 40

xB1 + xB2 = 10 + 20 = 30

xE1 + xB1 = 10 + 10 = 20

xE2 + xB2 = 30 + 20 = 50



∂y∂L = 9√

L= ω

p Thus Ld = ( p9ω )2 (2’)


9= 3 (2’)

Similarly, if ωmin



L

ω

3

9

1 9

unemployment


7

Econ 301Intermediate MicroeconomicsProf. Marek Weretka

Midterm 2 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (25, 20, 30, 25points)+Just For Fun question.

Problem 1. (25p) (Intertemporal Choice)Joseph earns m1= $10 when young and m2= $90 when old.a) Write down Joseph�s budget constraint (one inequality) and plot his budget set given interest rate

r = 200% in the graph. Find PV and FV of income and mark it in your graph (give two numbers).b) Joseph�s utility is given by U (C1; C2)= ln (c1)+

11+� ln (c2) where discount rate is � = 2: Using

magic formulas, �nd the optimal consumption plan and the optimal saving strategy (give three numbersC1; C2; S).

c) (Annuity) Joseph is contemplating leasing a car. Leasing would require three annual payments, each$1600; starting next year (after which you can keep your car). Alternatively, he can buy a car for $1; 500(payment this year). Which of the two alternatives should he choose if annual interest is r = 100%? Why?(calculate PV of the two alternatives)

d) (Perpetuity) Derive the (general) formula for PV of perpetuity.

Problem 2. (20p) (Uncertainty and Insurance)Trevor�s motorbike is worth 16 (thousand $). In an event (state of the world) of a crash, its value drops

to 0 (and, hence, the bike is a lottery with w = (w1; w2) = (16; 0)): Assume that the probability of a crashis equal to 1

2 :a) Find the expected value of the �bike� lottery (16; 0) (one number) and its certainty equivalent CE,

assuming Bernoulli utility function u(y) =py (one number). Is the certainty equivalent bigger or smaller

than the expected value? Why? (one sentence) .b) Write down the Von Neumann-Morgenstern (expected) utility function over lotteries U(C1;C2) for

Bernoulli utility function u(y) = ln y: Is Trevor risk averse, risk neutral or risk loving? (chose one+ onesentence)

c) Assume insurance premium = 12 and the utility function as in point b), �nd the optimal consumption

(C1; C2) and insurance coverage x (give three numbers, you can use Magic Formulas). Is Trevor fullyinsured? Why? (one sentence Hint: is insurance fair?)

Problem 3. (30p) (Edgeworth box and equilibrium)Jayden and Olivia are consume two types of commodities, ice cream x1 and burgers x2. Jayden is initially

endowed with !J = (9; 1) of the two goods and Olivia�s endowment is !O = (1; 9). The utility function ofJayden and Olivia is the same and is given by

U (x1; x2) =1

3lnx1 +

1

3lnx2:

a) Plot an Edgeworth box and mark the initial endowments.b) Find the competitive equilibrium (give six numbers) and depict the equilibrium in the Edgeworth

box.c) Verify that the allocation in the competitive equilibrium is Pareto e¢ cient (compare two numbers).d) Find analytically the contract curve (write down the condition for Pareto e¢ ciency and solve for a

contract curve line). Are the initial endowments located on this curve? (a yes or no answer)

Problem 4. (25p) (Producers)A producer has the following technology

y = K12L

12

1

a) Is MPL increasing, decreasing or constant (choose one) What are the returns to scale? (choose: IRS,DRS or CRS)

b) Given short-run level of capital �K = 1 derive the labor demand (formula) of a competitive �rm. Findthe equilibrium real wage rate if labor supply is given by Ls = 9 (one number).

c) Find the unemployment rate if the minimal (real) wage is wmin=p = 12 (one number+graph).

d) Find analytically a cost function if wK = wL = 2 (formula). Plot the cost function in the graph.

Just For FunGive a de�nition of the Pareto e¢ cient allocation. Prove that allocation is Pareto e¢ cient if and only if

"MRS" of consumers are equal (show the necessary and su¢ cient condition).

2


Midterm 2 (Group B)






c) (Annuity) Joseph is contemplating leasing a car. Leasing would require three annual payments, each$8; 000; starting next year (after which you can keep your car). Alternatively, he can buy a car for $7; 500(payment this year). Which of the two alternatives should he choose if annual interest is r = 100%? Why?(calculate PV of the two alternatives)






than the expected value? Why? (one sentence).b) Write down the Von Neumann-Morgenstern (expected) utility function over lotteries U(C1;C2) for





endowed with !J = (4; 6) of the two goods and Olivia�s endowment is !O = (6; 4). The utility function ofJayden and Olivia is the same and is given by

U (x1; x2) =1

5lnx1 +

1

5lnx2:





y = K12L

12

3







4


Midterm 2 (Group C)

















endowed with !J = (6; 14) of the two goods and Olivia�s endowment is !O = (14; 6). The utility functionof Jayden and Olivia is the same and is given by

U (x1; x2) = 3 lnx1 + 3 lnx2:





y = K12L

12

5







6


Midterm 2 (Group D)









to 0 (and, hence, the bike is a lottery with w = (w1; w2) = (100; 0)): Assume that the probability of acrash is equal to 1

2 :a) Find the expected value of the �bike�lottery (100; 0) (one number) and its certainty equivalent CE,







endowed with !J = (3; 17) of the two goods and Olivia�s endowment is !O = (17; 3). The utility functionof Jayden and Olivia is the same and is given by

U (x1; x2) = 5 lnx1 + 5 lnx2:





y = K12L

12

7


b) Given short-run level of capital �K = 1 derive the labor demand (formula) of a competitive �rm. Findthe equilibrium real wage rate wL

p if labor supply is given by Ls = 9 (one number).





8

Second Midterm SolutionsEcon 301 - Spring 2012

Maximum Points per Question

1a. = 3 2a. = 7 3a. = 3 4a. = 41b. = 10 2b. = 4 3b. = 15 4b. = 81c. = 5 2c. = 9 3c. = 6 4c. = 51d. = 7 3d. = 6 4d. = 8

Group A

Problem #1

A) The budget constraint maps consumption between the two time peri-ods. With the “young” period on the x-axis and the “old” period on they-axis, the budget constraint appears as

C1P1 + C2P2 ≤ m1 +m2P2

P2 =1

1 + r

C1P1 + C21

3≤ $10 + $90

1

3

=⇒ C1P1 + C21

3≤ $40

Future Value is the $90 that Joseph gets when he is “old” in addition to thevalue of the $10 he gets when he is “young” when he is old:

$10(1 + r) = $30

1

This means that FV = $120.

Present value is the $10 Joseph has when he is young plus the value when heis “young” of the $90 he will get when he is old:

$90(1

1 + r) + $10 = $40

This means that PV = $40

Figure 1: Budget Constraint

B) In terms of PV, the optimal demands for consumption are:

C1 =1

1 + 11+δ

× $10P1 + $90P2

P1

=⇒ C1 =3

4× $10 + $90P2

1

C2 =1

1+δ

1 + 11+δ

× $10P1 + $90P2

P2

=⇒ C2 =1

4× $10 + $90P2

P2

From the budget constraint we know that C1 + P2C2 ≤ 40, so combiningthis with the above equations we can confirm the value for P2 we previously

2

derived:

3

4× $10 + $90P2

1+ P2(

1

4× $10 + $90P2

P2

) = $40

$7.5 + $67.5P2 + $2.5 + $22.5P2 = $40

$90P2 = $30

=⇒ P2 =1

3

Plugging P2 into the magic formulas for C1 and C2, we get:

C1 =3

4×

$10 + $9013

1= $30

C2 =1

4×

$10 + $9013

13

= $30

Since Joseph is only endowed with $10 in the first period, but consumes $30,this means that he is borrowing (in present value) $20 from the future ($60in FV). This makes his savings vector S = (-$20,0).

C) The formula for an annuity of $1,600 lasting for three periods, startingnext period is:

$1, 600

r(1− (

1

1 + r

3

)) = $1, 400 < $1, 500

This means that the lease is a better option than the purchase since it costsless in terms of present value and available information (assuming you ignorethe value of the car remaining after three years).

D) 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 ]

3

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

(1− 1

1 + r)PV =

1

1 + rx

(1 + r

1 + r− 1

1 + r)PV =

1

1 + rx

(r

1 + r)PV =

1

1 + rx

PV =x

r

Problem 2A) To calculate the expected value it is necessary to find the value in

each state and multiply it by the probability that each state will occur. Thesummation is then:

EV = 0 ∗ π + 16 ∗ (1− π) = 8

The certainty equivalent is the amount of money that Trevor would take forcertain in order to avoid the gamble that would leave him with the expecta-tion of $8. To calculate this you must first find the utility that the gambleprovides, and then find the amount of money that would provide the sameutility with a 100% probability:

U(E(8)) =1

2

√0 +

1

2

√16

= 2

U(CE) = 2 =⇒√CE = 2

=⇒ CE = 4

Here we clearly see that the certainty equivalent, 4, is smaller than the ex-pected payoff from the gamble, 8. This makes sense because the concaveshape of Trevor’s utility function reflects his risk-averse preferences - mean-ing he is willing to take a smaller payout for certain than the one he wouldget in expectation to avoid the possibility of being left with no money at all.

B) The expected utility representation is:

U(C1, C2) =1

2lnC1 +

1

2lnC2

4

The natural log is a concave function, meaning that Trevor is risk averse.C) First, we calculate the formulas for consumption in each state of the world:

C1 = 16− xγC2 = (1− γ)x

Knowing that x, the amount of insurance, is the same in both states of theworld, we can solve both of the above equations for x and set them equal toone another:

x =16− C1

γ

x =C2

(1− γ)

=⇒ 16− C1

γ=

C2

(1− γ)

=⇒ C1 = 16− γ

1− γC2

This final equation is our budget constraint. Normalizing P1 to 1, this makesP2 = 1 as well. Plugging these values into the magic formulas for demandwe get:

C1 =1

2× 16

P1

C2 =1

2× 16

P2

C1 = C2 = 8

Going back to the first equation for C1, we now have:

8 = 16− 1

2x

=⇒ x = 16

Consumption in both states of the world is 8, so that means that Trevor isfully insured. Full insurance means that your consumption does not dependon the state of the world. Trevor has chosen to fully insure in this case be-cause the insurance premium, γ, is equal to the probability of loss, π.

5

Problem 3

A)

B) At an equilibrium, we need for MRSJ to equal MRSO and also for MRSJ

to be equal to P1

P2. We use the first identity to get the contract curve, the

second to calculate the slope of the budget line. Given the endowment point,we can follow the budget line away from the endowment point to find itsintersection with the contract curve, which is the equilibrium. Since the twoutility functions are symmetrical, we can solve for Jayden and Olivia both bysimply solving for Jayden. Analytically, this is done via the following equa-tions: First, we normalize P1 to equal 1 and use the Cobb-Douglas magicformulas to get the P2:

x1 =1

2× 3P1 + 7P2

P1

x2 =1

2× 3P1 + 7P2

P2

x1 + x2 = 3P1 + 7P2

=⇒ 1

2× 3P1 + 7P2

P1

+1

2× 3P1 + 7P2

P2

= 3P1 + 7P2

5P2 + 5 = 3 + 7P2

P2 = 1

6

MRSJ =MUJ

1

MUJ2

=1

3x11

3x2

=P1

P2

=⇒ x2x1

=P1

P2=⇒ x2 = x1

This last step, however, was extraneous expect to show that the contractcurve is the set of points where x1 = x2. Thus, our solution must satisfy thisidentity. Anyway, revisiting the Cobb-Douglas demands with P2 in hand, weget:

x1 =1

2× 3P1 + 7P2

P1

= 5

=⇒ xJ1 = xJ2 = xO1 = xO2 = 5 & P1 = P2 = 1

C) As we previously stated, at the competitive equilibrium, it is required

that MRSJ = MRSO which means thatxJ2xJ1

=xO2xO1

which is satisfied when

xJ1 = xJ2 = xO1 = xO2 = 5 since 55

= 55

D) Solving analytically for the contract curve requires knowledge of the totalendowment of each good in the economy. Here, we can see from the individual

7

endowments that there are 10 units of each good in the economy and that theedgeworth box depicting it is square. To solve for the slope of the contractcurve use the equations:

MRSJ =MUJ

1

MUJ2

=

13xJ11

3xJ2

=xJ2xJ1

= MRSO =MUO

1

MUO2

=10− xJ210− xJ1

=⇒ xJ2 = xJ1

By symmetry this is true for both individuals. Clearly the initial endowmentsare not located on the contract curve (no).

Problem 4A) The MPL is decreasing and there are constant returns to scale exhibitedby this production function.B) Labor demand can be calculated by setting MPL = w

p. Given that K is

fixed, this makes the equation:

1

2L− 1

2 =w

p1

2√L

=w

p√L =

p

2w

L∗ = (p

2w)2

9 = (p

2w)2

1

6=w

p

C)The unemployment rate is the ratio of the number of hours invidivdualsare under-employed relative to the equilibrium level of employment in a freemarket. Here, when wages are free to adjust due to supply and demandthe market clears at a wage real wage of 1

6and 9 hours of labor. When the

wage is constrained to be 12

the labor demand falls, while the supply remains

8

constant at 9 hours. This yields the following unemployment rate:

1

2√L

=1

2

2√L = 2

L = 1

=⇒ UnemploymentRate =(9− 1)

9=

8

9

D) First we must see that in order for the TRS = WK

WLit must be the case

that K = L:

TRS =12K− 1

2L12

12L− 1

2K12

=WK

WL

= 1

=⇒ K = L

plugging this result back into the production function to get costs as a func-tion of output, we see that:

y = K12 (K)

12

= K

=⇒ K = y

thus, the cost function is given by:

C(y) = WKK +WLL

= 8K + 8L

= 16K

(1)

Since above we learned that K = y, is must be that case that costs, as afunction of output, is given by the equation C(y) = 16y.

9

Group B

Problem #1


C1P1 + C2P2 ≤ m1 +m2P2

P2 =1

1 + r

C1P1 + C21

3≤ $20 + $180

1

3

=⇒ C1P1 + C21

3≤ $80

Future Value is the $180 that Joseph gets when he is “old” in addition tothe value of the $20 he gets when he is “young” when he is old:

$20(1 + r) = $60



$180(1

1 + r) + $20 = $80



C1 =1

1 + 11+δ

× $20P1 + $180P2

P1

=⇒ C1 =3

4× $20 + $180P2

1

C2 =1

1+δ

1 + 11+δ

× $20P1 + $180P2

P2

=⇒ C2 =1

4× $20 + $180P2

P2

10


From the budget constraint we know that C1 +P2C2 ≤ 80, so combining thiswith the above equations we can recover P2:

3

4× $20 + $180P2

1+ P2(

1

4× $20 + $180P2

P2

) = $80

$15 + $135P2 + $5 + $45P2 = $80

$180P2 = $60

=⇒ P2 =1

3


C1 =3

4×

$20 + $18013

1= $60

C2 =1

4×

$20 + $18013

13

= $60

Since Joseph is only endowed with $20 in the first period, but consumes $60,this means that he is borrowing (in present value) $40 from the future ($120in FV). This makes his savings vector S = (-$40,0).


$8, 000

r(1− (

1

1 + r

3

)) = $7, 000 ≤ $7, 500

11



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 =

1

1 + rx

(1 + r

1 + r− 1

1 + r)PV =

1

1 + rx

(r

1 + r)PV =

1

1 + rx

PV =x

r



EV = 0 ∗ π + 36 ∗ (1− π) = 18

The certainty equivalent is the amount of money that Trevor would take forcertain in order to avoid the gamble that would leave him with the expectationof $18. To calculate this you must first find the utility that the gambleprovides, and then find the amount of money that would provide the sameutility with a 100% probability:

U(E(18)) =1

2

√0 +

1

2

√36

= 3

U(CE) = 3 =⇒√CE = 3

=⇒ CE = 9

12

Here we clearly see that the certainty equivalent, 9 is smaller than the ex-pected payoff from the gamble, 18. This makes sense because the concaveshape of Trevor’s utility function reflects his risk-averse preferences - mean-ing he is willing to take a smaller payout for certain than the one he wouldget in expectation to avoid the possibility of being left with no money at all.


U(C1, C2) =1

2lnC1 +

1

2lnC2


C1 = 36− xγC2 = (1− γ)x


x =36− C1

γ

x =C2

(1− γ)

=⇒ 36− C1

γ=

C2

(1− γ)

=⇒ C1 = 36− γ

1− γC2


C1 =1

2× 36

P1

C2 =1

2× 36

P2

C1 = C2 = 18

13


18 = 36− 1

2x

=⇒ x = 36

Consumption in both states of the world is 18, so that means that Trevor isfully insured. Full insurance means that your consumption does not dependon the state of the world. Trevor has chosen to fully insure in this casebecause the insurance premium, γ, is equal to the probability of loss, π.Problem 3A)


to be equal to P1


second to calculate the slope of the budget line. Given the endowment point,we can follow the budget line away from the endowment point to find itsintersection with the contract curve, which is the equilibrium. Since the twounility functions are symetrical, we can solve for Jayden and Olivia bothby simply solving for Jayden. Analytically, this is done via the followingequations: First, we normalize P1 to equal 1 and use the Cobb-Douglas

14

magic forumlas to get the P2:

x1 =1

2× 4P1 + 6P2

P1

x2 =1

2× 4P1 + 6P2

P2

x1P1 + x2P2 = 4P1 + 6P2

=⇒ 1

2× 4P1 + 6P2

P1

+ P21

2× 4P1 + 6P2

P2

= 4P1 + 6P2

2 + 3P2 + 2 + 3P2 = 4 + 6P2 = 4 + 6P2

P2 = 1

MRSJ =MUJ

1

MUJ2

=1

5x11

5x2

=P1

P2

=⇒ x2x1

=P1

P2=⇒ x2 = x1


x1 =1

2× 4P1 + 6P2

P1

= 5

=⇒ xJ1 = xJ2 = xO1 = xO2 = 5 & P1 = P2 = 1



=xO2xO1


xJ1 = xJ2 = xO1 = xO2 = 5 since 55

= 55

D) Solving analytically for the contract curve requires knowledge of the totalendowment of each good in the economy. Here, we can see from the individualendowments that there are 10 units of each good in the economy and that theedgeworth box depicting it is square. To solve for the slope of the contract

15

curve use the equations:

MRSJ =MUJ

1

MUJ2

=

15xJ11

5xJ2

=xJ2xJ1

= MRSO =MUO

1

MUO2

=10− xJ210− xJ1

=⇒ xJ2 = xJ1


16


p. Given that K is


1

2L− 1

2 =w

p1

2√L

=w

p√L =

p

2w

L∗ = (p

2w)2

9 = (p

2w)2

1

6=w

p




the labor demand falls, while the supply remainsconstant at 9 hours. This yields the following unemployment rate:

1

2√L

=1

2

2√L = 2

L = 1


9=

8

9



that K = L:

TRS =12K− 1

2L12

12L− 1

2K12

=WK

WL

= 1

=⇒ K = L

17


y = K12 (K)

12

= K

=⇒ K = y


C(y) = WKK +WLL

= 5K + 5L

= 10K

(2)


Group C

Problem #1


C1P1 + C2P2 ≤ m1 +m2P2

P2 =1

1 + r

C1P1 + C21

2≤ $10 + $60

1

2

=⇒ C1P1 + C21

2≤ $40

18

Future Value is the $60 that Joseph gets when he is “old” in addition to thevalue of the $10 he gets when he is “young” when he is old:

$10(1 + r) + $60 = $80



$60(1

1 + r) + $10 = $40




C1 =1

1 + 11+δ

× $10P1 + $60P2

P1

=⇒ C1 =2

3× $10 + $60P2

1

C2 =1

1+δ

1 + 11+δ

× $10P1 + $60P2

P2

=⇒ C2 =1

3× $10 + $60P2

P2

19


2

3× $10 + $60P2

1+ P2(

1

3× $10 + $60P2

P2

) = $40

$6.66 + $40P2 + $3.33 + $20P2 = $40

$60P2 = $30

=⇒ P2 =1

2


C1 =2

3×

$10 + $6012

1= $

80

3

C2 =1

3×

$10 + $6012

12

= $80

3

Since Joseph is only endowed with $10 in the first period, but consumes $803

, this means that he is borrowing (in present value) $503

from the future($100

3in FV). This makes his savings vector S = (-$50

3,0).


$2, 400

r(1− (

1

1 + r

3

)) = $2, 100 < $2, 300



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 ]

20


(1− 1

1 + r)PV =

1

1 + rx

(1 + r

1 + r− 1

1 + r)PV =

1

1 + rx

(r

1 + r)PV =

1

1 + rx

PV =x

r



EV = 0 ∗ π + 64 ∗ (1− π) = 32


U(E(18)) =1

2

√0 +

1

2

√64

= 4

U(CE) = 4 =⇒√CE = 4

=⇒ CE = 16



U(C1, C2) =1

2lnC1 +

1

2lnC2

21


C1 = 64− xγC2 = (1− γ)x


x =64− C1

γ

x =C2

(1− γ)

=⇒ 64− C1

γ=

C2

(1− γ)

=⇒ C1 = 64− γ

1− γC2


C1 =1

2× 64

P1

C2 =1

2× 64

P2

C1 = C2 = 32


32 = 64− 1

2x

=⇒ x = 64

Consumption in both states of the world is 32, so that means that Trevor isfully insured. Full insurance means that your consumption does not dependon the state of the world. Trevor has chosen to fully insure in this case

22

because the insurance premium, γ, is equal to the probability of loss, π.Problem 3A)B) At an equilibrium, we need for MRSJ to equal MRSO and also for MRSJ

to be equal to P1


second to calculate the slope of the budget line. Given the endowment point,we can follow the budget line away from the endowment point to find itsintersection with the contract curve, which is the equilibrium. Since the twounility functions are symetrical, we can solve for Jayden and Olivia bothby simply solving for Jayden. Analytically, this is done via the followingequations: First, we normalize P1 to equal 1 and use the Cobb-Douglasmagic forumlas to get the P2:

x1 =1

2× 6P1 + 14P2

P1

x2 =1

2× 6P1 + 14P2

P2

x1P1 + x2P2 = 6P1 + 14P2

=⇒ 1

2× 6P1 + 14P2

P1

+ P21

2× 6P1 + 14P2

P2

= 6P1 + 14P2

3 + 7P2 + 3 + 7P2 = 6 + 14P2 = 6 + 14P2

P2 = 1

23

MRSJ =MUJ

1

MUJ2

=3x13x2

=P1

P2

=⇒ x2x1

=P1

P2=⇒ x2 = x1


x1 =1

2× 6P1 + 14P2

P1

= 10

=⇒ xJ1 = xJ2 = xO1 = xO2 = 10 & P1 = P2 = 1



=xO2xO1


xJ1 = xJ2 = xO1 = xO2 = 1 since 1010

= 1010

D) Solving analytically for the contract curve requires knowledge of the totalendowment of each good in the economy. Here, we can see from the individual

24

endowments that there are 10 units of each good in the economy and that theedgeworth box depicting it is square. To solve for the slope of the contractcurve use the equations:

MRSJ =MUJ

1

MUJ2

=

3xJ13xJ2

=xJ2xJ1

= MRSO =MUO

1

MUO2

=20− xJ220− xJ1

=⇒ xJ2 = xJ1



p. Given that K is


1

2L− 1

2 =w

p1

2√L

=w

p√L =

p

2w

L∗ = (p

2w)2

9 = (p

2w)2

1

6=w

p




the labor demand falls, while the supply remains

25

constant at 9 hours. This yields the following unemployment rate:

1

2√L

=1

2

2√L = 2

L = 1


9=

8

9



that K = L:

TRS =12K− 1

2L12

12L− 1

2K12

=WK

WL

= 1

=⇒ K = L


y = K12 (K)

12

= K

=⇒ K = y


C(y) = WKK +WLL

= 3K + 3L

= 6K

(3)


26

Group D

Problem #1


C1P1 + C2P2 ≤ m1 +m2P2

P2 =1

1 + r

C1P1 + C21

2≤ $10 + $100

1

2

=⇒ C1P1 + C21

2≤ $60

Future Value is the $100 that Joseph gets when he is “old” in addition tothe value of the $10 he gets when he is “young” when he is old:

$10(1 + r) + $100 = $120



$100(1

1 + r) + $10 = $60



C1 =1

1 + 11+δ

× $10P1 + $100P2

P1

=⇒ C1 =2

3× $10 + $100P2

1

C2 =1

1+δ

1 + 11+δ

× $10P1 + $100P2

P2

=⇒ C2 =1

3× $10 + $100P2

P2

27



2

3× $10 + $100P2

1+ P2(

1

3× $10 + $100P2

P2

) = $60

$6.66 + $66.6P2 + $3.33 + $33.3P2 = $60

$100P2 = $50

=⇒ P2 =1

2


C1 =2

3×

$10 + $10012

1= $40

C2 =1

3×

$10 + $10012

12

= $40

Since Joseph is only endowed with $10 in the first period, but consumes $40, this means that he is borrowing (in present value) $30 from the future($60 in FV). This makes his savings vector S = (-$30,0).


$2, 400

r(1− (

1

1 + r

3

)) = $2, 100 < $2, 300

28



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 =

1

1 + rx

(1 + r

1 + r− 1

1 + r)PV =

1

1 + rx

(r

1 + r)PV =

1

1 + rx

PV =x

r



EV = 0 ∗ π + 100 ∗ (1− π) = 50


U(E(50)) =1

2

√0 +

1

2

√100

= 5

U(CE) = 5 =⇒√CE = 5

=⇒ CE = 25

29



U(C1, C2) =1

2lnC1 +

1

2lnC2


C1 = 100− xγC2 = (1− γ)x


x =100− C1

γ

x =C2

(1− γ)

=⇒ 100− C1

γ=

C2

(1− γ)

=⇒ C1 = 100− γ

1− γC2


C1 =1

2× 100

P1

C2 =1

2× 100

P2

C1 = C2 = 50

30


50 = 100− 1

2x

=⇒ x = 100

Consumption in both states of the world is 50, so that means that Trevor isfully insured. Full insurance means that your consumption does not dependon the state of the world. Trevor has chosen to fully insure in this casebecause the insurance premium, γ, is equal to the probability of loss, π.Problem 3A)


to be equal to P1


second to calculate the slope of the budget line. Given the endowment point,we can follow the budget line away from the endowment point to find itsintersection with the contract curve, which is the equilibrium. Since the twounility functions are symetrical, we can solve for Jayden and Olivia bothby simply solving for Jayden. Analytically, this is done via the followingequations: First, we normalize P1 to equal 1 and use the Cobb-Douglas

31

magic forumlas to get the P2:

x1 =1

2× 3P1 + 17P2

P1

x2 =1

2× 3P1 + 17P2

P2

x1P1 + x2P2 = 3P1 + 17P2

=⇒ 1

2× 3P1 + 17P2

P1

+ P21

2× 3P1 + 17P2

P2

= 3P1 + 17P2

1.5 + 8.5P2 + 1.5 + 8.5P2 = 3 + 17P2 = 3 + 17P2

P2 = 1

MRSJ =MUJ

1

MUJ2

=5x15x2

=P1

P2

=⇒ x2x1

=P1

P2=⇒ x2 = x1


x1 =1

2× 3P1 + 17P2

P1

= 10

=⇒ xJ1 = xJ2 = xO1 = xO2 = 10 & P1 = P2 = 1



=xO2xO1


xJ1 = xJ2 = xO1 = xO2 = 1 since 1010

= 1010

D) Solving analytically for the contract curve requires knowledge of the totalendowment of each good in the economy. Here, we can see from the individualendowments that there are 10 units of each good in the economy and that theedgeworth box depicting it is square. To solve for the slope of the contract

32

curve use the equations:

MRSJ =MUJ

1

MUJ2

=

5xJ15xJ2

=xJ2xJ1

= MRSO =MUO

1

MUO2

=20− xJ220− xJ1

=⇒ xJ2 = xJ1

By symmetry this is true for both individuals. Clearly the initial endowmentsare not located on the contract curve (no).Problem 4A) The MPL is decreasing and there are constant returns to scale exhibitedby this production function.B) Labor demand can be calculated by setting MPL = w

p. Given that K is

33


1

2L− 1

2 =w

p1

2√L

=w

p√L =

p

2w

L∗ = (p

2w)2

9 = (p

2w)2

1

6=w

p




the labor demand falls, while the supply remainsconstant at 9 hours. This yields the following unemployment rate:

1

2√L

=1

2

2√L = 2

L = 1


9=

8

9



that K = L:

TRS =12K− 1

2L12

12L− 1

2K12

=WK

WL

= 1

=⇒ K = L

34


y = K12 (K)

12

= K

=⇒ K = y


C(y) = WKK +WLL

= 2K + 2L

= 4K

(4)


35


Midterm 2 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (25,30,25 and 20points).

Problem 1 (25p). (Uncertainty and insurance)You are an owner of a luxurious sailing boat, worth $10; that you use for recreation on Mendota lake.

Unfortunately, there is a good (50%) chance of a tornado in Madison (probability is equal to 12 ) that

completely destroys it. Thus, your boat is in fact a lottery with payment (0; 10).a) What is the expected value of the "boat" lottery? (give one number)b) Suppose your Bernoulli utility function is given by u(c) = c2. Give von Neuman-Morgenstern utility

function over lotteries U(C1;C2): (formula) Are you risk averse, neutral or risk loving? (two words). Findthe certainty equivalent (CE) of the "boat lottery" (one number). Which is bigger, CE or the expected valueof a lottery from a)? Why? (one sentence)

c) Your Bernoulli utility function changes to u(c) = ln c: Give von Neuman-Morgenstern utility function.(give a formula). Are you risk averse now?

d) You can insure your boat by buying insurance policy in which you specify coverage x: The insurancecontract costs � x where the premium rate is equal to = 1

2 : Find analytically and depict in the graphyour budget constraint. Mark the point that corresponds to no insurance.

e) Find optimal level of coverage x: Are you going to fully insure your boat? (one number and yes-noanswer). Depict optimal consumption plan on the graph.

f) Propose a premium rate for which you will only partially insure your boat. (one number)

Problem 2 (30p). (Edgeworth box, and equilibrium)Consider an economy with apples and oranges. Andy is initially endowed with !A= (0; 50) and Bob�s

endowment is !B= (50; 0):The utility function of both Andy and Bob is the same and given by

U (x1; x2)= 3 lnx1+3 lnx2

a) Plot the Edgeworth box and mark the allocation representing the initial endowment.b) Provide general de�nition of Pareto e¢ ciency (one sentence starting with: Allocation is Pareto e¢ cient

if ... ).c) Prove, that an allocation is Pareto e¢ cient if and only in such allocation satis�es MRSA=MRSB :

Start with necessity by showing that if the MRS condition does not hold then allocation is not Pareto e¢ cient.Then proceed to su¢ ciency by showing that if the condition MRS is satis�ed then indeed allocation is e¢ cient(use a graph and write two sentences for each of the two conditiions).

d) Find analytically a collection of all Pareto e¢ cient allocations (contract curve) and depict it in thegraph.

e) Find the competitive equilibrium (give six numbers).g) Give some other prices that are consistent with competitive equilibrium (give two numbers).f) UsingMRS condition verify that equilibrium allocation is Pareto e¢ cient and hence an invisible hand

of a free (and competitive) market guides sel�sh Andy and Bob to a socially optimal outcome.

Problem 3 (25p). (Short questions)a) Your sister has just promised to send you pocket money of $500 each month starting next month and

she will keep doing it forever. What is the present value of "having such sister" if monthly interest rate isequal to 5% (one number).

1

b) Sam is a hockey player who earns $100 when young and $0 when old. Sam�s intertemporal utilityis given by U(C1; C2) = ln (c1) +

11+� ln (c1) : Assuming � = r = 0 and using magic formulas �nd

optimal consumption plan and optimal saving strategy (give three numbers C1; C2; S). Does Sam smoothhis consumption? (yes/ no + one sentence) Is Sam tilting his consumption? (yes/ no + one sentence)

c) A production function is given by y = 2 �K3L

12 . Find analytically a short-run demand for labor (assume

�K = 1). Find analytically equilibrium real wage rate if labor supply is given by Ls= 16: Depict it in a gaph.d) You start you �rst job at the age of 21 and you work till 60; and then your retire. You live till 80.

Your annual earnings between 21� 60 are $100; 000 and interest rate is r = 5%. You want to maintain aconstant level of consumption. Write down an equation that allows to determine C (write down the equationbut you do not need to solve for C).

Problem 4 (20p). (Producers)Consider a producer that has the following technology

y = K14L

14 .

a) What returns to scale are represented by this production function? (choose between CRS, IRS orDRS; prove your statement with � argument).

b) Find analytically a (variable) cost function given wK= wL= 2. Plot it in the graph.c) �nd yMES and ATCMES if a �xed cost is F = 2:d) Find analitically a supply function of the �rm and show it in the graph.

Just for funUsing "secrets of happiness" show that if a �rm is maximizing pro�t by producing y�, it necessarily

minimizes the cost of production of y� (give two conditions for pro�t maximization and show that theyimply condition for cost minimization).

2


Midterm 2 (Group B)













U (x1; x2)= 5 lnx1+5 lnx2









3









y = K14L

14 .





4


Midterm 2 (Group C)













U (x1; x2)= 2 lnx1+2 lnx2









5









y = K14L

14 .





6


Midterm 2 (Group D)













U (x1; x2)= 8 lnx1+8 lnx2









7









y = K14L

14 .





8


Answer Keys to midterm 2 (Group A)

“X and Y (2pt).” means that you get 2 pts if you answered both X and Y, and no pts if you missed either (or both).

Problem 1. [Here I denote by T the state with tornado and by N the state without tornado, instead of 1 and 2.]

a) 0.5 · $10 + 0.5 · $0 = $5(2pt).b) With the Bernoulli utility function u(c) = c2, the v.N.M. expected utility function is U(CT , CN ) =

0.5C2T + 0.5C2

N (1pt). Since u(c) = c2 is a convex function, I am risk loving (2pt). The certaintyequivalent CE is the amount of sure money s.t. U(CE, CE) = CE2 = U(0, 10) = 50, i.e. CE = 5

√2

(2pt). CE is larger than EV, because I am risk loving (2pt).c) With the Bernoulli utility function u(c) = c2, the v.N.M. expected utility function is U(CT , CN ) =

0.5 lnCT + 0.5 lnCN (1pt). Yes, I’m risk averse (2pt), since u(c) = ln c is a concave function.d) As CT = (1−γ)x and CN = 4−γx with γ = .5, we obtain the budget constraint CT +CN = 10

(2pt). Its graph has the CT intercept on (CT , CN ) = (10, 0), the CN intercept on (CT , CN ) = (0, 10),and the slope -1 on the CT -CN plane (2pt). The endowment point should be plotted on (CT , CN ) =(0, 10) (1pt).

e) Now I should maximize the utility U(CT , CN ) = 0.5C2T +0.5C2

N on the constraint CT +CN = 10.The magic formula yields CT = (1/2) · (10/1) = 5 (1pt) and CN = (1/2) · (10/1) = 5 (1pt). Pluggingthis into CN = 4− γx, we obtain x = 10 (2pt). The optimal point should be plotted on (5, 5) (1pt).Yes, I am fully insured (1pt) since CT = CN .

f) e.g. γ = 1 (2pt). Actually I would be partially insured, i.e. CT < CN under any premium ratelarger than 0.5.

Problem 2. [Here I denote apple by 1, orange by 2, Andy by A and Bob by B. You could use another notation, as

long as you clarified it.]

a) The Edgeworth box should have length of 50 on each axis (1pt). The endowment is (50, 0)looked from A’s origin, i.e. (0, 50) from B’s origin (1pt). [This is a single point in an Edgeworth box. And,

it cannot be an origin.]

b) ... if none could not be better off (by another feasible allocation) unless anyone is worth off(2pt). [MRSA = MRSB : no point since it is just a mathematical equivalent property and not the definition.1]

c) [Need to prove the claim directly from the definition of Pareto efficiency. Arguing only slopes or tangency is

wrong. Besides, logical sequence (especially starting assumptions and ending conclusions) must be clarified. The curve’s

name, namely “an indifference curve”, should be clarified.]

Necessity (4pt): If MRSA 6= MRSB at an allocation x, both people’s indifference curves shouldcross each other at x and thus we can find a point between them. Because this point is above eachindifference curve looked from the people’s origin, this allocation is better than x for both and thusthe allocation x is not Pareto efficient. [The proof should start with MRSA 6= MRSB and end with Pareto

inefficiency of x. Graph is needed. On the graph, you need to specify another allocation that improves their utilities. If

you wrote two separate points and two separate curves for a single allocation, you misunderstand an Edgeworth box.]

Sufficiency (4pt): If MRSA = MRSB at an allocation x, both people’s indifference curves shouldbe tangent to each other at x and thus no point is below A’s indifferent curve looked from A’s origin,i.e. worse for A than x, or below B’s indifferent curve looked from B’s origin, i.e. worse for B, or belowboth. So any point (allocation) cannot be better than x for both people and x is Pareto efficient. [The

proof should start with MRSA = MRSB at x and end with Pareto efficiency of x. Graph is needed. On the graph,

you need to clarify who is worse off than x in each region defined by the two indifference curves.]

d) As we proved above, the Pareto efficiency is equivalent to MRSA = MRSB , given the feasibilityof the allocation xA

1 + xB1 = 50, xA

2 + xB2 = 50. So we solve

MRSA(xA1 , xA

2 ) =3/xA

1

3/xA2

=3/(50− xA

1 )3/(50− xA

2 )= MRSB(50− xA

1 , 50− xA2 ).

1A Pareto efficient allocation may not hold this equation, unless the preference is convex, monotone, and smooth.

Then we obtain xA1 = xA

2 [or xB1 = xB

2 ] (3pt). This is the equation for the contract curve. [You need to

clarify whose consumption it is.] Graphically it is the line starting from the origin of A with slope 1, i.e.the diagonal line connecting the two origins of the Edgeworth box (1pt).

e) Let the equilibrium price be (p1, p2). Then, Andy should maximize his utility UA(xA1 , xA

2 ) =3 lnxA

1 + 3 lnxA2 on the budget constraint p1x

A1 + p2x

A2 = 50p1. The magic formula yields his optimal

consumption bundle

xA1 =

12

50p1

p1= 25, xA

2 =12

50p1

p2= 25

p1

p2.

Bob should maximize his utility UB(xB1 , xB

2 ) = 3 ln xB1 + 3 ln xB

2 on the budget constraint p1xB1 +

p2xB2 = 50p2. The magic formula yields his optimal consumption bundle

xB1 =

12

50p2

p1= 25

p2

p1, xB

2 =12

50p2

p2= 25.

The feasibility (a.k.a. market clearing) of the allocation requires2

xA1 + xB

1 = 25 + 25p2

p1= 50, ∴ p2 = p1 6= 0.

Plugging this into the above optimal bundles, we obtain xA1 = 25 (2pt), xA

2 = 25 (2pt), xB1 = 25 (2pt)

and xB2 = 25 (2pt). The equilibrium price (p1, p2) can be any pair of two positive numbers as long as

p1 = p2: for example, p1 = 1, p2 = 1 (2pt). [No partial credit for only p1 or p2.]

f) As we argued, p1, p2 can be any pair of two positive numbers as long as p1 = p2 and differentfrom the answer in e): for example, p1 = 2, p2 = 2 (2pt).

g) At the equilibrium allocation ((xA1 , xA

2 ), (xB1 , xB

2 )) = ((25, 25), (25, 25)), the two’s MRSs are

MRSA(25, 25) =3/253/25

= 1, MRSB(25, 25) =3/253/25

= 1.

So we have MRSA = −1 = MRSB and thus this equilibrium allocation is Pareto efficient (2pt).[MRS must be calculated.]

Problem 3. a) PV = 100/(1.05) + 100/(1.05)2 + . . . = 10000 (dollars, 4pt).b) Sam should maximize his utility U = ln C1 + lnC2 on the budget constraint C1 + C2 = 200

(as C1 + S = 200, C2 = S.) The magic formula yields his optimal consumption bundle C1 = (1/2) ·(200/1) = 50 (2pt), C2 = (1/2) · (200/1) = 50 (2pt). Plugging this into C2 = S, we have S = 50 (2pt).Yes, he’s smoothing (1pt) as C1 = C2. No, he’s not tilting (1pt) as C1 = C2. [If you answered only either

one question and did not clarify which question you answered, you get no point.]

c) The production function y = 2K3L1/2 implies the marginal productivity of labor MPL =(1/2) · 2K3L−1/2 = K3L−1/2. In particular, MPL = L−1/2 at K = K = 1. Solving the secretof happiness MPL = L−1/2 = w/p, we find the short-run labor demand LD = (w/p)−2 where pis the product’s price and w is wage (4pt). [Thus w/p is the real wage rate. It is not enough to state only

the secret of happiness; the demand LD should be explicitly determined.3] Solving the demand-supply equalityLD = (w/p)−2 = 16 = LS , we obtain the equilibrium real wage w/p = 1/4 (2pt). The equilibriumpoint (L, w/p) = (16, 1/4) must be plotted on a graph (1pt).

d) (6pt.) The annual consumption C (thousand dollars) is determined from

1001.05

+ · · ·+ 1001.0540

=C

1.05+ · · ·+ C

1.0560∴

(1− 1

1.0540

)1001.05

=(

1− 11.05−60

)C

1.05.

[Further simplification gets full points.]

2We do not have to consider the market clearing of the other good 2: Walras’s theorem. Notice that if p1 = 0 thenp2/p1 =∞ and the equation does not hold; so we need p1 6= 0 too.

3Also I saw so manyy answers “LD = L−1/2”; this does not make sense at all, as it is read as the short-runlabor demand LD is the inverse of the square root of L and we must ask what is L. L = LD is the solution ofMPL = L−1/2 = w/p, but not a number on either side of this equation.

2

Problem 4. a) DRS (1pt). This is because F (λK, λL) = (λ1/4K1/4)(λ1/4L1/4) = λ1/2K1/4L1/4 =λ1/2F (K, L) < λ1/2F (K, L) [if λ > 1] (4pt). [Here F (k, l) is the output from K = k and L = l.]

b) The secret of happiness is

MPK

MPL=

0.25K−3/4L1/4

0.25K1/4L−3/4=

22

=wK

wL, ∴ K = L

To achieve the production of y = F (K, L), we need

y = F (K, K) = K1/2, ∴ K = L = y2

So the cost function is C = 2K + 2L = 2y2 + 2y2 = 4y2 (4pt).4 Graph should be drawn on the y-Cplane (1pt).

c) Solving MC(y) = 8y = (4y2 +2)/y = ATC(y), we obtain yMES = 1/√

2 (2pt) and ATCMES =ATC(yMES) = MC(yMES) = 4

√2 (2pt).5

d) (6pt for giving both the function and the graph.) The optimal supply should satisfy p = 8y∗ =MC(y∗), i.e. y∗ = p/8. But when p < ATCMES = 4

√2, the firm cannot get positive profit even from

the optimal supply and thus should quit the production.The supply function S(p) is therefore

S(p) =

{p/8 if p ≥ 4

√2

0 if p ≤ 4√

2.

On the y-p plane, the graph is y = p/8 (i.e. p = 8y) for p ≥ 4√

2 and y = 0 (a part of the verticalaxis) for p ≤ 4

√2.

Just for fun The secret of happiness for profit maximization is

MPK = pwK , MPL = pwL.

Here p is the product price, MPi is the marginal productivity of factor i, and wi is the price of factori. These two equations imply

MPK

MPL=

wK

wL;

i.e. the secret of happiness for cost minimization.6

4Or, you can think of maximization of Y = F (K, L) = K1/4L1/4 on the constraint 2K + 2L = c, thinking Y as avariable and c as a constant. Then the magic formula of Cobb-Douglas (utility) maximization implies K = (1/2)(c/2) =c/4 and L = (1/2)(c/2) = c/4. Then we obtain at the maximum Y = (c/4)1/4(c/4)1/4 = (c/4)1/2, i.e. c = 4Y 2. Thatis, when Y = y is given, the budget/cost C = 4y2 is needed to achieve this y at the optimum.

5Maybe ATCMES is easier to calculate from MC(yMES) than from ATC(yMES), though they should yield thesame number.

6So there’s a close link between maximization and minimization. This link is called duality and was a driving forceof mathematical economic theory during 1970s-80s: see Varian’s textbook for graduate and advanced undergraduate,Microeconomic Analysis. And, you will use it in undergraduate linear programming, like Computer Science 525: seeFerris, Mangasarian, and Wright, Linear Programming with MATLAB, SIAM-MPS, 2007.

3


Solutions to midterm 2 (Group B)



a) $2 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2

N (1pt). Risk loving (2pt). CE = 2√

2 (2pt). Larger thanEV, because I am risk loving (2pt). c) U(CT , CN ) = 0.5 lnCT + 0.5 lnCN (1pt). Yes, I’m risk averse(2pt). d) CT +CN = 4 (2pt). Graph is needed on the CT -CN plane and its position must be clarifiedwith slope and intercepts (2pt). Plot a point on (CT , CN ) = (0, 4) for endowment (1pt). e) CT = 2(1pt). CN = 2 (1pt). x = 4 (2pt). Plot a point on (2, 2) (1pt). Yes, fully insured (1pt). f) e.g. γ = 1(2pt). [Any number larger than 0.5 because we need CN > CT .]

Problem 2. [Here I denote apple by 1, orange by 2, Andy by A and Bob by B. You could use another notation,

as long as you clarified it.] a) The Edgeworth box should have length of 20 on each axis (1pt). Theendowment is (20, 0) looked from A’s origin, i.e. (0, 20) from B’s origin (1pt). [This is a single point in

an Edgeworth box. And, it cannot be an origin.] b) ... if none could not be better off (by another feasibleallocation) unless anyone is worth off (2pt). [MRSA = MRSB : no point since it is just a mathematical

equivalent property and not the definition.1]

c) [Need to prove the claim directly from the definition of Pareto efficiency. Arguing only slopes or tangency

is wrong. Besides, logical sequence (especially starting assumptions and ending conclusions) must be clarified. The

curve’s name, namely “an indifference curve”, should be clarified.] Necessity (4pt): If MRSA 6= MRSB atan allocation x, both people’s indifference curves should cross each other at x and thus we can finda point between them. Because this point is above each indifference curve looked from the people’sorigin, this allocation is better than x for both and thus the allocation x is not Pareto efficient. [The

proof should start with MRSA 6= MRSB and end with Pareto inefficiency of x. Graph is needed. On the graph, you

need to specify another allocation that improves their utilities. If you wrote two separate points and two separate curves

for a single allocation, you misunderstand an Edgeworth box.]




d) xA1 = xA

2 [or xB1 = xB

2 ] (3pt). [You need to clarify whose consumption it is.] The diagonal line connectingthe two origins of the Edgeworth box (1pt). e) xA

1 = 10 (2pt). xA2 = 10 (2pt). xB

1 = 10 (2pt). xB2 = 10

(2pt) p1 = 1, p2 = 1 (2pt). [p1, p2 can be any pair of two positive numbers as long as p1 = p2. No partial credit

for only p1 or p2.] f) p1 = 2, p2 = 2 (2pt). [p1, p2 can be any pair of two positive numbers as long as p1 = p2 and

different from your answer in e).] g) MRSA = −1 = MRSB and thus this equilibrium allocation is Paretoefficient (2pt). [MRS must be calculated.]

Problem 3. a) $2000 (4pt). b) C1 = 100 (2pt). C2 = 100 (2pt). S = 100 (2pt). Yes, he’ssmoothing (1pt). No, he’s not tilting (1pt). [If you answered only either one question and did not clarify

which question you answered, you get no point.] c) Demand: LD = (w/p)−2 where p is the product’s priceand w is wage (4pt). [Thus w/p is the real wage rate.] Equilibrium real wage: w/p = 1/4 (2pt). Thepoint (L, w/p) = (16, 1/4) must be plotted on a graph (1pt). d) (6pt.) The annual consumptionC (thousand dollars) is determined from

{1− (1.05)−40

}· 50/1.05 =

{1− (1.05)−60

}C/1.05. [Further

simplification gets full points.]

Problem 4. a) DRS (1pt). This is because F (tK, tL) = t1/2K1/4L1/4 = t1/2F (K, L) < tF (K, L)[if t > 1] (4pt). [Here F (k, l) is the output from K = k and L = l.] b) C = 4y2 (4pt). Graph is needed onthe y-C plane (1pt). c) yMES = 1/

√2 (2pt). ATCMES = 4

√2 (2pt). d) (6pt for giving both the

function and the graph.) The supply function S(p) is p/8 for p ≥ 4√

2, and 0 for p ≤ 4√

2. On they-p plane, the graph is y = p/8 (i.e. p = 8y) for p ≥ 4

√2 and y = 0 (a part of the vertical axis) for

p ≤ 4√

2.1A Pareto efficient allocation may not hold this equation, unless the preference is convex, monotone, and smooth.


Solutions to midterm 2 (Group C)



a) $3 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2
















d) xA1 = xA

2 [or xB1 = xB


1 = 20 (2pt). xA2 = 20 (2pt). xB

1 = 20 (2pt). xB2 = 20






{1− (1.05)−40

}· 40/1.05 =

{1− (1.05)−60

}C/1.05. [Further






2, and 0 for p ≤ 4√



p ≤ 4√



Solutions to midterm 2 (Group D)



a) $1 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2

N (1pt). Risk loving (2pt). CE =√















d) xA1 = xA

2 [or xB1 = xB


1 = 5 (2pt). xA2 = 5 (2pt). xB

1 = 5 (2pt). xB2 = 5






{1− (1.05)−40

}· 60/1.05 =

{1− (1.05)−60

}C/1.05. [Further






2, and 0 for p ≤ 4√



p ≤ 4√



Midterm 2 (Group A)

You have 70 minutes to complete the exam. The midterm consists of 4 questions (25+35+15+25=100points) + a bonus (10 "extra" points). Make sure you answer the �rst four questions before working on thebonus one!

Problem 1 (25p). (Labor supply)Eric�s total available time is 24h (per day). He works as a waiter with the wage rate w and he spends

his money on consuming New York Steaks C, that cost $p each:a) on a graph with leisure time (R) measured on the horizontal axis and consumption (C) on the vertical

one plot Eric�s budget set assuming w = 10; p = 2. Provide some economic interpretation of the slope ofthe budget line.

b) suppose his utility is given by

U(C;R) = R2�C

where R is leisure and C is consumption of New York Steaks. Find his optimal time at work (laborsupply LS), the relaxation time R and the steak consumption C as a function of w and p (parameters):Calculate the values of the three variables for w = 10; and p = 2:

c) on a graph with labor supply LS measured on the horizontal axis and real wage w=p on the verticalone plot the entire labor supply curve (marking the three points that you have found analytically); whatcan you say about the sensitivity (elasticity) of labor supply to changes in real wage rate? explain in 2 shortsentences.

Problem 2 (35p). (Edgeworth box - Irving Fisher interest rate determination)Consumption can take place in two periods: today (C1) and tomorrow (C2). Peter has income of $100

today and tomorrow. (hence his endowment is !P = (100; 100)): Amanda today�s income is $100 andtomorrow is $300 (!A = (100; 300)): They both have the same utility function

U i (C1; C2)= ln (C1)+ ln (C2)

a) mark the allocation corresponding to the endowment point in the Edgeworth boxb) argue whether the endowment allocation is (or is not) Pareto e¢ cient (use values of MRS at the

endowment point in your argument). Illustrate your argument geometrically in the Edgeworth Box from a)c) �nd analytically the equilibrium interest rate and allocation and show it in the Edgeworth box. (Hint:

Instead of working with "intertemporal" model, you can �rst �nd equilibrium prices p1 and p2, and thenuse the formula:

p1p2=1 + r

d) who among the two traders is borrowing and who is lending? How much? (one sentence + twonumbers)

e) argue that the "invisible hand of �nancial markets" works perfectly, that is, the equilibrium outcomeis Pareto e¢ cient. (one sentence, two numbers, use values of MRS)

f) �nd PV (in today�s $) , and FV (in tomorrow�s $) of Amanda�s income, given the equilibrium interestrate. (give two numbers)

Problem 3 (15p). (Short questions)Answer the following three questions a), b) and c)a) Consider a lottery that pays $100 when it rains and $36 when it does not, and both states are equally

likely��R = �NR =

12

�: Find the expected value of the lottery and the certainly equivalent of the lottery,

given Bernoulli utility function u(c) =pc. Which is bigger? Explain why. (two numbers+ one sentence)

1

b) Consider a pineapple tree that every year produces fruits worth $5000 (starting next year), forever.How much are you willing to pay for such a tree now, given the interest rate of 20%? (one number)

c) Find the constant payment x you have to make in three consecutive periods (one, two, and three),in order to pay back a loan worth $1400 taken in period zero, given that the interest rate is 100%? (onenumber)Problem 4 (25p). (Producers)Consider a producer that has the following technology

y = K14L

14

a) what returns to scale are represented by this production function? (choose: CRS, IRS or DRS andsupport your choice with a mathematical argument).

b) �nd analytically the level of capital (K), labor (L) and output (y) that maximizes pro�t, and thevalue of maximal pro�t, given p = 4 and wK = wL = 1:

c) �nd the average cost function AC (y) ; and plot it on a graph (prices are as in b). On the same graphshow geometrically the level of maximal pro�t from b) (Hint: for the second part, take the value y fromb)).Bonus Problem. (extra 10 points)Derive (not just give!) the formula for PV of annuity (explain each step, starting with deriving PV for

perpetuity).

2


Solutions to midterm 2 (Group A)

Problem 1 (25p). (Labor supply)a) The slope of the budget set is a real wage rate w=p that tells how many steaks Peter can get for every

hour he works.

b) This is a Cobb-Douglass utility function therefore we can �nd his optimal choice R; C using our"magic" formula, we have derived earlier in our class. The values of parameters are:

a = 2; b = 1;

and hence the relaxation time and consumption is

R =a

a+ b

m

p1=2

3

24w

w= 16

C =b

a+ b

m

p1=1

3

24w

p= 8

w

p

For w = 10 and p = 2 we have R = 16 and C=40. In such case the labor supply is given by

LS = 24�R = 8

c) The labor supply function is inelastic with respect to wp . The reason for that is that the substitution

e¤ect (higher real wage makes leisure more expensive relative to consumption encouraging work) is o¤set byincome e¤ect (the higher income makes leisure more attractive)

Problem 2 (35p).a)

b) The endowment allocation is not Pareto e¢ cient, as at this allocation the slopes of indi¤erence curves

MRSP =CP2CP1

=100

100= 1

MRSA =CA2CA1

=300

100= 3

and hence they do not coincide (see graph above).

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c) We normalize p2 = 1: The optimal consumption today is

CP1 =1

2

100p1 + 100

p1

CP1 =1

2

100p1 + 300

p1

Market clearing condition implies that

1

2

100p1 + 100

p1+1

2

100p1 + 300

p1= 200

orp1 = 2

and hencer = 100%

At this price consumption is given by

CP1 =1

2

2� 100 + 1002

= 75 and CA1 = 200� 75 = 125

and

CP2 =1

2

2� 100 + 1001

= 150 and CA2 = 400� 150 = 250

Hence allocation CP = (75; 150), CA = (125; 250) and interest rate r = 100% is an equilibrium.d) Savings are given by

sP = !P1 � CP1 = 100�1

2

200 + 100

2= 25

hence Peter is saving $25

sA = !A1 � CA1 = 100�1

2

500

2= �25

and Amanda is borrowing $25e)

MRSP =CP2CP1

=150

75= 2

MRSA =CA2CA1

=250

125= 2

The equilibrium allocation is Pareto e¢ cient as the indi¤erence curves are tangent (they have the same slopeMRS)

f)

PV = 100 +300

1 + 100%= 100 +

300

2= 250

FV = 100� (1 + 100%) + 300 = 200 + 300 = 500

Problem 3 (15p). (Short questions)a) Expected value of the lottery is

E (L) =1

2� 100 + 1

2� 36 = 68

The von Neuman Morgenstern lottery is

U =1

2

p100 +

1

2

p36 =

1

2� 10 + 1

2� 6 = 8

2

the Certainty equivalent is pCE = 8) CE = 64

CE < E (L) because the agent is risk averse, and hence is willing to accept lower payment for sure.b) You are willing to pay PV

PV =5000

0:2= 25000

c) Using annuity formula

1400 =x

1

1�

�1

2

�3!=7

8x) x =

8

71400 = 8� 200 = 1600

Problem 4 (25p). (Producers)a) Suppose � > 1: Then

F (�K; �L) = (�K)14 (�L)

14 = �

12K

14L

14 < �K

14L

14 = �F (K;L)

hence we have DRS.b) We use two conditions

MPK =wKp

MPL =wLp

which become14

1

4K� 3

4L14 =

1

41

4K

14L�

34 =

1

4

ImplyingK

L= 1) K = L

Plugging back in the two secrets of happiness

K� 34K

14 = K� 1

2 = 1) K = 1

L14L�

34 = L�

12 = 1) L = 1

The optimal level of production isy = K

14L

14 = 1

14 1

14 = 1

and pro�t� = 4� 1� 1� 1� 1� 1 = 2

c) Secret of happiness for cost minimization is

TRS =L

K=wKwL

= 1) K = L

hencey = K

14L

14 = K

12

henceK = L = y2

henceC (y) = 2y2

3

and

AC =C (y)

y= 2y

Bonus Problem. (extra 10 points)The present value of a perpetuity is

PV =x

1 + r+

x

(1 + r)2 +

x

(1 + r)3 + :::

=1

1 + r[x+ PV ]

Solving for PV gives

PV =x

r

For the asset that pays x up to period T we have to decrease the PV by the PV of a "missing payment"after T: The present value of this payment in $ from period T is xr and hence in $ from period zero it is�

11+r

�Txr : Subtracting this number from PV for perpetuity gives

PV =x

r��

1

1 + r

�Tx

r=x

r

"1�

�1

1 + r

�T#

which is the formula of the PV of annuity.

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Midterm 2 (Group A)


Problem 1 (20p). (Intertemporal choice)Frank works as a consultant. His income when young is $2000 (period 1) and $8000 when old (period

2), the interest rate is r = 100%:a) In the graph depict Frank�s budget set. Mark all the bundles on the budget line that involve saving

and the ones that involve borrowing. Find analytically PV and FV of income and show it in the graphb) Frank�s intertemporal preferences are given by

U(C1;C2) = lnC1 +1

1 + �lnC2

where the discount factor is � = 100%. Using the magic formula, �nd the optimal consumption plan(C1; C2) and how much Frank borrows or saves (three numbers).

c) Is Frank smoothing his consumption? (yes or no answer + one sentence)Problem 2 (30p). (Edgeworth box, and equilibrium)Consider an economy with apples and oranges. Peter is initially endowed with ten apples and 30 oranges

(!P = (10; 30)): Amanda�s endowment is !A = (30; 10) :a) Plot the Edgeworth box and mark the allocation representing the initial endowment.b) Describe the concept of Pareto e¢ ciency (one intuitive sentence). Peter and Amanda have the same

utility functionU i (C1; C2)= 3 ln (C1)+3 ln (C2) :

Verify whether the endowment allocation is (or is not) Pareto e¢ cient (use values ofMRS in your argument).Illustrate your argument geometrically in the Edgeworth Box from a).

c) Find analytically the competitive equilibrium (six numbers) and show it in the Edgeworth box. Findsome other prices that de�ne competitive equilibrium (two numbers).

d) Argue that competitive markets allocate resources e¢ ciently (give two numbers and compare them).

Problem 3 (20p). (Short questions)Answer the following three questions.a) The Bernoulli utility function is given by u(c) = c2 and two states of the world are equally likely.

Find the corresponding von Neuman Morgenstern (expected) utility function (give formula). Is such agentrisk neutral, risk loving or risk averse? (one sentence). Find the expected value and the certainty equivalentof a lottery

�10;p28�:(two numbers). Which is bigger and why (one sentence) (Hint: when calculating

expected value of the lottery, use thatp28 ' 5:3).

b) Derive the formula for perpetuity.c) You will live for 4 periods. You would like to maintain the constant level consumption throughout

your life C. How much can you consume if in the �rst three periods you earn $1500? The interest rate isr = 100%? (one number)


y = 8K14L

12

1

a) What returns to scale are represented by this production function? (choose between CRS, IRS orDRS; you do not have to prove it).

b) (Short run) Assume that �K = 1 and the �rm cannot change it in a short run. Derive a condition foroptimal demand for labor. Explain intuitively its economic meaning. (one sentence).

c) Suppose that labor supply is inelastic and given by Ls = 16h . Find analytically and on the graphthe equilibrium wage rate.

d) Find the unemployment rate with the minimal (real) wage given by wL=p = 4=3:(one number)e) Suppose wL = 1; wK = 2: Derive the cost function C(y); assuming that you can adjust both K

and L, and plot it on the graph. Relate the shape of your cost function to the returns to scale. (Hint: theconstants in this last questions are not round numbers)

Bonus Problem. (extra 10 points)Give examples of production functions with perfect complements and perfect substitutes that are char-

acterized by increasing and decreasing returns to scale.

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Solutions to midterm 2 (Group A)



and the ones that involve borrowing. Find analytically PV and FV of income and show it in the graph

PV = 2000 +8000

1 + 1= 2000 + 4000 = 6000

FV = (1 + 1)2000 + 8000 = 12000

b) Frank�s intertemporal preferences are given by

U(C1;C2) = lnC1 +1

1 + �lnC2


C1 =1

1 + 12

12000

2=2

36000 = 4000

C2 =12

1 + 12

12000

1=1

312000 = 4000

S = 2000� 4000 = �2000

Frank borrows �$2000c) Is Frank smoothing his consumption? (yes or no answer + one sentence).Yes, because C1= C2. This is because �= r.

Problem 2 (30p). (Edgeworth box, and equilibrium)Consider an economy with apples and oranges. Peter is initially endowed with �ve apples and ten oranges

!P = (10; 30): Amanda�s endowment is !A = (30; 10) :a) Plot the Edgeworth box and mark the allocation representing the initial endowment.

b) Describe the concept of Pareto e¢ ciency (one intuitive sentence). Peter and Amanda have the sameutility function

U i (C1; C2)= 2 ln (C1)+2 ln (C2) :

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The endowment allocation is not Pareto e¢ cient, as at this allocation the slopes of indi¤erence curvesare not tangent to each other

MRSP =CP2CP1

=30

10= 3

MRSA =CA2CA1

=10

30=1

3

and hence they do not coincide (see graph above).c) Find analytically the competitive equilibrium (six numbers) and show it in the Edgeworth box. Find

some other prices that de�ne competitive equilibrium (two numbers).We normalize p2 = 1

CP1 =1

2

10p1 + 30

p1

CA1 =1

2

30p1 + 10

p1

and market clearing condition gives

1

2

10p1 + 30

p1+1

2

30p1 + 10

p1= 40

From which one can �nd price p1= 1 = p2. Equilibrium consumption is CA1 = CB1 = 20 and C

A2 = C

B2 = 20.

Other prices: p1= p2= 2d) Argue that competitive markets allocate resources e¢ ciently (give two numbers and compare them).Allocation in competitive equilibrium is Pareto e¢ cient as MRS of both agents are the same

MRSP =CP2CP1

=20

20= 1

MRSA =CA2CA1

=20

20= 1

Problem 3 (20p). (Short questions)a) The Bernoulli utility function is given by u(c) = c2 and two states of the world are equally likely.


�10;p28�.(two numbers). Which is bigger and why (one sentence) (Hint: when calculating


Expected Utility function is given by

U (c1; c2) =1

2c21 +

1

2c22

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Agent is risk loving as Bernouli utility function is convex.

E(L) =1

210+

1

25:3 = 7:6

. Certainty equivalent can be found as

(CE)2= U

�10;p28�= 50 + 14 = 64

and henceCE = 8 > E(L)

This is because risk loving agent derives extra utility from uncertainty regarding the outcome.b) Derive the formula for perpetuity

PV =x

1 + r+

x

(1 + r)2 +

x

(1 + r)2 + ::: =

=x

1 + r+

1

1 + r

�x

(1 + r)+

x

(1 + r)+ :::

�=

=x

1 + r+

1

1 + rPV


PV =x

r

.c) You will live for 4 periods. You would like to maintain the constant level consumption throughout yourlife C. How much can you consume if in the �rst three periods you earn $1500? The interest rate is r=100%?(one number)

C

r

1�

�1

1 + r

�4!=

1500

r

1�

�1

1 + r

�3!15

16C = 1500

7

8C = 1400


y = 8K14L

12


DRSb) (Short run) Assume that �K = 1 and the �rm cannot change it in a short run. Derive a condition for

optimal demand for labor. Explain intuitively its economic meaning. (one sentence). (MPL =wkp )

wkp= 4L�

12

Last worker produces as much as he gets in terms of wage.c) Suppose that labor supply is inelastic and given by Ls= 16h . Find analytically and on the graph

the equilibrium wage rate.wkp= 4 (16)

� 12 = 1

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d) Find the unemployment rate with the minimal (real) wage given by wL=p = 4=3.(one number)With minimal wage rate the demand for labor is

4=3 = 4L�12 ) L = 9

and hence unemployment rate is

UR =16� 916

=7

16

e) Suppose wL= 1; wK= 2. Derive the cost function C(y), assuming that you can adjust both K and L,and plot it on the graph. Relate the shape of your cost function to the returns to scale. (Hint: the constantsin this last questions are not round numbers)

8K14L

12

TRS = �12

L

K= �2

1

and henceL = 4K

It follows that

K =

�1

16y

� 43

and

L = 4

�1

16y

� 43

It follows that

c (y) = 4

�1

16y

� 43

+ 2

�1

16y

� 43

= 6

�1

16y

� 43

The function is convex as we have DRS.


acterized by increasing and decreasing returns to scale.Perfect complements

y = [min (2K; 7L)]2 (IRS)

y = [min (2K; 7L)]12 (DRS)

Perfect substitutes

y = (2K + 7L)2 (IRS)

y = [2K + 7L]12 (DRS)

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Midterm 2 (Group B)


Problem 1 (25p). (Labor supply)Eric�s total available time is 24h (per day). He works as a waiter with the wage rate w and he spends

his money on consuming New York Steaks C, that cost $p each:a) on a graph with leisure time (R) measured on the horizontal axis and consumption (C) on the vertical

one plot Eric�s budget set assuming w = 100; p = 5. Provide some economic interpretation of the slope ofthe budget line.

b) suppose his utility is given by

U(C;R) = R3�C

where R is leisure and C is consumption of New York Steaks. Find his optimal time at work (laborsupply LS), the relaxation time R and the steak consumption C as a function of w and p (parameters):Calculate the values of the three variables for w = 100; and p = 5:

c) on a graph with labor supply LS measured on the horizontal axis and real wage w=p on the verticalone plot the entire labor supply curve (marking the three points that you have found analytically); whatcan you say about the sensitivity (elasticity) of labor supply to changes in real wage rate? explain in 2 shortsentences.

Problem 2 (35p). (Edgeworth box - Irving Fisher interest rate determination)Consumption can take place in two periods: today (C1) and tomorrow (C2). Peter has income of $100

today and $300 tomorrow. (hence his endowment is !P = (100; 300)): Amanda today�s income is $100 inboth periods (!A = (100; 100)): They both have the same utility function

U i (C1; C2)=1

2ln (C1)+

1

2ln (C2)

a) mark the allocation corresponding to the endowment point in the Edgeworth boxb) argue whether the endowment allocation is (or is not) Pareto e¢ cient (use values of MRS at the

endowment point in your argument). Illustrate your argument geometrically in the Edgeworth Box from a)c) �nd analytically the equilibrium interest rate and allocation and show it in the Edgeworth box. (Hint:

Instead of working with "intertemporal" model, you can �rst �nd equilibrium prices p1 and p2, and thenuse the formula:

p1p2=1 + r

d) who among the two traders is borrowing and who is lending? How much? (one sentence + twonumbers)

e) argue that the "invisible hand of �nancial markets" works perfectly, that is, the equilibrium outcomeis Pareto e¢ cient. (one sentence, two numbers, use values of MRS)

f) �nd PV (in today�s $) , and FV (in tomorrow�s $) of Amanda�s income, given the equilibrium interestrate. (give two numbers)

Problem 3 (15p). (Short questions)Answer the following three questions a), b) and c)a) Find the constant payment x you have to make in three consecutive periods (one, two, and three),

in order to pay back a loan worth $2800 taken in period zero, given that the interest rate is 100%? (onenumber)

1

b) Consider a lottery that pays $36 when it rains and $25 when it does not, and both states are equallylikely

��R = �NR =

12

�: Find the expected value of the lottery and the certainly equivalent of the lottery,

given Bernoulli utility function u(c) =pc. Which is bigger? Explain why. (two numbers+ one sentence)

c) Consider a pineapple tree that every year produces fruits worth $500 (starting next year), forever.How much are you willing to pay for such a tree now, given the interest rate of 10%? (one number)Problem 4 (25p). (Producers)Consider a producer that has the following technology

y = K16L

16

a) what returns to scale are represented by this production function? (choose: CRS, IRS or DRS andsupport your choice with a mathematical argument).

b) �nd analytically the level of capital (K), labor (L) and output (y) that maximizes pro�t, and thevalue of maximal pro�t, given p = 6 and wK = wL = 1:

c) �nd the average cost function AC (y) ; and plot it on a graph (prices are as in b). On the same graphshow geometrically the level of maximal pro�t from b) (Hint: for the second part, take the value y fromb)).Bonus Problem. (extra 10 points)Derive (not just give!) the formula for PV of annuity (explain each step, starting with deriving PV for

perpetuity).

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Problem 1 (25p). (Labor supply)a) The slope of the budget set is a real wage rate w=p that tells how many steaks Peter can get for every

hour he works.

b) This is a Cobb-Douglass utility function therefore we can �nd his optimal choice R; C using our"magic" formula, we have derived earlier in our class. The values of parameters are:

a = 3; b = 1;

and hence the relaxation time and consumption is

R =a

a+ b

m

p1=3

4

24w

w= 18

C =b

a+ b

m

p1=1

4

24w

p= 6

w

p

For w = 100; and p = 5 we have R = 18 and C = 120In such case the labor supply is given by

LS = 24�R = 6

c) The labor supply function is inelastic with respect to wp . The reason for that is that the substitution

e¤ect (higher real wage makes leisure more expensive relative to consumption encouraging work) is o¤set byincome e¤ect (the higher income makes leisure more attractive)

Problem 2 (35p).a)

b) The endowment allocation is not Pareto e¢ cient, as at this allocation the slopes of indi¤erence curves

MRSP =CP2CP1

=300

100= 3

MRSA =CA2CA1

=100

100= 1

and hence they do not coincide (see graph above).

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c) We normalize p2 = 1: The optimal consumption today is

CP1 =1

2

100p1 + 300

p1

CA1 =1

2

100p1 + 100

p1

Market clearing condition implies that

1

2

100p1 + 300

p1+1

2

100p1 + 100

p1= 200

orp1 = 2

and hencer = 100%

At this price consumption is given by

CP1 =1

2

2� 100 + 3002

= 125 and CA1 = 200� 125 = 75

and

CP2 =1

2

2� 100 + 3001

= 250 and CA2 = 400� 250 = 150

Hence allocation CP = (125; 250), CA = (75; 125) and interest rate r = 100% is an equilibrium.d) Savings are given by

sP = !P1 � CP1 = 100� 125 = �25hence Peter is borrowing $25

sA = !A1 � CA1 = 100� 75 = 25and Amanda is saving $25

e)

MRSP =CP2CP1

=250

125= 2

MRSA =CA2CA1

=150

75= 2

The equilibrium allocation is Pareto e¢ cient as the indi¤erence curves are tangent (they have the same slopeMRS)

f)

PV = 100 +100

1 + 100%= 100 +

100

2= 150

FV = 100� (1 + 100%) + 100 = 200 + 100 = 300

Problem 3 (15p). (Short questions)a) Using annuity formula

2800 =x

1

1�

�1

2

�3!=7

8x) x =

8

72800 = 8� 400 = 3200

b) Expected value of the lottery is

E (L) =1

2� 36 + 1

2� 25 = 18 + 121

2= 30

1

2

2

The von Neuman Morgenstern lottery is

U =1

2

p36 +

1

2

p25 =

1

2� 6 + 1

2� 5 = 11

2

the Certainty equivalent ispCE =

11

2) CE =

(11)2

4= 30

1

4

CE < E (L) because the agent is risk averse, and hence is willing to accept lower payment for sure.c) You are willing to pay PV

PV =500

0:1= 5000

Problem 4 (25p). (Producers)a) Suppose � > 1: Then

F (�K; �L) = (�K)16 (�L)

16 = �

13K

16L

16 < �K

16L

16 = �F (K;L)

hence we have DRS.b) We use two conditions

MPK =wKp

MPL =wLp

which become14

1

6K� 5

6L16 =

1

61

6K

16L�

56 =

1

6

ImplyingK

L= 1) K = L

Plugging back in the two secrets of happiness

K� 56K

16 = K� 2

3 = 1) K = 1

L16L�

56 = L�

23 = 1) L = 1

The optimal level of production isy = K

16L

16 = 1

16 1

16 = 1

and pro�t� = 6� 1� 1� 1� 1� 1 = 4

c) Secret of happiness for cost minimization is

TRS =L

K=wKwL

= 1) K = L

hencey = K

16L

16 = K

13

henceK = L = y3

henceC (y) = 2y3

3

and

AC =C (y)

y= 2y2

Bonus Problem. (extra 10 points)The present value of a perpetuity is

PV =x

1 + r+

x

(1 + r)2 +

x

(1 + r)3 + :::

=1

1 + r[x+ PV ]


PV =x

r

For the asset that pays x up to period T we have to decrease the PV by the PV of a "missing payment"after T: The present value of this payment in $ from period T is xr and hence in $ from period zero it is�

11+r

�Txr : Subtracting this number from PV for perpetuity gives

PV =x

r��

1

1 + r

�Tx

r=x

r

"1�

�1

1 + r

�T#

which is the formula of the PV of annuity.

4

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Midterm 2 (Group B)




and the ones that involve borrowing. Find analytically PV and FV of income and show it in the graphb) Frank�s intertemporal preferences are given by

U(C1;C2) = lnC1 +1

1 + �lnC2


c) Is Frank smoothing his consumption? (yes or no answer + one sentence)Problem 2 (30p). (Edgeworth box, and equilibrium)Consider an economy with apples and oranges. Peter is initially endowed with �ve apples and ten oranges

!P = (5; 10): Amanda�s endowment is !A = (10; 5) :a) Plot the Edgeworth box and mark the allocation representing the initial endowment.b) Describe the concept of Pareto e¢ ciency (one intuitive sentence). Peter and Amanda have the same

utility functionU i (C1; C2)= 2 ln (C1)+2 ln (C2) :


c) Find analytically the competitive equilibrium (six numbers) and show it in the Edgeworth box. Findsome other prices that de�ne competitive equilibrium (two numbers).

d) Argue that competitive markets allocate resources e¢ ciently (give two numbers and compare them).

Problem 3 (20p). (Short questions)Answer the following three questions.a) The Bernoulli utility function is given by u(c) = c2 and two states of the world are equally likely.


�2;p28�:(two numbers). Which is bigger and why (one sentence) (Hint: when calculating


b) Derive the formula for perpetuity.c) You will live for 4 periods. You would like to maintain the constant level consumption throughout

your life C. How much can you consume if in the �rst three periods you earn $3000? The interest rate isr = 100%? (one number)


y = 8K14L

12

1


b) (Short run) Assume that �K = 1 and the �rm cannot change it in a short run. Derive a condition foroptimal demand for labor. Explain intuitively its economic meaning. (one sentence).

c) Suppose that labor supply is inelastic and given by Ls = 16h . Find analytically and on the graphthe equilibrium wage rate.

d) Find the unemployment rate with the minimal (real) wage given by wL=p = 4=3:(one number)e) Suppose wL = 1; wK = 2: Derive the cost function C(y); assuming that you can adjust both K

and L, and plot it on the graph. Relate the shape of your cost function to the returns to scale. (Hint: theconstants in this last questions are not round numbers)


acterized by increasing and decreasing returns to scale.

2





and the ones that involve borrowing. Find analytically PV and FV of income and show it in the graph

PV = 4000 +16000

1 + 1= 4000 + 8000 = 12000

FV = (1 + 1)4000 + 16000 = 24000

b) Frank�s intertemporal preferences are given by

U(C1;C2) = lnC1 +1

1 + �lnC2


C1 =1

1 + 12

24000

2=2

312000 = 8000

C2 =12

1 + 12

24000

1=1

324000 = 8000

S = 4000� 8000 = �4000

Frank borrows �$4000c) Is Frank smoothing his consumption? (yes or no answer + one sentence).Yes, because C1= C2. This is because �= r.

Problem 2 (30p). (Edgeworth box, and equilibrium)Consider an economy with apples and oranges. Peter is initially endowed with �ve apples and ten oranges

!P = (5; 10): Amanda�s endowment is !A = (10; 5) :a) Plot the Edgeworth box and mark the allocation representing the initial endowment.

b) Describe the concept of Pareto e¢ ciency (one intuitive sentence). Peter and Amanda have the sameutility function

U i (C1; C2)= 2 ln (C1)+2 ln (C2) :

1

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The endowment allocation is not Pareto e¢ cient, as at this allocation the slopes of indi¤erence curvesare not tangent to each other

MRSP =CP2CP1

=10

5= 2

MRSA =CA2CA1

=5

10=1

2

and hence they do not coincide (see graph above).c) Find analytically the competitive equilibrium (six numbers) and show it in the Edgeworth box. Find

some other prices that de�ne competitive equilibrium (two numbers).We normalize p2 = 1

CP1 =1

2

5p1 + 10

p1

CA1 =1

2

10p1 + 5

p1

and market clearing condition gives

1

2

5p1 + 10

p1+1

2

10p1 + 5

p1= 15

From which one can �nd price p1= 1 = p2. Equilibrium consumption is CA1 = C

B1 = 7:5 and C

A2 = C

B2 = 7:5.

Other prices: p1= p2= 2d) Argue that competitive markets allocate resources e¢ ciently (give two numbers and compare them).Allocation in competitive equilibrium is Pareto e¢ cient as MRS of both agents are the same

MRSP =CP2CP1

=7:5

7:5= 1

MRSA =CA2CA1

=7:5

7:5= 1

Problem 3 (20p). (Short questions)a) The Bernoulli utility function is given by u(c) = c2 and two states of the world are equally likely. Find

the corresponding von Neuman Morgenstern (expected) utility function (give formula). Is such agent riskneutral, risk loving or risk averse? (one sentence). Find the expected value and the certainty equivalent of alottery

�2;p28�.(two numbers). Which is bigger and why (one sentence) (Hint: when calculating expected

value of the lottery, use thatp28 ' 5:3).

Expected Utility function is given by

U (c1; c2) =1

2c21 +

1

2c22

2

Marek Weretka

Pencil

Agent is risk loving as Bernouli utility function is convex.

E(L) =1

22+1

25:3 = 3:6

. Certainty equivalent can be found as

(CE)2= U

�2;p28�= 2 + 14 = 16

and henceCE = 4 > E(L)

This is because risk loving agent derives extra utility from uncertainty regarding the outcome.b) Derive the formula for perpetuity

PV =x

1 + r+

x

(1 + r)2 +

x

(1 + r)2 + ::: =

=x

1 + r+

1

1 + r

�x

(1 + r)+

x

(1 + r)+ :::

�=

=x

1 + r+

1

1 + rPV


PV =x

r

.c) You will live for 4 periods. You would like to maintain the constant level consumption throughout yourlife C. How much can you consume if in the �rst three periods you earn $1500? The interest rate is r=100%?(one number)

C

r

1�

�1

1 + r

�4!=

3000

r

1�

�1

1 + r

�3!15

16C = 3000

7

81

2C = 2800


y = 8K14L

12


DRSb) (Short run) Assume that �K = 1 and the �rm cannot change it in a short run. Derive a condition for

optimal demand for labor. Explain intuitively its economic meaning. (one sentence). (MPL =wkp )

wkp= 4L�

12

Last worker produces as much as he gets in terms of wage.c) Suppose that labor supply is inelastic and given by Ls= 16h . Find analytically and on the graph

the equilibrium wage rate.wkp= 4 (16)

� 12 = 1

3

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d) Find the unemployment rate with the minimal (real) wage given by wL=p = 4=3.(one number)With minimal wage rate the demand for labor is

4=3 = 4L�12 ) L = 9

and hence unemployment rate is

UR =16� 916

=7

16

e) Suppose wL= 1; wK= 2. Derive the cost function C(y), assuming that you can adjust both K and L,and plot it on the graph. Relate the shape of your cost function to the returns to scale. (Hint: the constantsin this last questions are not round numbers)

8K14L

12

TRS = �12

L

K= �2

1

and henceL = 4K

It follows that

K =

�1

16y

� 43

and

L = 4

�1

16y

� 43

It follows that

c (y) = 4

�1

16y

� 43

+ 2

�1

16y

� 43

= 6

�1

16y

� 43

The function is convex as we have DRS.


acterized by increasing and decreasing returns to scale.Perfect complements

y = [min (2K; 7L)]2 (IRS)

y = [min (2K; 7L)]12 (DRS)

Perfect substitutes

y = (2K + 7L)2 (IRS)

y = [2K + 7L]12 (DRS)

4


Midterm 2 (Group A)













U (x1; x2)= 3 lnx1+3 lnx2









1









y = K14L

14 .





2


Midterm 2 (Group B)













U (x1; x2)= 5 lnx1+5 lnx2









3









y = K14L

14 .





4


Midterm 2 (Group C)













U (x1; x2)= 2 lnx1+2 lnx2









5









y = K14L

14 .





6


Midterm 2 (Group D)













U (x1; x2)= 8 lnx1+8 lnx2









7









y = K14L

14 .





8


Answer Keys to midterm 2 (Group A)



a) 0.5 · $10 + 0.5 · $0 = $5(2pt).b) With the Bernoulli utility function u(c) = c2, the v.N.M. expected utility function is U(CT , CN ) =

0.5C2T + 0.5C2

N (1pt). Since u(c) = c2 is a convex function, I am risk loving (2pt). The certaintyequivalent CE is the amount of sure money s.t. U(CE, CE) = CE2 = U(0, 10) = 50, i.e. CE = 5

√2

(2pt). CE is larger than EV, because I am risk loving (2pt).c) With the Bernoulli utility function u(c) = c2, the v.N.M. expected utility function is U(CT , CN ) =

0.5 lnCT + 0.5 lnCN (1pt). Yes, I’m risk averse (2pt), since u(c) = ln c is a concave function.d) As CT = (1−γ)x and CN = 4−γx with γ = .5, we obtain the budget constraint CT +CN = 10

(2pt). Its graph has the CT intercept on (CT , CN ) = (10, 0), the CN intercept on (CT , CN ) = (0, 10),and the slope -1 on the CT -CN plane (2pt). The endowment point should be plotted on (CT , CN ) =(0, 10) (1pt).

e) Now I should maximize the utility U(CT , CN ) = 0.5C2T +0.5C2

N on the constraint CT +CN = 10.The magic formula yields CT = (1/2) · (10/1) = 5 (1pt) and CN = (1/2) · (10/1) = 5 (1pt). Pluggingthis into CN = 4− γx, we obtain x = 10 (2pt). The optimal point should be plotted on (5, 5) (1pt).Yes, I am fully insured (1pt) since CT = CN .

f) e.g. γ = 1 (2pt). Actually I would be partially insured, i.e. CT < CN under any premium ratelarger than 0.5.

Problem 2. [Here I denote apple by 1, orange by 2, Andy by A and Bob by B. You could use another notation, as

long as you clarified it.]

a) The Edgeworth box should have length of 50 on each axis (1pt). The endowment is (50, 0)looked from A’s origin, i.e. (0, 50) from B’s origin (1pt). [This is a single point in an Edgeworth box. And,

it cannot be an origin.]

b) ... if none could not be better off (by another feasible allocation) unless anyone is worth off(2pt). [MRSA = MRSB : no point since it is just a mathematical equivalent property and not the definition.1]

c) [Need to prove the claim directly from the definition of Pareto efficiency. Arguing only slopes or tangency is

wrong. Besides, logical sequence (especially starting assumptions and ending conclusions) must be clarified. The curve’s

name, namely “an indifference curve”, should be clarified.]

Necessity (4pt): If MRSA 6= MRSB at an allocation x, both people’s indifference curves shouldcross each other at x and thus we can find a point between them. Because this point is above eachindifference curve looked from the people’s origin, this allocation is better than x for both and thusthe allocation x is not Pareto efficient. [The proof should start with MRSA 6= MRSB and end with Pareto

inefficiency of x. Graph is needed. On the graph, you need to specify another allocation that improves their utilities. If

you wrote two separate points and two separate curves for a single allocation, you misunderstand an Edgeworth box.]




d) As we proved above, the Pareto efficiency is equivalent to MRSA = MRSB , given the feasibilityof the allocation xA

1 + xB1 = 50, xA

2 + xB2 = 50. So we solve

MRSA(xA1 , xA

2 ) =3/xA

1

3/xA2

=3/(50− xA

1 )3/(50− xA

2 )= MRSB(50− xA

1 , 50− xA2 ).

1A Pareto efficient allocation may not hold this equation, unless the preference is convex, monotone, and smooth.

Then we obtain xA1 = xA

2 [or xB1 = xB

2 ] (3pt). This is the equation for the contract curve. [You need to

clarify whose consumption it is.] Graphically it is the line starting from the origin of A with slope 1, i.e.the diagonal line connecting the two origins of the Edgeworth box (1pt).

e) Let the equilibrium price be (p1, p2). Then, Andy should maximize his utility UA(xA1 , xA

2 ) =3 lnxA

1 + 3 lnxA2 on the budget constraint p1x

A1 + p2x

A2 = 50p1. The magic formula yields his optimal

consumption bundle

xA1 =

12

50p1

p1= 25, xA

2 =12

50p1

p2= 25

p1

p2.

Bob should maximize his utility UB(xB1 , xB

2 ) = 3 ln xB1 + 3 ln xB

2 on the budget constraint p1xB1 +

p2xB2 = 50p2. The magic formula yields his optimal consumption bundle

xB1 =

12

50p2

p1= 25

p2

p1, xB

2 =12

50p2

p2= 25.

The feasibility (a.k.a. market clearing) of the allocation requires2

xA1 + xB

1 = 25 + 25p2

p1= 50, ∴ p2 = p1 6= 0.

Plugging this into the above optimal bundles, we obtain xA1 = 25 (2pt), xA

2 = 25 (2pt), xB1 = 25 (2pt)

and xB2 = 25 (2pt). The equilibrium price (p1, p2) can be any pair of two positive numbers as long as

p1 = p2: for example, p1 = 1, p2 = 1 (2pt). [No partial credit for only p1 or p2.]

f) As we argued, p1, p2 can be any pair of two positive numbers as long as p1 = p2 and differentfrom the answer in e): for example, p1 = 2, p2 = 2 (2pt).

g) At the equilibrium allocation ((xA1 , xA

2 ), (xB1 , xB

2 )) = ((25, 25), (25, 25)), the two’s MRSs are

MRSA(25, 25) =3/253/25

= 1, MRSB(25, 25) =3/253/25

= 1.

So we have MRSA = −1 = MRSB and thus this equilibrium allocation is Pareto efficient (2pt).[MRS must be calculated.]

Problem 3. a) PV = 100/(1.05) + 100/(1.05)2 + . . . = 10000 (dollars, 4pt).b) Sam should maximize his utility U = ln C1 + lnC2 on the budget constraint C1 + C2 = 200

(as C1 + S = 200, C2 = S.) The magic formula yields his optimal consumption bundle C1 = (1/2) ·(200/1) = 50 (2pt), C2 = (1/2) · (200/1) = 50 (2pt). Plugging this into C2 = S, we have S = 50 (2pt).Yes, he’s smoothing (1pt) as C1 = C2. No, he’s not tilting (1pt) as C1 = C2. [If you answered only either

one question and did not clarify which question you answered, you get no point.]

c) The production function y = 2K3L1/2 implies the marginal productivity of labor MPL =(1/2) · 2K3L−1/2 = K3L−1/2. In particular, MPL = L−1/2 at K = K = 1. Solving the secretof happiness MPL = L−1/2 = w/p, we find the short-run labor demand LD = (w/p)−2 where pis the product’s price and w is wage (4pt). [Thus w/p is the real wage rate. It is not enough to state only

the secret of happiness; the demand LD should be explicitly determined.3] Solving the demand-supply equalityLD = (w/p)−2 = 16 = LS , we obtain the equilibrium real wage w/p = 1/4 (2pt). The equilibriumpoint (L, w/p) = (16, 1/4) must be plotted on a graph (1pt).

d) (6pt.) The annual consumption C (thousand dollars) is determined from

1001.05

+ · · ·+ 1001.0540

=C

1.05+ · · ·+ C

1.0560∴

(1− 1

1.0540

)1001.05

=(

1− 11.05−60

)C

1.05.

[Further simplification gets full points.]

2We do not have to consider the market clearing of the other good 2: Walras’s theorem. Notice that if p1 = 0 thenp2/p1 =∞ and the equation does not hold; so we need p1 6= 0 too.

3Also I saw so manyy answers “LD = L−1/2”; this does not make sense at all, as it is read as the short-runlabor demand LD is the inverse of the square root of L and we must ask what is L. L = LD is the solution ofMPL = L−1/2 = w/p, but not a number on either side of this equation.

2

Problem 4. a) DRS (1pt). This is because F (λK, λL) = (λ1/4K1/4)(λ1/4L1/4) = λ1/2K1/4L1/4 =λ1/2F (K, L) < λ1/2F (K, L) [if λ > 1] (4pt). [Here F (k, l) is the output from K = k and L = l.]

b) The secret of happiness is

MPK

MPL=

0.25K−3/4L1/4

0.25K1/4L−3/4=

22

=wK

wL, ∴ K = L

To achieve the production of y = F (K, L), we need

y = F (K, K) = K1/2, ∴ K = L = y2

So the cost function is C = 2K + 2L = 2y2 + 2y2 = 4y2 (4pt).4 Graph should be drawn on the y-Cplane (1pt).

c) Solving MC(y) = 8y = (4y2 +2)/y = ATC(y), we obtain yMES = 1/√

2 (2pt) and ATCMES =ATC(yMES) = MC(yMES) = 4

√2 (2pt).5

d) (6pt for giving both the function and the graph.) The optimal supply should satisfy p = 8y∗ =MC(y∗), i.e. y∗ = p/8. But when p < ATCMES = 4

√2, the firm cannot get positive profit even from

the optimal supply and thus should quit the production.The supply function S(p) is therefore

S(p) =

{p/8 if p ≥ 4

√2

0 if p ≤ 4√

2.

On the y-p plane, the graph is y = p/8 (i.e. p = 8y) for p ≥ 4√

2 and y = 0 (a part of the verticalaxis) for p ≤ 4

√2.

Just for fun The secret of happiness for profit maximization is

MPK = pwK , MPL = pwL.

Here p is the product price, MPi is the marginal productivity of factor i, and wi is the price of factori. These two equations imply

MPK

MPL=

wK

wL;

i.e. the secret of happiness for cost minimization.6

4Or, you can think of maximization of Y = F (K, L) = K1/4L1/4 on the constraint 2K + 2L = c, thinking Y as avariable and c as a constant. Then the magic formula of Cobb-Douglas (utility) maximization implies K = (1/2)(c/2) =c/4 and L = (1/2)(c/2) = c/4. Then we obtain at the maximum Y = (c/4)1/4(c/4)1/4 = (c/4)1/2, i.e. c = 4Y 2. Thatis, when Y = y is given, the budget/cost C = 4y2 is needed to achieve this y at the optimum.

5Maybe ATCMES is easier to calculate from MC(yMES) than from ATC(yMES), though they should yield thesame number.

6So there’s a close link between maximization and minimization. This link is called duality and was a driving forceof mathematical economic theory during 1970s-80s: see Varian’s textbook for graduate and advanced undergraduate,Microeconomic Analysis. And, you will use it in undergraduate linear programming, like Computer Science 525: seeFerris, Mangasarian, and Wright, Linear Programming with MATLAB, SIAM-MPS, 2007.

3





a) $2 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2
















d) xA1 = xA

2 [or xB1 = xB


1 = 10 (2pt). xA2 = 10 (2pt). xB

1 = 10 (2pt). xB2 = 10






{1− (1.05)−40

}· 50/1.05 =

{1− (1.05)−60

}C/1.05. [Further






2, and 0 for p ≤ 4√



p ≤ 4√



Solutions to midterm 2 (Group C)



a) $3 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2
















d) xA1 = xA

2 [or xB1 = xB


1 = 20 (2pt). xA2 = 20 (2pt). xB

1 = 20 (2pt). xB2 = 20






{1− (1.05)−40

}· 40/1.05 =

{1− (1.05)−60

}C/1.05. [Further






2, and 0 for p ≤ 4√



p ≤ 4√



Solutions to midterm 2 (Group D)



a) $1 (2pt). b) U(CT , CN ) = 0.5C2T + 0.5C2

N (1pt). Risk loving (2pt). CE =√















d) xA1 = xA

2 [or xB1 = xB


1 = 5 (2pt). xA2 = 5 (2pt). xB

1 = 5 (2pt). xB2 = 5






{1− (1.05)−40

}· 60/1.05 =

{1− (1.05)−60

}C/1.05. [Further






2, and 0 for p ≤ 4√



p ≤ 4√


Econ 301


Prof. Marek Weretka

Midterm 2 (Group A)


For Fun question.

Problem 1. (40p) (Edgeworth box and intertemporal choice)

Consider economy with two periods (interpreted as �when young�and �when old�periods) and two consumers,

Gillian Murphy and Jeremy Krawczyk. Gillian Murphy is one of the top ballet dancers with lifetime income given

by !G = (50; 20). Jeremy Krawczyk is an econ Ph.D. student with income !J = (20; 50). Gillian and Jeremy have

identical utility functions given by

U (x1; x2) = lnx1 +1

2lnx2

a) Plot an Edgeworth box and mark the initial endowment point.

b) Write down the general de�nition of Pareto e¢ cient allocation (one sentence) and give the equivalent condition

in terms of MRS (give formula, you do not need to prove equivalence).

c) Derive the contract curve (write down conditions and solve for the curve) and depict it in the Edgeworth box.

d) Suppose Gillian and Jeremy can �trade� consumption in both periods at prices p1; p2. Find competitive

equilibrium (six numbers) and depict it in the Edgeworth box.

e) Using MRS condition verify that the equilibrium allocation obtained in point d) is Pareto e¢ cient.

f) Find equilibrium interest rate corresponding to competitive prices obtained in point d) (one number).

g) Very Hard: Suppose Gillian�s utility function is UG (x1; x2) = x1 + 2x2 and Jeremy�s utility is UJ (x1; x2) =

2x1 + x2: Without any calculations �nd (geometrically, in the Edgeworth box) the corresponding contract curve.

Hint: do not use the �MRS�conditon, argue from the de�nition of Pareto e¢ ciency.


A lottery pays $9 in a �winning�state and $1 otherwise. The probability of winning is 12 :

a) Find the expected value of lottery (1; 9).

b) Assume Bernoulli utility u(c) =pc:Write down the formula for Von Neumann-Morgenstern (expected) utility

function, U(c1; c2). Without any calculations sketch the indi¤erence curve map.

c) Find the expected utility from lottery (1; 9) (one number) and the utility derived from the expected value of

the lottery for sure (one number). Which of the two numbers is bigger and why? (one sentence)

d) Give answer to point c) assuming Bernouli utility u(c) = c2.

e) Find the certainty equivalent of lottery (1; 9) for two Bernouli utility function: u(c) =pc and u(c) =c2 (two

numbers).

1



y = 6K12L

12

a) Prove formally that the production function exhibits constant returns to scale (use ��argument).

b) Find analytically MPL and MPK. Are the two marginal products increasing, decreasing or constant?

c) Short run: Given stock of capital �K = 4 �nd labor demand (formula) of a competitive �rm. Find equilibrium

real wage rate if labor supply is given by Ls= 9 (one number). Find the unemployment rate if the minimal real wage

is wmin=p = 3 (one number+graph).

d) Long run: �nd cost functions given prices of inputs wK = 1 and wL = 4 (formula). Plot the cost function in

the graph.

e) Hard: Write down the two conditions for the pro�t maximization in long run. Demonstrate that these conditions

can be reduced to the condition for the cost minimization.

Problem 4 (20p). (Individual Supply and Entry)

Assume �xed cost F = 3 and (variable) cost function c(y) = 3y2:

a) Plot ATC curve. Explain why average cost becomes �in�nite�when the level of production is close to zero

and when it is very large (two sentences).

b) Find yMES and ATCMES (give two numbers).

c) Give the condition that determines optimal level of output for any level of price and provide the economic

interpretation (one sentence). Argue that with �xed cost the condition has to be suplemented with adadditional

(inequality) condition.

d) Write down supply function y(p). Plot the supply function, marking a price threshold for non-zero production.

e) Determine the number of �rms operating in the industry if demand is D(p) = 10� p, assuming free entry.(one number)

Just For Fun

Argue that with constant results to scale if for some K;L pro�t is strictly positive, then the pro�t can be made

arbitrarily large by appropriately scaling up the level of production.

2

Econ 301


Prof. Marek Weretka

Midterm 2 (Group B)


For Fun question.






U (x1; x2) = lnx1 +1

2lnx2









g) Very Hard: Suppose Gillian�s utility function is UG (x1; x2) = 2x1 + x2 and Jeremy�s utility is UJ (x1; x2) =

x1 + 2x2: Without any calculations �nd (geometrically, in the Edgeworth box) the corresponding contract curve.











numbers).

3



y = 4K12L

12







the graph.




Assume �xed cost F = 4 and (variable) cost function c(y) = y2:









Just For Fun



4

Econ 301


Prof. Marek Weretka

Midterm 2 (Group C)


For Fun question.




by !G =(15; 30). Jeremy Krawczyk is an econ Ph.D. student with income !J = (30; 15). Gillian and Jeremy have


U (x1; x2) = lnx1 +1

2lnx2





















numbers).

5



y = 2K12L

12




real wage rate if labor supply is given by Ls= 16 (one number). Find the unemployment rate if the minimal real

wage is wmin=p = 2 (one number+graph).


the graph.




Assume �xed cost F = 4 and (variable) cost function c(y) = 4y2:









Just For Fun



6

Econ 301


Prof. Marek Weretka

Midterm 2 (Group D)


For Fun question.






U (x1; x2) = lnx1 +1

2lnx2





















numbers).

7



y = 2K12L

12







the graph.




Assume �xed cost F = 4 and (variable) cost function c(y) = y2:









Just For Fun



8

Econ 301 Midterm 2 SolutionsGroup A

All graphs on last page of key.

Problem 1 (40p) Edgeworth Box and Intertemporal Choice

a) [6] See Figure 1 at end (ignore extra line in corners).

b) [8] An allocation is Pareto efficient if there is no other allocation that makes

any individual better off without making at least one individual worse off.

For agents A and B, the MRS condition is that MRSG = MRSJ .

c) [8] Contract curve conditions:

1) No waste: xJ1 + xJ1 = ωG1 + ωJ

1 = 70 and xG2 + xJ2 = ωG2 + ωJ

2 = 70

2) MRSJ = MRSG

MRSG = MRSJ ⇐⇒ 1/xG1

.5/xG2

=1/xJ

1

.5/xJ2⇐⇒ 2xG

2

xG1

=2xJ

2

xJ1

. Use condition (1) to

find2xG

2

xG1

=2xJ

2

xJ1

=2(70−xG

2 )

(70−xG1 )

. Then by algebra, xG2 = xG1 .

d) [8] Normalize p2 = 1. Then use an individual’s optimization condition−MRSG =p1p2

= p1. By the contract curve, −MRSG =2xG

2

xG1

=2xG

1

xG1

= 2, so p1 = 2.

Use these prices in the Cobb-Douglas magic formulas to find:

xG1 = (a

a+ b)(p1ω

G1 + p2ω

G2

p1) = (

1

3/2)(

2 ∗ 50 + 20

2) = 40 = xG2

xJ1 = (a

a+ b)(p1ω

J1 + p2ω

J2

p1) = (

1

3/2)(

2 ∗ 20 + 50

2) = 30 = xJ2

Alternatively, magic formulas as functions of p1 sum to endowments.

e) [6] MRSG = −2∗4040

equals MRSJ = −2∗3030

.

f) [4] Use p1p2

= 1 + r, so 2 = 1 + r ⇐⇒ r = 1.

g) [0] Contract curve will be on the two edges of the box where agents consume

the higher marginal utility good. Since they prefer different goods, interior

allocations can always be improved by trading low for high MU goods.

w_1

w_2

x_1

x_2

GM

JK

CC

CC

1

Problem 2 (20p) Uncertainty

a) [2] EV = 121 + 1

29 = 5

b) [4] U(CL, CW ) = 12

√CL + 1

2

√CW See Figure 2 below.

c) [5] U(1, 9) = .5√

1 + .5√

9 = 2

u(EV ) =√EV =

√5

The utility of the expected value is larger than the expected utility because the

bernoulli utility function√c is concave, making the agent risk averse.

d) [5] U(1, 9) = .512 + .592 = 41

u(EV ) = EV 2 = 52 = 25

The utility of the expected value is smaller than the expected utility because

the bernoulli utility function c2 is convex, making the agent risk loving.

e) [4] The certainty equivalent of a lottery is the payment delivered with certainty

that yields the same utility as the expected utility of the lottery.

1) u(CE) =√CE = 2 so CE = 22 = 4

2) u(CE) = CE2 = 41 so CE =√

41

Problem 3 (20p) Producers

a) [3] F (λK, λL) = 6(λK)1/2(λL)1/2 = 6λ1/2K1/2λ1/2L1/2 = λ6K1/2L1/2 = λF (K,L)→CRS

b) [3] MPL = 62K1/2L−1/2 so ∂MPL

∂L= 3−1

2K1/2L−3/2 < 0→ MPL decreases in L

MPK = 62K−1/2L1/2 so ∂MPK

∂L= 3−1

2K−3/2L1/2 < 0→ MPK decreases in K

c) [6] Now y = 12L1/2. To find labor demand, use the condition that MPL equals

the real wage, 6L−1/2 = wL

p. Solve for L to find LD = 36(wL

p)−2.

To find the equilibrium real wage, note that in equilibrium LS = LD, so 9 =

36(wL

p)−2. Solve for wl

p= 2.

Now the minimum wage is set above the equilibrium real wage, so we expect

unemployment. Find the number of workers hired by plugging the minimum

wage in to the labor demand function to find LD(wl

p= 3) = 4. Then the

unemployment rate isLS − LD

LS=

9− 4

9= 5/9. Graph in Figure 3.

d) [5] Need to eliminate K and L from typical equation TC = wKK + wLL

2

1) Solve for the optimal allocation across inputs using MPKMPL

= wK

wL= 1/4.

Gives allocation K∗ = 4L∗.

2) Using the allocation from (1), eliminate one input in production function

and solve for remaining input L = y/12

3) Using the allocation from (1), eliminate one input from TC = wLL+wKK

and then substitute in y using the expression from (2). This yields TC =23y. Graph is a line from origin of y-$ plane with slope 2/3.

e) [3] See lecture notes 16 slide 9. Profit Maximizing Conditions:

1) MPK = wK

p

2) MPL = wL

p

Take the ratio of conditions (1) and (2) to find the cost minimizing conditionMPK

MPL=wK/p

wL/p=wK

wL

.

Problem 4 (20p) Individual Supply and Entry

a) [4] ATC = 3/y + 3y. Graph in Figure 4.

When y is close to zero, the FC/y term dominates and gets arbitrarily large

because fixed costs are being averaged over a small level of output. When y is

large, the FC/y term goes to goes to zero and the expression converges to the

line cost = 3y, which is increasing in y because costs per unit increase due to

diminishing returns.

b) [4] Use either ∂ATC/∂y = 0 or ATC = MC to find ymes = 1 and ATCmes =

ATC(ymes) = 6

c) [4] The firm wants to set y so that p = MC because for price-takers this

maximizes the profit function by producing goods as long as MR ≥MC. With

a fixed cost, the firm only wants to operate if it is at least breaking even, so it

only produces y > 0 if p∗ ≥ pBE = pmes = ATCmes.

d) [4] Produce so that p = MC = 6y or y = p/6 if producing at all. Supply

function is overlapping red line in Figure 4.

p(y) =

{p/6 if p ≥ 6

0 if p < 6

3

e) [4] With free entry we know profits will go to zero, so firms produce ymes = 1

and sell at pBE = 6. By the demand curve, QD = D(6) = 10 − 6 = 4. In

equilibrium QD = QS, so QS = 4. Since all firms are identical, QS = Nymes, so

the number of firms N = QS/ymes = 4.

Figure 1: #1 (a), (c), (d)

w_1G w_1

w_2G

w_2

x_1

x_2

GM

JK

Endowment

Equilibrium

w_2J

w_1J

Figure 2: Indifference Curves #2.b

100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

C_lose

C_win

Figure 3: Cost Functions #3 (c) and (d)

0L

w/p L_s

L_d

L*

(w/p)*

min wg

unemployment

Figure 4: Cost and Supply Curves#4 (c) and (d)

0

y

$

MC

ATC

y^mes

ATC^mes

4

Group B









1 = 90 and xG2 + xJ2 = ωG2 + ωJ

2 = 90

2) MRSJ = MRSG


.5/xG2

=1/xJ

1

.5/xJ2⇐⇒ 2xG

2

xG1

=2xJ

2

xJ1


find2xG

2

xG1

=2xJ

2

xJ1

=2(90−xG

2 )

(90−xG1 )




2

xG1

=2xG

1

xG1

= 2, so p1 = 2.


xG1 = (a

a+ b)(p1ω

G1 + p2ω

G2

p1) = (

1

3/2)(

2 ∗ 60 + 30

2) = 50 = xG2

xJ1 = (a

a+ b)(p1ω

J1 + p2ω

J2

p1) = (

1

3/2)(

2 ∗ 30 + 60

2) = 40 = xJ2


e) [6] MRSG = −2∗5050


.

f) [4] Use p1p2

= 1 + r, so 2 = 1 + r ⇐⇒ r = 1.




w_1

w_2

x_1

x_2

GM

JK

CC

CC

5


a) [2] EV = 121 + 1

24 = 2.5

b) [4] U(CL, CW ) = 12

√CL + 1

2


c) [5] U(Q,Q) = .5√

1 + .5√

4 = 1.5

u(EV ) =√EV =

√2.5



d) [5] U(Q,Q) = .512 + .542 = 8.5

u(EV ) = EV 2 = 2.52 = 6.25





1) u(CE) =√CE = 1.5 so CE = 1.52 = 2.25

2) u(CE) = CE2 = 8.5 so CE =√

8.5



b) [3] MPL = 42K1/2L−1/2 so ∂MPL

∂L= 2−1



∂L= 2−1





p)−2.


36(wL


p= 2.

Here the minimum wage is set above the equilibrium real wage, so we expect



p= 3) = 4. Then the


LS=

9− 4



6


= wK

wL= 2. Gives

allocation 2K∗ = L∗.


and solve for remaining input K = y/4√

2


and then substitute in y using the expression from (2). This yields TC =2√2y. Graph is a line from origin of y-$ plane with slope 2√

2.


1) MPK = wK

p

2) MPL = wL

p


MPL=wK/p

wL/p=wK

wL

.


a) [4] ATC = 4/y + y. Graph in Figure 4.

When y is close to zero, the 4/y term dominates and gets arbitrarily large



line cost = y, which is increasing in y because costs per unit increase due to



ATC(ymes) = 4







p(y) =

{p/2 if p ≥ 4

0 if p < 4

7



equilibrium QD = QS, so QS = 12. Since all firms are identical, QS = Nymes,

so the number of firms N = 12/2 = 6.

Figure 1: #1 (a), (c), (d)

w_1G w_1

w_2G

w_2

x_1

x_2

GM

JK

Endowment

Equilibrium

w_2J

w_1J


100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

C_lose

C_win


0L

w/p L_s

L_d

L*

(w/p)*

min wg

unemployment


0

y

$

MC

ATC

y^mes

ATC^mes

8

Group C









1 = 45 and xG2 + xJ2 = ωG2 + ωJ

2 = 45

2) MRSJ = MRSG


.5/xG2

=1/xJ

1

.5/xJ2⇐⇒ 2xG

2

xG1

=2xJ

2

xJ1


find2xG

2

xG1

=2xJ

2

xJ1

=2(45−xG

2 )

(45−xG1 )




2

xG1

=2xG

1

xG1

= 2, so p1 = 2.


xG1 = (a

a+ b)(p1ω

G1 + p2ω

G2

p1) = (

1

3/2)(

2 ∗ 15 + 30

2) = 20 = xG2

xJ1 = (a

a+ b)(p1ω

J1 + p2ω

J2

p1) = (

1

3/2)(

2 ∗ 30 + 15

2) = 25 = xJ2


e) [6] MRSG = −2∗2020


.

f) [4] Use p1p2

= 1 + r, so 2 = 1 + r ⇐⇒ r = 1.




w_1

w_2

x_1

x_2

GM

JK

CC

CC

9


a) [2] EV = 124 + 1

29 = 6.5

b) [4] U(CL, CW ) = 12

√CL + 1

2


c) [5] U(Q,Q) = .5√

4 + .5√

9 = 2.5

u(EV ) =√EV =

√6.5



d) [5] U(Q,Q) = .542 + .592 = 97/2 = 48.5

u(EV ) = EV 2 = 6.52 = (13/2)2 = 169/4





1) u(CE) =√CE = 2.5 so CE = 2.52 = 6.25

2) u(CE) = CE2 = 97/2 so CE =√

97/2



b) [3] MPL = 22K1/2L−1/2 so ∂MPL

∂L= 1−1



∂L= 1−1



the real wage, 4L−1/2 = 42

= 2. Solve for L to find LD = 16(wL

p)−2.


16(wL


p= 1.

Now the minimum wage is set above the equilibrium real wage, so we expect



p= 2) = 4. Then the


LS=

16− 4



10


= wK

wL= 2. Gives

allocation 2K∗ = L∗.


and solve for remaining input K = y/(2√

2)


and then substitute in y using the expression from (2). This yields TC =

8K = 4√2y = 2

√2. Graph is a line from origin of y-$ plane with slope 2

√2.


1) MPK = wK

p

2) MPL = wL

p


MPL=wK/p

wL/p=wK

wL

. See lecture notes 16 slide 9.









ATC(ymes) = 8





d) [4] Produce so that p = MC = 8 or y = p/8 if producing at all. Supply function

is overlapping red line in Figure 4.

p(y) =

{p/8 if p ≥ 8

0 if p < 8

11





Figure 1: #1 (a), (c), (d)

w_1G w_1

w_2G

w_2

x_1

x_2

GM

JK

Endowment

Equilibrium

w_2J

w_1J


100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

C_lose

C_win


0L

w/p L_s

L_d

L*

(w/p)*

min wg

unemployment


0

y

$

MC

ATC

y^mes

ATC^mes

12

Group D









1 = 140 and xG2 + xJ2 = ωG2 + ωJ

2 = 140

2) MRSJ = MRSG


.5/xG2

=1/xJ

1

.5/xJ2⇐⇒ 2xG

2

xG1

=2xJ

2

xJ1


find2xG

2

xG1

=2xJ

2

xJ1

=2(140−xG

2 )

(140−xG1 )




2

xG1

=2xG

1

xG1

= 2, so p1 = 2.


xG1 = (a

a+ b)(p1ω

G1 + p2ω

G2

p1) = (

1

3/2)(

2 ∗ 100 + 40

2) = 80 = xG2

xJ1 = (a

a+ b)(p1ω

J1 + p2ω

J2

p1) = (

1

3/2)(

2 ∗ 40 + 100

2) = 60 = xJ2


e) [6] MRSG = −2∗8080


.

f) [4] Use p1p2

= 1 + r, so 2 = 1 + r ⇐⇒ r = 1.




w_1

w_2

x_1

x_2

GM

JK

CC

CC

13


a) [2] EV = 124 + 1

29 = 6.5

b) [4] U(CL, CW ) = 12

√CL + 1

2


c) [5] U(Q,Q) = .5√

4 + .5√

9 = 2.5

u(EV ) =√EV =

√6.5



d) [5] U(Q,Q) = .542 + .592 = 97/2 = 48.5

u(EV ) = EV 2 = 6.52 = (13/2)2 = 169/4





1) u(CE) =√CE = 1.5 so CE = 1.52 = 2.25

2) u(CE) = CE2 = 8.5 so CE =√

8.5


a) [3] F (λK, λL) = 2(λK)1/2(λL)1/2 = 2λ1/2K1/2λ1/2L1/2 = λQK1/2L1/2 = λF (K,L)→CRS

b) [3] MPL = 22K1/2L−1/2 so ∂MPL

∂L= 1−1



∂L= 1−1





p)−2.


9(wL


p= 1.

Here the minimum wage is set above the equilibrium real wage, so we expect


wage in to the labor demand function to find LD(wL

p= 2) = 9/4. Then the


LS=

9− 9/4



14


= wK

wL= 1/2.



and solve for remaining input L = y/(2√

2)


and then substitute in y using the expression from (2). This yields TC =

4L = 2√2y =√

2y. Graph is a line from origin of y-$ plane with slope√

2.


1) MPK = wK

p

2) MPL = wL

p


MPL=wK/p

wL/p=wK

wL

.


a) [4] ATC = 4/y + y. Graph in Figure 4.

When y is close to zero, the 4/y term dominates and gets arbitrarily large



line cost = y, which is increasing in y because costs per unit increase due to



ATC(ymes) = 4







p(y) =

{p/2 if p ≥ 4

0 if p < 4

15


and sell at pBE = 4. By the demand curve, QD = D(pBEQQ) = 20−4 = 16. In



Figure 1: #1 (a), (c), (d)

w_1G w_1

w_2G

w_2

x_1

x_2

GM

JK

Endowment

Equilibrium

w_2J

w_1J


100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

C_lose

C_win


0L

w/p L_s

L_d

L*

(w/p)*

min wg

unemployment


0

y

$

MC

ATC

y^mes

ATC^mes

16

Group E









1 = 480 and xG2 + xJ2 = ωG2 + ωJ

2 = 280

2) MRSJ = MRSG


.5/xG2

=1/xJ

1

.5/xJ2⇐⇒ 2xG

2

xG1

=2xJ

2

xJ1


find2xG

2

xG1

=2xJ

2

xJ1

=2(280−xG

2 )

(480−xG1 )

. Then by algebra, xG2 = 712xG1 .



2

xG1

=2 712

xG1

xG1

= 76, so p1 = 7/6.


xG1 = (a

a+ b)(p1ω

G1 + p2ω

G2

p1) = (

1

3/2)(

76∗ 400 + 80

76

) = 656021

and xG2 = 712∗ 6560

21

xJ1 = (a

a+ b)(p1ω

J1 + p2ω

J2

p1) = (

1

3/2)(

76∗ 80 + 200

76

) = 352021

and xJ2 = 712∗ 3520

21


e) [6] MRSG =−2 ∗ 6560

21712∗ 6560

21

equals MRSJ =−2 ∗ 3520

21712∗ 3520

21

.

f) [4] Use p1p2

= 1 + r, so r = 1/6.

g) [0] Contract curve will be the two optimal proportion lines and the outside

edges near them. Graph below not to scale.

w_1

w_2

x_1

x_2

GM

JK

CC

17


a) [2] EV = 121 + 1

29 = 5

b) [4] U(CL, CW ) = 12

√CL + 1

2


c) [5] U(1, 9) = .5√

1 + .5√

9 = 2

u(EV ) =√EV =

√5



d) [5] U(1, 9) = .512 + .592 = 41

u(EV ) = EV 2 = 52 = 25





1) u(CE) =√CE = 2 so CE = 22 = 4

2) u(CE) = CE2 = 41 so CE =√

41


a) [3] F (λK, λL) = 12(λK)1/2(λL)1/2 = 12λ1/2K1/2λ1/2L1/2 = λ12K1/2L1/2 =

λF (K,L)→ CRS

b) [3] MPL = 122K1/2L−1/2 so ∂MPL

∂L= 6−1


MPK = 122K−1/2L1/2 so ∂MPK

∂L= 6−1





p)−2.


144(wL


p= 4.

Now the minimum wage is set below the equilibrium real wage, so it has no

bite and we do not expect unemployment. Then the unemployment rate is 0.

Graph in Figure 3.



= wK

wL= 1/4.


18


and solve for remaining input L = y/24


and then substitute in y using the expression from (2). This yields TC =13y. Graph is a line from origin of y-$ plane with slope 1/3.


1) MPK = wK

p

2) MPL = wL

p


MPL=wK/p

wL/p=wK

wL

.









ATC(ymes) = 10







p(y) =

{p/10 if p ≥ 10

0 if p < 10




so the number of firms N = QS/ymes = 10.

19

Figure 1: #1 (a), (c), (d)

w_1G w_1

w_2G

w_2

x_1

x_2

GM

JK

Endowment

Equilibrium

w_2J

w_1J


100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

C_lose

C_win


0L

w/p L_s

L_d

L*

(w/p)*

min wg


0

y

$

MC

ATC

y^mes

ATC^mes

20

Econ 301 Intermediate Microeconomics Prof. Marek Weretka ...

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