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 301 Intermediate Microeconomics Prof. Marek Weretka...

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