1/2

Managerial Economics Formulas1 Elasticity of Demand

,

%

%

%

%

%

%

x x x

p

x x x

x x x

I

D

xX Y

x y

Y y x

Q dQ P E

P dP Q

Q dQ P E

I dI I

dQQ P E

P dP Q

  



  



  



Price, Marginal Revenue and Elasticity

1 1 1

1 1P

P

MR P P MR E

E

                 

Consumption Theory

X XY

Y

X X X Y

Y Y X Y

X Y

MU MRTS

MU

MU P MU MU

MU P P P

X P Y P I



  

 

Calculus Rules: Derivatives

 If , 0 dy

y a then dx

 

 If 1

, b bdY

y ax then abx dx

  

 If     , dy dy dz

y u x z x then dx dx dx

   

 If    . , then dY dZ dU

y u x z x u z dx dx dx

  

 If  

  2 . .

,

du dz z u

u x dy dx dxy then z x dx z



 

At Maximum 2

2 0

Y

X

 

 and at minimum

2

2 0

Y

X

 



Production Theory ( )

L L L

L K L L

L K K K

L K

TP d TP AP MP P MP W

L dL

MP MP MP P

P P MP P

TC L P K P

  

  

 

1 Ibrahim Elsaify, Applications in Managerial Economics, 2011. All Rights Reserved.

2/2

Economies of Scale

( , )

( , )

Q f L K

Q f L K

Q Q IRS

 







 

Economies of Scope

, ) , 0) , )

) ) , ) , 0

, )

( ( (0

( ( (

(

x y x y

x y x y

x y

Economies of Scope

Also S Economies of Scope

TC Q Q TC Q TC Q

TC Q TC Q TC Q Q

TC Q Q

  

    

Equilibrium Conditions:

Market Equilibrium: Q Q

d s 

Firm Equilibrium: MR = MC where

( ) ( )

d TR d TC MR and MC

dQ dQ  

where TR TC TR P Q   

Cost Functions

( ) ( )

AC AVC AFC

TC TVC TFC

TC TVC TFC

Q Q Q

d TC d TVC MC

dQ dQ

 

 

 

Market Concentration a. N-Firm Concentration Ratio (N-Firm CR)

1 2 3 4 4 1 2 3 4

: i i

T T

S S S S S C W W W W Where W

S S

        

b. Herfindahal-Herchman Index (HHI)

1

2 2 2 2

10, 000 0 10, 000

1 1 1 1 1 : .....

1 -

n

i

i

HHI W where HHI

HHI N N N N N N

Number equal size firms in the market HHI





      

 

     



With N equal size firms

c. Lerner Index and Markup Factor

Markup Factor

1

1

1

1 P

E

P MC P L P MC Markup Factor

P L MC 



     



d. Rothchild Index Market Elasticity of Demand

Firm Elasticity of Demand

MK

F

RI 

 

Applications in Managerial Economics_Class R

10/17/2014 Page 1 of 32

Applications in Managerial Economics1

Ibrahim Elsaify, Ph.D. (Draft 2014)

1 I developed the applications help my MBA Managerial Economics (ECO642) students at Goldey Beacom College.

This is a work-in-progress and will change to include new applications or revisions of existing ones. It may also

contain typos and simple errors. I will appreciate any feedback or corrections you have, as this will enhance the

document and it more useful for future groups.

Applications in Managerial Economics_Class R

10/17/2014 Page 2 of 32

Market Equilibrium, Elasticity and Government Intervention Answered Applications 1. The U.S. Department of Agriculture is interested in analyzing the domestic market for wheat. The USDA's

staff economists estimate the following equations for the demand and supply curves:

20 - 2

- 1

d

w w

s

w w

Q P

Q P





Quantities are measured in millions of bushels; prices are measured in dollars per bushel.

a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW.

b. Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply.

c. At what price is the price elasticity of demand equals to zero, infinity, and one?

d. At a price floor of $9 per bushel, calculate surplus, CS, PS, and SW. Compare with free market outcomes, and calculate DWL. If government purchased the surplus, how would this change the outcome?

e. Suppose that the USDA's staff economists expand their estimation of the demand curve to include the price

of corn ( ) cP and the income of consumers ( ) I as follows:

= 20 - 2 3 .1dw w cQ P P I 

i. Is wheat a normal or an inferior good?

ii. What is the relationship between wheat and corn? Answer

* * a. At equilibrium: 20 - 2 - 1 3 21 7 7 1 6

3(6) $9

2 6(6)

PS= =$18 2

SW=9+18=$27

* 7

b. 2 2.33 (Demand is elsatic) * 6

* 7

1 1.2 (Sy * 6

d

s

d s Q Q P P P P Q

w w w w w w w

CS

dQ P

dp Q

dQ P

dp Q





          

 

    

  

     

     

pply is elsatic)

c. 0 at 0, at 10, 1 at 5

s d d. Surplus=Q -Q

@p=9 @p=9

d Q =20-2(9)=2

@p=9

s Q =9-1=8

@p=9

Surplus=8-2=6

1(2) CS= =$1

2 6+8

PS=2 =$14 2

SW=1+14=15

DWL=27-15=$12

If gov

P P P        

     

ernment purchased the surplus, CS stays the same at $1. However PS increases to:

8(8) PS= =$32

2 SW=1+32=$33

d e. An increas in P increases Q Corn and wheat are substiututes.

c w

An increas in I increases d

Q Wheat is a normal good. w

Applications in Managerial Economics_Class R

10/17/2014 Page 3 of 32

2. The United States Government uses wheat as an effective tool of foreign aid. Therefore, the government is committed to maintain a strategic production level that far exceeds the domestic market demand. In the meantime, this policy insures support of farmers. The USDA's staff economists estimate the following equations for the demand and supply curves of wheat:

42 - 3

- 2

d

w w

s

w w

Q P

Q P





Quantities are measured in millions of bushels; prices are measured in dollars per bushel.

a Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply. c. At what quantity is the price elasticity of demand equals to zero, one and infinity. d. f a price floor of $12/unit is imposed, what will be the imbalance in the market? Calculate CS, PS,

and SW and DWL. e. If government purchased all additional supply at the price floor of $12/unit, calculate PS, and cost to

the government. Answer

1

* * a. At equilibrium: 42 - 3 - 2 4 44 11 11 2 9

3(9) $13.5

2 9(9)

PS= =$40.5 2

SW =13.5+40.5=$54

* 11

b. 3 3.67. Demand is elasitic * 9

* 11

1 * 9

d

s

d s Q Q P P P P Q

w w w w w w w

CS

dQ P

dp Q

dQ P

dp Q





          

 

    

 

     



2

1 2

1.22. Supply is elasitic

. 0 42, 0, 1 21

s d d. Surplus=Q -Q

@p=12 @p=12

2(6) CS= =$6

2 10+4

PS=6 =$42 2

SW =6+42=48

DWL=SW 54-48=$6

e. If government purc

10 6 4

-SW =

c at Q at Q at Q  



      

    

  

     

hased the surplus, CS stays the same at $6. However PS increases to:

10(10) PS= =$50

2 SW=6+50=$56

The Cost to the govenment=4(12) $48

Applications in Managerial Economics_Class R

10/17/2014 Page 4 of 32

3. The elected officials in a west coast university town are concerned about the "exploitative" rents being charged to college students. The town council is contemplating the imposition of a $350 per month rent ceiling on apartments in the city. An economist at the university estimates the demand and supply curves as:

5600 - 8

4 - 500

Q P

Q P





Where P = monthly rent, and Q = number of apartments available for rent. For purposes of this analysis, apartments can be treated as identical.

a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b. Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply. c. At what quantity does the price elasticity of demand equal to zero, infinity, and 1? d. What quantity will eventually be available if the rent ceiling is imposed? Calculate shortage, CS, PS, SW

and DWL. Answer

 

 

* *

1

. : 5600 - 8 4 - 500 6100 12 1, 533.3

(700 508.33) $146, 946.68

2 (508.33 -125)

1533.33 $293,885.69 2

146, 946.68 293,885.69 $440,8

6100 508.3

12

1533.33

d s a At equilibrium Q Q P P P P Q

w w

CS

PS

SW

        

  

 

  



32.37

* 508.30

. 8 2.65 Demand is elastic * 1, 533.3

* 508.30

4 1.325 Supply is elastic * 1, 533.3

. 0 5, 600, 0, 1 2,800

. @ 350

4(350) -500 900

d

s

dQ P b

dp Q

dQ P

dp Q

c at Q at Q at Q

s d Q

p





  

    

  

      



     

     

 

 

2

1 2

@ 350 @ 350

(587.50 - 350) 350 $264, 375

2 (350 -125)(900)

$101, 250 2

264, 3751 101, 250 365, 625

440,832.37 - 365, 625 $75, 207.37

[5600 8(350)] 900 1,900

900

-

d s Q Q

p p

CS

PS

SW

DWL SW SW

Shortage  

  

 

  

 

     



Applications in Managerial Economics_Class R

10/17/2014 Page 5 of 32

4. U.S. Department of Agriculture is interested in analyzing the domestic market for wheat. The USDA's staff economists estimate the following equations for the demand and supply curves:

29 - 2

- 1

d

w w

s

w w

Q P

Q P





Quantities are measured in millions of bushels; prices are measured in dollars per bushel. a Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply c At what quantity does the price elasticity of demand equal to zero, one, and infinity d. Suppose that the USDA's staff economists expand their estimation of the demand curve to include

the price of corn ( ) cP and the income of consumers ( ) I as follows:

20 - 2 3 - .1dw w cQ P P I 

i. What is the relationship between wheat and corn? ii. Is wheat a normal or an inferior good?

Answer

*

*

*

*

1

a. At equilibrium:

29 - 2 - 1

3 30 10

10 1 9

10 b. 2 2.22. Demand is elastic

9

9(14.5 10) $20.25

2 9(10 1)

$40.5 2

$60.7520.25 40.5

d s

w w

w w

w w

w

d

Q Q

P P

P P

Q

dQ P

dp Q

CS

PS

SW





 

   

   

    

  

  

  

     

*

*

10 1 1.11. Supply is elastic

9

c. 0 at Q 29

at Q 0

1 at Q 14.5

d. An increas in increases Corn and wheat are substiututes.

An increas in decreases Wheat is a

d

c w

d

w

s dQ P

dp Q

P Q

I Q









  

 

  

 





     

n inferior good.

Applications in Managerial Economics_Class R

10/17/2014 Page 6 of 32

Unanswered Applications 5. The supply and demand curves for corn are as follows:

3750 - 725

920 690

d

s

Q P

Q P



 

Where Q = millions of bushels and P = price per bushel.

a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW.

b. The government has imposed a $2.50 per bushel support price. How much corn will the government be forced to purchase?

c. Calculate the change in consumer surplus that would occur under the support program. Answer

Applications in Managerial Economics_Class R

10/17/2014 Page 7 of 32

6. The severe heat wave of last summer affected global demand and supply for wheat. The United States government uses wheat as an effective tool of foreign aid. Therefore, the U.S. Department of Agriculture is interested in analyzing the domestic market for wheat to insure support of farmers and plentiful supply of wheat for domestic market and as a policy tool. The USDA's staff economists estimate the following equations for the demand and supply curves:

41 - 2

- 1

d

w w

s

w w

Q P

Q P





Quantities are measured in millions of bushels; prices are measured in dollars per bushel.

a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b. Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply. c. At what quantity is the price elasticity of demand equals to zero, one and infinity. d. If a price floor of $16/unit is imposed, what will be the imbalance in the market? Calculate CS, PS, and

SW and DWL.

Answer

Applications in Managerial Economics_Class R

10/17/2014 Page 8 of 32

7. To prevent the spread of Avian Flu, a pharmaceutical company has made a breakthrough in preventing the virus from infecting the workers coming in direct contact with potentially infected birds. The marketing department of the company estimated the demand on the product, which is a latex complete outerwear, to be:

75 -

2 - 60

d

s

Q P

Q P





Quantities are measured in thousands of gears and prices are measured in dollars per gear.

a Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b Calculate price elasticity of demand and supply at equilibrium, and categorize demand and supply. c At what quantity is the price elasticity of demand equals to zero, one and infinity. d If a price ceiling of $35/unit is imposed, what will be the imbalance in the market? Calculate CS, PS, and

SW and DWL. e Calculate the cost to subsidize suppliers to keep the price at $35/unit and supply the market with all

required gears at that price.

Answer

Applications in Managerial Economics_Class R

10/17/2014 Page 9 of 32

Consumption Theory (Optional) Answered Applications

1. An individual consumes products and X Y and spends $250 per time period. The prices of the two goods are $3 per unit for X and $2 per unit forY . The consumer in this case has a utility function expressed as:

( , ) 0.5 U X Y XY

a. Express the budget equation mathematically.

b. Determine the values of X and Y that will maximize utility of the consumer.

c. Calculate the total utility at its maximum.

d. Suppose that a tax of $1/unit is levied on good X. How will this change the consumer’s optimum choice? How much will be the tax proceeds?

e. Suppose that an equivalent lump sum tax is imposed instead of the unit tax. What will be the utility maximizing combination?

f. Which of the two taxes will the consumer prefer? Explain your answer carefully. Answer

yx

x y

* *

* * *

d. With $1 tax

a. 3 2 250

MUMU b. At equilibrium:

P P

.5 .5 3

3 2 2 3

Substituting into budget line 3 2 250 2

41.67 6 250 41.67 3 62.5

2

c. .5 .5(41.67)(62.5) 1, 302

X Y

Y X X Y

X X

X X Y

U X Y

 



   

  

      

  

           

   

yx

x y

* *

per unit on , the new budget line will be:

4 2 250

MUMU .5 .5 At equilibrium: 2

P P 4 2

Substituting into budget line 4 2 2 250

8 250 31.25 2 31.25 62.5

Tax proceeds will b

X

X Y

Y X Y X

X X

X X Y

 

    

  

      

yx

x y

e $31.25e. With a lump sum tax equals to $31.25

e. With a lump sum tax of $31.25, the new budget line will be:

3 2 250 31.25 218.75

MUMU .5 .5 3 At equilibrium:

P P 3 2 2

Subst

(same as b)

X Y

Y X X Y

   

    

* *

* * *

* * *

3 ituting into budget line 3 2 218.75

2 3(36.46)

6 218.75 36.46 54.69 2

f. with per unit tax .5 .5(31.25)(62.5) 976.56

with lump sum tax .5 .5(36.46)(54.69) 997

Therefore

X X

X X Y

U X Y

U X Y

  

      

  

  

     

, the consumer will prefer a lump sum tax.

Applications in Managerial Economics_Class R

10/17/2014 Page 10 of 32

2. Anthony Bradford is an undergraduate student at Siena College in Albany, New York. Anthony has a discretionary spending income of $300 a semester. Anthony’s favorite activities are going to the local

theater at Cross Gates Mall ( X ) or hanging out with friends at the local Starbucks (Y ). Going to local theater costs Anthony $15 and hanging out with friends at Starbucks costs him $10. Anthony’s utility function can be modeled as:

( , ) 3 U X Y XY

a Determine the equilibrium values of X, Y, and total utility. b Suppose that City of Albany imposed 20% tax on theater tickets, at the same time local Starbucks offered a

10% discount to Siena College students. How will this change Anthony’s optimum choice? c Compare Anthony’s utility after the change to his original utility.

Answer

yx

x y

* *

* * *

b. With 20% tax on , and 10% d

a. 15 10 300

MUMU At equilibrium:

P P

3 3 3

15 10 2 3

Substituting into budget line 15 10 300 2

10 30 300 10 3 15

2

3 3(10)(15) 450

X

X Y

Y X X Y

X X

X X Y

U X Y

 



   

  

      

  

     

     

   

yx

x y

* *

*

iscount on Y, the new budget line will be:

18 9 300

MUMU 3 3 At equilibrium: 2

P P 18 9

Substituting into budget line 18 9 2 300

36 300 8.3 2 8.3 16.67

c. after change in p

X Y

Y X Y X

X X

X X Y

U

 

    

  

      

* * rices = 3 3(8.3)(16.67) 415X Y  

Applications in Managerial Economics_Class R

10/17/2014 Page 11 of 32

3. An individual consumes products and X Y and spends $300 per time period. The prices of the two goods are $3 per unit for X and $2 per unit forY . The consumer in this case has a utility function expressed as:

( , ) 0.5 U X Y XY

a. Express the budget equation mathematically. b. Determine the values of X and Y that will maximize utility of the consumer. c. Calculate the total utility at its maximum. d. Suppose that a tax of $1/unit is levied on good X. How will this change the consumer’s optimum choice?

How much will be the tax proceeds? e. Suppose that an equivalent lump sum tax is imposed instead of the unit tax. What will be the utility-

combination? f. Which of the two taxes will the consumer prefer? Explain your answer carefully.

Answer

yx

x y

* *

* * *

d. With $1 tax per unit on

a. 3 2 300

MUMU b. At equilibrium:

P P

.5 .5 3

3 2 2 3

Substituting into budget line 3 2 300 2

50 6 300 50 3 75

2

c. .5 .5(50)(75) 1875

X

X Y

Y X X Y

X X

X X Y

U X Y

 



   

  

      

  

     

     

   

yx

x y

* *

, the new budget line will be:

4 2 300

MUMU .5 .5 At equilibrium: 2

P P 4 2

Substituting into budget line 4 2 2 300

8 300 37.50 2 37.5 75

Tax proceeds will be $37.5

e. With a

X Y

Y X Y X

X X

X X Y

 

    

  

      

yx

x y

lump sum tax equals to $37.50, the new budget line will be:

3 2 300 37.5 262.50

MUMU .5 .5 3 At equilibrium:

P P 3 2 2

3 Substituting into budget line 3 2 262.50

2

6

(same as b)

X Y

Y X X Y

X X

X

   

    

  



     

* *

* * *

* * *

3(43.75) 262.50 43.75 65.63

2 f. with per unit tax .5 .5(37.5)(75) 1, 406.25

with lump sum tax .5 .5(43.75)(65.63) 1, 435.55

Therefore, the consumer will prefer a lump sum tax.

X Y

U X Y

U X Y

     

  

  

Applications in Managerial Economics_Class R

10/17/2014 Page 12 of 32

Unanswered Applications 4. Adam Wilkins is an undergraduate student at John Carroll University, which is located at University

Heights, a suburb on the east side of Cleveland, Ohio. Adam has a discretionary spending income of $300

a semester. Adam favorite activities are going to the local theater at University Circle ( X ) or hanging out

with friends at the local Starbucks (Y ). Going to local theater costs Adam $15 and hanging out with friends at Starbucks costs $10. Adam’s utility function can be modeled as:

( , ) 2 U X Y XY

a. Determine the equilibrium values of X, Y, and total utility. b. Suppose that City of Cleveland imposed 20% tax on ticket theater, at the same time local Starbucks offered

a 10% discount to JCU students. How will this change Adam’s optimum choice? c. Compare Adam’s utility after the change to his original utility

Answer

Applications in Managerial Economics_Class R

10/17/2014 Page 13 of 32

Production Theory Answered Applications

1. The production function of a process that uses two inputs, labor ( )L and capital ( )K , is:

4 Q LK

Where Q represents output per day (tons), labor costs $200 per unit and capital costs $1,000 per unit. The goal of the firm is to produce 10,000 tons daily.

a. Determine returns to scale of the production process, and its elasticity with respect to labor and capital. b. Determine the least-cost combination of L and K, and calculate total cost. c. Carefully analyze the effects on labor productivity and AC if cost of capital declined to $800 a unit.

Answer a. This plant exhibits increasing returns to scale since increasing both inputs 10% increases output 21%. Also,

the sum of powers is greater than 1.

1

1

s L

s K









b. The equilibrium condition for profit maximization is:

L K

2

* *

L K

MP MP 4K 4L = = L=5K

200 1000

Substituting 5K for L in Q=4LK, where Q=10,000:

10,000= 4K(5K)=20K * = 500 22.36

5 5(22.36) 111.80

Total Cost=L.P +K.P =111.80(200)+22.36(1000)=$44,720

L KP P

K

L K

 

 

   

L K

2

* *

L K

c. At equilibrium: MP MP 4K 4L

= = L=4K 200 800

Substituting 4K for L in Q=4LK, where Q=10,000:

10,000= 4K(4K)=16K * = 625 25

4 4(25) 100 Total Cost=L.P +K.P =100(200)+25(800)=$40,000

Labor Pro

L KP P

K

L K

 

 

   

L@ 1,000

L@ 800

@ 1,000

@ 8

ductivity: 10,000

AP 89.45 111.8

10,000 AP 100

100 Therfore, a decrease in the price of capital increaes labor productivity.

Average cost: 44,720

AC $4.4789 10,000

AC

K

K

K

K

P

P

P

P

Q

L Q

L

TC

Q









  

  

  

00

40,000 $4.0

10,000 Therfore, a decrease in the price of capital decreases cost per unit. This is consistent with the increase in labor productivity.

TC

Q   

Applications in Managerial Economics_Class R

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2. The production function of a process that uses two inputs, labor ( )L and capital ( )K , is: 2 2 4 Q L K

Where Q represents output per day (tons), labor costs $200 per unit and capital costs $1,000 per unit.

a. Determine returns to scale of the production process, and its elasticity with respect to labor and capital. b. If the goal of the firm is to produce 10,000 tons daily, determine the least-cost combination of L and K, and

calculate total cost. c. If cost were limited to $5,000, calculate the equilibrium combination of L and K, and the level of production. d. Carefully analyze the effects on labor productivity and average cost in b and c above. Analyze your answer

carefully.

Answer a. This plant exhibits increasing returns to scale since the sum of powers is greater than 1.

2

2

s L

s K









b. The equilibrium condition for profit maximization is:

2 2

2 2

2 2 4

* *

8 8 5

200 1000

5 4 , 10,000 :

10,000 4 (5 ) 100 * 10 3.16

5 5(3.16) 15.81

. . 15.81(200) 3.16(1000) $6,322.27

.

L K

L K

L K

MP MP LK L K L K

P P

Substituting K for L in Q L K where Q

K K K K

L K

Total Cost L P K P

c A

    

 

    

   

    

*

2 2 @ 5000

L

$5,000 :

5000 200 * 1,000 *

5 5000 200(5 ) 1,000

2.5

* 5(2.5) 12.5

4(2.5) (12.5) 3,906.25

d. Labor Productivity and Average Cost:

10,000 AP 632.51

15.81

6,322.

TC

t C

L K

Substituting L K K K

K

L

Q Units

Q

L

AC





 

   

 

  

 

  



L@TC=$5,000

27 $0.63

10,000

3,906.25 AP 312.5

12.5

5,000 $1.28

3,906.25

Therfore, limiting the cost to $5,000 decreased productivity and increased average cost. This is due to IRS.

Q

L

AC



  

 

Applications in Managerial Economics_Class R

10/17/2014 Page 15 of 32

3. The production function of a process that uses two inputs, labor (L) and capital (K), is:

Q = 10LK

Where Q represents output per day (tons), labor costs $200 per unit and capital costs $800 per unit. The goal of the firm is to produce 16,000 tons daily.

a. Determine returns to scale of the production process, and its elasticity with respect to labor and capital. b. Determine the optimum combination of L and K, and calculate total cost. c. Carefully analyze the effects on labor productivity and average cost if cost of capital increased to $1,000

per unit. Answer a. This plant exhibits increasing returns to scale since sum of powers is greater than 1.

1

1

s L

s K









b. The equilibrium condition for profit maximization is:

2

* *

10 10 4

200 800

4 10 , 16,000 :

16,000 10 (4 ) 40 * 400 20

4 4(20) 80

. . 80(200) 20(800) $32,000

. :

L K

L K

L K

L

L

MP MP K L L K

P P

Substituting K for L in Q LK where Q

K K K K

L K

Total Cost L P K P

c At equilibrium MP

P

    

 

    

   

    

2

* *

10 10 5

200 1000 4 10 , 16,000 :

16,000 10 (5 ) 50 * 320 17.89

5 5(17.89) 89.44 . . 89.44(200) 17.89(1000) $35,778

Pr

K

K

L K

MP K L L K

P Substituting K for L in Q LK where Q

K K K K

L K Total Cost L P K P

Labor oducti

    

 

    

        

@ 800

@ 1000

@ 800

@ 1,000

: cos :

16,000 200

80 16,000

178.89 89.44

, .

32,000 $2

16,000 35,

K

K

K

K

L P

L P

P

P

vity Average t

Q AP

L Q

AP L

Therfore an increase in the price of capital decreaes labor productivity

TC AC

Q TC

AC Q









  

  

  

  776

$2.24 16,000

An increase in the price of capital increases cost per unit. This is consistent with a decrease in labor productivity.



Applications in Managerial Economics_Class R

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Unanswered Applications 4. The production function of a process that uses two inputs, labor (L) and capital (K), is:

Q = 5LK

Where Q represents output per day (tons), labor costs $200 per unit and capital costs $1,000 per unit. The goal of the firm is to produce 10,000 tons daily.

a. Determine returns to scale of the production process, and its elasticity with respect to labor and capital. b. Determine the least-cost combination of L and K, and calculate total cost. c. Carefully analyze the effects on labor productivity and average cost of capital costs declined to $800 per unit.

Answer

Applications in Managerial Economics_Class R

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5. The design unit at Silicon Graphs, which is specialized in printing electronic circuits, estimates its production

function as: 2 2 Q LK

Where Q represents number of units produced per day, ( )L is labor which costs $200 per unit and ( )K is

capital which costs $400 per unit.

a. Determine returns to scale of the production process, and its elasticity with respect to labor and capital.

b. If the goal of the firm is to produce 16,000,000 units a day, determine the least-cost combination of L and K, and calculate total cost.

c. If cost were limited to $75,000 calculate the equilibrium combination of L and K, and the level of production.

d. Carefully analyze the effects on labor productivity in b and c above. Answer

Applications in Managerial Economics_Class R

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Cost Structure Answered Applications

1. A firm's total cost function is given by the equation:

2TC = 125 + 10 Q + 5Q

a. Determine the level of output that minimizes AC. b. Prove that MC=AC when AC is at its minimum. c. How much is TC, FC and VC when MC=AC?

Answer

2 TC = 125 + 10 Q + 5Q

TC 125 a. AC= 10 5

Q

125 2 *5 0 5 125 5 2

125*b. AC = 10 5(5) 60@Q 5 5

10 10

* 10 10(5) 60@Q 5

MC=AC at its minimum (desired).

2*c. TC =125 10(5) 5(5) 300@Q 5

FC

Q Q

dAC Q Q

dQ Q

dTC MC Q

dQ

dTC MC

dQ

  

       

  

  

   



  

=125

2 VC =10(5) 5(5) 175

*@Q 5  



Applications in Managerial Economics_Class R

10/17/2014 Page 19 of 32

2. A firm’s total cost function is given by the following equation:

2TC = 256+20Q +4Q

a. Determine the level of output that minimizes AC. b. Prove that AC =MC when AC reaches its minimum. c. How much is TC, FC and VC when MC=AC?

Answer

256 20 +4

. 4

* 4 0 8

2

*. 84@ 8

20 8

* 84@ 8

min ( ).

*. @ 8

2

256 20

256

2256 20(8) 4(8) 256 416 672

TC Q Q

TC a AC Q

Q

dAC Q

dQ

b AC Q

dTC MC Q

dQ

MC Q

MC AC at its imum desired

c TC Q

Q

Q

FC FC VCVC

 

  

   



  



 

   



 

 

Applications in Managerial Economics_Class R

10/17/2014 Page 20 of 32

3. Stylish Accessories is a professional designer and manufacturer of men’s wallets. Its total cost function is estimated to be:

2 3TC = 100Q -20Q + 2Q

a. Determine the level of output that minimizes AC. b. Prove that AC =MC when AC reaches its minimum. c. How much is TC, FC and VC when MC=AC? d. If the firm can sell all what it wants at a price of $100/wallet, calculate the level of sales that

maximizes the firm profit, and calculate that profit. e. Calculate the minimum price acceptable to firm to start production.

Answer

2

2

2

2 3 100Q-20 2

2 . 100 20 2

* 20 4 0 5

*. 100-20(5) 2(5) 50@ 5

100 40 6

100 40(5) 6(5) 50* @ 5

MC=AC at its minimum (desired).

*. @ 5 2 3100(5)-20(5) 2(5)

TC Q Q

TC a AC Q Q

Q

dAC Q Q

dQ

b AC Q

dTC MC Q Q

dQ

MC Q

c TC Q

VC

 

   

    

  

   

    



 

2 3

250

0

250

. 100

202 * : 100 100 40 6

3

100(7) 100(7) 20(7 ) 2(7

. .

2 100 - 20 2

7

) $294

0 20 4At a minimum:

FC

VC

d MR

At Equilibrium MR MC Q Q Q

TR TC

e Minimum acceptable price Min AVC

TVC AVC Q Q

Q

dAVC

dQ











      

     



  





    *

2

0 5

100 - 20(5) 2(5) $50

Q Q

Minmum AVC

  

  

Applications in Managerial Economics_Class R

10/17/2014 Page 21 of 32

4. A firm's total cost function is given by the equation:

000 0 2TC 4 5Q 1 Q  

a Write an expression for each of the following cost concepts:

 Total Fixed Cost

 Average Fixed Cost

 Total Variable Cost

 Average Variable Cost

 Average Total Cost

 Marginal Cost

b Determine the quantity that minimizes average total cost. c Demonstrate that the relationship between marginal cost and average cost holds.

Answer

2

2

2

4,000

2 5 10

2 5 10

5 10

4,000 5 10 4,000 5 10

5 20

4000 * 10 0

. Relationship: M

4,000

. 400 20

. 4000 5 10

TFC

AFC

Q Q

Q Q AVC Q

Q

Q Q AC Q

Q Q

dTC MC Q

dQ

dAC Q

dQ Q

b

TFC

Q Q

TVC

b

a TC Q Q







   

    

  

    







 

  

C=Min. AC

4,000 5 10(20) 405*

@ 20 20

5 20(20) 405* @ 20

MC=AC at its minimum (desired).

AC Q

MC Q

   

   





Applications in Managerial Economics_Class R

10/17/2014 Page 22 of 32

Unanswered Applications 5. Fashion Frames is a professional designer and manufacturer of photo frames. Its total cost function is

estimated to be: 2 3TC = 50Q -10Q + Q

a Determine the level of output that minimizes AC. (Round to whole digit) b Prove that AC =MC when AC reaches its minimum. c How much is TC, FC and VC when MC=AC? d If the firm can sell all what it wants at a price of $50/Frame, calculate the level of sales that maximizes

the firm profit. Calculate that profit. e Calculate the minimum price acceptable to firm to start production.

Answer

Applications in Managerial Economics_Class R

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Market Structure and Regulations 1. In a competitive market in long-run equilibrium, the market supply and demand are:

30 0.50

100 - 1.5

P Q

P Q

 



where P is dollars per unit and Q is level of production and sales in hundreds of units per day. A typical firm in this market has a marginal cost of production expressed as:

2.5 15MC q 

a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b. Determine the level of sales by a typical firm and its PS. c. If an output tax is imposed on ONE firm's output such that the ONE firm has a new marginal cost

(including the tax) of 5 15MC q   what will be the firm's new level of production and PS?

d. If all firms in this industry were bought by a monopolist, calculate monopolist equilibrium P, Q, MUF, CS, PS, and SW

e. As a policy maker, what would you do to restore competitive equilibrium outcome? Answer

* * a. At equilibrium: D=S 100 1.5Q=30+.5Q 2Q=70 Q 35 100 1.5(35) $47.5

52.5(35) 17.5(35) $918.75 $306.25

2 2

918.75 306.25 $1,225

*. Firm's Equilibrium: MR=MC (MR is the market price P )

P

CS PS

SW CS PS

b

        

   

    

*2.5+15q=47.5 3

35 12

3

45(3) Firm PS = $67.5

2

'. Firm's Equilibrium after the tax: MR=MC

*47.5=5+15q 2.83 less than quantity produced before tax.

(47.5 5)(2.83) after the tax

q

Number of firms

c

q

Firm PS

  

 



  

  $60.14 ' $7.3

2

d. Monopolist Equilibrium: MR=MC (MC is market Supply)

Monopolist faces market demand curve, therefore its MR curve is donward sloping with twice the slop

MR=1

Tax reduced the firm s PS by 



 

00-3Q

* *100 3Q=30+.5Q 70=3.5Q Q 20 100 1.5(20) $70m

70 1.75

40

(100 70)(20) (70 30) (70 40) $300 20 $700

2 2

300 700 $1,000

1,225 1,000 $225

. We can restore com

Pm

P MUF

MC

CS PSm m

SWm

DWL SW SWCompetition m

e

        

  

       

  

    

petition outcome by setting the price at its competitive level of $47.5

Applications in Managerial Economics_Class R

10/17/2014 Page 24 of 32

2. In a competitive market with small, equal-size firms in long-run equilibrium, the market supply and demand are:

P = 30 + 0.5Q

P = 75 - Q

Where P is dollars per unit and Q is level of production and sales in hundreds of units per day. A typical firm in this market has a marginal cost of production expressed as:

MC = 3.0 + 14q

Where q is the firm’s level of production. a. Calculate the free market equilibrium values of P, Q, CS, PS, and SW. b. Determine a typical firm’s equilibrium quantity. How many firms are in this market? c. If all firms in this industry were bought by a monopolist, calculate monopolist equilibrium P, Q,

MUF, CS, PS, and SW d. As a policy maker, what can you do to regulate the monopolist and restore competition outcome? e. If an output tax is imposed on ONE firm's output such that the ONE firm has a new marginal cost

(including the tax) of MC = 5 + 14q , what will the firm's new level of production and PS be after

the tax is imposed? Compare with pretax levels. Answer

* * a. At equilibrium: D=S 75 Q=30+.5Q 45 1.5Q Q 30 75 30 $45

30(30) $450

2

30(15) $225

2

450 225 $675

*. Firm's Equilibrium: MR=MC (MR is the market price P )

30*3+14q=45 3 # Firms = 1 3

P

CS

PS

SW CS PS

b

q

         

 

 

    

     0

. Monopolist Equilibrium: MR=MC (MC is market Supply)

Monopolist faces market demand curve, therefore its MR curve is donward sloping with twice the slop

MR=75-2Q

*75-2Q=30+.5Q 45=2.5Q Q 18m

Firms

c

Pm



    

 

* 75 18 $57

39 1.46

(75 57)(18) $162

2

(57 39) (57 30) 18 $405

2

162 405 $567

675 567 $108

. We can restore competition outcome by setting the price at its competitiv

P MUF

MC

CSm

PSm

SWm

DWL SW SWCompetition m

d

  

  

  

    

  

    

' *

*

e level of $45

. Firm's Equilibrium after the tax: MR=MC (MR is the market price P )

45=5+14q 2.86 less than quantity produced before tax.

(45 5)(2.86) after the tax $57.2

2

before th

e

q

Firm PS

Firm PS

  

  

(45 3)(3) e tax $63

2 Therfore, the tax reduced the firm's PS by $5.8

  

Applications in Managerial Economics_Class R

10/17/2014 Page 25 of 32

3. A competitive market with the following demand and supply curves:

2 - 60

75 -

S

d

Q P

Q P





where P is dollars per unit and Q is quantity produced and sold in the market in hundreds of units. A typical firm in this market has a marginal cost of production expressed as:

3.0 14MC q 

Where q is the quantity produced by a typical firm.

a. Determine the market equilibrium P, Q, CS, PS, and SW. b. Determine a typical firm equilibrium quantity and number of firms. Calculate a typical firm PS.

c. If an output tax is imposed on one firm changing its MC to 10 14tMC q  what will be the taxed

firm equilibrium Q and PS? Comment on your result. d. If all firms in this industry were bought by a monopolist, calculate monopolist equilibrium P, Q, CS,

PS, and DWL. e. As a policy maker, what would you do to restore competitive equilibrium outcome?

Answer *

a. At equilibrium: 2 - 60 75 - 3 135 $45

* 2(45) 60 30

30(30) $450

2

15(30) $225

2

450 225 $675

* *. Firm's Equilibrium: MR=MC (MR is the market price P ) 45=3+14q 3

Number of Firms i

D S P P P P

Q

CS

PS

SW CS PS

b q

      

  

 

 

    

  

 *Q 30

n the Industry= 10 firms * 3

(45 3)(3) Firm PS $63

2

' *. Firm's Equilibrium after the tax: MR=MC (MR is the market price P )

*45=10+14q 2.5 less than quantity produced before tax.

Firm PS after the

q

c

q

 

  

  

(45 10)(2.5) tax $43.75

2

Therfore, the tax reduced the firm's PS by $19.25

d. Monopolist Equilibrium: MR=MC (MC is market Supply). MR=75-2Q

* *75-2Q=30+.5Q 45=2.5Q Q 18 75 18 $57m

(75 57)(18) $16

2

Pm

CSm

  



       

  

 

2

(57 39) (57 30) 18 $405

2

162 405 $567

675 567 $108

. We can restore competition outcome by setting the price at its competitive level of $45.

PSm

SWm

DWL SW SWCompetition m

e

    

  

    

Applications in Managerial Economics_Class R

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4. To encourage investment in Satellite Radio in its early stage of development, a firm was given monopoly power over the whole market for a specific period of time. The demand and total cost function are estimated to be:

2

P = 900 - 3Q

TC = 200 + 100Q+Q

Where P is dollars per unit and Q is level of production and sales.

a. Determine the market equilibrium price, quantity, profit, MUF, and social welfare under monopolist. b. In a later stage, the market will be opened for competition. It is expected that a large number of identical

small firms will be active in the market, and will act like perfectly competitive firms. Determine price, quantity, and social welfare under perfect competition. (Hint: competitive supply is MC). Compare results with monopolist.

c. If the number of firms in the competitive market is decreased to eight identical firms, determine MC of a representative firm, calculate each firm production and PS.

Answer

a. Monopolist Equilibrium: MR=MC

Monopolist faces market demand curve, therefore its MR curve is donward sloping with twice the slop

MR=900-6Q

* *900-6Q=100+2Q 800=8Q Q 100 900 3(100) $600m

* *

Pm

TR TC



       

 

 

* 600(100) [200 100(100) 100(100)] $39,800

600 2

300

(900 600)(100) $15,000

2

(600 100) (600 300) 100 $40,000

2

15,000 40,000 $55,000

. :

900 3Q=100+2Q 80

P MUF

MC

CSm

PSm

SWm

b Competitive Market Equilibrium D S

    

  

  

    

  



   * *0=5Q Q 160 900 3(160) $420

(900 420)(160) $38,400

2

(420 100)(160) $25,600

2

38,400 25,600 $64,000

. If the number of firms deacreased to eight firms:

' :

MC=100+2Q 8 100

P

CSc

PSc

SW CS PSc

c

Firm s MC

Q q MC f

     

  

  

    

    16

*Q 160*q = 20 8 8

PS 25600cFirm PS = $3200 8 8

600 . 2

300

q

units

P d MUF

MC



 

 

  

Applications in Managerial Economics_Class R

10/17/2014 Page 27 of 32

Price Discrimination Answered Applications 1. American Tire and Rubber Company sells identical radial tires under the firm's own brand name and to

discount stores for private labeling. Marginal cost is a constant $10 per tire, regardless of the sub-market in which the tire is sold. The firm has estimated the following demand curves for each of the markets:

1 1

2 2

P = 70 - 0.0005 Q (brand name)

P = 20 - 0.0002 Q (private label).

Quantities are measured in thousands per month and price refers to the wholesale price. Marginal cost is a constant $10 per tire. a. Find the equilibrium price, quantity, consumer surplus and produce surplus in each market. b. Calculate the elasticity of demand at equilibrium in each market. c. If the Company cannot price discriminate between the two markets, calculate equilibrium quantity,

price, CS, and PS. Compare with your answer in (a).

Answer

1

1 1

* * 1 1 1

1

1

2 2

* 2 2

a. Equilibrium in first market: MR

70 .001

70 .001 10 60,000 70 .0005(60,0000) $40

(70 40)(60,000) $900,000

2

(40 10)(60,000) $1,800,000

20 .0004

20 .0004 10 25,000

MC

MR Q

Q Q P

CS

PS

MR Q

Q Q



 

        

  

  

 

      *2

2

2

1 1 1 1 1

2 2 2 2 2

20 .0002(25,000) $15

(20 15)(25,000) $62,500

2

(15 10)(25,000) $125,000

. Easticity of Demand

1 1 4 1 10 40 1 1.33

3

1 1 1.5 1 10 15 1 3

.5

. Witho

P

CS

PS

b

MR P E E E

MR P E E E

c

  

  

  

              

   

              

   

1

1 2

1

2 2

* *

ut price-discrimination: Q=Q

140,000 2000

100,000 5000

240,000 7000 34.29 .0001429 34.29 .0002858

equilibrium: MR=MC

34.29-.0002858Q=10 Q 84,975 34.29 .0001429(84,974) $22.

Q

Q P

Q P

Q P P Q MR Q

At

P



 

 

        

       14

(34.29 22.14)(84,975) $516,223.12

2

(22.15 10)(84,975) $1,031,596.5

Under price discrimination, the sum of CS in both markets is $962,500, which is more than CS without price

discrimination. Simil

CS

PS

  

  

arlry, PS under price discrimination ($1,925,000) is more than PS with a single price.

Applications in Managerial Economics_Class R

10/17/2014 Page 28 of 32

2. Thatcher Park of upstate New York has a low demand during work days (Market 1), but on Saturday and Sunday demand increases (Market 2). The demand functions are estimated to be:

1 1

2 2

Market 1: P = 4 - 0.001Q

Market 2: P = 22.6 - 0.01Q

Q is the number of cars entering the park each day. The marginal cost of running the park is the same on weekdays and weekends:

MC = 1 + 0.004Q

In order to control crowds, the park's management uses peak-load pricing which is a form of price discrimination. This pricing policy controls crowds and makes sure the park is self-supporting.

a. Find the equilibrium P, Q, and TR in each market. b. Calculate CS, PS, and elasticity of demand at equilibrium in each market. c. If the park cannot price-discriminates, calculate P, Q and TR. Compare with results in a.

Answer

1 1 1

* * *

1 1 1 1 1

2 2 2

2

a. Equilibrium in first market: MR 4 .002

3 4 .002 1 .004 500 Seat 4 .001(500) $3.5 3.5(500) $1, 750

.006

Equilibrium in second market: MR 22.6 .02

22.6 .02 1 .004

MC MR Q

Q Q Q P TR

MC MR Q

Q Q

   

             

   

    * * *

2 2 2 2

1 1

1 1

1

21.6 900 Seat 22.6 .01(900) $13.6 13.6(900) $12, 240

.024

. CS, PS and Easticity of Demand at Equilibrium:

(4 3.5)(500) (3.5 1) (3.5 3) $125 and (500) $750

2 2

1 1 3 3.5 1

Q P TR

b

CS PS

MR P E

         

       

         

1

1

2 2

2 2 2

2 2

1 2

1

1 3.5 7

0.5

(22.6 13.6)(900) (13.6 1) (13.6 4.6) $4, 050 and (900) $9, 720

2 2

1 1 13.6 1 4.6 13.6 1 1.51

9

. Without price-discrimination: Q=Q

4, 000 1000

E E

CS PS

MR P E E E

c Q

Q P

     

       

         



 

     

           

1

2 2

* *

2260 100

6260 1100 5.6 .0009 5.6 .0018

equilibrium: MR=MC 5.6 .0018 =1 .004 Q 809 Seats $4.87

793(4.89) $3,877.77

Yes. Thatcher Park benifits from price discrimination beacause

Q P

Q P P Q MR Q

At Q Q P

TR

 

        

      

 

TR is higher.

Applications in Managerial Economics_Class R

10/17/2014 Page 29 of 32

3. Cleveland Orchestra leaves its Severance Hall in Cleveland and operates from its summer home in Akron, Ohio during the summer. The summer home is a magnificent park with outdoor seating (Market 1) and a covered pavilion (Market 2) and The demand functions are estimated to be:

1 1

2 2

Market 1: P = 7 - 0.001Q

Market 2: P = 22.6 - 0.01Q

Q is the number of cars entering the park each day. The marginal cost of running the park is the same on weekdays and weekends:

MC = 1 + 0.004Q

In order to control crowds, the shrewdly-run Orchestra uses peak-load pricing which is a form of price discrimination. This pricing policy controls crowds and insures Orchestra is self-supporting.

a. Find the equilibrium P, Q, and TR in each market. b. Calculate CS, PS, and elasticity of demand at equilibrium in each market. c. If the Orchestra cannot price discriminate, calculate P, Q and TR. Does Cleveland Orchestra

benefit from price-discrimination? Answer

1 1 1

* * *

1 1 1 1 1

2 2

a. Equilibrium in first market: MR 7 .002

6 7 .002 1 .004 1, 000 Seat 7 .001(1, 000) $6 6(1000) $6, 000

.006

Equilibrium in second market: MR 22.6 .02

MC MR Q

Q Q Q P TR

MC MR Q

Cleveland Orchestra

   

             

    2

* * *

2 2 2 2 2

1 1

1

21.6 22.6 .02 1 .004 900 Seat 22.6 .01(900) $13.6 13.6(900) $12, 240

.024

. CS, PS and Easticity of Demand at Equilibrium:

(7 6)(1, 000) (6 1) (6 5) $500 and (1000) $3, 000

2 2

Q Q Q P TR

b

CS PS

MR P

             

       

 1 1 1 1

2 2

2 2 2

2 2

1 2

1 1 6 1 5 6 1 6

1

(22.6 13.6)(900) (13.6 1) (13.6 4.6) $4, 050 and (900) $9, 720

2 2

1 1 13.6 1 4.6 13.6 1 1.51

9

. Without price-discrimination: Q=Q

E E E

CS PS

MR P E E E

c Q

Q

        

       

         



           

           

11

2 2

* *

7, 000 1000

2260 100

9260 1100 8.42 .001 8.42 .002

equilibrium: MR=MC 8.42 .002 =1 .004 Q 1, 236 Seat $7.18

1, 236(7.18) $8,879

Yes. Cleveland Orchestra benifits from price discr

P

Q P

Q P P Q MR Q

At Q P

TR

 

 

        

      

 

imination beacause TR is higher.

Applications in Managerial Economics_Class R

10/17/2014 Page 30 of 32

4. The local zoo has hired you to assist them in setting admission prices. The zoo's managers recognize that there are two distinct demand curves for zoo admission. One demand curve applies to those ages 12 to 64, while the other is for children and senior citizens. The two demand and marginal revenue curves are:

PA = 9.6 - 0.08QA MRA = 9.6 - 0.16QA PCS = 4 - 0.05QCS MRCS = 4 - 0.10QCS

Where PA = adult price, PCS = children/senior citizens price, QA= daily quantity of adults, and QCS = daily quantity of children and senior citizens. Crowding is not a problem at the zoo, so that the managers consider marginal cost to be zero.

a. If the zoo decides to price discriminate, what should the price and quantity be in each market? Calculate

total revenue in each sub-market. b. What is the elasticity of demand at the quantities calculated in (a) for each market. c. If the zoo cannot price discriminate, what will be the equilibrium price and quantity? Can price

discrimination increase total revenue?

1 * *

1 1 1 1 1

1

* * 2 2 2 2 2

2

1 1 1

a. Equilibrium in first market: MR

9.6 .16 0 0.16 9.6 60 9.6 .08(60) $4.8 60(4.8) 288

4 0.1 0 0.1 4 40 4 .05(40) $2 40(2) 80

. Easticity of Demand

1 1

MC

MR Q Q Q P TR

MR Q Q Q P TR

b

MR P E



            

            

  

 1

1

2 2 2 2 2

* *

*

1 4.8 0 4.8 1 1

4.8

1 1 2 1 0 2 1 1

2

. 120 12.5 80 20 200 32.5 6.15 .03

6.15 .06

: 6.15 .06 0 102.5 3.08

102.5(

A SC A SC

E E

MR P E E E

c Q Q Q P P Q P P Q

MR Q

At Equilibrium MR MC Q Q P

TR

           

      

               

           

  

       

 3.08) 315.7 A CSTR TR  

Applications in Managerial Economics_Class R

10/17/2014 Page 31 of 32

Managerial Economics Formulas2 Elasticity of Demand

,

%

%

%

%

%

%

x x x

p

x x x

x x x

I

D

xX Y

x y

Y y x

Q dQ P E

P dP Q

Q dQ P E

I dI I

dQQ P E

P dP Q

  



  



  



Price, Marginal Revenue and Elasticity

1 1 1

1 1P

P

MR P P MR E

E

                 

Consumption Theory

X XY

Y

X X X Y

Y Y X Y

X Y

MU MRTS

MU

MU P MU MU

MU P P P

X P Y P I



  

 

Calculus Rules: Derivatives

 If , 0 dy

y a then dx

 

 If 1 , b b

dY y ax then abx

dx

 

 If     , dy du dz

y u x z x then dx dx dx

   

 If    . , dY dz du

y u x z x u z dx dx dx

   then

 If  

  2

. .

,

du dz z uu x dy dx dxy then

z x dx z



 

At Maximum 2

2 0

Y

X

 

 and at minimum

2

2 0

Y

X

 



Production Theory ( )

L L L

L K L L

L K K K

L K

TP d TP AP MP P MP W

L dL

MP MP MP P

P P MP P

TC L P K P

  

  

 

2 Ibrahim Elsaify, Applications in Managerial Economics, 2011. All Rights Reserved.

Applications in Managerial Economics_Class R

10/17/2014 Page 32 of 32

Economies of Scale

( , )

( , )

Q f L K

Q f L K

Q Q IRS

 







 

Economies of Scope

, ) , 0) , )

) ) , ) , 0

, )

( ( (0

( ( (

(

x y x y

x y x y

x y

Economies of Scope

Also S Economies of Scope

TC Q Q TC Q TC Q

TC Q TC Q TC Q Q

TC Q Q

  

    

Equilibrium Conditions:

Market Equilibrium: Q Qd s

Firm Equilibrium: MR = MC where

( ) ( )

d TR d TC MR and MC

dQ dQ  

where TR TC TR P Q   

Cost Functions

( ) ( )

AC AVC AFC

TC TVC TFC

TC TVC TFC

Q Q Q

d TC d TVC MC

dQ dQ

 

 

 

Market Concentration a. N-Firm Concentration Ratio (N-Firm CR)

1 2 3 4 4 1 2 3 4 :

i i

T T

S S S S S C W W W W Where W

S S

        

b. Herfindahal-Herchman Index (HHI)

1

2 2 2 2

10,000 0 10,000

1 1 1 1 1 : .....

1 -

n

i

i

HHI W where HHI

HHI N N N N N N

Number equal size firms in the market HHI





      

 

     



With N equal size firms

c. Lerner Index and Markup Factor

Markup Factor

1

1

1

1 P

E

P MC P L P MC Markup Factor

P L MC 



     



d. Rothchild Index Market Elasticity of Demand

Firm Elasticity of Demand

MK

F

RI 

 