Question #1

RussCo manufactures pesticides that can be sold directly to retail outlets or to a wholesale firm

for further processing and eventual sale as a completely different product. The demand

function for each of these markets is:

Retail Outlets: P1 = 55-2Q1

Wholesale Company: P2=40-Q2

What are the prices charged and what are the quantities sold in the respective markets.

Phillips’s total cost function for the manufacture of this product is TC=10+8(Q1+Q2)

A. Determine RussCo's total profit function? (4 points)

B. What are the profit-maximizing price and output levels for the product in the two markets?

(4 Points)

C. At these levels of output, calculate the marginal revenue in each market? (4 points)

D. What are RussCo’s total profits if the firm is effectively able to charge different prices in

the two markets? (4 points)

E. Calculate the profit-maximizing level of price and output if RussCo is required to charge the

same price per unit in each market. What are RussCo’s profits under this condition? (4 points)

Sol

a) Retail Outlets: P1 = 55-2Q1

Wholesale Company: P2=40-Q2

Profit = Total Revenue( TR) – Total Cost (TC)

TR= P*Q

Hence , TR = P1Q1+P2Q2 = (55-2Q1)*Q1 + (40-Q2)*Q2

= -2Q12 – Q2

2 + 55 Q1 +40Q2

PROFIT = TR- TC

= -2Q12 – Q2

2 + 55 Q1 +40Q2 – [ 10 + 8(Q1+Q2) ]

TOTAL REVENUE (TR) = -2Q12 – Q2

2 + 55 Q1 +40Q2

TOTAL COST (TC) = 10 + 8(Q1+Q2)

PROFIT FUNCTION= -2Q12

– Q22+ 47Q1+32Q2-10

b) Retail Outlets:

Profit maximum when:

d (-2Q12

– Q22+ 47Q1+32Q2-10)= 0

dQ1

-4Q1+47=0 => Q1=11.75 unit

Profit-maximizing price , put Q1=11.75 in P1 = 55-2Q1

P1=31.5

Wholesale market

Profit maximum when:

d (-2Q12

– Q22+ 47Q1+32Q2-10)= 0

dQ1

-2Q2+32=0 => Q2=16 unit

Profit-maximizing price , put Q2=16 in P2=40-Q2

P2=24

d (profit) = 0 (i.e. differentiation of profit wrt to quantity is 0, then we get maxima)

dQ1

Profit maximizing Price = $ 31.5

Output is 11.75 units

d (profit) = 0 (i.e. differentiation of profit wrt to quantity is 0, then we get maxima)

dQ2

c) Marginal Revenue(MR)

MR = d (TR)

dQ

TR= P1Q1+P2Q2

MR = d (P1Q1+P2Q2) = P1 (Retail Outlets)

dQ1

MR = d (TR) = P2 (Wholesale market)

dQ2

MR = d (P1Q1+P2Q2) = P2

dQ2

d)

PROFIT FUNCTION= -2Q12

– Q22+ 47Q1+32Q2-10

When we charged differently in both the markets

Total Profit = -2Q12

– Q22+ 47Q1+32Q2-10 ( put values of Q1,Q2)

= - 2 (11.75)2 – 16

2 + 47*11.75 + 32*16 – 10

= $522.125

Profit maximizing Price = $ 24

Output is 16 units

Hence , Marginal revenue at 11.75 units is P1=$31.5 = MR

Hence , Marginal revenue at 16 units is P2=$24 = MR

e)

Profit-maximizing level of price and output when Price are same i.e. P1=P2=P

TR= P1Q1+P2Q2= P(Q1+Q2)

TC= 10 + 8(Q1+Q2)

PROFIT = P(Q1+Q2) - 10 - 8(Q1+Q2)

For maximizing profit, differentiation of Profit wrt Q1 = 0

For maximizing profit = d [(P-8) (Q1+Q2) – 10] = P-8 =0 (Retail Outlets)

dQ1

P=$ 8

For maximizing profit, differentiation of Profit wrt Q2 = 0

For maximizing profit = d [(P-8) (Q1+Q2) – 10] = P-8 =0 (Wholesale Market)

dQ2

P=$ 8

Retail Outlets: 8 = 55-2Q1=> Q1=23.5

Wholesale Company: 8 =40-Q2=> Q2=32

PROFIT = (P-8)(Q1+Q2) – 10

= (8-8) (23.5+32)- 10 = -10

(-) minus sign indicates that there is loss

Total Profit of the firm = $522.125

PROFIT = (P-8)(Q1+Q2) – 10

RussCo’s Profit = -10; the firm is better off shutting down. It’s a LOSS

Question #2

Kashian Airlines has determined that the price elasticity of demand for two customer segments

(Coach and Business Class) is -1.35 and -2.50. Based on their expectations of profitability,

Kashian realizes the price of a Coach seat should be $175 (one way). How much should

Kashian charge for a Business Class ticket.? (20 Points)

Sol

Price Elasticity of Demand means percentage change in quantity demanded due to percentage

change in price of the product (holding constant all the other determinants of demand, such as

income).

Ed=% Change in quantity demanded/% Change in price=(Q2-Q1)/Q1/(P2-P1)/P1=

P1 - Price before change

P2 - Price after change

Q1 - Quantity before change

Q2 - Quantity after change

Ed- Price elasticity of demand

The above formula usually yields a negative value, due to the inverse nature of the relationship

between price and quantity demanded, as described by the "law of demand". For example, if

the price increases by 5% and quantity demanded decreases by 5%, then the elasticity at the

initial price and quantity = −5%/5% = −1.

In the above case as the price elasticity of Coach Segment is -1.35 that means that if the price

of coach fair increase by 1% then the quantity will be reduced by 1.35%. In similar case of

business segment the price elasticity is -2.5 , that shows that if the price of business fair

increase by 1% then the quantity will be reduced by 2.5%.

So we can say that Piece elasticity ep is directly proportional to original price [ other things

remain constant)

[ ep1/ ep2] = Pcoach/Pbusiness

1.35/2.5 = $175/ Pbusiness

Pbusiness= $ 324.07

Price of business class will be $ 324.07 charged by Kashian Airlines

Question #3

In 2000, the town of Brother’s Bay in Door County Wisconsin had a more-or-less free market

in boat services. Any adult citizen could provide boat services as long as the drivers and the

boats satisfy certain safety standards. Suppose that the marginal cost per trip of a boat ride is

constant, where MC = $5, and that each boat has a capacity of 20 trips per day.

If the demand function for boat rides was Qd = 1200 – 20P, where demand is measured in rides

per day. Assume that the industry is perfectly competitive.

A. What is the competitive equilibrium price per ride?(4 points)

B. What is the equilibrium number of rides per day? (2 Points) How many boats will

there be in equilibrium? (2 Points)

C. In this competitive market, what is the aggregate profit?(4 points)

D. In 2005, the town board of Brother’s Bay created a boat licensing board and issued a

license to each of the existing boats. The board stated that it would continue to adjust

the boat fares so that the demand for rides equals the supply of rides, but no new

licenses will be issued in the future. In effect, all profit would be turned over to the

township for licenses. How many licenses would be sold? (3 Points)

E. In 2010, costs had not changed, but the demand curve for boat rides had become Qd =

1220 – 20P. What was the equilibrium price of a ride in 2010? (3 Points)

F. In 2010, how much money would each current boat license owner be willing to pay to

prevent any new licenses from being issued? (2 Points)

Sol#a

Competitive equilibrium Price per ride

Qd = 1200 – 20P

P = (1200- Qd)/20

Total Revenue = P*Q

TR=60Q – Q2/20

At equilibrium:

MR=MC=P= d (TR) = $ 5

dQ

60- Q/10 = 5 => Q= 550 rides per day

Competitive equilibrium Price per ride

=> 1200 – 20P= 550

P= $32.5

Sol # b

Equilibrium rides per day = 550

Number of boats in equilibrium = 550 / 20 = 27.5 = 27 (approx).

Sol# c

Aggregate profit = TR – TC = Total Revenue – Total Cost

As MC *Q = TC

(60- Q/10)*Q = 60 Q – Q2

/10

Aggregate profit = Total Revenue – Total Cost

= (60Q – Q2/20) – (60 Q – Q

2 /10)

Sol#d

As the question is talking about the licenses in the equilibrium (boat fares so that the demand

for rides equals the supply of rides)

Profit generated = $ 15125

But this profit is given to township for licenses

Hence Number of licenses sold will become = Number of boat = 27

Sol # e

Demand curve for boat rides had become Qd = 1220 – 20P.

Cost has not changed = $ 5 = MC

P = (1220- Q)/20

TR=P* Q

TR=61Q – Q2/20

MC=61-Q/10

5=61-Q/10

Q= 560 rides per day

equilibrium Price ,

Profit = Q2/20

PROFIT ( Q=550) = 5502/20= $ 15125

P = (1220- Q)/20 = $ 33

Sol# f No of boats now = 560/20= 28

Profit per boat now = (Q2/20)*(1/28)= $560

Question 4

Suppose that the demand curve for apples is given by Qd = 140- 5P, where Qdis the

number of pounds demanded per year and p is the price per pound. The supply of apples

can be described by Qs = 40 + 3P, where Qs is the number of pounds provided.

A What is the equilibrium price? (Hint: At the equilibrium, quantity demanded and

quantity supplied are equal, Qd = Qs.) (2 Points)

B What is the equilibrium quantity supplied and demanded? (2 Points)

C Calculate the consumer surplus at the equilibrium price. (3 Points)

D Calculate the producer surplus at the equilibrium price. (3 Points)

E Calculate the total surplus at the equilibrium price. (4 Points)

F Now suppose that the government imposes a tax of $8 per each pound sold, paid by

the consumers,. Inthis case, what are the price and the consumer surplus? (6 Points)

Sol#4

Equilibrium price is when Qd = Qs ( i.e. quantity demanded and quantity supplied are equal)

140- 5P= 40 + 3P

P= $12.5

b) Equilibrium quantity supplied and demanded will be equal

Put P= $ 12.5 in

Qd = 140- 5P or Qs = 40 + 3P

Qd=Qs= 40+ 3*12.5 = 77.5 unit

c)& d) & e)

Consumer surplus is the difference between the total amount that consumers are willing

and able to pay for a good or service (indicated by the demand curve) and the total amount

that they actually pay (the market price).

Equilibrium Price = $ 12.5

Quantity supplied= Quantity demanded = 77.5 unit

License owner have to pay = $ 560

Producer surplus is the difference between what producers are willing and able to supply

a good for and the price they actually receive. The level of producer surplus is shown by

the area above the supply curve and below the market price.

Total surplus = Consumer surplus + Economic surplus.

Sol c)

CONSUMER

SURPLUS

Equilibrium point

D (77.5, 12.5)

F (0, 28)

DEMAND CURVE Q=140-5P

QUANTITY----------

B (0,-13.33)

Supply curve

Q=40+3P

<<< -------------------- P

RIC

E 0 (0, 0)

P=$ 12.5

E(0, 12.5)

A(140,0)

PRODUCER SURPLUS

C(140,0)

Sol c)

CONSUMER SURPLUS = ½* ED*EF = ½(77.5)(28-12.5) = $600.625

d)

PRODUCER SURPLUS = ½* ED*EB = ½(77.5)(12.5+13.33) = $161.43

e)

TOTAL SURPLUS = CONSUMER SURPLUS + PRODUCER SURPLUS

= $(600.625+ 161.43)

TOTAL SURPLUS = $ 762.062

f) When $8 tax is added equilibrium will change

Hence Price becomes = Equilibrium Price=$ 12.5 + $8 = $ 20.5=Pnew

We know that Qd = 140- 5P

Pnew= (140-Qd)/5

=>$20.5 = (140-Qd)/5

=> Qd = 37.5 pounds

Consumer surplus = ½ * (28-20.5)*37.5 = $ 159.375

Price = $ 20.5

Consumer surplus = $ 159.375

Question #5

A. Does anyone have a dominant strategy? (6 points)

B. What is the Nash Equilibrium? (6 points)

C. What is the socially optimal solution (at what point is total profit maximized)? (6

points)

D. How would a negotiated solution lead to this socially optimal solution (technically

this is called a Coase Solution—however, the path to this solution is straight-

forward)? (1 point)

E. If you had to call in a mediator to negotiate this socially optimal solution, how

much would they charge and why? (1 point)s

Solution

a) A dominant strategy will be one which will be successful or optimal for a firm

regardless of what others do, that is , no matter what strategy the rival firms adopt .

Lets us illustrate this dominant strategy in this case (duopoly) in the choice of whether to

―advertise’ or not . In this case deciding in favor of advertising by a firm to promote its sales

and hence profits or deciding not to advertise are the two strategies. Thus ADVERTISING and

NOT ADVERTISING are the two strategies between which each firm has to make a choice.

We assume that there are two firms A and B which have o make a choice between two

strategies.

The above diagram which is the outcome is presented in form of a payoff matrix .It should be

noted that outcomes or profits made by a firm by adopting a strategy is influenced by the

choice of a particular strategy by the rival firm.

It will be seen from the payoff matrix

If both firms adopts ― ADVERTISING‖

A PROFIT =$ 4

B PROFIT =$ 4

A advertise , B not

A=$20

B=$1

B advertise , A NOT

A=$1

B=$20

If both don’t adopt advertising

A=$10

B=$10

Since, both firms are symmetric in their corresponding strategies of advertising or not

advertising and hence both have a case of not having dominant strategy with respect to each

other.

b)

Nash Equilibrium

We have seen that sometimes both firms don’t have dominant strategies still they achieve

equilibrium in the adoption of strategies. The application of Nash Equilibrium is quite relevant

here.

In the above matrix FIRM A has no dominant strategy we reached the conclusion that the

equilibrium state is reached when FIRM A adopts strategy of advertising, given that the firm B

will choose the strategy of advertising, that is Firm A is making the best choice , given the

choice by its rival firm B and Firm B is choosing the best strategy given the strategy of Firm

A.

c.] Optimal Solution is when both did not advertise and each earn a profit of $ 10 in this case ,

which is the best optimal solution for both the firm.

d.] In law and economics, the Coase solution describes the economic efficiency of an

economic allocation or outcome in the presence of externalities. The theorem states that

if trade in an externality is possible and there are no transaction costs, bargaining will lead to

an efficient outcome regardless of the initial allocation of resources , hence in this case of firm

A and B , an negotiated solution will lead to socially optimal solution.

e.] Intermediate will be called in and suppose he may charge 1% of $ 10 i.e. $ 1 which is our

assumption to reach a optimal solution.