Chapter 8: Managing in Competitive, Monopolistic, and MonopolisticallyCompetitive Markets

In this chapter we characterize the optimal price, output and advertising decisions of managers under three market structures: (1) perfect competition; (2) monopoly; and (3) monopolistic competition.

PERFECT COMPETITION

The key five assumptions for perfect competition are:1. There are many small buyers and sellers in the market.2. Firms’ products are homogeneous (identical or perfect substitues).3. Buyers and sellers have perfect information of output, price and quality.4. There are no transaction costs (traveling costs from one store to another).5. In the long run there is free entry and exit in and from the market.

The first four assumptions imply that single sellers are too small to have a perceptible influence on the price. Each seller is a price taker and the price or inverse demand equation for the firm is a constant. The second assumption implies that the products are perfect substitutes because they are identical.Since in the 4th assumption there are no transaction costs (e.g.: cost of traveling to a store), then if one firm charges a higher price consumers would not shop at that firm. Assumption (5) implies if the industry experiences a positive profit, new firms will enter the market and the market price drops and the economic profit shrink until it becomes zero (profit pays the opportunity costs for the owner). Similarly, if there are sustaining losses in the market firms are free to leave and price would move up, losses shrink and the firms earn zero profit. This implies that in the long run the perfectively competitive firm earns zero or normal economic profit.

An example of perfect competition that fits the five assumptions above is agriculture (e.g.: corn, wheat, pork, beef, etc.). Another example is the catfish farm industry in the US. There are 2,000 small catfish farmers in the US. Another example is the T-shirt retailers in the US.

Demand at the Market and Firm Levels

The (output and demand) for the firm and the industry are represented by (Q, Df ) and

(Qm, D), respectively, as shown below:

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Market (industry) demand for corn shows how much corn all consumers will buy at each

possible price in the market. Market demand (for corn) is downward sloping because

consumers as a group buy more (corn) at each lower price.

The individual firm sells additional corn at the same price (i.e., it is a price taker and the

price is constant or the firm’s demand curve is a horizontal line).

Short run output decisions: (One decision)

To maximize profit in the short run, the manager must take fixed costs as given and use

the market price and variable cost to determine the optimal output level. (Q*). Perfect

competition is the easiest market structure for mangers to make decisions. They only

have to determine the optimal output level Q*, given the market-determined price.

Maximizing Profit in the Short-Run

This leads to determining the profit-maximizing output Q*. The plant size (K) is fixed and there is a fixed cost because this is the short run.

Let R be total revenue which is defined by P*Q where P is constant. Then profit is

Profit = R – TC (where TC = VC + FC) or

π = R – TC = total profit

The marginal profit per additional unit of output is:

∆π / ∆Q = (∆R/∆Q) – (∆TC/∆Q) = MR – MC

If MR > MC then firm should increase output (Q↑)

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(Typical FIRM) (Corn INDUSTRY)Q Qm

D

S

PP

Horizontal line

$4$4Df

If MR < MC then firm should decrease output (Q↓)

If MR = MC then there is no change in Q. This output is called equilibrium output (or

the profit-maximizing output) and will be referred to by Q*. This rule MR = MC is

called the first profit-maximizing rule (output choice Q*).

We can examine profit maximization under perfect competition using two approaches: the total approach and the marginal approach.

The total approachAs noted above, total profit is given byп = total revenues – total cost = P*Q – C(Q).

Fig 8-2 Revenue, Costs, and Profits for a Perfectly Competitive Firm

In Fig. 8-2, total revenue under perfect competition is a straight line originating from the

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origin because the price is constant (R= P-*Q). The cost function is generally a cubic

equation. In this figure, the profit or loss at any output level is the vertical difference

between sales revenues(R) and the cost function (C(Q)). The maximum vertical difference

or maximum profit is located where the slope of the cost function equals to the slope of

the total revenue or MR = MC (or slope of TR = Slope of C(Q)).

This profit maximization rule (output choice) determines the firm’s equilibrium level Q*

that maximizes profit. This is the 1st profit-maximization rule.

Under perfect competition, it can be rewritten as P = MC because total Revenue is linear.

That is, ∆R / ∆Q = ∆(P*Q)/ ∆Q = P*∆Q/ ∆Q = P.

The Marginal Approach

An alternative approach to the total approach is the marginal approach as depicted by

Fig. 8-3. This approach applies the same 1st profit maximization rule but also uses the

average and marginal costs instead of the total cost because in the short run part of the

cost is fixed and that does not influence optimal decisions. Under this approach we will

look at three cases of profit maximization.

Case 1: Firm earning a positive profit in S/R.

First draw the two average cost curves and the MC curve going through the minimums of

the averages. Then determine output Q* where MR = MC o

Fig. 8-3: Profit Maximization under Perfect Competition

S/R Profit Maximization Rule:

4

MC

Pe = AR

ATCAVC

d-Curve

AT*C

Pe

P

qQ*0

A

B

MC = MR but as mentioned before because MR = P then this rule can be rewritten as

MC = P

Pe A

π = rectangle =

ATC* B

This rectangle gives the maximum (total) profit. It is given by its base (Q*) times the

height [Pe –ATC*] which is the profit per unit, where ATC* = TC / Q* or C (Q*) / Q*.

That is, this profit area equals to

Q*[Pe - {(C (Q*) / Q*}] = Pe *Q* - C(Q*)= total revenue – total cost

Note again that [Pe – ATC] is the profit per unit of output.

TR = Pe A

0 Q* TC = ATC* B

0 Q*

Example 1: A watch-making firm. Suppose:

TC = 100 + Q2 → MC = ∆TC / ∆Q =0 + 2Q2-1 = 0 + 2Q = 2Q (MC is a straight line

starting from the origin). FC = $100 and VC = Q2 and AVC = Q2/Q = Q (AVC is also a

straight line but with a lower slope than MC). Note ATC = 100/Q + Q2/Q = 100/Q + Q

Pe = $60 (the firm is a price-taker working under perfect competition)

First profit-maximization rule: P = MC

$60 = 2Q* o

Q* = $60/2 = 30 units.

Profit = R - TC = P*Q* - TC= ($60)*(30) – {100 + (30)2} = $800.

Profit = $800. In this example, MC and AVC are linear (see graph below)

PS = Profit + FC = $800 + $100 = $900. (NOTE: PS = TR-VC= TR - (TC-FC)

Or PS = TR-TC + FC = Profit + FC, which is PS = TR - VC.

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Price Costs

Profit

Example 2 VC = 3Q + Q2 (FC is unknown and is not needed for determining Q* but

we cannot determine profit)

MC = ∆VC / ∆Q = 1*3Q1-1 +2Q2-1= 3 + 2Q Pe = $9 (constant for perfect competition)

Set Pe = MC

$9 = 3 + 2Q* or 6 = 2Q*

Q* = 6/2 = 3 units Then PS = TR –VC = $9*3 – (3*3+ (3)2) = $27 - $18 = $9

Demonstration 8-1 (maximizing profits)

Suppose the total cost function of a firm operating under perfect competition is

given by

C(Q) = 5 + Q2

And the market price is $20 per unit. What price should the manager charge?

What’s the level of output that maximizes profit (Q*)? How much is the profit?

(Hint: MC = 0 + 2Q2-1 = 2Q)

Answer: The firm’s price is the market price ($20) because the firm is a price-taker.

Set P = MC (1st profit maximization rule) and solve for Q:

$20 = 2Q* Q* = 10 units.

The maximum profit is

π = P*Q* - TC = (20) (10) – (5+102) = 200 – 5 – 100 = $95

Case 2: Firm Earning a Loss in S/R. should it shut down?

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MC =2Q

AVC = Q

qQ*

Pe = 60

Profit

MC

AVC

ATC

ATC = (100/Q) + Q

ATC*

At point A, set P = MC → Q*

π = rectangle

ATC* B Pe A

>

0 Q* 0 Q*

Loss = ATC* B

Pe A

In case 2, the firm produces at a loss in the short run. Should this firm shut down? Here,

Loss < FC. The firm covers part of the fixed cost (FC = Q**AFC).

Since, ATC* B ATC* B

>

AVC* C Pe A

If it produces, it will cover part of the fixed cost (how much is covered?), if it shuts down

it will incur all of the FC. Since Loss < FC then there is no shut down.

Demonstration 8-2 (minimizing losses with linear MC and AVC equations)

Suppose the cost function of a perfectly competitive firm is given by

C(Q) = 100 + Q2

7

qQ*

ATC*Pe

AVC*

Loss

0

B

CA

MinAVC

d-curve

where FC = $100, VC = Q2 and the market price is $10.

What level of output Q* should the firm produce to maximize profits or minimize

losses? What’s the level of profit or loss? Should the firm produce or shut down?

Answer: Equilibrium condition:

P = MC

10 = 2Q* Q* = 5 units.

Profit = P**Q* – TC = ($10) (5) – (100+52) = -$75 (loss)

The firm should not shutdown because

Loss < FC

$75 < $100

or ALTERNATIVELY Pe ≥ AVC = VC / Q* = Q2/Q or 10 ≥ (52) / 5 = 5

(which is equivalent to Loss < FT). The firm should not shutdown.

Case 3. Losses with Shut Down Rule:

If P < min AVC or loss > FC → Q* = 0 the firm should shut down.

The firm should shut down because the loss is greater than FC (that is, Loss > FC).

How to show that those two shutdown conditions are equivalent?

Let P < min AVC. Multiply both sides by Q:

Q*P < Q*AVC. Substitute in revenue and VC:

R < VC.

Substitute for VC as the difference between TC and FC:

R < TC – FC. Rearrange:

FC < TC - R

FC < Loss or

Loss > FC (shut down), which is equivalent to P < min AVC.

The Short-Run and Industry Supply Curves

The supply curve for a firm describes how much output a firm will produce at each price

level during a given period of time. This can be derived from the two profit max rules.

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How? See below.

For a perfectly competitive firm the price P* is determined in the market by intersection

of market supply and demand. The firm’s equilibrium output Q* in the short run is

determined by the two rules:

1. The profit-maximization rule;

P = MC

2. The shutdown rule if there is a loss;

whether P ≥ min AVC or loss < FC (no shutdown and Q* is positive)

If P0 = min AVC then Q* can be positive or zero.

If P < min AVC then Q* = 0 or shutdown. (Loss > FC).

If P1 > min AVC then; Q1* > 0 no shutdown. (Loss < FC).

If P2 > min AVC then; Q2* > 0. (positive profit).

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SP

Qq

AVC

MC = S/R Supply curve

ATC

Qq

0 Q0 Q1 Q2

P2

P1

P0

Min AVC

Competitive Firm’s S/R Supply Curve

Fig. 8-6: Short run supply curve for a perfectly competitive firm

The firm’s supply curve in the short run is the cross-hatched portions of the vertical axis

below P0 and the marginal cost curve above min AVC. This is the supply graph in the

previous graph for the perfectly competitive firm’s supply in the short-run. It is a cost

curve.

The market (or industry) supply is closely related to the supply curve of the

individual firms in a perfectly competitive industry. The market supply is the horizontal

sum of the marginal costs (above min AVC) of all firms and it determines how much total

output will be produced at each price.

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Fig 8-7: The Market Supply Curve

Fig 8-7 illustrates the relation between a typical individual firm’s supply curve (MCi)

and the market supply curve (S) for an industry that has, say 500 firms. Suppose when

the price is $12 each firm produces, say, 1 unit of output. At $12 the industry total

quantity is 500 units. What would be the industry output if the price is $12 and each firm

produces 2 units? Suppose when P =$15, each firm produces 2 units. The market supply

is much flatter than the individual firm’s supply, depending on the number of firms in the

industry.

Long-Run Decision [normal or zero economic profit)

In the long-run there is a free entry into the competitive market if there is a positive

profit. There is also an exit if losses exist. In the case of free entry, the market supply (S0)

shifts to the right to (S1) if there are more firms entering. It shifts to the left to (S2) if

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Fig 8-8 Entry and Exit: The Market and Firm’s Demand

firms exit. At the firm level, the horizontal demand curve will also shift. In the case of

positive profits, the firm’s demand curve (Df) shifts from P0 to P1. This decline in the

price will shrink economic profit to zero in the case of entry. In the case of exiting the

market as a result of negative profit, the increases in the price reduce losses to zero.

Thus, under either way economic profit under perfect competition in the long run is zero

(normal economic profit). That is,

(Pe –AC)*Q* = Profit per unit * Q* - 0 (which is equivalent to Total

Revenue = Total Cost).

For economic profit to be zero, P = AC (or R = TC) as well as Pe = MC.

For those two conditions to be satisfied the demand line or price line which is

horizontal must be tangent to the min AC curve. Thus the long-run competitive

equilibrium is characterized by the following conditions:

1. Pe = MC

2. Pe = min AC ( or zero econ profit)

See Fig 8-9.

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Fig. 8-9: Long-Run Competitive Equilibrium

Monopoly

A monopolist is a sole producer who sells a product that does not have a close substitute.

When one thinks of a monopoly, it is important to specify the relevant market. Is the

market local, regional or national? A utility company is a local monopoly in a city.

People in this city must buy their electricity from this company or move to another city.

Monopoly does not mean a large firm. A gas station in an isolated small town is a

“small” monopoly. Because the monopolist is the sole seller, it has a monopoly power

over the price. It can restrict output to increase the price over MC. Moreover, the demand

curve for the monopolist’s product is the market demand. That means Df = Dm (firm’s

demand = market demand) and thus the demand curve has a negative slope (see Fig 8-

11). If the monopolist sets the price too high, consumers do not have to buy the product.

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Fig 8-11: The Monopolist’s Demand

Sources of Monopoly Power:

There are sources of monopoly power that constitute a barrier to entry in the market.

These sources include economies of scale and scope, cost complementarities, patents and

other legal barriers.

Economies of Scale:

This means average cost decreases when output increases. For many companies there is

a certain range of output like [0 - Q*] in Fig 8-13 where economies of scale exist. Any

output above Q* generates diseconomies of scale. If one firm exists in this market and

produces say Qm to meet the demand, the ATC is ATCm. In this case the sole firm is

making a profit because P > ATCm. If another firm enters and both firms share the output

(Qm / 2 for each) then ATC for each one at Qm / 2 is [ATC(Qm / 2)] which is higher then

the price. Both firms will earn a loss. This will deter the second firm from entering the

market.

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Fig 8-12 Economies of Scale and Minimum Prices.

The number of firms that are able to fully exploit economies of scale depends on the size

of the total market demand and the technology of the product. A study found for a plant

to fully exploit economies of scale, it should produce at its minimum ATC. This study

examined this issue for twelv e industries in six countries. It was found that this number of

firms varies from industry to another and from one country to another. For example, there

are many plants that can fully exploit the economies of scale in the shoe industry, giving

rise to a more competitive market in this industry. In the refrigerator industry the number

is very small, implying an oligopolistic market structure.

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Economies of Scope:

If economies of scope exist, then it is easier and cheaper to produce two outputs Q1 and

Q2 jointly in one firm than to produce them in two separate firms. Efficient production

requires that the two outputs be produced in one firm. In this case the existence of

economies of scope encourages building larger companies instead of small ones. This in

turn gives greater access to capital markets, otherwise large capital can be a barrier.

Cost Complementarity:

When the marginal cost of producing one product decreases when production of another

product is increased, then this encourages the establishment of multi-product firms. Such

firms have large capital requirements which, discourages other firms from entering the

market. This cost complementarily can be a barrier to entry.

Patent and other Legal Barriers:

The above sources of monopoly power are technological in nature. This legal source has

to do with government regulations and policies which, for example, may grant a

monopoly power for only one public utility in a specific city. Other examples include

patents, trademarks and copyright protection. See INSIDE BUSINESS 8-3.

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Maximizing Profits under Monopoly in Contrast to Perfect Competition:

The manager under monopoly maximizes profit by choosing both equilibrium

price Pe and equilibrium quantity Qe, knowing that it has monopoly power (i.e., P > MC).

The monopolist maximizes profit by setting

MR = MC.

Then it determines Qe and Pe.

Marginal Revenue: Formula

The general relation between price and MR is given by

MR = [Px(1+E)]/E,

where E is the direct price elasticity of demand, %ΔQ / %ΔP = (ΔQ / ΔP)*(P / Q),

(and E must be elastic). This relationship shows that MR is less than the price. For

example if demand is elastic, MR is positive but less than P (e.g., E = -2 then MR =

[P{(1-2)/-2}] = ½ P). If the elasticity is unitary, (EC = -1) then MR = [Px{(1-1)/1 }] = 0

and TR is at its maximum, which is less than the price since the price is positive. When

demand is inelastic (say E = -1/2) and the price is positive then MR = [Px{(1-1/2)/-1/2}]

= -P. This is less than the positive price. We can summarize the relationship between the

price and MR in Fig. 8-13 as follows. When demand is relatively elastic, MR is positive,

and when it is inelastic MR is negative. Moreover, when demand is unitary elastic, MR is

zero. The implication of this relation is that the monopolist will not operate in the output

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range where demand is inelastic because this means the contribution to the total revenue

is negative or MR < 0. When demand is elastic an increase in output and a decrease in

price are associated with an increase in total revenue. On the other hand, when demand is

inelastic, an increase in output and a decrease in price are associated with a decline in

total revenue. Finally, when demand is unitary total revenue is at its maximum and MR =

0 (TR maximization). When demand is zero ( P = 0), total revenue R is zero. Since the

price changes when quantity changes, then total revenue (= P*Q) is not linear but is

concave.

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Fig 8-13 Elasticity of Demand and Total Revenues

Formula: Deriving MR from Linear Inverse Demand Equations:

Since demand has a negative slope under monopoly, that is, changes in quantity affect the

price, and then the price is a function of quantity.

P = P (Q).

This is called an inverse demand function where the price is a function of output and is

not a constant like under perfect competition. The most common form of this inverse

demand function is the linear inverse demand.

P = a – bQ,

where (a) is the constant and (-b) is the (inverse) slope = ΔP / ΔQ. It can be shown that

MR for the inverse linear demand can be written as MR = a – 2bQ. That is,

Slope of MR = 2 * slope of (inverse) demand.

Graphically, this result implies that MR curve divides the interval on the horizontal axis

between zero and where the demand curve hits this axis into half.

Demonstration 8–4: Determining type of price elasticity from MR.

Suppose the inverse demand function is given by P = 10 – 2Q.

What is the MR equation?

MR = 10 – 2 * 2Q = 10 – 4Q

What is the maximum price a monopolist can sell if output = 3 units?

P = 10 – 2*(3) = $4.

What is MR associated with 3 units of output?

MR = 10 – 4*(3) = -2.

The third unit reduced total revenue by $2.

Is demand elastic or inelastic at Q = 3? Since MR is negative, then demand is -------.

The Output Decision:

Both the price and total cost are functions of output under monopoly. Then profit can be

written as:

п = R(Q) – C(Q).

where R(Q) is total revenue and C(Q) is total cost.

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The monopoly profit-maximization rule is MR (Qm) = MC (Qm), which can be solved for

monopoly profit-maximizing output Qm. Output Qm can then be inserted in the inverse

price equation, giving rise to the monopoly price rule: Pm = P(Qm).

Demonstration 8-5: profit maximization under monopoly

Suppose TC = 50 + Q2 → MC = 2Q is a straight line.

Suppose P = 40 – Q (price is a function and not a constant) → MR = 40 – 2Q (twice

the slope of P).

R = PQ = (40 – Q)*Q = 40Q – Q2 (that is, multiply P by Q)

1. For equilibrium, set MR = MC

40 – 2Qm = 2Qm

QM = 40/4 = 10 units →

Plug Qm into Pm = 40 – 10 = $30.

Calculate profit?

Profit = Pm* Qm – TC = $30*10 – [50 + (10)2] = $300 - $150 = $150

The General Case for Profit Maximization: The Marginal Approach

Here we skip the total approach for profit maximization to concentrate on the marginal

approach.

As mentioned above, for monopoly one sets

MR = MC

and solves for Qm.

Then it substitutes Qm into the inverse demand equation to solve for Pm.

In the graph below, total profit is: Output* unit profit

Profit = Qm *(Pm – ATCm)

where (Pm – ATCm ) = unit profit, or

Profit = TR – TC = Pm*Qm -TC

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Fig 8-15: Profit Maximization under Monopoly

Absence of Supply Curve under Monopoly

Monopoly does not have a supply curve because this curve is usually derived from

equilibrium points formed by equating P and MC. Under monopoly, equilibrium is

determined from having MR = MC and P > MR.

Monopolistic Competition

Examples: fast-food, toothpaste (see handout), soap, shampoo, cold medicine, etc.

Characteristics:

Monopolistic competition has three key characteristics:

1) Each firm competes by selling differentiated products. The differentiated products

are highly substitutable but are not perfect substitutes like under perfect

competition (i.e. the cross price elasticity of demand between the products of the

firms is positive and high but not infinite). Crest is different from Colgate, Aim,

and Close-up… etc. Therefore, because of differentiation there is consumer

loyalty on part of some consumers. Consumers are willing to pay 25¢ to 50¢

more (but may be not a 1$). Therefore, Proctor & Gamble has some but limited

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monopoly power. However, some of the customers may move to the substitutes.

Therefore, advertising is important under monopolistic competition.

2) The demand curve is downward sloping but is fairly price elastic. The demand

elasticity for crest is –7. Thus, because of its limited monopoly power, P&G

charges a price that is higher than marginal cost but not much higher.

3) There is free entry and exit. It’s easier and cheaper to introduce, new brands of

toothpaste than to start new models of cars. The latter requires large capital and

technology to realize economies of scale. The free entry and exit implies that

economic profit under monopolistic competition is zero (normal).

Equilibrium in the short run and the long run: Like in monopoly, firms under

monopolistic competition have monopoly power and, thus, they face a downward

sloping demand curve. Therefore, MR < P. The profit maximization rule is

MR = MC.

In the short run the firm can earn a positive economic profit as shown in Fig. 8-18.

Fig. 8-18: Profit Maximization under Monopolistic Competition

If there is a positive profit, there will be an entry into this market and prices should

drop. This will shift both demand and MR curves of the individual firm down, and

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profit will shrink until it becomes zero ( TR= TC or P = ATC) as shown by the

tangency between the new inverse demand P and ATC curve (Fig. 8-19).

Fig. 8-19: Effect of Entry on Monopolistically Competitive Firm’s Demand

Like in perfect competition, because of free entry and exit firms under monopolistic

competition earn zero economic profit in the L/R. The point where MR=MC should

correspond to the point where the demand curve is tangent to the ATC curve to realize

zero profit.

The Long run

The positive profit will induce entry by other firms who introduce competing brands.

The incumbent firm will lose some market share and the demand curve will shift

down. ATC and MC may also shift when more firms enter the market. Assume no

shift in those cost curves. The DLR will shift down until it becomes tangent to the long

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run AC corresponding to where MR=MC. In this case the profit is zero. We have two

rules for the long run under monopolistic competition:

1. MR = MC (The 1st profit-max rule)

2. P = ATC > min ATC zero profit (because R = TC). See fig.8-20. This

condition is different from the long run condition for perfect competition P = min

ATC.

Fig. 8-20: Long-Run Equilibrium under Monopolistic Competition

Implication of Product Differentiation: Advertising

As mentioned above, monopolistically competitive firms differentiate their products in

order to have some control over the price. In this case, the products are not perfect

substitutes, and this makes the demand less than perfectly elastic. The implication of this

is that some consumer won’t switch when the prices go up within a limit, while others are

willing to switch. To keep the other consumers from switching to the substitutes, firms

under monopolistic competition spend a lot of money on advertising. There are two kinds

of advertising under monopolistic competition.

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ACLR = P*LR

Q*LR MRLR

DLR

ACMC

1) Comparative Advertising: This involves campaigns designed to differentiate a given

firm’s brand from brands sold by competing firms. Comparative advertising is common

in the fast–food industry, where firms such as McDonalds attempt to simulate demand for

their hamburgers by differentiating them from competing brands. This may induce

consumers to pay a premium for a particular brand. This additional value for a brand in

the price is called brand equity.

2) Niche Marketing: Firms under monopolistic competition frequently introduce new

products. The products could be totally “new” or “new improved”. Firms can also

advertise a product that fills special needs in the market. This advertising strategy targets

a special group of consumers. For example “green marketing” advertise “environmentally

friendly” products to target the segment of the society that is concerned with the

environment. The firm packages a product with materials that are recyclable.

These advertising strategies can bring positive profits in the short–run. In the long–run

other firms will mimic their strategy and reduce profits to zero.

Optimal Advertising Decisions

Optimal advertising is determined by the following formula

Formula: The profit maximizing advertising-to-sales ratio.

A/R = [(EQ, A) / - (EQ, P)] > 0,

where A is expenditure on advertising and R is sales revenue. Note: A/R is a positive

fraction because (EQ, P) is already negative and multiplied by a minus).

EQ, A = %ΔQ / %ΔA = (ΔQ / ΔA)*(A/Q)

is advertising elasticity of demand, and

EQ, P = %ΔQ / %ΔP = (ΔQ / ΔP)*(P / Q),

is the own–price direct elasticity of demand, which is negative.

If EQ, P = - ∞ (demand is perfectly price elastic under perfect competition), then

A/R = 0. That is, the optimal advertising-to-sales ratio is zero for the perfectly

competitive firm.

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The more elastic the demand with respect to own price (i.e., products are less

differentiated and more substitutable), the lower the optimal advertising-to-sales

ratio. This is a case of more competition than less, and there is not much need for

advertising.

The more elastic the demand with respect to advertising, the higher the optimal

advertising- to-sales ratio.

Demonstration 8-8

Suppose Corpus Industries operates under monopolistic competition and produces a

product at a constant MC. Suppose the demand for its product is estimated with a log

linear equation and the elasticities are:

EQ, P = - 1 (price elasticity of demand)

EQ, A = + 0.2 (advertising elasticity of demand)

To maximize revenue what portions of revenue should this firm spend on advertising?

Answer:

A/R = EQ, A / - EQ, P = [+0.2 / - (-1.0)] = (+0.2 / +1.0) = 0.2 = +20% of total sales.

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Chapter 9: Basic Oligopoly Models

This chapter discusses managers’ decisions under five different oligopolistic

market structures: Sweezy, Cournot, Stackelberg, Bertrand and Collusion. Comparison of

the outcomes in these different oligopolistic situations reveals the following. The highest

market output is produced under Bertrand oligopoly, followed by Stackelberg, then

Cournot, and finally collusion. Profits are highest for the Stackelberg leader and the

colluding firms, followed by Cournot, then the Stackelberg follower. Bertrand

oligopolists earn the lowest level of profits.

CONDITIONS FOR OLIGOPOLY

Examples of Oligopoly: Steel industry, airline industry and auto industry.

An Oligopoly is a market structure where there are few large firms in an industry. No

explicit number is required. However, the number is usually between two and ten firms.

If there are two firms, then the market structure is called duopoly. The product under

oligopoly can be homogeneous (steel) or differentiated (airlines travel). The manager has

a more difficult job in making decisions under oligopoly than under other market

structures. Under oligopoly there is firm rivalry and interdependence in decision making.

A manager, before it lowers the price of its product, it should consider the impact of the

lower price on the other firms in the industry.

THE ROLE OF BELIEFS AND STRATEGIC INTERACTIONS

The optimal decision whether to increase or decrease the price depends on how

the manager believes other managers in the industry will respond. If other managers

lower the price in reaction to this firm’s lowering the price, this firm will not increase its

sales much. In Figure 9.1, the reference point is B where the price is Po. The demand

curve D1 is the demand when other firms match any price change. If the manager of a

certain firm lowers his/her price, and the other firms in the market match this price

decrease, then the quantity will not increase much as given by D1. But if they don’t match

the price decrease then the manager can sell more as given by D2. Thus, the match D1 is

more inelastic than the no-match D2 , or D2 is more elastic than D1.

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If the manager increases the price and the other firms match, the firm’s sales will not

decline much. So the matching demand curve will be D1 .But if they do not match the

price increase, the firm will lose some market share and its demand will be the non-

matching D2. The only difficulty for the firm manager to make decisions is determining

whether or not rivals will match price changes.

Demonstration 9-2 (The kink Demand):

Thus if, for example, other firms match price reductions (D1) and do not match price

increases (D2) then the oligopoly effective demand is kinked as given by ABD1 as in Fig.

9-1. This assumption gives rise to what is known as the kinked demand curve ABD1.

Fig 9-1: A Firm’s Demand Depends on Actions of Rivals

Then the kinked demand is given by the two segments defined by A, B and D1.

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PROFIT MAXIMIZATION FOR OLIGOPOLY SETTINGS:

We will examine profit maximization under four alternative assumptions on how rivals

respond to price or output changes.

Sweezy Oligopoly:

An industry is characterized as Sweezy oligopoly if

1. There are few firms serving many customers.

2. The firms produce differentiated products.

3. *Each firm believes that rivals will respond to price reductions (effective D1) but

will not respond to price increases (effective D2) (ABD1 is kinked demand as in

Demonstration 9-2). This assumption represents the kinked demand curve.

4. Barriers to entry exist.

In Fig. 9-2, the kinked demand curve that fits assumption 3 is given by ABD1. If the price

is below P0 then the demand is the match demand D1, while if the price is above P0, then

the demand is the no-match D2. The corresponding MR to the kinked demand is ACEF.

The Kinked Demand Curve

P0

Price

Q0

B

Q

No Match

Match

D1

D2

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Fig 9-2: Sweezy Oligopoly

Profit maximization occurs when MR = MC. Let us for simplicity assume that MC is

linear (or straight line). If marginal cost is MC1 then profit maximization occurs at point

E and the price is P0. If MC is MC0 the profit maximization occurs at point C and the price

is P0. Note that if MC moves between points E and C (called the MR gap) there will be no

change in the equilibrium price P0. This model is good in explaining that firms avoid

price wars and thus prefer price stability by keeping the price at P0 even if MC changes

(however, within a limited range). This model is criticized for not explaining how the

firm arrived at point B in the first place. Nevertheless, the Sweezy model shows that

strategic interactions among firms in terms of prices and the managers’ beliefs on how

other firms would react to their price increases and decreases has a profound effect on

pricing decisions.

30

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The kinked demand is given by ABC and beyond as shown above. The corresponding gapped MR curve is depicted below. If the MC curve passes through the MR gap, modest shifts, upward or downward, in this curve will not change the industry price or the firms output. The Figure below (the cost cushion) shows the shifts in the MC1 curve to the MC2

and MC3 curves without a change in output or price (price stability). Recall, the 1st profit maximization rule requires thatMR = MC q* p*

Example: if the Match D1 is given by P1 = 15 – 2.5Q1 and the no match D2 is given by P2 = 10 –

0.5Q2, how do you determine the current or reference Q0 and P0 at point A of the Kink? Can you derive

MR1 and MR2? Can you calculate the MR gap ?

Answer: Set D1 = D2 and solve for the current or reference Q0 (=2.5) and P0 (=$8.75). Then substitute

Q0 in the respective marginal revenues (MR1 = 15 – 2*2.5Q1 (=$2.5?) and MR2 = 10 – 2*0.5Q2 (=$7.5)

to calculate the MR gap. Recall, the slope of the MR equation is twice the slope of the inverse demand

equation. To find MC in the gap and profit maximization point, substitute Q0 into the MC equation.

Cournot Oligopoly

An industry is a Cournot oligopoly if

1. There are few firms serving many customers.

2. The products are either differentiated (e.g. automobile) or homogenous (steel).

3. *Each firm believes that rivals will hold their outputs constant if it changes its

own output (naïve belief). Note that decision variables are outputs and not prices.

4. Barriers to entry exist.

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Thus, in contrast to Sweezy oligopoly which uses prices, the firm under Cournot

oligopoly believes that its output decisions have no effect on rivals output.

Reaction Functions in Cournot Oligopoly

To make matters easier suppose there are two firms. In this case, the market structure is a

duopoly. To determine the optimal output level, firm 1 will equate its MR1 to its MC1, and

firm 2 equate MR2to its MC2. The MR1 and MR2 equations are derived from the inverse

market demand equation.

P = a – b(Q1 + Q2)= a – bQ1 -b Q2 (note: output is homogenous there is one

P)

MR1 is derived by multiplying the slope b of Q1 by 2.

MR1 = a – 2*bQ1 - b Q2

Firm 1’s marginal revenue MR1 is affected by firm 2’s output (Q2), as well as by its own

Q1. The greater firm 2’s output, the lower is the marginal revenue of firm 1. In this case,

firm 1’s profit-maximizing output depends on firm 2’s output level Q2 and its Q1. Set MR1

= MC1 and solve for Q1 as a function of Q2. This relationship between firm 1’s profit-

maximization output Q1 and firm 2’s output Q2 is called a reaction function of firm 1.

The same applies to firm 2 setting MR2 = MC2 where

MR2= a – bQ1 – 2*b Q2

and deriving its reaction function which specifies Q2 as a function of Q1.

Therefore, a reaction function for firm 1 is its profit-maximizing output (Q1) as a function

of firm 2’s output (Q2). That is,

Q1 = r1(Q2),

where r1 is a “reaction function of”.

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Similarly, the reaction function of firm 2 is its profit- maximizing output as a function of

firm 1’s output. That is,

Q2 = r2(Q1).

Graphically, the reaction functions for a duopoly are given in Fig 9.3 where firm

1’s output is measured on the horizontal axis and firm 2’s output on the vertical axis.

Q1 = r1(Q2), and Q2 = r2(Q1).

Fig 9.3: Cournot Reaction Functions and Adjusting to Equilibrium

If firm 2 produces a zero output, then firm 1 is a monopoly and its profit- maximizing or

optimal output is Q1M. The greater firm 2’s output in Firm 1’ reaction function is, the

lower firm 1’s profit-maximizing output. For example, if the firm 2’s output is Q*2 then

the profit-maximizing output for firm 1 is Q*1.

Similarly, if firm 1’s output is zero, then firm 2 is a monopoly and its profit-

maximizing output is Q2M. Firm 2’s profit maximizing-output will go down if firm 1’s

output in firm 2’s reaction function increases. What is the firm 2’s profit maximizing

output when firm 1’s output is Q*1? It is Q*2.

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Equilibrium in Cournot Oligopoly

Graphically, we will describe how the duopoly reaches the equilibrium point (E)

based on movements along the two reaction functions. Suppose firm 1 produces Q1M.

Inserting this output into firm 2’s reaction function (by assumption 3), then this firm’s

profit-maximizing output corresponds to point A on the r2 reaction function.

On the other hand, given the positive output for firm 2 in the reaction function of

firm 1, then firm 1’s profit maximizing-output will correspond to point B. Given firm 1’s

output corresponding to point B in firm 2’s reaction function, then firm 2’s profit-

maximizing output will correspond to point C. Given this output in firm1’s reaction

function, firm 1’s output corresponds to point D. Then this will continue until it leads to

point E. where the two reaction functions intersect.

Therefore, equilibrium in Cournot oligopoly is determined by the intersection of the two

reaction functions which determine Q*1 and Q*2.

Formula: Marginal Revenues for Cournot Duopoly

Suppose for a Cournot duopoly with a homogenous product, inverse demand function is

P = a – b(Q1 + Q2)

(we sum up the two outputs because the product is assumed to be homogeneous).

Since the slope of MR is twice that of price then

MR1 = a – bQ2 – 2bQ1 (only slope of Q1 is doubled)

and

MR2 = a – bQ1 – 2bQ2 (only slope of Q1 is doubled)

Marginal products depend on own and the other firm’s outputs.

Formula: Reaction Functions for Cournot Duopoly

Suppose the inverse demand function is linear

P = a – b(Q1 + Q2),

and the cost functions with no fixed costs are

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C1(Q1) = c1*Q1 (the cost function is linear starting from the origin and c1 is MC1)

C2(Q2) = c2*Q2 (where c2 is MC2)

To derive the reaction function for firm 1, set MR1 = MC1 and solve for Q1 as a function

of Q2.

a – bQ2 – 2bQ1 = c1 (divide both sides by 2b and solve for Q1), we have

a/2b – 1/2Q2 – c1/2b = Q1.(combine the two constant terms a/2b and – c1/2b)

Q1 = r1(Q2) = (a - c1) / 2b – 1/2Q2 [please remember this formula]

Similarly for the reaction function of firm 2, set and solve for Q2 as a function of Q1.

MR2 = MC2.

a – bQ1 – 2bQ2 = c2 (divide both sides by 2b and solve for Q2)

Q2 = r2 (Q1) = (a - c2) / 2b – 1/2Q1 [please remember this formula]

To find the Cournot equilibrium (Q1*, Q2*) for this duopoly, substitute Q2 into the

reaction function Q1 = r2(Q2) and solve for Q*1. Then substitute Q*1 into Q2 = r2(Q1) and

solve for Q*2. The Cournot equilibrium is (Q1*, Q2*).

SEE THE SOLVER TEMPLATE FOR THE SOLUTION OF LINEAR Cournot case on

the website.

Demonstration 9-4. (Remember in this example c2 = 0 and c1 =0)

Suppose:

The inverse market demand function is:

P = 10 – Q1 – Q2 where a =10 and b =1.

The firms’ cost functions are:

C1(Q1) = 0 where C1(Q1) is total cost and MC1 is assumed to be c1 =0

C2(Q2) = 0 where c2 = 0. Same as above

The long way for both firms:

Then derive the two marginal revenues

MR1 = 10 – Q2 - 2Q1 (twice the slope of inverse demand for Q1)

MR2 = 10 – Q1 - 2Q2 ((twice the slope of inverse demand for Q2)

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Long way for Firm 1

Next for firm 1, set

MR1 = MC1

10 – Q2 - 2Q1 = c1

10 – Q2 - 2Q1 = 0

(MC1 or c1 is assumed to be zero in this example. Please keep in mind that c1 = 0 is a

special case) and then solve for Q1 = r1(Q2) which implies that Q1 = (10-0)/)– 0.5Q2

(Remember: Firm 1’s reaction function which is Q1 = [(a - c1) / 2b – 1/2Q2 ].

The Formula way for Firm 1: use the above formula and P = 10 – Q1 – Q2 where

a =10 and b =1. Here c1 is assumed to be zero.

Q1 = (a - c1)/2b – 0.5Q2 = (10 - 0)/2 – 0.5Q2 = (10/2) – 1/2Q2 , where a=10, b = 1 and c1 =

0.

Long way for Firm 2.

Similarly, for firm 2 set MR2 = MC2 (the long way)

10 – Q1 - 2Q2 = c2

10 – Q1 - 2Q2 = 0 where c2 = 0

and divide both side by 2 and then solve for Q2 = r2(Q1):

Q2 =(10-0)/2 – 1/2Q1 (Firm 2’s reaction function).

Formula way for Firm 2: Use the formula

Q2 = (a - c2)/2b – 0.5Q1 = (10-0)/2 -1/2Q1 where c2 = 0.

To find the Cournot equilibrium point, substitute Q2 into Q1

Q1 = 10/2 – ½*Q2 = 10/2 -1/2(10/2 -1/2Q1) = 10/2 -10/4 +1/4Q1

Q1 = (20 – 10)/4 + 1/4Q1 . Then move the last term to the left,

3/4 Q1 = 10/4

solve for Q1* = (10/4)*(4/3) = 10/3= 3.33 units.

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To solve for Q2*, substitute Q1* into the Q2*reaction function

Q2 = (a - c2)/2b – 0.5Q1

Q2 =10/2 – ½(3.33)

and solve for Q2*.

Q2* = 10/3= 3.33 units.

The result is Cournot equilibrium (Q1*, Q2*) = (3.3, 3.33)

Calculate the market price where the output is homogenous:

P* = 10 – Q1* – Q2* = 10 -10/3 -10/3 = 10 -20/3 = (30 -20)/3 = $10/3

Calculate market quantity Q* = Q1* + Q2* = 20/3 units.

See the detailed continuation of the example solved above in the pages below

(you may skip the second part because it is redundant):

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5. To calculate profits for firm 1 define

π1 = P**Q1* - c1Q1* = ($10/3)* (10/3) – 0* (10/3) = $100/9

Profit of firm 2 can be defined similarly.

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π 2 = P**Q2* - c2Q2* = $100/9

Profit in Cournot oligopoly: Isoprofit curve.

Each firm has its own isoprofit curves given by the equation P*Qi – TC = π i where πi is

constant. Each level πi of gives an isoprofit curve. Each curve includes combinations of

outputs of both firms. For firm 1 the closer the curve to Q1M the greater the profit. In Fig

9-4, the points F, A and G for example have the same profit because profit is constant

along the single isoprofit line πi0 and so on. The formula for the isoprofit line for firm 1 is:

(a –bQ1 –bQ2)*Q1 = π1 which is a constant. Solve for Q1 as a function of Q2. Repeat it.

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Fig. 9-4: Isoprofit Curves for Firm 1

The Isoprofit curve π2 is associated with greater profit than π1 and so on. (Why?)

The chosen point should be on the intersection of the isoprofit curves with the

respective reaction function line because the reaction functions come from profit

maximization. This is also where the isoprofit curves reach their peaks, given the outputs

of the other firm. For example, for given output of firm 2, say Q*2, if we move

horizontally away from the peak point C in Fig. 9.5 we will be on lower and lower

isoprofit curves along the way compared to ΠC1 . For a given output of firm 2, say Q*2,

compare isoprofit curves associated with points A, B and D with that associated with

point C which lies on the reaction function of firm 1 in Fig 9-5.

Fig 9-5 Best Response to Firm 2’s Output

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Similarly, Fig. 9-6 illustrates the isoprofit curves increase in value as they approach QM2.

Fig. 9-6: Firm’s 2 Reaction Function and Isoprofit Curves

Now we can bring Figures 9-5 and 9-6 together in one graph to determine Cournot

equilibrium and profits for the two firms. The two isoprofit lines ΠC1 and ΠC

2 through

point C, where the two reaction functions intersect, represent the maximum profits for

firm1 and firm 2 as in Fig 9-7. Thus, the equilibrium in a Cournot duopoly is given by

point C which defines (Q*1 and Q*2). At this point, the two isoprofit lines also intersect.

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Fig 9-7 Cournot Equilibrium and Profits

Changes in Marginal Costs under Cournot

In a Cournot oligopoly, the effect of a change in marginal cost is very different than in a

Sweezy model. Suppose the firms are initially in Cournot equilibrium (Q*1 and Q*2) at

point E in Fig. 9-8 below. Now suppose that Firm 2’ MC declines. Then for this firm 2

MR2 = MC2.

The reaction function of firm 2 is

Q2 = (a - c2)/2b – 1/2Q1

This means that this function will shift up and intersect firm 1’s reaction function at a

higher output for firm 2 and lower output for firm 1. What will happen to profits of both

firms? (hint: compare new profit level to that of monopoly or add the isoprofit curves).

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Fig. 9-8 Effect of a decline in Firm 2’s Cost under Cournot

Stackelberg Oligopoly

The industry under this oligopoly has the following assumptions:

1. There are few large firms serving many customers.

2. The products can be differentiated or homogenous.

3. *In an oligopoly there is a leader (firm 1) and a follower (firm 2).

4. There are barriers to entry.

In this oligopoly, the leader acts first and determines its output, knowing the reaction

of the follower to its output decision. It has the first mover advantage. In this case, the

leader maximizes profit, given the follower’s reaction function which depends on the

leaders output. The follower maximizes profit given the leaders output Q1 as is the case in

Cournot oligopoly. Thus, the follower’s reaction function is given by Q2 = r2 (Q1).

For example, suppose the inverse market demand equation is given by the linear

function

P = a - b(Q1 + Q2) where output is homogenous.

and firm 2’s cost function is C1= c1Q2 and C2 = c2Q2 where c2 is MC2.

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Firm 2, the follower, sets MR2 = MC2. That is,

a - bQ1 – 2bQ2 = c2 (move 2bQ2 to the left-hand side)

2bQ2 = a - bQ1 – c2 (next divide both sides by 2b)

Q2 =( a-– c2)/2b - 1/2Q1 (Remember: follower’s reaction function).

This follower firm solves for Q2 as a function of Q1, which is its Cournot reaction

function:

Q2 = r2(Q1) = (a - c2)/2b –[(1/2)Q1.

Next Firm 1, the leader, knows this reaction function and plugs it into its profit equation

in place of Q2 after substituting the inverse demand equation for P below:

π1 = P.Q1 – c1Q1= [a - b(Q1 + Q2)]*Q1 – c1Q1 (substitute follower’s reaction

function for Q2 below)

π1 = {a – b[Q1 + (a - c2)/2b –(1/2)Q1]}*Q1 – c1Q1 (multiply things out)

π1 = aQ1 – bQ12 - b(a - c2)/2b)Q1 + b(1/2)Q1

2 – c1Q1

π1 = aQ1 – bQ12 – (a - c2)/2)Q1 + b(1/2)Q1

2 – c1Q1

It then maximizes profit with respect to its own output Q1 by taking the derivative of π1

with respect to Q1.

dπ1 / dQ1 = a – 2bQ1 – (a -c2)/2 + bQ1 – c1 = 0.

Combine the constant terms together by combining a and c2:

dπ1 / dQ1 = (a + c2)/2 – c1 – 2bQ1 + bQ1 = (a + c2)/2 – c1 – bQ1 = 0.

Solve for Q1 by dividing by b. That is, Q*1 = (a + c2)/2b - c1/b

At the end for this linear case, Firm 1 has the following value for its output Q1:

Q*1 = (a + c2 - 2c1)/2b [remember this formula for the leader]

The final step is to plug Q1* into the reaction function for Q2 above (see page 335 of the

text and the Solver template on my Website or in your CD).

Example on Stackelberg Oligopoly (Demonstration 9-6)

Suppose that the inverse demand function for a homogenous Stackelberg Oligopoly is

given by:

P = 50 –Q1 - Q2 where a = 50 and b = 1.

And the cost functions are given by

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C1(Q1) = 2Q1 (where MC1 = c1 = 2)

C2(Q2) = 2Q2 (where MC2 = c2 = 2)

1. Determine firm 2’s reaction function. This is a Cournot.

Set MR2 = MC2 which is 50 – Q1 - 2Q2 = 2 and solve for Q2 as a function of Q1.

OR use the formula for firm 2’s reaction function directly:

Q2*

= (a - c2)/2b – 1/2Q1*: (follower’s reaction function)

Then Q2 = [(50 - 2)/2] – 1/2Q1 = 24 – ½ Q1

2. What is Firm 1’s output Q1* that maximizes profit

Q1* = (a + c2 - 2c1)/2b (Leader’s reaction function)

Q1* = (50 + 2 - 4)/2 = 24 units

3. Derive the follower’s output Q2*

Q2* = (a - c2)/2b – (1/2)Q1

* = (50 – 2)/2 – (1/2)Q1* = 24 -1/2(24) = 12 units

4. Calculate the market price

P* = 50 – Q1* – Q2* = 50 -24 -12 = $14.

5. DETERMINE FIRM 1’S PROFIT.

π1 = TR1 - TC1= P*Q1* – c1Q1

* = $14*24 – $2*24 = $336 - $48 = $288 (leader’s profit)

Firm 2’s profit can be determined the same way.

π2 = P*Q2* – c2Q2* = 14*12 - $2*12 = $168 – $24 = $144 (follower’s profit)

Bertrand Oligopoly

1. There are few large firms selling to too many customers.

2. The products can be identical or differentiated.

3. *The firm sets the price (not the output) that maximizes profit, given the price of

the rival firm. (This is different from the kinked demand curve of Sweezy).

4. *Consumers have perfect information and there are no transaction costs.

5. There are barriers to entry.

Suppose first firm 1 charges the monopoly price (initially one firm). The consumers have

perfect information, there are no transaction costs and the products are identical . Firm 2

enters. If firm 2 slightly undercuts the monopoly price and since consumers know all

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prices, they would switch to firm 2’s output (because of identical product, perfect

information and no transaction costs). In this case firm 2 would capture the whole market.

Therefore, firm 1, finding itself with no customers, would retaliate by undercutting firm

2’s lower price, thus recapturing the entire market. Then there is a price war under

homogeneous product Bertrand with perfect information and no transaction cost. When

would this “price war” end?

When each firm charges a price equal to MC, P1 = P2 = MC. No firm would choose to

lower this price below MC because it would make a loss. This is know as the “Bertrand

Trap”. This is like perfect competition but the solution variable is the price (not output)

and the profit is zero.

In short, this type of Bertrand oligopoly, would lead to a situation where firms

charge a price equal to MC and earn zero economic profit. Then the equilibrium is found

by setting

P1 = P2 = MC

and then solving for Q1* and Q2* and P*.

In any oligopoly with differentiated products including Bertrand, each firm has

monopoly power over its brand loyal customers and it can charge a price higher than MC

and earn positive economic profit. Fig 9-14 illustrates Bertrand equilibrium with

differentiated products.

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Fig 9-14: Reaction Functions and Equilibrium in a differentiated Bertand

Then, in contrast to Cournot oligopoly, the reaction functions of Bertrand oligopoly with

differentiated products are for prices and have positive slopes. When P2 is theoretically

zero, the minimum price for firm 1 is P1min. This is a (high) price that firm 1 charges to its

brand loyal customers who won’t switch to firm 2’s product despite firm 1’s higher price.

As P2 increases, so does P1. Bertrand equilibrium in this case is given by point A.

Reminder: Bertrand oligopoly is different from Sweezy which has match and no

match demand curves.

Collusion

Finally, we will determine the collusive outcome, which results when the firms choose

output to maximize total industry profits. This model is similar to the monopoly model as

explained in Fig. 8-15 in chapter 8. When firms collude, total industry output is the

monopoly level, based on the industry or market inverse demand curve. Since the inverse

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market demand curve, which is result of summing up horizontally the outputs of all firms

in the industry at each price, is

P = 1,000 – Q = 1,000 – (Q1 + Q2), where Q = Q1 + Q2 (sum means homogenous

output).

The associated industry or market MR is

MR = 1,000 – 2Q = 1,000 – 2(Q1 + Q2 ) (double the slopes for both Q1 and Q2)

Notice that this MR function assumes the firms act as a single profit-maximizing firm,

which is what collusion is all about.

Assume total cost for the ith firm is:

TCi = ciQi = 4Qi, where ci = $4 = MCi and i = 1,2 (assume identical MCs).

Setting industry MR equal to MCi (which is equal to $4 as shown above) yields

Market MR = MCi

1,000 - 2Q* = 4, where Q = Q1 + Q2

1,000 - 4 = 2Q*

Then total industry output Q* is:

996 = 2Q*

Q* = 996/2= 498 units.

Thus, total industry output under collusion is 498 units, with each firm producing half of

the market share:

Q1* = 0.5Q* = 249 units

Q2* = 0.5Q*= 249 units.

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The industry’s price is:

P* = 1,000 – Q* = 1,000 – 498 = $502.

Since each firm had 50% of total output.

Each firm earns profits = TRi – TCi = P**Q1

* – c1Q1 = P** 0.5Q* – $4*(0.5Q*)

= $502*249 - $4*249

= $124,002.

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Chapter 10

Game Theory: Inside Oligopoly

In this chapter, we will continue the discussion on managerial decisions in presence of

strategic interaction and interdependence. We will develop tools using game theory that

will assist future managers in making decisions in oligopolistic markets.

Summary

There should be a distinction between one-move games and repeated games. There also

should be a difference between one-move, competing games and one-move, coordination

games.

1. If each of the two players in a simultaneous-move, one-shot game has a dominant

strategy, those strategies constitute a Nash equilibrium.

2. If player 1 has a dominant strategy, while player 2 does not, then the optimal

strategy for player 1 is his dominant strategy. The best strategy for player 2

should be the strategy with highest payoff given player 1’s optimal strategy (both

in the same cell).

3. If the simultaneous-move game is a one-shot game and there is no tomorrow, the

collusion will not be sustained as a Nash equilibrium. Each player will cheat. In

this case. Nash equilibrium will not have the highest payoffs.

4. If player 1 has a dominant strategy and player 2 does not, player 2’s secure

strategy should correspond to player 1’s dominant strategy (in the same cell).

5. Suppose the simultaneous game is a one-shot game. Suppose each of the two

diagonal cells of the two players is identical but, those numbers in each cell are

not. Suppose the two off-diagonal cells have identical cells but their numbers are

lower than the numbers in the diagonal cells. Then the game has two Nash

equilibriums, which are the diagonal cells. If the game is infinitely repeated then

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it is possible for collusion to be the Nash equilibrium (check the condition for

sustainable collusion).

Overview of Games and Strategic Thinking

In a game, managers are players and the plans of managers are strategies. The

payoffs are the profit or losses that result from the strategies. Due to strategic

interdependence among firms, one player’s payoff depends on this player’s strategy and

those of the other players.

In a simultaneous-move game, each player makes decisions without the

knowledge of other player’s decisions (an example of this game is the Bertrand duopoly

game). In a sequential move game one player makes a move after observing the other

players’ move (e.g.: chess, Tic-tac-toe, checkers and Stackelberg oligopoly). If the

underlying game is played once, it’s a one-shot game. If the underlying game is played

more than once, it’s a repeated game. First, we will study the foundation of game. We

will begin with the study of simultaneous-move, one-shot games.

Simultaneous Move, One Shot Games

Such games are important to managers operating in an environment of

interdependence. Let us examine the general theory which is used in analyzing managers’

decision in these games. First, strategies are decision rules that describe players’ actions.

Second, normal-form representation of a game includes the players, the players’ possible

strategies and the possible payoffs. To understand these concepts let us look at Table 10-

1. There are two players: A and B who are engaged in a situation of strategic interaction.

You could think of the two players as managers of two firms competing in a duopoly.

Player A has two possible strategies: “Up” and “Down”; while B has also two possible

strategies: “Left” and “Right”.

Table 10-1: A Normal Form Game: Dominant Strategies

Player A

Player B

Strategy Left Right

Up 10,20 15,8

Down -10,7 10,10

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Each cell in the matrix above represents payoffs for the two players. For example,

the cell “Up” for player A and “Left” for player B contains player A’s payoff equal to 10

and player B’s payoff equal to 20. The game is a simultaneous move, one shot game, the

players make only one decision and they make it at the same time without any conditions.

One shot implies that there is no future between the two players

What is the optimal strategy for a player in a simultaneous move, one shot game?

We characterize “optimal” by a situation that involves a dominant strategy. A strategy is

dominant if it results in the highest payoff for a player regardless of what the opponent

chooses. In Table 10-1, assume player B chooses “Left”, then find the highest payoff for

player A over both his/her strategy (UP=10). Similarly, fix player B’s strategy at “Right”

and let player A choose the highest payoffs over both his/her strategies (UP=15). Then

the dominant (optimal) strategy for payoffs is “UP”. If a player has a dominant strategy

he/she will play it.

Principle: If a player has a dominant strategy, he/she will play it.

In some games a player may not have a dominant strategy (see below).

Demonstration 10-1.

In Table 10-1 above, does player B have a dominant strategy? (Hint: move row-wise

to look for B’s dominant strategy).

The answer is No. Note that if player A chooses “UP”, the best choice for player B

would be “LEFT” since the payoff 20 is better than the payoff 8 she would earn by

choosing “RIGHT”. But if Player A chooses “DOWN” the best choice by player B

would be “RIGHT”, since 10 is better than 7 she would realize by choosing “LEFT”.

The best choice for B depends on what player A does. Thus, player B does not have

a dominant strategy.

What should a player do in the absence of dominant strategy? One possibility is to play a

secure strategy: a strategy that guarantees the highest payoff given the worst possible

scenario (max-min). This situation is not an optimal strategy; it just maximizes the payoff

of the “worst case scenario”.

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Demonstration 10-2

What is the secure strategy for player B in Table 10-1?

Answer. If player A’s strategy is fixed at “UP”; then the payoff’s for B are (20 and

8). Choose 8. If A’s strategy is fixed at “DOWN” then the B’s payoff’s are (7 and

10). Choose 7. Then Players B’s worst case scenario is (8, 7). The best worst case for

B is then 8 (right). Thus the secure strategy by player B is “RIGHT’. This strategy is

a max-min strategy.

Shortcomings of Secure Strategy

1. It is a conservative strategy that should be considered only if you have a good

reason to be extremely risk-averse.

2. It does not take into consideration the optimal (dominant) strategy of the rival;

and thus, it may prevent the player (manager) with the secure strategy from

earning a significantly higher payoff. If player B reasons that in such a game

player A will choose the dominant strategy and that player will therefore choose

“Up”, then player B will earn 20 by choosing “Left” instead of “Right” that brings

8. So if the rival has a dominant strategy, the other player should anticipate that

the rival will use it.

Nash Equilibrium

This equilibrium represents a condition in which each player does the best he/she can,

given the decision of the other player. In other words, no player can improve his/her

payoff by unilaterally changing his strategy, given the other players’ strategies. No player

can improve his/hr payoff without hurting the other player. In Table 10-1, given that

player A chooses dominant strategy “UP”, the Nash equilibrium for player B is to take

this dominant strategy as given and choose strategy “LEFT” which gives 20 units of

payoff’s compared to for “RIGHT”. Similarly, if player B chooses “LEFT” Nash

equilibrium for Player A is “UP” which gives 10 units of payoffs.

Application of One-Short Games (look for dominant strategies first)

An application of simultaneous move, one-shot game is Bertrand duopoly (ZERO

PROFITS). Table 10-2 has two players with two possible strategies: to charge high price

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or low price. The collusion is both charge high price and cheating is one charges the low

price. The two number cells are the profits for firm A and firm B. For example, in the cell

corresponds for low price for firm B and high price for firm A, the first number (-10) is a

loss for firm A, and the second number (50) is the profit of firm B.

Table 10-2: A Pricing Game (Bertrand Duopoly)

Firm

A

Firm B

Price Low High

Low 0,0 50,-10

High -10,50 10,10

In a one shot play of the game, the Nash equilibrium strategies are for each firm to charge

low price. Why? Because if firm B charges high price, firm A will make 50 by charging

the low price which is better than the 10 it will make by charging a high price. Similarly,

if firm B charges the low price, firm A will charge the low price and make zero payoff

which is higher than (-10) that firm A will make by charging the high price. This is also a

dominant strategy for firm A. Thus firm A will always charge low price regardless of

firm B’s decision. The same argument goes for firm B which should charge the low price

regardless of what firm A will choose. This is also the dominant strategy for firm B. The

outcome of the game is both firms charge the low price and earn zero profit in a Bertrand

duopoly.

Profits under Nash equilibrium (0, 0) are less than under collusion (10, 10). If the

firms collude both would charge the High price and make 10 profits for each of them.

This makes the Nash equilibrium inferior to the collusion. This result is called a dilemma.

But collusion is illegal and if the firms colluded secretly, one firm may cheat by charging

the low price and make the other firm’s customers switch to it. In this case the firm that

did not cheat will suffer from a loss (-10). The manager of this firm that did not cheat

either has to reveal to the shareholders that he colluded but did not cheat and

consequently suffered a loss. This will bring him to jail. The second alternative is to

explain nothing for making a loss and in this case he will be fired. Then this manger will

cheat in one shot games. The situation can be different under repeated games.

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Demonstration 10-4: Advertising and Dominant Strategies

Firms advertise in order to entice customers from other competing companies.

Suppose there are two firms: A and B, and two strategies: to advertise or not to

advertise as illustrated in Table 10-3.

Table 10-3: An Advertising Game

Firm

A

Firm B

Strategy Advertise Do not

Advertise

Advertise $4,$4 $20,$1

Do not

Advertise

$1,$20 $10,$10

The profit maximizing strategies for both firms are to “advertise” to cancel each

other advertising out. These to-advertise strategies are dominant strategies for both

firms. For each player “TO ADVERTISE” brings more money than the “DO NOT

ADVERTISE”, regardless of what the strategy of the other player, because of

cheating. Thus if both “advertise” each will make $4. Note that if both collude and

agreed to “Do not advertise” each will make more money ($10). But collusion does

not work in one-shot games. If one cheats and “advertises” it will make $20 and the

one that “did not advertise” will make $1. In one-shot game, the game is over right

after it is played and there is no chance for punishment. So collusion (10, 10) does

not work. Here advertising brings more money. The advice is to advertise.

Coordination Games

In the previous games, the firms were competing in the sense, what one firm’s gains are

at the expense of the other firm. In coordination games, firms find it more profitable to

coordinate their actions and do like wise. An example of coordination games is two

producers of electric appliances.

Each firm has a choice of producing one of two types of outlets: 120 volt, two prong

outlets; and 90 volt, four prong outlets. If those firms coordinate and produce likewise,

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then in this case consumers do not have to spend more money on wiring their houses with

different outlets and will have more money to buy appliances. If they do not, then that

will make consumers spend less to buy the appliances because in this case they have to

spend more money in wiring their houses. Let us assume that the two firms’ profits are

given by the matrix in Table 10-4.

Table 10-4: A Coordination Game

Firm

A

Firm B

Strategy 120 volt 90 volt

120 volt $100,$100 $0,$0

90 volt $0,$0 $100,$100

Given firm B’s strategy of producing 120 volt or 90 volt outlets, firm A would

maximize profit by choosing to profit by matching B’s chosen strategy. In this case firm

A’s profit would be 100 compared to zero profit by not coordinating. Similarly, given

firm A’s strategy of choosing either 120 volt or 90 volt, firm B would maximize profit by

matching A’s chosen strategy. In this case, the firms must match each other. In this

coordinating game, there are two Nash equilibriums: an equilibrium of $100 profit for

both firms choosing 120 volt outlet, and another equilibrium with $100 profit for both

choosing 90 volt outlets. In this case each firm should guess what the other firm is going

to do. If the firm has no clue of the other firm’s choice, then this firm will have a very

tough decision. If the firms cannot talk and coordinate, then the government can set up

standards requiring all firms to operate on, for example, 120 volt. In this case, there are

no incentives to cheat. Coordination games are different from competing (advertising)

game in Table 10-3.

INFINITELY REPEATED GAMES

In the simultaneous move, one shot games, collusion is not very likely because games

are played only once and punishment is too late. There is today but no tomorrow, if one

firm cheats in such unrepeated games the profits from cheating exceed those from

collusion. However, in reality, firms compete every week, every year over and over again

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and forever. Thus, the games are repeated. In the case when games are repeated infinitely

it is possible under certain conditions that collusion will stick (i.e., be the solution).

Theory

When a repeated game is played, players receive payoffs during each repetition of the

game. Payoff received today has a higher time-value than payoff that will be received

tomorrow. The future payoffs must be discounted and we must compare the present

values of the future payoffs to today’s value of the current payoff. In case of cheating, we

have to compare the value of current one-time profit from cheating plus present value of

Nash payoffs (no more cheating after this and we will have the present value of the Nash

payoff for each game after that ) with the present value of the stream of profits from

cooperation or collusion over infinite time.

Present Value (PV)

PVfirm = π 0 + π1 / (1+i) + π2 / ( 1+i)2 + -----+ πT / ( 1+i)T

where π 0 is profit today, π 1 profit a year from now, π T is from T years from now, and i is

the interest rate or discount rate and [1/ (1+i)] is the discount factor and also the term of

the series. If the period is not infinite and profit is constant (π I = π ), then the above series

can be written as:

PVfirm = π / ( 1+i)0 + π / ( 1+i) + π / ( 1+i)2 + -----+ π / ( 1+i)T for T repeated games..

PVfirm = π[1 + 1/ (1+i) +1/ (1+i)2 + 1/ (1+i)3 +…+ 1 / ( 1+i)inf]

= π * t =0 t= inf 1 / ( 1+i)t for infinitely repeated games.

If the period is infinite and profit πi is constant, then the series is expressed as

PVfirm =

The series converges or has a limit because it’s typical term or element

(1/1+i) is a fraction. The converging limit of this series is1/[1- ELEMENT OF SERES] =

1/[1-1/(1+i)] = (1+ i) /i. Substitute this limit into PVfirm equation above, we have

PVfirm = [(1+i)/i] *π

This is the term which we will use for cheating to compare the profit from cheating today

(plus PV of Nash payoffs if non zero in the future) with the present value, PVfirm ,of

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streams of current and future profits from collusion or cooperati on over the life span of

the firm.

Table: Infinitely Repeated Games

Firm

A

Firm B

Price Low High

Low 10, 10 70,-40

High -40,70 50,50

PVfirm (cheat) = Cheating Payoff at current period + PV(Nash Payoffs)

PVfirm (cheat) = 70 /( 1+i)0 + 10 / (1+i) + 10/ (1+i)2 + -----+ 10 / (1+i)Inf where 70 is one

time payoff from cheating. Break 70 into 60 and 10 because 10 is Nash and is needed for

convergence for a series with a constant which 10 in this case)

PVfirm (cheat)= 60 + 10(1+i)0 + 10 / (1+i) + 10/ (1+i)2 + … + 10 / (1+i)Inf

= 60 + 10[ ]

= 60 + 10(1+i)/i where (1+i)/i is the limit and 10 is Nash payoff.

PVfirm (Collusion) = 50 + 50/ (1+i) + 50/ (1+i)2 + … + 50 / (1+i)Inf

= 50[ ] = 50(1+i)/i

You can plug the value for i in both PV equations and then compare.

Supporting Collusion with Trigger Strategies

As mentioned above, collusion in infinitely repeated games is possible under certain

conditions. Firms enter into a collusion agreement based on past plays of the firms. If a

firm deviates and cheats then there will be a deviation from past plays. In this case other

firms will use trigger strategies that are intended to punish the deviation. The punishment

means that other firm will punish the cheater by doing exactly what he did, they would

lower the price (they go to Nash if exists). If every firm relies on trigger strategies

collusion will last if

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current profit from cheating (plus discounted non zero future Nash profits if exist)

< PV of future streams of profits under collusion.

as explained in the PV equations above.

Table 10-8 is an example for that with Nash payoffs are zeros (Bertrand).

.

Table 10-8: A Pricing Game That is Repeated (Bertrand)

Firm

A

Firm B

Price Low High

Low 0,0 50,-40

High -40,50 10,10

If both firms collude in this repeated game, then the stream of future profits for each firm

is 10. If one player cheats, while the other one sticks to the collusion agreement, the

cheater will make 50, while the non cheater would make -40. If the collusion breaks

down and both firms recourse to low prices (or Nash) then each will make zero profits in

the future for this Bertrand oligopoly.

The PV Approach:

Suppose firm A cheats, while firm B does not. The game is over after one play. Then

PVCheatFirm A = current $ cheating payoff + disct’d Nash1 + disct’d Nash1 +… = $50 + 0 +

0 +--- = $50

(Note: in this example Nash exists and equals zero). If firm A does not cheat and

“cooperates” in this repeated game, the present value is

PVCoopFirm A = 10 + 10/ ( 1+i) + 10 / ( 1+i)2 +10 / ( 1+i)3

+ …. = 10(1+i) / i

where i is the interest rate. Thus, there is no incentive for Firm A to cheat if:

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PVCheatFirm A ≤ PVCoop

Firm A

PVCheatFirm A = 50 + 0 + 0+ … ≤ 10 (1+i) / i = PVCoop

Firm A

50 ≤ 10 (1+i)/i or

50/10 ≤ (1+i)/I (divide both sides by 5)

5 ≤ (1+i)/i (multiply both sides by i)

5i ≤ 1+ i

5i -i ≤ 1

4i ≤ 1 (subtract one from both sides)

Or i ≤ ¼ (no cheating).

Solve for i. (In this case i = 25%)

If i < 25%, Firm A will lose more in present value by cheating than it will gain by

cooperating. The solution is COOPERATION. In general, the lower the interest rate is,

the more likely that conclusion will persist, and vice versa.

More generally, we can write the principle for sustaining collusion in terms of one shot-

game payoffs without using preset values as follows.

(ΠCheat - ΠCoop) / (ΠCoop – ΠN ) ≤ 1/i ( no cheating)

Or (ΠCheat - ΠCoop) ≤ 1/i (ΠCoop – ΠN)

where ΠCheat is the maximum one-shot payoff if the player cheats, ΠCoop is the one-shot

cooperative or collusive payoff and ΠN is the one-shot Nash equilibrium. If any player

cheats, the trigger strategy is to punish the player by choosing the Nash one-shot

equilibrium strategy forever after. Apply this condition to example 10-8 when i = 10%.

(50 – 10)/(10 - 0) < 1/0.1 = 10 for no cheating

4 < 10.

(collusion is more profitable). Each firm can earn a payoff of 10.

Intuitively, the above condition for sustaining collusive or cooperative outcomes is that

“provided the one-time gain from breaking the collusive agreement (cheating) is less than

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the present value of what would be given up by cheating (cooperation), players find it to

their interest to live up to the agreement”

The lower the interest rate, the more likely that conclusion will persist, and vice versa.

Demonstration 10-6: The Principle to the Sustainability Approach

Use the information in Table 10-8 and apply the above principle to the sustainability

of collusive agreements when i = 40% (higher than before). Check if cooperation or

collusion will persist over cheating.

(50 – 10)/(10 - 0) < ? 1/0.4 = 2.5 for no cheating (cooperation)

40/10 > 2.5

(cheating and no collusion because interest rate is very high))

The other PV approach requires that we check if

PVCheatFirm A ≤ PVCoop

Firm A

50 + 0 + … ? 10 + 10/(1.4) + 10/(1.4)2 + 10/(1.4)3 +

…. where 0.40 = i (very high).

50 ? 10*(1+i) / i

50? 10*[1.4 / 0.4]

50 > 35 (cheat no collusion).

Since the matrix in table 10-8 is symmetric each firm has the incentive to cheat.

(You can also use the principle of collusion sustainability stated above here)

Factors Affecting Collusion in Pricing Games (increases in monitoring costs reduce

incentives to collude).

1. Number of Firms (Remember the Heinz case study)

Collusion is easier when there are fewer firms rather than many. If there are five firms

in the market, each firm must be monitored four times by its rivals. Total number of

monitoring in the industry is 5*4 = 20 total firms monitored. The cost of monitoring

reduces the gains to colluding.

2. Firm Size

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It is easier for a large firm with 20 outlets to monitor a small firm with one outlet. The

large firm must monitor 1 store, but the small firm must monitor 20 stores).

3. History of the Market

Firms may not meet to collude but they can reach an understanding of the way the

game has been played over time. Thus, the firms reach a tacit coordination. So they

accomplish collusion indirectly by learning from past experience.

4. Punishment Mechanism

Punishing a rival has a cost. If a firm posts a single price to all its current and

potential customers then if it punishes its rival by lowering the price it must lower it

on all the customers including those of the rival. This results in high punishment cost

for the firm. But if this firm charges different prices to different customers, it can just

lower the prices for the rival’s customers. In this case the cost of punishment for the

firm is lower.

FINITELY REPEATED GAMES

There are two types of finitely repeated games: one type that has a known final

end period; and the second that has an uncertain or unknown final or end period

Finite Games with Uncertain end of period.

These games, although finite, is similar to infinite games. The firms know that

their product has a finite lifespan and some day they will become obsolete but they do not

know when. Thus, the end of the finite period is uncertain. Suppose both firms played the

game today. They know that the probability that the game will end tomorrow is Ө and

that it will continue until tomorrow is (1- Ө). Note that 0< Ө <1. For the next two days,

the probability it will end is Ө * Ө = Ө 2 and the probability that it will continue for two

days is (1- Ө)*(1- Ө) = (1- Ө)2. The probability it will end in three days is Ө3 and will

continue for three days is (1- Ө)3 and so on. Let us apply this type of games to Table 10-

10.

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Table 10-10: A Pricing Game that is Finitely Repeated

Firm

A

Firm B

Price Low High

Low 0,0 50,-40

High -40,50 10,10

If there is collusion and both firms adopt high price strategies, the payoff for each of

them is 10. This will continue until the product terminates and the game ends. Assume

interest rate is zero (no discounting) for simplicity. Then (PV of) payoff from

cooperation for firm A is:

PV П firm A Coop = 10 + (1-Ө)10 + (1- Ө)2 10 + (1- Ө)310 ….=10/ Ө

The term of seres is (1- Ө) and the limit of the series is 1/(1- term of series)] =[1/(1- (1-

Ө)) = [1/ Ө])

(Footnote: this equation is similar to the equation of collusion for the infinitely repeated

games with (1- Ө) are replacing [1/(1+ i)] as the term of series. In both games they

receive the same benefits. We assume i =0).

Note that if the games will end or terminate tomorrow and that Ө =1 then the pay

from collusion is 10 (Nash is zero in this example). This is a one-shot game. If the firm

cheats, then the relationship between the payoff from cheating and that from collusion

which assumes that the game will continue is:

PV П firm A Cheat = 50 + 0 + …> PV П firm A Coop = 10/1 = 10.

(that is greater than the payoff from collusion in a one shot game). In this case there is no

incentive for firms to collude. Since the matrix in Table 10-10 is symmetric (compare the

off-diagonal cells) firm B will have the same thing and there will be no cooperation. On

the other hand, if Ө is a small fraction such that

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PV П firm A Coop = > PV П firm A Cheat = 50 (which is the cheating payoff) + PV

(Nash for future payoffs)

10/ Ө > 50 + 0 + 0 + …

0r

10/ Ө > 50, (where Ө = 10/50 = 0.20).

then the firms will cooperate and collude. More precisely, if Ө < 0.20 (i.e., 1/5) then the

firms will cooperate and collude.

We can conclude by saying that the lower Ө and the higher (1- Ө), the more likely the

firms will cooperate and collude.

Demonstration 10-7: Billboard Advertising Game

Suppose two cigarette manufacturers repeatedly played the following simultaneous

–move billboard advertising game as illustrated in Table 10-11.

Table 10-11: A Billboard Advertising GameF

irm A

Firm B

Strategy Advertise Do not

Advertise

Advertise 0,0 20,-1

Do not

Advertise

-1,20 10,10

In this table, if both companies cooperate and “DO NOT ADVERTISE” (collusion)

each will earn $10, while if they both “ADVERTISE” (Nash) each will make zero. If

one advertise and the other does not (cheating), the one that advertised makes $20

and the one that does not make -$1. Assume there is a 10% chance that the

government will ban (end) cigarette sales in any given year, can the firms “collude”

by agreeing not to advertise? Note that 1-Ө = 0.9.

If firm A cooperates and doesn’t cheat it can expect to earn:

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PV П firm A Coop = 10 + (1-Ө)10 + (1- Ө)2 10 + (1- Ө)310 ….=10/ Ө

PV П firm A Coop = 10 + (.9)10 + (.9)2 10 + (.9)310 +…. = 10/.10 = $100.

Since $20 < $100 the firm has no incentive to cheat (that is the solution is collusion).

The incentives for firm B are the same. Thus, firms can collude by using this type of

trigger strategy which involves punishing the cheating firm by charging a lower

price until the game ends.

Repeated Games with a Known Final Period: The End-of-Period Problem

Suppose a game is repeated some known number of times with strategies and payoffs as

supposed in Table 10-12.

Table 10-12: A pricing Game

Firm

A

Firm B

Price Low High

Low 0,0 50,-40

High -40,50 10,10

Let us assume for simplicity the game is repeated twice (two one-shot games) and the

players know the game will end in period two. This means after the game is played twice

there is no tomorrow (at the end of the second period). At that time there are no trigger

strategies and no punishments even if player A cheats. The two-shot game is really played

as a one-shot game twice. Player A kept charging the high price. In this case since there is

no tomorrow. Player A can charge a low price in the second period and player B cannot

punish him/her. In fact player A would be happy if player B continues charging the high

price in the second. In this case player A if charges the low price it will earn 50. But

player B knows that player A will charge the low price and thus B will do likewise. This

means this two-shot game will end in the first period and will not go to the second or end

period in this example. Nash equilibrium in this two-shot game is to charge low price in

each period. The game is played as two one-shot games and each player will earn zero

profit in each of the two periods.

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In that collusion will not work even if the game is played three, four, 1000 times. This

type of “backward unraveling” continues until the players realize no effective punishment

can be used during any period. The key reason is that each player knows that promises of

cooperation will be broken any time because the period has an end and then there is no

tomorrow. So the solution is low prices with zero profits.

Demonstration 10-8

Suppose firms A and B will play the game in Table 10-12 twice. Assume that firm A’s

strategy is to charge high price each period provided that firm B (the opponent never

charged a low price in any previous period. Assume interest rate = 0.

1. How much will firm B earn?

2. How much firm A earn.

Answer: Since firm A will also charge a high price each period, the opponent firm B will

be able to trick firm A in the second period because in this period the game will end.

Firm A will stick to its strategy for the first and second periods because it will not

discover B’s cheating until the second period, and at that time it will be too late to punish

firm B. Then firm B will charge a high price in the first period and earn 10 and charge a

low price and earn 50 in the second period for a total of 60 (this is better than cooperating

and charging higher price in each period for a total profit of 10 + 10 in the two periods).

Correspondingly, Firm A will earn 10 in the first period and make a loss of 40 in the

second period, for a total loss of 30 in the two periods. Since each player knows when the

game will end and trigger strategies will not enhance profits.

Applications of the End-of-Period Problem

End of period problem arises when workers know precisely when a repeated game will

end. In the final period, there is no tomorrow and there is no way to punish a player for

doing something wrong in the last period. Here is an implication of the end-of-period

problem for managerial decisions.

Resignations and Quits

Workers weigh the benefits from shirking with the cost of being fired. If the benefits are

less than the costs, workers will find it in their interest to work hard. If the worker

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announces today that he/she will quit tomorrow than there is no reason for the worker to

work hard because the threat of being (the trigger strategy) fired has no bite.

What can the manager do to overcome the end-of-period problem? He can fire the

worker today but legally this may not be feasible. Moreover, there is a more fundamental

reason why the manger should not adopt this policy. To avoid being fired on

announcement, workers will not announce their plans of quitting until the end of the day

and in this case they get to work longer than if they announce their plans. Consequently,

the manager will not solve the end-of-period problem but instead he/she will be

continuously be surprised by worker resignations.

A good strategy is to give the workers some rewards for good work that extend

beyond the termination of employment with the firm. In this case the worker will not take

advantage of the end-of-period problem. But if the worker takes advantage of the end-of-

the period problem the manager, being well connected, can punish the worker by

informing potential employers about it.

Multistage Games

These games differ from the class of simultaneous games one-shot infinitely

repeated games in the sense that timing is very important for multistage games. In

particular, multistage games permit players to make sequential rather than simultaneous

decisions.

Theory

In order to understand how multistage games differ from one shot and infinitely

repeated games. We need to introduce the extensive form of a game. An extensive- form

game summarizes who these players are, the information sets available to those players at

each stage, the strategies available to the players, the order of moves and the payoffd

from the alternative strategies.

Fig. 10-1 depicts the extensive form of a game assume that there are two players:

A and B; and that player A is the first mover and player B is the second mover. Each

player has two strategies: Up and Down. The numbers at the end of branches in this

figure are the players payoffs since player A is the first mover the first number is that

players payoff and the second number is player B’s payoff.

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(Fig. 10)

(0, 0)

(6, 20)

(5, 5)

(10, 15)

A

B

B

Down

Down

Down

Up

Up

Up

69

In Fig. 10-1, player A moves first, and once this player moves, it’s player B’s

turn. If player A chooses Up and player B makes the same Up move, then the payoff for

A and B, respectively, are (10,15). But if player B moves in the other direction and

chooses the Down strategy then their respective payoffs are (5, 5). As in simultaneous-

move games, each player’s payoff depends on both player’s actions. This is the similarity

between these types of games. For example, if the first move of player A is Down and

player B chooses Up then player A’s payoff is (0), but if B chooses Down player A’s

payoff is (6). There is important difference between the sequential and simultaneous

types of games. Since player A is the first mover in this case, this player cannot make

decisions based on player B’s moves, but player B gets to make decision after player A.

Thus, there is no conditional “if” in player A’s strategy.

Let’s see how strategies work in this game. Suppose the strategies are: player B

chooses Down if player A chooses Down. What is the best strategy for A? The best

strategy for A is Down because in this case A will make 6, which is better than 5. Given

that player A chooses Down, does player B have an incentive to change his strategy? The

answer is NO. Choosing Down instead of Up, B earns 20 instead of 0. Since neither

player has an incentive to change his/her strategies then there is a Nash equilibrium

associated with those strategies.

Player A: Down;

Player B: Down if player A chooses Up, and Down if player A chooses Down. (player B

threatens to play Down all the time).

The payoff: (6, 20)

Is this a reasonable game? Why doesn’t A choose Up and make 10 instead of choosing Down and making 6? The answer is in the way B’s strategy is formulated. If A chooses Up, B threatens to choose Down all the time. In this case A will make 5 instead of 6. Should A believe B’s threats? If B chooses Down it will make 5. What to make out of all this? There are two Nash equilibria in this game.

Nash Equilibrium: As explained above when B threatens to play Down all the time.Nash Equilibrium: When A finds that B’s threats are not credible.

70

Player A: UpPlayer B: Up if player A chooses Up and Down if player A chooses Down.

Player B will have to chooses Up if A chooses Up. In this case, the neither player has an incentive to change his/her mind. The second Nash equilibrium is more reasonable because B’s threats are not credible in the sense that A can choose Up and this will force B to choose Up and NOT Down because it will have a lower payoff (5 instead of 15) if it follows upon its threat to choose Down.

71

Chapter 11

Pricing Strategies for Firms with Market Power

In this chapter we deal with pricing strategies of firms that have some market

power: firms in monopoly, oligopoly and monopolistic competition. As we learned in

chapter 8, firms in perfect competition are price takers and they don’t have a pricing

strategy of their own. This chapter goes as far as providing practical advice on

implementing pricing strategies for those firms with market power, typically using

information that is readily available to managers, including publicly available

information such as the price elasticity of demand.

The optimal pricing strategies for firms with market power vary depending on the

underlying market structure and the instruments (e.g., advertising) available. To account

for that, this chapter presents more sophisticated pricing strategies that enable a manger

to extract greater profits from the consumers.

BASIC PRICING STRATEGIES

We will first look at the very basic pricing strategy which relies on single or

uniform pricing. This strategy uses the profit-maximizing rule: MR=MC to derive the

optimal price. This rule is then mathematically manipulated to provide a rule of thumb

that makes use of the markup to arrive at the price.

Review of the Basic Rule of Profit Maximization

Firms with market power can restrict output to charge a higher price; thus they

have a downward-sloping demand curve. In this case the price is different from marginal

revenue. The profit-maximizing rule for firms with market power is given by

MR = MC.

This rule is first solved for the equilibrium output which in turn is substituted in the

inverse demand equation to solve for the optimal or equilibrium price as was illustrated in

chapter 8. Managers of large firms may have research department that have economists

who can estimate demand and cost functions and apply this rule and to solve for optimal

price and output

72

Demonstration 11-1

Suppose the inverse demand equation is given by

P = 10 -2Q (downward sloping demand = market power)

and the cost function is

C(Q) = 2Q.

Determine the profit-maximizing output and price.

Answer: Recall MR has twice the slope of the price in this case.

Then

MR =10 – 4Q.

Set MR = MC

10-4Q* = 2

Solve for Q*. Then Q* = 2 units. Plug Q* into the inverse demand equation

P* = 10 -2Q* = $6.

A Simple Pricing Rule for Monopoly and Monopolistic Competition

Some small firms such as retail clothing stores do not hire economists to estimate

their demand and cost functions. They can, however, rely on publicly available

information such as information on price elasticity of demand (see chapter 7 for estimates

of price elasticity for different industries). We can derive a rule of thumb from the profit-

maximization rule and estimate the price with minimal or crude information and still be

consistent with profit-maximization.

Formula: Marginal Revenue for a firm with Market Power (Monopoly and Monopolistic

Competition):

MR = P[(1+Ef)/Ef] where Ef = %∆Q/%∆p = (∆Q/∆P)*P/Q

where Ef is the firm’s own direct price elasticity of demand. Substitute this in the profit-

maximization rule

P[(1+Ef)/Ef] = MC

73

Solve for the price:

P = [Ef /(1+Ef)]MC

or

P = (K)MC

where K = Ef /(1+Ef) can be viewed as the profit maximization (optimal factor)

markup factor.

Example: The clothing store’s best estimate of elasticity is -4.1 and this is known. Thus,

the optimal markup is

K = -4.1/(1- 4.1) = 1.32.

Then the optimal price

P = (K)MC = 1.32*MC

(That is, 1.32 times marginal cost).

The manger should note two things about this price elasticity: First, the more

elastic the price is, the lower the markup factor and the price (if Ef = -infinity, then K= 1

and P = MC as is the case in perfect competition); the lower MC is, lower the price.

Demonstration 11-2

Suppose the manger of a convenience store competes in a monopolistically

competitive market and buys Soda at a price of $1.25 per liter. Chapter 7 reports

that the price elasticity of demand for the typical grocery is -3.8. The manger of this

convenience store believes that demand is slightly more elastic than -3.8. Let the

price elasticity of the convenience store is -4. What is the profit maximizing price for

this store?

P = [-4/(1-4)]MC = 1.3 MC

A Simple Pricing Rule for Cournot Oligopoly

Strategic interaction is an important issue in Cournot oligopoly. Each firm

maximizes profit taking into account of the output of the rival firms in the industry. It

believes that the output of the rivals will stay constant. The maximization rule is the same

as in the monopoly case,

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MR = MC.

But under Cournot monopoly, MR depends on the firm’s output and on the rivals’ output

as well. Each oligopolistic firm uses this rule to derive its interaction functions in which

its own output depends on the rivals’ outputs. Then the interaction functions are used to

determine the profit-maximizing outputs (Q1*, Q2*)

Fortunately and similar to monopoly, a simple markup pricing rule can be used in

Cournot oligopoly when the oligopolistic firms have identical cost structures and

producing similar products. Suppose the industry consists of N firms with each firm

having identical cost structures and produces similar products. In this case we can use

the markup pricing rule for monopoly and monopolistic competition to derive a pricing

formula for a firm in a Cournot Oligopoly. First, it can be shown that if products are

similar then

Ef = N*EM

where Ef is the price elasticity of demand for the typical firm, EM is the industry’s price

elasticity of demand and N is the number of firms in the industry. Recall that the markup

pricing rule under monopoly and monopolistic competition is given by

P = [Ef /(1+Ef)]MC

where MC is the individual firm’s marginal cost. Upon substitution for Ef from above, the

profit maximizing price for a firm under Cournot is given by:

P = [NEM /(1+NEM)]*MC (rule of thumb pricing under Cournot)

Demonstration 11-3

Suppose a Cournot industry has three firms, with market elasticity Em equal -2 and the

individual firm’s MC is $50. What is the firm’s profit maximizing price under Cournot

oligopoly

P = {(3)(-2)/[1+(3)(-2)] }*$50 = $60

75

STRATGIES THAT YIELD EVEN GREATER PROFITS

These are strategies that can be implemented under monopoly, monopolistic competition

and oligopoly by which the manager can earn a profit greater that it can get using the

single pricing rule (MR = MC) whether directly or through a pricing formula. These

strategies which include: price discrimination, two–part pricing, block pricing and

commodity bundling, are appropriate for firms with various cost structures and degrees of

market interdependence.

Extracting Surplus from Consumers

All the above four strategies aim at extracting consumer surplus and turn it into profit for

the producers.

I. Price Discrimination

Price discrimination is the practice of charging different prices to different consumers for

the same good or service sold. There are three types of discrimination; each requires that

the manager have different types of information about consumers.

First- degree price discrimination (perfect price discrimination)

This type of prices discrimination amounts to charging each customer the maximum price

it is willing and able to pay. This price is called the reservation price.

Definition: Reservation Price: The maximum price the customer is willing to pay (e.g. P1

and P2 ), which is greater than or equal to the actual price.

Actual P

P

Q

D

P1

P2

76

If monopoly single pricing strategy is used and the monopoly price is P*M, then

consumer surplus (CS) in the graph below is the yellow triangle above the P*M-

line and below the D-curve.

If 1st degree price discrimination is practiced then: Consumer surplus (rectangle area) =

0, (because the price is the maximum price the consumer is willing to pay).

Fig. 11-1a below shows the firms’ total (operating) profit (CS + PS) when the firm

charges the maximum price. It is the area below the demand curve and above the MC

curve up to Q*M. Note that the area below the MC curve and below the price line P*M up

to the quantity Q*M is the producer surplus (PS).

First-degree price discrimination is also called perfect price discrimination because it

requires identifying the reservation price for each consumer under alternative quantities.

This is not possible in the real world.

CS

PC

MC

P*M

Q*M MR

M

77

Fig. 11-1 First and Second Degree Price Discrimination

Second Degree Price Discrimination (discrimination based on quantity)

This type of price discrimination leaves the consumer with some consumer surplus. Thus

relative to the first degree price discrimination, the total profit under the second degree is

lower. This discrimination practice is based on giving discount for buying extra quantities

of the good.

In Fig. 11-1b, the firm charges the consumers $8 a unit for the first two units. In this case

it extracts [1/2*(8-5)*2= $3] of the consumer surplus which would have gone to the

consumers under single pricing. It also extracts some more by charging $5 per unit of on

the units from 2 to 4. This is an additional extraction of CS. The firms cannot extract all

consumer surpluses; some consumer surplus will be left to the consumers under the 2nd

degree-price discrimination.

Example: Electric companies: it works by charging different prices for different

quantities or blocks of the same good or service (KWH). This is the case of natural

monopoly (economies of scale) where both AV and MC curves are declining all the way.

Natural Monopoly: MR = MCBreakeven: P = AC or TR = TC 78

Graph: Natural monopoly with second-degree price discrimination.

Fig. 11-1(b) above shows how much of the consumer surplus is extracted by the firm

when the second-degree practice is used.

Third-Degree Price Discrimination

Customers are divided into few groups with a separate demand curves or elasticities for

each group. This is the most prevalent form of price discrimination.

Example: Airline fares: Airline passenger tickets are divided into groups 1st class fare,

regular unrestricted economy fare, and restricted economy fare.

How are customers divided into groups?

Some characteristic is used to divide consumers’ into distinct groups: willingness to pay,

Identity can be readily established (ID ….etc)

What price to charge each group?

Given whatever total output is produced, this total output is allocated among the groups

based on the profit maximization rule 1.

1. MR1 = MR2 = --- = MRN

That is, prices should be designed as a result of equating MRS and read off their

corresponding demand curves.

Q3Q1

P3

P2

PM*

P1

MR D

MC

Q2QM*

1st block 2nd block 3rd block

AC

Break even

EM

79

If for example MR1 > MR2 output should be shifted from group 2 to group 1 (because the

first group is adding more to total revenue), this will lower P1 and increase P2 until that

MR1 = MR2

2. Determination of total output (Q*) is by equating MRT = MCT

Where MRT is the horizontal sum of all groups MRi , i = 1,…, n. That is, fix MRi at a

certain level then add up the corresponding quantities Q1, Q2,. ..,Qn. Then repeat this

process by fixing MRi at a different level and so on. You will get MRT.

Then equate MR1 = MR2 = --- = MRN = MCT to divide the total output among the n

customer groups.

Where MCT is the marginal cost of total output.

If MRi > MCT for all groups i, then profit will increase by increasing total output and

lowering prices.

MRi < MCT then profit will increase by decreasing total output and increasing prices.

This continues until MRi = MCT for all groups i = 1,…., n.

Suppose there are two groups

Group 1 Group 2Total

output

Q1 Q2

QT = Q1 + Q2

P1 P2

Total cost function C = C (QT)

TR1 = P1Q1

TR2 = P2 Q2

π = P1Q1 + P2Q2 – C(QT) (profit)

Q1 will increase until incremental profit ∆π / ∆Q1= 0

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∆π /∆Q1 = ∆ (P1Q1) / ∆Q1 – ∆C / ∆Q1 = 0 which means

MR1 – MC = 0

this implies that

MR1 = MC

Similarly Q2 will increase until incremental profit ∆π / ∆Q2 = 0

MR2 = MC

Putting these relationships together

MR1 = MR2 = MC (which is the condition allocating total output Q* among the two

groups).

This is the condition for profit maximization under third degree monopoly.

Monopolists practicing this price discrimination may find it easier to think in terms of

the relative prices that should be charged to each group and to relate these prices to

elasticity.

Recall MR1 = P1 + P1(1 / EP1D1) = P1(1+1/EP1

D1)

Recall MR2 = P2 + P2(1 / EP2D2) = P2(1+1/EP2

D2)

Note that Ep11 /(1+Ep1

D1) = ( 1 +1/EP1D1)

This can be rewritten as

P1[(1+E1)/E1] = MC

P2[(1+E2)/E2] = MC

Therefore from 1st profit max ruler under 3rd price discrimination:

MR1 = MR2

P1(1+1/EP1D1) = P2(1+1/EP2

D2)

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P 1 = [1+(1/EP2D2 )]

P2 [1+(1/EP1D1)]

The higher price will go to the consumers with the lower elasticity.

Example: EP1D1

= - 2 (lower elasticity)

EP2D2

= - 4 (higher elasticity).

P1 / P2 = (1-1/4) / (1-1/2) = 1.5

Or P1 = 1.5P2

Demonstration 11-4

Local monopoly is near campus. Let MC =$6 per pizza.

During the day only students eat there, while at night faculty members eat. If

student’s elasticity of demand is -4 and of faculty is -2, what should be the pricing

policy be to maximize profit?

Answer:

The faculty has more elastic demand

P1[(1+E1)/E1] = MC

P2[(1+E2)/E2] = MC

Let L =lunch or day pizza, and D = Dinner pizza.

PL[(1-4)/-4] = $6

PD [(1-2)/-2] = $6

Then PL =$8 (more elastic )and PD =$12 (less elastic)

II. Two-Tier (Part) Pricing

With two-part pricing, the firm charges a fixed fee for the right to purchase its goods,

plus a per-unit charge for each unit purchased. This pricing policy is commonly used by

athletic and night clubs. As is the case with price discrimination, the purpose of this

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policy is to enhance the seller’s profit by extracting consumer surplus from consumers.

Similar to the first-degree price discrimination, this two-part pricing strategy allows firms

to extract the entire consumer surplus. To address this pricing strategy, we first present

the case of profit maximization by a firm with market power (say monopoly) and

estimate its profit based on using a single pricing policy. Then we use the two-part

pricing policy and estimate the profit for this policy. In this example, we will show how

the two-part pricing gives higher profit.

Fig. 11-2: Comparison of Standard Monopoly Pricing and Two-Part Pricing

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Fig.11-2(a) gives the profit maximization for a firm with market power using

single pricing which based on the rule:

MR =MC.

Suppose that the demand curve is given by

Q = 10- P.

Then the inverse demand is given by

P = 10 –Q

and, thus,

MR = 10 – 2Q.

Suppose that the total cost function is given by:

C(Q) = 2Q,

which implies that MC = 2 (in this case MC = AC and constant).

The firm’s equilibrium output and price based on single pricing are determined by

10 -2Q = 2.

Then Q* = 8/2= 4 units and P* = $6.

Total profit = (P – MC)*MC = (6 – 2)*4 = $16

Consumer surplus = (1/2)*(10 -6)* 4 = $8

Now let us use the two-part pricing strategy. Suppose the demand function in Fig. 11-2

(a) be for a single consumer. The firm can use the following two-part pricing strategy: the

fixed initiation fee for the right to purchase units $32 and that the price per unit is $2.

This situation is depicted in Fig. 11-2(b).With a price of $2 per unit, the consumer will

purchase

Q = 10 – P = 10 -2 = 8 units.

The consumer surplus with 8 units is

CS = (1/2)*(10 - 2)*8 = $32.

To implement this pricing strategy, the firm can charge a fixed initiation fee (whether as

membership fee or an entrance fee) of $32. This fee will extract the entire consumer

surplus.

Note that at $ 2/ unit, revenue will equal cost (net of fixed cost). That is,

(Variable) Profit = (P – MC)*Q = (2-2)*8 = $0.

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But the firm receives $32 as a fixed payment which is greater than the $18 profit which

receives by charging a single price

Demonstration 11-5

Suppose the total demand for golf services is Q = 20 – P and MC =$1. The total

demand function is based on individual demands of 10 golfers. What is the optimal

two part pricing strategy for this golf services firm? How much profit will the firm

earn?

Answer:

The optimal per unit charge is marginal cost. At this price, 20-1 = 19 rounds of golf

will be played each month. The total consumer surplus received by all 10 golfers at

this price is thus: ½[(20-1)19] = $180.50

Since this is the total consumer surplus enjoyed by all 10 consumers, the optimal

fixed fee is the consumer surplus enjoyed by an individual golfer ($180.50/10 =

$18.05 per month). Thus, the optimal two part pricing strategy is for the firm to

charge a monthly fee to each golfer of $18.05, plus greens fee of $1 per round. The

total profits of the firm thus are $180.05 per month, minus the firm’s fixed costs.

III. Block Pricing

Here the seller packs units of the same product and sells them as one package. The

consumer is faced with buying either the whole package or none of it. An example of this

practice is selling eight rolls of toilet paper or 12–pack of soda. The seller will assign a

value to the package that covers the cost as well as the consumer surplus.

Example: Suppose an individual consumer’s demand is given by

Q = 10 – P

The inverse demand is expressed as

P = 10 – Q

Let the cost be C(Q) = 2Q.

Then P = MC

10 – Q = $2

85

Q = 8 units.

In this case, the firm will sell eight units. (see Fig. 11 – 3; Block pricing).

The cost of buying the eight consumer is $16 and the CS = ½ (10-2)*8 = $32

Total value of the eight units = 16+32 = $48

Fig. 11-3: Block pricing

Then the profit maximizing price for the package of eight units = $48

Demonstration 11-6

Suppose a consumer’s (inverse) demand for gum produced by a firm with market

power is given by

P = 0.2 – 0.04 Q

And the marginal cost is zero. What price should the firm charge for a package

containing five pieces of gum?

Answer

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When Q = 5, P = 0.2 – 0.04 * (5) = 0

When Q = 0, P = $0.2 . The linear demand is graphed in Fig. 11-4 (optimal Block

Total

Pricing with zero marginal cost)

Value of the five units = C5

= ½ ($0.2 - $0) * 5 = $ 0.50

The firm extracts all consumer surplus and charges a price if $0.50 for a package of

five pieces.

IV. Commodity Bundling

Travel bundle may include “airfare, hotel, car rental, meals”. A computer bundle may

include “computer, printer, scanner, software …”. This pricing practice is different from

block pricing because under bundle pricing the goods or the services are not the same,

while they are identical under price discrimination because under bundling the sellers

know that for different consumers, price the components of the bundle differently but

cannot identify them into groups. Because of this lack of information the profit under

bundling is usually less than under price discrimination.

Suppose the manager of a computer firm knows there are two groups of consumers who

value its computers and monitors differently. Table 11-1 shows the maximum prices the

two groups would pay for a computer and a monitor.

Table 11-1: Commodity Bundling

Consumer Valuation of Computer Valuation of Monitor

1 $2000 $200

2 $1,500 $300

The manager does not know the identity of those two groups, and thus cannot practice

price discrimination. Suppose the cost is constant and equals to zero to simplify matters.

The manager can separately sell one computer and total profit equals

TR – TC = 2,000 – 0 = $2,000

If it sells it at $1,500, then

87

TP = 3,000 – 0 =3,000

Moreover, it can also sell monitors separately. At $300 it can sell one. At $200, it can sell

two and then total profit equals

= $3,000 + 2 * $200 = $3,400

If the manager bundles the computers and the monitors and sell them at $1,800 a bundle

then

Total profits = 2 * $1,800 = 3,600

which $200 more than selling the computers and the monitors separately. Thus

commodity bundling can hence profit.

Demonstration 11-7

Suppose there are three purchasers of a new car that has the following valuations of two

options: air conditioner and power brakes.

Consumer Air Conditioner Power brakes

1 $1000 $500

2 $800 $300

3 $100 $800

Suppose the costs are zero

1. If the manager knows the valuations and consumer identities what is the optimal

pricing strategy?

Profit from consumer 1 = 1,000 + 500 = 1,500

Profit from consumer 2 = 800 + 300 = 1,100

Profit from consumer 3= 100 + 800 = 900

Total Profit = $3,500

2. Suppose the manager does not know the identities of the buyers. Hoe much will

the firm make if the manager sells brakes and air conditioners for $800 each but

offers a special options, package (power brakes and an air conditioner) for $1,100.

Consumer 1 and 2 will buy the bundle

Profit = 2 * $1,100

Consumer 2 will buy power brakes at $800

Total Profit = $3,000

88

Chapter 12: The Economics of Information

In the previous chapters it was assumed that consumers and firms operate in an

environment of perfect information and certainty whether in terms of prices, output,

income etc. But the real world is far from that. In this chapter we will examine consumers

and firms’ behavior under imperfect information and uncertainty. We demonstrate means

(piecing, advertising etc.) by which managers can cope with uncertainty.

Mean and Variance.

Under uncertainty the variable is random and has possible outcome and their

respective probability of occurrence. All this information about a variable can collapse

into a single number which is the mean or the expected value.

Mean

Example: Offshore oil exploration company

Event : Oil exploration.

Possible outcomes: success or failure of finding oil.

Payoffs: Price of the stock of this company in cases of success and failure

Outcomes Probabilities Payoffs

Success 1/2 $ 40 / Share

Failure 1/2 $ 20 / Share

Expected value = Payoff 1* q1 + Payoff 2 * q2 = (40)*1/2 + (20)*1/2 = $30.

In general, suppose event = X

X1 = Payoff 1; X2 = Payoff 2; ….. ; Xn = Payoff N

Expected values : E(X) = q1*X1 + q2*X2 + …… + qn*Xn

Variability (Risk):

89

The mean or the risk is not enough to convey all the information about a random

variables. Some variables may have the same mean but the outcomes are deviated far

from their mean . In other words the two variables have different spread or risk.

Risk is measured by variance or the standard deviation. If the event has two

possible outcomes (X1, X2) then the variance can be written as

σ2 = Pr1 * (X1 – E (X) )2 + Pr2 * (X2 – E(X) )2

where E(X) = q1*X1 + q2*X2 is the expected value or weighted average for X1 and X2.

Example: Suppose the two events have the same expected income (E (X)) but different

risks. These events are two different sales jobs.

Job A : a commission job with two possible outcomes.

Job B : a salaried job with two possible outcomes.

EVENT OUTCOME 1 OUTCOME 2

Income q1 Income q1

Job A $ 2,000 0.5 $ 1,000 0.5

Job B $ 1,510 0.99 $ 510 0.01

In job B, $510 is a severance pay if the company that offers this job goes out of business.

Then the expected incomes for these jobs are:

E (XA) = 0.5*XA1 + 0.5*XA

2 = 0.5* (2000) + 0.5*(1000) = $1,500.

E (XB) = 0.99*XB1 + 0.01*XB

2 = 0.99*(1510) + 0.01*(510)= $1,500.

The Variances are σ2A = q1*(X1

A – E(XA) )2 + q2*(X2A – (XA) )2 =

= 0.5*( 2000 – 1500)2 + 0.5*( 1000 – 1500 )2 = $250,000.

The Standard deviation is σA = 500.

σ2B = q1*( X1

B – E (XB) )2 + q2*( X2B – E(XB) )2 =

= 0.99*(1510 – 1500) 2 + 0.01* (510 – 1500)2 = $9,900.

The standard deviation for this job is σB = $ 99.5.

Thus, job A is much riskier than job B.

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Demonstration 12-2

The manager of XYZ company is introducing a new product that will yield $1,000 in

profits if the economy does not go into recession. If the recession occurs, then the

company will lose $4,000. If economists project that there is a 10% chance the

economy will go into a recession how risky is the introduction of the new product.

Answer

E(Profit) = 0.9*(1,000) + 0.1*(-4,000) = $500

σ2 = 0.9*(1,000 – 500)2 + 0.1*(-4,000 -500)2

σ2 = 0.9*(500)2 + 0.1*(-4,500)2

σ2 = 2,250,000

σ2 = √2,250,000 = $1,500

Uncertainty and Consumer Behavior

We will see how the presence of uncertainty affects both consumers and managers.

Risk Aversion

People may have different tastes for the same set of risky prospects, and thus they exhibit

different preferences for these prospects. Suppose F represents the uncertain prospects

associated with buying 100 shares of stock F, and G is the uncertain prospect of buying

100 shares of stock G. Because attitude and preferences among consumers differ, a risk

averse person prefers a sure amount of $M to a risky prospect with an expected value of

$M.

A risk-loving individual prefers a risky prospect with an expected value of $M to the

same amount of $M.

A risk neutral individual is indifferent between a risky prospect with an expected value of

$M and a sure amount of $M.

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Managerial Decisions with Risk Averse Consumers

Here are some examples of how risk aversion affects managerial decisions.

Product Quality

The analysis of risk can be used to show how uncertainty about product quality affect

consumer’s behavior and how managers can deal with it. There is risk associated with

buying new products. If risk averse individual is faced with the new product Y and the

regular product X and views these two products to be of the same quality, he will buy the

regular product X.

The manager has two primary tactics to induce the risk averse consumers to buy the new

product.

1. The manager may lower the price of the new products. For example, he can give

free samples (where the price is zero).

2. The manager can use comparative advertising to convince the consumer that the

new product is of better quality than the regular brand. If consumers are

convinced they may buy the new product.

Chain Stores

Risk aversion may explain that it is in a firm’s best interest to be part of a chain store

instead of remaining independent. The type and quality of products offered by national

chains are certain. Example, imagine a family driving through a small town and looking

for a restaurant to eat. In this town there are two restaurants to eat: a local diner and a

national hamburger chain. The family is uncertain about quality of the food of the diner,

but it is more sure about the food of the national chain. It would choose to eat at the

national chain.

The same applies to retailing outlets, gas stations, etc.

Insurance

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People buy insurance on their automobiles and homes. They give a small amount of

money to avoid loosing a huge sum if a catastrophic event occurs. Here buying insurance

represents choosing the “sure thing” over the risky prospect of a catastrophic event.

Uncertainty and the Firm

As uncertainty affects consumer behavior and managers must account for that uncertainty

also effects the managers’ input/output decisions.

Risk Aversion (and the firm)

Just as consumers have different preferences regarding different risky prospects, so does

the manager of the firm.

1. A manager who is risk neutral is interested in maximizing expected profits. The

variance of profits does not have an effect on his/her decision.

2. A manager is risk averse if he/she prefers the project that has a lower risk with a

lower expected value to the project that has higher risk and expected value.

When a manager faces a decision to choose among risky projects, it is important to

evaluate the risks and expected returns of the projects and then to document this

evaluation. Risky projects may have bad outcomes and that could get the managers fired.

The manager is not likely to get fired if he/she provide evidence that based on the

available information the decision was sound. A convenient way to do this is to use mean

variance analysis as shown below.

Demonstration 12-3

Suppose a risk averse manager is considering two options: expanding the market

for bologna and expanding the market for caviar. Suppose that there is a 90%

chance of an economic boom and 10% of a recession. Suppose also, there is a risk

free alternative (say, a treasury bill). The manager can have a joint project that

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combines bologna and caviar. The four projects and their payoffs during boom and

recession are given below. What should the manager do? Why?

Project Boom (90%) Recession (10%) Mean Standard Deviation

Bologna -$10,000 $12,000 -$7,800 6,600

Caviar 20,000 -8,000 17,200 8,400

Joint 10,000 4,000 9,400 1,800

T-Bill 3,000 3,000 3,000 0

Answer

The manager should not invest in T-bill because the lowest payoff for joint project is

greater than 3,000 which is the payoff for T-bills. Moreover, risk averse and risk

neutral managers will not invest in a project with negative expected payoff. This will

eliminate the bologna project. This will leave the manager with two projects: the

caviar and joint projects. Which of these two projects, which have different

expected values and risk, would the manager invest? It all depends on his/her

preferences toward risk.

The payoffs associated with the joint project above reveal the importance of

diversification. By investing in multiple projects the manager, can reduce the

systematic risk.

Diversification also reveals why shareholders are risk neutral. They want managers

to maximize the value of the firm without a regard to risk. This is because

shareholders diversify in different stocks. We know that diversification diversifies

systematic risk away.

Producer Search

Producers search for low prices of inputs when there is uncertainty regarding input prices,

firms employ optimal search strategies.

Demonstration 12-4

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Profit Maximization Under Uncertainty

Under certainty profit is

π = PQ – C (Q)

Then first profit maximization rule is

MR = MC

And under perfect competition this rule is:

P = MC

which is solved for Q*.

Under uncertainty, demand is uncertain and thus total revenue is uncertain. The firm

maximizes expected profit.

E π = EP*Q – C(Q)

where EP = q1 * p1 + q2 * p2 +….. is the expected price and qi is the expected price and qi

is the ith probability. Then first-profit maximization rule is EMR = M.

Where EMR is the expected marginal revenue under perfect competition, this rule is EP =

MC Which can be solved for Q0?

Demonstration 12-5

Suppose the perfectly competitive firm Appleway must determine how much to

produce before the actual price is unknown. The firm knows the expected price.

There is a 10% probability that the market price is $2 and 70% that the market

price $1 when the apple juice hits. Then the expected price is

EP = 0.1 * ($2) + 0.7 * (1) = 0.6 + 0.70 = $1.30

Suppose the cost function is given by

C = 200 + 0.0005 Q2

How much should this firm produce to maximize expected profit? What are the

expected profits of this firm?

Answer

Set EP = MC

Then $1.30 = 0.001Q

Q* = 1,300 gallons

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Expected Profit = EP*Q – 200 – 0.0005Q2

= (1.30) (1,300) – 200 – 0.0005 (1,300)2

= 1,690 – 200 – 0.0005 (1,300)2

= $645

Uncertainty and the Market

The presence of uncertainty may have a strong impact on the ability of the markets to

efficiently allocate resources because it creates problems with the market.

Asymmetric Information

A situation that exists when some people in the market has more information than other.

The people with the least information may choose not to participate in the market. For

example, if a person has a box and she knows it has $10. These people who do not have

this information will not accept to buy the box from her because she will not sell the box

at a loss.

In the stock market when some traders have insider information and others do not, there

is a asymmetric information. In extreme cases asymmetric information can lead to the

destruction of stock markets if asymmetric information continues to exist.

Asymmetric information between consumers and the firm can affect the firm’s profit.

Suppose the firm invests heavily and produces a superior product. If the consumer does

not have this information, it will not buy this superior product.

Asymmetric information may also affect many managerial decisions including hiring

workers (workers know this abilities better then managers do), issuing credit to

consumers (consumers know their credit abilities). This is why companies spend a lot of

money checking on individuals and their backgrounds.

There are two specific manifestations (types) of asymmetric information: adverse

selection and moral hazard. Those two concepts are difficult to distinguish.

Adverse selection arises when an individual has hidden characteristics

(characteristics that she knows but unknown by the other proxy in an economic

transaction). For example, the job applicant knows his own abilities, but the

employee does not. The workers abilities thus reflect a hidden characteristic.

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Moral hazard generally takes place when one proxy takes hidden actions (actions

that it knows another party cannot observe). For example, if a manager cannot

monitor (observe hidden action) then the workers effort represent a hidden action.

Adverse Selection

A situation where individuals have hidden characteristics and in a selection process

results in a pool of individuals with economically undesirable characteristics.

Example1: An industry with firms that allow 5 days of paid sick leave. One firm decides

to allow 10 days. If the workers have hidden characteristics (the firm cannot distinguish

between healthy and unhealthy workers). This increase in the monthly sick leave will

mostly attract unhealthy workers. Healthy workers are not interested in this policy.

Example2: A pool of poor drivers may have adverse selection. This pool includes two

types of poor drivers: those who have bad driving habits, and those who had a string of

bad luck. If the insurance companies increase the insurance premium on this pool, only

drivers with bad habits would accept to pay the higher premium but those who had bad

luck won’t accept to pay the higher premium. Then the insurance company will end up

with the bad drivers and in this selection there is adverse impact. The insurance company

should not increase the premium but should refuse to insure the bad drivers. There are

insurance companies who specialize in bad drivers and they ask them to pay a high

premium.

Moral Hazard

A situation where one party to a contract takes a hidden action that will benefit him/her at the expense of another party is called a moral hazard.

Example 1: The principal agent problem. In this case the principal (the owner) offers the

agent (the manager) a contract (a salary + benefits) to do certain tasks. Since the manager

will receive the salary, and his/her behavior is unobservable by the owner, he/she has

incentives to work less (hidden actions). The reduced effort may result in reducing profit.

One way to mitigate this moral hazard by the owner is to monitor the behavior of the

manager (taking away the hidden action). Another way is to compensate the manager

based on his performance.

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Example 2: Health insurance: Insurance companies are vulnerable to the moral hazard

problem. The probability of a loss depends on the hidden efforts expended by the insured

to avoid the loss. This moral hazard exists. When individuals are fully insured they have

a reduced incentive to put forth effort to avoid a loss.

Signaling and Screening

Managers and other market participants can use signaling and screening to mitigate some

of the problems that arise when one party to a transaction has hidden characteristics.

Signaling is an attempt by an informed party to send an observable indicator of his/her

hidden characteristic to an uninformed party. Thus signal must be observable.

Example of “observable indicators” in the product markets are that companies send

signals such as money back guarantees, free trial, labeling that indicates the product has

won a “special award” or the manufacturer has been in business since say, 1933.

In the labor markets, the signal takes the form that the job applicant graduated from a

certain prestigious school. If the productivity of the job seeker is unobservable that will

lead to lower salaries for both the productive and unproductive workers. In this case

productive workers should find ways to provide information to the manager that reveals

that they are indeed productive. How can productive workers send the right signals to the

manager that they are productive? Talk is cheap. Unproductive workers should not easily

mimic the signal.

Screening

The uninformed party can use screening to reduce the effects of hidden characteristics.

Example: In this job market, the manager can use a self-selection device to distinguish

between peoples’ skills.

Example: Two people with different characteristics are applying for a job in a company.

One applicant is an administrator and the other is a salesman. The manager can use a self-

selection device to fit the two workers to the right job. The device may stipulate that the

manager’s job will pay $20,000 and the salesman job pays 10% of total sales. The second

worker who identifies his characteristic to be a salesman he will ask for the salesman job.

He knows he is a salesman and can generate a million dollars in sales. Then this salesman

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compensation is $100,000 (10%*1Mn), which is higher than the $20,000 job. The

manager knows that his ability does not fit the salesman job. He will go for the

administrative job.-

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