Thursday, October 13: Short and Long Run...

Amherst College Department of Economics Economics 54 Fall 2005

Thursday, October 13: Short and Long Run Equilibria Equilibrium in the Short Run

The equilibrium price and quantity are determined by the market demand and short run market supply curves.

Short Run Market Supply Curve

The short run market supply curve is the horizontal sum of each individual firm’s short run supply curve.

Individual Firm’s Short Run Supply Curve

Profit Maximization: MR = SMC Marginal Revenue: The change in the firm’s total revenue resulting from a one unit change in the quantity of output produced. Short Run Marginal Cost: The change in the firm’s total cost resulting in a one unit change in the quantity of output produced. Profit = TR − STC Use calculus to find the profit maximizing quantity of output:

dProfit

dq = dTRdq −

dSTCdq = 0

dTRdq =

dSTCdq

MR = SMC

Short Run Marginal Cost Curve Shape: The short run marginal cost curve is upward sloping.

Marginal Revenue and Perfect Competition: MR = P

MR = dTRdq =

d(P×q)dq since TR = P × q

= P + q dPdq

A perfectly competitive industry is composed of a large number of small firms. A single firm’s production decisions will not significantly affect the price. Consequently, each firm is a price taker; each firm takes the price as a given, as a constant:

q dPdq = 0;

therefore, MR = P q

If P=1.00

If P=2.00

SMC

MR=2.00

MR=1.00

Profit maximizing quantityif P=1.00 if P=2.00

P

Q

SS

D

P*

Q*

Market

2

Shutdown in the Short Run: P < SAVC It is advantageous for a firm to shut down in the short run whenever the price is less than short run average variable costs.

Individual Firm’s Short Run Supply Curve: The firm’s marginal cost curve as long as price exceeds average variable cost; if price is less than short run average variable costs, the firm shuts down and produces nothing.

Short Run Average Total Cost Curve • Shape: The short run average total cost curve is

U-shaped. Short Run Average Total Cost Curve and Short Run Marginal Cost Curve

• Location: The short run average total cost curve intersects the marginal cost curve at minimum short run average total cost.

Profits in the Short Run

Profit = TR − STC

= P×q − SATC×q since TR = P×q and SATC = STC

q

= (P − SATC) × q

Now, it is easy to determine sign of profit by comparing price and short run average total cost: P < SATC P = SATC P > SATC ↓ ↓ ↓ P − SATC < 0 P − SATC = 0 P − SATC > 0 ↓ ↓ ↓ Profit < 0 Profit = 0 Profit > 0

Market Equilibrium, the Typical Firm, and Its Profit

Since P < SATC at the profit maximizing quantity of output, profit is negative. Questions: How will the typical firm react in the long run?

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*Q*

Market Typical Firm

SATC

As a consequence of perfect competition, the horizontal axes have different scales.

q

MC

ATC

MC, ATCq

SMC SS

P

if P<SAVC

if P>SAVCSAVC

3

Long Run Cost Curves The shape of a firm’s long run cost curves depends on returns to scale: Increasing returns Constant returns Decreasing returns to scale to scale to scale ↓ ↓ ↓ LATC curve is LATC curve is LATC curve is downward sloping horizontal upward sloping

Short Run and Long Run Average Total Costs

A short run average total cost curve (SATC) lies above the long run average total cost curve (LATC) except for a single point at which they just touch. The point where they touch represents the quantity at which the firm is able to use its cost minimizing combination of capital and labor. There is one short run average total cost curve for each amount of capital.

LATCSMC and SATC for different amounts of K

q Short Run Versus Long Run

Short Run Long Run ↓ ↓ Firms can vary labor, but Firms can vary not capital; capital is fixed. both labor and capital. Firms cannot enter or exit an industry. Firms can enter or exit an industry.

Unlike in the short run, in the long run firms can

• vary the amount of capital it uses; • enter or exit the industry.

LATC

q

ConstantIncreasing Decreasing

Returns To Scale

4

The Long Run: Entry, Exit, and the Sign of Profit Claim: As a consequence of the fact that economic profit includes both explicit and implicit costs, the sign of profits is key when deciding whether firms will enter or exit an industry in the long run. To justify this claim we shall review the argument given in your introductory course. Consider an owner operated firm. What are the explicit and implicit costs of such a firm? Explicit costs are the costs that come to mind immediately. The wages and salaries the firm pays to employees, the firm’s electricity bill, telephone bill, heating bill, etc. In general, the explicit costs are what the owner of the firm must pay out to others. Implicit costs are more subtle; implicit costs are opportunity costs. In the case of an owner operated firm, implicit costs include the salary that the owner could earn if he/she did not operate the firm and worked for someone else.

Total Costs = Explicit Costs + Implicit Costs ↓ ↓ What the owner Income the owner pays to others could earn if he/she worked for someone else

Now, let us do a little algebra: Profits = Total Revenues − Total Costs = Total Revenues − (Explicit Costs + Implicit Costs) = (Total Revenues − Explicit Costs) − Implicit Costs Income the owner Income the owner actually earns when he/she could earn if he/she operates the firm worked for someone else

Total revenues less explicit costs equals the income the owner takes home when he/she operates the firm. Implicit costs equal the income the owner could earn if he/she worked for someone else. Profit compares the income the owner actually earns when he/she operates the firm with the income he/she would earn if he/she worked for someone else:

Profit < 0: Exit in the long run - income owner earns when operating the firm is less than the income he/she could earn by working for someone else. There is an incentive for firm owners in this industry to exit in the long run.

Profit > 0: Entry in the long run – income owners of firm in this industry exceeds what they could earn by working for someone else. This lures entrepreneurs into the industry. There is an incentive for entrepreneurs to enter the industry.

Profit = 0: Long run equilibrium – there is no incentive for exit or entry.

Long Run Behavior: Price and Minimum Long Run Average Total Cost Claim: To determine what happen to an industry in the long run we compare the price and minimum long run average total cost:

If Price < Minimum long run average total cost: exit occurs in the long run causing the price to rise.

If Price > Minimum long run average total cost: entry occurs in the long run causing the price to fall.

If Price = Minimum long run average total cost: a long run equilibrium exists.

5

First, consider the case in which the price falls short of minimum long run average total cost.

Note that since the price falls short of minimum long run average total cost, the long run average total cost curve is always above the price:

First, note that the firm is presently earning negative profits because short run average total cost exceeds the price:

Profits = TR − STC = (P − SATC)×q This is not the worst part of the news, however. Recall how the short and long run average total cost curves are related: a short cost average total cost curve lies above the long run average total cost curve except at a single level of production when the short run curve touches the long run curve. Regardless of how the firm adjusts the amount of capital it uses in the long run, the short run average total cost curve will lie above the price. Consequently, the firm will always earn negative profits because

Profits = TR − STC = (P − SATC)×q Since P is always less than SATC profits are always negative. Negative profits cause firms to exit the industry. As firms exit the short run market supply curve shifts to the left causing the price to rise:

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*

NB: As a consequence of perfect competition, the scale of the horizontal axes are different.

Q*

Market Typical Firm

LATC

P

Q

SS

D

P* P*

SATCSMC

MR=P*

q


Market Typical Firm

LATC

MR=P**P**P**

SS’

6

When price falls short of minimum long run average total cost the following occurs in the long run:

• regardless of how firms in the industry adjust the amount of capital they use, they will earn negative profits;

• the negative profits causes firms to exit in the long run; • the exit of firms shifts the short run market supply curve to the left

causing the equilibrium price to rise toward minimum long run average total cost.

Second, consider the case in which the price exceeds minimum long run average total cost.

Note that since the price exceeds minimum long run average total cost, the long run average total cost curve dips below the price:

Recall that the profits earned by the firm at the present time depend on the price and short run average total cost. In the above diagram, price is less than short run average total cost, profits are negative. Although at the present time the firm is incurring losses, the long run outlook is anything but bleak. Remember that unlike the short run, in the long run the firm can

• vary the amount of capital it uses; • enter or exit the industry.

Claim: by changing the amount of capital, the firm can move to a short run average total cost curve that dips below the price. To justify this claim recall how the short run and long run curves are related: a short cost average total cost curve lies above the long run average total cost curve except at a single level of production when the short run curve touches the long run curve. Since the long run average total cost curve dips below the price the firm can change the amount of capital it uses so that its short run average cost curve dips below the price.

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC

7

Since the short run average total cost curve dips below the price, it is now possible for the firm to earn positive profits:

Profit = TR − STC = (P − SATC)×q Question: How much output will the firm now produce? Answer: The quantity of output that maximizes profit. The profit maximizing quantity of output is determined by marginal revenue and short run marginal cost. Note that the short run marginal cost curve and the short run average total cost curve intersect at minimum short run average total cost. q** is the profit maximizing level of output. At the profit maximizing level of output the firm is earning positive profits because the price exceeds short run average total costs. Where do we stand now? When the price exceeds minimum long run average total cost, a firm that is already in the industry can vary the amount of capital it uses to enable it to earn a positive profit. What will happen next? Entrepreneurs will recognize that there are positive profits to be earned in this industry. Consequently entry will occur. As firms enter the short run market supply curve shifts to the right because the short run market supply curve is the horizontal sum of each individual firm’s short run supply curve. The rightward shift of the short run market supply curve causes the equilibrium price to fall.

P

Q

SS

D

P* P*

SATC

SMC

MR=P*

q


Market Typical Firm

LATC

q**

P

Q

SS

D

P* P*

SATC

SMC

MR=P*

q


Market Typical Firm

LATC

SS’

MR=P**P** P**

8

When price exceeds minimum long run average total cost the following occurs in the long run:

• firms already in the industry will adjust the amount of capital they use so that they can earn positive profits;

• the positive profits lead to the entry of new firms in the long run; • the entry of new firms shifts the short run market supply curve to the

right causing the equilibrium price to fall toward minimum long run average total cost.

Long run equilibrium

Summary: what have we shown thus far? • If the price falls short of minimum long run average total costs, firms will

exit the industry in the long run causing the price to rise toward minimum long run average total cost.

• If the price exceeds minimum long run average total costs, new firms will enter the industry in the long run causing the price to fall toward minimum long run average total cost.

Since the price

• rises in the long run whenever the price is less than minimum long run average total cost,

and • falls in the long run whenever the price is greater than minimum long

run average total cost a long run equilibrium requires that the price to equal minimum long run average total cost. What amount of capital will the firm now use? Recall how short and long run average total cost curves are related: a short run average total cost curve lies above the long run average total cost curve except at a single level of production when the short run curve touches the long run curve. The best the firm can do is to adjust its amount of capital so that it earns a profit of 0:

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC

9

Note that • it is impossible for the firm to adjust the amount of capital it uses so

as to earn a positive profit; • at the present time even though the firm is maximizing its profit, it is

earning a profit of 0 because price equals short run average total cost.

Consequently, there is no incentive for new firms to enter the industry and also no incentive for firms already in the industry to exit. We have a long run equilibrium. Note that when a long run equilibrium is achieved, the price equals minimum long run average total cost.

Long Run Supply Curve The long run supply curve provides the answers to the following series of hypothetical questions:

If the price were _____, how much output would firms produce in the long run when the industry had achieved a long run equilibrium; that is, how much output would be produced after the industry achieves a long run equilibrium?

Constructing the long run supply curve - constant cost industry First, begin with at a long run equilibrium as we described above. This provides us with one point on the long run supply curve:

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC

Point on LR Supply Curve

10

How do we find another point on the long run supply curve? Suppose that the market demand curve shifts to the right:

The equilibrium price increases for P* to P** which causes the firm’s marginal revenue curve to shift up from MR to MR’. The firm’s profit maximizing quantity of output increases from q* to q**. Note that the price now exceeds short run average total cost. The firm is earning positive profits. What will happen in the long run? The positive profits lure entrepreneurs into the industry; that is, firms will enter the industry. Entry shifts the short run market supply curve to the right and the equilibrium price falls.

So let us review what is happening here: Demand Curve Shifts Rightward ↓ Price Increases ↓ Price > LATC ↓ Profits Can Earn Positive Profits

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC


D’

P** P**

q**

MR’

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*

NB: As a conequence of perfect competition, the scale of the horizontal axes are different.

Q*

Market Typical Firm

LATC


D’

P** P**

q**

MR’

SS’

MR’’P***

11

Consequently, Firms Enter ↓ SS Shifts Rightward ↓ Price Falls This process will continue until the price returns to minimum long run average total cost when the new long run equilibrium is established. So it looks like the price will return to its original level, P*:

It looks like that the long run supply curve is just a horizontal line: This is true in a constant cost industry. In the analysis provided above we implicitly assume that the industry exhibited constant cost; that is, we implicitly assumed that as firms entered, the costs of the firms in the industry were not affected. Consequently, as firms entered, the cost curves of the firms in the industry did not shift. Summary: the long run supply curve is horizontal in a constant cost industry. The price must equal minimum long run average total cost to be in long run equilibrium. Since entry and exit does not cause the cost curves to shift in a constant cost industry, entry and exit will not change minimum long run average total cost. Consequently, the long run supply curve must be horizontal in a constant cost industry.

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC


D’

P**

q**

MR’

SS’

MR’’P***

SS’’

12

Constructing the long run supply curve - increasing cost industry Some industries do not exhibit constant costs, however. As entry occurs often costs are driven up. For example, if new firms enter the beer industry, the demand for brew masters will increase which in turn increases the wage paid to brew masters. Higher wages for brew masters increases the costs of beer firms. These industries are called increasing cost industries. We shall follow the same strategy as before to construct the long run supply curve of an increasing cost industry. Suppose that the demand curve shifts to the right.

Just as in the constant cost case, price rises and firms can earn positive profits. Demand Curve Shifts Rightward ↓ Price Increases ↓ Price > LATC ↓ Firms Can Earn Positive Profits Again, just as before firms enter and the short run supply curve shifts to the right and the price falls. But in an increasing cost industry something else also begins to happen. As firms enter the costs of the firms are driven up and consequently, the long run average total cost curve shifts up Firms Enter ⎯⎯→ LATC Shifts Up ↓ SS Shifts Rightward ↓ ↓ Price Falls Minimum LATC Increases

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*


Q*

Market Typical Firm

LATC


D’

P** P**

q**

MR’

13

So, while the price is falling, minimum long run average total cost is increasing. Recall that a long run equilibrium is achieved whenever price equals minimum long run average total cost. The industry will now achieve a long run equilibrium “sooner” and the price will be greater than P*. Therefore in an increasing cost industry, the long run supply curve will be upward sloping.

P

Q

SS

D

P* P*

SATCSMC

MR=P*

qq*

NB: As a conequence of perfect competition, the scale of the horizontal axes are different.

Q*

Market Typical Firm

LATC


D’

SS’

MR’’P*** P**LS


Tuesday, October 18: Consumer and Producer Surplus Consumer and Producer Surplus

Economists use consumer and producer surplus to obtain quantitative estimates of how consumers and firms will be affected by an event. Consumer surplus measures the net benefit households receive from the purchase and consumption of a good; producer surplus the net benefit firms receive from the production and sale of a good. Consequently, changes in consumer and producer surplus indicate how households and firms are affected.

Consumer Surplus Consumer surplus measures the net benefit households receive from the purchase and consumption of a good. Geometrically, consumer surplus is the area beneath the market demand curve that lies above the price. To justify this we shall first argue that the height of the market demand curve reflects the value that households place upon the consumption of each unit of output. To do so, consider the six students who are enrolled in intermediate microeconomics and how much each would be willing to pay for an A in the course:

Potential Buyer Willingness to Pay Andy $225 Kate 175 Dan 140 Liz 75 Meg 60 Darce 25

Willingness to pay equals the maximum amount that a potential buyer will pay for an A; that is, willingness to pay is the benefit that a potential buyer would enjoy from consuming an A. Andy is willing to pay $225 for an A: if the price exceeds $225, Andy would not buy an A, but if the price were $225 or less, Andy would purchase one. Similarly, Kate would not buy an A if the price exceeded $175, but would buy one if the price were $175 or less. We can construct the market demand curve for A’s from information about each potential buyer’s willingness to pay.

Recall that a market demand curve for A’s answers a long series of hypothetical questions:

If the price were _____, how many consumers would purchase an A? If the price exceeds $225, no one would buy an A:

If the price exceeds $225, the quantity demanded would equal 0. If the price equals $225, however, Andy is now willing to buy an A:

If the price equals $225, the quantity demanded would equal 1.

P

Q

P*

Q*

D

2

If the price is less than $225, but exceeds $175, Andy is the only consumer who would buy an A:

If the price is less than $250, but exceeds $175, the quantity demanded would equal 1.

If the price equals $175, not only Andy, but also Kate is now willing to buy an A:

If the price equals $175, the quantity demanded would equal 2. By continuing this logic, we can produce the following table:

Quantity Demanded If Price >250 0 No one buys If Price ≤ 250 and > 175 1 Only Andy buys If Price ≤ 175 and > 140 2 Andy and Kate buy If Price ≤ 140 and > 75 3 Andy, Kate, and Dan buy If Price ≤ 75 and > 60 4 Andy, Kate, Dan, and Liz buy If Price ≤ 60 and > 25 5 Andy, Kate, Dan, Liz, and Meg buy If Price ≤ 25 6 All buy

Using this table, we can draw the market demand curve: We have now justified our claim that the height of the demand curve reflects the benefits households enjoy from the consumption of each unit of the good. Andy receives $225 of benefits from the consumption of an A, Kate $175 of benefits, Dan $140 of benefits, etc. Next, we want to calculate the net benefit that each consumer receives from the purchase and consumption of the good. Clearly, the net benefit depends on the price of the good. For each consumer who purchases the good, the net benefit or the surplus equals the benefit he/she enjoys less the price he/she must pay.

50

100

150

200

250

1 2 3 4 5 6Quantity of A’s

Price of an AAndy

Kate Dan Liz Meg Darce

3

If the price were $150, Andy and Kate would buy an A. Andy would enjoy a surplus of $75 ($225-$150); Kate a surplus of $25 ($175-$150). If the price were $150, total consumer surplus would equal $100, the sum of Andy’s and Kate’s surplus. Geometrically, the total net benefit, the consumer surplus, equals the area beneath the market demand curve that lies above the price, $150. If the price were $100, Andy, Kate, and Dan would buy an A. Andy would enjoy a surplus of $125; Kate a surplus of $75; and Dan a surplus of $40. If the price were $100, total consumer surplus would equal $240. Geometrically, the total net benefit, the consumer surplus, equals the area beneath the market demand curve that lies above the price, $100. If the price were $50, Andy, Kate, Dan, Liz, and Meg would buy an A. Andy would enjoy a surplus of $175; Kate a surplus of $125; Dan a surplus of $90; Liz a surplus of $25; Meg a surplus of $10. If the price were $50, total consumer surplus would equal $425. Geometrically, the total net benefit, the consumer surplus, equals the area beneath the market demand curve that lies above the price, $50. Surplus if Price equals Buyer Willingness to Pay $150 $100 $50 Andy $225 $75 $125 $175 Kate 175 25 75 125 Dan 140 - 40 90 Liz 75 - - 25 Meg 60 - - 10 Darce 25 - - - Consumer Surplus $100 $240 $425

50

100

150

200

250


Price of an AAndy: $75

Kate: $25

50

100

150

200

250


Price of an AAndy

KateDan

50

100

150

200

250


Price of an AAndy: $125

Kate: $75Dan: $40

Liz: $25Meg: $10

4

Summary of Consumer Surplus

• The height of the demand curve reflects the benefit households enjoy from consuming each unit of the good.

• The benefit households enjoy from consuming each unit of the good (the height of the demand curve) less the expense households incur to purchase each unit (the price) is the surplus households enjoy.

• Geometrically, consumer surplus equals the area beneath the demand curve that lies above the price.

• Consumer surplus reflects the (net) benefit households enjoy from purchasing and consuming the good.

Producer Surplus - Short Run

Producer surplus in the short run reflects the net benefit firms enjoy from producing and selling the good. Geometrically, producer surplus is the area above the short run market supply curve that lies below the price. To justify this we shall first argue that the height of the market supply curve reflects the costs firms incur from producing each unit of output. Recall that

• the short run market supply curve is the horizontal sum of each individual firm’s short run supply curve;

• geometrically, an individual firm’s short run supply curve is its short run marginal cost curve.

P

Q

P*

Q*

SS

P

QD

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

QQ

PPShort Run Market Supply for BeerFirm A Firm B

SSSSA SSB

SMCA SMCB

5

If the price were $1.50, suppose that Firm A produced 800 units and Firm B 600 units; consequently, the quantity supplied is 1400 units.

How much does it cost to produce the 1400th unit? Claim: it costs $1.50 to produce the 1400th unit, an amount just equal to the height of the short run market supply curve. We can justify this claim easily. One of the firms must produce the 1400th unit; suppose that Firm B produces the 1400th unit. Note that the 1400th unit in the market is the 600th unit produced by Firm B. How much does it cost Firm B to produce its 600th unit? The answer to this question is straightforward: an amount just equal to Firm B’s short run marginal cost:

Verbal definition: Short run marginal cost equals the change in the firm’s short run total cost resulting from a one unit change in output.

NB: Marginal cost only reflects variable cost because fixed costs never change. Consequently, the short run supply curve only reflects variable costs.

What is Firm B’s short run marginal cost for producing its 600th unit. Look at Firm B’s short run marginal cost curve on the diagram above. Its short run marginal cost of producing the 600th unit is just $1.50. We have shown what we set out to prove. We have justified our claim: the height of the market supply curve reflects the costs firms incur from producing each unit of output. The net benefit or surplus a firm enjoys from the production and sale of a unit equals the benefit it derives less the costs it incurs. The benefit derived from the sale of a unit just equals the price. Therefore, the gap between the price and the short run market supply curve reflects the net benefit firms enjoy from the production and sale of a unit of output. In total, producer surplus is just the area above the short run market supply curve that lies above the price. Summary of Producer Surplus in the Surplus

• The height of the short run supply curve reflects the costs firms incur in the short run from producing each unit of output.

• The revenues firms receive from selling each unit of output, the price, less the costs firms incur in the short run from producing each unit, the height of the supply curve, is the surplus firms enjoy.

• Geometrically, producer surplus in the short run equals the area above the short run supply curve that lies below the price.

• Producer surplus reflects the (net) benefit firms enjoy from producing and selling the good.

P

QD

If P = 1.50

QQ

PPShort Run Market Supply for BeerFirm A Firm B

SSSSA SSB

SMCA SMCB

800 600 1400

1.50 1.50

P*

Q*

P

Q

SS

6

Producer Surplus and Profits First, review the definition of profits; profit equals total revenue less short run total cost:

Profit = TR − STC Recall that in the short run we can divide total costs into two components: short run fixed costs and short run variable costs; that is, short run total costs equal short run variable costs plus short run fixed costs

Profit = TR − (SVC + SFC) Simplifying,

Profit = TR − SVC − SFC Next, recall that the height of the short run market supply curve reflects the costs of producing each unit of output. Does the height of the short run market supply curve reflect fixed or variable costs? Clearly, the answer is variable costs; the height of the supply curve tells us by how much costs rise when the firm produces each unit of output. Therefore, the area beneath the short run supply curve reflects the short run variable costs of all firms:

Area Beneath the Supply Curve = SVC of all firms What does P*×Q* equal? P*×Q* equals the total revenues collected by all firms. Focusing attention on the diagram to the right, recall that producer surplus equals the area above the short run supply curve that lies below the price. Geometrically, producer surplus is just the area of the P*×Q* rectangle less the area beneath the short run supply curve:

Area Beneath Producer Surplus = P*×Q* − the Supply Curve TR SVC = Collected by − Incurred by All Firms All Firms

Now, recall that

Profit = TR − STC = TR − SVC − SFC

Hence, Profit TR SVC SFC Earned by = Collected by − Incurred by − Incurred by All Firms All Firms All Firms All Firms

Substituting in for producer surplus:

Profit SFC Earned by = Producer Surplus − Incurred by All Firms All Firms

P

Q

P*

Q*

SS

7

Welfare Maximizing Quantity of Output From the perspective of society as a whole, how much output should be produced? Economists take two approaches to answer this question:

• consumer and producer surplus approach; • Pareto efficiency approach.

Both approaches end up with the same conclusion, but they use different means to get there. We shall discuss the “surplus” approach here; in the last two weeks of the course, we shall consider the Pareto approach. Equilibrium in the Market

In equilibrium, the quantity demanded equals the quantity supplied. We shall argue that the equilibrium quantity maximizes the welfare of society as a whole. To do so, recall that the height of the market demand curve and short run market supply curves provides important information:

• The height of the market demand curve reflects the benefit households enjoy from consuming each unit of the good.

• The height of the short run market supply curve reflects the costs firms incur from producing each unit of the good.

Equilibrium Quantity Is the Welfare Maximizing Quantity

We shall argue that the equilibrium quantity is the welfare maximizing quantity in two steps:

• if less than the equilibrium quantity were produced, society as a whole would be made better off by producing more;

• if more than the equilibrium quantity were produced, society as a whole would be made better off by producing less.

First, suppose that less than the equilibrium quantity were produced. Consider all the units between this quantity and the equilibrium quantity. Since the height of the market demand curve exceeds the height of the short run market supply curve, the benefits household would receive from the consumption of each of these units exceeds the costs firm would incur from producing them. From the standpoint of society as a whole these units should be produced. In other words, if less than the equilibrium quantity were produced, society as a whole would be made better off by producing more. Second, suppose that more than the equilibrium quantity were produced. Consider all the units between this quantity and the equilibrium quantity. Since the height of the short run market supply curve exceeds the height of the market demand curve, the costs firms are incurring from the production of each of these units exceeds the benefits households are receiving from consuming them. From the

P

Q

P*

Q*

D

SS

P

QQ*

D

SS

P

QQ*

D

SS

8

standpoint of society as a whole these units should not be produced. In other words, if more than the equilibrium quantity were produced, society as a whole would be made better off by producing less.

Producer Surplus - Long Run Producer surplus in the long run reflects the returns earned by the inputs (labor, capital, etc.). Geometrically, producer surplus in the long run is the area above the long run supply curve that lies below the price. To justify this we shall review what we learned about the long run supply curve. Recall the shape of the long run supply curve depends on what happens to input prices when entry or exit occurs. Constant cost industry: horizontal long run supply curve

In a constant cost industry, the cost of inputs is not affected by entry or exit; that is, entry and exit does not affect the returns to inputs. Question: how will an increase in demand affect long run producer surplus? For a constant cost industry, long run producer surplus is unaffected. Since the long run supply curve is horizontal producer surplus is 0 before and after the increase in demand. Constant Cost Industry ↓ Entry Does Not Affect Input Prices ↓ Returns to Inputs Unaffected ↓ Long Run Producer Surplus Unaffected

P

Q

P*

Q*

LS

P

Q

P*

Q*

LS

Q**

D

D’

9

Increasing cost industry: upward sloping long run supply curve

The cost of inputs is driven up by entry; consequently, entry increases the returns to inputs. Question: how will an increase in demand affect long run producer surplus? For an increasing cost industry, long run producer surplus increases. Since the long run supply curve is upward sloping, an increase in demand will increase producer surplus. Increasing Cost Industry ↓ Entry Increases Input Prices ↓ Returns to Inputs Increases ↓ Long Run Producer Surplus Increases

Supply Elasticities

Recall that when we calculate elasticites we are always concerned with percent changes. Short Run Price Elasticity of Supply (E SS

Q,P )

Verbal Definition: E SSQ,P reflects how sensitive the short run quantity supplied of a

good is to the good’s price.

E SSQ,P = the percent change in the short run quantity supplied resulting

from a 1 percent change in the good’s price

= percent change in short run quantity supplieded

percent change in the good's price

Substituting in the expressions for the percent changes:

E SSQ,P =

ΔQQ × 100

ΔPP × 100

Simplfying,

E SSQ,P =

ΔQQ

ΔPP

= ΔQΔP

PQ

Taking limits as ΔP approaches 0:

E SSQ,P =

dQdP

PQ

P

Q

P*

Q*

LS

D’

D

P**

Q**

10

Long Run Price Elasticity of Supply (E LSQ,P )

Verbal Definition: E LSQ,P reflects how sensitive the short run quantity supplied of a

good is to the good’s price.

E LSQ,P = the percent change in the long run quantity supplied resulting

from a 1 percent change in the good’s price

= percent change in long run quantity supplieded

percent change in the good's price

Following the same logic as before

E LSQ,P =

dQdP

PQ


Thursday, October 20: Tax Incidence and Monopoly Application of Consumer and Producer Surplus - Tax Incidence

Before Tax Equilibrium In equilibrium,

Quantity Demanded = Quantity Supplied.

Effect of a Tax: Two Notions of the Price When a tax is imposed, two notions of the price emerge: one notion from the viewpoint of the households and a second from the viewpoint of the firms. These two notions of the price differ by the amount of the tax. To understand this, consider the gasoline tax. The legal incidence of the gasoline tax is borne by the firm; that is, the firm is legally obligated to send the gasoline tax revenue to the government. In Massachusetts, the Federal and state tax now totals about $.40 per gallon. For every gallon of gasoline the firm sells, it must sent $.40 to the government. How is the price seen through the eyes of the firm, PF, related to the price seen through the eyes of the household, PH. The price seen from the viewpoint of the firm is just $.40 less than the price seen from the viewpoint of the household because for each gallon sold, the firm must send $.40 of what the household pays to the government:

PF = PH − .40 In general,

PF = PH − t where t = the tax rate Illustrating the New Equilibrium

How can we illustrate the new equilibrium? First, note that two conditions must be met at the new equilibrium

• PF = PH − t • Quantity Demanded = Quantity Supplied

Second, note that the quantity demand and the quantity supplied are sensitive to different notions of the price:

• Quantity demanded is sensitive to the price as seen from the viewpoint of the household, PH;

• Quantity supplied is sensitive to the price as seen from the viewpoint of the firm, PF.

To illustrate the new equilibrium, start at the old equilibrium and then reduce the quantity by moving to the left until the vertical gap between the market demand curve and market supply curve equals the amount of the tax, t. This quantity equals the new equilibrium quantity, Q**. The point on the market demand, PH ** , is the new equilibrium price from the viewpoint of the households; the point on the market supply curve, PF ** , is the new equilibrium price from the viewpoint of the firm.

P

Q

P*

Q*

D

SS

P

QD

SS

t

PH

PF

**

**

Q**

Sensitive to PF

Sensitive to PC

2

To argue that this illustrates the new equilibrium, we must show that • P F ** = PH ** − t

and • quantity demanded = quantity supplied

First, it is easy to show that PF ** = PH ** − t. This follows directly from the fact that the vertical gap between the market demand and market supply curve equals t. The above diagram clearly illustrates that

P F ** = PH ** − t

Second, we must show that quantity demanded equals quantity supplied. Recall that the quantity demanded is sensitive to the price seen from the viewpoint of the household and the quantity supplied is sensitive to the price seen from the viewpoint of the firm: Household’s viewpoint Firm’s viewpoint ↓ ↓ Price = PH ** Price = P F **

↓ ↓ Quantity demand = Q** Quantity supplied = Q** Both the quantity demand and the quantity supplied equal Q**.

How does the tax affect households, firms, and the government?

Households and firms are hurt by the tax; the government is helped.

• Since the price paid by households rises as a consequence of the tax, consumer surplus falls. The shaded red area in the left diagram illustrates the fall in consumer surplus; that is, the shaded red area reflects how much consumers are hurt by the tax.

• Since the price received by firms falls as a consequence of the tax, producer surplus falls. The shaded blue area in the center diagram illustrates the fall in producer surplus; that is, the shaded blue area reflects how much firms are hurt by the tax.

• The government is helped by the tax, since it collects more revenues. Question: how much revenue does the government collect from this tax? Answer: t×Q**. The shaded green area in the right diagram reflects the tax revenue collected from the tax; that is, the shaded green area reflects how much the government is helped by the tax.

P

QD

SS

Q**

P P

SS SS

D DQ Q

Q** Q**

t

Tax Revenue = t × Q**PH ↑ Consumer Surplus ↓ PF ↓ Producer Surplus ↓

3

The amount by which households and firms are hurt by the tax exceeds the amount by which the government is helped. The net harm done to society as a whole is called the excess burden or dead weight loss of the tax. The shaded area to the right illustrates the excess burden.

Producer Surplus - Long Run Producer surplus in the long run reflects the returns earned by the inputs (labor, capital, etc.). Geometrically, producer surplus in the long run is the area above the long run supply curve that lies below the price. To justify this we shall review what we learned about the long run supply curve. Recall the shape of the long run supply curve depends on what happens to input prices when entry or exit occurs. Constant cost industry: horizontal long run supply curve

In a constant cost industry, the cost of inputs is not affected by entry or exit; that is, entry and exit does not affect the returns to inputs. Question: how will an increase in demand affect long run producer surplus? For a constant cost industry, long run producer surplus is unaffected. Since the long run supply curve is horizontal producer surplus is 0 before and after the increase in demand. Constant Cost Industry ↓ Entry Does Not Affect Input Prices ↓ Returns to Inputs Unaffected ↓ Long Run Producer Surplus Unaffected

P

Q

P*

Q*

LS

P

Q

P*

Q*

LS

Q**

D

D’

P

QD

SS

t

PH

PF

**

**

Q**

4

Increasing cost industry: upward sloping long run supply curve

The cost of inputs is driven up by entry; consequently, entry increases the returns to inputs. Question: how will an increase in demand affect long run producer surplus? For an increasing cost industry, long run producer surplus increases. Since the long run supply curve is upward sloping, an increase in demand will increase producer surplus. Increasing Cost Industry ↓ Entry Increases Input Prices ↓ Returns to Inputs Increases ↓ Long Run Producer Surplus Increases

P

Q

P*

Q*

LS

D’

D

P**

Q**

5

Monopoly Preview: Review of Key Points

Profit Maximization: Profits = TR − STC To find the profit maximizing quantity of outputs, differentiate profits with respect to quantity and set the derivative equal to 0:

dProfitdq =

dTRdq −

dSTCdq = 0

Clearly, to maximize profits, dTRdq =

dSTCdq

MR = SMC Marginal Revenue

Verbal definition: Marginal revenue equals the change in the firm’s total revenue resulting from a one unit change in output. Calculus definition: Marginal revenue equals the derivative of total revenue with respect to quantity:

MR = dTRdq

Total revenue equals price times quantity: TR = P×q. Now, apply the rules of differentiation,

MR = dTRdq = P + q

dPdq

What does dPdq equal? That is, how does the quantity of output the firm produces

affect the price. There are two polar cases: Perfect competition Monopoly ↓ ↓ Large number of small firms One large firm ↓ ↓ No single firm’s production The monopoly produces a decisions can significantly quantity and charges a price that affect the price lies on the market demand curve ↓ ↓ Each firm takes the price as To sell more output, the given; that is, each firm considers monopolist must reduce the price to be a constant the price ↓ ↓

dPdq = 01

dPdq < 0

Thus far, we have focused on the perfect competitive case. Now, we shall turn to monopoly. We have not yet justified the statements we make about monopoly.

1 We mentioned before this argument “cheats” a little; we shall address this shortly.

6

Notation First, a word about the notation we use. We have been consistent with our use of “Q” and “q”. We have used upper case Q to represent quantity in the entire market; we have used lower case q to represent production by a single firm. In the case of a monopoly, however, there is no need to differentiate between lower and upper case because there is only one firm in the industry. Consequently, in the monopoly case we can use upper case Q and lower case q interchangeably. Also, traditionally there is little distinction made between the short run and the long run when discussing monopoly. The reason for this is that for whatever reason if an industry is a monopoly, entry has not occurred. If we initially have a monopoly and entry occurs, suddenly we do not have a monopoly any more. Therefore, we typically drop references to short and long run when we discuss monopoly.

Monopoly, the Market Demand Curve, and Market Clearing Claim: A monopoly will always choose a market clearing price; that is, a monopoly will always produce a quantity and charge a price that lies on the market demand curve; that is, a monopoly will always clear the market. To justify this claim in two steps:

• First, we shall argue that a profit maximizing monopolist would never choose a price that would create a shortage; that is, a monopolist would never operate at a point below the market demand curve.

• Second, we shall argue that a profit maximizing monopolist would never choose a price that would create a surplus; that is, a monopolist would never operate at a point above the market demand curve.

Why would a monopoly never operate at a point below the market demand curve? To explain why, suppose that it did. A profit maximizing firm would not be content with this situation. The firm could raise the price to the point on the market demand curve lying directly above and still sell all the quantity of output it was producing. Since the firm produces the same amount of output, total costs would not be affected. At the higher price, the firm would now collect more revenues. With higher revenues and the same costs, the firm’s profits would rise. Why would a monopoly never operate at a point above the market demand curve? To explain why, suppose that it did. A profit maximizing firm would not be content with this situation. The firm is not able to sell all the output it is producing. Why incur the costs of producing something when it cannot be sold? The firm could reduce the quantity of output to the point on the market demand curve lying directly to the left with no change in the total revenues it collects. Since the firm is producing less, costs would fall. With the same total revenues and lower costs, the firm’s profits would rise.

P

QD

P

QD

P

QD

7

Since a profit maximizing monopoly would never • create a shortage by operating at a point below the market demand

curve or

• create a surplus by operating at a point above the market demand curve

there is only one alternative left. The monopoly will always choose a market clearing price by operating at a point on the market demand curve. How will the quantity of output the monopoly produces be related to the price it charges? When the monopoly produces more, it must lower the price to sell the additional output; that is, when the monopoly produces more, it must charge a lower price to remain on the market demand curve:

Q↑ P↓ We can express this relationship in terms of a derivative:

dPdQ < 0 where

dPdQ is the derivative of the market demand curve

Monopoly, Marginal Revenue, and Market Demand

What does this suggest about marginal revenue for a monopoly? Recall that

MR = dTRdQ where TR = P×Q

Clearly,

MR = P + q dPdQ

Since dPdQ < 0, whenever the quantity of output is positive marginal revenue is less than

price: MR < P whenever Q>0

Let us summarize what we have just learned. For a given quantity of output,

• the monopolist will charge a price that lies on the demand curve

and • marginal revenue will be less than this

price (whenever the quantity of output is positive).

We can now draw a monopoly’s marginal revenue curve.

P

QD

MR

PriceMarginal Revenue < Price

8

Monopoly’s Profit Maximizing Quantity Now,

Profit = TR − TC To find the profit maximizing quantity of outputs, differentiate profits with respect to quantity and set the derivative equal to 0:

dProfitdQ =

dTRdQ −

dTCdQ = 0

Clearly, to maximize profits, dTRdQ =

dTCdQ

MR = MC The profit maximizing quantity of output is the quantity of output at which marginal revenue equals marginal cost.

What price will the monopoly charge? Remember that the monopoly will always operate at a point on the market demand curve. So, once we determine the profit maximizing quantity of output, we look to the market demand curve to determine the price. How much profit is the firm now earning? The above diagram does not provide the information that we need to determine how much profit the monopoly firm is earning. Recall that profit depends on the relationship between price and average total cost.

ATC = TCQ

Multiplying by q: ATC × Q = TC

Since TR = P×Q and TC = ATC×Q, we can substitute for TR and TC in the profit equation:

Profit = TR − TC = P×Q − ATC×Q

Next, factor a q from both terms: Profit = (P − ATC)×Q

Recall that when we draw the average total cost curve, we must be certain that it intersects the marginal cost curve at minimum average total cost. If the diagram to the right is accurate, the monopoly would be earning a positive profit.

What is “bad” about a monopoly? Excess Profits - The Popular Notion

Most people believe a monopoly is bad because it earns obscenely high profits. While economists recognize that it is possible for a monopoly to earn high positive profits, it need not always be true. In other words, the view that the existence of a monopoly guarantees high profits is simply incorrect. To understand why, recall that

Profit = (P − ATC)×Q

P

QD

MR

MCPM

QM

P

QD

MR

MCATCATC

for q* units

PM

QM

P

QD

MR

MCATC

ATCfor q* units

PM

QM

9

If when maximizing profits, the price falls short of average total cost; in the case, a monopoly will be incurring losses. The above diagram illustrates this. Consequently, since the existence of a monopoly does not guarantee high profits, high profits cannot be the reason that a monopoly is bad.

Monopoly Leads and Excess Burden (Dead Weight Loss) “What If” Question: If the monopoly, for whatever reason, believed that it was one of many firms in a perfectly competitive industry, what quantity and price would result?

In a perfectly competitive industry, • the equilibrium quantity and price are determined by the market

demand and market supply curve. • the market supply curve is the horizontal sum of each firm’s

individual supply curve; • an individual firm’s supply curve is its marginal cost curve.

If the monopoly really believed that it was one of many firms in a perfectly competitive industry, its individual supply curve would be its marginal cost curve. Since there is only one firm in a monopoly, the market supply curve would be the monopoly’s individual supply curve. Consequently, the market supply curve would be the monopoly’s marginal cost curve. The competitive price would be PC and competitive quantity would be QC.

Comparing the monopoly and competitive case.

The monopoly firm is better off in the monopoly case because its profits are maximized when QM units are produced. Since more than QM units are produced in the competitive case, profits in the competitive case must be less than profits in the monopoly case. Households are better off in the competitive case. The competitive price, PC, is less than the monopoly price, PM. Households would clearly prefer to purchase the good at a low rather than a high price.

How is society as a whole affected? The firm benefits from monopoly while households are hurt. We need to measure the magnitude of the benefit experienced by the monopoly firm and the harm done to households. Consumer and producer surplus allows us to do this.

P

QD

MR

PM

QM

PC

QC

MC"S"

MC

P

PC

QC

DQ

"S"

10

P

QD

MR

PM

QM

PC

QC

MC MC

P

PM

PC

QM QC

MRD

Q

Consumer surplus Producer surplus

Competition

"S" "S"

P

QD

MR

PM

QM

PC

QC

MC MC

P

PM

PC

QM QC

MRD

Q

Consumer surplus Producer surplus

Monopoly

"S" "S"

P

QD

MR

PM

QM

PC

QC

MC MC

P

PM

PC

QM QC

MRD

Q

Consumer surplusProducer surplus

From Competition to Monopoly

LostGained Lost

"S" "S"

Some, but not all of the loss in consumer surplus is compensated by a gain in producer surplus. In net, there is a loss. In other words, the gain experience by the monopoly firm is less than the loss experienced by households. This represents the excess burden or dead weight loss of a monopoly.

P

QD

MR

PM

QM

PC

QC

MC"S"


Tuesday, October 25: Perfect Competition versus Monopoly – Marginal Revenue Revisited

Review of Key Points Monopoly

• The monopolist will always clear the market – a monopolist will always produce a quantity and charge a price that lies on the market demand curve

• Marginal revenue of a monopolist will be less than this price.

Marginal Revenue

Verbal definition: Marginal revenue equals the change in the firm’s total revenue resulting from a one unit change in output. Calculus definition: Marginal revenue equals the derivative of total revenue with respect to quantity:

MR = dTRdq where TR = P×q

Therefore,

MR = P + q dPdq

What does dPdq equal? That is, how does the quantity of output the firm produces

affect the price. Two polar cases: Perfect competition Monopoly ↓ ↓ Large number of small firms One large firm ↓ ↓ No single firm’s production The monopoly produces a quantity decisions can significantly and charges a price that lies on affect the price the market demand curve ↓ ↓ Each firm takes the price as To sell more output, the given; that is, each firm considers monopolist must reduce the price to be a constant the price ↓ ↓

dPdq = 0

dPdq < 0

↓ ↓ MR = P MR < P Perfect competition and monopoly differ because of the relationship between marginal revenue and price is different. But as we shall see shortly, the argument we have used until now to claim that marginal revenue equals the price “cheats” a little.

P

QD

MR

PriceMarginal Revenue < Price

2

Monopoly’s Profit Maximizing Quantity and Price To maximize profits,

MR = MC The profit maximizing quantity of output is the quantity of output at which marginal revenue equals marginal cost.

What price will the monopoly charge? Remember that the monopoly will always operate at a point on the market demand curve. So, once we determine the profit maximizing quantity of output, we look to the market demand curve to determine the price.

Welfare Implications: How is society as a whole affected?

From the perspective of society as a whole, monopoly results in too little production. Each unit of output between QM and QC: Height of > Height of MC Demand Curve (“Supply”) Curve ↓ ↓ Households’ Monopoly’s Benefits of > Costs of Consuming Producing Each Unit Each Unit

The monopolist does not produce more than QM units because its profit would decrease if it did so.

Aside: Price Discrimination

Thus far we have been implicitly assuming that a monopoly charges the same price to all its customers. At times, however, a monopolist can charge different prices to different customers.

Perfect Price Discrimination (First Order)

In this case, a monopolist would sell each unit of output for the maximum amount that buyers were willing to pay. The total revenue collected by a perfect price discriminating monopolist equals the area beneath the demand curve. To understand why, recall your classmates who are willing to buy an A in our course:

Potential Buyer Willingness to Pay Andy $225 Kate 175 Dan 140 Liz 75 Meg 60 Darce 25

Willingness to pay equals the maximum amount that a potential buyer will pay for an A. Suppose that I receive permission to sell 3 A’s. I would sell the first to Andy for $225, the second to Kate for $175, and the third to Dan for $140. Geometrically, this is just the area beneath the market demand curve:

P

QD

MR

MCPM

QM

50

100

150

200

250


Price of an AAndy

Kate Dan Liz Meg Darce

3

Question: What is marginal revenue for a perfect price discriminating monopoly? To answer this question, recall the verbal definition of marginal revenue:

Verbal definition: Marginal revenue equals the change in the firm’s total revenue resulting from a one unit change in output.

When a perfect price discriminating monopoly sells an additional unit of output, the revenues collected from the units sold previous do not change since it is still charging all other customers the same prices. Consequently, marginal revenue for the price discriminating monopoly equals the price at which the unit is sold or rather the height of the demand curve.

To maximizing profit, the marginal revenue and marginal cost is crucial. Consequently, the profit maximizing perfect price discriminating monopolist will continue to produce more output as long as the demand curve lies above the marginal cost curve. It will produce QC units of output, the competitive quantity of output.

Question: When could a monopoly act as a perfect price discriminator? Only if

• the monopolist has detailed knowledge of each consumers demand function. • arbitrage were impossible; that is, only if the product cannot be resold. If resale is

possible, the scheme would break down: consumers placing a low value on the good would buy it for a low price and then would undercut the monopoly by selling the good to consumers placing a high value for a price a little lower than the monopoly would charge. For example, If A’s could be resold, the scheme would break down. I would sell an A to Dan for $140. Dan could then turn around and offer to sell it to Andy for a little less than $225, say $220. But Dan and Andy are made better off by this. Dan turns a quick profit of $80 and Andy is paying $220 for his A rather than $225.

Price Discrimination (Third Order): Market Separation Airlines regularly practice this type of price discrimination by charging customers lower round trip prices if they stay over on Saturday night or booking flights two weeks in advance. In doing so, the airlines are trying to separate two types of customers: business and pleasure. Business travelers typically do not work on weekends and often must travel on short notice; consequently, a Saturday night stay and advance booking does a good job of separating these two types of customers. Why do the airlines do this? The demand of business travelers is less elastic than pleasure travelers. Consequently, the price discriminating monopoly can push the price paid by business travelers higher and loss less business; by doing so, it increases its profits. Note that this type of price discrimination could occur if arbitrage is impossible.

Price Discrimination (Second Order): Price Schedule – For Example, Two-Part Tariffs

Instead of charging customers a flat per minute charge for each call, telephone companies often use a two-part tariff. A customer pays a fee to gain access to the network and then a per minute charge for each call made. Presumably, a firm would adopt a two-part tariff only if it would increase profits by doing so. Again, note that this type of price discrimination could occur if arbitrage is impossible.

Price Discrimination and Profits

The possibility of price discriminate provides the firm with more flexibility because it can always pursue a nondiscriminatory policy by charging a single price. Consequently, whenever we observe price discrimination we can conclude that a firm’s profits are greater with discrimination than without it.

P

QD

QC

MC"S"

4

Oligopoly Thus far, we have considered two polar cases: perfect competition and monopoly. In the perfectly competitive case, we have a large number of small firms. In the monopoly case, we have one large firm. Oligopolies are cases between these two extremes; in a oligopoly we have a few moderately sized firms. Many interesting industries are in fact oligopolies; consequently, it is important to learn more about them. We shall begin by consider how a firm’s marginal revenue is related to the number of firms in the industry by starting with a monopoly and then increasing the number of firms. Claim: As there are more and more firms in an industry, a firm’s marginal revenue approaches the price. To justify this claim, begin by considering Apple Computer Company in 1980. While there were some other “fringe” companies producing computers in 1980, Apple was, for all practical purposes, the “only game in town.” IBM had not yet introduced its first PC. Furthermore, companies like Dell, Gateway, etc. had not even be established. For simplicity, assume that Apple had a monopoly on the production of personal computers in 1980. Question: What do we know about the quantity of computers Apple produced, the price Apple charged, and the market demand curve for PC’s? Answer: Since Apple was a monopoly, it produced a quantity and charged a price that lied on the market demand curve for computers. In 1980, Apple charged about $3,000 per PC and produced about 500 PC’s per day. Summary: On October 25, 1980

q P TR = P×Q Oct 25 500 3000 3000×500

Next, review the verbal definition of marginal revenue:

Marginal Revenue = equals the change in the firm’s total revenue resulting from a one unit change in output.

Question: What experiment could we perform on October 26 to determine Apple’s marginal revenue? Answer: We could ask Apple to increase production on October 26, 1980 by 1 unit, from 500 to 501, and then calculate the change in its total revenue from the previous day.

3,000

4,000

P ($/PC)

250 500

D

Slope = −2

Q (PC’s per day)

3,500

5

One firm - Initially, Apple produces 500 computers q P TR = P×Q Oct 26 501 2998 2998×501 = 2998×(1 + 500) = 2998 + 2998×500 Oct 25 500 3000 3000×500 = 3000×500 Marginal Revenue = 2998 + 2998×500 − 3000×500 = 2998 + (2998−3000)×500 = 2998 + −2×500 ↓ ↓ Direct Effect of Indirect Effect of Additional Unit Lower Price

↓ ↓ Gain in TR from the Loss in TR from sale of the additional the lower price unit itself: the price

Question: How would the calculations change if instead of one firm, there were two firms in the personal computer industry each producing 250 computers? Two firms - Initially, each produces 250 computers

Quantity Firm 1 Other Total Price Firm 1’s Total Revenue Oct 26 251 250 501 2998 2998×251 2998×(1+250) = 2998 + 2998×250 Oct 25 250 250 500 3000 3000×250 = 3000×250 Marginal Revenue = 2998 + 2998×250 − 3000×250 = 2998 + (2998−3000)×250 = 2998 + −2×250 ↓ ↓ Direct Effect of Indirect Effect of Additional Unit Lower Price

↓ ↓ Gain in TR from the Loss in TR from sale of the additional the lower price unit itself: the price Note that it is the total quantity of computers produced that determines the price. Just as before when a total of 501 computers are produced, the price must equal $2998 to clear the market. When there are two firms rather than one, the indirect effect of the lower price is smaller because Firm 1 produces 250 rather than all 500 computers. The revenue Firm 1 uses from the $2 lower price is only $2×250 rather than $2×500; that is, from the perspective of Firm 1, the revenue lost is $500 rather than $1,000.

6

Question: How would the calculations change if instead of one or two firms, there were fifty firms in the personal computer industry each producing 10 computers? Fifty firms - Initially, each produces 10 computers

Quantity Firm 1 Other Total Price Firm 1’s Total Revenue Oct 26 11 490 501 2998 2998×11 2998×(1+10) = 2998 + 2998×10 Oct 25 10 490 500 3000 3000×10 = 3000×10 Marginal Revenue = 2998 + 2998×10 − 3000×10 = 2998 + (2998−3000)×10 = 2998 + (−2)×10 ↓ ↓ Direct Effect of Indirect Effect of Additional Unit Lower Price

↓ ↓ Gain in TR from the Loss in TR from sale of the additional the lower price unit itself: the price

When there are fifty firms rather than one or two, the indirect effect of the lower price is even smaller because Firm 1 produces only produces 10 computers. The revenue lost from the $2 lower price is only $2×10 or $20.

Summary: • For a monopoly firm, marginal revenue is less than price; • As the number of firms increases, a firm’s marginal revenue becomes closer and

closer to the price; in the limit, marginal revenue equals the price.

7

We can also obtain this result more formally by using calculus. Review: Chain Rule

y depends x depends on x on z

dydz =

dydx ×

dxdz

Let

N = total number of firms in the industry q1 = quantity produced by firm 1 q2 = quantity produced by firm 2 . . . qN = quantity produced by firm N Q = total quantity produced in the industry Q = q1 + q2 + . . . + qN

What do we want to show? Let Firm 1 be a typical firm; that is, assume that Firm 1 produces the “average” for all the firms in the industry:

q1 = QN

We want to show that as the number of firms increases, typical firm’s marginal revenue will approach the price; that is, we want to show:

lim N→∞ MR1 = P

What do we know?

MR1 = dTR1

dq1 where TR1 = Pq1

= P + q1dPdq1

Since Firm 1 is typical,

MR1 = P + QN

dPdq1

What about dPdq1

? Well what does the price, P, depend on? The market must

clear; therefore, the quantity demanded must equal the quantity supplied. Consequently, since Q equals the total quantity supplied, the point (P,Q) must lie on the market demand. Therefore, P depends on Q in the way described by the market demand curve. Hence,

dPdQ = slope of the market demand curve < 0

8

How can we use what we know about dPdQ to say something about

dPdq1

? Let us

review the chain rule y depends x depends on x on z

dydz =

dydx ×

dxdz

P depends Q depends on Q on q1

dPdq1

= dPdQ ×

dQdq1

What does dQdq1

equal? We know that

Q = q1 + q2 + . . . + qN Therefore,

dQdq1

= 1 + dq2

dq1 + . . . +

dqN

dq1

Cournot assumption: each firm takes the production of other firms as given:

dq2

dq1 = . . . =

dqN

dq1 = 0.

Hence, given the Cournot assumption, dQdq1

= 1 and from the chain rule we have:

dPdq1

= dPdQ

Now, let us summarize:

MR1 = P + q1 dPdq1

Since Firm 1 is assumed to be typical, q1 = QN ; also, given the Cournot

assumption, dPdq1

= dPdQ . Therefore, substituting for q1 and

dPdq1

:

MR1 = P + QN

dPdQ

As the number of firms increases, that is, as N approaches infinity, the second term approaches 0:

lim N→∞

QN

dPdQ = 0

Consequently, as the number of firms, N, approaches infinity, Firm 1’s marginal revenue approaches the price:

lim N→∞ MR1 = P

So, why does marginal revenue equal the price for a perfectly competitive

industry? It is not the case that dPdQ equals 0, as we suggested before. Instead, as

9

the number of firms increases the share of the market one firm captures gets

smaller; that is, as N approaches infinity, QN approaches 0.


Thursday, October 27: Oligopoly, Collusion, Cheating, and Game Theory How Do Firms in an Oligopoly Behave? Two Polar Cases

In general, we cannot predict how firms in an oligopoly will behave; that is, in general we cannot determine the price and quantity that will emerge in an oligopolistic industry. We can describe two polar cases, however:

• Quasicompetitive case. • Shared monopoly case;

Typically, we would expect reality to be somewhere between these two polar cases. We shall discuss these cases in the framework of a two firm oligopoly, a duopoly. Once we understand a duopoly, we can extend easily the analysis to three firms, four firms, etc. Quasicompetitive Case

In the quasicompetitive case, each firm in the oligopoly acts as though it were one of a very large number of small firms; that is, each firm acts as though it were in a perfectly competitive industry. Let us review: In a perfectly competitive industry:

• the price and quantity are determined by the market demand and market supply curves;

• the market supply curve is the horizontal sum of each individual firm’s supply curve;

• an individual firm’s supply curve is its marginal cost curve.

To determine the price and quantity in the perfectly competitive case we need only superimpose both the demand and supply curve on the market graph:

PQC = quasicompetitive price QQC = quasicompetitive quantity q1QC and q2QC are the quantities produced by Firm 1 and Firm 2

P

Q

If P = .50

If P = 1.00

If P = 1.50

If P = 2.00

PPFirm 1 Firm 2

MC1 MC2"S1" "S2"

ΣMC"S"

q1 q2

Market

P

Q

D

PPFirm 1 Firm 2

MC1 MC2"S1" "S2"

ΣMC"S"

q1 q2 QQC

PQC

q2QCq1QC

Market

2

Shared Monopoly Case In the shared monopoly case, the firms in the oligopoly maximize their joint profits; that is, the firms collude, coordinate their decisions, and act as though they are different divisions of a single monopoly firm. To analyze this case, first let us calculate joint profits:

Firm 1’s Profit = Firm 1’s Total Revenues − Firm 1’s Total Costs = TR1 − TC1 = P×q1 − TC1 Firm 2’s Profit = Firm 2’s Total Revenues − Firm 2’s Total Costs = TR2 − TC2 = P×q2 − TC2 Joint Profits = (P×q1 + P×q2) − (TC1 + TC2) = P×(q1 + q2) − (TC1 + TC2) = P×Q − (TC1 + TC2) where Q equals the total quantity of output produced in the industry. That is, Q equals the sum of what Firm 1 and Firm 2 produce individually:

Q = q1 + q2 The shared monopoly’s total revenue, TRSM, is just the product of the price and the total quantity produced:

TRSM = P×Q We can now write the expression for joint profits by substituting TRSM for P×Q:

Joint Profits = TRSM − (TC1 + TC2) Let MRSM equal the marginal revenue of the shared monopoly

MRSM = dTRSM

dQ = P + QdPdQ

The shared monopoly’s marginal revenue curve looks like the “standard” monopoly’s marginal revenue curve. Recall that the monopolist

• always produces a quantity and charges a price that lies on the market demand curve;

• marginal revenue is less than the price.

Consequently, the shared monopoly’s marginal revenue curve lies below the market demand curve.

Claim: to maximize joint profits, the following two conditions must be met:

• total production must each the sum of what each firm produces: Q = q1 + q2

• the shared monopoly’s marginal revenue must each marginal cost for each firm:

MRSM = MC1 and MRSM = MC2

P

Q

DMRSM

Shared Monopoly

dP

dQ< 0

3

The first condition is obvious. Since there are only two firms in the industry, total production must equal the sum of what each of the two firms produces individually. To justify the second condition, we shall argue that if it were not true, then the oligopoly can always increase its profits. Suppose that the shared monopoly’s marginal revenue did not equal firm 1’s marginal costs:

MRSM ≠ MC1 For example, suppose that

MRSM = 1.50 and MC1 = 1.00. It is easy to argue that the oligopoly’s joint profits would increase if Firm 1 produced one additional unit of output. When Firm 1 produces one additional unit of output:

Joint Profits = TRSM − (TC1 + TC2)

↑ $1.50 ↑ $1.00 because because MRSM = 1.50 MC1 = 1.00

Joint profits rise by $.50. In general, whenever MRSM > MC1, joint profits would increase if Firm 1 produced more output. On the other hand, if MRSM < MC1, point profits would increase if Firm 1 produced less output. Consequently, to maximize joint profits the shared monopoly’s marginal revenue must equal Firm 1’s marginal costs:

MRSM = MC1 Using the same logic, we can show that to maximized joint profits the shared monopoly’s marginal revenue must equal Firm 2’s marginal costs:

MRSM = MC2 The following diagram illustrates how shared monopoly would maximize its joint profits: PSM = shared monopoly price QSM = shared monopoly quantity q1SM and q2SM are the quantities produced by firm 1 and firm 2

P

Q

D

PPFirm 1 Firm 2

MC1 MC2 ΣMC

q1 q2 QSM

PSM

q1SM q2SM

Market

MRSM

MRSM

4

Now, we can show that the diagram meets the two joint profit maximization conditions we just derived.

• total production must each the sum of what each firm produces: QSM = q1SM + q2SM

This occurs as a consequence of the way we constructed the ΣMC (or “S”) curve on the market graph. The ΣMC or S curve on the market graph is the horizontal sum of the MC (or S) curves of each firm.

• the shared monopoly’s marginal revenue must each marginal cost for each firm:

When total production is QSM and firm 1 produces q1SM: MRSM = MC1

When total production is QSM and firm 2 produces q2SM: MRSM = MC2

5

Conflicting Interests of the Firms in an Oligopoly Conflicting Interests: Collective versus Individual

• It is in the collective interest of the firms in an oligopoly to coordinate their production decisions, act as a shared monopoly, and abide by a collusive agreement that maximizes their joint profits.

• On the other hand, if such a collusive agreement were in place, it would be in the individual interest of each firm to cheat on the agreement by producing more output than the agreement allows (assuming that the other firms would not react -- Cournot assumption). By cheating, a firm would increase its individual profits.

The first point is self-evident: maximizing joint profits is clearly in the collective interest of the firms. How can we show that the individual interests of the firms conflict with their collective interest? To do so, we shall begin with a claim. Claim: when a shared monopoly agreement is in place, a firm has an incentive to produce more output than the agreement allows.

Strategy to justify this claim: Focus attention on Firm 1 and then show that when the shared monopoly agreement is in place,

MR1 > MC1 Whenever MR1 > MC1, the Firm 1 can increase its profits by producing more; that is, Firm 1 would have an incentive to produce more than the agreement allows. What do we know? When the agreement is in place:

MRSM = MC1 Therefore, if we can show that

MR1 > MRSM, the claim will be justified. We could justify this by appealing to something we learned in the last class, the relationship between a firm’s marginal revenue and the number of firms:

• for a monopoly firm, marginal revenue is less than price; • as the number of firms increases, a firm’s marginal revenue increases

and becomes closer and closer to the price; in the limit, marginal revenue equals the price.

This is such an important result, however, we shall show directly why it is true. What do MR1 and MRSM equal? Marginal revenue is the derivative of total revenue with respect to quantity. MR1 equals the derivative of Firm 1’s total revenue with respect to Firm 1’s quantity, q1. MRSM equals the derivative of the shared monopoly’s total revenue with respect to the total quantity produced, Q. That is,

TR1 = P× q1 TRSM = P×Q where Q = q1 + q2

MR1 = dTR1

dq1 MRSM =

dTRSM

dQ

= P + q1dPdq1

= P + QdPdQ

6

Let us begin by focusing on the equation for MR1. What does dPdq1

equal? To

answer this question, review the chain rule: y depends x depends on x on z

dydz =

dydx ×

dxdz

Now, the price, P, depends on the total quantity produced, Q, because the market must clear. P and Q are related in the way described by the market demand curve. The total quantity produced, Q, depends on q1; the total quantity produced is the sum of what each firm produces individually. Therefore, P depends Q depends on Q on q1

dPdq1

= dPdQ ×

dQdq1

We know that dPdQ < 0. So, what does

dQdq1

equal? This is easy to answer because

Q = q1 + q2; therefore,

dQdq1

= 1 + dq2

dq1

Now, consider the Cournot assumption: each firm assumes that the other firm’s level of production is constant. Since Firm 1 assumes that Firm 2’s level of production is constant; consequently,

dq2

dq1 = 0

Therefore,

dQdq1

= 1 and dPdq1

= dPdQ

Now, return to the equations for MR1 and MRSM:

MR1 = P + q1dPdq1

MRSM = P + QdPdQ

↓ since dPdq1

= dPdQ

= P + q1dPdQ

Compare the expressions for MR1 and MRSM:

MR1 = P + q1dPdq1

MRSM = P + QdPdQ

7

Now, recall that Q = q1+q2; clearly, q1 is less than Q. Since dPdQ is less than 0, we are

subtracting less from the price when we calculate MR1, than when we calculate MRSM. Consequently, MR1 is greater than MRSM. More formally, q1 < Q

↓ since dPdQ < 0

q1 dPdQ > Q

dPdQ

↓ adding P to both sides of the inequality

P + q1 dPdQ > P + Q

dPdQ

↓ from the equations for MR1 and MRSM MR1 > MRSM Now, recall that when the shared monopoly agreement is in place: MRSM = MC1 Putting the two pieces together, MR1 > MRSM = MC1 Clearly, MR1 > MC1

When the shared monopoly agreement is in place, MR1 > MC1; consequently, Firm 1’s profits will increase when it produces more. Firm 1 has an incentive to cheat on the agreement by producing more than the agreement allows. All this assumes that the Cournot assumption is realistic, however. That is, we are assuming that when Firm 1 increases its production, Firm 2 will not react.

Game Theory Revisiting the Cournot Assumption

Cournot assumed that each firm would take the production decisions of all other firms as a given. Is this realistic? If one firm decides to expand production, is it reasonable to believe that other firms will not react? In many cases, the answer to this question is no. How then can we analyze the interaction of firms? Game theory provides a very rich way to model these interactions.

Prisoners' Dilemma Game No doubt the most famous game in the field of game theory is the prisoners’ dilemma. Two individuals, Adam and Beth, are arrested by the police on the suspicion that they have conspired to commit a serious crime. In reality they are guilty, but the evidence that the authorities have is not definitive. Only a confession would provide enough evidence to prove that a felony was committed; otherwise, the prosecution must settle with a misdemeanor conviction. The district attorney decides to interrogate Adam and Beth separately and offer each the following deal:

• “If neither you nor your partner confesses, you both will be convicted of a misdemeanor and spend five years in prison.”

• “If you confess and your partner does not, we will drop the charges against you in exchange for your testimony implicating your partner; you will spend no time in prison, but your uncooperative partner will serve twenty years.”

• “If both you and your partner confess, you both will be convicted of a felony; but since you both cooperated, you will be sentenced to only eight years.”

8

The possible outcomes of this game are illustrated to the right. Adam's actions determine the row in which the outcome will lie; if she does not confess, the outcome will be in the top row and if she does confess it will be in the bottom row. Beth's actions determine the column. It is very easy to understand how each individual ranks the alternative outcomes: Ranking Outcome 1 No prison time 2 5 year sentence 3 8 year sentence 4 20 year sentence Dominant Strategy: A player has a dominant strategy whenever the player is better off by choosing the same strategy regardless of what the other player does.

Does Adam have a dominant strategy? Yes, confess is Adam’s dominant strategy.

• If Beth does not confess, Adam must choose between the upper left and lower left cells. Adam is better off by confessing; he would spend no time in prison as opposed to 5 years.

• If Beth does confess, Adam must choose between the upper right and lower right cells. Adam is better off by confessing; he would spend 8 years in prison as opposed to 20 years.

Does Beth have a dominant strategy? Yes, confess is Beth’s dominant strategy.

• If Adam does not confess, Beth must choose between the upper left and upper right cells. Beth is better off by confessing; she would spend no time in prison as opposed to 5 years.

• If Adam does confess, Beth must choose between the lower left and lower right cells. Beth is better off by confessing; she would spend 8 years in prison as opposed to 20 years.

Recall that Adam and Beth are being interrogated separately. Consequently, they cannot communicate. In this case, we would expect each to confess. Note that this situation is stable. Neither player has an incentive to change his/her strategy assuming that the other player does not change strategies. This type of equilibrium is called a Nash equilibrium.

Beth does Beth does not confess confess Adam does Adam 5 years Adam 20 years not confess Beth 5 years Beth 0 years Adam does Adam 0 years Adam 8 years confess Beth 20 years Beth 8 years

Beth does Beth does not confess confess Adam does Adam 5 years Adam 20 years not confess Beth 5 years Beth 0 years Adam=2 Beth=2 Adam=4 Beth=1 Adam does Adam 0 years Adam 8 years confess Beth 20 years Beth 8 years Adam=1 Beth=4 Adam=3 Beth=3

9

Nash Equilibrium: A cell is a Nash equilibrium whenever neither player has an incentive to change strategies assuming that the other player will not change strategies.

Note that a Nash equilibrium is somewhat similar to Cournot’s assumption: each player assumes that the other player does not change his/her behavior. Is the upper left cell a Nash equilibrium? That is, is the cell in which both Adam and Beth do not confess a Nash equilibrium? No, for two reasons.

• Assuming that Adam does not confess, Beth is better off by confessing thereby moving to the upper right cell; by doing so Beth spends no time in prison as opposed to 5 years.

• Similarly, assuming that Beth does not confess, Adam is better off by confessing thereby moving to the lower left cell; by doing so Adam spends no time in prison as opposed to 5 years.

Is the upper right cell a Nash equilibrium? That is, is the cell in which Adam does not confess and Beth does confess a Nash equilibrium? No.

• Assuming that Beth does not confess, Adam is better off by confessing thereby moving to the lower left cell; by doing so Adam spends 8 years in prison as opposed to 20 years.

Is the lower left cell a Nash equilibrium? That is, is the cell in which Beth does not confess and Adam does confess a Nash equilibrium? No.

• Assuming that Adam does not confess, Beth is better off by confessing thereby moving to the lower left cell; by doing so Beth spends 8 years in prison as opposed to 20 years.

Is the lower right cell a Nash equilibrium? That is, is the cell in which Beth does confess and Adam does confess a Nash equilibrium? Yes.

• Assuming that Adam does confess, Beth is not better off by moving to a different cell; if Beth did so by not confessing, she would spend 20 years in prison rather than 8.

• Assuming that Beth does confess, Adam is not better off by moving to a different cell; if Adam did so by not confessing, he would spend 20 years in prison rather than 8.

Summary

Assuming that the other player does not change his/her strategy what will Adam do? Beth do? Upper left cell change change Upper right cell change no change Lower left cell no change change Lower right cell no change no change

The prisoners’ dilemma game has a single Nash equilibrium.

10

Note that a Nash equilibrium is a self-reinforcing phenomenon. If both players assume that the other’s actions will not change and, given this assumption, neither has an incentive to change strategies, then neither will change strategies. Next, note that from the perspective of both Adam and Beth, the upper left cell is preferable to the Nash equilibrium. Both would be better off if neither of them confessed than they are at the Nash equilibrium. Might there be a way for them to move to this mutually preferable cell?



Tuesday, November 1: Oligopoly and Game Theory Review of Prisoners’ Dilemma

Adam and Beth commit a serious crime and are arrested by the police. In reality, they are guilty, but the evidence that the authorities have is not definitive. Only a confession would provide enough evidence to prove that a felony was committed; otherwise, the prosecution must settle with a misdemeanor conviction. The district attorney decides to interrogate Adam and Beth separately and offer each the following deal:

• “If neither you nor your partner confesses, you both will be convicted of a misdemeanor and spend five years in prison.”

• “If you confess and your partner does not, we will drop the charges against you in exchange for your testimony implicating your partner; you will spend no time in prison, but your uncooperative partner will serve twenty years.”

• “If both you and your partner confess, you both will be convicted of a felony; but since you both cooperated, you will be sentenced to only eight years.”

Dominant strategy: A player has a dominant strategy whenever the player is better off by choosing the same strategy regardless of what the other player does.

Nash equilibrium: A specific cell represents a Nash equilibrium whenever neither player has an incentive to change strategies assuming that the other player will not change strategies.

Next, note that from the perspective of both Adam and Beth, the upper left cell is preferable to the Nash equilibrium. Both would be better off if neither of them confessed than they are at the Nash equilibrium. Might there be a way for them to move to this mutually preferable cell?


2

Credible Threat: a threat is credible if a player has an incentive to carry out the threat if put in a position to do so.

Recall that Adam and Beth were interrogated separately. What if they were able to communicate, however? If they could communicate, they can threaten each other. Adam’s threat: “Beth, if you do not confess, I will not confess either; but Beth, if you do confess, I will confess also.”

Adam’s threat is credible. If Beth confesses, Adam has an incentive to confess. It is better for him to spend 8 years in prison as opposed to 20.

Beth’s threat: “Adam, if you do not confess, I will not confess either; but Adam, if you do confess, I will confess also.”

Beth’s threat is also credible. If Adam confesses, Beth has an incentive to confess. It is better for her to spend 8 years in prison as opposed to 20.

These two credible threats effectively eliminate the off-diagonal cells. Now Adam and Beth must choose between the two remaining cells. Clearly, it is better for both of them not to confess: by not confessing, each spend 5 rather than 8 years in prison. Note the importance of communications and monitoring here. The players must be able to communicate with each other; otherwise, they would not be able to issue threats. Furthermore, each player must be able to monitor the other to ensure that he/she is not “double crossed.” The irony is that credible threats allow each player to end up better off than he/she would be otherwise.


3

Oligopoly as a Prisoners’ Dilemma We can illustrate the problem faced by the firms in an oligopoly using game theory. To do so, review the two polar cases of oligopolistic behavior:

Quasicompetitive case: each firm in the oligopoly acts as though it were one of a very large number of small firms; that is, each firm acts as though it were in a perfectly competitive industry. Shared monopoly case: the firms in the oligopoly maximize their joint profits; that is, the firms collude, coordinate their decisions, and act as though they are different divisions of a single monopoly firm.

Also, recall that the firms in an oligopoly face conflicting interest:

• It is in the collective interest of the firms in an oligopoly to collude, coordinate their actions, and act as a shared monopoly. By doing so, they maximize their collective profits.

• On the other hand, if such a collusive agreement were in place, it would be in the individual interests of each firm to cheat on the agreement by producing more output than the agreement allows.

We capture all this in the following game played in a two firm oligopoly. Each firm can either abide by the shared monopoly agreement that maximizes their joint profit. When they do so, each earns a profit of 40 for a total of 80 joint profits. On the other hand, if each firm cheats, their joint profit will be less; suppose that when they both cheat, each earns a profit of 30 for a total of 60 joint profits. What if one firm cheats and the other does not? We know that when both firms abide by the agreement, each individual firm has an incentive to cheat. Therefore, when one firm cheats and the other does not, the cheating firm would earn more profits than it would if it abided by the agreement. In our example, the cheating firm earns 50 rather than 40 profits. Note that the abiding firm must earn profits of less than 30 because the joint profits of the two firms must be less than 80, the joint profits when both firm abide by the agreement that maximizes their joint profits. We have assumed that when one firm cheats and the other abides, the abiding firm earns profits of 20. It is easy to see how firm A and firm B rank the four cells. Compare the rankings of the four cells with the rankings in the prisoners’ dilemma. They are the same. The prisoners’ dilemma illustrates the conflicting interests facing the firms of an oligopoly.

Firm B Firm B abides cheats Firm A Profits for A: 40 Profits for A: 20 abides Profits for B: 40 Profits for B: 50 Firm A Profits for A: 50 Profits for A: 30 cheats Profits for B: 20 Profits for B: 30

Firm B Firm B abides cheats Firm A Profits for A: 40 Profits for A: 20 abides Profits for B: 40 Profits for B: 50 A=2 B=2 A=4 B=1 Firm A Profits for A: 50 Profits for A: 30 cheats Profits for B: 20 Profits for B: 30 A=1 B=4 A=3 B=3

4

A Noneconomic Application of Game Theory: The Yom Kippur War On October 6, 1973, Egypt and Syria, armed with Soviet weapons, coordinated a surprise attack on Israel. Since the war started during the Jewish holiday Yom Kippur, the war became known as the Yom Kippur War. Egypt and Syria caught the Israelis off guard and initially experienced some success. After nearly a week of fighting the tide turned, however. Israel had launched a very successful counteroffensive. On October 22, the UN Security Council called for a ceasefire that was ignored. The Egyptian Third Army was in particularly dire straits. It had retreated to the Sinai peninsula and in the process had all but exhausted its food and water. Without water, the Third Army could not survive in the Sinai desert. Not only was the Egyptian Third Army in jeopardy, but many Egyptians also feared that the rapidly advancing Israeli troops would cross the Suez Canal and seize Cairo. Facing these threats Egyptian President Anwar Sadat asked the Soviets to send military troops to Egypt to extricate Sadat and his country from this impending disaster. The Soviet Union quickly agreed because it had long sought a military presence in the Middle East. The Soviets suggested to the United States that both superpowers send troops to implement the UN cease-fire. Furthermore, the Soviets informed the U.S. that they would not tolerate an Israeli victory and would send their troops even if the U.S. sent none. On October 25, the United States placed all U.S. military forces throughout the world on a precautionary alert, an action not taken for more than a decade. Subsequently, the Soviets changed their mind. Instead of sending troops the Soviets pursued a diplomatic initiative. The two superpowers cooperated; Israel and Egypt were pressured to accept a ceasefire and the Egyptian Third Army to received much needed food and water. Time line

October 6: Egypt and Syria attack Israel; initially Egypt and Syria enjoy success October 13: The tide turns and Israel now begins to gain ground October 22: Israel dominating - Egyptian Third Army in dire straits UN calls for ceasefire - Israel ignores call October 23: Soviets announce their intention of sending troops October 25: U.S. places all military forces on precautionary alert October 27: Soviets change their mind and pursue a diplomatic initiative Outcome: Israel accepts the ceasefire Egypt allowed to supply troops with food and water, but no arms

We will now use game theory to gain a better understanding of the war. In doing so, we shall model the events that occurred. Whether implicitly or explicitly we use models every day of our lives. Models always involve simplifying assumptions; these assumptions are made to allow us to focus on the most important features of the problem at hand. While it certainly sounds trite to say that the world is a very complex place, it is nonetheless true. It is impossible to account for all aspects of any real world problem. We ignore those factors which we believe are of secondary importance in order to focus on the critical ones. In the discussion that follows, we will implicitly make many simplifying assumptions to gain a better understanding of the most important aspects. When Sadat requested Soviet military assistance, the Soviet leadership was faced with two broad alternatives: intervene militarily in the war by sending Soviet troops to oppose the Israelis or pursue a diplomatic initiative to impose a cease-fire. The United States also had to decide whether to go along with the Soviet actions or take measures to resist them.

5

Note that the U.S. determines the row of the outcome and the Soviets the column. If the Soviets sent troops and the United States went along, nearly all agree that the Soviets would have stopped the Israeli drive across the Sinai peninsula. The Soviets would have introduced their troops into the Middle East, an objective they had long sought. If the Soviets sent troops and the United States resisted their action, a superpower confrontation would ensue. On the other hand, if the Soviet Union pursued the diplomatic option and the U.S. cooperated, it was believed that the U.S. could pressure Israeli to accept a ceasefire and allow the re-supply of the Third Army; in fact, this was the eventual outcome. Lastly, if the U.S. decided to frustrate a Soviet diplomatic initiative by not applying pressure to Israel, an outright Israeli victory over Egypt was very likely. It was widely believed that only U.S. pressure or Soviet troops would prevent Israel from exploiting its dominant military position. How did the Soviets and the U.S. rank each cell? It is easy to understand the Soviet rankings. The Soviets first choice would be to have its troops in the Middle East. The Soviets had spent much time, energy, and resources courting the Arab world. The presence of troops would give the Soviets even more influence in this area. The Soviets worst outcome was an Israeli victory. Egypt and Syria were considered allies by the Soviets; the Soviets did not want their allies to endure a humiliating defeat. Lastly, the Soviets no doubt preferred a cease-fire above a confrontation. A confrontation would involve the possible use of nuclear weapons, something the Soviets did not want. What about the U.S. rankings? The table to the right reports on the Soviet perception of the U.S. rankings. The rankings of the first and second choices are clear. The U.S. would prefer an Israeli victory to all other outcomes because the U.S. and the Israelis were allies. Also, the U.S. preferred a cease-fire to the other two outcomes. But how did the U.S. rank Soviet troops and confrontation. The Soviets had good reason to believe that the U.S. would prefer Soviet troops in the Middle East to a superpower confrontation. President Nixon was embroiled in the Watergate scandal. The American President was under attack domestically and the Soviets concluded that the Nixon administration was too weak to risk a superpower confrontation. Consequently, the Soviets informed the U.S. that it was sending troops.

USSR uses USSR sends diplomacy troops US Cease-fire Soviet troops goes along US Israeli victory Confrontation resists

Soviet perception Ranking of U.S. rankings 1 Israeli victory 2 Cease-fire 3 Soviet troops 4 Confrontation

Ranking USSR rankings 1 Soviet troops 2 Cease-fire 3 Confrontation 4 Israeli victory

6

If Soviet perceptions had been correct, they would have had a dominant strategy: send troops. The Soviet Union would have been better off by sending troops, regardless of what the U.S. did. Once the Soviets sent troops, the U.S. would have gone along rather than resist, if Soviet perceptions had been correct. The Soviet Union would have been able to achieve its first choice by sending troops because the top right cell would have been a Nash equilibrium. Was the Soviet perception in fact correct? In reality, the Soviets miscalculated the American ranking. After the Soviets informed the Nixon administration that it was preparing to send combat troops, President Nixon consulted with Congressional leaders. Nixon ordered American military forces throughout the world on a precautionary alert. This was designed to be a signal to the Soviets. U.S. domestic concerns were temporarily set aside; American leaders of both parties united to tell the Soviets that the U.S. was not going to sit idly by while Soviet troops were introduced troops into the Middle East. The alert accomplished its purpose; the Soviets realized that they had miscalculated American resolve. The worst outcome from the U.S. perspective was the introduction of Soviet troops, not a superpower confrontation; so we now have a new game. The Yom Kippur War is actually a prisoners’ dilemma game. The U.S. and the Soviets each have a dominant strategy: The U.S. is better off by resisting regardless of what the Soviets do and the Soviets are better off by sending troops regardless of what the U.S. does. The lower right cell is a Nash equilibrium. But, just as before, note that both the U.S. and the Soviets prefer the upper left. Could credible threats allow them to achieve this outcome? What actually occurred? The leaders of both countries issued threats and furthermore, these threats were credible:

USSR uses USSR sends diplomacy troops US Cease-fire Soviet troops goes along US=2 USSR=2 US=3 USSR=1 US Israeli victory Confrontation resists US=1 USSR=4 US=4 USSR=3

Actual Ranking U.S. rankings 1 Israeli victory 2 Cease-fire 3 Confrontation 4 Soviet troops


7

US’s threat: “If you pursue diplomacy, we will go along with it and cooperate; but if you send troops we will force a nuclear confrontation.”

The U.S.’s threat is credible. If the Soviets send troops, the U.S. has incentive to resist because the U.S. prefers a nuclear confrontation to Soviet troops in the Middle East. The U.S. threat is credible.

USSR’s threat: “If you go along and cooperate, we will pursue diplomacy; but if you resist our efforts, we will send troops to the Middle East.”

The USSR’s threat is credible. If the U.S. resists, the Soviets has an incentive to send troops because the Soviets prefer a nuclear confrontation to an Israeli victory. The Soviet threat is credible.

The diplomatic lines between Washington and Moscow were very busy communicating these threats. Both the U.S. and Soviets recognized that these threats were credible; the credible threats made the off-diagonal cells irrelevant. The top left hand cell is now the better choice for both sides. This is in fact what occurred. The Soviets pursued its diplomatic initiative and the U.S. cooperated by pressuring Israel. A cease-fire was implemented; the Egyptian Third army was resupplied and the Soviets did not send troops to the region.

The Game of Chicken Two teenage drivers speed toward each other. Each is driving down the middle of the road. Some say that this game is a test of will, but perhaps it would be more accurate to call it a test of stupidity. As the drivers speed toward each other, each driver must decide on a strategy: either swerve or do not swerve. The driver who chickens out and swerves is the loser. The driver who does not swerve is the winner. Let us consider the possible outcomes of a game of chicken that is being played by two teenagers Adam and Beth. If both players swerve, the game ends in a draw. If one player does not swerve and the other does swerve, the non-swerving player is the winner and the swerving player the loser. If both players do not swerve, there is a fatal collusion. It is easy to determine how the players rank the cells. Winning is the best outcome, a draw second best, and a loss third best. A collision is ranked lowest by both players.


Beth does Beth does swerve not swerve Adam does Draw Adam loses swerve Adam=2 Beth=2 Beth wins Adam=3 Beth=1 Adam does Adam wins Collision not swerve Beth loses Adam=4 Beth=4 Adam=1 Beth=3

8

Dominant Strategies Does Adam have a dominant strategy? No.

If Beth swerves, Adam is better off by not swerving. If Beth does not swerve, Adam is better off by swerving.

Does Beth have a dominant strategy? No. If Adam swerves, Beth is better off by not swerving. If Adam does not swerve, Beth is better off by swerving.

Nash equilbrium Does this game have a Nash equilibrium? Actually, it has two. To understand why, consider each cell and see if each player has an incentive to change strategies:

Assuming that the other player does not change his/her strategy what will Adam do? Beth do? Upper left cell change change Upper right cell no change no change Lower left cell no change no change Lower right cell change change

Both the upper right and the lower left cells are Nash equilibria.

Credible Commitments How can we predict the outcome of this game? Can either player make a credible threat? For example, is the following statement made by Adam credible?

Adam: “I will not swerve regardless of what you do, Beth.” This is not a credible threat because if Beth does not swerve, we know that Adam would be better off by swerving; that is, we know that Adam prefers to lose the game to being killed in a collision. Similarly, Beth cannot issue a credible threat either. To make such a statement credible, Adam would have to do something that would make it impossible for him to swerve. A threat will not work in this case because a simple threat is not credible. A player would have to take some action that would give him no choice that to carry through on what he stated he would do. When a player does so, the player has made a credible commitment. In this example, it is difficult to conjure up a credible commitment. Perhaps a player could tear his steering wheel off and hold it outside the window for the other player to see. If he did so, it would be impossible for him to swerve. But this is a little far fetched. To learn more about credible commitments, we shall consider a more realistic example.

Beth does Beth does swerve not swerve Adam does Draw Adam loses swerve Adam=2 Beth=2 Beth wins Adam=3 Beth=1 Adam does Adam wins Collision not swerve Beth loses Adam=4 Beth=4 Adam=1 Beth=3

9

An Illustration of the Game of Chicken: Airbus and Boeing At the present time, both Airbus and Boeing are contemplating the development of a new generation of wide body aircraft that would be nearly twice as large as a Boeing 747. The development costs for such a new generation of aircraft would be very high. Airbus and Boeing must make a decision: develop or not develop. Let us investigate the ramifications of their decisions.

• If neither develops each continues to produce the current generation of aircraft; assume that if neither develops each would earn a profit of 40.

• If one firm develops and the other does not, the firm which develops will benefit greatly from many new customers. These will allow the developing firm to recoup the high development costs and then some. Consequently, the developing firm would see higher profits; suppose that the developing firm profits rose to 50. The non-developing firm would lose many customers, however; its profits would fall. Suppose that the non-developing firm’s profits fall to 20.

• If both develop, both incur the high development costs. Neither would be able to recoup its development costs, however. The profits of both would fall; suppose that their profits would fall to 5.

Based on the profits, it is easy to understand how each firm would rank the cells. Note that this is just the game of chicken. Neither firm has a dominant strategy and there exist two Nash equilibria. Obviously, Airbus prefers the lower left Nash equilibrium while Boeing prefers the upper right one. What type of credible commitment could Airbus or Boeing make to affect the outcome of the game? Boeing could construct a new assembly line that could only be used be used to manufacture the new generation of wide body aircraft. This would be a credible commitment. Airbus would know that Boeing was going ahead with development; Airbus would recognize that it was better off by not developing. Could government play a role in affecting the outcome? Suppose that the Common Market agrees to give Airbus a subsidy of 20 only if it develops the new generation of aircraft. This changes the game considerably; Airbus’s rankings of the upper and lower right cells is reversed. Airbus now has a dominant strategy: develop. With the subsidy, Airbus is better off by developing regardless of what Boeing does. There is now only one Nash equilibrium; the lower right cell. Note that this benefits Airbus greatly.

Boeing does Boeing does not develop develop Airbus does Profits for A: 40 Profits for A: 20 not develop Profits for B: 40 Profits for B: 50 A=2 B=2 A=3 B=1 Airbus does Profits for A: 50 Profits for A: 5 develop Profits for B: 20 Profits for B: 5 A=1 B=3 A=4 B=4

Boeing does Boeing does not develop develop Airbus does Profits for A: 40 Profits for A: 20 not develop Profits for B: 40 Profits for B: 50 A=2 B=2 A=3 4 B=1 Airbus does Profits for A: 50 70 Profits for A: 5 25 develop Profits for B: 20 Profits for B: 5 A=1 B=3 A=4 3 B=4

Date post:	04-Apr-2020
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Thursday, October 13: Short and Long Run...

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