bapa 500 managerial economics

8/18/2019 Bapa 500 Managerial Economics

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UNIVERSITY OF BRITISH COLUMBIAFACULTY OF COMMERCE & BUSINESS ADMINISTRATION

STRATEGY AND BUSINESS ECONOMICS DIVISION

BAPA 500:MANAGERIAL ECONOMICS

Instructor: Tom Ross

PURPOSE OF THIS COURSE:

Managerial Economics involves the study of how to direct scarce resources

in the way that most efficiently achieves a managerial goal. The coursewill attempt to give students a facility with the techniques of modernmicroeconomics sufficient to enable them to be better managerial decision-makers. Topics covered include:

1. Pricing in Competitive Markets2. Consumer Behaviour 3. Costs and Production4. Monopoly and Monopsony Pricing

5. Pricing Strategies6. Market Structure and Game Theory7. Organization of the Firm8. Efficiency and Market Failure

Key Concepts include:

1. Identifying Goals and Recognizing Constraints

2. Economic vs. Accounting Profits3. Understanding Incentives4. Understanding Markets -- Competitive, Monopoly and Other 5. The Time Value of Money6. Marginal Analysis


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BAPA 500: Managerial Economics

2

Questions

Decision making really only gets hard when resources are scarce, but thatis exactly when economic analysis is useful. For this reason, a solidfoundation in microeconomic theory is helpful for making almost all business decisions -- here are just a few examples:

1. How should you compensate your employees?

2. Of all the inputs you need to produce your product, which should you

buy and which should you make for yourself?

3. The two major suppliers of an essential input are proposing a merger. How would this affect you and what can you do about it?

4. You notice that competitors are taking away some customers with acheaper, lower quality product. How should you respond?

5. The government is reducing a tax on your product. How should youadjust your price?

6. Another firm threatens to enter into a lucrative market that you haveheld all to yourself. What steps can you take to deter entry?

7. You have regular battles with governments at the federal and provincial levels about conduct in your market. Why are theyhounding you and what can you do about it?

8. An investment proposal you are considering involves significantcosts in the first two years, with the possibility of very high earnings beginning in about five years. How do you decide whether the project is a good idea for your company?


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Topic 1: Pricing in Competitive Markets

An Ancient Question: Is the price of a good determined by the cost of producing that good or by the inherentusefulness or value of that good to users?

For competitive marketsØØ the intersection of demand and supply.

Key assumptions of the Competitive Model:

A.1 There are so many buyers and sellers that no individual buyer or seller has any influence on price. All agents are "price-takers"

A.2 Homogeneous products.

A.3 Full information.

A.4 Free entry and exit.

The Demand Curve:

The Demand Curve relates the total quantity demanded of a good to the

price charged for that good. The Law of Demand tells us that this

should be a downward-sloping curve.

Recall that the location of the demand curve will depend upon many

things, such as incomes of buyers, prices of substitutes and complements,advertising level, product quality, tastes etc. If any of these other variableschange, the demand curve will shift.


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The Demand Function:

The Demand Function, Q d (P,Z), relates the total quantity demanded of

a good to the price charged for that good and to other relevant variables.

e.g. consider the effects on the demand curve of:1. Heavier advertising and promotion by many firms in the market.2. Rising incomes of buyersØØ Normal vs. inferior goods.3. The development of an important substitute product.4. An increase in the price of a complementary good.

Willingness-to-Pay and Consumer's Surplus

Typically, consumers derive a net benefit from consumption, even after they have paid their bills. Consumer's surplus is the difference between themaximum one would be willing to pay for a good and the amount oneactually does pay for the good. Graphically, the willingness-to-pay isillustrated as the area under the demand curve up to the quantity inquestion. Consumer's surplus is illustrated as the area below an individual's

demand curve but above the price actually paid. The surplus earned by allconsumers can be represented by a similar area below the market demandcurve and above the market price.

X

P

D

X*

P* Willingness-to-pay

Consumer’s Surplus


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The Supply Curve:

The Supply Curve relates the total quantity of a good offered for sale by

all sellers to the price charged for that good.

Just as with Demand, the location of the supply curve can depend upon ahost of factors such as taxes, factor prices and the state of technology.

The Supply Function:

The Supply Function, Q s (P,W), relates the total quantity of a good

offered for sale by all sellers to the price charged for that good and to

other relevant variables.

e.g. consider the effects on the supply curve of increases in the minimumwage, the adoption of a new technology, the imposition of newenvironmental regulations, the entry of new firms

Producer's SurplusThe excess of revenues received over those absolutely necessary to lead

suppliers to provide the quantity in question. Illustrated by the area belowthe market price and above the supply curve.

X

P S

X*

P*

Producers’ Surplus

Total SurplusThe sum of producers' and consumers' surpluses is typically called "TotalSurplus" and is frequently used as a measure of the market's efficiency.


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Q

P

D

Qc

Pc

PS

CS S

Equations - Linear

Demand P a bQ= −

Supply P c dQ= +

Equilibrium a bQ c dQ− = +

( )b d Q a c+ = −

Qa c

b d

c=

−

+

P ad bc

b d

c=

+

+


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Competitive Pricing Demonstrated:

The Oral Double Auction

Three questions:

1. Does theory correctly predict price?2. Does theory correctly predict quantity?3. Is the market efficient?

Typically, these markets do conform to the predictions of the competitivemodel, even with relatively few players. This seems independent of shapeof demand and supply, though the path to convergence of price (i.e. is itfrom above or below) can depend upon shapes. (convergence from above

if consumer's surplus is greater than producer's)


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Class #1 Problems:

1. Let the market demand curve be: X = 20 - 2Px + 10Py (where Y is some other good and Py its price -- currently stable at Py = 2)Let the market supply curve be: X = 3Px - 15Solve for competitive equilibrium price (Px*), quantity (X*), consumers' surplus (Z*),and producers' surplus (S*)

2. Continuing with the market in question 1 -- assess the impact on price and quantityof the following government actions:

2.1 A tax imposed on sellers of 1 per unit.2.2 A price ceiling of Pc = 8 .

Also, how could you determine if these policies were socially beneficial?

2.3 What would the effect on price be of an increase in the price of Y to Py = 4?

3. An individual has a demand for a particular kind of T-shirt given by X = 5 - Px/2.Your company is thinking about selling the shirts in packages of two. What would

the demand curve look like for the first one and two packages -- i.e. what would thistypical buyer be willing to pay for one package and for two packages?

4. What would happen to (and explain why):

4.1 the price of cars if OPEC succeeded in dramatically increasing the price ofoil?

4.2 the price of domestic automobiles if the Voluntary Export Restraints on Japanese manufacturers were tightened?

4.3 the price of beef if new research showed it to be a much less important cause

of health problems than previously believed?


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An Aside on Marginal Analysis and Derivatives

Much of economic analysis involves looking for optimal values of somevariable and this usually means understanding the benefits and costs of small changes in that variable.

Assume we must make a decision about some variable x, call it output, andour objective is to maximize some value π(x), call this profit. Then wewant to consider increasing x as long as this increases π and stop when πno longer rises. We look for a stationary value of π.

The change in profit produced by a one unit change in x is called themarginal profit and can be denoted Mπ. It is also the derivative of the profit function: dπ/dx or π'. If we write profit as revenues R(x) minuscosts C(x): i.e. π(x) = R(x) - C(x)then we see that marginal changes in profit will equal marginal changes inrevenue minus marginal changes in costs: Mπ = MR - MC

Small changes in quantity lead to changes in revenues and costs. If MR >

MC then the expansion yields more new revenues than costs and it is profitable. If MR < MC then it yields more costs than revenues and it isnot profitable. Profit maximization will occur when MR=MC.

Notice:1. The slope of the total curve is the marginal curve.2. The marginal revenue curve will typically be steeper than the demand

curve. e.g. For linear demand with slope = -b

p = a - bxTR = px = (a-bx)xMR = a - bx - bx = a - 2bx →→ slope = -2b (twice as steep)


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See the slope of total revenue = marginal revenue

π

x

MC

MR,MC

TRx

MR

{

$

x*

TC

Profit is maximized where MR = MC.


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Topic 2: Consumer Behaviour

Purpose of this lecture is to learn more about how consumers make thechoices that we represent with demand curves -- so we can better understand how and why curves have the shapes they do and react tochange they way they do.

Model: An simplified and sometimes abstract representation of something more complex.

Model of Consumer Choice

The consumer has preferences which she applies to make choices fromfeasible sets. It is her choice that we observe and her choice is the productof both her tastes (preferences) and the constraints she faces or her opportunities.

Assumed Properties of Consumer Preference

Notation: Let A, B & C represent hypothetical bundles of commodities.Let the preference relation be denoted š

(so A š B means "A is at least as good as B")

Property 1: Completeness -- For any two bundles, A & B, either

A™ B, B ™ A, or A ~ B.

Property 2: Transitivity -- If A š B and B š C, then A š C .

Property 3: Continuity -- the set of bundles not better than a particular bundle is closed


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Utility Functions

Debreu showed that if preferences satisfy Properties 1 - 3, we can represent preferences with a utility function. (But not unique!)

Additional Frequently Assumed Properties

Property 4: More is Better (Nonsatiation) -- If A has at least as much

of every good as B and more of some good, then A ™ B.

Property 5: Convexity or Diminishing Marginal Rate of Substitution-- a consumer indifferent between A and B would strictly prefer a linear combination C = αA + (1-α)B to either Aor B .

Property 6: Differentiability -- the utility function is differentiable.

Results:

Properties 1-3 give us a utility function that we can describe with familiesof (noncrossing) indifference curves. Property 4 tells us that utility isincreasing in quantity and that indifference curves must therefore havenegative slope. Property 5 tells us that indifference curves must be convexto the origin and Property 6 tells us that we can determine the slope of theindifference curve from the ratio of marginal utilities.


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Indifference Curve:A curve that defines the combinations of two or more goods that give theconsumer the same level of satisfaction (utility). Each level of utility hasits own associated indifference curve. e.g.

Food

Clothing

u1

25

2

4

10

u0

u2

More formally, if there are two goods, x and z, and a utility functionU(x,z), the indifference curve associated with a utility level of 10 would besimply given by: 10 = U(x,z). So if U = x + z, the indifference curvewould be the straight line: 10 = x + z . (Question -- what property abovewould these preferences not strictly satisfy?)

Marginal Rate of Substitution:The rate at which a consumer can substitute one good for another and stillmaintain the same level of utility. This is the (absolute value of the) slopeof the indifference curve. Formally, we derive it by total differentiation of the utility function:

dU = Uxdx + Uzdz set = 0, so dz/dx u* = - U x/Uz

where Ux = ∂U/∂x & Uz = ∂U/∂z , the marginal utilities of x and z

Note:MRS is not constant -- depends on which curve you are on, andwhere you are on that curve.


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Preferences and Indifference CurvesWhat are the general shapes of indifference curves associated with thefollowing preferences:

1. U = 3x + 2z

2. U = min {x,z}

3. U = xz

4. U = 10(xz)2


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Constraints

Consumers have limited resources (income, time) with which to purchasetheir preferred bundles. These resources define what set of bundles arefeasible. Consider the budget constraint:

Pxx + Pzz ≤ M where M is income.

This defines the shaded feasible region below:

x

z

M/Pz

M/Px

The boundary of the feasible set is given by

Pxx + Pzz = M , or z = M/Pz - (P x/Pz)x .

and nonsatiation tells us that consumers will always choose to be on that boundary. Note that the slope of the budget line will be equal to thenegative of the ratio of prices -- this is a sort of "market rate of substitution" between the goods.

Consider the effect on the budget set of:1. A change in income M .2. A change in either (or both) price.


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Consumer Equilibrium -- Tastes meet Constraints!

The consumer wants to achieve the highest level of utility feasible. Thismeans that she wants to get on the highest indifference curve that still hassome point common with the feasible set. In general, this impliessomething like:

z

• A

B •

C •

x

u*

Points A, B, and C all exhaust the full budget, but at point A, theindifference curve associated with utility u* is just tangent to the budgetline. No higher indifference curve is feasible. At A, the slope of theindifference curve (the MRS) is just equal to the slope of the budget line(the price ratio).

The consumer equilibrium is achieved when the consumer selects the bundle that yields her the greatest utility among those bundles that lie inher budget set.


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Comparative Statics

1. Changes in income:

x

z

M1/Px

• B

• A

M0/Px

u1

u0

If the quantity purchased of a good rises with increases in income, we callit a normal good. If the quantity purchased falls with increasing income,it is called an inferior good.

2. Changes in the price of one good:

Changing a single price rotates the budget line. It rotates out if the price

fell, in if the price rose. Think about the effect of the price change on thequantity demanded of that good, and on the quantity demanded of theother good. The law of demand suggests that price and quantity areinversely-related.

x

z

M/Px0

• A• B

M/Px1


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Substitutes: Goods for which an increase (decrease) in the price of one leads to an increase (decrease) in consumption of theother. e.g. different brands of some product

x

z

M/Px0

• A

• B

M/Px1

Here z is a substitute for x.--------------------------------------------------------------------------------------

Complements: Goods for which an increase (decrease) in the price of one leads to a decrease (increase) in consumption of theother. e.g. pretzels & beer

x

z

• B

• A

M/Px0 M/Px

1

Here, z is a complement of x.


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Lesson: It is very important for managerial decision makers to be awareof the substitutes-complements relationships involving the products theysell. You must be able to anticipate the effects of changes in the pricesothers charge for their products on the demand for your products, and youmust be able to recognize how prices on each of your goods affects thequantities purchased of your other goods.

Substitution and Income Effects of a Price Change

When a price changes, there are really two effects on the quantity

purchased of that good.

Suppose the price of x increased -- then we would observe

1. Consumers substitute toward z because on the margin it is a cheaper way to get utility. -- Substitution Effect

2. Consumers have less real income due to the higher prices and

therefore adjust consumption to compensate -- Income Effect

A = Initial BundleB = Bundle if change in price compensatedC = Final BundleA to B →→ substitution effectB to C →→ income effect

x

z

• C• A

M/Px0 M/Px

1

• B


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Deriving Individual Demand Curves

We simply solve for the optimal amount of x purchased for all different prices of x, holding other prices and income constant.

x

Dx

Px

x

z

x2

Px3

x1 x3

Px3

Px2

Px1

Px2 P x

1

U2 U 1

U3


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Analytical Solutions

You can solve for the demand analytically by maximizing utility subject toa budget constraint. To do this you can use Lagrangean techniques or usesubstitution. To use the latter, simply use the budget line to substitute for one of the goods; e.g. z = M/Pz - (P x/Pz)x

To maximize U = xz subject to Pxx + Pzz = M

sub in for z, then maximize U = x[M/Pz - (P x/Pz)x]

f.o.c.: Ux = 0 = x[-P x/Pz] + [M/Pz - (P x/Pz)x]

Result: x* = M/(2Px) & z* = M/(2Pz)

Market Demand

The Market Demand Curve simply sums up Individual Demand Curves

horizontally: i.e. for two buyers G and H

x

px

5 7 10

DH x

px

5 15

DG

10

8

4

x

px

5 12 25

Dmarket

+ =


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Applications of the Framework:

1. Labour-Leisure Choice

2. Cash vs. Gifts in Kind

3. Quantity Discounts


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Class #2 Problems:

1. What will the quantity demanded of x and z be by a consumer with income 100,facing prices Px = 2, Pz = 1 with a utility function U = 4x + z ?

2. If x and z are perfect complements with U = min {x,y} and the price of x rises by 1,how much must the seller of z lower its price if it wishes to sell the same quantity asbefore? Why?

3. Consider the example above in which U = xz.

3.1 Are x and z substitutes or complements?

3.2 The income elasticity of the demand for a good is the % change in the quantitydemanded due to a 1% increase in income (holding prices fixed). What is theincome elasticity of demand for x and what is it for z?

3.3 Harder: The own price elasticity of demand is the % change in quantitydemanded of a good due to a 1% change in its price. In differential form this can berepresented by: ex = -(∂x/∂Px)(Px/x). What are the own price elasticities of demand

for x and for z?

3.4 What would the demand curves for x and z look like if we changed the utilityfunction to U = 15 + 4(xz)½ ? Hint: you do not have to grind this out to answer thequestion correctly.

4. An employer is considering offering his workers overtime pay to encourage them towork extra hours and increase output. Illustrate a case in which the overtime offerdoes induce workers to work more hours and illustrate a second case in which the

overtime pay does not lead to more hours being supplied by each worker.


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Topic 3: Costs and Production

The purpose of this lecture is to get behind the supply curve, to develop a better understanding of how costs and production opportunities determinethe prices at which firms will supply to the market.

Types of Productive Efficiency

1. Static Efficiency: finding the least cost way to produce a given set of products with current technology and inputs.

2. Dynamic Efficiency: finding the least cost way to produce productsover time, given the evolving state of technology and changingavailability of inputs

This lecture is about static efficiency.Typically a firm should pursue both, but they may conflict. How?

Production Functions

Firms have costs because they employ factors of production like labour,machinery, energy etc. They use factors as inputs into a production process which we represent as a production function, e.g. if there were justtwo inputs, capital (K) and labour (L) we might write:

Q = F(K,L)

Production function: defines the maximum amount of output(s) that can be produced with a given set of inputs.

Note: The PF does not tell you the cost of producing the output.


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Sample Production Functions - What are the functional forms?

1. It takes one person with one shovel one hour to dig a hole. What isthe production function for holes dug per hour? Let L = the number of people, S = number of shovels, and Q = output.

2. A certain routine production process can be done either by a person(L) or a machine (K). Machines and people do not work together butthey can work side-by-side. Machines are twice as productive -- theycan make two units per hour.

Answers

1. Q = min {L,S}

Note: the form Q = min {aK, bL} is called the "Leontief production function" or "fixed-proportions productionfunction".

2. Q = 2K + L

Note:the form Q = aK + bL is called a "linear production function".


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Variable vs. Fixed Factors

In the short run some factors are fixed and the manager can only makedecisions about the levels of other, variable, factors. If capital is fixed wemight write:

Q = f(L) = F(K*,L)

Key concepts in measuring productivity:

Total Product: Usually just the total output achieved from a givenset of inputs.

Average Product: The average product of a factor is output produced per unit of that factor.

e.g. APL = Q/L AP K = Q/K

Marginal Product: The change in total output due to the addition of one more unit of an input.

e.g. MPL = ∆Q/∆L MPK = ∆Q/∆K

or with small changes: MPL = ∂Q/∂L MPK = ∂Q/∂K

While average and marginal products can in principle go up or down,it is generally expected that the marginal product will begin to fallafter some point:

The Law of Diminishing (Marginal) Returns

Question: What are the marginal products of the inputs in the Leontief and linear production functions?


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A Production Function (from B-B, Table 5-1):

K* L ∆∆L Q MPL APL

2 0 - 0 - -

2 1 1 76 76 76

2 2 1 248 172 124

2 3 1 492 244 164

2 4 1 784 292 196

2 5 1 1,100 316 220

2 6 1 1,416 316 2362 7 1 1,708 292 244

2 8 1 1,952 244 244

2 9 1 2,124 172 236

2 10 1 2,200 76 220

2 11 1 2,156 -44 196

Marginal Returns to Labour are:

(i) Increasing: from 0 to 5 units of labour

(ii) Decreasing: from 6 units upwards

(iii) Negative: at 11 units and (probably) higher

Questions:1. When is labour productivity at its maximum?2. How do you decide how much labour to employ?3. What is the relationship between AP and MP?


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A picture:

Total,

average,& marginal

product

L

Increasing

marginal

returns (MR)

Decreasing MR Negative MR

Q

APL

MPL

How much of an input to use?

Again this is a "marginal benefit vs. marginal cost" type of decision. Mustcompare the benefit of one more unit of input to the cost.

MB: the benefit of one more unit of an input is simply the revenue youwill earn from the extra output you get. For a competitive (i.e. price-taking) firm, this is just the market price of the output

time the number of extra units of output. This is called thevalue of the marginal product (VMP).

i.e. for labour: VMPL = P(MP L) where P is the price of output

MC: For a price-taking firm, the cost of one more unit of an input issimply the price of that unit (call it w).

Rule for Profit Maximization: H ire more if VMP L > w , fewer if VMP L< w. Stop when VMP L = w as long as VMP L is decreasing.


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Back to the Production Function Table:

L Q P MPL VMPL

0 0 3 - -

1 76 3 76 $228

2 248 3 172 516

3 492 3 244 732

4 784 3 292 876

5 1,100 3 316 948

6 1,416 3 316 9487 1,708 3 292 876

8 1,952 3 244 732

9 2,124 3 172 516

10 2,200 3 76 228

11 2,156 3 -44 -132

Thus, the VMP curve gives us the firm's (derived) demand for the input,since for any given input price it tells us how much of that input we want:

L

VMPL

L*

w*

VMPL


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The Classic Production Function:

The Cobb-Douglas Production Function is given by:

Q = K aL b

This functional form allows for substitution between inputs (unlikeLeontief) but at varying rates (unlike linear).

Marginal Products with a Cobb-Douglas production function:

MPL = ∂Q/∂L = bK aL b-1 (= bQ/L)

MPK = ∂Q/∂K = aK a-1L b (= aQ/K)

Notice that the marginal product of each input:

(i) goes down as more of that input is added (diminishing returns) (if a


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Returns to scale:

If you increase all inputs by the same proportion, does output go up by thesame, a smaller or a greater proportion?

i.e. if s > 1 will F(sK, sL) > = or < sF(K,L) ?

If greater: increasing returns to scale If smaller: decreasing returns to scale If same: constant returns to scale

What kind of returns to scale are exhibited by the following productionfunctions:

1. Q = min {aK, bL}

2. Q = aK + bL

3. Q = K aL b

For Cobb-Douglas -- if we increase both K and L by a factor of s weget:

(sK) a(sL) b = s a+bK aL b

so returns to scale depends on whether a+b>1 or


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Varying Both Inputs

Isoquants: Defines the combinations of inputs that yield the samelevel of output.

L

K

22

4Q = 8

6

10

Q = 5

Q = 3

Equation of Q* isoquant: Q*=F(K,L) -- e.g. if F(K,L) = 5K + 3L,isoquant for Q = 10 would be:

10 = 5K + 3L or K = 2 - .6L

Marginal Rate of Technical Substitution (MRTS):The rate at which a producer can substitute between two inputs andmaintain the same level of output. (The slope of the isoquant.)This is typically not constant along an isoquant or between isoquants.

MRTSKL = MPL/MPK

To see this, recognize that ∆Q = ∆L(MPL) + ∆K(MPK )

For this to be equal to 0: (∆K/∆L)Q* = MRTS KL = MP L/MPK


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Cost Minimization:

A firm cannot maximize its profits without minimizing the costs of producing whatever quantity it chooses to produce. So we need to knowwhich point on the isoquant involves the lowest cost.

Isocost Line:A line that represents the combinations of inputs that will cost the producer the same total amount.

e.g. C = wL + rK or K = C/r - (w/r)L

where w is the wage rate for labour, r the cost of capital. With constantinput prices, isocost curves are straight lines with slope = -w/r.

L

K

C = $1,000

C = $1,800

e.g. if w = 2 and r = 5, the isocost line for $1,000 would be:

$1,000 = 2L + 5K or K = 200 - .4L

Like budget lines, isocost lines rotate if individual prices change andisocost lines for larger cost levels are just parallel shifts outward.


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Costs Meet Technological Possibilities

To determine how to most cheaply produce any level of output we have tofirst know what combinations of inputs can produce that level (i.e. what isthe isoquant?) and then we need to find the least cost of those alternatives.In other words we need to find the point on the isoquant that is on thelowest isocost line:

L c1 c 2 c 3

A

QD

B

c3 > c2 > c1

K

Notice that the least cost point represents a tangency of the isoquant withthe isocost line so we know the two slopes are equal.

slope of isoquant (MRTS) = MPL/MPK = w/r = slope of isocost

Notice that this is the same as saying MPL/w = MPK /ri .e. the last dollar spent on each input should be equally productive.


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More Formally:

To minimize costs but still produce Q* we can use constrainedoptimization to

minimize C = wL + rK s.t. Q*=F(K,L)

Notice that we could have also maximized the output from any given cost(just like consumer problem). Then we can substitute the isocost line in just like the budget line:

Max Q = F(C/r - wL/r, L)

to maximize set dQ/dL = 0: ∂F/∂K (-w/r) + ∂F/∂L = 0

or: MPL/MPK = w/r as above.

Questions:

1. Suppose all inputs are perfectly divisible. For your firm the currentMPK is 8 and you pay $4 per unit of capital. The current MP L is 3 and w =$1. How could you move the last dollar spent on inputs to get more outputfor the same cost? How much more output would you get? How muchcould you save if you wanted to produce the same level of output?

2. If the government takes steps to raise the wages of some workers, willthat necessarily raise the incomes of those workers? Why might it not?


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Question:How will the cost-minimizing bundle of inputs change as the price of oneinput rises? Hint: This is like the substitution effect from consumer theoryand we expect it to be negative -- a higher input price leading to less use of that input.

L

K

Q*

A

B

L0 L1

Obviously, the extent to which the producer can switch in response toinput price changes will depend upon the technological possibilities for substitution. Think about Leontief production functions and linear

production functions.

L

K

Q2 L

K

Q1

Q2

Q1


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The Cost Function

Now we see how firms can choose inputs to minimize the cost of producing any given quantity Q. We can summarize this process of costminimization with the Cost Function: C(Q). The cost function gives thetotal cost associated with the production of Q units assuming costs areminimized as above.

Short-Run vs. Long-Run Costs

In the short-run some inputs quantities are fixed to the firm (e.g. ??),

leaving the firm less flexibility to minimize costs.

Fixed Costs (FC): Those costs that do not change with changes inoutput. While most costs can be varied in the longrun, many important costs cannot be varied quickly.

Variable Costs (VC(Q)): Costs that change with changes in output. In

the long run (almost) all costs are variable.

A related concept:

Sunk Cost: A cost that is forever lost once it is paid. If you can avoidall fixed costs by shutting down, then there are no sunk costs. If you must incur all fixed costs even if you shutdown (e.g. a lease that cannot be broken) then all fixedcosts are sunk. Sunk costs are very important

strategically.


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Short-Run Cost Function: Gives the minimum possible cost of producing each level of output whenvariable factors are employed in thecost-minimizing way.

Obviously, we can have different short-run cost functions as we allowdifferent periods for inputs to vary.

Total Costs: C(Q) = FC + VC(Q)

Illustrating the cost curves:

Q

Costs

FC

FC

C

VC

Average and Marginal Curves:(See tables 5-4 and 5-5 for a numerical example deriving these costs.)

Average Fixed Cost AFC = FC/Q

Average Variable Cost AVC = VC(Q)/Q

Average Total Cost ATC = C(Q)/Q

Marginal Cost MC = ∆C(Q)/ ∆Q = dC/dQ for small ∆(the cost of producing one more unit of output)


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Relating these measures of costs to each other:

1. C(Q) = FC + VC(Q)

2. ATC = AFC + AVC

3. MC is also = ∆VC(Q)/∆Q

4. When MC > ATC, ATC must be rising, when MC < ATC, ATC must be falling.

5. When MC > AVC, AVC must be rising, when MC < AVC, AVC must be falling.

6. ATC and AVC get closer as Q goes higher.

AFC

Q

Costs

AFCAVC

ATC

MC


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Long-Run Cost Function

Obviously, since in the long run we can adjust the use of all inputs, in thelong-run we can generally produce with lower cost. So long-run costs areless than or equal to short-run costs and equal only if the variables fixed inthe short-run happen to be fixed at exactly their cost-minimizing levels.

Long-Run Average Cost Curve:A curve that defines the minimum average cost of producing the variouslevels of output allowing for the optimal selection of all factors of production.

Average

Costs

SRAC1

Q Q3 Q2 Q1

LRAC

SRAC2

SRAC3


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Class #3 Problems

1. Cubic Cost Function: C = f + aQ + bQ2 + cQ3

BB claim that this is a good approximation to most cost functions. Thereis some sense in this -- the cubic form allows the function to take manyshapes as the parameters f,a,b,c change.

For this function: derive the expressions for ATC, AVC, AFC, MC

2. Consider the production function Q = KL. With this function howmuch will you spent on labour relative to what you spend on capital if youwant to minimize costs?

3. Consider the cost function C = 64 + 7Q + 4Q2. At Q = 3, what areATC, AVC, AFT and MC? At what quantity is ATC at a minimum?

4. A hint of things to come. Suppose you have a cost function of the formC = 20 + 10Q + 5Q2 and the current market price is 50. You are acompetitive firm that is too small to materially influence market price.

4.1 How much would you choose to produce and sell?4.2 At that rate of output, what would your revenues, costs and profits be?4.3 What output would you produce if fixed costs went up to 50?And what if they went up to 100?


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Profit Maximisation for a Competitive Firm:

Total Revenue = price x quantity sold = pQ

Total Cost = C(Q)

Profit = total revenue - total cost = pQ - C(Q)

In the graph below, profits are the vertical distance between the totalrevenue curve and the total cost curve. This distance will be greatest whenthe slopes of the two curves are equal --

i.e. Profit maximisation implies that the added (marginal) revenue fromone more unit sold -- which is just equal to the price for acompetitive firm-- should be set equal to the added (marginal) cost of producing that unit

Q

Revenues,

Costs

C(Q)TR (Q)

π {


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Supply Curve of a Competitive Firm:

Consider the effect on profits of a price-taking firm increasing productionfrom 0 -- its marginal benefit is its marginal revenue which here is price.Its marginal cost is its MC. So as long as P>MC it should expand, if P


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Illustrating a profit maximizing firm:

Costs

Q Q*

Profits

AVC

ATCP*

ATC(Q*)

MC

Should shut down if that is more profitable. It will be more profitable toshut if p


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Elasticity of Supply:

This measures the sensitivity of supply to changes in price. It is defined asthe percentage increase in quantity supplied as a result of a 1% increase inmarket price:

Elasticity is related to slope but is not equal to the slope -- it is unit-free.

elastic supply ⇒ large changes in quantity for a given price change

inelastic supply ⇒ small changes in quantity for a given price change

Q

P SI

SE

Q

P

Elastic Supply Inelastic Supply

se

Q

Q

p

p

=Q

p p

Q≡

∆

∆

∆

∆


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Shifting Supply Curves:Anything that alters marginal costs will change supply curves, but changesin fixed costs can also affect supply by altering the entry and exit points:

Q

P SS1

Q

P

S0

S1 starts here

S0 starts here

Higher wage rates Increase in cost of office space

Market Supply in a Competitive IndustryThis is simply the horizontal summation of all the individual supplycurves:

p

12 18

S2

Q1

p

10 25

S1

QM

p

22 43

Smarket

+ =

Firm 1 Firm 2 Market

Q2

Entry will move market supply outward as a new firm supply is added.


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Economies of ScaleAn important property of long run ATC curves is whether they are upwardor downward sloping -- that is, does expanding production raise or lower unit costs.

Economies of Scale: These exist whenever long-run average total costs decline as output increases.

Diseconomies of Scale: These exist whenever long-run average total costs increase as output increases.

ATC

Economies

Q Q*

ATC

Minimum Efficient Scale(MES)

Diseconomies

This is clearly related to the concept of "returns to scale" discussed earlier -- but they are not exactly the same thing. In general:

increasing returns to scale ⇒ economies of scale

decreasing returns to scale ⇒ diseconomies of scale

constant returns to scale ⇒ constant average cost

Question: when would these relationships not necessarily hold?


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Multiproduct Cost Functions

Many (maybe most) firms produce not just one product, but a variety of products. While they may be closely related, but they differ in some waysincluding costs. The multiproduct cost function defines the cost of producing different combinations of outputs assuming all inputs are usedefficiently.

e.g. a telephone company produces Q1 units of local calls and Q 2 units of long-distance calls at a cost given by: C(Q1,Q2) .

Such cost functions are not necessarily the result of simple addition of thecosts of good 1 and the costs of good 2:

Economies of Scope: Economies of scope exist when the cost of producing a group of products together is lessthan the total cost of producing themseparately.

i.e. C(Q1,Q2) < C(Q1,0) + C(0,Q2)

e.g. suppose local and long distance services share a common fixed costand have a cost function like:

C(Q1,Q2) = F + aQ1 + bQ 2

then to produce the outputs separately would cost:

C(Q1,0) + C(0,Q2) = F + aQ1 + F + bQ 2 = 2F + aQ 1 + bQ 2

Economies of Scope are an important determinant of the scope of the

firm: the set of products it produces and sells.


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Primary Sources of Economies of Scope

1. Common fixed costse.g. shared facilities like phone lines

2. Cost complementaritiesThese exist when the marginal cost of producing one output isreduced when the output of another product is increased.

i.e. ∆MC1(Q1,Q2)/∆Q2 < 0

e.g. leading examples come from "joint products"doughnuts and doughnut holescattle hides and beef

A Cost Function that allows for both:

The Quadratic Multiproduct Cost Function

C(Q1,Q2) = F + aQ1Q2 + (Q 1)2 + (Q 2)

2

SoMC1(Q1,Q2) = aQ2 + 2Q 1 and MC 2(Q1,Q2) = aQ1 + 2Q 2

Results:

1. Cost complementarities if a < 0

2. Economies of scope if F - aQ1Q2 > 0.

To test your understanding, try Demonstration Problem 5-11 on p. 193.


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Unit Cost Measures with Multiproduct Firms

Question: What does "average cost" mean for a multiproduct costfunction?

With some costs common to a set of products, there is no good way todefine average cost. There is no precise way to allocate the common costsacross products -- any allocation is arbitrary. Still it is useful to have somemeasures of unit costs -- and in regulatory industries where prices arecontrolled it is often required.

Fully Distributed Average Cost

Average Incremental Cost

Marginal Cost

11 1

12

2 2

2

FDAC = F + VC ( Q )

Q FDAC =

(1- )F + VC ( Q )

Q

α α

11 2 2

1

AIC =C( Q ,Q ) - C(0,Q )

Q

11 2

11 2 1 2 MC =

C( Q ,Q )Q

C( Q +1,Q ) - C( Q ,Q )∂∂

≈


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Topic 4: Competition, Monopoly,Monopsony

and Monopolistic Competition

Equilibrium in Competitive Markets:

Recall the assumptions:1. There are many buyers and sellers and each is small relative to the

market.2. Homogeneous products.3. Full information.4. No transaction costs.5. Free entry and exit.

Assumptions 1-4 make all buyers and sellers price-takers. Assumption 5guarantees that there will be no profit opportunities left for entrants andthat active firms will not be losing money in equilibrium.

To be a price-taker is to perceive that you face a flat demand curve (if youare a seller) or a flat supply curve (if you are a buyer).

Again we see that equilibrium is obtained when the quantity demandedequals the quantity supplied.

QM

P S

QF

P

D

Firm Market

DF


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If the equilibrium price allows sellers to make positive economic profits,we can expect entry. This entry will shift the supply curve outward,lowering price. We can expect price to fall until there are no economic profits left to be made.

Long run competitive equilibrium:

Q

P S0

Q

P

D

S1

QM Q F

Pe

ATCMC

For the Market For the typical firm

Two Important Properties of this Equilibrium:

1. Price = Marginal Cost (for all firms)This has nice efficiency properties since the price is really the socialmarginal value of the good and marginal cost the social cost of thelast unit. To produce less or more would lead to social losses.Consumers' plus producers' surplus is maximised.

2. Price is at the minimum of average cost Meaning that firms are making zero economic profits and alleconomies of scale have been exhausted.


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Monopoly

We often speak of monopoly power whenever firms have some controlover price but literally ---

Monopoly: A market structure in which a single seller serves an entiremarket for a good that has no close substitutes.

So a monopolist is not a price-taker -- it recognises that it can influence the price at which it sells by changing the quantity it puts on the market. Itsees the market demand as its own demand.

The term monopoly power is often applied when a firm has some controlover price (not necessarily a strict monopoly) and can raise price abovecost and make positive economic profits.

Some Sources of Monopoly Power

1. Economies of Scale (e.g. Natural Monopoly)

2. Government -- licenses, patents etc.3. Collusion

Examples

• "Natural" monopolies in electricity, telecommunications, gas• patented medicines• marketing boards in agricultural products

• cartels in electric lamps, sugar, hotels, and recently: fax paper, compressed gas


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So optimal Q comes where this MR equals MC

Q

P MC

P*

D

MC=MR

P*

Q*

MR

We can either think of the monopolist as choosing Q* or as choosing price p* -- they are equivalent here.

If we draw in the other cost curves, we can illustrate profit.

Q

P

P*

D

ATC

P*

Q*

ATC

MR

MC

Note: The monopolist does not, not, not have a supply curve!

Question: What are the profit maximizing output and price if p= 170-4Q and C=10Q?


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Marginal Revenue and Elasticity of Demand:

Note that we can write MR

so MR is only positive when elasticity of demand is greater than 1 -- so amonopolist never prices in the inelastic region of its demand curve.

A Simple Expression for Monopoly Markups

set MR = MC ⇒ p + Q( ∆ p/∆Q) = MC

we can rearrange this expression to yield:

Therefore the monopoly markup is simply the inverse of the elasticity of demand -- the more elastic the demand the lower the markup.

p + Q p

Q = p[1 +

Q

p

p

Q ] = p[1-

1

e ]

∆

∆

∆

∆

p - MC

p =

p

Q Q

p =

1

e

∆

∆


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Marginal Revenue and Elasticity with a Linear Demand Curve

Let p = a - bQ so we know that MR = a - 2bQ

Linear demand or not, we know that TR is maximized when MR = 0. Andwe know that when MR = 0 elasticity = 1.

With linear demand MR = 0 when Q = a/(2b) which is exactly themidpoint of the demand curve (verify this for yourselves). Thus we have:

TotalRevenue

Q

P

MaximizedRevenues

a/(2b)

Elastic Demand

e > 1

a/b

Unitary Elasticity

e = 1

Inelastic Demand

e < 1

a/2b a/b


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Monopoly Allocation of Production Across Plants

Let Q = Q1 + Q 2 be the total output from two plants

How does the monopolist decide to allocate production between plants if each has its own marginal costs MC1(Q1) and MC2(Q2) ?

Think of this as the monopolist producing two outputs with the samemarginal revenue -- so it will want to set:

MR = MC1 = MC 2

To understand this, think about how you could produce the same outputfor less total cost if MC1 > MC 2 .


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Monopsony

The buyer counterpart to monopoly. Typically one buyer facing acompetitive supply situation. Buyer recognizes that the more it buys thehigher the price it will have to pay for each. Let p(Q) be the (inverse)supply curve giving the price it will have to pay per unit for Q units.Assume buyer a firm that puts a value on each unit of v (it can resell at v sohas a horizontal demand at v). In other words its marginal benefit of a unitis v and total benefit from Q units will be vQ.

Its profit from buying Q units will be: π = vQ - Qp(Q)

maximize profit where the MB = MC i.e. v = Q(∆∆p/∆∆Q) + p > p

so it does not go to the competitive point where p = v

⇒⇒ it "underbuys"

Q

$

Qc

S

Q*

V

P*

MC to Monopsonist


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Monopolistic Competition

Conditions:1. Many buyers and sellers.2. Each seller produces a differentiated product.3. Free entry and exit.

Examples of such industries?

Results:

(i) Sellers have some control over price, though demand can be veryelastic.

(ii) Free entry/exit means profits go to (or very near) zero.

Short Run EquilibriumThis is the same as monopoly -- firm sets MR=MC. But positiveeconomic profits will lead to entry. This will shift each brand's demand

curve (and MR) inward leading to lower profits.

Output

Brand i

$

P0

Di

Q0

MR i

ATC

Profits ofBrand i

MC


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Long Run EquilibriumEntry continues until brands are just breaking even. Notice that thiscannot be at the bottom of the ATC curve.

Qi

$

Pi*

Di

Qi*

ATCi

MR i

MCi

Properties of this Monopolistically Competitive Equilibrium

1. p > MC

2. p = ATC but ATC not at its minimum (excess capacity result)

3. 1. and 2. suggest inefficiency but we have "variety" and this is agood thing -- whether or not the outcome is inefficient will dependupon how much variety is valued -- can end up with too many or toofew brands

Note also that firms in these kinds of industries have incentives toundertake marketing activities (e.g. advertising) while those in

competitive and monopoly industries do not. If some firms are better at this than others they could be left with some monopoly profits -- atleast in the short run.


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Class #4 Problems:

1. Suppose the MC curve for a competitive firm is MC = 100 + 20Q.Write down the expression for the market supply curve (i.e. Q = f(p)) if there are 100 such firms. (Assume there are no fixed costs.)

2. Suppose your company assembles bikes. To assembly one bike youneed one hour of labour and one bike kit. The labour costs you w per hour and each bike kit costs k. What is your supply curve? Illustrate. Illustratewhat happens to your supply curve if you give your workers a raise?

3. A monopolist faces a demand curve P = 2,000 - 25Q and a total costcurve of C = 500Q. What are the profit maximizing price and quantity?What will its profits be?

4. If a monopolist has a demand with an elasticity = 5, what will itsmarkup over marginal costs be? If MC = 4, what is price?

5. A trickier one. You are a monopolist operating two plants with the

following MC curves: MC1 = 10 + Q 1 and MC 2 = 2Q 2 . Illustrate thetwo MC curves graphically. If you want to produce 80 units at the lowestcost, how should you allocate production between the two plants? (Hint: You need two linear equations to solve for your two unknown quantities,Q1 and Q 2. One equation comes from the condition for optimal allocationdiscussed in the lecture, the other comes from the condition that Q1 + Q 2 =80.)

6. Monopolistic competition: Suppose ATC = 4 + (x-10)2, market

demand is given by P = 1000 - 8X, and all individual firm demand curveshave slope = -8 too. What P and x will the individual firm set in zero profitequilibrium?


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Introduction to Game Theory

Outline

1. Usual Assumptions

• self-interested, rational agents

• agents capable of the most sophisticated reasoning

• structure of the game is understood by players

2. Terms

• game, player, strategy, payoffs, best response function

3. Categorizing Games

• Cooperative vs. Noncooperative

• sequential vs. Simultaneous move

• one-shot vs. Repeated, static vs. Dynamic

4. Solving Games

• dominance arguments (dominant strat., dominated strat.)

⇒ prisoner’s dilemma• Nash equilibrium

⇒ Cournot output game

• existence of Nash equilibrium

• multiple Nash equilibria

⇒ battle-of-the-sexes

⇒ coordination games

5. Extensive Form Games - the importance of timing


6. Credible and Incredible Threats – subgame perfection



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Topic 5: Introduction to Game Theory

Consider the following three decision problems:

1. VHS v. BETA and Standardization

2. Pulling your money out of a troubled bank

3. Bidding on a big contract

What do they have in common:

To make a good decision, you have to take into account the(unknown) actions of others. And these "others" are decision-makers like you.

What is "Game Theory"?:

Game Theory might be called "Interactive Decision Theory". It isthe study of the behaviour of decision makers ("players") whosedecisions affect each other. The analysis is typically from arational rather than psychological or sociological viewpoint.

Sometimes we say that it is the study of decision-making in"strategic" situations.


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Some Usual Assumptions:

! self-interested, "rational" agents! agents capable of the most sophisticated reasoning! structure of the game is understood by players

Glossary of Terms

Game:

a situation of strategic interdependence between 2 or more players -- the payoffs to each depending on that player's own choices (or strategy) and onthe choices of other players

Player:

an agent participating in the game; making choices and receiving payoffs(generally assumed to be rational i.e. self-interested)

Strategy:

a possible action or move available to some player; in games in which players

must make more than one move it is a vector of actions specifying what moveis to be taken in any eventuality. Think of a strategy as a set of instructionsthat you could leave for your lawyer to carry out without any further consultation with you.

Strategy Space or Set:

the set of all possible strategies for a given player -- e.g. S1 is the set for

player 1. (The term "strategy space" is sometimes reserved for the big set of all possible strategy choices by all players. i.e. if player one can choose

strategy A or B his strategy set will be {A,B}, if player two can choose C or D, her strategy set will be {C,D}; the strategy space of the game will be {AC,AD, Bc, BD})

Payoffs:

the reward or punishment result of a game; should be measured in utility butwe often use dollars.


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Payoff function:

a function that maps the strategies selected by the n players into a vector of

payoffs for all players, i.e. B(s1,s2,..sn) = (B1,B2,...Bn) . Will also write B1(s)as the payoff function for player 1.

Best Response (Reply) Function:

a function that gives, for a particular player, her best action as a function of the actions of all other players. In Cournot Oligopoly we call this the"reaction function".

Repeated Game:

generally a game in which some simple game is played more than once (ofteninfinitely many times); the simple game is called the "stage" game or the"constituent" game -- a game repeated infinitely is called a "supergame"

Rules of the Game:

a set of information that includes who the players are, who knows what andwhen, all the possible moves by all players, who moves when, and all the possible payoffs -- these rules are generally assumed to be common

knowledge


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Definition: A strategy for player i is a dominated strategy if there

exists some other strategy available to player i that yields her higher

payoffs for all possible strategies chosen by other players. ("weakly

dominated" allows some ties)

We can expect that dominated strategies wi l l not be played, so we

shoul d el iminate any dominated strategies from consideration, and

go on doing so successively.

If there are only two strategies and one is dominant, the other must be

dominated, but it is possible to have dominated strategies with no

dominant strategy. For example:

Player 2

Player 1C R

T 10,10 15,5 0,3

M 5,15 8,8 5,6

B 4,12 7,15 2,2

No dominant strategies, but -- B is dominated by M for Player 1 and R

is dominated by L and C for Player 2. Rule these out and you are left

with reduced game:

Player 2

Player 1L C

T 10,10 15,5

M 5,15 8,8

Now T dominates M and L dominates C, so we have outcome (T,L)Y

(10,10)


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Solving Games 2: Nash Equilibrium

Having exhausted the simple avenues of looking for dominant strategies

or ruling out dominated ones, the next thing to do is to look for an

equilibrium of the game.

Definition:A Nash Equil ibrium (NE) occurs when each player chooses

her strategy to maximize her payoff, assuming all the

other players are playing their NE strategies.

I nterpretations of Nash Equil ibr ium

1. Each player chooses her payoff-maximizing strategy given aconjecture about the strategies to be selected by all other players.When all the conjectures are correct, we are in a Nash equilibrium.

2. Nash equilibrium is outcome of nonbinding pre-play communicationor "cheap talk". That is, it is a self-enforcing agreement.

3. Nash equilibrium is result of a lot of jockeying around by agents --adjusting to each other's strategy choices -- NE is where this processsettles down. This is the old Cournot equilibrium story and it is rather silly.


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A simple example:

Player 2

Player 1C R

T 6,3 4,4 4,8

M 8,9 2,7 5,1

B 3,8 5,4 2,3

No dominant or dominated strategies. Simple NE in pure strategies at(M,L).

Best response functions:

R 1(L)=M R

1(C)=B R

1(R)=M & R

2(T)=R R

2(M)=L R

2(B)=L

Problems with Nash Equil ibr ium

1. Existence -- will one always exist?

2. Multiple Equilibria -- what if there are more than one?


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Problem 1: Existence of Nash Equilibrium

Do all games have NE? Consider the monitoring game below. Worker

must decide to work or shirk. Would prefer to shirk but knows that if

he gets caught he will be fired. Manager must decide to monitor or not

-- monitoring is costly but she knows it is the only way to detect

shirking:

Worker

Work Shirk

Monitor -1 , 1 1 , -1

Manager Shirk 1 , -1 -1 , 1

There is no pure strategy equilibrium here. But a NE will exist in

mixed strategies. If Manager expects Worker to play Work with

probability and Shirk with probability (1- ), the expect return to

playing Monitor and Don't for Manager is:

ER 1(Monitor) = (-1) + (1- )(1) = 1 - 2

ER 1(Don't) = (1) + (1- )(-1) = 2 -1

Note that ER 1(H)=ER

1(T) when = ½

So if Worker mixes 50-50, Manager is indifferent between Head and

Tail. Since game is basically symmetric, same is true for Worker.

So we have a Mixed Strategy Nash Equilibrium with each playing each

strategy with probability = .5.

Other examples from sports: pitching to a batter

serving to a tennis opponent


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An Aside on another kind of strategy:

Maximin Strategies

A conservative alternative to profit-maximizing strategies – player A’s

maximin strategy is the one that guarantees A the best outcome if the

other player plays the strategy that is worst for A. See this example

adapted from the book:

Player BLeft Right

Top 1,1 0 , 2

Player A Bottom -1000 , 0 2 , 2

If Player A chooses Top, the worst that can happen is that B chooses

Right and she gets 0. If Player A does bottom, the worst that can

happen is that Player B chooses left and A gets -1000. We see that Top

is the maximin strategy – it gets the maximum of the minimum payoffsin each row.

Similarly, the maximin strategy for B is to play Right.

So if they both play maximin strategies we go to Top, Right. Is this the

Nash Equilibrium?

Maximin strategies become more attractive if you are worried about the

rationality of the other player.


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Returning to the question of existence:

Theorem: (Nash) Every game with a finite number of players and

finite number of strategies will have at least one

Nash Equilibrium (though it may involve mixed

strategies).


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Extensive Form Games: On the importance of timing and sequence

How would these games be played differently if one player moved first

and that player's move was observed by the other? Recall the BOS :

Wife

Ballet Boxing

Ballet 2,6 0,0

Husband Boxing 0,0 6,2

or a coordination game:

Player 2

Join Network Don't Join

Join Network 10, 10 2,5

Player 1 Don’t Join 5,2 5,5

or a sequenti al bargaini ng game -- Players 1 and 2 must agree on a split

of $1. Contrast the outcomes if they must pick shares simultaneously to

what would happen if Player 1 could make a take-it-or-leave-it offer to

Player 2.


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Sometimes outcomes that would not even be Nash Equilibria of the

simultaneous move game can be Nash Equilibria of a sequential move

version of the same game.

A simplified Cournot output game --

Two firms choose output - three levels: Low, Medium or High

e.g.

Firm 2

Firm 1 Low Medium High

Low 80,80 40,90 30, 70

Medium 90,40 50,50 20,40

High 70, 30 40, 20 10,10

Questions:

1. What is Nash Equilibrium with simultaneous moves (i.e.

Cournot)?

2. What is Nash Equilibrium with sequential moves if 1 moves first

(e.g. Stackelberg)?

3. What is joint profit-maximizing?


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Extensive Form of Stackelberg example:

Recall normal f orm --

Firm 2

Firm 1Low Medium High

Low 80,80 40,90 30, 70

Medium 90,40 50,50 20,40

High 70, 30 40, 20 10,10


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Credible & I ncredible Threats: Subgame Perfection

While in a NE all players are acting rationally given the strategies of all

others, some NE can be supported by "unreasonable" beliefs about play

off the equilibrium path. Some would eliminate from consideration NE

based upon such beliefs or on "incredible threats".

Extensive form of Entry Game:

Entrant can decide to come in or stay out. Incumbent can decide to be

tough on entrant (if entrant chooses to enter) or to be soft.

Notice: 1. Two NE: (In, Soft) and (Out, Tough).

2. Incumbent would like entrant to expect it to be tough, to

deter entry -- but is this a reasonable belief for entrant?

3. Solving the game backwards -- backward induction --

reveals that if it enters, entrant should expect incumbent to

be soft. Being tough an "incredible" threat. This eliminates

second NE, leaving (In, Soft).

Lesson for Entrant: Do not believe incredible threats if you know your

rival to be rational. But you can still worry about

the "crazy" ones.


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Lessons for I ncumbent:

1. Could try to take advantage of entrant's imperfect information.

2. Can take actions to make threats credible -- e.g. excess

capacity.

Definition:

A subgame perfect equilibrium (SPE) is a NE in which beliefs are such

that players expect NE play in every subgame of the game (i.e. even

those off the equilibrium path).


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Repeated Games

Recall Prisoner's Dilemma:

Player 2

Defect Cooperate

Defect 5 , 5 10 , 0

Player 1 Cooperate 0 , 10 8, 8

Questions:

1. What does backward induction tell us would happen if this

game were played finitely many (e.g. 10) times?

2. What if it were repeated infinitely many times?

and consider:

Selten' s Chain Store Paradox

Incumbent faces potential (sequential) entry from a number of

entrants, one in each of its markets. Payoffs in each market:

Incumbent

Accommodate Fight

Enter 40 , 50 -10 , 0

Entrant Don’t Enter 0 , 100 0 , 100

Generally, NE to stage game will also be NE in repeated game, but

more equilibria might be possible as well.


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F ini tely Repeated Prisoner 's Di lemma

!! Only NE to stage game is (Defect, Defect)

!! Even tit-for-tat breaks down (tit-for-tat strategy: start by

cooperating in the first period, then what ever your rival did

in a period, you do the same in the next period)

!! Using backward induction, we see that only NE to repeated

game is (Always Defect, Always Defect)

!! There is experimental evidence that some players will

cooperate -- what might explain this?

imperfect information about rival's motives or type


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The Last Per iod (or End of Period) Problem

The lesson here is that in a repeated game of finite length you must

worry about the final period. The "last period problem" arises because

in the last period agents have no concern about the future of the

relationship and will therefore take actions that hurt the other player

but for which there is no chance to punish them.

When we see a last period coming we have reason to be careful:

1. If you are a tourist trying to get an honest estimate and work

from an auto mechanic. (Lesson: Do not admit you are a

tourist!)

b. If you are an employer who must dismiss a worker. (Lesson:

After dismissal, you might not want to let him/her back into

the office. You might also want to foster a continuing

relationship of some sort.)

3. If you have been cooperating with some potential rivals on

the bidding for a set of projects, and the last project has just

been put up for bid. (Lesson: you might want to look for

contractual ways to protect yourself.)


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Supergames: I nf ini tely Repeated Prisoner 's Dilemma

Now backward induction is no help, since there is no last period. It

turns out that a large number of new subgame perfect equilibria might

now become possible; some of them using credible threats to support

cooperation.

Equilibria:

For both players to Defect in every period is still a SPE. But consider

an attempt to support cooperation by employing the following

strategies:

Grim Strategy: Play Cooperate unti l the other player defects, then play

Defect for ever after.

The Grim Strategy is an example of a "Trigger Strategy":

Trigger Strategy: A strategy that is contingent on the past plays of

players in a game. A trigger strategy usually

involves taking the same action in every stage of a

repeated game until some other player takes an

action that "triggers" a different action by the

first player.


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This is credible (if somewhat harsh), but is it sufficient to deter

cheating?

At any point, the payoffs from defecting BB(D) and from

cooperating BB(C) will be (let = 1/(1+r) be the discount

factor):

BB(D) = 10 + 5 +25 +

35 + .... = 5 + 5/(1- )

BB(C) = 8 + 8 +28 +

38 + .... = 8/(1- )

Therefore, cooperation can be supported [i.e. BB(C) BB(D)] if 0.4.

More generally, will support cooperation if:

> (BBDC - BBCC)/(BBDC - BBDD) or,

equivalently if r < (BBCC - BBDD)/(BBDC - BBCC)

Factors af fecting abil ity to collude:

1. Information and monitoring costs

2. Number of firms - more difficult to monitor many firms, to

find the right firm to punish, and it is

more tempting to cheat the smaller

your market share.

3. Punishments possible - how can you punish cheaters

4. History of market - is there a spirit of cooperation


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Bottom L ine on Cooperation in Repeated Games:

In an infinitely-repeated game (or one with an uncertain end

point) it is frequently possible to sustain cooperation that would

not be sustainable in a one-shot setting. Credible punishments

come from threats to the continuing relationship.

Role of Reputations

With reputations, a game that involves a sequence of

one-shot games against different rivals can look like a

repeated game. Firms can strive to build reputations

for toughness, or honesty etc. that link the play of the

one-shots together.


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Class #5 Problems:

1. Consider the following game:

Firm 1Firm 2

Left Centre Right

Top 5,4 3,1 9,6

Middle 8,3 2,4 6,5

Bottom 1,7 7,2 4,3

1.1 Identify any dominant strategies and any dominated strategies.1.2 Solve using dominance arguments.

2. Consider the following game:

Firm 1Firm 2

Left Centre Right

Top 9,9 3,10 6,4

Middle 5,1 6,4 7,5

Bottom 8,4 5,3 2,2

2.1 Find the Nash Equilibrium for the game in which players movesimultaneously.2.2 Find the Nash Equilibrium for the game in which Firm 1

moves first.


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6.1 Cournot Oligopoly

Probably the most famous model of oligopoly. The assumptions behindthis model:

1. There are relatively few firms serving many customers.2. Firms produce either homogenous or differentiated products.3. Each firm makes its own output decision given its belief about theoutput decisions of its rivals.4. Barriers to entry exist, so positive profits can persist.

Take careful note of assumption 3. This is different from the text which

includes an alternative old-fashioned assumption that game theorists have persuaded us to abandon.

In the language of game theory, this is a simultaneous move game inwhich quantities are the strategic variables, payoffs are determined by thequantities chosen, the market demand curve and cost function, and theequilibrium is a Nash equilibrium.

recall the simple discrete choice version of this from our previous lecture:

Firm 1Firm 2

Low Medium High

Low 80,80 40,90 30, 70

Medium 90,40 50,50 20,40

High 70, 30 40, 20 10,10

Here the Cournot Equilibrium would be for each firm to choose to producea "Medium" level of output. Sometimes called the "Cournot-NashEquilibrium".But now we want to understand more fully this game:


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x1 = R 1(x2) known as the "reaction function". Notice that a gametheorist would simply refer to this as a best-response function.

So, Firm 1's reaction function tells it how much to produce given aconjecture about x2. Firm 2 has a similar function.

x2 = R 2(x1)

Equilibrium:

We have Cournot-Nash Equilibrium when both firms have correctconjectures. This means that both are on their reaction functions -- i.e. thatwe are at the intersection of the reaction functions.

At this point, each firm is maximizing profits given the output of the other firm.

X1

X2

A

XM X c

XM

Xc

R 1 R 2

Here, XM represents the monopoly level of output.


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And XC represents the competitive level of output under the assumptionthat marginal costs are constant.

At A, each firm is choosing the profit-maximizing level of output, giventhe output level chosen by the other firm.

More formally, for the linear case:

f. o. c. d

dx = 0 (a - c) - 2 bx - bx = 0

x = a - c

2b - x

2 and (by symmetry) x = a

- c

2b - x

2

which comb ined yield : x = x = a - c

3b

Note that this impli es X

= 2(a - c)

3b

compare to monopoly: X = a - c

2b and com petitive: X =

a - c

b

For n Cour not oligop olists lik e these we get:

x = a - c

(n +

1

1 1 2

1 2

2 1

1 *

2 *

*

M C

1

Π =

∴

1)b and X = (

n

n + 1 ) (

a - c

b )

So, the Cournot model with these assumptions predicts:


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(i) a total output and price somewhere between competitive and

monopoly levels, and(ii) output and price levels that will approach competitive levels

as the number of firms grows.

An Example with Numbers:

Let P = 242 - 4(x1 + x2) and C(xi) = 2xi for i = 1 and 2

Consider Firm 1's problem:

π1(x1, x2) = [P(X) - c]x1 = [242 - 4x1 - 4x2 - 2]x1

now:dπ1/dx1 = [240 - 4x1 - 4x2] + x1(-4) = 0

this yields the reaction function: x1 = 30 - x2/2 (i)

and another like it for Firm 2: x2 = 30 - x1/2 (ii)

Confirm: X = 30 is the monopoly output and X = 60 thecompetitive output

Put (ii) into (i) to get: x1 = 20x2 = 20, so total X = 40

At X = 40, P = 82.

So we have:


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6.2 Stackelberg Oligopoly

Same conditions as the Cournot model except that one firm (the"leader") chooses its output before all others and the other firms(the "followers") observe and take this output choice as given whenthey choose their output levels to maximize their profits.

This is a simple sort of dynamic game. To solve we use backwardinduction and so begin at the second stage when followers arechoosing their outputs. Assume there is only one follower (Firm 2)for simplicity.

What describes how the follower will choose x2 given any initialchoice of x1 by the leader? Answer: the reaction function derivedabove. The only difference is that the output of Firm 1 is hereknown with certainty and is not just a conjecture. So Firm 2'sproblem is very similar. But Firm 1's is different. Since it can movefirst and it knows how its choice will influence the follower via thereaction function, it can incorporate the reaction function in its

optimization.Now it can maximize: π1(x1, x2) = [P(x1 + R2(x1)) - c]x1

It will be inclined to choose a higher level of output than a Cournotplayer.

Why? Because a Cournot player balances the marginal revenueof an increase in its quantity against the MC of that

increase assuming the other firm will not change output. But a Stackelberg leader knows that if it raises output,the follower will contract a little (i.e. the reaction functionhas a negative slope) so marginal revenue will be higher. This encourages greater output by the leader and less bythe follower.


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In our simple linear example above in which

P = 242 - 4(x1 + x2) and C(xi) = 2xi for i = 1 and 2

we then get x1 = 30 and x2 = 15, so that X = 45 and P = 62.

Illustrating the Stackelberg equilibrium:

X1Leader

X2Follower

R 2

C-N S

Lesson: Commitment can be profitable!

Questions: How can firms make commitments like these?What if the firm could change its output level easily?

What if the follower did not observe the leader's move?


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An Application to Foreign Trade: The Effects of Quotas or VERs

Consider a market in which domestic producers were leaders andforeign producers were followers. We begin with the standardStackelberg outcome at point A of figure 1.

Q

Domestic

QForeign

R F

A

But now a quota on imports or a VER is imposed at Qf*. Now theleader might choose a point like B. Notice that at B, the domesticindustry must be better off (or it could have chosen A again), butsince they are getting more market share the foreign firms are better

off too.Question: Who is worse off?

Q

Domestic

QForeign

R F

A

B

VER = Qf *


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6.3 Bertrand Oligopoly

Sometimes it seems more natural to think of price as the firms'strategic variable. The Bertrand Oligopoly model assumes:

1. Few firms serving many customers.2. Firms produce identical or differentiated products.3. Each firm chooses its price to maximize its profits given aconjecture about the prices charged by rivals. Equilibriumobtains when these conjectures are all correct. i.e. NashEquilibrium in prices!

4. Perfect information, no transaction costs.5. Barriers to entry exist.

Solve this game for identical products, a constant MC of c anda Bertrand Duopoly:

If Firm 1 believes that Firm 2's price will be P2*, what priceshould Firm 1 choose, knowing that with full information, all

buyers will go to the lowest price firm? Answer:

If P2* > PM (the monopoly price) then P1 = PM.

If P2* < PM then P1 = P2* - εε (i.e. slightly undercut Firm 2)


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But Firm 2 feels the same way -- wants to undercut Firm 1. Wheredoes this end?

Answer: P1B = P2B = c

Prices equal marginal costs!!! Only here will the incentive to undercutyour rival end.

So the Bertrand-Nash Equilibrium with identical productsinvolves marginal cost pricing -- even with only two firms!

Quest i on: Do you w ant t o be a Stackelberg-t ype leader in a

pr i ce game?


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Bertrand Duopoly with Differentiated Products

A key condition that led us to derive marginal cost pricingabove was the assumption of homogeneous products. Let usrelax that -- now, products may be good substitutes for eachother but not perfect. A slightly lower price will attract somesales from the other firm, but not all.

e.g. suppose demand looked

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