Price Discrimination

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Industrial Organization Lecture Notes

Professors Emmanuelle Auriol & Bertrand Gobillard. Working copy (2011).

These notes follow Professor Emmanuelle Auriol and Professor Bertrand Gobillard’s course “Industrial Organization” for the Master 2 program at the Toulouse School of Economics. These notes currently lack the graphs and fig- ures given in lecture. (Where a figure should appear in the notes, one will see a remark such as “See Figure #.#”.) Any errors in the notes that follow are entirely my own. Please contact shwolffsan@gmail.com with corrections, expan- sions, or improvements.

As of Saturday, 28 May 2011, these notes have been revised through the end of Lecture 10 (some passages will have to be revisited).

SHW (28 mai 2011)

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1 Introduction 4

1.1 Globalization . . . . 4

1.2 Coordination and the Role of Government . . . . 5

1.2.1 Industrial Structure of the EU . . . . 5

1.2.2 Firm Size . . . . 6

1.2.3 Returns to Scale . . . . 6

2 Monopoly 8 2.1 The Mono-Product Monopoly . . . . 8

2.2 Cartels . . . . 10

2.2.1 The Social Cost of Cartelization: Posner (1976) . . . . 10

2.2.2 Antitrust Law . . . . 12

2.3 Multi-Product Monopoly . . . . 12

2.3.1 Independent Demand and Separable Costs . . . . 12

2.3.2 Dependent Demand and Separable Costs . . . . 13

2.4 Multi-Period Model . . . . 14

2.5 Independent Demand and Nonseparable Costs . . . . 15

2.6 Organization of a Firm — M versus U Model . . . . 16

2.7 Quality . . . . 16

2.7.1 Musa-Rosen Model . . . . 17

2.7.2 Hotelling Model (1929) . . . . 18

2.7.3 Monopoly Incentive to Offer Quality . . . . 18

2.7.4 Example: Longevity of Light Bulb . . . . 23

2.7.5 Overprovision of Quality . . . . 24

2.8 Experience Attributes . . . . 26

2.8.1 The Lemons Model . . . . 26

2.8.2 Unique Purchase of an Experience Commodity . . . . 27

2.9 Credence Attributes . . . . 29

2.9.1 The Model . . . . 30

3 Price Discrimination 34 3.1 Perfect Price Discrimination . . . . 34

3.1.1 Third-Degree Price Discrimination . . . . 35

3.2 Second-Degree Price Discrimination . . . . 39

3.2.1 Quality . . . . 46

4 Oligopoly 48 4.1 The Coase Conjecture . . . . 48

4.1.1 The Single-Consumer Model . . . . 48

4.1.2 Two-Period, Two-Type Model . . . . 49

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4.1.3 More General Case . . . . 52

4.2 Basic Oligopoly Theory . . . . 55

4.2.1 Price Setting and Residual Demand . . . . 55

4.2.2 Reaction Functions, Strategic Complements and Substitutes 57 4.2.3 Competition Games . . . . 57

4.2.4 Linear Cournot Model . . . . 58

4.3 Bertrand Competition . . . . 61

4.3.1 The Basic Model . . . . 62

4.3.2 Bertrand Competition with Entry Costs . . . . 62

4.3.3 Imperfect Information and Search Costs . . . . 63

4.3.4 Uninformed Consumers: A Simple Model . . . . 63

4.3.5 Sequential Search: The Diamond Paradox . . . . 65

4.3.6 Nonsequential Search . . . . 66

4.3.7 Product Differentiation . . . . 68

4.3.8 Tacit Collusion . . . . 70

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Introduction

Lecture 1

mardi 22 mar 2011 The field of industrial organization studies the structure of firms and markets,

the boundaries between firms and markets, and the strategic interactions among firms.

1.1 Globalization

How did the world change in the last half of the twentieth century?

• There have been no global wars since the fall of the Berlin Wall in 1989.

• The rapid development of information technology has facilitated commu- nication and decreased transportation costs.

• The construction of the integrated market of the European Union.

Does globalization mean the end of war? Extrastate (colonial) and interstate conflict (mostly) yes, but intrastate conflict no.

Following the end of the Cold War, many autocracies (governments run by a powerful person) fell, while anocracies (governments that exist nominally but exercise little real power; anocracy literally means “the rule of nothing”) and democracies (governments in which power ultimately rests in the citizens) rose.

Who benefits from globalization? Poor countries are big winners. After the Cold War, exports from developing economies to the rest of the world rose relative to exports from developed economies. (Note that developed economies still export a large amount, and in many cases they have increased exports rel- ative to before globalization.) Furthermore, exports from developing countries shifted toward manufactured goods. Globalization allows developing countries to specialize more easily in sectors in which they have a comparative advantage.

Among developing economies, the economic development of Asian countries out- strips the economic development of countries in Latin America and Africa.

Is the US a big importer? Relative to its size, not really: The US is large enough to be able to produce much of what it needs internally. As a general rule, small countries tend to be more open, relative to their size, for the simple reason that they cannot produce all of what they need internally, and hence must trade.

As another general rule, rich countries tend to have service-oriented economies.

Why is globalization good? The number of people below the poverty line of

$1 per day has fallen substantially between 1970 and 2000; see the graphs of the

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distribution of individual income. Absolute poverty falls as a result of globaliza- tion; as the world becomes more open to trade, poorer countries can specialize in labor-intensive production. Modern-day famines are almost always the result of poor policy. But globalization has a downside: the rise of globalization has brought a rise in inequality.

1.2 Coordination and the Role of Government

There are seven billion people in the world economy. An economy this large clearly raises the issue of coordination.

Complex systems are almost always decentralized and auto-organized. This general rule applies equally well to living organisms as to economies! One prob- lem any complex system must deal with is “crazies”. Living organisms, for example, must deal with renegade cancer cells. Economies have tried the central planning experiment, and it did not work (incentive issues). In the decentralized world, prices serve as signals. General equilibrium theory is therefore a very useful tool.

One question economists ask is what is the optimal size for the economic analog of the “brain”, i.e. the government that controls the decentralized econ- omy. Rich countries tend to have big governments. In Sweden, for example, the government comprises nearly 60% of the economy. In the United States, the government makes up a little more than 30% of the economy. Poor countries tend to have weaker, smaller governments, usually less than 20% of GDP. For comparison, the average size of government of OECD countries, as a proportion of GDP, is 45%.

On one hand, then, large government is associated with a strong economy.

(Note that we say nothing about causality in this statement.) On the other hand, beyond a certain level, government size is negatively correlated with economic growth. This suggests there is an optimal size for the government (a Laffer-type curve, if you will).

What roles does government serve in the modern state? Some of the major roles of modern government are listed below.

• Government provides essential public goods (ex. education, legal system, military, etc.).

• Government can help address externalities (by assigning polluting rights, for example).

• Government builds and maintains infrastructure (roads, public buildings, etc.).

• Government regulates and sometimes runs non-competitive industries.

• Government engages in macroeconomic policies and redistribution.

1.2.1 Industrial Structure of the EU

90% of firms have fewer than 10 employees. These firms produce 1/5 of total added value and provide 30% of employment. For these firms, laissez-faire is often the most-effective policy.

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10% of firms have more than 10 employees. These larger firms produce 4/5 of total added value and provide 70% of employment. These firms have a disproportionally large impact on the economy. For these firms, regulation and anti-trust is often critical.

1.2.2 Firm Size

Why are some firms big and other firms small? Economics is a bridge between politics and laws of nature. As an illustration, consider politicians who attempt to make unemployment “vanish” by reducing the legal work-week from 40 hours to 35 hours. Their thinking is that by reducing the number of hours any given employee is allowed to work, they will cause firms to hire additional workers to make up the difference. But this naive thinking ignores individual responses to the policy change.

As a second example, consider politicians who try to make the energy in- dustry “competitive” by opening the market. The number of firms in the in- dustry subsequently falls. While competition often results in outcomes that are friendlier to consumers, this outcome is far from guaranteed. The illustration of the energy industry shows why it is important to have a clear understanding of market and firm structure.

1.2.3 Returns to Scale

Returns to scale captures how output changes as result of a proportional change in all inputs. There are three possibilities.

Constant returns to scale (CRS). If a firm exhibits constant returns to scale, then when all inputs are scaled by some positive scalar α, the resulting output is scaled by α as well. Mathematically, if y is a vector of inputs and f (y) represents the firm’s production function (i.e. its output), then under CRS we have

f (αy) = αf (y).

Increasing returns to scale (IRS). If a firm exhibits increasing returns to scale, then scaling all inputs by α > 1 results in a proportionally greater increase in output. Mathematically,

f (αy) > αf (y).

If a firm exhibits IRS, then its marginal cost is less than its average cost. To see this, let C(·) denote the firm’s cost function. For any δ > 0, IRS implies

C((1 + δ)y) < (1 + δ)C(y).

Subtracting C(y) from both sides, dividing through by y, and moving the δ factor to the left, we have

C((1 + δ)y) − C(y) < δC(y) C((1 + δ)y) − C(y)

δy < C(y) y .

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Taking limits as δ → 0, we obtain

δ→0lim

C((1 + δ)y) − C(y)

δy = C0(y) < C(y)

y = ¯C(y),

where by definition C0(·) is the marginal cost and ¯C(·) is the average cost. We can further show that increasing returns to scale implies decreasing marginal cost, simply by taking second derivatives of the above.

Decreasing returns to scale. A firm exhibits decreasing returns to scale if scaling all inputs by α > 1 results in a proportional increase in output that is less than α:

f (αy) < αf (y).

Decreasing returns to scale does not exist in real life; we can always cut produc- tion in half in two identical firms. When decreasing returns to scale is observed in real life, it is usually due to a fixed input.

Under the assumption of perfect competition, firms behave as price takers.

Under the assumption of (global) IRS, marginal-cost pricing leads to negative profit (minimum profit); the profit-maximizing solution occurs at a corner, either q = 0 or q = +∞.

IRS and Market Structure

• Natural monopoly

• Oligopolistic structure

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Monopoly

2.1 The Mono-Product Monopoly

A monopolist firm produces a single good and sets its price. Let D(p) denote the quantity demanded when the price is p, and let C(·) denote the firm’s cost function. The firm seeks to maximize its profits (revenue minus costs):

maxp π(p) = D(p)p − C(D(p)).

This maximization problem is commonly referred to as the firm’s profit-maxi- mization problem, or PMP.

The elasticity of demand sets a limit on the monopoly power. To explore the relationship between monopoly power and the elasticity of demand, we compute the solution to the monopolist’s PMP. The first-order condition of the PMP is found by taking the first derivative of the profit function with respect to the choice variable (here, price) and setting the result equal to zero:

π0(p) = D0(p)p + D(p) − D0(p)C0(D(p)) =

set0. (2.1)

Rearranging this expression, we can obtain

p − C0(D(p)) = −D(p)

D0(p). (2.2)

Dividing both sides of this equation by p, we obtain p − C0(D(p))

p = − D(p)

pD0(p). (2.3)

Recall that the price elasticity of demand, denoted εD,t, is defined as the percent- age change in quantity demanded divided by the percentage change in price:1

εD,p:= %∆D(p)

%∆p = D0(p)/D(p)

1/p = pD0(p) D(p) .

1In lecture, the price elasticity of demand was implicitly defined with a minus sign:

εD,p:= −%∆D(p)

%∆p .

To be consistent with other literature, we use the definition given above in the text, without the minus sign. With this definition (no minus sign), the price elasticity of a normal good is nonpositive, since the quantity demanded is a decreasing function of price. To avoid writing minus signs everywhere, when economists discuss price elasticity, they often (implicitly) take its absolute value.

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The right side of (2.3), then, is simply the negative inverse of the price elasticity of demand. The left side of this equality — price minus marginal cost, divided by price — is referred to as the Lerner index, denoted L:

L := p − C0(D(p))

p .

Note that we can rewrite (2.3) as

L = − 1 εD,p,

showing that the Lerner index equals the negative inverse of the price elasticity of demand.

Computing the second derivative of the firm’s profit function, we have π00(p) = D00(p)p + 2D0(p) − D00(p)C0(D(p)) − (D0(p))2C00(D(p)).

The C00(·) term could be positive, negative, or zero (viz., the cost function could be convex, concave, or linear, respectively). To ensure that the first-order con- dition yields a maximum, we need to require that the firm’s profit function is concave, i.e. that π00(p) < 0:

π00(p) = D00(p) p − C0(D(p)) + 2D0(p) − (D0(p))2C00(D(p)) ≤ 0. (2.4) Plugging (2.2) into the second-order condition (2.4) yields

−D00(p)D(p)

D0(p) + 2D0(p) − (D0(p))2C00(D(p)) ≤ 0.

Quantity Approach

We analyze the same profit-maximation problem a second way, using the “indi- rect” quantity approach. In particular, we suppose that the monopolist chooses quantities instead of prices. Define y := D(p), so that p(y) = D−1(y). The monopoly’s PMP can then be written

maxy π(y) = p(y)y − C(y).

The first-order condition is

π0(y) = p0(y)y + p(y) − C0(y) =

set0.

A little algebra allows us to rewrite this equation as p(y) − C0(y)

p(y) = −yp0(y)

p(y) . (2.5)

This result is equivalent to (2.3) above. The expression on the left side is the Lerner index, price minus marginal cost divided by price. The right side is again the negative inverse of the price elasticity of demand.2 Since for most goods

2By definition, the price elasticity of demand is given by εD,p= %∆y

%∆p = 1/y

p0(y)/p(y) = p(y) yp0(y). Hence

1 εD,p

= −yp0(y) p(y) , precisely the expression on the right side of equation (2.5).

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(Giffen goods and Veblen goods are exceptions) the price elasticity of demand is negative, the right side of equation (2.5) is positive. Hence (2.5) implies that in equilibrium, the monopolist’s price is strictly larger than its marginal cost.

Recall that for the first-order condition to yield the profit-maximizing quan- tity, we need to ensure that the second-order condition π00(y) ≤ 0 holds.

Review

Lecture 2

jeudi 24 mar 2011 Recall that last lecture we were introduced to the Lerner index L:

L := p − C0

p = − 1

εD,p

.

We saw how the price elasticity of demand limits the power of the monopoly.

Recall also that we stressed the importance of the second-order condition:

2p0(y) + p00(y)y − c00(y) ≤ 0 ∀y.

Only if when the second-order condition holds does the first-order condition yield a maximum.

2.2 Cartels

Historically, globalization has followed the following path.

• First wave: 1870 – 1914.

• Retreat: 1914 – 1950.

• Second wave: 1950 – 1980.

• Third wave: 1980 – present.

Large cartels emerged with the first wave of globalization. The goal of a cartel is to allow the multiple firms within the cartel to behave as a monopoly, and hence extract monopoly surplus from the consumers. Cartels face two major problems:

(1) coordination, and (2) enforcement among the constituent members. Cartels are bad (in an economic sense) because they create a welfare loss to society as a whole. (Remark: If firms can extract all consumer surplus, this is good in terms of efficiency, though almost surely not in terms of equality.) The social cost of a cartel has been estimated by Posner (1976), to whose analysis we now turn.

2.2.1 The Social Cost of Cartelization: Posner (1976)

Assume that before the cartel forms, there is perfect competition, with price equal to marginal cost. After the cartel forms, assume that the price moves to the monopoly price. Following Posner, we will further assume that demand is linear with respect to price. (In the following equations, BC denotes “before cartel”, AC “after cartel”, and M “monopoly”.)

Demand: p(y) = A − By Before Cartel: pBC ≡ C0(yBC) yBC After Cartel: pAC≡ pM yAC.

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We expect that following the formation of the cartel, the quantity produced will fall and the price will rise. Since demand is linear, the deadweight loss (DWL) resulting from the formation of the cartel is equal to the area of the triangle whose base equals the reduction in output due to the cartel (i.e. customers who are no longer served) and whose height equals the increase in price:

DW L = 1

2(yBC− yAC)(pAC− pBC).

Restoring First-Best

One solution to the cartel-monopoly distortion of quantity and price might be (cue economist’s autoresponse) taxes and subsidies. Suppose the cartel or monopoly faces a per-unit tax of amount t. The firm’s (or cartel’s) profit- maximization problem is then

maxp π(p) = pD(p + t) − C(D(p + t))

A benevolent social planner would like to set a tax t that induces the monopolist or cartel to provide the socially optimal quantity. This is an incentive problem, so we solve it backward: First solve what the firm will do given a tax of size t, then determine the tax t that yields the desired quantity.

The first-order condition (with respect to the firm’s decision variable, p; the tax t is chosen by the government and is taken as given by the firm) is

π0(p) = D(p + t) + pD0(p + t) − D0(p + t)C0(D(p + t)) =

set0 Adding and subtracting tD0(p + t) and grouping terms, we have

D(p + t) − tD0(p + t) + D0(p + t)p + t − C0(D(p + t)) = 0.

If the firm does marginal-cost pricing, then the term p + t − C0(D(p + t)) in the second set of square brackets must be zero. For the equality to hold, the term in the first set of square brackets must be zero as well, implying that

D(p + t) = tD0(p + t).

Isolating t, we obtain an implicit expression for the optimal tax t:

t = D(p + t) D0(p + t) < 0.

Since the quantity demanded D(·) is nonnegative and decreasing with price (i.e.

D0(·) < 0), we conclude that the tax t that restores first-best is negative. That is, the government should subsidize the cartel or monopoly. Of course, for this solution to yield a maximum, we need to ensure that the second-order condition holds.

Example: The medicinal drug industry is a good example of an industry in which government intervenes with taxes and subsidies to induce marginal-cost pricing. Once an effective drug is developed, government wants people to use it at marginal cost, not at the monopoly price. Hence government subsidizes the medicine industry.

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2.2.2 Antitrust Law

In the US, antitrust law began with the Sherman Act in 1890. In the EU, the first antitrust law took effect in 1956 (at the beginning of the second wave of globalization, post-WWII).

2.3 Multi-Product Monopoly

As the name implies, a multi-product monopoly produces several goods. Assume that the monopoly produces l goods, which we will index by k = 1, . . . , l. The demand for good k can, in general, depend on the prices of all l goods, and is given by

yk= Dk(p1, . . . , pk).

Denote the firm’s cost function by C(y1, . . . , yk). Note that this functional form allows for externalities.

The monopolist’s profit-maximization problem is max

(p1,...,pk)π(p) =

l

X

k=1

pkDk(p) − C(D1(p), . . . , Dl(p)).

The first-order conditions (with respect to price pk for k = 1, . . . , l) are

∂π

∂pk = Dk(p) + pk

∂Dk(p)

∂pk +X

j6=k

pj

∂Dj(p)

∂pk

l

X

h=1

∂C(D1(p), . . . , Dl(p))

∂Dh

∂Dh(p)

∂pk =

set0. (2.6) We will consider two cases: The special case in which the demand for each good is independent of the others (so that all cross-derivatives ∂Dj/∂pk with j 6= k are zero), and the more general case in which demands can be dependent. To simplify analysis, in both cases we will assume that costs are separable.

2.3.1 Independent Demand and Separable Costs

When demand for each good is independent, Dk(p) ≡ Dk(pk) for all k. Separable costs implies that C(D1, . . . , Dl) = Pl

k=1Ck(Dk). Thus in this case, equation (2.6) reduces to

Dk(pk) + pkD0k(pk) + 0 − Ck0(Dk)Dk0(pk) = 0.

Rearranging terms and dividing through by pk, we obtain L := pk− Ck0(Dk)

pk

= − Dk(pk)

pkDk0(pk) = − 1 εDk,pk

for all k = 1, . . . , l; as usual, L denotes the Lerner index and εDk,pk denotes the price elasticity of demand. From this equation, we see that the percentage mark- up (the difference between monopolist price and marginal cost, normalized by the price) is greater on consumers with lower price elasticity. This result agrees with economic intuition: The less sensitive a consumer’s quantity choice is to changes in price (i.e. the lower the consumer’s price elasticity of demand), the more the monopolist can increase the price without losing significant sales.

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2.3.2 Dependent Demand and Separable Costs

We retain the assumption of separable costs, so C(D1, . . . , Dk) =Pl

k=1Ck(Dk).

However, we now allow for dependent demand, so the cross-derivatives ∂Dj/∂pk may be non-zero. In this case, equation (2.6) reduces to

Dk(p) + pk∂Dk(p)

∂pk +X

j6=k

∂Dj(p)

∂pk pj

l

X

h=1

Ch0(Dh)∂Dh(p)

∂pk = 0.

Isolating terms with index k, we obtain Dk(p) + ∂Dk(p)

∂pk pk− Ck0(Dk) = −X

j6=k

∂Dj(p)

∂pk pj− Cj0(Dj) .

Isolating the pk− Ck0(Dk) term and then dividing through by pk to obtain the Lerner index on the left side, we obtain

pk− Ck0(Dk)

pk = − Dk(p)/pk

∂Dk(p)/∂pk X

j6=k

∂Dh(p)/∂pk

pk(∂Dk(p)/∂pk)pj − Cj0(Dj)

= − 1

εDk,pk X

j6=k

∂Dh(p)/∂pk

pk(∂Dk(p)/∂pk)pj − Cj0(Dj) . (2.7) If good k is a normal good, then pk∂Dk(p)/∂pk is negative. Assuming the existence of anti-dumping laws, we also have pj− Cj0(Dj) ≥ 0. So the sign of the sum on the right side of (2.7) is determined by the sign of ∂Dj(p)/∂pk.

If j and k are substitute goods, the cross-derivative ∂Dj(p)/∂pkis positive, so the sum in equation (2.7) is negative. Hence the Lerner index of the commodity k will be higher than the negative inverse of the price elasticity of demand of commodity k. Thus this model argues against horizontal integration, as it harms consumer welfare.

If j and k are complement goods, the cross-derivative ∂Dj(p)/∂pkis negative, so the sum in (2.7) is positive. Hence the Lerner index of commodity k is less than the negative inverse price elasticity of demand of commodity k. Thus this model predicts that vertical integration improves consumer welfare when goods are complements.

Note that there is no question that vertical integration is better for the firm.

What we wanted to know is whether this integration is also better for consumers.

The case of complement goods relates to the concept of double marginalization, a phenomenon arising when two monopolies exist in a vertical stream, with one monopoly supplying an input to the other. The term “double marginalization”

refers to the fact that the two monopolies each apply a monopolist’s marginal markup to their output (the intermediate good in the case of the first monopolist, and the final good in the case of the second). Thus when the final good reaches the consumer, the price is higher than it would be if the two monopolies were integrated into a single firm, as we illustrate below.

Example: Double Marginalization

Consider two firms in a vertical supply stream. Firm 2 takes as its only input the good produced by firm 1. Suppose that firm 1 has a constant marginal cost

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C10(y) = C (so that its cost function is linear, C1(y) = K + Cy). Firm 1 chooses the monopoly price pM1 that solves

p1− C

p1 = − 1 εD1,p1.

Suppose that firm 2 also has a constant marginal cost equal to the price firm 1 charges for its good: C20(y) = pM1 . Firm 2 chooses the price p2 that solves

p2− pM1

p2 = − 1 εD2,p2.

Everybody — produces and consumers — are better off when these two firms are merged.

2.4 Multi-Period Model

Consider the following two-period model. A monopolist produces and sells a sin- gle good in each of two time periods. Assume that if a consumer tries a product in the first period, she will also choose to consume in the second period. Let p1 denote the price charged by the monopolist in the first period; the quantity demanded is then y1= D1(p1), and the cost to the monopolist is C(y1). The de- mand in the second period depends on the price in both periods: y2 = D2(p1, p2).

The cost to the monopolist of producing y2 is C(y2). Assume that ∂D2/∂p1 < 0;

if the firm increases the price in period 1, its demand in period 2 will fall. The firm’s total profit over both time periods is given by

π(p1, p2) = p1D1(p1) − C(D1(p1)) + δ [p2D2(p1, p2) − C(D2(p1, p2))] , where δ captures the degree of forward-lookingness of the firm manager. (Es- sentially, δ is a discount factor.) The first-order conditions are

∂π

∂p2

= D2(p1, p2) + p2∂D2

∂p2

∂D2

∂p2

C20 =

set0

∂π

∂p1 = D1(p1) + (p1− cm)D0(p1) + δp2− C20 ∂ D2(p1, p2)

∂p1 =

set0.

Solving, we obtain p2− C20

p2

= − 1

εD2,p2 (2.8)

p1− C10 p1

= − 1 εD1,p1

− δ∂D2(p1, p2)/∂p1

∂D1(p1)/∂p1

p2− C20 p1

. (2.9)

Let’s analyze equation (2.9) a bit. The final fraction, (p2−C20)/p1, is non-negative by the anti-dumping assumption. The partial derivative ∂D1(p1)/∂p1is negative.

Hence the sign of the second term on the right in equation (2.9) depends on the sign of the partial derivative in the numerator. Assume ∂D2(p1, p2)/∂p1 < 0.

Then we can make the following claim.

Proposition: p01(δ) < 0.

 Remark: It is not always good for a firm to price at marginal cost. Example of the airline industry.

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2.5 Independent Demand and Nonseparable Costs

Lecture 3

mardi 29 mar 2011 We now turn to a model of independent demand and dependent cost, a model

that captures “learning by doing”.

A monopolist produces a single good in each time period t.3 Assume that the demand for this good in time period t is a function of the time t price only;

in particular, the demand for the good is independent of the price in all other time periods:

yt= Dt(pt).

Costs of production, however, are assumed to be nonseparable: the cost of pro- ducing the good in time period t depends not only on the amount Dt produced in this period, but also on the amounts produced in all preceding periods:

Ct(D1, . . . , Dt).

It is this assumption of nonseparable costs that captures “learning by doing”.

The monopoly’s profit-maximization problem is (I don’t think the following is accurate; see footnote)4

maxpt

πt(pt) = ptDt(pt) − Ct(D1, . . . , Dt).

Let yt= Dt(pt) denote the quantity produced in period t. Under the assumptions of independent demand and nonseparable costs, the first-order condition is

Dt(pt) + pt

∂Dt(pt)

∂pt ∂Ct(D1, . . . , Dt)

∂yt

∂Dt(pt)

∂pt =

set0.

Notice that if ∂Ct(D1, . . . , Dt)/∂yk < 0, there is a positive externality from today’s production on future production: producing more today reduces the monopoly’s cost of production in future periods. Hence, intuitively, we might expect the monopolist to set lower prices in earlier periods, in order to sell more and take advantage of this positive externality.

Two-Period Model

Consider a model with just two time periods, T = 2. The costs of production are given by C1(y1) and C2(y1, y2), respectively; the demands are y1 = D1(p1) and y2 = D2(p2). The learning-by-doing assumption implies that ∂C2/∂y1 < 0:

increasing production in period 1 decreases the cost of production in period 2. Denote the geometric time-discount factor between periods by δ. The net present value (as valued in period 1) of the monopoly’s total profits in the first two periods is given by

π(y1, y2) = π1(y1) + δπ2(y1, y2)

= p1(y1)y1− C1(y1) + δ [p2(y2)y2− C2(y1, y2)] .

3I believe a complete statement of the model would include the number of time periods T , or the monopolist’s belief about the number of time periods it will be producing.

4What the monopolist really wants to maximize is the (discounted) sum of profits over all relevant time periods t.

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The monopolist seeks to maximize this profit. The first-order conditions are

∂π

∂y2

= δ



p2(y2) + p02(y2)y2∂C2(y1, y2)

∂y2



set=0 (2.10)

∂π

∂y1

= p1(y1) + p01(y1)y1− C10(y1) − δ∂C2(y1, y2)

∂y1

set=0. (2.11) Solving (2.10), we obtain

p2(y2) −∂C2∂y(y1,y2)

2

p2(y2) = 1 εp2,y2

.

Similarly, solving (2.11), we obtain p1(y1) − C10(y1)

p1(y1) = − 1 εp1,y1 + δ

∂C2(y1,y2)

∂y1

p1(y1) .

The expression on the left side of this equation is the Lerner index. In the second term on the right side of this equation, the discount factor δ > 0 by assumption, ∂C2(y1, y2)/∂y1 < 0 under the learning-by-doing assumption, and p1(y1) > 0 since the monopoly will never choose to set a nonpositive price. Hence the second term is negative, so the Lerner index is less than the absolute value of the price elasticity of demand. Concluding remark?

2.6 Organization of a Firm — M versus U Model

Until the 1920s, large firms were organized in a unitary structure: one leader at the top controlled everything. In the 1920s, firms began moving toward a multidivision structure in which there were several divisions within a firm, each with its own CEO, its own marketing department, etc. As a result of the move toward a multidivision structure, the oversight of each division (or branch) became easier. General Motors was one of the first major firms to make this transition.

2.7 Quality

When discussing competitive firms, we rarely discuss quality. The reason is that, by definition, a competitive firm makes the same product as its competitors.

Globalization and innovation have had the consequence that in today’s world, it is much harder to control the quality of the product we consume. As an example, how can you be sure the tomato you are eating has not been genetically modified?

And how can you be sure the dress you are wearing wasn’t made by child labor?

Quality is hard to measure because it is (almost always) multidimensional.

To illustrate this, consider a mobile phone. The “quality” of the mobile phone depends on the quality of the connection, the quality of the user support, the quality of the phone’s design, and the quality of the available calling plans, just to name a few.

One important aspect of quality is when the quality of a good is observable:

before purchase, after purchase, or never? A search attribute is an aspect of quality that is observable before purchase. A experience attribute is an

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aspect of quality that the consumer learns through use, i.e. after purchase.5 A credence attribute is an aspect of quality that the consumer can never learn.

Another question is whether an aspect of quality is verifiable in court. If yes, then agents can contract on this aspect of quality; we say it is a verifiable attribute. If not, it is a non-verifiable attribute.

2.7.1 Musa-Rosen Model

There is a continuum of consumers of type θ ∈ [0, ¯θ), where ¯θ ∈ R+∪{+∞}. (For simplicity in the analysis that follows, we will assume that ¯θ = +∞.) Denote the probability density function of consumer types by f (θ) and the corresponding cumulative distribution function by F (θ).

Consumers choose to purchase either one unit or no units of the good. The firm selects the quality level s of the good and the price p. The utility of consumer type θ is given by

Uθ(s, p) =

s − p

θ if consumer purchases 1 unit, 0 otherwise.

Denote the marginal consumer type by θL. By definition, the marginal consumer is indifferent between consuming and not consuming, so s − pθ = 0. Solving for θ, we find θL= ps.6

All consumers with type θ greater than the marginal consumer’s type θL= p/s will choose to consume the good, since Uθ = s − pθ > 0 whenever θ > θL. Therefore, the fraction of consumers who choose to consume — i.e., the demand for the good — is given by

D(p, s) = Z +∞

θL

dF (θ)

= F (+∞) − F (θL)

= 1 − F (p/s). (2.12)

(If we want the number of consumers who choose to consume, rather than the fraction, multiply this demand by N .)

We will make the natural assumptions that demand is decreasing in price,

∂D/∂p < 0, and increasing in quality, ∂D/∂s > 0. Denoting the quantity demanded by q := N × D(p, s) and solving (2.12) for the price p, we obtain

p(q, s) = sF−1

 1 − q

N



. (2.13)

(N.B. We divide q by N as shown since q is defined as the number, not the fraction, of consumers.)

5One might wonder whether experience attributes can become search attributes if consumers share their information about goods, a possibility that is increasingly easy thanks to the in- ternet. While information sharing (not to mention the associated incentive issues) is certainly relevant, quality would still be an experience attribute for the first users of a good. In addi- tion, goods for which preferences are highly subjective — the efficacy of a certain medicine, for example — would exhibit experience attributes that information sharing would not easily convert into search attributes.

6If ¯θ = +∞, then θL= p/s is sure to fall in the support of θ. If instead ¯θ is finite, we should check explicitly that θLfalls in the support.

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2.7.2 Hotelling Model (1929) Linear and circular variations.

Consider a linear Hotelling model with exogenous localization (i.e., the firm locations are given). In particular, take the unit line segment with Firm 1 at x = 0 and Firm 2 at x = 1; both firms sell an identical good of quality ¯s. Let there be linear transportation costs given by tx, where x is the distance from the firm to the consumer. The utility of a consumer of type x (i.e., a consumer at position x) is given by

Ux(p1, p2) =

¯

s − p1− tx if purchase from Firm 1,

¯

s − p2− t(1 − x) if purchase from Firm 2,

0 otherwise.

To complete the model, we would need to describe the density of consumers over the interval.

If both firms are active and the market is covered (i.e. all consumers decide to purchase from one of the firms), then by definition the marginal consumer ˜x obtains the same utility whether she chooses to purchase from Firm 1 or from Firm 2:

¯

s − p1− t˜x = ¯s − p2− t(1 − ˜x).

Solving for ˜x, we find

˜

x(p1, p2) = p2− p1+ t

2t .

Different cases to consider: covered market in which both firms are active, covered market in which only one firm is active, non-covered market. See Figures 3.1, 3.2, and 3.3, respectively.

2.7.3 Monopoly Incentive to Offer Quality

Consider a commodity with a search attribute; vertical differentiation. Let us return to the Musa-Rosen model in subsection 2.7.1. Assume that quality is costly, so that the monopoly’s production cost C(q, s) satisfies Cs = ∂C/∂s > 0.

We will further assume that Cq = ∂C/∂q > 0, Css= ∂2C/∂s2≥ 0, Cqq ≥ 0, and Csq≥ 0.

The firm’s inverse demand, as found in (2.13), is given by p(q, s) = sF−1(1 − q),

such that p(q, s) satisfies ps(q, s) > 0 and pss(q, s) ≤ 0.

The Monopoly Computation The monopolist solves

maxq,s πm(q, s) = p(q, s)q − C(q, s).

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The first-order conditions to this profit-maximization problem are

∂πm

∂q = p(q, s) + pq(q, s)q − Cq(q, s) =

set0 (2.14)

∂πm

∂s = ps(q, s)q − Cs(q, s) =

set0. (2.15)

Solving (2.15), we obtain

Cs(q, s) = ps(q, s)q,

Thus, in a Musa-Rosen model in which each consumer buys either one unit or no units of the good, the monopolist sets the marginal cost of increasing quality equal to the marginal revenue from increasing the quality.

Quality, Continued

Lecture 4

jeudi 31 mar 2011 Recall that we are analyzing a model in which the monopoly chooses both quan-

tity q and quality s. Solving the first-order conditions (2.14) and (2.15), we obtain, respectively,

p(q, s) − Cm(q, s)

p(q, s) = −qpq(q, s) p(q, s)

qps(q, s) = Cs(q, s). (2.16) The first equation is the now-standard result that the Lerner index equals the negative inverse of the price elasticity of demand: L = −1/εD,p. The second equation states that the incentive to offer quality is related to the marginal consumer’s marginal propensity to pay for quality.

Let’s see how the monopolist’s choice compares to the first-best. The social optimum maximizes total social welfare:

maxq,s W (q, s) = Z q

0

p(x, s) dx − C(q, s). (2.17) Adding and subtracting the price, we obtain the equivalent formulation that the social optimum maximizes the surplus of the marginal consumer plus the monopoly’s profit:

maxq,s W (q, s) = Z q

0

p(x, s) dx − p(q, s)q + [p(q, s)q − C(q, s)]

= Smc+ π.

Working with expression (2.17) will be nicest. The first-order conditions for this maximization problem are

∂W

∂q = p(q, s) − Cq(q, s) =

set0 (2.18)

∂W

∂s = Z q

0

ps(x, s) dx − Cs(q, s) =

set0. (2.19)

Comparing this last condition to (2.16) above, we see that the monopoly’s incen- tives yield a different solution than the first-best: sm(q) 6= s(q). Rearranging (2.19) and multiplying and dividing the left side by q, we obtain

q Z q

0

ps(x, s) dx

q = Cs(q, s).

Figure

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References