There is no universally accepted definition for price discrimination (PD). In most cases, you may consider PD as: producers sell two units of the same physical good at diﬀerent prices (, either to the same consumer or to diﬀerent consumers.) Note that:

• There is no PD if the price diﬀerence reflects the costs of serving;

• One cannot infer PD does not happen when diﬀerentiated products are sold to diﬀerent consumers.

It is useful to classify, according to Pigou (1920), price discriminations by first, second, and third degree PD:

• First degree (1^{◦}) PD: producers capture the whole consumer surplus;

• Second degree (2^{◦}) PD: in the case of incomplete information, producers use selfse-
lection devices to extract consumer surplus;

• Third degree (3^{◦}) PD: there exist direct signals about demands, and producers use
this signal to price-discriminate.

### 3.1 First degree (Perfect) PD

One necessary condition for the 1^{◦} PD is complete information: producers know the de-
mands. Given this, how can a monopolistic producer engage in 1^{◦} PD? Consider a market
with n identical consumers and (market) demand function D(p). Denote the inverse demand
as: p = P (q).

• The monopoly can oﬀer a tariﬀ T (q) such that:

T (q) =

⎧⎪

⎨

⎪⎩ Z q

0 P (x)dx/n − Z q

0

P (x)dx/n

if q = q^{c}/n
otherwise,
where > 0, and q^{c}is the competitive market demand.

• Consider an aﬃne pricing schedule (or a two-part tariﬀ): T (q) = A + pq. Note
that when A = 0, T (q) is a linear tariﬀ. Let S^{c}=

Z q^{c}

0 [P (x) − p^{c}]dx, and oﬀer:

T (q) =

⎧⎨

⎩

S^{c}/n + p^{c}q
0

if q > 0
if q = 0,
where p^{c} is the competitive market demand.

• When consumers have diﬀerent demands:

T (q) =

⎧⎨

⎩

S_{i}^{c}+ p^{c}q
0

if q > 0 if q = 0.

• No that 1^{◦} PD has the same price and quantity as in perfect competition.

### 3.2 Third degree (multimarket) PD

Based on some direct signal (exogenous information), a producer can divide consumers into m groups. It is then as if there are m “independent” markets. Note that for the markets to be truly independent, we need to assume:

• No arbitrage between groups;

• The producer cannot price discriminate within the group.

Hence the monopolistic producer charges a linear tariﬀ for each group. That is, the monopoly solves:

p1,pmax2,...,pm

Xm

i=1piDi(pi) − C³X^{m}

i=1Di(pi)´ .

Recall the multiproduct monopoly’s problem (independent demands, separable costs), we know that pi are determined according to the inverse elasticity rule:

pi− C^{0}(q)
pi

= 1 εi

,
where εi = −Di^{0}(pi)pi/Di(pi).

• Some examples:

— Low-price discount to first time magazine subscribers;

— Student and senior citizen discounts;

— Legal and medical service bills;

— Goods in poor countries that do not reflect transportation costs and import taxes, etc.

• Compared to uniform pricing in all markets, 3^{◦} PD makes the monopoly and con-
sumers in high-elasticity markets better oﬀ, while it makes consumers in the low-
elasticity markets worse oﬀ. How about total social surplus? Let C³X

iqi

´

= c ·³X

iq_{i}´

, and consumer surplus S_{i}(p_{i}) = R_{∞}

p D_{i}(x)dx. We can write the change

in social surplus when monopoly practices 3^{◦} PD with {p1, ..., p_{m}} and uniform price
with p:

∆W =X

i[Si(pi) − S^{i}(p)] +X

i(pi− c)q^{i}−X

i(p − c)qi. Note that:

Si(pi) ≥ S^{i}(p) + S^{0}_{i}(p)(pi− p),
since: S_{i}^{00}(·) = −D^{0}i(·) > 0, or S(·) is convex. Hence,

S_{i}(p_{i}) − Si(p) ≥ −qi[p_{i}− c − (p − c)],
or,

S_{i}(p_{i}) − Si(p) + (p_{i}− c)qi− (p − c)qi≥ (pi− c)qi− (pi− c)qi = (p_{i}− c)∆qi.
Hence, we have the lower bound for the social surplus change:

∆W ≥X

i(p_{i}− c)∆qi.
Similarly,

Si(p) ≥ S^{i}(pi) + S_{i}^{0}(pi)(p − p^{i}),
following the same reasoning, we have the upper bound:

∆W ≤X

i(p − c)∆q^{i}.

• Thus, a necessary condition for the 3^{◦} PD to be preferred socially is that it raises
total output. The elimination of the 3^{◦} PD can be dangerous if it reduces outputs
significantly (for example, it leads to the closure of markets). Sometimes it can be a
win-win situation (Pareto improvement) to allow firms to price-discriminate.

### 3.3 Second degree PD (arbitrage and screening)

Note that the absence of direct signals empowers consumers with the ability to engage in
personal arbitrage. Perfect PD is, in general, not possible. However, even when there is no
exogenous information (direct signals) for the monopolistic producer to tell the consumers
apart, a monopoly can still extract some consumer surplus. Let’s first go through a common
way of 2^{◦} PD — the two-part tariﬀ.

3.3.1 Two-part tariﬀs

A two-part tariﬀ, T (q) = A + pg, oﬀers a (continuum) menu of bundles {T, q} located on a straight line. Uniform pricing is a special case of two-part tariﬀ when A = 0. Suppose the consumers have the following preferences:

U =

⎧⎨

⎩

θV (q) − T 0

if she pays T for quantity q otherwise;

where: V (0) = 0, V^{0}(·) > 0, and V^{00}(·) < 0. Let V (q) = ^{1−(1−q)}_{2} ^{2} and there be two groups of
consumers: one with taste parameter θ_{1} and the other taste parameter θ_{2}. Let the fraction
of consumers with θ = θ1 be λ, and θ1 < θ2. The monopolistic producer has a constant
marginal cost c < θ_{1}. Solving the consumer’s problem:

maxq . θiV (q) − (A + pq), we have:

qi = Di(p) = 1 − p/θ^{i},
and consumer surplus Si(p) except for T as:

S_{i}(p) = θ_{i}V (D_{i}(p)) − pDi(p) = (θ_{i}− p)^{2}
2θi

. The aggregate demand at price p can be expressed as:

D(p) = λD_{1}(p) + (1 − λ)D2(p)

= 1 − p[λ/θ1+ (1 − λ)/θ2]

≡ 1 − p/eθ.

• Perfect PD: charging a price p^{1} = c and Ai= ^{(θ}^{i}_{2θ}^{−p)}^{2}

i . Hence profit Π1:
Π1 = λ(θ_{1}− c)^{2}

2θ1 + (1 − λ)(θ_{2}− c)^{2}
2θ2

.

— This is the highest profit that a monopoly can get;

— Welfare is optimal.

• Monopoly pricing (linear tariﬀ): the monopoly solves:

maxp . (p − c)D(p).

And, the monopoly price is: p2 = (c + eθ)/2.

— The monopoly may want to give up consumers with lower θ, and serves only θ_{2}
consumers. This cannot be optimal if:

D_{1}(c + θ_{2}
2 ) ≥ 0,
or,

θ1≥ c + θ_{2}
2 .
Let’s assume this inequality holds.

• Two-part tariﬀ: A = S^{1}(p). Now, the monopoly solves:

maxp . S1(p) + (p − c)D(p).

And, the monopoly price is: p_{3} = cθ_{1}/(2θ_{1}− eθ).

Compare these results, we have:

Π1 ≥ Π^{3} ≥ Π^{2},
and (under the assumption: θ_{1}≥ (c + θ2)/2),

c = p_{1}< p_{3}< p_{2}.

Note again, ignoring the redistributive concerns, welfare (social surplus) is higher under the two-part tariﬀ than that of the linear one, because: the marginal price is lower.

• In general, for any linear tariﬀ T (q) = pq with p > c, there exists a two-part tariﬀ T (q) = ee A +pq such that, if consumers are oﬀered the choice between T and ee T , both types of consumers as well as the monopoly are made better oﬀ.

— As it turns out it is tricky to demonstrate this algebraically. We’ll get into the idea in just a minute. For now, a figure helps a lot...

— Consider c <p < p, and ee A = (p − ep)D2(p).

— Type-θ1 consumers prefer the linear tariﬀ, while type-θ2 consumers choose the
two-part tariﬀ. The change in the monopoly’s profit (from type-θ_{2} consumers)
is:

(1 − λ)[ eA +pDe _{2}(p) − pDe 2(p) − c(D2(p) − De 2(p))]

= (1 − λ)[ep(D_{2}(p) − De 2(p)) − c(D2(p) − De 2(p))]

= (1 − λ)(ep − c)(D2(p) − De 2(p)) > 0.

3.3.2 Tie-in Sales and 2^{◦} PD

Consider a consumer who enjoys one unit of basic (tying) good together with q complemen- tary (tied) goods, and derive surplus: θV (q) − T (q). The producer produces the basic good at cost c0, and the complementary good at cost c per unit. Note that c0 becomes a fixed cost per customer served. It plays no role in the pricing decision of the complementary good.

• Suppose tie-in sales are not allowed, and there are competitive firms willing to provide
the complementary good at price c. If the producer serves both types of customers,
it sets the price of the basic good to S_{1}(c), the surplus of the type-θ_{1} consumers.

• If tie-in sales are allowed, we have already shown that the producer would charge a price p > c for the complementary good, and S1(p) < S1(c) for the basic good.

But note that: here, the tie-in sale reduces welfare as long as the producer serves both types of consumers. But the prohibition of tied-in sales may result in only type-θ2consumers

are served (if (1 − λ)S2(c) ≥ S1(c)). In this case, tie-in sales are not necessary detrimental.

3.3.3 Non-linear tariﬀs

Again, studying a figure helps us to draw the following conclusions:

1. Type-θ1 (low-demand) consumers derive no net surplus; type-θ2 (high-demand) con- sumers derive a positive surplus;

2. The incentive compatibility (personal-arbitrage) constraint for the high-demand consumers are binding;

3. The high-demand consumers purchase the social optimal quantity, q_{2} = D_{2}(c); the
low-demand consumers purchase a suboptimal quantity, q1 < D1(c). This is the
so-called “absence of distortion at the top”.

• The monopoly’s profit is:

Π^{m}= λ(T_{1}− cq1) + (1 − λ)(T2− cq2).

It chooses two bundles: (T_{1}, q_{1}) and (T_{2}, q_{2}) to maximize Π^{m}. There are two kinds of
constraints:

— IR: θ_{1}V (q_{1})−T1 ≥ 0. The IR (individual rationality) constraints require the con-
sumers to buy. Note that if the IR constraint for type-θ1 consumers is satisfied,
type-θ_{2} consumers are automatically willing to buy, because: they can always
choose to buy q1 at T1 and get net surplus θ2V (q1) − T^{1} > 0.

— IC: θ2V (q2) − T^{2} ≥ θ^{2}V (q1) − T^{1}. The IC (incentive compatibility) constraints
are used to prevent personal arbitrage (type-θ_{2} consumers choose the bundle
designed for type-θ1 consumers). Since the idea is to induce type-θ2 consumers
to “reveal” their high demands, thus, the IC constraint for type-θ_{1} consumers is
most likely irrelevant (not binding). We can proceed taking into account only
type-θ_{2} consumers’ IC constraint and check later if the IC constraint for type-θ_{1}
consumers is satisfied.

• When Π^{m} is maximized, it must be the case that both constraints (the IR of type-θ_{1}
and the IC of type-θ_{2}) are binding; i.e., T_{1} = θ_{1}V (q_{1}), and θ_{2}V (q_{2})−T2= θ_{2}V (q_{1})−T1.
Hence,

T_{2} = θ_{2}V (q_{2}) − θ2V (q_{1}) + T_{1}= θ_{2}V (q_{2}) − (θ2− θ1)V (q_{1}).

— Observe that: Type-θ1 consumers surplus is entirely appropriated;

— Note also, T_{2} < θ_{2}V (q_{2}) so that type-θ_{2} consumers enjoy net surplus: (θ_{2} −
θ1)V (q1) > 0.

• Using T1 and T_{2}, we can write down the monopoly’s problem as:

max

(q1,q2)λ[θ_{1}V (q_{1}) − cq1] + (1 − λ)[θ2V (q_{2}) − (θ2− θ1)V (q_{1}) − cq2].

The FOCs are: ⎧

⎨

⎩

θ1V^{0}(q1) = c/³

1 −^{1−λ}_{λ} ^{θ}^{2}_{θ}^{−θ}_{1} ^{1}´

;
θ2V^{0}(q2) = c.

• Observe that: 0 < 1−^{1−λ}_{λ} ^{θ}^{2}_{θ}^{−θ}_{1} ^{1} < 1 when: λ ∈ (^{θ}^{2}_{θ}^{−θ}_{2} ^{1}, 1). Hence, as long as both types
of consumers are served (λ is not too small), θ_{1}V^{0}(q_{1}) > c (the quantity consumed by
type-θ_{1} consumers is suboptimal).

• Note also: the FOCs ⇒ V^{0}(q1) > V^{0}(q2), since V (·) is concave, we have q^{1}< q2.

• Finally, check the IR constraint for type-θ^{1} consumers. θ1V (q2) − T^{2} = θ1V (q2) −
θ_{2}V (q_{2}) + (θ_{2}− θ1)V (q_{1}) = (θ_{2}− θ1)[V (q_{1}) − V (q2)] < 0 = θ_{1}V (q_{1}) − T1.

3.3.4 Quality Discrimination

So far, we consider the monopoly discriminates among consumers with diﬀerent tastes by oﬀering diﬀerent quantities of the same good at diﬀerent prices. The same analysis applies to the situation that the monopoly discriminates among consumers with diﬀerent tastes for quality by oﬀering diﬀerent qualities at diﬀerent prices.

Consumers have preference U = θs − p (if buying), where: s is the quality, and p = p(s).

The producer produces s with cost c(s), where: c(·) is increasing and convex.

• Let q ≡ c(s): the cost of quality s;

• Let V (q) ≡ c^{−1}(q) = s: the quality obtained at cost q.

Then,

U = θV (q) − p(V (q)) = θV (q) − ep(q),

where: p(q) ≡ p(V (q)). Moreover, by construction, the monopoly’s cost function is lineare in q. Therefore, it should be clear now that quality discrimination is identical to the above analysis. The monopolistic producer uses lower-quality goods as a market-segmentation device.