Industrial Organization: Markets and Strategies
Paul Belle‡amme and Martin Peitz
published by Cambridge University Press
Part II. Market power
Exercises & Solutions
Exercise 1 Monopoly with quality choice
Consider a monopolist who sells batteries. Each battery works for h hours
and then needs to be replaced. Therefore, if a consumer buys q batteries, he
gets H = qh hours of operation. Assume that the demand for batteries can be
derived from the preferences of a representative consumer whose indirect utility
function is v = u(H) pq, where p is the price of a battery. Suppose that u
is strictly increasing and strictly concave. The cost of producing batteries is
C(q) = qc(h), where c is strictly increasing and strictly convex.
1. Derive the inverse demand function for batteries and denote it by P (q).
2. Suppose that the monopolist chooses q and h to maximize his pro…t. Write
down the …rst-order conditions for pro…t maximization assuming that the
problem has an interior solution, and explain the meaning of these condi-
tions.
3. Write down the total surplus in the market for batteries (i.e., the sum
of consumer surplus and pro…ts) as a function of H and h. Derive the
…rst-order conditions for the socially optimal q and h assuming that there
is an interior solution. Explain in words the economic meaning of these
conditions.
4. Compare the solution that the monopolists arrives at with the social op-
timum. Prove that the monopolist provides the socially optimal level of
h. Give an intuition for this result.
Solutions to Exercise 1
1. The inverse demand for batteries is obtained by solving the following problem:
max u(qh) pq:
q
The …rst-order conditions for this problem can be written as
P (q) = hu0 (qh)
which is the inverse demand function for batteries.
2. The monopolist’s maximization problem is given by
max qP (q) qc(h);
q;h
1
,where P (q) is given in part 1. The …rst-order conditions for an interior solution are:
h(u0 (H) + Hu00 (H)) = c(h) and
q(u0 (H) + Hu00 (H)) = qc0 (h)
where h(u0 (H) + Hu00 (H)) is the marginal revenue from selling an extra battery.
First, let us interpret the expressions inside the brackets: u0(H) is the revenue from
selling an extra hour of operation, while u00 (H) is the discount in the price per hour
of operation that the monopolist must give in order to induce consumers to buy an
extra unit. This discount has to be multiplied by H since the discount is given to all
inframarginal units. Hence, u0 (H) + Hu00 (H) is the marginal revenue from selling
an extra hour of operation. Since the monopolist sells hours of operation in packets
of h units each (a battery provides h hours of operation), u0 (H) + Hu00 (H) has to
be multiplied by h to give the marginal revenue from selling an extra battery. c(h) is
the marginal cost of batteries. Therefore the …rst equation is the familiar monopoly
pricing condition that says that at the optimum, the monopolist produces up to the
point where his marginal revenue equals his marginal cost. Similarly, the second
equation indicates that at the optimum, the marginal revenue from extending the life
of each battery by one hour must equal to marginal cost (note that the marginal cost
has to be multiplied by the number of batteries that the monopolist produces since
the cost of each one of them increases by c0 (h)).
3. The total surplus in the market for batteries is given by u(H) qc(h). The
…rst-order conditions for the social optimum (assuming that an interior solution to
this problem exists) are
hu0 (H) = c(h) and
qu0 (H) = qc0 (h)
where hu0 (H) is the marginal utility of consumers from having an extra battery. It
equals the consumers’ willingness-to-pay for an extra hour of operation multiplied
by h which is the number of hours of operation they get when they buy an extra
battery. The right-hand side of the …rst equation is the marginal cost of producing
an extra battery; the …rst equation states that, at the social optimum, the marginal
utility of consumers has to be equal to the marginal cost of production. The second
equation says that, at the social optimum, the marginal utility of consumers from
having batteries which provide one more hour each, has to be equal to the marginal
cost of extending the life of each battery by one hour.
4. To compare the solutions in parts 2 and 3, we divide the …rst-order conditions
for the monopoly problem by one another:
h c(h) c(h)
= 0 , c0 (h) = :
q qc (h) h
This equation shows that the monopolist produces h by equating the marginal cost of
h with the average cost of h. This implies in turn that h is produced at the minimum
average cost.
By dividing the …rst-order conditions for social optimum by one another we get the
same condition which implies that the monopolist chooses the socially optimal level
2
, of h. Moreover, since the condition that determines h is independent of q , it follows
that this result is true even though the monopolist provides too little quantity.
Exercise 2 Price competition [included in the 2nd edition of the book]
Consider a duopoly in which homogeneous consumers of mass 1 have unit
demand. Their valuation for good i = 1; 2 is v(fig) = vi with v1 > v2 . Marginal
cost of production is assumed to be zero. Suppose that …rms compete in prices.
1. Suppose that consumers make a discrete choice between the two products.
Characterize the Nash equilibrium.
2. Suppose that consumers can now also decide to buy both products. If
they do so they are assumed to have a valuation v(f1; 2g) = v12 with
v1 + v2 > v12 > v1 . Firms still compete in prices (each …rm sets the price
for its product— there is no additional price for the bundle) Characterize
the Nash equilibrium.
3. Compare regimes from parts (1) and (2) with respect to consumer surplus.
Comment on your results.
Solutions to Exercise 2
1. Nash equilibrium given by p1 = v1 v2 , p2 = 0, 1 = v1 v2 and 2 = 0.
2. Nash equilibrium given by p1 = v12 v2 , p2 = v12 v1 , 1 = v12 v2 and
2 = v12 v1 .
3. Consumer surplus in 1) is CSa = v2 , whereas it is given by CSb = v1 +v2 v12
in 2). As v12 > v1 by assumption, consumer welfare is strictly greater in 1) than in
2). In 2) the nature of competition changes because consumers have positive valuation
to buy both products; this relaxes competition.
Exercise 3 Cournot competition
Two …rms (…rm 1 and …rm 2) compete in a market for a homogenous good
by setting quantities. The demand is given by Q(p) = 2 p. The …rms have
constant marginal cost c = 1.
1. Draw the two …rms’ reaction function. Find the equilibrium quantities
and calculate equilibrium pro…ts.
2. Suppose now that there are n …rms where n 2. Calculate equilibrium
quantities and pro…ts.
Solutions to Exercise 3
This is the standard Cournot model, just in case students have forgotten about their
intermediate micro class.
Exercise 4 Equilibrium uniqueness in the Cournot model
3
Paul Belle‡amme and Martin Peitz
published by Cambridge University Press
Part II. Market power
Exercises & Solutions
Exercise 1 Monopoly with quality choice
Consider a monopolist who sells batteries. Each battery works for h hours
and then needs to be replaced. Therefore, if a consumer buys q batteries, he
gets H = qh hours of operation. Assume that the demand for batteries can be
derived from the preferences of a representative consumer whose indirect utility
function is v = u(H) pq, where p is the price of a battery. Suppose that u
is strictly increasing and strictly concave. The cost of producing batteries is
C(q) = qc(h), where c is strictly increasing and strictly convex.
1. Derive the inverse demand function for batteries and denote it by P (q).
2. Suppose that the monopolist chooses q and h to maximize his pro…t. Write
down the …rst-order conditions for pro…t maximization assuming that the
problem has an interior solution, and explain the meaning of these condi-
tions.
3. Write down the total surplus in the market for batteries (i.e., the sum
of consumer surplus and pro…ts) as a function of H and h. Derive the
…rst-order conditions for the socially optimal q and h assuming that there
is an interior solution. Explain in words the economic meaning of these
conditions.
4. Compare the solution that the monopolists arrives at with the social op-
timum. Prove that the monopolist provides the socially optimal level of
h. Give an intuition for this result.
Solutions to Exercise 1
1. The inverse demand for batteries is obtained by solving the following problem:
max u(qh) pq:
q
The …rst-order conditions for this problem can be written as
P (q) = hu0 (qh)
which is the inverse demand function for batteries.
2. The monopolist’s maximization problem is given by
max qP (q) qc(h);
q;h
1
,where P (q) is given in part 1. The …rst-order conditions for an interior solution are:
h(u0 (H) + Hu00 (H)) = c(h) and
q(u0 (H) + Hu00 (H)) = qc0 (h)
where h(u0 (H) + Hu00 (H)) is the marginal revenue from selling an extra battery.
First, let us interpret the expressions inside the brackets: u0(H) is the revenue from
selling an extra hour of operation, while u00 (H) is the discount in the price per hour
of operation that the monopolist must give in order to induce consumers to buy an
extra unit. This discount has to be multiplied by H since the discount is given to all
inframarginal units. Hence, u0 (H) + Hu00 (H) is the marginal revenue from selling
an extra hour of operation. Since the monopolist sells hours of operation in packets
of h units each (a battery provides h hours of operation), u0 (H) + Hu00 (H) has to
be multiplied by h to give the marginal revenue from selling an extra battery. c(h) is
the marginal cost of batteries. Therefore the …rst equation is the familiar monopoly
pricing condition that says that at the optimum, the monopolist produces up to the
point where his marginal revenue equals his marginal cost. Similarly, the second
equation indicates that at the optimum, the marginal revenue from extending the life
of each battery by one hour must equal to marginal cost (note that the marginal cost
has to be multiplied by the number of batteries that the monopolist produces since
the cost of each one of them increases by c0 (h)).
3. The total surplus in the market for batteries is given by u(H) qc(h). The
…rst-order conditions for the social optimum (assuming that an interior solution to
this problem exists) are
hu0 (H) = c(h) and
qu0 (H) = qc0 (h)
where hu0 (H) is the marginal utility of consumers from having an extra battery. It
equals the consumers’ willingness-to-pay for an extra hour of operation multiplied
by h which is the number of hours of operation they get when they buy an extra
battery. The right-hand side of the …rst equation is the marginal cost of producing
an extra battery; the …rst equation states that, at the social optimum, the marginal
utility of consumers has to be equal to the marginal cost of production. The second
equation says that, at the social optimum, the marginal utility of consumers from
having batteries which provide one more hour each, has to be equal to the marginal
cost of extending the life of each battery by one hour.
4. To compare the solutions in parts 2 and 3, we divide the …rst-order conditions
for the monopoly problem by one another:
h c(h) c(h)
= 0 , c0 (h) = :
q qc (h) h
This equation shows that the monopolist produces h by equating the marginal cost of
h with the average cost of h. This implies in turn that h is produced at the minimum
average cost.
By dividing the …rst-order conditions for social optimum by one another we get the
same condition which implies that the monopolist chooses the socially optimal level
2
, of h. Moreover, since the condition that determines h is independent of q , it follows
that this result is true even though the monopolist provides too little quantity.
Exercise 2 Price competition [included in the 2nd edition of the book]
Consider a duopoly in which homogeneous consumers of mass 1 have unit
demand. Their valuation for good i = 1; 2 is v(fig) = vi with v1 > v2 . Marginal
cost of production is assumed to be zero. Suppose that …rms compete in prices.
1. Suppose that consumers make a discrete choice between the two products.
Characterize the Nash equilibrium.
2. Suppose that consumers can now also decide to buy both products. If
they do so they are assumed to have a valuation v(f1; 2g) = v12 with
v1 + v2 > v12 > v1 . Firms still compete in prices (each …rm sets the price
for its product— there is no additional price for the bundle) Characterize
the Nash equilibrium.
3. Compare regimes from parts (1) and (2) with respect to consumer surplus.
Comment on your results.
Solutions to Exercise 2
1. Nash equilibrium given by p1 = v1 v2 , p2 = 0, 1 = v1 v2 and 2 = 0.
2. Nash equilibrium given by p1 = v12 v2 , p2 = v12 v1 , 1 = v12 v2 and
2 = v12 v1 .
3. Consumer surplus in 1) is CSa = v2 , whereas it is given by CSb = v1 +v2 v12
in 2). As v12 > v1 by assumption, consumer welfare is strictly greater in 1) than in
2). In 2) the nature of competition changes because consumers have positive valuation
to buy both products; this relaxes competition.
Exercise 3 Cournot competition
Two …rms (…rm 1 and …rm 2) compete in a market for a homogenous good
by setting quantities. The demand is given by Q(p) = 2 p. The …rms have
constant marginal cost c = 1.
1. Draw the two …rms’ reaction function. Find the equilibrium quantities
and calculate equilibrium pro…ts.
2. Suppose now that there are n …rms where n 2. Calculate equilibrium
quantities and pro…ts.
Solutions to Exercise 3
This is the standard Cournot model, just in case students have forgotten about their
intermediate micro class.
Exercise 4 Equilibrium uniqueness in the Cournot model
3
