Sustainable Pricing in a Durable Goods Monopoly∗
Takeki Sunakawa†
The Ohio State University and Bank of JapanMay 2012

Abstract
Ausubel and Deneckere [1989. Reputation in bargaining and durable goods monopoly.
Econometrica 57, 511-531] showed that the monopoly prot is approximately sustainable
when agents are patient enough and the length of time is small enough. This paper considers
the case in which these conditions do not hold, and examines the short-run dynamics, whichare, to my knowledge, neglected in previous studies. This paper nds that the sustainablepricing is intermediate between the monopolistic pricing and the Markov-perfect pricing,
and the monopolist quotes the price between the monopolistic price and marginal costs
for every period until the market is satiated. At a low discount factor, the sustainablepricing converges to marginal costs, but very slowly. At a suciently high discount factor,
the speed of convergence becomes arbitrarily slow, which is consistent with Ausubel and
Deneckere's result. This paper also considers the eect of introducing depreciation andstochastic marginal cost into the model.

Keywords : Durable goods monopoly; reputation; policy function iteration method
JEL codes : C63; D21; D42∗
First draft: May 2011. Word count: 5605 words. This paper is a part of my dissertation at the Ohio State
University. I am grateful to William Dupor for continuous support and guidance. I also appreciate comments
and discussions from Paul Evans, Aubhik Khan, Jim Peck, and Paulina Restrepo-Echavarria, Julia Thomas,
as well as seminar participants at International Conference on Computing in Economics and Finance 2011 andMidwest Economic Theory and International Trade Meetings 2012. Any remaining errors are mine. The views
expressed, herein, are solely mine and should not be interpreted as those of Bank of Japan.
†
Correspondence: Department of Economics, The Ohio State University, 410 Arps Hall, 1945 N High Street,
Columbus OH 43210, USA. Email: takeki.sunakawa@gmail.com.

1

1

Introduction

Let's consider a monopolist which oers prices of innitely durable goods to an innite mass of
consumers over time. Assume that in period 0, the monopolist quotes the monopolistic pricewith markup over marginal costs, and sell goods to consumers less than the amount at which
goods are sold in perfect competition. Also assume that after period 1, no transaction occurs.
This pricing strategy is

not time-consistent; the monopolist has an incentive to oer lower prices

than the precommitted price, and sell more goods after period 1. The monopolist continues tohave such an incentive until the market is satiated. Also, consumers form expectations that theprices will be lower, and postpone their purchases in period 0.
Based on this consideration, Ronald Coase (1972) conjectured that the monopolist quotes
the price equal to the competitive price (i.e., marginal costs) for every period. This is known
as the Coase conjecture; the monopolist does not have monopolistic power. Economists have
tried to nd out how reasonable this conjecture is. Stokey (1981) showed that there is a uniquetime-consistent Markov-perfect equilibrium. As the length of time between each timing of pricesetting approaches zero, the monopolist immediately satiates the market at the competitive
price. Stokey supported the Coase conjecture in the Markov-perfect equilibrium in a continuoustime setting.

On the contrary, Ausubel and Deneckere (1989, hereafter AD) considered thesubgame-perfect equilibrium (SPE), and found that the monopolistic price is approximatelysupported as the best SPE if the agents (both the monopolist and consumers) are patient enoughand the length of time is small enough. If the monopolist deviates from the optimal strategy,
the economy switches to the Markov-perfect equilibrium as dened in Stokey (1981) forever.
The monopolist's reputation in a long-run relationship with consumers makes it possible to sellgoods at higher prices. AD showed that the Coase conjecture does not necessarily hold even ina durable goods market; a monopoly

is a monopoly (AD, p. 512).

However, in the case that the agents are not patient enough, and building a long-run relationship is dicult, it is not clear what kind of pricing strategy is optimal. Also, the short-run
dynamics of prices and quantities in the best SPE, which are, to my knowledge, neglected inprevious studies, are of natural interest because the nature of the problem is dynamic.
In this paper, I solve the monopolist's problem utilizing the idea of Chari and Kehoe's (1990)
Sustainable Plan, which has been used in macroeconomics to solve time-inconsistency problems.
I also examine the short-run dynamics by a version of policy function iteration methods with2

participation constraints (Kehoe and Perri, 2002).
There are three main ndings. First, the sustainable pricing is intermediate between themonopolistic pricing and the Markov-perfect pricing. The monopolist quotes the price between
the monopolistic price and marginal costs for every period until the market is satiated. Second,at a low discount factor β , the sustainable pricing converges to marginal costs, but very slowly;
the monopolist oers lower prices over time and continues to sell until the market is satiated.
Reputation has the eect like an adjustment cost. Third, at a suciently high discount factor

β , the speed of convergence becomes arbitrarily slow, which is consistent with AD's result.
Chari and Kehoe's (1990) seminal paper proposed a concept of sustainable plans, which is
optimal for the strategic player's dynamic decision making when the strategic player cannot
commit to future plans. The equilibrium concept is very close to subgame perfect equilibria in
the theory of repeated games between a strategic player and an innite mass of small agents,
and the authors characterized the sustainable equilibrium using game theory; they used Abreu's(1988) technique of using the worst sustainable equilibrium to characterize the entire set of thesustainable equilibrium. The equilibrium condition is summarized into an inequality, which iscalled the sustainability constraint.
In time-inconsistency problems, computations are often dicult because I have to deal withdynamic incentive constraints with which the Bellman's principle of optimality does not hold.
Marcet and Marimon (1998; 2011) developed the recursive saddle point method to consider such
dynamic incentive constraints. Kehoe and Perri (2002) applied the method for a two-country
model with endogenous incomplete markets. They used a version of policy function iteration
methods to solve for the policy function, which is also used in this paper. It is closely related
to the method dealing with occasionally binding constraints (e.g., Christiano and Fisher, 2000),
because the incentive constraints are occasionally binding.
The rest of paper proceeds as follows. In the next section, I explain the basic setup andeach pricing strategy in turn.

In Section 3, I show the numerical result.

Section 4 includes

some extensions: depreciation and shocks to marginal costs. Finally Section 5 concludes. Some
proofs and the details of computational method are shown in the appendix.

3

2

Setup and Three Pricing Strategies

I consider three pricing strategies in turn; the monopolistic pricing, the Markov-perfect pricing (Stokey, 1981), and the sustainable pricing (Ausubel and Deneckere, 1989). The setup iscommon among these dierent pricing strategies.
Time is discrete and innite; t = 0, 1, 2, ..., ∞. The monopolist produces innitely durable
goods and sell them to a continuum of identical consumers. Consumers can resell goods in the
future, or a rental market is available. The dynamic demand equation is given by

pt = f (qt ) + βpt+1 , ∀t ≥ 0.
where f (0)> 0 and f 0 (q) < 0.

(1)

Durable goods are stock, and the price at period t is the∞discounted sum of future resale values. The monopolist chooses the sequence of {pt }t=0 (and∞hence {qt }t=0 given (1)) so as to maximize its discounted sum of future prots

R0 ≡

∞
X

β t (pt − c)(qt − qt−1 ),

(2)

t=0
subject to (1).

2.1

Monopolistic Pricing

I begin with the situation that the monopolist can commit to future prices.

I call it themonopolistic pricing, because the monopolist quotes the price associated with the quantity
determined at the level with which the marginal cost is equal to the marginal revenue in every
period.

The monopolistic pricing is optimal in terms of the monopolist's prot in period 0.

However, the monopolistic pricing is not time-consistent without such commitment technologies.
The monopolist has an incentive to deviate from the original plan after period 1.
Let λt be the Lagrange multiplier associated with (1).

If the monopolist can commit to

future prices and quantities, the rst-order necessary conditions (FONCs) of the maximization4

problem of (2) subject to (1) are

qt − qt−1 = λt − λt−1 ,(3)

pt − c − β(pt+1 − c) + λt f 0 (qt ) = 0,(4)

pt − βpt+1 − f (qt ) = 0.

(5)

There are two state variables in the dynamic system. One is the amount of goods sold until theend of period t − 1, qt−1 , and the other is the Lagrange multiplier in period t − 1, λt−1 , whichshows the monopolist's commitment in the past. It is natural to assume that no goods are soldand no commitment is made before period 0. Given the initial conditions q−1 = λ−1 = 0, by
solving equation (5) backward, I have qt = λt for all t ≥ 0. A positive amount of goods soldalso implies the monopolist's commitment in the past, because the monopolist has to commit
to a particular level of prices to sell their goods to consumers who otherwise postpone theirpurchase. Equations (4) and (5) together with qt = λt become(1 − β)c = f (qt ) + qt f 0 (qt ),∞X
pt =β i f (qt+i ),

(6)
(7)

i=0
where equation (6) shows that average marginal cost c̄ = (1 − β)c (c = c̄ + βc̄ + ...) is equal0

to the marginal revenue f (qt ) + qt f (qt ) in periods t ≥ 0. Equation (6) determines the amountof goods sold, and equation (7) sets prices. Because qt = q
monopolist sets the monopolistic price pt

m is constant by equation (6), the= pm constant over time by equation (7), and alltransactions occur in period 0. The monopolist sells the amount of goods q0 = q

m and earn all

m in period 0, and no prots after that, R = 0 for all t ≥ 1.
t

prots R0 = R0

This pricing strategy is, however,not time-consistent. If the monopolist can reset prices inperiod 1, the monopolist also can reset commitment made in the past, by setting λ0 = 0. Given

q0 > 0 and λ0 = 0, I have qt − q0 = λt for all t ≥ 1. Then
(1 − β)c = f (q̃t ) + q̃t f 0 (q̃t ) − q0 f 0 (q̃t )(8)0holds. Comparing equations (6) and (8), the additional term −q0 f (q̃t ) > 0 shifts the marginal
revenue curve toward right. This yields p̃t < pt and q̃t > qt for all t ≥ 1, and R̃1 = (p̃1 − c)(q̃1 −5

q0 ) > 0; the monopolist has an incentive to quote lower prices than the prices committed in thepast, sell more goods, and earn positive prots in period 1 by such a deviation. The new plan∞{p̃t , q̃t }∞t=1 is not consistent with the original plan {pt , qt }t=0 .
Figure 1 also shows that the monopolist continues to have such an incentive until the market
is satiated. An irresponsible monopolist, which abandons the price committed in the past in
every period, quotes lower prices in every period to sell more goods. The curve of the marginal
revenue in equation (8) continues to shift toward right, and given a constant marginal cost, theamount of goods sold until the end of period t gets higher, until the market is satiated at q = qcwith which the price is equal to the marginal cost.

2.1.1 Analytical solutions in the LQ framework
As in the most of previous studies, I assume a linear demand function of f (q) = a − bq and aquadratic prot function so that the problem is one in the linear-quadratic (LQ) framework.
Then I can analytically solve for the monopolistic price, quantity and protpt = pm = (a + c̄)/(2 − 2β),∀t ≥ 0,qt = q m = (a − c̄)/(2b),∀t ≥ 0,R0 = R0m = (a − c̄)2 /[4b(1 − β)],Rt = 0,The monopolist quotes p

m which is higher than the competitive price pc = c, and sell q m which

is lower than the competitive quantity q

2.2∀t ≥ 1.

c = 2q m = (a − c̄)/b in period 0.

Markov-Perfect Pricing

Stokey (1981) showed that there is a unique Markov-perfect equilibrium in the monopolist's
problem.

1 The problem is dened as a recursive form:

W (qt−1 ) = (pt − c)(qt − qt−1 ) + λt [f (qt ) + βp(qt ) − pt ] + βW (qt ).
1

Stokey originally solved the nite-horizon dynamic programming and set the initial period innite past.

6qt−1 is the only natural state variable, because the monopolist cannot commit to future prices
2 The future price p

and there is no commitment made in the past in every period.

t+1 = p(qt ) is

also function of qt . Even though the monopolist cannot commit to future prices, the monopolist
considers the eect of changing the amount of goods on future prices. The FONCs are∂pt : qt − qt−1 = λt ,

(9)



∂qt : pt − c − βW 0 (qt ) + λt f 0 (qt ) + βp0 (qt ) = 0,(10)∂λt : pt − βp(qt ) − f (qt ) = 0.

(11)

0

Also, W (qt ) = −(pt+1 − c) holds by the envelope theorem. Additional amount of goods sold decreases future prots, which the monopolist could earn otherwise in the next period. Equations(9)-(11) becomec̄ = f (qt ) + qt f 0 (qt ) − qt−1 f 0 (qt ) + βp0 (qt )(qt − qt−1 ).

(12)

0

There are two additional terms to the marginal revenue as seen in equation (6). −qt−1 f (qt ) >0 appears as in the case of the monopolistic pricing seen in equation (8).

The monopolist

cannot control future prices by commitment, which shifts the marginal revenue curve towardright, decreases prices, and increases quantities. The Markov-perfect pricing is time-consistent,
because the monopolist quotes lower prices than the monopolistic pricing to sell more amountof goods, so that the monopolist's strategy in the current period is consistent with the one in
the next period. The monopolist also internalizes the eect of changing the amount of goods0

on future prices; βp (qt )(qt − qt−1 ) < 0 diminishes the marginal revenue and increases prices.

0

Note that, p (qt ) in equation (10) implies a diculty to solve for the policy functions p(qt−1 )0

and q(qt−1 ), because I need to know p(qt−1 ) to obtain p (qt−1 ) and solve for the policy functions(Krusell et al, 2002). In the LQ framework, the guess and verify method is easily applied to
avoid this problem.

2

If I consider such a commitment, the monopolist's problem becomes non-recursive, because the constraint
have a future variable pt+1 ; Marcet and Marimon (2011) denes the saddle point problem to consider this typeof the problem. I solve for the monopolistic pricing by solving the sequential problem, instead of solving therecursive problem, as I have done in the previous section. To solve for the sustainable pricing, I extensively use
the saddle point problem. See section 2.3.

7

2.2.1 Analytical solutions in the LQ framework
I assume a linear function of f (q) = a − bq . Then I haveqt − qt−1 = λt ,pt − βpt+1 − a + bqt = 0,
pt − c − β(pt+1 − c) + [−b + β(∂pt+1 /∂qt )]λt = 0.
The system has the steady state

pc = c,q c = (a − c̄)/b.

At the steady state, prices and quantities are at the associated competitive level. If the solutionof the dynamic system is stable, prices and quantities converge to the steady state.

I cananalytically solve for the stable solution, by using the undetermined coecient method to guessand verify the policy functions

pt = τp + τpq qt−1 ,qt = τq + τqq qt−1 ,where τqq = [1 − (1 − β).5 ]/β ∈ (0, 1), τcpq = −bτqq /(1 − βτqq ) < 0, τq = (1 − τqq )q > 0, and2 − 2τ + 1 = 0.
τp = pc − τpq q c > 0. τqq ∈ (0, 1) is the stable root of the quadratic equation βτqq
qq
The solution is the Markov-perfect equilibrium, because qt−1 is the only natural state variable.
It is also unique, because the quadratic equation has only one stable root and the other root is
not stable (Blanchard and Kahn, 1980).

2.2.2 Comparison of the prot functions in the LQ framework
In the LQ framework, I can also analytically solve for the prot of the Markov-perfect pricing,which is a function of the amount of goods sold, qt−1

2
W (qt−1 ) ≡ Rt = µ + µq qt−1 + µqq qt−1
.

8

(13)

Then I can prove

Proposition 1. W (qt−1 ) > 0 and W 0 (qt−1 ) = −(p(qt−1 ) − c) < 0 for qt−1 ∈ [0, qc ). Also,
W (q c ) = W 0 (q c ) = 0.

Proof. See Appendix A.1.
Proposition 1 shows that the monopolist can always earn positive prot by the Markovperfect pricing when the amount of goods already sold are between zero and the competitive
level. The prot is decreasing and zero at the competitive level. This result suggests that, if
there are no exogenous shocks, the amount of goods monotonically converges to the competitive
level. I can also analytically solve for the prot of the monopolistic pricing in period t for any
values (qt−1 , λt−1 ), by solving the FONCs of the monopolistic pricing (5)-(5) and the prot3function (2)

m
m 2
m 2
m
Rm (qt−1 , λt−1 ) = µm + µm
q qt−1 + µλ λt−1 + µqq qt−1 + µλλ λt−1 + µqλ qt−1 λt−1 ,Note that, R(14)m (0, 0) = µm = Rm (see Appendix A.2), and as shown in the previous section,0

qt = λt for all t holds and there is no dynamics after period 0 in the monopolistic pricing.
Still, the prot function is important to consider the monopolist's incentive to deviate fromthe current pricing in periodt, given the amount of goods sold and the past commitment(qt−1 , λt−1 ). I can prove

Proposition 2. D1 Rm (qt−1 , λt−1 ) < 0 for qt−1 ∈ [0, qc ] and D2 Rm (qt−1 , λt−1 ) < 0 for λt−1 ∈[0, q c ]. Also, Rm (qt−1 , λt−1 ) ≤ 0 for λt−1 ∈ [q c − qt−1 , q c ].

Proof. See Appendix A.2.
Proposition 2 shows that as qt−1 or λt−1 is higher, the monopolist more likely deviate fromthe current pricing, because the monopolist can earn positive prot by the Markov-perfectpricing wheneverqt−1 ∈ [0, q c ) (Proposition 1).

When

λt−1 > q c − qt−1 (or equivalently,

qt−1 > q c − λt−1 ), the prot from the monopolistic pricing is always non-positive.

Thus,

considering propositions 1 and 2, I can show that

Corollary 3. Rm (qm , qm ) < W (qm ), i.e., the monopolistic pricing is not sustainable for anyβ ∈ (0, 1).
3

Note that in the monopolistic pricing, qt−1 = λt−1 = 0 holds without deviation.
9

Proof. Note that q m = q c /2. Then W (q m ) > 0 holds from proposition 1 and Rm (q m , q m ) ≤ 0
holds from proposition 2.

Corollary 3 is consistent with AD's result.
discount factor β such that [ε, π(SPE), where πε, there exists a∗ − ε] is in the set supported by subgame perfect equilibria∗ denotes the monopoly prot. As β → 1, ε becomes arbitrary small to support

the monopoly prot as the best SPE. πto ε > 0.

AD showed that for every∗ is only approximately supported as long as β < 1 up

4 In the present analysis, Figure 2 shows max{Rm (qt−1 , λt−1 ), −W (qt−1 ), 0} on thestate space (qt−1 , λt−1 ). In the region where the value takes zero, the monopolistic pricing is
not sustainable, because the monopolist can earn positive prot by deviating to the Markovperfect pricing as Rm (qt−1 , λt−1 ) ≤ W (qt−1 ) holds. Corollary 3 shows that it is always true

for any discount factor β∈ (0, 1) at qt−1 = λt−1 = q m in the LQ framework.

As β

→ 1,

the gain by deviation becomes smaller. This makes the monopolistic pricing approximately
sustainable. The discount factor is important for the monopolist's incentive to deviate, because
the long-term relationship is more important as β → 1 as usual in a repeated environment.
Figure 2 also shows that even if λt−1

< q c − qt−1 , the monopolist may deviate from themonopolistic pricing to the Markov-perfect pricing. If the discount factor β is low, the promises
made in the past is too much for the monopolist to keep. The region where both λt−1 < q
and R

m (q

c −q

t−1

t−1 , λt−1 ) < W (q) hold shrinks as β → 1, because the value of commitment is higher

as β → 1. When Rm (qt−1 , λt−1 ) ≥ W (qt−1 ) holds, a pair (qt−1 , λt−1 ) is credible in a reputationcmechanism. Also, at (qt−1 , λt−1 ) = (q , 0), the monopolist does not have incentive to deviate forcany β ∈ (0, 1), because W (q ) = R

m (q c , 0) = 0. The competitive level of prices and quantities

are supported by SPE, but it is the worst SPE as the monopolist may be better o in otherequilibria.

In the next section, I consider the dynamics in such equilibria supported by themonopolist's reputation among consumers.

2.3

Sustainable Pricing

AD considered the subgame perfect equilibrium (SPE) in durable goods monopoly. AD formulated the dynamic game between the monopolist and an continuum of innitely-lived consumers.
As usual in repeated games, the trigger strategy takes a form of punishment to the monopo-

4

AD originally considered discount rate r and the length of time z to have discount factor β = (1 + rz)−1 ,and then made z arbitrary small.

10

list for deviating from the optimal strategy, and the trigger strategy gives the monopolist an
incentive to keep its promises in a long-term relationship with consumers.

The monopolist's

reputation among consumers makes it possible to sell goods at higher prices. The monopolist's
deviation from the current pricing damages its reputation, but such a deviation is o the equilibrium.

The monopolist's strategy is summarized into the sustainability constraint (Abreu,

1988; Chari and Kehoe, 1990), in which the continuation value is greater than the one from thedeviation.

2.3.1 Pricing Game
The monopolist quotes the pricept at t = 0, 1, 2, ....5

A continuum of identical innitely-

6 In each period, the monopolist quotes the price rst, then

lived consumers purchases goods.

consumers who have not purchased goods decide whether or not to buy. Both the monopolist
and consumers discount the future by a common discount factor β .

7History in period t is dened as the prices and the quantities up to period t − 1.

h0 = ∅and ht = (ht−1 , pt−1 , qt−1 ) for all t > 0. The monopolist's strategy is given by pt = σt (ht ) andcontingent plans for any future histories. Consumers are price takers, and their decision ruleis given by qt = τt (ht , pt ) and contingent plans for any future histories. Then, I can dene asustainable equilibrium as follows.

Denition 4. A sustainable equilibrium of the model is a pair (σ, τ ) of a monopolist's strategy
and consumers' reaction to the strategy such that(i) given the pricing strategy σ and current history (ht , pt ), the continuation of consumers'
reaction τ satises

pt = βσt+1 (ht+1 ) + f [τt (ht , pt )],t≥0

σs (hs ) = βσs+1 (hs+1 ) + f [τs (hs , σs (hs ))], s ≥ t + 1
5
AD originally used t = 0, z, 2z, ..., nz, ... where z is the time interval. I use t instead of n as the index oftime.
6Here I assume consumers are identical so that I can ignore consumers' distribution whose status of purchasing
goods depends on the history as well; only qt−1 summarizes the information of those who have already purchased
goods among identical consumers. AD originally imposed measurability restrictions on consumers who havedierent reservation prices, so that the the monopolist's and consumers' strategies are a function of consumers'
distribution rather than a quantity.
7
AD originally used e−rt ≈ (1 + r)−t as the discount factor.

11

for all possible future histories induced by σ .
(ii) given consumers' reaction τ and and current history ht , the continuation of the pricing
strategy σ solves

max(σ̃s )s≥t∞
X

β s−t [σ̃s (hs ) − c][τs (hs , σ̃s (hs )) − τs−1 (hs−1 , σ̃s−1 (hs−1 ))]

s=t
s.t.

σ̃s (hs ) = β σ̃s+1 (hs+1 ) + f [τs (hs , σ̃s (hs ))], for all s ≥ t,

for all t ≥ 0 and for all possible future histories induced by (σ̃s )s≥t .

2.3.2 Sustainability Constraint
In the sustainable pricing, the monopolist sets prices so that∞X

β s−t (ps − c)(qs − qs−1 ) ≥ W (qt−1 ),

(15)

s=t
holds for all t ≥ 0.

8 (15) is called the sustainability constraint. As long as this inequality holds,

the monopolist has no incentive to deviate from the current pricing. The sustainable pricing isdened as a strategy that species to continue the current pricing as long as it has been adaptedin the past. Such a pricing is sustained by the monopolist's reputation among consumers who
utilize the Markov-perfect pricing as a trigger strategy.
I use the following propositions (see Appendix for proofs):

Proposition 5. The Markov-perfect equilibrium is the worst sustainable equilibrium.
Proposition 5 shows that W (qt−1 ) actually attains the lowest value among the outcome ofsustainable equilibrium. The worst sustainable equilibrium value in the sustainability constraint
ensures the best sustainable equilibrium to be included in the set of all sustainable equilibria.

Proposition 6. Any pair (p, q) of contingent sequences of prices and quantities is the outcomeof the sustainable equilibrium if and only if (i) the pair (p, q) satises (1) in every period t ≥ 0
and (ii) the inequality (15) holds in every period t ≥ 0.
Proposition 6 also shows that any arbitrary sequence is an outcome of a sustainable equilibrium if and only if Equations (1) and (15) are satised for t ≥ 0. The entire set of outcome

8

There is no temporary gain from deviation because the monopolist moves rst.
12

induced by the sustainable equilibrium is characterized by this proposition.

The sustainablepricing is characterized by the outcome of the best sustainable equilibrium; the monopolist

9 The

∞

chooses {pt , qt }t=0 so as to maximize (2) subject to Equations (1) and (15) for t ≥ 0.
sustainable pricing is obtained by solving for the constrained ecient allocation.

Note that,the sustainable pricing is a strategy which assumes the absence of commitment technologies,
whereas, the outcome of the sustainable pricing assumes the presence of commitment technologies.
Let ϕt ≥ 0 be Lagrange multiplier associated with the sustainability constraint (15), andlet the sum of the Lagrange multiplier Ψt = Ψt−1 + ϕt ≥ 1, given q−1 = λ−1 = 0 and Ψ−1 = 1.
Then the recursive form of the saddle point problem as in Marcet and Marimon (2011) is writtenas

V (qt−1 , xt−1 ) = (pt − c)(qt − qt−1 ) − (xt − zt xt−1 )pt + xt f (qt )−(1 − zt )W (qt−1 ) + (β/zt+1 )V (qt , xt ),where xt

= λt /Ψt ≥ 0, zt = Ψt−1 /Ψt ∈ (0, 1] and Vt = Rt /Ψt . xt is normalized Lagrange

multiplier λt , and xt > 0 shows the monopolist's commitment in the past.

zt is the ratio of

the sum of Lagrange multiplier ϕt , and zt < 1, which is equivalent to ϕt > 0, shows that thesustainability constraint is binding in the current period. The FONCs are∂pt : (qt − qt−1 ) − (xt − zt xt−1 ) = 0,
∂qt : (pt − c) + xt f 0 (qt ) + (β/zt+1 )D1 V (qt , xt ) = 0,
∂xt : −pt + f (qt ) + (β/zt+1 )D2 V (qt , xt ) = 0.
Also, the envelope theorem yields

D1 V (qt−1 , xt−1 ) = −(pt − c) − (1 − zt )W 0 (qt−1 ) ≤ 0,
D2 V (qt−1 , xt−1 ) = zt pt > 0.
9
One might wonder if other fallback equilibria were chosen with a somewhat less harsh punishment. As notheory seems to be available on how fallback equilibria are chosen, my discussion here assumes that both themonopolist and consumers choose the Markov-perfect pricing for sure if, at all, the monopolist abandons thecommitment.

13

The rst condition shows that by increasing marginal amount of goods sold, the monopolist loses

00

future prot −(pt − c) ≤ 0, but outside option −(1 − zt )W (qt−1 ) ≥ 0. Note that W (qt−1 ) ≥−(pt − c) holds in the sustainable pricing, because the monopolist quotes higher prices in thesustainable pricing than the Markov-perfect pricing, in which0W 0 (qt−1 ) = −(pt − c) holds;0

therefore, I have D1 V (qt−1 , xt−1 ) = −(pt − c) − W (qt−1 ) + zt W (qt−1 ) ≤ 0. The monopolist's
prot is decreasing in the amount of goods as in the Markov-perfect equilibrium.

However,the monopolist can utilize the outside option to earn more prot than in the Markov-perfectpricing, when the sustainability constraint is binding, (1 − zt ) > 0.
The second condition shows that, by increasing marginal amount of commitment, the monopolist can sell marginal amount of goods and earn pt (Note that Vt = Rt /Ψt ; zt pt = Ψt pt /Ψt−1
is a normalized value in period t − 1). The commitment to future prices yields more prots forthe monopolist, if such a commitment is feasible.
Combining the FONCs and the envelope theorem, I haveqt − qt−1 = xt − zt xt−1 ,(pt − c) + xt f 0 (qt ) − (β/zt+1 ) (pt+1 − c) + (1 − zt+1 )W 0 (qt ) = 0,
−pt + f (qt ) + βpt+1 = 0,
Rt ≥ W (qt−1 ) ⊥ zt ≤ 1.
The last condition is the complementary slackness condition, as the sustainability constraint is
occasionally binding. Summarizing FONCs, I have

(1 − β)c = f (qt ) + qt f 0 (qt ) − (qt−1 − zt xt−1 )f 0 (qt )−1−β(1 − zt+1 )zt+1[pt+1 − c + W 0 (qt )].

0

(16)

0

where pt+1 − c + W (qt ) = zt+1 W (qt ) − D1 V (qt , xt ) > 0 is the dierence of marginal value
between the current pricing and the Markov-perfect pricing.

The discretion and outside-option eect in marginal revenue
following three dierent optimality conditions in each pricing strategies:

14

Let me consider the(Monopolistic)(1 − β)c = f (qt ) + qt f 0 (qt ) −q0 f 0 (qt ),| {z }+(Markov-perfect)0(1 − β)c = f (qt ) + qt f (qt ) −qt−1 f 0 (qt ) +βp0 (qt )λt ,|{z} | {z }−+(Sustainable)0(1 − β)c = f (qt ) + qt f (qt ) −(qt−1 − zt xt−1 )f 0 (qt )
|
{z
}
+

−1
−β(1 − zt+1 )zt+1[pt+1 − c + W 0 (qt )],

|
0

{z

}

−

0

0

where f (qt ) < 0, p (qt ) < 0, and pt+1 −c+W (qt ) > 0. The rst equation is for the monopolistic
pricing, the second equation is for the Markov-perfect pricing, the third equation is for the0sustainable pricing each. Note that the marginal revenue is ∂(pq)/∂q = p + qp (q) in the staticcase. In the dynamic case I consider here, with the monopolist's commitment to future prices,0

the marginal revenue is f (qt ) + qt f (qt ).

However, if such commitment technologies are notavailable, there are two changes in the marginal revenue. One is the discretion eect which
lowers prices, and the other is the outside-option eect which highers prices.
The discretion eect stems from the fact that the monopolist quotes lower prices withoutcommitment as consumers purchase more goods. In the monopolistic pricing, if the monopolist

0deviates and quotes lower prices, the monopolist obtains additional marginal revenue −q0 f (qt ).
In the Markov-perfect pricing, such a commitment is not feasible, and the marginal revenue

0

increases in every period by −qt−1 f (qt ), as the monopolist quotes lower prices and sells moregoods. In the sustainable pricing, the past commitment zt xt−1 weakens such an eect, and themonopolist can quotes higher prices than in the Markov-perfect pricing, as long as xt−1 > 0.

0

In other words, the marginal revenue decreases by zt xt−1 f (qt ) < 0 compared to the Markovperfect pricing.
The outside option eect is related to the future prices in the Markov-perfect pricing, or theoutside option in the sustainable pricing, which shifts the marginal revenue curve toward left
and the monopolist can quote higher prices. In the Markov-perfect pricing, marginal changes in0quantity sold in period t decreases the future prot by βp (qt )(q t − qt−1 ) < 0. The monopolist
internalizes the eect and quotes higher prices to sell less goods today. Also, in the sustainablepricing, marginal changes in quantity decreases the future value by

c + W 0 (qt )] < 0.

−1β(1 − zt+1 )zt+1[pt+1 −

Note that, as long as the sustainable constraint in the next period is not15binding,eect.

zt+1 = 1, the monopolistic pricing is sustainable, and this term does not come in

However, once the sustainability constraint is binding, the monopolist internalize theeect of marginal changes in quantity in the outside option, and quotes higher prices as well as
in the Markov-perfect pricing.

2.3.3 Solving for the Policy Functions Numerically
The solution can be obtained only numerically. By using some auxiliary variables, I have

Rt = (pt − c)(qt − qt−1 ) + βRt+1 ,pt = f (qt ) + βpt+1 ,pt − c + xt f 0 (qt ) = βθt+1 ,zt θt = pt − c + (1 − zt )W 0 (qt−1 ),qt − qt−1 = xt − xt−1 ,Rt ≥ W (qt−1 ),where Rt is the prot in period t and θt summarizes forward terms.

Let the state variabless = (q−1 , x−1 ), then I have a state-space representation
R(s) = (p(s) − c)(q(s) − q) + βR[q(s), x(s)],p(s) = f (q(s)) + βp[q(s), x(s)],p(s) − c + x(s)f 0 (q(s)) = βθ[q(s), x(s)],z(s)θ(s) = p(s) − c + (1 − z(s))W 0 (q−1 ),q(s) − q−1 = x(s) − z(s)x−1 ,R(s) ≥ W (q−1 ).
I can solve this system by policy function iteration method with occasionally binding constraints
as in Kehoe and Perri (2002).
The dynamic Lagrangean used here corresponds to the recursive saddle point method considered in Marcet and Marimon (1998; 2011). Messner and Pavoni (2004) pointed out that Marcet
and Marimon's method may yield non-optimal and non-feasible solutions when the problem is
not strongly concave. This criticism also may apply to the analysis here because the Markov-

16

perfect equilibrium has an endogenous state variable and the sustainable constraints consideredhere may not be strongly concave. To check the optimality, I update the policy function slowly
by randomly choosing updating grids. I also check the feasibility after the algorithm converges.

3

Numerical Results

3.1

Parametrization

In the LQ framework, a = b = c = 1 are arbitrarily chosen for just demonstration purposes.

β = {0.6, 0.95} are chosen for the case of impatient agents and the case of patient agents each.
Grids of (q−1 , x−1 ) are divided into 51 × 51 = 2601 points and linear interpolation is used toapproximate values between grids.

3.2

Short-run Dynamics

Figure 3 shows the short-run dynamics in the model.
with βFirst, in the case of impatient agents= 0.6, the sustainable pricing is intermediate between the monopolistic pricing andthe Markov-perfect pricing. In the monopolistic pricing, price and quantity immediately jump
to the monopolistic level after period 1, and maintain the level after that, even though themonopolist has incentive to deviate. In the Markov-perfect pricing, price and quantity quickly
converge to the steady state until the market is satiated. This is consistent with Stokey's result
when the length of period is small enough (note that I xed the length of time to one). The
sustainable pricing also converges to the steady state, but very slowly; the monopolist oers
lower prices over time and continues to sell until the market is satiated. The monopolistic priceis not sustainable for any β ∈ (0, 1) (Corollary 3), so the sustainable price is lower than themonopolistic price for every period. In the case of patient agents with β = 0.95, the sustainablepricing and the monopolistic pricing are almost equivalent, but the sustainable price is slightly
lower than the monopolistic price.
The binding pattern of the sustainability constraint makes the short-run dynamics of thesustainable pricing distinct from the others. Figure 4 shows the policy function of z(q, x), theratio of the sum of Lagrange multiplier on the sustainability constraint. Note that z(q, x) <1 shows that the sustainability constraint is binding.

The triangle formed by hyperplanes

q−1 = q c , x−1 = q c and x−1 = q c − q−1 is the area of state space (q, x) where z(q, x) < 1,which corresponds to Proposition 2. In the case of impatient agents with β = 0.6, the state17

of the economy (q, x) jumps to near (qthe monopolistic pricing.

m , q m ) in period 0, where q m is the quantity sold in

After period 1, z(q, x) < 1 holds until the market is satiated; themonopolistic pricing is not sustainable. To keep the sustainability constraint with equality, themonopolist lowers prices and sells more goods.

In the case of patient agents with β = 0.95,

the state of the economy (q, x) jumps to near (qm , q m ), but much closer than in the case with

β = 0.6, as the area where z(q, x) < 1 shrinks as β → 1. After period 1, z(q, x) < 1 holds, but
the speed of convergence is much slower than in the case with β = 0.6, because z(q, x) is close
to one. As β → 1, the speed of convergence becomes arbitrarily slow, which is consistent with
the AD's result.
What is the mechanism behind the result? The spread between q and x and the Lagrange
multiplier z are the key to understand the mechanism, which are shown in Figure 4. Let the
spread between q and x be denoted by ∆t ≡ qt − xt = qt−1 − zt xt−1 . Then I have

∆t =



∆t−1

if zt = 1,∆
t−1 + (1 − zt )xt−1

if zt ∈ (0, 1).

The more often the sustainability constraint is binding, the larger the spread is. By substituting

xt = qt − ∆t ≥ 0 into the FONCs, I have
(1 − β)c = f (qt ) + qt f 0 (qt ) − ∆t f 0 (qt )−1[(pt+1 − c) + W 0 (qt )].
−β(1 − zt+1 )zt+1(17)

As the sustainability constraint is binding, the spread becomes larger, and the marginal revenue
shifts toward right as in Figure 1. This is an analogy to the case when the monopolist can reset
price in every period in the monopolistic pricing. However, in this case, the sustainable pricing
is time consistent. Note that if qt = q

c = ∆ and x = 0, then (1 − β)c = f (q ) holds in (17);ttt

The sustainable pricing converges to the marginal cost. The speed of convergence is governedby the law of motion of ∆t and the binding pattern of the sustainability constraint.
Figure 4 shows that in the case of impatient agents with β = 0.6, the spread q − x becomeslarger as time goes by after period 1. The sustainability constraint is binding, i.e., zt < 1, until
the market is satiated. In the case of patient agents with β = 0.95, the spread remains small,it becomes larger very slowly though.

18

4

Extensions

In this section, I consider two extensions to the basic setup: depreciation and stochastic marginalcost. Now suppose that the purchased goods are separated at rate δ and the marginal cost is
stochastic. Then equation (1) becomes

pt = f (qt ) + β(1 − δ)Et pt+1 .

(18)

The future price is stochastic, and discounted by not only the discount factor β , but also one
minus the depreciation rate

(1 − δ).

The marginal cost

ct follows an exogenous stochastic

process:log ct+1 = (1 − ρ)c + ρ log ct + εt+1 ,where εt+1 ∼ N (0, σ(19)2 ) is the disturbance to the marginal cost, which follows a normal distribu-

tion with the standard deviation σ . The monopolist chooses the sequence of {pt } (and hence{qt } given (18) and (19)) so as to maximize its discounted sum of expected future prots
R0 ≡ E0∞X

β t (pt − ct )(qt − (1 − δ)qt−1 ),t=0subject to (18) and (19) for all t ≥ 0, given q−1 = 0 and c−1 = c. Note that the amount of
newly purchased goods in period t is qt − qt−1 + δqt−1 > 0.

4.1

Depreciation

Karp (1996) showed that, in a continuous-time setting, depreciation of durable goods yields multiple Markov-perfect equilibria and the Coase conjecture does not necessarily hold. In a discretetime setting considered in the present paper, the equilibrium is unique, but the Markov-perfectpricing no longer converges to the marginal cost. Depreciation erodes the Coase conjecture, because the monopolist can make a positive prot, and the price is higher than the marginal costin the steady state. The sustainable pricing is intermediate between the monopolistic pricing
and the Markov-perfect pricing, and the sustainable pricing does not converge to the Markovperfect pricing in the steady state. As depreciation rate δ is higher, the monopolistic pricing ismore likely sustainable; for any discount factor β ∈ (0, 1), there is a positive depreciation rateδ ∈ (0, 1] which makes the monopolistic pricing sustainable.

19

4.2

Stochastic Marginal Cost

Fabinger, Gopinath, and Itskhoki (2011) studied price markup dynamics with stochastic marginal
cost, and found an incomplete pass-through in the Markov-perfect pricing. Although the shock
to marginal cost itself may generate an interesting dynamics, when I consider stochastic marginal
cost in the sustainable pricing, there are other interesting issues through the working of the sustainability constraint. For example, when the shock to marginal cost occurs, the outside option
is worse, and the monopolist can quotes higher prices based on reputational power. Even without the shock realization, the possibility of the shock in the future diminishes the monopolist's
expected future prot more intensely in the Markov-perfect pricing than in the monopolistic
pricing; the possibility of the future shock also makes the monopolist easier to maintain his
reputation.

5

Concluding Remarks

In this paper, the short-run dynamics of the sustainable pricing in a durable goods monopoly is
studied. By taking advantage of reputation among consumers, the monopolist can oer higher
prices in the sustainable pricing than in the Markov-perfect pricing. The sustainable pricing
adjusts toward the Markov-perfect pricing, but the speed of adjustment is slow. Reputation has
the eect like an adjustment cost. Reputation also makes the monopolist more protable in thesteady state, if durable goods depreciates over time. In the future research, I will examine theeect of marginal cost shocks more carefully, especially focusing on the incomplete pass-through.

A

Appendix

A.1

Proof of Proposition 1

First, I solve for the coecients of the policy functions by using undetermined coecient10

methodpt = τp + τpq qt−1 ,qt = τq + τqq qt−1 ,10

The detail of calculation for the coecients are available upon request.

20

whereτqq = [1 − (1 − β).5 ]/β ∈ (0, 1), τpq = −bτqq /(1 − βτqq ) < 0,
τp = pc − τpq q c > 0,

τq = (1 − τqq )q c > 0,and τqq

2 − 2τ + 1 = 0. Given the∈ (0, 1) is the stable root of the quadratic equation βτqq
qq

policy functions, I can solve for the quadratic prot function analytically2
W (qt−1 ) = µ + µq qt−1 + µqq qt−1
,

where

µ = −τpq (q c )2 /2 > 0, µq = τpq q c < 0, µqq = −τpq /2 > 0.
0

Then W (0)= µq < 0 and W 00 (q) = 2µqq > 0 for all q , W 0 (q c ) = µq + 2µqq q c = 0, andW (q c ) = µ + µq q c + µqq (q c )2 = 0 hold; therefore, W 0 (qt−1 ) < 0 for qt−1 ∈ [0, q c ) also holds.
A.2

Proof of Proposition 2

As in the case of the Markov-perfect pricing, I can solve for the coecients of the policy functionsmm
pt = τpm + τpq
qt−1 + τpx
λt−1 ,
mm
qt = τqm + τqq
qt−1 + τqx
λt−1 ,mm
λt = τxm + τxq
qt−1 + τxx
λt−1 ,

wherem = −[b/(1 − β)]/2, τ m = [b/(1 − β)]/2,
τpm = pm , τpqpxτqm = q m ,m = 1/2,
τqqm
τqx= −1/2,

τxm = q m ,m = −1/2,
τxq

m
τxx
= 1/2.

21

Given the policy functions, I can solve for the quadratic prot function analytically

m
m 2
m 2
m
Rt = µm + µm
x λt−1 + µq qt−1 + µxx λt−1 + µqq qt−1 + µxq λt−1 qt−1 ,whereµm = [b/(1 − β)](q m )2 > 0,µm
x = 0,mmµm
q = −2(p − c)τqq < 0,

m m
m
m
m m
µm
xx = τpq τqq < 0, µqq = −τpq τqq > 0, µxq = 0.

Then D1 R

m (q

m
m
m
m
m
c
t−1 , λt−1 ) = µq + 2µqq qt−1 = −2τqq [(p − c) + τpq qt−1 ] ≤ 0 for qt−1 ∈ [0, q ] andc
D2 Rm (qt−1 , λt−1 ) = 2µm
xx λt−1 ≤ 0 for λt−1 ∈ [0, q ] hold. Then,

m 2
m
c
2
Rm (q, q c − q) = µm + µm
q q + µqq q + µxx (q − q) ,m
cc 2
= µm + µm
q q − µqq (−2q q + (q ) ),
c 2
m
m c
= [b/(1 − β)](q m )2 − µmqq (q ) + (µq + 2µqq q )q,m m c 2mm c
= [b/(1 − β)](q m )2 + τpq
τqq (q ) − 2τqq[(pm − c) + τpqq ]q.

Note that pm = (a + (1 − β)c)/(2 − 2β), q m = (a − (1 − β)c)/(2b) and q c = 2q m . Then,m c(pm − c) + τpq
q = pm − c − [b/(1 − β)]q c /2,= 0,m m c 2[b/(1 − β)](q m )2 + τpq
τqq (q ) = [b/(1 − β)](q m )2 − [b/(1 − β)](q c )2 /4,= 0.
Therefore, R

A.3

m (q, q c − q) = 0 for all q holds.

Proof of Proposition 5

See Ausubel and Deneckere (1989).

A.4

Proof of Proposition 6

See Ausubel and Deneckere (1989).

22

References
Abreu, D. (1988):  On the Theory of Innitely Repeated Games with Discounting,

Econo-

metrica, 56(2), 383396.
Ausubel, L. M.,

and R. J. Deneckere (1989):  Reputation in Bargaining and Durable

Goods Monopoly,

Blanchard, O. J.,

Econometrica, 57(3), 511531.

and C. M. Kahn (1980): The Solution of Linear Dierence Models

under Rational Expectations,

Econometrica, 48(5), 13051311.

and P. J. Kehoe (1990):  Sustainable Plans, Journal of Political Economy,

Chari, V. V.,

98(4), 783.

Christiano, L.,

and J. Fisher (2000):  Algorithms for Solving Dynamic Models with Occa-

sionally Binding Constraints,

Journal of Economic Dynamics and, 24(8), 11791232.

Coase, R. (1972): Durability and Monopoly,

Fabinger, M., G. Gopinath,

Journal of Law and Economics, 15(1), 143149.

and O. Itskhoki (2011): Price Dynamics for Durable Goods,

Mimeo.

Karp, L. (1996): Depreciation Erodes the Coase Conjecture,European Economic Review,40(2), 47390.

and F. Perri (2002):  International Business Cycles with Endogenous Incom-

Kehoe, P. J.,

plete Markets,

Marcet, A.,

Econometrica, 70(3), 907928.

and R. Marimon (1998): Recursive Contracts, EUI Working Papers, ECO

1998/37.

Marcet, A.,

and R. Marimon (2011):  Recursive Contracts, EUI Working Papers, ECO

2011/15.

Messner, M.,

and N. Pavoni (2004):  On the Recursive Saddle Point Method On the Re-

cursive Saddle Point Method, IGIER Working Paper n 255, Bocconi University.

Stokey, N. L. (1981):  Rational Expectations and Durable Goods Pricing,

of Economics, 12(1), 112.
23

The Bell JournalFigure 1: Demonstrating time inconsistency.

( )

̅

24

Figure 2: Dierence of value functions.

25

Figure 3: Short-run dynamics.

a. the case of impatient agents: β = 0.6
price
1.8
1.6
1.4
1.2
1
0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

12

14

16

18

20

quantity
0.5
0.4
0.3
0

2

4

6

8

10
profit

0.2
0.1
0
0

2

4

6

8

10

b. the case of patient agents: β = 0.95
price
10
8
6
4
2
0

2

4

6

8

10
quantity

0.8
0.6
0.4
0.2
0

2

4

6

8

10
profit

4
2
0
0

2

4

6

8

10

26

Figure 4: Policy function of Lagrange multipliers.

a. the case of impatient agents: β = 0.6

b. the case of patient agents: β = 0.95

Notes: xt−1 is normalized Lagrange multiplier, and xt−1 > 0 shows the monopolist's promise in thepast. zt is the ratio of the sum of Lagrange multiplier, and zt < 1 shows the sustainability constraint isbinding.
27

Figure 5: Short-run dynamics of Lagrange multipliers.

a. the case of impatient agents: β = 0.6
qt−1 and xt−1

zt

0.7

1
qt−1
xt−1

0.6

0.95

0.5
0.9
0.4
0.85
0.3
0.8
0.2

0.75

0.1

0
0

10

20

30
time

40

50

60

0.7
0

10

20

30
time

40

50

60

40

50

60

b. the case of patient agents: β = 0.95
qt−1 and xt−1

zt

0.7

1
qt−1
xt−1

0.6

0.95

0.5
0.9
0.4
0.85
0.3
0.8
0.2

0.75

0.1

0
0

10

20

30
time

40

50

60

0.7
0

10

20

30
time

Notes: xt−1 is normalized Lagrange multiplier, and xt−1 > 0 shows the monopolist's promise in thepast. zt is the ratio of the sum of Lagrange multiplier, and zt < 1 shows the sustainability constraint isbinding.

28