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Dynamic consistency and monopoly.


In a seminal article Coase [1972] shows that a monopolist which sells a durable product faces a credibility or commitment problem with buyers. The firm has little incentive to account for the loss in capital value of the existing stock in any future period since the stock is held by buyers and not the firm. Thus the firm wishes to announce a future price that is sub-optimal when the period is reached, that is, the solution is dynamically inconsistent. If consumers are rational they recognize the firm's incentive to re-optimize in the future and, consequently, will incorporate this expected behavior into their purchasing behavior. Unless the firm can credibly convince (commit to) consumers it will take their losses into account when setting a future price, a durable goods monopolist may lose its market power and be forced to price at or near its marginal cost.

This commitment problem and its associated impact on imperfectly competitive durable goods manufacturers is commonly referred to as the Coase conjecture. In essence, the conjecture argues that the durability of the firm's product is the root cause of any commitment or time consistency problems. Indeed a rather large number-of studies have attempted to sort out the various conditions under which Coase conjecture holds and have analyzed a variety of ways firms may mitigate its commitment or time consistency problem.' Manufacturers in the aircraft, automobile, computer, and copier industries have often been discussed as likely candidates for Coase conjecture type commitment problems. However, all of these studies typically presume that if a product is non-durable there will likely be no commitment or time consistency pricing problem. For example, Bond and Samuelson [1987, p. 57] state that:

"recent work by Bulow [1982] and Stokey [1981] demonstrates that the monopoly sellers of a perfectly durable good face special problems not found with nondurable goods."

More recently Colangelo [1995, p. 269] indicates that a durable goods monopolist can avoid any commitment problems by renting, stating that a "monopolist would then become like a seller of a non-durable good" and Purohit [1995, p. 70] states that:

"a firm that markets a durable experiences a problem not experienced by a counterpart who markets a non-durable."

This paper argues that manufacturers of non-durable products may face a similar type of commitment problem. (2) Moreover, this paper shows that the time consistency problems in durable goods, non-durable goods and government planning models conceptually all originate from the same source, namely, the assumption of market power under intertemporal choice.

The authors use a simple two-period model where consumers maximize utility subject to a wealth constraint. In each period consumers can purchase a non-durable product from a monopolist for consumption in that period or they can purchase an asset which provides future consumption possibilities. In this setting they show that a price-setting monopolist faces a time consistency or commitment problem in that a non-durable goods monopolist will wish to announce a period two price that is sub-optimal when the period is reached. The structure of the paper is as follows. In the next section a basic two-period model is developed and results are examined. Section three contains the authors' concluding remarks.


Suppose that consumers maximize their utility over a two-period horizon subject to their available wealth. (3) The representative consumer of interest can choose a consumption level of a non-durable good [x.sub.t] in period t as well as an asset level [A.sub.t]. (4) The asset provides a one-period rate of return equal to [gamma], implying these consumers can provide for future consumption by holding the asset (saving). If the rate of return is positive it implies that the nominal value of a unit of the asset actually appreciates over time rather than remaining constant. (5) The consumer's wealth constraint in period one and two are given by:

[w.sub.1] = [A.sub.1] + [p.sub.1] [x.sub.1] , (1)

[w.sub.2] = (1 + [gamma])[A.sub.1] = [A.sub.2] + [p.sub.2] [x.sub.2] , (2)

where [p.sub.1] and [p.sub.2] are the first and second period prices of the non-durable consumption good. (6) In addition, the consumer's wealth in period one [w.sub.1] > 0 is assumed to be exogenously specified.

Both (1) and (2) indicate that consumers can use their available wealth in each period to invest in an asset or purchase the non-durable consumption good. However, we know that in the second (final) period a rational consumer of this type will use all of their available wealth on consumption since there is no benefit from investments in [A.sub.i] in the final period. Consequently, consumers will optimally set [A.sub.2] = 0 and purchase as many units of the consumption good [x.sub.2] as possible. Thus:

[w.sub.2] = (1+[gamma])[A.sub.1] = [p.sub.2][x.sub.2], (2')

governs the consumer's second period behavior.

Let [U.sub.t]([x.sub.t]) represent the twice continuously differentiable utility function in each period with derivatives [U'.sub.t]([x.sub.t]) > 0 and [U".sub.t]([x.sub.t]) < 0. This allows that the consumer may have a different utility function and in each period but that diminishing marginal returns of consumption always prevails. By recursively substituting from (1) and (2') and rearranging terms we can write the representative consumer's constrained utility maximization problem as:

[Max.sub.x1] U = [U.sub.1]([x.sub.1]) + [beta][U.sub.2]([x.sub.2]) , (3)

subject to [x.sub.2] = (1 + [gamma])([w.sub.1] - [p.sub.1][x.sub.1])/[p.sub.2]

where 0 [less than or equal to] [beta] [less than or equal to] 1 is the one period discount factor. The solution to (3) will yield the consumer's optimal demand functions in each period. (7) Clearly, while the demand for the good in period one depends upon the price in each period as well as the initial endowment, the demand for the good in period two will depend upon the price in each period, the initial endowment and the amount wealth carried over to period two.

A non-durable goods monopolist will wish to maximize discounted profit [pi]:

[pi] = [pi]([x.sub.1], [x.sub.2], [p.sub.1], [p.sub.2]) , (4)

constrained by the demand for the good in period one and two:

[x.sub.1] = [x.sub.1]([p.sub.1], [p.sub.2], [w.sub.1]) , (5)

[x.sub.2] = [x.sub.2]([p.sub.1], [p.sub.2], [x.sub.1], [w.sub.1]) . (6)

Note that (6) indicates the demand in period two depends in part upon the amount of wealth carried over from period one which is inversely related to the quantity of the good purchased in period one. (8)

In period one, assuming an interior solution, the first-order conditions for the optimal values for [p.sub.1] and [p.sub.2] are respectively:

([partial][pi]/[partial][x.sub.1] + [partial][pi]/[partial][x.sub.2] [partial][x.sub.2]/[partial][x.sub.1]) [partial][x.sub.1]/[partial][p.sub.1] + [partial][pi]/[partial][x.sub.2] [partial][x.sub.2]/[partial][p.sub.1] + [partial][pi]/[partial][p.sub.1] = 0 , (7)

([partial][pi]/[partial][x.sub.1] + [partial][pi]/[partial][x.sub.2] [partial][x.sub.2]/[partial][x.sub.1]) [partial][x.sub.1]/[partial][p.sub.2] + [partial][pi]/[partial][x.sub.2] [partial][x.sub.2]/[partial][p.sub.2] + [partial][pi]/[partial][p.sub.2] = 0 . (8)

However, once period two is reached the first-order condition for the optimal choice of [p.sub.2] is:

[partial][pi]/[partial][x.sub.2] [partial][x.sub.2]/[partial][p.sub.2] + [partial][pi]/[partial][p.sub.2] = 0 . (9)

Note, once period two is reached [partial][x.sub.1]/[partial][p.sub.2] = 0. This result leads to the following proposition:

Proposition: Non-durable goods monopolists may face credibility, commitment, or time consistency pricing problems. An announced future price may be sub-optimal when the period is reached even if the firm's product is non-durable.

In contrast to the standard two period durable good monopolist model, the committed firm's second period price shown by (8) indicates that the firm does not necessarily wish to commit to high period two prices. By comparing (8) and (9) we see that the difference is due to the existence of the additional term containing [partial][x.sub.1]/[partial][p.sub.2] in (8). If this term is non-zero, m general the solutions of (8) and (9) are time inconsistent. In other words, the non-durable goods monopolist may wish to announce or commit to a period two price which is sub-optimal when the period is reached if its choice of this price influences consumers' current buying behavior. Intuitively, the additional term in (8) captures the impact (on firm profits) of changes in second period price that involve changes in first period consumer demand. If [partial][x.sub.1]/[partial][p.sub.2] [not equal to] 0, the firm can influence consumer demand in period one by announcing or committing to a second period price. If, for exam ple, [partial][x.sub.1]/[partial][p.sub.2] < 0 over the relevant range the firm may be able to induce consumers to purchase a larger amount of the non-durable product in period one by committing to a relatively low second period price.

The commitment problem here is due to the consumer's intertemporal wealth constraint. Consumers decide on the amount of first and second period consumption based on their real wealth in each period that depends, among other things, upon the price the firm charges for the non-durable consumption good. As a result, consumer demand in the first period depends on the firm's prices in both periods. By announcing, for example, a low second period price the firm may be able to induce consumers to purchase a larger amount of the non-durable product in the first period since the real purchasing power in period two is increased ceteris paribus. Thus the firm may in period one wish to announce a relatively low period two price, but when it reaches the second period this price will be sub-optimal and the firm will set a different price in period two (for example, set a price above the announced price), Of course, if consumers are rational they will not believe that the firm will maintain the announced price when the seco nd period is reached. This indicates that non-durable good manufacturers may face the same type of time consistency or commitment problem (with buyers) as durable good manufacturers. Hence, one should not necessarily assume that non-durable goods producers have no commitment problems.

This indicates that it is the intertemporal linkage of consumer demands which is the root cause of any commitment or time consistency problems for imperfectly competitive firms, not the durability of the product per se. Even non-durable goods monopolists may wish to commit themselves to prices which are sub-optimal when viewed at some later date. Thus the so-called durable goods monopoly commitment problem may actually occur in non-durable goods industries as well. (9)

The driving factor for the main proposition above is the intertemporal linkage of demands with this linkage being the ability of the consumer to save in the first period by purchasing an asset rather than the non-durable good. Our model suggests that any sort of dynamic linkage may cause commitment or credibility problems for imperfectly competitive firms. The good does not have to be durable for these sorts of problems to exist. Equation (8) and (9) show that if a firm can influence a consumer's current behavior as they change future prices they may wish to announce future prices which are not credible. Consumers would rationally expect the firm to set the actual price in period two according to (9) regardless of the announced price unless the firm could credibly bind itself to this future price through, for example, contracting. This is the same sort of commitment problem first noted by Coase [1972] for durable goods monopolists.

One can also draw other parallels to the durable goods literature. Note, for example, that if the firm is forced to price at its true marginal cost in period one this removes one of the additional terms in (8). As in the durable goods literature our model suggests (although it is not as analytically convenient to show since we are assuming price-setting rather than quantity-setting behavior) there is no time consistency problem if firms set in each period price equal to marginal cost. Also, similar to the results in a durable good model, if consumers' expectations result in a credibility problem for the firm, profits will tend to be lower since these expectations impose additional constraints upon firm behavior.

In general, the model suggests that there are two components which are necessary for commitment or time consistency problems of the type found above: (1) market power and (2) intertemporally linked demands.

However, in order for the time consistency problem to have a significant impact, it may be necessary that expenditures upon the consumption good or service in question must be relatively large with respect to consumers' total wealth. Some possible goods or services provided in imperfectly competitive markets satisfying this criteria are, air travel, medical services and, perhaps most exactly, legal services.

Consider a typical lawsuit regarding breach of contract. This legal action could be divided into two periods. The first period would be the pre-trial period during which the case is researched, papers are filed, pre-trial hearings occur and pre-trial settlement may occur. In the second period the trial occurs, appeals are filed and post-trial settlement may occur. In the first period, an attorney will usually outline the cost of legal services the client is expected to incur in total. Thus the attorney will outline the legal fees in order to prepare the case for trial (the legal fees for period one) and the legal fees for trial and possible appeal (the anticipated legal fees for period two). Clearly, given the dynamic consistency problems, in period one the attorney may wish to state a price (cost of legal fees) in period two lower than the optimal price once period two arrives in order to affect the clients decision in period one, that is, initiate the legal action. (10) One solution to this problem may be t he use of a contingency contract. Once a contingency contract is signed in period one, the legal costs to the client will be a fixed proportion of the total settlement that tends to lessen the time consistency problem presented above. (11)

Concluding Remarks

In summary, the authors' two-period model suggests that Coase conjecture type commitment problems may occur in a much broader class of dynamic models than what is commonly assumed. Time consistency, credibility, and commitment effects may be relevant questions for study in both durable and non-durable goods monopolistic industries. Indeed, even monopolistic service providers, such as lawyers, may face non-trivial Coase commitment problems. This potential time consistency problem is not apparent in the standard durable good monopoly models [for example, Bulow 1982] under pure renting since the only intertemporal linkage found in these models is in the monopolist's profit maximization problem. Hence, it is often claimed in the durable goods literature that commitment problem of a durable good manufacturer is solved when the firm is a pure renter. However, the authors show that if the intertemporal choice on the part of consumers is included in the model, even a pure renting firm as well as a nondurable good man ufacturer face potential commitment problems.

The results indicate that it is not the durability of the product per se which implies time consistency and commitment problems, instead it is the intertemporal linkage of consumer behavior. In the model, the intertemporal linkage results from the ability of consumers to make purchases today or to save for the future by purchasing an asset. If imperfectly competitive firms can alter buyers' expectations by announcing future prices (and, consequently, affecting consumers' current purchasing and saving behavior) we are immediately faced with time consistency and commitment issues irrespective of the product's durability. Thus one cannot necessarily assume that non-durable good manufacturers do not have commitment or time consistency problems. Furthermore, many studies address the strategies and instruments (for example, best-price provisions or planned obsolescence schemes) the durable good producers may use to lessen their commitment problem with buyers. The model suggests that similar sorts of strategies and instruments may be used by non-durable goods manufacturers.

Finally, time consistency problems are not limited to imperfectly competitive firms and there is a large literature examining the time consistency problems of government planning. This discussion illustrates that the potential time consistency problems of imperfectly competitive firms and government planning are both due to the combined effects of intertemporal choice and market power.


(1.) See Stokey [1981], Gui et al. [1986], Kahn [1986], Bagnoli et al. [1989], Ausubel and Deneckere [1989], Olsen [1992], and Goering [1993] for a sampling of studies addressing the impact of the Coase conjecture and the conditions under which it is likely to hold. In terms of firm strategies to reduce or mitigate its commitment problem, Bulow [1982; 1986] argues that a selling firm may reduce the durability of its product, that is, practice "planned obsolescence" to help mitigate its commit problem with buyers. Butz [1990] examines best-price provisions and shows how a selling durable products may reduce its commitment problem.

(2.) Dudey [1996] also shows that non-durable goods manufactures may face time consistency issues. Dudey analyses a model without intertemporal consumption choice but instead assumes that the monopolist posts a sequence of prices prior to the final delivery period (that is, the seller opens the market multiple times before the final delivery of the non-durable good). This type of forward contracting model differs markedly from our analysis in which the time consistency problem arises due to intertemporal consumption of a non-durable good. Indeed, Dudey notes in footnote six [p. 473] that in his model "The absence of discounting means that delayed consumption cannot be a source of inefficiency in the model." In the authors' model of intertemporal consumption choice this is, of course, the driving force behind the firm's time consistency problem. Each model illustrates a potential, albeit very different, source for time inconsistency in non-durable goods frameworks. See also Kydland and Prescott [1977] for a ge neral paper on time consistency.

(3.) The two-period assumption makes the problem tractable and is a natural one since many of the durable goods models which focus on commitment and time consistency issues are two-period models (for example, Bulow (1982; 1986], Bond and Samuelson [1987], Colangelo [1993], and Purohit [1995]). Note, also that Bulow's [1986; 1982] framework allows product durability to be entered explicitly, which is another reason why we have utilized this approach. Much of the previous literature does not explicitly model durability. For example, Gul et al. [1986], does not utilize a durability parameter or choice variable in their model. Product durability is entered by their assumption that consumer only purchase at most one unit. In other words, in their model, when a unit is purchased it lasts forever and consumers have no desire to buy another (no repeat purchases are allowed). The authors have not used this rather indirect approach in their analysis.

(4.) For expositional ease the authors let the phrase non-durable good denote any product or service that is entirely consumed in a given period (units produced today do not remain for future use). Also, the unit value of the asset [A.sub.t] is normalized as the numeraire price.

(5.) Obviously, the model is a partial equilibrium finite horizon model. The authors do not attempt to endogenously determine the rate of return [gamma] on the asset A, nor do they analyze the role (demand) for money in this setting. Thus this model can be criticized, as can all partial equilibrium models (for example, conventional static consumer choice models), on these grounds. However, the authors do show that the rate of return on the asset does not influence their main result on the existence of commitment and time consistency problems with monopolistic non-durable goods producers. What is important in this model is the existence of dynamic linkage in consumer consumption levels.

(6.) Note that (2) implies that the consumer's wealth in period two is derived solely from any assets purchased or held in period one. Hence the model abstracts from any income the consumer may be able earn by supplying their labor services. This assumption could be relaxed without influencing the results. In addition, note that the model assumes the consumer's wealth depends upon the non-durable goods monopolist's price. In particular, as discussed in section four, this would hold for a variety of service oriented industries, such as travel, legal, and medical care services.

(7.) The authors assume an interior solution. An appendix containing a complete analysis of the utility maximization problem may be obtained from the authors.

(8.) It is interesting to note that this model is mathematically equivalent to the model presented in Kydland and Prescott [1977] which examines the time consistency problem inherent in government planning.

(9.) The reason why previous durable goods monopoly literature such as Bond and Samuelson [1987], Colangelo [1993] and Purohit [1995] make statements which appear to limit the commitment problem only to durable goods is that the intertemporal linkage in these models is limited to a firm's intertemporal profit function and the effect product durability has on the future stock of the good. Hence, consumers do not make an intertemporal choice in these models.

(10.) There are other reasons for this type of pricing behavior by attorneys, vis-a-vis, sunk costs are sunk. However, the type of time inconsistency discussed here will also contribute to this type of behavior.

(11.) Of course, there may be other causes of contingency agreements such as risk pooling. Also while the attorney's stated pricing policy for period two may understate the actual price in period two in an attempt to increase the number of law suits initiated in period one, the execution of a contingency agreement (which obviates the consistency problem) may actually further increase the number of law suits initiated.


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Author:Goering, Gregory E.; Pippenger, Michael K.
Publication:Atlantic Economic Journal
Date:Jun 1, 2003
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