Durable Goods Monopoly with Endogenous Quality


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1 Durable Goods Monopoly with Endogenous Quality JongHee Hahn Keele University (Preliminary) March 30, 2004 Abstract This paper examines the dynamic pricing problem of a durablegood monopolist when product uality is endogenous. It is shown that the relationship between the firm s uality choice and the timeinconsistency problem crucially depends on how the unit production cost varies with uality. The monopolist may use uality as a strategic commitment device to eliminate the timeinconsistency problem. Also, it may have incentives to choose a uality higher or lower than the optimal commitment level. This contrasts with the planned obsolescence literature where durable goods monopolists reduces durability (often regarded as a measure of uality) to mitigate the timeinconsistency problem. Keywords: Durable Goods, Quality, Timeinconsistency. JEL classification: D42, L2. Introduction An important proposition in economic literature on durablegood monopolies is that a seller faces a problem of time inconsistency (Coase (972), Bulow (982), and others). The problem arises because durable goods sold in the future affect the future value of units sold today, and in the absence of the ability to commit to future prices the monopolist does not internalize this externality. There have been numerous investigations on the robustness of the result in the Department of Economics, Keele University, Keele, Staffs ST5 5BG, UK.
2 literature. Bond and Samuelson (984) show that depreciation and replacement sales reduce the monopolist s tendency to cut price. Kahn (986) shows a similar result when the monopolist faces an upwardsloping marginal cost schedule. Also, there have been many studies analyzing means and practices durablegood monopolists can employ to overcome or mitigate the timeinconsistency problem, notably leasing (Bulow, 982), planned obsolescence (Bulow, 986; Waldman, 993), contractual provisions such as bestprice provisions or mostfavoredcustomer clauses (Butz, 990), and more recently productline extension using uality differentiation (Hahn, 2002; Inderst, 2002; Takeyama, 2002). However, one of the aspects that received little attention in this literature is how the timeinconsistency result is affected when uality of product is chosen endogenously by the firm, as in real world durable goods markets. In the standard durablegoods monopoly model, uality of product is exogenous and therefore its effect on firm s strategic behavior is ignored. We analyze a durablegood monopoly model in which the firm is allowed to choose uality as well as prices. Themainuestionstobeaskedinthisnewsetupare How does endogenizing uality affect the timeinconsistency problem? How does the timeinconsistency problem, if it is present, affect the monopolist s optimal choice of uality? We consider a threeperiod dynamic game where a singleproduct monopolist first decides on the uality of good in period 0, and then chooses the price of the good seuentially over periods and 2. The focus of our analysis is to investigate how the monopolist s decision on product uality is related to the timeinconsistency problem, and how the optimal uality differs from the one chosen under commitment to future prices. We show that the relationship critically depends on how the unit product cost varies with uality. The main findings of the analysis are as follows. With endogenous uality the durablegoods monopolist may be naturally immune to the timeinconsistency problem. This happens when the average cost at the optimal commitment uality is high (i.e. the elasticity of scale of uality is small) enough for the firm to find selling the goods to lowvaluation consumers in period 2 unprofitable. This result is uite different from the one observed in the standard durablegoods monopoly model with an exogenous uality and a constant unit cost. In the standard model the firm does not face timeinconsistency problems if the absolute level of the constant unit cost is sufficiently high (see Bulow, 982 for example), 2
3 while here the relevance of the timeinconsistency problem depends on the shape of the unit cost function of uality. For instance, there is no timeinconsistency problem if the elasticity of scale of uality is sufficiently small even if the unit product cost itself is high. As the average unit cost at the optimal commitment uality gets smaller (i.e. the elasticity of scale of uality increases) the timeinconsistency problem becomes relevant (i.e. the firm has the incentive to sell the good to lowvaluation consumers in period 2). The firm s optimal reaction to the problem differs depending on the level of the elasticity of scale of uality. If it is relatively small (but still high enough for time consistency to be relevant), the firm finds it optimal to eliminate the timeinconsistency problem by choosing a uality higher than the optimal commitment level (therefore increasing the average unit cost of production). So, the monopolist uses uality as a strategic device of committing to no future sales, similar to the wellknown burninghisfactory story in the standard durablegoods monopoly. Here, the firm also increases price up to a level where the euilibrium demand is smaller than the one in the commitment regime. As the elasticity of scale of uality increases further, increasing uality up to the point where the timeconsistency problem disappears is too costly and the firm chooses to sell the goods in period 2. But, still the monopolist may have incentives to increase or decrease uality relative to the optimal commitment level. With secondperiod sales, the firm s decision on uality depends on the levels of the two marginal types and the relative importance (distribution) of the first and secondperiod demands. The secondperiod marginal type is more sensitive to the change of the average unit cost than the firstperiod marginal type, i.e. the secondperiod marginal type decreases more rapidly than the firstperiod marginal type as the average unit cost gets smaller. So, if the elasticity of scale of uality is very large (i.e. the average unit cost is sufficiently small at the optimal commitment uality), the euilibrium secondperiod marginal type is too low so that the firm chooses a uality lower than the optimal commitment level. If the elasticity of scale of uality is in an intermediate level, however, the firm increases uality, partially mitigating the timeinconsistency problem. Our result is in contrast with the planned obsolescence literature (Bulow, 986; Waldman, 993) in which durable goods monopolists tends to reduce durability (often regarded as a measure of product uality) to mitigate the timeinconsistency problem. This result is also reminiscent of the reverse AverchJohnson effect established by Bulow (982), i.e. durablegood monopolists may invest less in fixed costs in order to keep their marginal costs high (a signal of lower future output and thus high future prices). 3
4 2 The model We incorporate a uality dimension into the classical durablegoods monopoly model of Bulow (982). There is a monopolist selling durable goods over two periods. The good is characterized by a onedimensional uality index 0. The unit cost of production is constant and is given by c(), where c(0) = 0,c 0 ( ) > 0, and c 00 ( ) > 0. There is no economies of scale or learningbydoing over time. Before production begins (say, in period 0) the firm chooses uality of the good () it will sell in periods and 2. We assume constant fixed costs of production independent of uality, and also assume that the fixed cost corresponding to producing goods of a single uality (such as costs of setting up production lines) is too large so that the firm has no incentives to produce multiple ualities or alter the uality of the good later. So, we confine ourselves to a singleuality monopolist, i.e. the firm is constrained to produce goods of the single uality determined in period 0 and the uality choice is virtually irreversible. On the demand side, there are a large number of consumers who live two periods. The total number of consumers is normalized to. Consumers have unit demands, and each consumer is indexed by a type parameter θ. The perperiod (nondiscounted) gross surplus a typeθ consumer derives from the consumption of a good of uality is given by v(θ) =θ. So, the marginal utility of uality is linear and increasing in type parameter θ. Consumertype is private information. The good purchased and used during period can be used again in period 2 without depreciation. After period 2 the good becomes obsolete or is replaced by a new product. Consumers purchase if they are indifferent between buying and not buying. We assume that there is no upgrade or technological innovation during the time horizon considered. Consumers and the firm have a common discount factor denoted δ [0, ]. All agents have complete and perfect information (except consumer type), and have perfect foresight on future outcomes. The solution concept we are using is subgame perfection. 3 Two types of buyers Consumers come in two types: θ {θ l,θ h } where θ h >θ l >c 0 (0). There is a continuum of consumers for each type with mass µ for type θ h and µ for type θ l. So, consumers act as a 4
5 price taker and all the bargaining power is given to the firm. First, consider the benchmark case where the firm can commit to future prices. The firm has two options. If it sells to only typeh consumers at a high price the total profit is π h max : µ[( + δ)θ h c()], and the optimal uality h is given by the first order condition, ( + δ)θ h = c 0 ( h ). () If it sells to both types of consumers at a low price the total profit is π l max and the optimal uality l is given by :[(+δ)θ l c()], ( + δ)θ l = c 0 ( l ). (2) Note that both profit functions are concave, and therefore h and l given above are the true optimal uality in each case. The firm will choose to serve typeh consumers only if π h π l,i.e. µ[( + δ)θ h h c( h )] ( + δ)θ l l c( l ), (3) and choose to serve both types of consumers otherwise. It is easily observed that h > l from the convexity of c( ). In order to highlight the timeconsistency issue in the classical durablegoods monopoly pricing, we will focus on the cases where condition (3) holds, i.e. the firm optimally chooses the intertemporal sales rather than the immediate market clearing and achieves the optimal commitment profit π h with the optimal uality h. Condition (3) simply says that the marginal utility of uality is sufficiently differentiated between the two types of consumers and or the proportion of typeh consumers is sufficiently large relative to typel consumers. Next, consider the case where the firm cannot commit to future prices. Let us first define e such that θ l e = c(e), which denotes the level of uality where the price the firm can charge for a typel consumer is just eual to the unit cost of production. Note that if the firm chooses e The assumption of a continuum of buyers rules out the perfectly discriminating euilibria proposed by Bagnoli, Salant, and Swierzbinski (989, 995). See von der Fehr and Kühn (995) for more details on how the relative commitment power between the seller and buyers affects the euilibrium outcome. 5
6 it does not have incentives to sell in period 2 and therefore does not face the timeinconsistency problem. But, for < e it is optimal for the firm to sell to type l consumers in period 2, which constrains the firstperiod price the firm can charge according to the typeh consumers intertemporal incentive constraint, ( + δ)θ h p δ(θ h θ l ). Given, thefirm s profit isgivenby ( µ[θ h + δθ l c()] + δ( µ)[θ l c()] if <e π() = µ[θ h + δθ h c()] if e. The profit function is discontinuous and nondifferentiable at = e, but it is differentiable and concave elsewhere. 2 In fact, it jumps up at = e. 3 Now we examine the firm s optimal choice of product uality without commitment. First, suppose that h e, i.e. θ l is small relative to θ h and or the unit cost of production is sufficiently large at the optimal commitment uality (subject to condition (3)). Then, from the concavity of the profit function (except for = e) andthefactthat limπ() <π(e) and lim e e π0 () <π 0 (e), it is optimal for the firm to choose the optimal commitment uality h. In this case, there is no timeinconsistency problem since it is optimal for the firm not to serve typel consumers in period 2 (θ l h <c( h )), and therefore the firm can achieve the optimal commitment outcome even without commitment power. Next, suppose that h < e, i.e. θ l is close to θ h (subject to condition (3)) and or the unit cost of production is sufficiently small at the optimal commitment uality. Then, the concavity of the profit function in [e, ] implies that lim e+ π0 (e) < 0. Also, from condition (), the definition of e, the convexity of c( ), and the fact that θ h >θ l we have lim e π0 () = µ[θ h + δθ l c 0 (e)] + δ( µ)[θ l c 0 (e)] < 0. So, we need to compare π(e) andπ() where is such that µ[θ h + δθ l c 0 ()] + δ( µ)[θ l c 0 ()] = 0, ( 2 π 0 µ[θ h + δθ l c 0 ()] + δ( µ)[θ l c 0 ()] if <e () = and µ[θ h + δθ h c 0 ()] if e ( π 00 µc 00 () δ( µ)c 00 () if <e () = µc 00 () if e. 3 lim π() =µ[θ he + δθ l e c(e)] <π(e) =µ[θ h e + δθ h e c(e)]. e 6
7 and the optimal uality is given by =argmax{π(e),π()}. Here, two ualitatively different euilibria can emerge. If the optimal uality is determined at = e > h, the demand is exactly the same as in the commitment case, but the firm, by increasing uality, commits itself to not serving typel consumers in period 2 and effectively avoids the timeinconsistency problem. 4 This strategic commitment, however, comes with costs: the optimal profit π(e) is smaller than the commitment profit π h. If the optimal uality is determined at =, on the other hand, the firm chooses to serve typel consumers in period 2 and therefore the total demand increases. Let us compare with h in this case. Recall that h satisfies condition (). Evaluating the derivative of the profit function π( ) at = h,wehave π 0 ( h )=µ[θ h + δθ l c 0 ( h )] + δ( µ)[θ l c 0 ( h )] < 0. So, it is clear that < h,i.e. thefirm decreases uality in this case. This is because the marginal revenue of uality is smaller than in the commitment regime due to the expanded demand and the timeconsistency constraint. With the general cost function c( ) itisabitcumbersometo characterize the exact situations under which the firm chooses a particular level of uality. But, from the following numerical example we can clearly see that it depends on the structure of the unit cost function of uality, more specifically the elasticity of scale of the technology with respect to uality, which measures how the unit cost of production varies with uality. 5 Example : Suppose that θ h =,θ l = 2,δ =,µ = 2, and c() =α (α>). Note that the parameter α is the elasticity of scale of uality of the cost function. 6 From the firstorder conditions () and (2), we find that h =( α 2 ) α and l =( α ) α. The average cost of uality is k( h )= 2 α at h and k( l )= α at l, which are decreasing in α. So, we can regard α as a (indirect) measure of the average cost of uality at the commitment euilibrium. Given that α >, condition (3) is always satisfied, i.e. ( 2 ) α α 2 for all α >. Also, we have e =( 2 ) α. The condition for no timeinconsistency problem (h e) reduces to 4 This demand rigidity is in fact an artifact of the two type model and is not generally true. In the next section, using a continuous type model we characterize how the (firstperiod) demand is affected by the firm s attempt of changing uality in order to avoid the timeconsistency problem. 5 In the present context, the elasticity of scale of the technology measures the percent increase in cost due to a one percent increase in uality. 6 Note that α also represents the degree of homogeneity of the cost function. 7
8 ( 2 α ) α ( 2 ) α,whichholdsforα 4. If α>4, then h < e and we need to compare π(e) and π() where =( α ) α <h =( 2 α ) α < e, i.e. π(e) = 3 2 ( 2 ) α α and π() =(α )( α ) α α, and it turns out that the optimal uality is e for α and for α Hence, we have three different euilibria depending upon the parameter α. Forα 4, the cost function increases more steadily as uality increases, leading to a relatively high level of unit production cost at the optimal commitment uality. In this case, there is no timeinconsistency problem and the firm can achieve exactly the same outcome as in the commitment regime. For 4 α , the timeinconsistency problem becomes relevant but the firm increases uality relative to the optimal commitment level in order to avoid the timeinconsistency problem. For α , the firm finds it more profitable to accept the timeinconsistency problem and serve typel consumers in period 2, but decrease uality as the marginal revenue of uality is smaller than the commitment regime. 4 A continuumoftypes case In this section we generalize our analysis to a situation with a continuum of consumer types. For simplicity, we assume that there is a continuum of consumers with unit mass, and the type parameter θ is uniformly distributed on the unit interval [0, ]. 4. Euilibrium with commitment Suppose the firm can commit to prices it will charge in the future. Then, the firm s optimal pricing policy is to sell the goods in period at its (twoperiod) monopoly price and sell nothing in period 2. Given uality and price p, consumers of type θ such that ( + δ)θ p will buy the good in period and others will not. It is convenient to solve the firm s profitmaximization problem in two stages. We first find the optimal marginal type (which is euivalent to choosing the optimal price) for a given uality, and the optimal uality is determined later. Given, the firm chooses θ in order to maximize profits: max : ( θ)[( + δ)θ k()] θ subject to 0 θ, where k() = c() is the average cost of uality at the given. The corresponding optimal price is then determined by the euation ( + δ)θ = p. The profit 8
9 function is concave in θ, and therefore from the firstorder condition the optimal marginal type is given as θ () = 2 + for 0 <k() +δ. 7 k() 2(+δ) (4) Plugging in the optimal marginal type in (4), the firm s uality choice problem is as follows: max : 4(+δ) [( + δ) k()]2. Note that the profit function is uasiconcave (singlepeaked) for the relevant region where 0 <k() +δ. Then, from the firstorder condition the optimal uality is given by k( )+2 k 0 ( )=+δ (5) = 2c 0 ( ) c( ) =+δ. Robustness of the timeconsistency problem: In order to examine the relevance of the timeinconsistency problem when the firm can not commit to future prices, we consider the firm s incentive to sell the goods in period 2 after serving consumers of type greater than or eual to θ (the optimal marginal type at the commitment euilibrium) in period. The marginal consumers willingnesstopay for the good for the secondperiod consumption is h i v(θ )=θ = 2 + k( ) 2(+δ). The firm would not face the timeinconsistency problem even without commitment if the euilibrium uality under commitment is chosen so that serving consumers of type θ θ in period 2isnotprofitable, i.e. v(θ ) c( )= k( ) +δ, (6) which simply says that the average cost of uality is sufficiently large at the commitment euilibrium level of uality. This clearly shows that when uality is endogenous the relevance of the timeinconsistency problem in durable goods monopoly crucially depends on how the unit production cost varies with uality, and in some cases a durablegood monopolist is naturally immune to the timeinconsistency problem. For example, for c() =β α (α>,β > 0) condition (6) holds if 7 We ignore the cases of k() +δ in which the firm gets zero profits. 9
10 α +δ, which does not depend upon the scaling index β. This means that the condition for no timeinconsistency problem hinges on the shape of the unit cost function of uality (i.e. the elasticity of scale of uality) rather than its absolute level. It should be noted that this result is uite different from the one observed in the standard (exogenousuality) durablegoods monopoly model with a constant unit cost where the firm does not face timeinconsistency problems if the constant unit cost is sufficiently large. For example, in the twoperiod durablegoods model of Bulow (986) there is no timeinconsistency problems if the constant marginal cost c is greater than +δ. But, here with endogenous uality the firm is immune to the timeinconsistency problem when the average cost of uality is larger than +δ at the optimal commitment uality, i.e. the elasticity of scale of uality is sufficiently large, even if the unit cost of production is very low (a low β). 4.2 Euilibrium without commitment We now consider the monopolist s choice of uality when the firm cannot commit to future prices. The firm problem is to choose uality in period 0 and a seuence of the firstperiod and secondperiod prices p and p 2 in periods and 2 to maximize total profits, given consumers rational expectation about secondperiod outcomes. Given and (p,p 2 ), θ [0, ] denotes the type of consumers who are indifferent between buying in the first period and waiting to buy in the second period, i.e. ( + δ)θ p = δ(θ p 2 ), and similarly θ 2 [0,θ ] denotes the type of consumers who are indifferent between buying in the second period and buying nothing, i.e. θ 2 p 2 =0. We find a subgame perfect Nash euilibrium by solving the problem backward. It is useful to express the monopolist s profits in terms of θ and θ 2 rather than p and p 2. First, given and θ the firm s secondperiod problem is max : (θ θ 2 )[θ 2 k()] θ 2 subject to θ 2 θ.sincek( ) is monotone increasing (from the convexity of c( )), for a given we can treat k() as a constant unit cost in solving the problem at this stage. The solution is ( θ2(, θ for θ k() θ )= 2 [θ + k()] for θ >k(). 0
11 Given the secondperiod euilibrium outcome, in period the firm chooses θ maximize total profits: π() max : {( θ )[θ + δθ2 k()] + δ(θ θ2)[θ 2 k()]} θ subject to 0 θ. Again we focus on the cases where 0 <k() +δ. The solution is given by 2+δ θ() 4+δ + 2( δ) 2+δ 4+δ k() for 0 <k() = k() for 2+δ +δ k() π() = 2 + 2(+δ). +δ k() for k() +δ Plugging in the optimal marginal types, the total profit is then rewritten as 4(4+δ) {4[ ( δ)k()][(2 + δ) 2 (4 + δ 2 )k()] 2 + δ[2 + δ (2 + 3δ)k()] 2 } δ[ k()]k() for 2+δ for 0 <k() 2+δ k() +δ. 4(+δ) [( + δ) k()]2 for +δ k() +δ The profit functionπ() is continuous and differentiable except at = k ( +δ )and = k ( 2+δ )(k ( ) is the inverse function of k( )) where it is continuous but nondifferentiable. Also, note that it is uasiconcave in each of the three regions. Differentiating the profit function where it is possible, we have π 0 () = 4(4+δ) 2 {4[ ( δ)k()][(2 + δ) 2 (4 + δ 2 )k()] + δ[2 + δ (2 + 3δ)k()] 2 + k 0 ()[(32 24δ +32δ 2 +0δ 3 )k() 2(4 + δ)(4 + δ 2 )]} δ{[ k()]k()+k 0 ()[ 2k()]} for 2+δ for 0 <k() < 2+δ <k() < +δ. 4(+δ) [( + δ) k()][( + δ) k() 2k0 ()] for +δ <k() +δ We now analyze how the monopolist optimal choice of uality is affected by the lack of commitment power, and how it is related to the structure of the unit cost function. +δ First, if k( ) < +δ, i.e. the average cost of uality is sufficiently high at the optimal commitment uality, the monopolist does not face the timeconsistency problem and
12 can achieve exactly the same outcome as in the commitment regime. Note that this condition for no timeinconsistency problem is identical to condition (6) derived in the previous subsection. 2+δ Second, if k( ) < +δ, i.e. the average cost of uality at the optimal commitment uality is in an intermediate range, the derivative of the profit function evaluated at = is always positive in the region, i.e. π 0 ( )= δ 2 [(δ +) ( + 2δ)k( )] > 0, where the use was made of condition (5). This means that it is optimal for the monopolist to commit to not selling in period 2 by increasing uality from the optimal commitment level. With a continuum of consumer types we can now see how this strategic commitment using uality affects the euilibrium demand, an aspect that has not been analyzed in the previous discrete type case. Since the convexity of c( ) implies that θ () =k() isincreasingin, the firstperiod demand is smaller than the euilibrium demand in the commitment regime. Increasing both uality and the firstperiod marginal type immediately implies an increase of the firstperiod price relative to the optimal price in the commitment regime. So, in this case the monopolist finds it profitable to increase uality as well as price up to a level where the firstperiod demand is smaller than the euilibrium demandinthecommitmentregimeinorder to avoid the timeinconsistency problem. Last, if 0 <k( ) < 2+δ, i.e. the average cost of uality is sufficiently low at the optimal commitment uality, increasing uality to commit to not selling in period 2 is too costly and therefore the firm finds it more profitable to accept the timeinconsistency problem and sell to some low types of consumers in period 2. But, the monopolist may wish to increase or decrease uality relative to the optimal commitment level. Let denote the optimal uality in this case, where is given by the firstorder condition, 4[ ( δ)k()][(2 + δ) 2 (4 + δ 2 )k()] + δ[2 + δ (2 + 3δ)k()] 2 + k 0 ()[(32 24δ +32δ 2 +0δ 3 )k() 2(4 + δ)(4 + δ 2 )] = 0. Evaluating the first derivative of profit function at = we have π 0 ( )= 5δ3 4(4+δ) [k( ) 5 ]. So, given the uasiconcavity of the profit function the monopolist will decrease uality if 0 < k( ) < 5 and increase uality if 5 k( ) < 2+δ. The intuition for this result is as follows. 2
13 The firm s decision on uality basically depends on the levels of the two marginal types and the relative importance (distribution) of the first and secondperiod demands. In this case of 0 <k( ) < 2+δ, the two marginal types are given as and θ = 2+δ 4+δ + 2( δ) 4+δ k() θ 2 = 2 [θ + k()] = 2+δ 2(4+δ) + 6 δ 2(4+δ) k(). Note that the secondperiod marginal type is more sensitive to the level of the average unit cost than the firstperiod marginal type. 8 This implies that the secondperiod marginal type decreases more rapidly than the firstperiod marginal type as the average unit cost gets smaller. In the former case, with a relatively small average unit cost the euilibrium secondperiod marginal type is too low so that the optimal uality is determined lower than the optimal commitment level. In the latter case, however, the average unit cost is relatively high and therefore the firm increases uality, partially mitigating the timeinconsistency problem. Combining the above results leads to the following proposition. Proposition Compared with the commitment euilibrium, the monopolist i) decreases uality if 0 <k( ) < 5, ii) increases uality if 5 k( ) < 2+δ with secondperiod sales (still subject 2+δ to the timeinconsistency problem), iii) increases uality if k( ) < +δ without secondperiod sales (avoiding the timeinconsistency problem), and iv) offers the same uality as in the +δ commitment case if k() < +δ. Example 2: Suppose c() = α (α>). From condition (5), the optimal commitment uality is given by =( 2α +δ ) α, which initially decreases and then increases after some critical value of α (e.g. α = for δ = ). The average unit cost at the optimal commitment uality is then given by k( )= +δ 2α, which is monotone decreasing for α>. Recall that α is the elasticity of scale of uality. So, the average unit cost of production at the optimal commitment uality is higher when the cost function is more elastic to uality change. If 0 < +δ 2α < 6+5δ 5 α> 2, the monopolist decreases uality. If 5 +δ 2α < 2+δ 4+6δ+3δ2 2(2+δ) < α 6+5δ 2+δ 2,thefirm increases uality but chooses to sell the good in period 2. If 2α +δ < 8 For instance, for δ = (no time discounting) the firstperiod marginal type (θ 2)isconstantindependentof the average unit cost. 3
14 +δ 4+6δ+3δ2 +δ<α 2(2+δ),thefirm increases uality in order to commit to no sales in period +δ 2 (avoiding the timeinconsistency problem). Finally, if 2α +δ < +δ α<+δ, the firm is free from the timeinconsistency problem and offers the optimal commitment uality, achieving the same outcome as in the commitment case. The following figure exhibits the range of parameters corresponding to each of the above four cases. α 6 5 Decrease uality with the timeinconsistency problem 4 3 Decrease uality with the timeinconsistency problem 2 Increase uality to eliminate the timeinconsistency problem No timeinconsistency problem δ Fig : The optimal uality without commitment. 5 Concluding remarks This paper has examined how a firm s choice of uality interacts with the timeinconsistency problem in a durablegood monopoly framework. It has been shown that the relationship depends on the structure of unit production function of uality. Durablegoods monopolists may have incentives to raise uality in order to eliminate or ameliorate the timeconsistency problem, and the euilibrium level of uality can be higher or lower than the one in the commitment regime. The basic model can be extended in several lines. The most demanding is probably is to extend the model to a finite (but more than two period) or infinite horizon setup. Also interesting is to examine the robustness of the result to allowing the firm to produce multiple ualities or alter the uality of the good later at some fixed costs. 4
15 References [] Bagnoli, M., S. Salant, and J. Swierzbinski (989), DurableGoods Monopoly with Discrete Demand, Journal of Political Economy 97, [2] Bagnoli, M., S. Salant, and J. Swierzbinski (995), Intertemporal SelfSelection with Multiple Buyers, Economic Theory 5, [3] Bond, E. and L. Samuelson (984), Durable Good Monopolies with rational Expectations and Replacement Sales, RAND Journal of Economics 5, [4] Bulow, J. (982), Durable Goods Monopolists, Journal of Political Economy 90, [5] Bulow, J. (986), An Economic Theory of Planned Obsolescence, Quarterly Journal of Economics 5, [6] Butz, D. (990), Durable Good Monopoly and Best Price Provisions, American Economic Review 80, [7] Coase, R. (972), Durability and Monopoly, Journal of Law and Economics 5, [8] Hahn, J. (2002), damaged durable Goods, Keele University, mimeo. [9] Inderst, R. (2002), Why Durable Goods May Be Sold Below Costs, University College London, mimeo. [0] Kahn, C. (986), The Durable Goods Monopolist and Consistency with Increasing Costs, Econometrica 54, [] Takeyama, L. (2002), Strategic Vertical Differentiation and Durable Goods Monopoly, JournalofIndustrialEconomics50(), [2] von der Fehr, N.H. and K.U. Kühn (995), Coase versus Pacman: Who Eats Whom in the DurableGoods Monopoly?, Journal of Political Economy 03, [3] Waldman, M. (993), A New Perspective on Planned Obsolescence, Quarterly Journal of Economics 58,
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