Opening Secondary Markets: A Durable Goods Oligopoly with Transaction Costs

Transcription

1 Opening Secondary Markets: A Durable Goods Oligopoly with Transaction Costs Jiawei Chen Department of Economics UC-Irvine Susanna Esteban Department of Economics Universidad Carlos III de Madrid Matthew Shum Department of Economics Johns Hopkins University February 27, 2007 Preliminary and Incomplete Draft Abstract In this paper, we introduce a model of durable goods oligopoly, in which consumers face lumpy costs of transacting in the car market. With transactions costs, the timing of purchases becomes important on the demand side, and consumers buy and sell in the automobile market infrequently in order to avoid incurring the lumpy adjustment costs. We present simulation results from simple monopoly and duopoly versions of the model, and discuss the empirical methodology for structural estimation of this model. Having modeled transaction costs provides us with a direct measure of the separate effects of secondary markets and durability on firms behavior. For a quantified example, making the good durable decreases profits for the monopolist by 14.78%; adding frictionless secondary markets increases them by 31.26%. Keywords: Secondary Markets, Durable Goods, Oligopoly, Transaction Costs, Automobile Industry The authors can be contacted by at jiaweic@uci.edu (Chen), sesteban@eco.uc3m.es (Esteban) and mshum@jhu.edu (Shum). 1

2 2 1 Introduction In this paper, we consider the effects of secondary markets on the producers of durable goods. Are secondary markets good or bad for producers? The existing literature has returned ambiguous answers to this question. As has been pointed out in the literature going back to Coase (1972), durability per se erodes a monopolist s market power, and profits, due to its inability to commit to keeping prices high (equivalently, keeping production low) in the future. Consider a market with heterogenous consumers: having sold to the higher valuation consumers, the monopolist is left with a market of lower valuation types to whom it can only sell if it lowers prices. Forward looking consumers, anticipating the price decrease, delay their purchases and force the monopolist to start decreasing prices earlier on. In the limit, if the good is infinitely durable, does not depreciate, and the monopolist can make instantaneous price adjustments, price converges immediately to the marginal cost of production. In such a setting, as pointed out by Liang (1999), secondary markets refrain the firm from overproducing in the future. Higher valuation types who might own a used product can now sell it in the secondary market and return to the primary market. Thus, the secondary market works by improving the efficiency in the allocation of new and used products and allows the monopolist to sell to higher types in every period. The monopolist has less incentives to drop prices in the future, which can lead to higher prices and, hence, higher profits. Therefore, although durability erodes profits for the firm, the ability to trade the product in a competitive secondary market mitigates (at least partially) the decrease in profits. If the secondary market will be active, increasing durability might have an ambiguous effect on profits. Indeed, the planned obsolescence literature (eg. Swan (1985) and Bulow (1986)) has pointed out how the firm may have an incentive to reduce the durability of its product, as a means of shrinking the secondary market. The existing literature has examined these issues in models which differed in their assumptions about consumer preferences and the nature of durability, for instance which makes it difficult to compare results across the models. In this paper, we introduce a dynamic equilibrium durable goods model with secondary markets and consumer transactions costs and use this model to study the effects of secondary markets on firms profits. In our model, we use consumer transactions costs as a measure of the level of activity or

3 3 trade in secondary markets, with larger values of the transaction costs leading to an endogenously smaller or less active secondary market. The existing literature on secondary markets (eg. Rust (1985) and Esteban and Shum (2004)) has for the most part considered the case of zero transactions costs, so that secondary markets are completely active. This has mainly been for tractability reasons, as the demand functions with positive transactions costs are very nonlinear step functions. At the opposite extreme, when transactions costs approach infinity, no consumers will ever trade, which corresponds to the setting originally consider in the Coase literature. The advantage of our setting is that, by modifying the value of transaction costs, we can close down the secondary market and obtain the effects of secondary markets separate from those of durability. (In contrast, in the planned obsolescence literature, closing down the secondary market isdone at the expense of reducing durability.) We construct a model of durable goods oligopoly, in which consumers face lumpy costs of transacting in the car market. By allowing for these transactions costs, we build upon the work in Esteban and Shum (2004), in which a dynamic equilibrium model of durable goods oligopoly was considered. In that paper, a convenient and tractable linear-quadratic specification was derived due to some simplifying assumptions, an important one of which was that consumers faced zero transaction costs in buying or selling in the automobile market. By introducing transactions costs, nontrivial dynamics result on the demand side. Specifically, with transactions costs, the timing of purchases becomes important on the demand side, and consumers will buy and sell in the automobile market infrequently in order to avoid incurring the lumpy adjustment costs. At the individual consumer level, the demand functions display the (S,s) -type of nonlinearities, and resemble the demand functions considered in much of the macro literature on durables (including Eberly (1994), Attanasio (2000), Stolyarov (2002), and Adda and Cooper (2000)). More recently, the empirical literature on durable goods demand has also consider the case where consumers have transactions costs (eg. Gowrisankaran and Rysman (2006), Schiraldi (2006)). Most closely related to our paper is Porter and Sattler (1999), who derive and test the properties of an overlapping generations durable goods monopoly model with secondary markets and transactions costs under the assumption that firms can commit to a production sequence. In our model, firms are not able to commit, and we analyze the time-consistent Markov-perfect equilibrium paths. This difference is significant, because if firms are able to commit, then the Coase problem disappears and one benefit of secondary markets that of limiting the overproduction by the firm is no longer relevant. As far as we are aware,

4 4 our model is the first Markov perfect equilibrium model of a durable goods industry with transactions costs and secondary markets. 1 Results to date show that a secondary market is desirable for durable goods producers in that it outweighs the competitive pressure resulting from the durability of the product. Simulations quantify the gains of an active frictionless secondary market: profits for a durable goods monopolist increase by 31.26% when transactions costs are decreased from a high level (one which closes down the secondary market) to zero (indicating frictionless secondary markets). Furthermore, when we compare the case of no durability to durability with zero transactions costs, the firm s profits increase by 11.86%, whereas when we compare the case of no durability to durability with high transactions costs, the firm s profits decrease by 14.78%, suggesting that secondary markets make durability more desirable to the firm. One caveat of the model is that it might understate the gain in allocative efficiency resulting from active secondary markets. In our model, consumers are ex-ante identical except for their current durable good endowment, but are hit with iid preference shocks each period. Each period, consumers with high preference shocks will trade their used products to consumers hit with lower preference shocks, and upgrade to a new product. Since these preference shocks are iid, however, consumers are ex-ante identical, and there is no intertemporal allocative role of the secondary market in that consumers might decide to make one purchase or the other based on their persistant valuations. In the next section, we set up the model, and describe the dynamic equilibrium of the model. In Section 2, we describe the computational algorithm, and discuss simulations from a simple monopoly version of the model. In Section 3, we discuss the existing results regarding the effects of secondary market obtained by varying the size of consumer transactions costs. In Section 4, we briefly discuss the results from a duopoly version of the model. 2 Model There are J products in the market (this includes both new and used cars). Assume that the per-period utility offered by each product j is α j. Let all the available cars be indexed j = 1,..., J. Let I {1, 2,..., J} denote the subset of indexes which are the indexes of the new cars 1 Huang, Yang, and Anderson (2001) considered a framework similar to ours, but the equilibrium derivation appears to be non-standard.

5 5 produced by the firms in the market. Assume for simplicity that each firm produces only one car model. With some abuse of notation, let i = 1,..., I index the firms (as well as the models they produce). Let r t ( {1,..., J}) denote the index of the car owned in at the beginning of period t by some consumer. Also, let r t = 0 denote the index of no car ownership (or ownership of a car that has died). We set α 0 = 0 and p 0t = 0, for all periods t. Also, let v(r t ) ( {1,..., J}) denotes next period s index of a car which is currently indexed r t (with the convention that v(0) = 0). Also, let w(j) denote the previous period s index of a car which currently has index j. That is, consider three years t, t + 1, t + 2: then the indexes for a car which has index j in period t + 1 is r t = w(j), r t+1 = j, r t+2 = v(j). In what follows, we consider both the firm and consumer problems in partial equilibrium. Afterwards, we consider equilibrium. 2.1 Consumer problem Consumers are are differentiated in two dimensions. First, we allow consumers to vary in their taste for quality; second, a consumer s utility from a particular product choice is perturbed every period by idiosyncratic utility shocks which are independent over time. If the consumer l keeps the car she has in period t, she gets utility α rt + ɛ lrtt. If she sells and purchases a car with index j (where j r t ) instead, she gets utility α j + γ l (p rtt p jt k) + ɛ ljt. γ l measures consumer l s marginal utility of money, which varies across consumers. We assume that there are a finite number (L) of consumer types, so that γ l {γ 1,..., γ L }. Let π 1,..., π L denote the population shares of the L consumer types (with l π l = 1). (In another abuse of notation, we use l to index both consumer types (of which there are a finite number) as well as consumers, of which there are a continuum. k denotes the transaction cost: we assume that k is k > 0 if the transaction involves the buying or selling of a used car, and is zero otherwise. The vector ɛ lt (ɛ l1t,..., ɛ ljt ) are the idiosyncratic shocks of consumer l for period t. Importantly, we assume that selling and buying decisions are made at the beginning of each period, before depreciation, so that buying a car with index s t in period t yields car utility of α st.

6 6 Let K lit denote the share of the population who are type l and who own a i-ranked car at the beginning of period t, with i {0, 1,..., J} and l {1,..., L}. Let the vector K t = (K 10t,..., K 1Jt, K 20t,..., K L0t,..., K LJt ). Similarly, let B t denote the vector of used car production in period t, with elements K lit, i {1,..., J} \ I, and l {1,..., L}. We put the consumer s problem in a dynamic programming framework. The state variables are r t, the prices p t {p 1t,..., p Jt }, and the consumer s shocks ɛ lt. Let the control variable be denoted s t {0,..., J}. Then the current utility in period t is: u(r t, s t, p t ; γ l, ɛ lt ) = α st + 1(s t r t ) γ l (p rt p st k) + ɛ lstt ũ(r t, s t, p t ; γ l ) + ɛ istt. The second term represents the disutility incurred when the consumer sells her r t -indexed car and buys the s t -indexed car, which includes the transactions cost term k. The consumer s Bellman equation is: V (r t, p t ; γ l, ɛ lt ) = max s t [ u(rt, s t, p t ; γ l, ɛ lt ) + βe rt+1, p t+1, ɛ it+1 r t, p t,s t, ɛ lt V (r t+1, p t+1 ; γ l, ɛ lt+1 ) ]. The expectation is taken over the transition equations for the state variables, which are the index of the current car, r t, as well as the prices p t. For the state variable r t, the transition is deterministic given s t : r t+1 = { v(r t ) v(s t ) if s t = r t otherwise. The Bellman Eq. (1) also has expectations over future prices p t. In equilibrium, prices will depend on the firms past, current, and future car production. For the time being, however, we will work in partial equilibrium, and only assume that consumers beliefs about the evolution of prices are captured in a conditional distribution G(p p). The consumer s optimal policy function will be a function of the state variables: s t = s (r t, p t, γ, ɛ lt ; G), where we have also included the price beliefs G as an argument to emphasize the dependence of consumers optimal choices on these beliefs. (1) Note that, given transaction costs, the optimal car choice is state-dependent, because the choice of s t depends explicitly on r t, the index of the currently-owned car.

7 7 Define Ṽ (r, p, γ) E ɛv (r, p, γ, ɛ). Then the Bellman equation in (1) above can be written as [ ]} Ṽ (r, p, γ) = E ɛ {max ũ(r, s, p, γ) + ɛ is + βe r s, p r, p,sṽ (r, p, γ). (2) Deriving aggregate demand functions Consider the demand for car j in period t. We make the assumption that each idiosyncratic shock ɛ ljt is i.i.d. across (l, j, t). With this assumption, for all consumers who currently have car ranked r t and have type γ, the proportion of them who will switch to car j is: 1 [s (r t, p t,, γ, ɛ lt ; G) = j] df ( ɛ lt ), for all j {0, 1,..., J}. If we further assume that each ɛ ijt is distributed extreme-value (with CDF F (ɛ) = exp( exp( ɛ))), then this expression becomes ( ) exp ũ(r t, j, p t ; γ) + βeṽ (r t+1 = v(r t ), p t+1 ; γ) Q j (r t, p t ; γ, G) = ( ) (3) J j =0 exp ũ(r t, j, p t ; γ) + βeṽ (r t+1 = v(r t ), p t+1 ; γ) where the expectation is taken over p t+1. (If r t = j, then Q j (r t, p t, p t, γ) denotes the proportion of type-γ consumers having car j in period t who do not buy another one.) The assumptions that the ɛ ljt s are i.i.d. is very important here, because it implies that F ( ɛ r t ) = F ( ɛ), for all r t. If these shock are persistent over time, this is no longer true, and we would have to carry around the joint distribution of (r t, ɛ lt ) as a state variable. However, the presence of different γ s across consumers implies that even when the ɛ s are assumed to be i.i.d. across time, we would need to carry around the joint distribution of (r t, γ l ) as a state variable. The discreteness of (r t, γ l ) is important, because it implies that the joint distribution is finite-dimensional, so that the state space is also finite-dimensional. 2 Our assumptions on heterogeneity can be contrasted with those in Porter and Sattler (1999). The model in that paper is a vertically differentiated model, where the distribution of γ is continuous and uniform, and there are no ɛ errors. It is similar to the pure vertical model in Berry and Pakes (1999). However, the pure vertical differentiation model is not so convenient for computational reasons, as the market shares may be discontinuous functions of the model parameters. 2 If γ is continuous, the approach of Krusell and Smith to approximate the infinite-dimensional joint distribution of (r, γ) by its moments may be helpful.

8 8 With the extreme-value assumptions on the error terms, Eq. (2) can be written as { J ( Ṽ (r, p; γ) = log exp ũ(r, s, p; γ) + βe r, p r, p;γ,sṽ (r, p ; γ)) } (4) s=0 which is a functional equation that one could iterate over to solve for the expected value function Ṽ (r, p; γ). This is as in Rust (1987). The total demand for car j in period t is ( ) D Bt j, p t ; G = M where M is the population size. l J π l j =0,j j,j I K lj t Q j (j, p t ; γ l, G) (5) Note that the demand for car j does not include the consumers who currently own car j (ie. l j), who decide to keep their car and not sell it. Also, given our assumption that consumers make their car purchase decision at the beginning of periods, while depreciation occurs at the end of periods, the summation in the aggregate demand (5) does not include the new cars (j I), because no one owns a new car at the beginning of any period (ie. consumer who had a new car in the previous period now have a used car at the beginning of the current period). Also define ( ) H Bt j, p t ; G = M π l K ljt Q j (j, p t ; G), j = 1,..., J, j I, (6) l the consumers for whom r t = j, and who keep their car in period t instead of switching to another one. Finally, define the supply for each used car in period t as S j ( Bt, p t ; G) = M J π l l j =0,j j K lj t Q j (j, p t ; γ l G), j {1,..., J} \ I which are the number of consumers who own car j at the beginning of period t, who choose another car j j. 2.2 Firm problem We assume that firms choose quantities. To formulate the firm problem, we must define the inverse demand functions. For each new car i I, define the inverse demand function

9 ( ) g Bt i, p t, i, q it ; G, which is the inverse of the aggregate demand function D i ( ) with respect to p it, evaluated at q it. ( p t, i denotes all the prices except the price for car i, p it.) For each new car model i I, firm i s static profits are Π i (q it, B t ) q it (g i ( B t, p t, i, q it ; G) c i ). (7) The Bellman equation is W i ( B t ) = max q it Π i (q it, B t ) + βeb t+1 B t,q it W i ( B t+1 ). (8) [ ] Firm i s equilibrium production strategy would be a state-contingent function q it = q i ( B t ), for i I, which solves the RHS of Eq. (8). The prices will be determined by market-clearing conditions period-by-period. For new cars i I, the prices must satisfy the following market-clearing conditions: q i ( Bt ) = D i ( Bt, p t ; G). (9) The prices p j, j = 1,..., J, j I, for the used cars in period t, must satisfy the market clearing condition S j ( B t, p t ; G) = D j ( B t, p t ; G) where D j ( B t, p t ) is defined in Eq. (5) above. These market-clearing conditions will define the equilibrium prices p jt, for all i = 1,..., J. Let ρ( B t ) denote the full J-dimensional vector of prices derived by solving this J-dimensional nonlinear equation system jointly Equilibrium In rational expectations equilibrium, consumers beliefs about the price evolution, summarized by the conditional distribution G, must correspond to the actual evolution. In equilibrium, consumers will recognize the functional dependence of prices on the state vector B t, so that the consumer problem in equilibrium will also have B t as the state variable, rather than prices. In what follows, we use an asterisk ( ) to denote equilibrium functions. Let q ( B t ) denote the equilibrium policy function, p ( B t ) denote the equilibrium pricing functions. Let h ( B t ) 3 Note that, if we were to solve the new-car market-clearing conditions (9) on an equation-by-equation basis, for arbitrary q and fixed values of p t, j, then we get the inverse demand functions g j B t, p t, j, q; G which we defined at the beginning of this section.

10 10 denote the equilibrium transition vector for B t, and G the equilibrium transition for prices. We will describe these functions in more detail here. Since, in equilibrium, the evolution of prices is driven by evolution of B t, we can rewrite the consumer problem to have B t be the state vector, in place of p t., and with the transition h( B t, q ( B t )) governing equilibrium beliefs about the evolution of B t, and prices each period given by p ( B t ). The Bellman equation for the new consumer problem would be V (r t, B ) [ t ; γ, ɛ it = max u(r t, s t, p (B t ); γ, ɛ it ) + βev (r t+1, )] B t+1 ; γ, ɛ it+1 s t (10) where the consumers beliefs over the evolution of Bt will be given by the equilibrium transition function h( B t, q ( B t )), which will be endogenously determined in the consumer problem, as we will show later. In order to describe the market-clearing conditions which determine the equilibrium price sequences p ( B t ), we still need to define firms inverse demand functions as follows. Consider firm i, and consider the production vector q i ( B t ; q), which sets q it = q and q i t = q i ( B t ) for all other firms i i. Corresponding to this production vector would be the transition function h( B t, q i ( B t ; q)), which gives the values of B t+1 given that all other firms besides firm i produce according to the policy function q ( B t ), but firm i produces q. The prices p i ( B t ; q) which would result at state B t, and production vector q t = q i ( B t ; q)) are given by the market clearing conditions: ( q i = D Bt i, p t ; h( B t, q i ( B ) t ; q)), i I ( S Bt j, p t ; h( B t, q i ( B ) ( t ; q)) = D Bt j, p t ; h( B t, q i ( B ) t ; q)), j {1,..., J} \ I. The locus of p i s generated by solving the mkt-clearing conditions for different values of q it = q (while holding all q i t fixed at q i ( B t )) yields the inverse demand function for firm i, which we denote p( B t, q). In the above market-clearing conditions, the demand curve D j ( B t, p t ; h) is defined, analogously to Eq. (5), as ( ) D Bt j, p t ; h = M l J j =0,j j,j I K lj tπ l Q j (j, B t, p t ; γ l, h) and Q j (r, p t ; γ l, h) denotes the state r-dependent probability of choosing product j, which (11)

11 11 is defined (analogously to Eq. (3)) as ( Q j (r t, B exp ũ(r t, j, p t ; γ) + βe h Ṽ (v(j), ) B t+1 = h; γ) t, p t ; γ, h) = ( J j =0 exp ũ(r t, j, p t ; γ) + βṽ (v(j ), B ). t+1 = h; γ) The supply function S j ( B t, p t ; h) is defined analogously. Given this notation, we note that the transition function h( B t, q) is defined implicitly by the requirement that the function must yield correct predictions of next period stocks, given optimizing consumer behavior. That is, for all used products j {1,..., J}\I, the (l, j)-th element of h( B t, q) is K l,j,t+1 = π l K l,j,tq w(j) (j, p( B t, q); γ l, h( B t, q)) j (this is the share of product j in period t + 1 among type k consumers, which is just equal to the demand for car w(j) in period t among type k consumers). In the above, the marketclearing prices p( B t, q) are the prices consistent with the market-clearing conditions (11), and the transition h( B t, q). In equilibrium: 1. q ( B t ) solves firm problem: q i ( B t ) = argmax q q ( p i ( B t, q) c i ) + βew ( B t+1 ) where expectation of B t+1 is according to h( B t, q i ( B t ; q)). 2. Consumer value functions are given by Ṽ (r, B; γ) = E ɛ {max s [ ũ(r, s, p ( B); γ) + ɛ is + βṽ (r, h( B, q ( B t )); γ) ]}. 3. Equilibrium prices p ( B t ) satisfy the market-clearing conditions qi ( B ( t ) = D Bt i, p t ; h( B t, q ( B ) t )), i I ( S Bt j, p t ; h( B t, q ( B ) ( t )) = D Bt j, p t ; h( B t, q ( B ) t )), j {1,..., J} \ I. 4. The transition B t+1 = h( B t, q ( B t )) is consistent with consumer maximization, and the equilibrium prices: K l,j,t+1 = π l j K l,j,tq w(j) (j, p ( B t ); γ l, h( B t, q ( B t )))

12 12 3 Monopoly example In this section, we focus on a simplified monopoly version of the model, and present some computational results which illustrate features of the Markov equilibrium paths. Consider the example where there is only one firm, producing a car that lasts for two periods. Thus J = 2, with v(1) = 2 and v(2) = 0. Assume that the firm chooses quantity, and the prices adjust to clear the market. Consider the monopoly model with two-period goods as above, but amended to have two consumer types (ie. L = 2). The state vector for this model will be B t = K 2t (K 12t, K 22t ) The firm s Bellman equation is W ( K [ 2t ) = max Π(q t, p t ) + βw ( ] K 2t+1 ) q t (12) s.t. q t M l (1 K 2t ), p t = p(q t, K 2t ), and K l2t+1 = h( K 2t, q t ), where p t = p(q t, K 2t ) is the solution to the following system of 2 equations: { = D 1 ( K 2t, p t, h( K 2t, q)) q t S 2 ( K 2t, p t ) = D 2 ( K 2t, p t, h( K 2t, q)) (13) Eq. (3) can be written as Q j (r t, K 2t p t, K 2t, γ, h) = ( exp ũ(r t, j, p t, γ) + βṽ (v(j), h( K ) 2t, q, γ) ( J j =0 exp ũ(r t, j, p t, γ) + βṽ (v(j ), h( K ), (14) 2t, q), γ) and the system of equations in (13) becomes { q t /M l π l i=0,1 K l2t Q i (2, K 2t p t, K l2t, γ l, h) = l π l i=0,2 K lit Q 1 (i, K 2t p t, K 2t, γ l, h) = l π l i=0,1 K lit Q 2 (i, K 2t, p t, K 2t, γ, h). (15) 3.1 Algorithm for model with consumer heterogeneity The main difference is that the state vector is bigger now: Bt is two-dimensional. Also, now the consumer value function V has an additional argument, which is γ. An additional difficulty is that the transition function h( B t, q) must be iterated and solved for as part of the consumer problem.

13 13 Make initial guesses of firm policy function q 0 ( K 2t ) and consumer value functions Ṽ 0 (r t, K 2t, γ l ), for l = 1, 2. First solve consumer problem. Start with a guess of h 0 ( B t, q). By the end of the consumer loop, we will obtain values for h( ) consistent with q 0 and V 0, as well as an iterated guess for the consumer value functions V 1. Given q 0 ( B t ), and the guess of consumer value functions V 0, and h 0, we can solve for market clearing prices p( B t ) using (15). Given these market clearing prices, we can evaluate the choice probabilities in Eq. (14). Then we can evaluate next period stocks as l = 1, 2. K l2t+1 = π l j=0,1 K ljt Q 2 (j, B t, p t ( B t ), γ l, h 0 ), Set h 01 = K 2t+1, and repeat previous two steps until you get a convergent value for h function. Call this h 1. Using h, evaluate the consumer Bellman equation { J Ṽ 1 (r, K ( 2, γ l ) = log exp ũ(r, s, p( K 2 ), γ l ) + βṽ 0 (v(r), h( K 2, q 0 ( K ) } 2 )), γ l ) s=0 for types l = 1, 2 to get Ṽ 1 (r t, K 2t, γ l ). Solve the firm s problem to obtain q 1 ( K 2t ). Make initial guesses of q 00 ( K 2t ) and W 00 ( K 2t ). Solve for the firm s optimal policy q 01 ( K 2t ) using the firm s Bellman equation (16) in (12), the transition function h 1 ( K 2, q) derived from the consumer problem, and the inverse demand function p t = p(q t, K 2t ) as the solution to the system of equations in (15). Update the firm s value function W 01 ( K 2t ) according to the firm s Bellman equation. The loop terminates if the relative changes from q 00 ( K 2t ) to q 01 ( K 2t ) and from W 00 ( K 2t ) to W 01 ( K 2t ) are both below a pre-specified tolerance. Otherwise update the guesses and iterate on.

14 14 Set q 1 (K 2t ) = q 01 (K 2t ). The algorithm terminates if the relative changes from Ṽ 0 (r t, K 2t ) to Ṽ 1 (r t, K 2t ) and from q 0 (K 2t ) to q 1 (K 2t ) are both below a pre-specified tolerance. Otherwise update the guesses and iterate on. Note that the algorithm consists of three loops: two inner loops (one for the consumer s problem and one for the firm s problem) and an outer loop. 3.2 Approximation Approximate the firm s policy function as n q (K 2 ) d i φ i (K 2 ), i=1 where φ 1, φ 2,..., φ n are Chebychev polynomials and d 1,..., d n are a set of coefficients to be determined. Ṽ (r, K 2 ) and W (K 2 ) are approximated analogously. 3.3 Parameterization We use the following parameter values in our baseline model: 4 The effects of secondary markets on firm profits: preliminary results In this section, we consider results from simulations which address the question raised at the very beginning of this paper: what are the effects of secondary markets on durable goods producers? As we discussed earlier, we use the level of transactions costs to measure the activity in the secondary market, with higher transactions costs inducing a less active secondary market. Table 1 shows the effect of durability and secondary markets, both individually and combined, on profits. Making the good durable the effect of durability implies a change in profits for the firm of 14.78% relative to the nondurable good case. Adding to durability no frictions the effect of secondary markets and thus making the secondary market most

15 15 Table 1: Effects of Durability and Secondary Markets (c = 0) Not durable Durable Secondary markets (S = 0) New car production New car price Profit Used car transactions n.a % Durability a % Secondary markets (S = 0) b New car production New car price Profit % Durability & Secondary markets (S = 0) c. New car production New car price Profit a Relative to the Not durable numbers. b Relative to the Durability numbers. c Secondary markets numbers relative to Not durable numbers

16 16 active, increases profits for the firm by 31.26%. Overall, if we simply compare no durability to durability with zero transaction costs, we obtain a net gain of 11.86% in profit. Thus, the cause for the increases in profit is per se the secondary market. Notice that if we were quantify the effects of secondary markets by eliminating durability, we would understate the effect: 11.86% relative to the true 31.26% increase. This results illustrate clearly the separate effects of secondary markets and durability on firm s behavior. Durability erodes market power by standard Coasian arguments. Secondary markets mitigate this erosion by improving allocative efficiency and refraining the firm s incentives to overproduce. The preliminary results show that the gains for the firm in reducing frictions in the secondary market are large. Nonetheless, the qualitative results in Table 1 are sensitive to the marginal cost of production. Intuitively, a positive marginal cost of production raises the profitability of producing a durable good. On the one hand, durability brings the consistency problems for the firm and is responsible for the previously described decrease in profit. On the other, by producing a durable good, the firm can accrue revenues from sales in multiple periods, since the new car price is the capitalized value of the asset, without having to manufacture the good in each one of these periods. If the cost is high, see Table 2, the savings in cost may outweigh the competitive dynamic arising from durability. Table 2: Effects of Durability and Secondary Markets (c = 1) Not durable Durable Secondary markets (S = 0) New car production New car price Profit Used car transactions n.a To understand the marginal effects of secondary markets on firm s behavior, we build two statistics. Let y(s) be an industry variable, e.g., price, profit, etc, for a give level of transaction costs S 0. Let S 0 0, which corresponds to he secondary market being most active. Let S max be the level of transaction costs that close down the secondary market. Then compute: y(s+ ) y(s) y(s) S η y (S), for all S [0, S max ], (17) as the change in y from a marginal increase in transaction costs. That is, η y (S) is the

17 17 elasticity of y to a change in transaction costs. Our second statistic directly imputes changes in y to changes in used car transactions the level of activity in the secondary market. That is, we compute η y (S) x U (S+ ) x U (S) x U (S) υ y (S), (18) where x U are secondary market transactions. This latter measure the impact of the secondary market on market variables, and is an elasticity of y to changes in used car transactions. 5 Estimating the model Here we consider how the monopoly model could be estimated. How we estimate depends on what data is observed. Here I assume that in each period t, we observe (p t, K 1t, K 2t ): prices, and the ownership/purchases of new and used cars, for each type of consumers. We assume that we can classify consumers into the different type groups. For simplicity, assume there are only two groups. For example, we could consider two types of consumers, those with income > $50, 000, and those with income < $50, 000. Then π 1 and π 2, the population proportion of type 1 and type 2 consumers, is observed. In order to confirm that a classification scheme makes sense, we could do a simple test for consumer heterogeneity. We check whether the percentage of new cars owned is different across the proposed classification groups: if so, then there is heterogeneity across the classification groups. Intuitively, because the prices of new and used cars are identical across the two groups of consumers, heterogeneity in WTP to quality γ should be reflected in heterogeneity across groups in propensity to own new vs. used cars (which differ in quality). If we find no differences between the groups, then we should go back and redefine the consumer groups. The estimation will proceed in three steps. First, because we observe K 12t and K 22t, we observe all the state variables. Hence, we can estimate the h, the equilibrium transition function for the observed state variables. This is crucial, since with knowledge of h, we can simulate both the consumer and firm dynamic optimization problems. Second, we estimate the parameters related to the consumer problem. On the consumer side, we observe K ljt, for each consumer type l, and each vintage j. Given knowledge of the

18 18 transition function h, we can solve the consumers dynamic choice problem (Eq. (10)) and obtain the choice probabilities Q j (r t, B t, p t ; γ, h). For each l and j, then, we can estimate by matching the observed stocks to the predicted stocks: K ljt = K l,j,t 1Q j (j, B t, p t ; γ l, h) j where the LHS is observed, and the RHS can be computed given parameter values and the solution to the consumer problem. Note that in this way, we could estimate the consumerrelated parameters (the γ s and the α s) separately from the supply side. Third, we estimate the parameters related to the supply side. In each period t, we observe q t, the new production that period. Given our knowledge of the equilibrium transition function h and the consumer parameters from the previous steps, we can solve the firm problem to derive q(k 12t, K 22t ), the equilibrium production function of the firm. Estimation of the firm-level parameters (the cost parameters) can be based on matching observed to predicted production: q t = q(k 12t, K 22t ). References Adda, J., and R. Cooper (2000): Balladurette and Jupette: A Discrete Analysis of Scrapping Subsidies, Journal of Political Economy, 108, Attanasio, O. (2000): Consumer Durables and Inertial Behavior: Estimation and Aggregation of (s,s) Rules, Review of Economic Studies, 67, Berry, S., and A. Pakes (1999): Estimating the Pure Hedonic Discrete Choice Model, Manuscript, Yale University. Bulow, J. (1986): An Economic Theory of Planned Obsolescence, Quarterly Journal of Economics, 101, Coase, R. (1972): Durability and Monopoly, Journal of Law and Economics, 15, Eberly, J. (1994): Adjustment of Consumers Durables Stocks: Evidence from Automobile Purchases, Journal of Political Economy, 102, Esteban, S., and M. Shum (2004): Durable Goods Oligopoly with Secondary Markets: the Case of Automobiles, Forthcoming, RAND Journal of Economics.

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