Instrument-free Identification and Estimation of Differentiated Products Models using Cost Data

Transcription

1 Instrument-free Identification and Estimation of Differentiated Products Models using Cost Data David P. Byrne Masayuki Hirukawa Susumu Imai Vasilis Sarafidis February 27, 2014 Abstract In this paper, we propose a methodology for jointly estimating the demand and cost functions of differentiated goods oligopoly models when demand and cost data are available. The method deals with the endogeneity of prices to unobserved product quality in the demand function, as well as the endogeneity of output to unobserved cost shocks in firms cost functions. Our method does not, however, require instruments to do so. We establish identification, consistency and asymptotic normality of our estimator. Using Monte-Carlo experiments, we demonstrate that it works well in situations where instruments are correlated with the error term in the demand and cost functions, and where the standard IV approach results in bias. We are grateful to seminar participants at the University of Melbourne, University Technology Sydney, and the University of New South Wales for comments. Department of Economics, University of Melbourne, byrned@unimelb.edu.au Department of Economics, Setsunan University, hirukawa@econ.setsunan.ac.jp Business School, University of Technology Sydney, susumu.imai@uts.edu.au Department of Econometrics and Business Statistics, Monash University, vasilis.sarafidis@monash.edu 1

2 1 Introduction Since Berry (1994) and Berry, Levinsohn and Pakes (1995), logit models and random coefficient logit models with unobserved heterogeneity (hereafter, the BLP model) have been the main tool in the estimation of the differentiated goods oligopoly models. As Berry and Haile (2012) and others have pointed out, so long as there are instruments available, based on which moment conditions can be constructed, demand functions can be specified in a fairly flexible manner. The key issue then, as far as the asymptotic properties of the resulting method of moments estimator is concerned, is whether the instruments are valid or not. Commonly used instruments include cost shifters (e.g. market wages), market structure instruments (e.g. the number of products of rival firms) and the price of a product in other markets. While there has been some research assessing the robustness of the BLP estimation algorithm (Dube et. al., 2012), there are no results with regards to the validity of the BLP specification itself, which relies on the validity of the BLP instruments. This issue is of major concern in this paper. We develop a new methodology that simultaneously estimates the parameters of the demand and cost function for the differentiated products models using both demand and cost data. The procedure deals with both the endogeneity of prices in demand estimation and the endogeneity of output in the cost function estimation, but does not require any instruments, neither on the demand side nor cost side. That is, the entire estimation procedure is instrument-free. For demand estimation, we exploit the first order condition of profit maximization, where firms choose price and quantity to equalize marginal revenue to marginal cost. For cost estimation, an important challenge is how to recover the marginal cost function when, as is typically the case, only data on total cost are available. We do not use conventional methods for cost function estimation because they impose strong restrictions on the cost structure, in particular the separability of the cost shock in the cost function and the exogeneity of output. Instead, we propose a semiparametric sieve method which jointly 2

3 estimates a pseudo-cost function and the demand parameters. More specifically, the pseudo-cost function expresses cost as a function of output, input prices and marginal revenue. In doing so, we make the common assumption in the empirical IO literature that firms maximize profits, which implies that marginal revenue equals marginal cost in equilibrium. We exploit this property and use marginal revenue as a control function for the cost shock in estimating the cost function. This is similar to the approach of Olley and Pakes (1995) and Levinsohn and Petrin (2005), with the difference being that the control function is a function of variables in the data and the parameters. It turns out that marginal revenue works as a proper control function only when the demand parameter is at its true value; in that case, the pseudo-cost function has the best fit in terms of explaining the data. Therefore, we jointly estimate the control function and the pseudo-cost function, where the pseudo-cost function is non-parametrically specified as a sieve function of output, input price and marginal revenue. Hence, it is an application of the sieve estimator by Chen (2007) and Bierens (2013). Furthermore, we show how we can reconstruct the cost function from the estimated pseudo-cost function. The resulting cost function estimate is semi parametric in the sense that it uses the marginal revenue recovered based on the parametric BLP demand model, however no parameter restrictions are imposed on the cost function. Related literature Previous studies estimating statistical differentiated goods monopoly and oligopoly models have augmented their analyses with cost data. For instance, Nevo (2001) and Slade (2009) use aggregate, industry-level marginal cost estimates from the ready-to-eat cereal and beer industries in order to validate the marginal cost that is inferred from the supply-side first order conditions of the BLP model. Other papers have estimated cost using cost data for a single market. McManus (2007) uses product-level marginal cost data for a single market for coffee to evaluate whether firms engage in strategic non-linear pricing. Houde (2012) estimates gas stations marginal 3

4 cost functions using the BLP model s inferred marginal costs and wholesale gas prices for a single market. 1 Recent studies by Crawford and Yurukoglu (2012) and Byrne (2013) on the cable television directly incorporate cost data in estimating the BLP model s supplyside parameters across many markets. All previous applications of the BLP differentiated products framework that use cost data to estimate the model s cost parameters rely on the standard instrument-based approach for estimating demand. The cost functions are then estimated separately in a second step, after the demand parameters have been estimated. In contrast to this prior work, we develop an integrated methodology that uses similar data to what these authors used, but that does not require instruments neither on the demand nor supply side. To do so, we rely on the common assumption in the empirical IO literature that firms profit maximize. Furthermore, virtually all research on differentiated products markets to date imposes strong functional form assumptions on the supply-side model, namely constant marginal cost. As we show below using Monte-Carlo experiments, our methodology permits a fullyflexible cost function estimation that is subject only to some reasonable monotonicity conditions. In addition, we allow for measurement error in the cost function as well as omitted variables. The latter feature of our approach should be particularly attractive for researchers since it is often the case that only limited cost data are available. The only paper that we can find that exploits first order conditions to estimate demand parameters is Smith s (2004) study of multi-store competition and mergers in the UK supermarket industry. He estimates a demand model using consumer-level choice data for within-store products. He does not, however, have product-level price data. To overcome this missing data problem Smith uses data on national price-cost margins for firms in his 1 We would be remiss not to mention related papers on homogeneous products oligopoly markets that have also incorporated cost data in their analyses. These include Genesove and Mullin (1998), Wolfram (1999), Clay and Troesken (2003), and Kim and Knittel (2003). All of these authors validate the supplyside modelling assumptions with information on costs. In homogeneous products markets the cost data requirement is simplified substantially by the fact that all firms can reasonably be assumed to face the same marginal costs. 4

5 sample to identify the price coefficient in the demand model that rationalizes the observed mark-ups. Our study differs considerably in that we focus on the more common situation where a researcher has price data and aggregate market shares. Indeed, we directly build on the general BLP framework, whereas Smith develops a clever identification strategy that is quite unique to his empirical and industrial setting. This paper is organized as follows. In Section 2, we specify the differentiated goods demand model and highlight instrument-based identification challenges. In Section 3, we discuss identification of our model. In Section 4, we propose a semi parametric sieve estimator and analyze its asymptotic properties. In Section 5, we present Monte-Carlo results that illustrate the effectiveness of our estimator in environments where standard approaches to demand and cost estimation yield biased results. 2 Differentiated Products Models and their IV Estimation In this section we describe standard approaches to estimating the demand and cost parameters in differentiated products markets and discuss their pitfalls. We first analyze demand, and then supply. 2.1 Demand estimation Consider the following standard differentiated products demand model. Consumer i in market m gets the following utility from consuming product j: u ijm = x jmβ + αp jm + ξ jm + ɛ ijm, (1) where x j is a K 1 vector of observed product characteristics, p jm is price, ξ jm is the unobserved product quality (or demand) shock that both firms and consumers account for in making their decisions, and ɛ ijm term is an idiosyncratic taste shock. Denote the 5

6 demand parameter vector by θ = [β, α]. Suppose there are m = 1... M isolated markets that have respective market sizes Q m. 2 Each market has j = 0... J m products whose aggregate demand across individuals is q jm = s jm Q m, where q jm denotes output and s jm denotes the market share. In the case of the simple Berry (1994) logit demand model which assumes ɛ ijm has a logit distribution, the aggregate market share for product j in market m is s jm (θ) = s (X m, p m, ξ m, j; θ) = exp ( ) x jmβ + αp jm + ξ jm Jm j=0 exp ( x jm β + αp ) = jm + ξ jm exp (δ jm ) Jm j=0 exp (δ jm), (2) where p m = [p 0m, p 1m,..., p Jmm], a (J m + 1) 1 vector, X m = [x 0m, x 1m,..., x Jmm], a (J m + 1) K matrix, ξ m = [ξ 0m, ξ 1m,..., ξ Jmm], and δ jm = x jmβ + αp jm + ξ jm is the mean utility for product j. Following standard practice, we label good j = 0 the outside good resulting from not buying one of the j = 1,..., J m inside goods. To identify the level and scale of utility in the model, we normalize the outside good s product characteristics, price and demand shock to x 0m = 0, p 0m = 0, ξ 0m = 0 for all m. In the case of BLP, the demand structure is identical except that one allows the price coefficient to be a random variable. That is, consumer i has price coefficient α i, where α i has a distribution function F (.; θ α ), where θ α is the parameter of the distribution. 3 Conditional on α i, the probability consumer i purchases product j is identical to that provided by the market share formula in equation (2). The aggregate market share is 2 With panel data, the m index can correspond to a market-period. 3 We can also allow for random product characteristics coefficients for each of the k = 1... K product characteristics (e.g., allow for β ik s). The standard parametric assumption on each of these distributions is also i.i.d normal. 6

7 obtained by integrating over the distribution of α i, i.e. ˆ s jm (θ) = s (X m, p m, ξ m, j; θ) = α i ( ) exp x jm β + α i p jm + ξ jm Jm j=0 exp ( x jm β + α )df (.; θ α ). (3) ip jm + ξ jm Often the distribution of α i is assumed to be normal, implying that θ α consists of the first two moments of the distribution, the mean and the variance, θ α = [α, σ α ], and θ = [β, α, σ α ]. Estimation Given θ and data on market shares, prices and product characteristics, we can solve for the vector δ m = [δ 0m, δ 1m,..., δ Jmm] through market share inversion. This involves finding the value of δ m that solves s(δ m, θ) s m = 0, where s(δ m, θ) = (s (δ m (θ), 0; θ),..., s (δ m (θ), J m ; θ)) i.e. s (δ m (θ), j; θ) s jm = 0, for j = 0,..., J m, (4) and therefore, δ m (θ) = s 1 (s m ; θ), (5) where s m = (s 1m,..., s Jmm). That is, we find the vector of mean utilities that perfectly align the model s predicted market shares to those observed in the data. For example, under the simple logit model we can easily recover the demand shocks for product j using its market share and the share of the outside good, δ jm = log(s jm ) log(s 0m ). In the random coefficient case, there is no such closed form formula for market share inversion. Instead, BLP propose a contraction mapping algorithm that recovers the unique 7

8 δ m that solves (4) under some regularity conditions. Using the inferred values of δ jm for all products and markets, we can construct a GMM estimator for θ by assuming the following population moment conditions are satisfied at the true value of the demand parameters, θ 0 E[ξ jm (θ 0 )z jm ] = 0, where ξ jm (θ) = δ jm x jmβ αp jm and z jm is an L 1 vector of instruments. These typically include the market characteristics x jm, market-level demographic variables, and so on. They also include excluded instruments from the demand model that are used to correct for the endogeneity of p jm to ξ jm, which we discuss in a moment. For estimation, we can construct the sample analogue to the population moment conditions, 1 ξ jm (θ)z jm N jm j,m where N jm is the number of product-market observations. Using these sample moment conditions, it is straightforward to obtain the GMM estimator of θ, ˆθ. Identification An endogeneity problem naturally arises in differentiated product markets since firms tend to charge higher prices if their products have higher unobserved product quality. Researchers use a variety of excluded demand instruments to overcome this issue, though each type of instrument has its pitfalls. Cost shifters are often used as price instruments. This is inline with traditional market equilibrium analysis which identifies the demand curve from shifts in the supply curve caused by cost shifters. Popular examples are input prices, w jm. However, one cannot rule out the possibility that the exclusion restriction of cost shifters in the demand function does not hold. Input prices, like wages, may also affect the demand of the product in 8

9 the same local market through changes in consumer income. Changes in other input prices such as gasoline or electricity could reasonably be expected to affect both firms and consumers choices. Even further, higher input prices may induce firms to reduce product quality. In instances where cost shifters are likely to satisfy exclusion restrictions, they are often weak instruments. For example, if one assumes input prices are exogenously determined in some external market (such as the labor market), then all firms will face the same input prices. Therefore, cost shifters may not have sufficient within-market variation across firms to identify the demand parameters. 4 In the absence of cost shifters, researchers often use product characteristics of rivals products or market structure characteristics like the number of firms as price instruments. One naturally would worry, however, just like prices, these variables are endogenous with respect to unobserved demand shocks. Indeed, Crawford and Yurukoglu (2012), Fan (2012), Byrne (2013), and others have documented that product characteristics, like prices, are strategic choices made by firms that depend on demand shocks. A final commonly-used set of instruments are the prices of product j in markets other than m (Nevo, 2001; Hausman, 1996). The identifying assumption is that the demand shock for product j in market m is uncorrelated with product j s prices in other markets. The strength of these instruments come from common cost shocks for product j across markets that creates cross-market correlation in product j s prices. These instruments are invalid, however, if there is spatial correlation in demand shocks across markets. Regional demand shocks, for example, could generate such correlation. 5 Unfortunately, there is little evidence of whether such cross-market correlation in demand shocks is problematic for identifying demand parameters in practice. 4 These concerns are similar to those from the productivity literature in empirical IO. See, for example, Olley and Pakes (1996). 5 Firm, market, and year fixed effects are typically included in the set of instruments when panel data are available. So the exclusion restriction fails if the innovation in the demand shock in period t for product j is correlated across markets. 9

10 2.2 Supply and cost function estimation Supply The cost of producing q jm units of product j is assumed to be a strictly convex function of output, input price w jm, and a cost shock ω jm. That is, C jm = C (q jm, w jm, ω jm ; τ ), (6) where C jm is the total cost of producing product j in market m, C( ) is a positive, strictly increasing, convex and continuously differentiable function, and τ is a cost parameter vector. Given this cost function and the demand model above, we can write firm j s profit function as π jm = p jm s (X m, p m, ξ m, j; θ) Q m C (s (X m, p m, ξ m, j; θ) Q m, w jm, ω jm ; τ ), (7) Keeping with the standard differentiated products model, we assume that firms act as differentiated Bertrand price competitors. Therefore, the optimal price and quantity of product j in market m is determined by the first order condition (F.O.C.) that equates marginal revenue and marginal cost 6 ] 1 [ s (Xm, p m, ξ p jm + s m, j; θ) jm p }{{ jm } MR jm = C (q jm, w jm, ω jm ; τ ) q jm } {{ } MC jm, (8) where p jm is a vector that contains the prices of all other firms in market m other than the firm j. Notice that the marginal revenue function can be written as MR jm = MR (X m, p m, ξ m, j; θ). It follows from the inversion in (4) and the definition of ξ jm (θ) = δ jm x jmβ αp jm, 6 Here, we are assuming there is one firm for each product. We will relax this later. 10

11 ξ m is fundamentally a function of X m, p m, s m and θ. Therefore, marginal revenue is also a function of those variables. That is, MR jm = MR(X m, p m, s m, j, θ) = MR jm (θ), where MR jm (θ) is shorthand for MR(X m, p m, s m, j, θ). Below, we will extensively use the first order condition (8), and two important aspects of MR jm : it is only a function of data and parameters (and not unobservables), and it depends on market shares and not quantities. Cost function estimation Like with demand estimation, there are important endogeneity concerns with standard approaches to cost function estimation. Suppose, for example, that we were to estimate the following log-linearized version of the cost function in equation (6) by OLS: log(c jm ) = τ 1 q jm + τ 2 w jm + ω jm. In doing so, we would face an endogeneity issue that output q jm would potentially be negatively correlated with the cost shock ω jm. Traditionally, researchers have either ignored this problem or tried to find instruments for quantities. Similar to the inversion procedure in demand, the unobserved cost shock satisfies: C jm = C (q jm, w jm, ω jm ; τ ) ω jm (τ ) = C 1 (q jm, w jm, C jm ; τ ). (9) So in principle, one can estimate the cost function parameters by using the excluded demand shifters as instruments for quantities. Denote the vector of cost instruments by z jm. We can estimate τ assuming that the following population moments are satisfied at 11

12 the true value of the cost parameters τ 0 : E [ω jm (τ 0 ) z jm ] = 0. The difficulty of this approach is that typically valid instruments such as demand shifters (e.g., market demographics) affect all firms. This implies they often do not generate sufficient within-market across-firm variation in equilibrium prices to identify the cost parameters. A further difficulty with identification arises from the fact that one often wants to use a higher order polynomial to flexibly estimate the cost function. This implies, however, that potentially implausibly many excluded demand shifters are required to estimate all the parameters in the cost function. 3 Instrument-free identification and estimation Instead of using instruments, we estimate the demand and cost parameters directly from the first order conditions. That is, we exploit restrictions implied by the oligopolistic firms first order condition in (8) to identify the demand and cost function parameters. In the above Berry (1994) or BLP specification, marginal revenue can be written in the usual form [ ] 1 s (Xm, p m, ξ MR jm (θ) = p jm + s m, j; θ) jm p jm Notice that marginal revenue is a function of price and market share, which both do not appear in the marginal cost function from equation (8). We also use this for identification. In particular, we use observations that have different market shares in different markets (e.g. s jm s j m ), but the same levels of output q jm = s jm Q m = q j m = s j m Q m. It is important to note that at no point do we explicitly or implicit use market size Q m as an instrument. We allow for the market size variable to be correlated with demand or cost shocks, but we do not allow it to be a deterministic function of such shocks. It must have variation that is independent to the variation in these shocks. In other words, for the 12

13 same level of output, differences in market size imply differences in market share. These differences can be used to separately identify the demand and cost function parameters. The idea that data on total costs can be used to help identify demand parameters is not immediately obvious, and perhaps even counterintuitive. Before developing our general identification results, we first make this idea more concrete through an example based on the simple Berry (1994) logit model. Suppose that < α < 0, and consider a pair of firms (Q m, w jm, s jm, p jm ) and (Q m, w j m, s j m, p j m ) whose demand shocks are ξ jm, ξ j m. Under the logit specification we have s jm = exp (αp jm + ξ jm ) J j=0 exp (αp jm + ξ jm ), s j m = exp (αp j m + ξ j m ) J j=0 exp (αp jm + ξ jm ). We maintain that C (.) is positive, strictly increasing, convex and continuously differentiable. Going forward, we drop τ since we will treat C( ) as non-parametric. We further assume that the solution of the F.O.C. for profit maximization is at the global optimum. Suppose this pair of observations satisfies Q m Q m, w jm = w j m, q jm = s jm Q m = q j m = s j m Q m and C jm = C j m.7 For such a pair we claim that ω jm = ω j m. Suppose this is not the case, for instance ω jm > ω j m. Then, from the strict monotonicity of the cost function in terms of the cost shock ω C (q jm, w jm, ω jm ) = C (q j m, w j m, ω jm) > C (q j m, w j m, ω j m ), contradicting C jm = C j m. Similarly for ω jm < ω j m. Therefore, ω jm = ω j m. As a result, the marginal cost of the two observations is the same. That is, MC (s jm Q m, w jm, ω jm ) = MC (s j m Q m, w j m, ω j m ). 7 We have not included the product characteristics here. Recall that if x jm is uncorrelated with ξ jm, then we can use the E[ξ jm (θ 0 )x jm ] = 0 moment conditions to identify β. 13

14 Therefore, for those two data points, because marginal revenue equals marginal cost, their marginal revenues must be the same. For instance, in the case of the logit model, p jm + 1 (1 s jm ) α = p j m + 1 (1 s j m ) α. Notice that since Q m Q m, s jm s j m and thus, for bounded negative α, p jm p j m. It then follows that α is identified from such pair of data points from 1 α = p jm p j m [ 1 1 s jm 1 1 s j m ]. (10) Of course, in practice one is unlikely to find a pair that exactly satisfies Q m Q m, w jm = w j m, q jm = s jm Q m = q j m = s j m Q m and C jm = C j m However, a similar argument can be made for pairs that satisfy the above equality relationship approximately, and such pairs can be found even if market size is correlated with the demand and cost shocks. The above example highlights the importance of the variation of market size Q m for identification. If all the data came from a single market, then q jm = q j m implies s jm = s j m, and thus α could not be identified from (10). Two issues are likely to arise in practice with this estimation strategy. Firstly, suppose there exists two pairs of firms where Q m Q m, w jm = w j m, q jm = s jm Q m = q j m = s j m Q m and C jm = C j m for both pairs but yet provide two very different estimates of α. This would immediately lead a practitioner to conclude that the model is misspecified since, if the model is correct, it is impossible to have two such pairs of markets that deliver quite different α estimates. This issue arises because the specification of the model is too strong. It allows for no discrepancies in the estimate of α across market pairs, except for that which is due to the within-pair distance of the variables. Secondly, it is widely accepted that cost data are measured with error. 8 If measurement error is allowed, then one needs to have appropriate instruments to identify the model 8 For a discussion of this issue see, for example, Wang (2003) 14

15 even if the endogeneity of output is assumed away. To handle both issues, we explicitly introduce an additive measurement error in the cost function. To identify the demand parameters, we impose the restriction that the measurement error is mean zero i.i.d. across products, markets, and time, and is independent from q jm, w jm, s jm, p jm, the demand shock ξ jm and the cost shock ω jm. Under these assumptions, the measurement error is orthogonal to the true cost Cjm, and the observed cost is C jm = Cjm + η jm. (11) 3.1 Pseudo cost function Definition 1 A pseudo-cost function is defined to be P C(q jm, w jm, MC jm ), where recall that MC jm denotes marginal cost for product j in market m. Next, we state and prove a lemma that relates the cost function to the pseudo-cost function. The lemma shows that given output and input prices, marginal cost, if observable, can be used as a proxy for the cost shock. Because we assume profit maximization, marginal revenue equals marginal cost, and thus, when the parameters of the demand functions are at their true values, marginal revenue is observable from the demand data and can be used as a proxy for the cost shock. Before proving the lemma, we formally state two key assumptions of the supply-side model and marginal cost function: Assumption 1 Within each market, each firm takes as given other firms strategies as given, and chooses prices to equate marginal revenue and cost. It is also assumed that the global optimum for profit maximization is attained at this point. Assumption 2 The marginal cost function is strictly increasing and continuous in ω. 15

16 Lemma 1 Suppose that Assumptions 1, 2 are satisfied. Then, C (q jm, w jm, ω jm ) = P C (q jm, w jm, MR jm (θ 0 )), and the pseudo-cost function is increasing in marginal revenue. Proof. First, we show that C (q jm, w jm, ω jm ) = P C (q jm, w jm, MC jm ). Note that because MC is an increasing function of ω jm given q jm and w jm, there exists an inverse function on the domain of MC (q jm, w jm, ω jm ) such that ω jm = ω (q jm, w jm, MC jm ). This implies that we can use (an unspecified function of) q jm, w jm and MC jm, ω (q jm, w jm, MC jm ), to control for ω jm. Substituting this control function for ω jm into the cost function, we obtain the pseudo-cost function from Definition 1 C (q jm, w jm, ω jm ) = P C (q jm, w jm, MC jm ). From the F.O.C. we know that marginal revenue must equal marginal cost when the demand parameters are at their true values, θ 0. We can therefore substitute MR jm (θ 0 ) in for MC jm in the pseudo-cost function C (q jm, w jm, ω jm ) = P C (q jm, w jm, MC jm ) = P C (q jm, w jm, MR jm (θ 0 )). Finally, because ω jm is an increasing function of MC jm given q jm, w jm, and because MR jm (θ 0 ) = MC jm under the profit maximization assumption at θ 0, P C is also an increasing function of M R. 16

17 This lemma allows us to use the pseudo-cost function instead of the cost function in estimation. The advantage in doing so is that the former is strictly a function of data and parameters, whereas the latter depends on the unobservable cost shock ω. 3.2 Proposed estimator We now present our estimator and prove identification. It chooses demand and cost parameters to fit the pseudo-cost function to the cost data using a non-parametric sieve regression estimator (Chen, 2007; Bierens, 2013). Suppose (q jm, w jm, MR jm ) comes from a compact finite dimensional Euclidean space, W. Then if P C (q jm, w jm, MC jm ) is a continuous function over W, from the Weierstrass Theorem it follows that the function can be approximated arbitrarily well by an infinite sequence of polynomials. That is, P C (q jm, w jm, MR jm (θ 0 )) = γ l ψ l (q jm, w jm, MR jm (θ 0 )) (q jm, w jm, MR jm (θ 0 )) W, l=1 (12) where γ 1,..., γ is a set of sieve coefficients, and ψ 1 ( ),..., ψ ( ) are the basis functions for the sieve. Polynomials and splines are often used as basis functions in practice. Our estimator is derived from the approximation in (12). It is useful to introduce some additional notation before formally defining it. Let M be the number of markets in our sample. Define the sieve space for a set of L M basis functions as { H M = h : h = l L M γ l ψ l (q jm, w jm, MR jm (θ)), h H < B M, (q jm, w jm, MR jm ) W for some bounded constant B M > 0 with L M such that H M H M+1... H, and where the pseudo-norm, h H, is defined as }, h H = E H h (q jm, w jm, MR jm ). 17

18 The subscript in L M makes explicit the fact that the complexity of the sieve is increasing in the sample s number of markets. The corresponding space of sieve coefficients is similarly defined as { G M = γ : } γ l ψ l (q jm, w jm, MR jm ) H M. l L M At the true values of θ and γ = [γ 1,..., γ LM ], the sieve approximation error is minimized. That is 9 [θ0, γ0] = arg min (θ,γ) Θ G E [C jm h (q jm, w jm, MR jm (θ))] 2, (13) where Θ is the demand parameter space. Our estimator solves the sample analogue to (13), given a sample of M markets [ˆθM, ˆγ M ] = arg min (θ,γ) Θ G 1 M [C jm h (q jm, w jm, MR jm (θ))] 2. (14) j,m Given θ, the choice of the compact set (q jm, w jm, MR jm (θ)) could be somewhat arbitrary. However, we can construct the compact set as follows. Let W be the compact set of variables q jm, w jm, MR jm. We can make the lower bound of q jm, q arbitrarily close to 0, and the upper bound q arbitrarily large. Similarly for the lower and upper bound of w jm, w and w. Then, given (q jm, w jm ) in the compact set, for any MR jm [ MR, MR ], there exists a cost shock that MR jm = MC (q jm, w jm, ω jm ). Furthermore, for any MR jm [ MR, MR ], there exists ξ m such that MR jm (θ) = MR (X m, p m, s m, ξ m ; θ) Identification We now prove identification of the estimator. First, we state the assumption that the demand function identifies the marginal revenue in the population space. Assumption 3 The marginal revenue identifies the demand function parameter in the 9 This is equivalent to stating that the sieve approximation error is minimized at the true values of the demand parameters and sieve basis functions, [θ 0, h 0 ] = arg min E [C jm h (q jm, w jm, MR jm (θ))] 2. (θ,h) Θ H 18

19 compact set W. That is, if θ θ 0, then there exists two measurable sets C, C C = C = ( ) {(q jm, w jm, X m, p m, s m ) : (q jm, w jm ) Ã, Xm, p m, s m B, } j, m ( ) } {(q jm, w jm, X m, p m, s m ) : (q jm, w jm ) Ã, Xm, p m, s m B, j, m, where Ã, B and B are all measurable sets. Furthermore, 1. B B = ; ( ) ( ) 2. P rob C > 0, P rob C > 0; 3. ( q jm, w jm, MR ) ) jm (θ) W, (q jm, w jm, MR (θ) W, ( q jm, w jm, MR ) ( ) jm (θ 0 ) W, q jm, w jm, MRjm (θ 0 ) W. 4. For any (q jm, w jm, X ) m, p m, s m C, ) there exists (q jm, w jm, Xm, p m, s m C such that MR jm (θ) = MRjm (θ) and MR jm (θ 0 ) MRjm (θ 0 ) and vice versa. 10 Assumption 4 Measurement error η jm is i.i.d(0, ση), 2 ση 2 <, and is independent from (q jm, w jm, X m, p m, s m ) j, m. Proposition 1 Suppose Assumptions 1-4 hold. Then, the proposed sieve-pseudo-cost estimator in equation (14) identifies θ 0. Proof. Denote in shorthand MR jm (θ) to be MR (X jm, p jm, s jm ; θ). Recall from above that for each firm the observed cost is C jm = C jm + η jm 10 This assumption on the non-linearity of the marginal revenue function from the demand system. 19

20 for firm/product j in market m, and η jm is the measurement error. Denote the sieve function of q jm, w jm and MR jm as ψ (q jm, w jm, MR jm (θ), γ) = γ l ψ l (q jm, w jm, MR jm (θ)). l=1 Then, E[(C jm ψ(q jm, w jm, MR jm (θ), γ)) 2 q jm = q, w jm = w, MR jm (θ) = MR] = E[(Cjm ψ(q jm, w jm, MR jm (θ), γ)) 2 q jm = q, w jm = w, MR jm (θ) = MR] +2E[(Cjm ψ(q jm, w jm, MR jm (θ), γ))η q jm = q, w jm = w, MR jm (θ) = MR] +E[η 2 q jm = q, w jm = w, MR jm (θ) = MR] = E[(Cjm ψ(q jm, w jm, MR jm (θ), γ)) 2 q jm = q, w jm = w, MR jm (θ) = MR] + ση. 2 Now, from Lemma 1 and the Weierstrass Theorem, there exists an infinite sequence γ 0 = {γ 0l } l=1 such that C jm = P C (q jm, w jm, MR jm (θ 0 )) = ψ(q jm, w jm, MR jm (θ 0 ), γ 0 ) = γ 0l ψ l (q jm, w jm, MR jm (θ 0 )) l=1 for the compact space (q jm, w jm, MR (θ 0 )) W. Therefore, ( E C jm = 0 + ση. 2 2 γ 0l ψ l (q jm, w jm, MR jm (θ 0 ))) q jm = q, w jm = w, MR jm (θ 0 ) = MR l=1 Let ν = ( q jm, w jm, vec( X (. m ), p m, s m) and ν = q jm, w jm, vec( Xm ), p m, s m) Without loss of generality, we omit the market subscript m. From the assumption on the identification of the marginal revenue, if θ θ 0, ν C, and its corresponding ν C such 20

21 that ( ) ( ) MR jm (θ) MR Xm, p m, s m, j, θ = MR Xm, p m, s m, j, θ MRjm (θ) = MR, MR jm (θ 0 ) MRjm (θ 0 ). Hence, for any γ G, l=1 γ l ψ l (q jm, w jm, MR ) jm (θ) = l=1 ) γ l ψ l (q jm, w jm, MRjm (θ) but because the true pseudo-cost function is strictly increasing in MR and MR jm (θ 0 ) MR jm (θ 0 ), l=1 Therefore, either or l=1 l=1 γ 0l ψ l (q jm, w jm, MR ) jm (θ 0 ) γ 0l ψ l (q jm, w jm, MR ) jm (θ 0 ) ) γ 0l ψ l (q jm, w jm, MRjm (θ 0 ) l=1 l=1 l=1 ) γ 0l ψ l (q jm, w jm, MR (θ0 ). γ l ψ l (q jm, w jm, MR ) jm (θ) ) γ l ψ l (q jm, w jm, MRjm (θ) or both. Now, consider the former case. Then, because of continuity of l=1 γ 0lψ l and l=1 γ lψ l, there exists a set B (q jm, w jm ) measurable with respect to the conditional ( ) probability P (. q jm, w jm ) and P B (qjm, w jm ) q jm, w jm > 0, where ν B (q jm, w jm ) and for any element b B (q jm, w jm ) l=1 γ 0l ψ l ( q jm, w jm, MR jm ( b, θ0 )) l=1 ) γ l ψ l (q jm, w jm, MR jm ( b, θ) 21

22 Similarly for latter case. That is, there exists an open set B (qjm, w jm ) measurable with ( ) respect to the conditional probability P (. q jm, w jm ), and P B (qjm, w jm ) q jm, w jm > 0, where ν B (qjm, w jm ) and for any element b B (qjm, w jm ) l=1 This implies that γ 0l ψ l ( q jm, w jm, MR jm ( b, θ0 )) E [( l=1 l=1 l=1 γ 0l ψ l (q jm, w jm, MR ) jm (θ 0 ) γ l ψ l (q jm, w jm, MR ) ) 2 jm (θ) γ l ψ l ( q jm, w jm, MR jm ( b, θ) ) B (q jm, w jm ) > 0, and E [( l=1 ) γ 0l ψ l (q jm, w jm, MRjm (θ 0 ) ) γ l ψ l (q ) 2 jm, w jm, MRjm (θ) B (qjm, w jm ) > 0, l=1 Therefore, integrating over q, w and MR, we obtain that for any γ which makes P C( ) to be strictly increasing or strictly decreasing in MR, for θ θ 0, 22

23 ( ) 2 E Cjm γ l ψ l (q jm, w jm, MR jm (θ)) l=1 ( = E γ 0l ψ l (q jm, w jm, MR jm (θ 0 )) l=1 [ E (qjm,w jm ) E ) 2 γ l ψ l (q jm, w jm, MR jm (θ)) l=1 [( γ 0l ψ l (q jm, w jm, MR jm (θ 0 )) l=1 2 γ l ψ l (q jm, w jm, MR jm (θ))) B ( ) ( ) (q jm, w jm ) dp rob B (qjm, w jm ) I B (qjm, w jm ) l=1 ( +E (qjm,w jm ) E γ 0l ψ l (q jm, w jm, MR jm (θ 0 )) l=1 ( ) ( dp rob B (qjm, w jm ) I B (qjm, w jm ), B )] (q jm, w jm ) = > 0 2 γ l ψ l (q jm, w jm, MR jm (θ))) B (qjm, w jm ) l=1 Therefore, E [ (C jm ψ (q jm, w jm, MR jm (θ), γ)) 2] σ 2 η, with equality only holding for θ = θ 0, ψ (q jm, w jm, MR jm (θ 0 )) = P C (q jm, w jm, MR jm (θ 0 )). Surprisingly, it turns out that in fitting the pseudo-cost function to the cost data, we identify the demand parameters. In Section 3.3, it is demonstrated that given a nonparametric cost function, the marginal revenue function can be identified nonparametrically. It would be natural to estimate the demand by exploiting the nonparametric identification directly. Actually, we advocate sieve nonlinear least squares ( NLLS ) regression estimation, which relies on the functional form of the demand. Why do we resort to this strategy? There are a few rationales. First, fully nonparametric estimation of the marginal revenue function is infeasible because of the number of regressors that must be included. In short, due to the curse of dimensionality, we are 23

24 obliged to maintain parametric specification on the demand side. Second, sieve NLLS estimation using the pseudo-cost function is preferred, because it is built on the assumption that the first order condition MR = MC holds exactly for each firm. A more straightforward usage of the pairwise identification argument is to construct the following pairwise differenced estimator. θ JM = argmin θ Θ j,m i,m :(j,m ) (j,m) [ (C d jm C d j m ) 2 W h ( q d jm q d j m, wd jm w d j m, MR jm(θ) MR j m (θ))] where W h ( q d jm q d j m, wd jm w d j m, MR jm(θ) MR j m (θ)) ( ) ( ) K hq q d jm qj d m Khw w d jm wj d m KhMR (MR jm (θ) MR j m (θ)) k,n k,n :(k,n ) (k,n) K ( ) ( ) h q q d kn qk d n Khw w d kn wk d n KhMR (MR kn (θ) MR k n (θ)) By construction, the first order condition is not assumed to hold exactly for any firm. To put it differently, each estimator loses efficiency because the equilibrium condition is not imposed explicitly. The sieve NLLS estimator can be viewed as a constrained estimator with the constraint MR jm = MC jm for each (j, m), which is thought to attain efficiency gain. As we will see later, the sieve NLLS estimator is more flexible in dealing with the data issue such as multi-branch firm, etc. than the pairwise differenced estimator. In the above approach, we are using a control function approach, where we estimate the pseudo-cost function, using the marginal revenue as a control function for cost shock, only when the demand parameters are at the true value. Then, it turns out that the pseudo-cost function has the best fit to the data. In the above approach, we can consider the issues of endogeneity in a similar way to the issue of endogeneity in cost function estimation. Recall that in estimating cost functions, an endogeneity issue arises because output q jm is correlated with the cost shock ω jm. 24

25 There are two potential ways to deal with this endogeneity problem: find an instrument that is orthogonal to ω jm, or a control function approach that finds a variable that is a function of ω jm. With our estimator, the right hand side of (14) is minimized only when the demand parameters are at their true value θ 0 so that the computed marginal revenue equals the true marginal revenue, i.e., the marginal cost, and thus works as a control function for the supply shock ω. If θ θ 0, then using the false marginal revenue adds noise, which increases the right hand size of the sum of squared residual in (14). In that sense, we are adopting a pseudo-control function approach to avoid using instruments. As the above makes clear, a by-product of this approach is we can also obtain the true demand parameter θ Identification of marginal revenue It is important to note that Assumption 3 is a high level assumption; it is not necessarily satisfied in all demand models. For example, if the marginal revenue is a linear function ( ) of θ, then for any positive constant a > 0 if we set θ = aθ 0, then MR Xm, p m, s m, θ = ( ) MR Xm, p m, s m, θ implies ( ) ( ) MR Xm, p m, s m, θ = amr Xm, p m, s m, θ 0 = MRjm (θ) = a MRjm (θ 0 ). Hence, if the marginal revenue is a linear function of θ then Assumption 3 is violated. The remaining issue is whether standard differentiated products demand models satisfy Assumption 3. Next, we prove that both logit model and the BLP model of demand satisfy Assumption 3. Lemma 2 Assumption 3 is satisfied for the logit model and the BLP model of demand. Proof. 25

26 It is easy to show that the Berry (1994) logit demand model from (2) satisfies Assumption 3. Suppose that for p jm, p j m and s jm, s j m α α 0 p jm + [ 1 (1 s jm ) α = p j m + 1 (1 s j m ) α α = 1 1 p jm p j m 1 s jm 1 1 s j m ]. Then, for α α 0, 1 α 0 p jm p j m [ 1 1 s jm 1 1 s j m ], and p jm + 1 p j (1 s jm ) α m (1 s j m ) α. 0 Therefore, the price coefficient satisfies Assumption 3. Thus, the price coefficient is identified. Recall that the above analysis abstracted from having product characteristics, x jm. Without further restrictions β is not identified: once α = α 0 is identified, then A jm = x jmβ+ξ jm is identified, as we can see from equation (4) and (5), but we cannot separately identify x jmβ and ξ jm. The additional restriction that identifies β 0 could be the following orthogonality condition, ξ jm = log (s jm ) log (s 0m ) x jmβ α 0 p jm ; E [ξ jm x jm ] = 0 By assuming this orthogonality condition we are following standard practice in the literature (Berry (1994), BLP, and most applications assume the exogeneity of product characteristics). Like with all differentiated products markets studies, the validity of this assumption will depend on the industry context and time horizon of a given study. Next, we prove that the random coefficient BLP model also satisfies Assumption 3. First, we consider the monopoly case. First, denote As before, consider the pair (s, p, A) and (s, p, A ) that satisfy the share equation is 26

27 ˆ ( ) ˆ ( ) exp (A + pα) 1 α µα exp (p (α + A/p)) 1 α µα φ dα = φ dα = s [1 + exp (A + pα)] σ α σ α α [1 + exp (p (α + A/p))] σ α σ α α ˆ α ( ) exp (p (α + A /p )) 1 α µα φ dα = s [1 + exp (p (α + A /p ))] σ α σ α and [ˆ MR = p + [ˆ = p + α α p exp [p (α + A/p)] [1 + exp [p (α + A/p)]] 2 α 1 φ σ α p exp [p (α + A /p )] [1 + exp [p (α + A /p )]] 2 α 1 σ α φ ( α µα σ α ( α µα σ α ) dα] 1 s ) ] 1 dα s Now, denote η = µ α /σ α, a = A/p, a = A /p, and ã = a/σ α, ã = a /σ α, and α = α/σ α, α = α /σ α,. Now, we let ˆ ˆ α α p exp [p (α + A/p)] [1 + exp [p (α + A/p)]] 2 α σ α φ ( α µα σ α ( exp (p (α + ãσ α )) 1 α µα φ 1 + exp (p (α + ãσ α )) σ α. For large p, p, σ α ) dα = ) ˆ dα = α ˆ α ( ) p exp [p (α + ãσ α )] α α [1 + exp [p (α + ãσ α )]] 2 φ η dα σ α σ α B (ã, p, η, σ α ) exp (pσ α ( α + ã)) 1 + exp (pσ α ( α + ã)) φ ( α η) d α A (ã, p, η, σ α). ˆ s = A (ã, p, η, σ α ) = I ( α ã) φ ( α η) d α+o ( p 1) = 1 Φ ( ã η)+o ( p 1) = Φ (ã + η)+o ( p 1) α ˆ s = A (ã, p, η, σ α ) = I ( α ã ) φ ( α η) d α + O ( p 1) = 1 Φ ( ã η) + O ( p 1) α = Φ (ã + η) + O ( p 1) and B (ã, p, η, σ α ) = ãφ ( ã η) + O ( p 1) = ãφ (ã + η) + O ( p 1) B (ã, p, η, σ α ) = ã φ ( ã η) + O ( p 1) = ã φ (ã + η) + O ( p 1) 27

28 MR = p [1 + [ ãφ (ã + η) + O ( p 1)] ] [ 1 s = p 1 + [ ã φ (ã + η) + O ( p 1)] ] 1 s Let η = µ α /σ α, η 0 = µ α0 /σ α0. Suppose that η η 0 > γ > 0 for some small γ > 0, and consider small δ > 0. Now, let s = Φ ( δ + η 0 ). Then, for (µ α, σ α ), ã = Φ 1 ( s + O ( p 1)) η = δ + η 0 η + O ( p 1). Furthermore, let s = Φ ( C + η 0 ). where for positive constant C > 0. Then, ã = Φ 1 ( s + O ( p 1)) η = C + η 0 η + O ( p 1). Then, p, p can be chosen such that p [1 + [ ãφ (ã + η) + O ( p 1)] ] [ 1 s = p 1 + [ ã φ (ã + η) + O ( p 1)] ] 1 s That is, [ ] p 1 + [[δ η 0 + η + O (p 1 )] φ ( δ + η 0 + O (p 1 )) + O (p 1 )] 1 s [ ] = p 1 + [[C η 0 + η + O (p 1 )] φ ( C + η 0 + O (p 1 )) + O (p 1 )] 1 s. for sufficiently large p and p, the bracketed terms on the LHS and RHS are constants, except for small O (p 1 ), O (p 1 ). Thus, given η η 0 > γ > 0, and for sufficiently small δ > 0, for sufficiently large p and p, B and B are bounded away from zero, and positive. Thus, both 1 + B 1 s and 1 + B 1 s are positive and bounded. Hence, given large p, there exists large p such that LHS > RHS and also p such that LHS < RHS. Because RHS is a continuous function of p, by intermediate value theorem, the above equality holds for some large p. Furthermore, because both 1 + B 1 s and 1 + B 1 s are positive and bounded for arbitrary small δ > 0, p/p is bounded and bounded away from zero as well. 28

29 Now, we change µ α to µ α0 and σ α to σ α0. Then, ã 0 = Φ 1 ( s + O ( p 1)) η 0 = δ + O ( p 1). ã 0 = Φ 1 ( s + O ( p 1)) η 0 = C + O ( p 1). Then, LHS = p [1 + [( δ + O ( p 1)) φ ( δ + η 0 + O ( p 1)) + O ( p 1)] ] 1 s [ RHS = p 1 + [( C + O ( p 1)) φ ( C + η 0 + O ( p 1)) + O ( p 1)] ] 1 s where [(δ + O (p 1 )) φ ( δ + η 0 + O (p 1 )) + O (p 1 )] 1 s can be made arbitrarily large by making δ > 0 small enough. On the other hand, as before, [(C + O (p 1 )) φ ( C + η 0 + O (p 1 )) + O (p 1 )] 1 s is still bounded and bounded. Thus, for the equality to hold, p/p = O (δ) has to hold, which is not the case. Next, consider the case: η 0 η > γ > 0 for some γ > 0, Let s = Φ ( δ + η) and s = Φ ( C + η). Then, ã = Φ 1 ( s + O ( p 1)) η = δ + O ( p 1). ã = Φ 1 ( s + O ( p 1)) η = C + O ( p 1). Thus, MR = p [1 + [[ δ + O ( p 1)] φ ( δ + η + O ( p 1)) + O ( p 1)] ] 1 s [ = p 1 + [[ C + O ( p 1)] φ ( C + η + O ( p 1)) + O ( p 1)] ] 1 s 29

30 Then, for the equality to hold, p/p = O (δ) has to hold. Now, we change µ α to µ α0 and σ α to σ α0. Then, ã = Φ 1 ( s + O ( p 1)) η 0 = δ + η η 0 + O ( p 1). ã = Φ 1 ( s + O ( p 1)) η 0 = C + η η 0 + O ( p 1), and MR = p [1 + [[ δ + η η 0 + O ( p 1)] φ ( δ + η + O ( p 1)) + O ( p 1)] ] 1 s [ = p 1 + [[ C + η η 0 + O ( p 1)] φ ( C + η + O ( p 1)) + O ( p 1)] ] 1 s. Then, because δ + η η 0 + O (p 1 ) is now bounded with positive γ > 0 and with positive and small δ > 0, p/p = O (δ) would not make RHS and LHS to be equal. Next, consider the case where that η = η 0. Then, recall that ˆ A (ã, p, η, σ α ) = α exp (pσ α ( α + ã)) φ ( α η) d α. 1 + exp (pσ α ( α + ã)) Then, ˆ A (ã, p, η, σ α ) = σ α α exp (pσ α ( α + ã)) 2 p ( α + ã) φ ( α η) d α [1 + exp (pσ α ( α + ã))] Furthermore, ˆ B (a, p, η, σ α ) = α = 0 if ã = η < 0 if ã < η > 0 if ã > η ( ) p exp [p (α + ãσ α )] α α [1 + exp [p (α + ãσ α )]] 2 φ η dα σ α σ α 30

31 Now, consider a change from σ α to σ α0. Forα = α + ãσ α dσ ˆ ( ) p exp [p (α + ãσ α (1 + dσ))] α α α [1 + exp [p (α + ãσ α (1 + dσ))]] 2 φ η dα σ α σ α ˆ ( ) ( ) p exp [p (α + ãσ α )] α α = α [1 + exp [p (α + ãσ α )]] 2 ãdσ φ ãdσ η dα σ α σ α ˆ ( ) p exp [p (α + ãσ α )] α = B (ã, p, ησ α ) ã dα dσ ˆ + α α [1 + exp [p (α + ãσ α )]] 2 φ p exp [p (α + ãσ α )] ã [1 + exp [p (α + ãσ α )]] 2 α (α ησ σα 3 α ) φ B (ã, p, η, σ α ) (ã + η) +ã2 (ã + η) φ (ã + η) = ãφ +O ( p 1) = σ α σ α σ α ˆ ) B (a, p, η, σ α ) = Thus, α p exp [p (α + a)] α [1 + exp [p (α + a)]] 2 φ σ α ( α σ α η η σ α ( ) α η dα dσ σ α ãφ (ã + η) σ α ( 1 + ã (ã + η))+o ( p 1) dα = ãφ (ã + η) + O ( p 1) 1 + B (a, p, σ α ) 1 A (a, p, σ α ) = 1 + [ ãφ (ã + η) + O ( p 1)] 1 Φ (ã + η) Now, For large p, ˆ A (ã, p, σ α ) = σ α α pσ α exp [pσ α ( α + ã)] α + ã [1 + exp [pσ α ( α + ã)]] 2 φ ( α η) d α σ α A (ã, p, σ α ) σ α = O ( (pσ α ) 1) Therefore, change from σ α to σ α0 only results in small change in ã,i.e. ã 0 = ã + O ( (pσ α ) 1). B (ã, p, η, σ α ) (ã + η) +ã2 (ã + η) φ (ã + η) = ãφ +O ( p 1) = σ α σ α σ α ãφ (ã + η) σ α ( 1 + ã (ã + η))+o ( p 1) Then, for let ã = η/2 η 2 /4 + 1 < 0. B (ã, p, σ α ) σ α = 0 31

32 . Then, for sufficiently large p, 1 + B (ã 0, p, σ α0 ) 1 s = 1 + B (ã, p, σ α ) 1 s Similarly, let ã = C for C < η, and let s = A ( C, p, η, σ α ). Then, for large p, A (ã, p, η, σ α ) σ α = O (p ). Therefore, ã 0 = ã + O (p 1 ). Then, as before, choose ã such that Φ (ã + η) ã φ (ã + η) = C 1. for some positive C > 0. B (ã, p, η, σ α ) = ã φ (ã + η) ( 1 + ã (ã + η)) + O ( p 1) σ α σ α Then, if we set ã 0, ã η 2 ± η 2 B (ã, p, σ α ) σ α and thus, for sufficiently large p, p p [ 1 + B (ã, p, η, σ α ) 1 s ] p [ 1 + B (ã 0, p, η, σ α0 ) 1 s ] B (ã, p, η, σ α ) B (ã, p, η, σ α0 ) p [ 1 + B (ã, p, η, σα ) 1 s ] p [ 1 + B (ã 0, p, η, σ α0 ) 1 s ] = 1 + O ( p 1) 32

33 , thus, p [ 1 + B (ã 0, p, η, σ α0 ) 1 s ] p [ 1 + B (ã 0, p, η, σ α0 ) 1 s ] p [ 1 + B (ã, p, η, σα ) 1 s ] p [ 1 + B (ã, p, η, σ α ) 1 s ] Next, we show local identification, that is the case when the difference µ α0 /σ α0 µ α /σ α = η 0 η is small. That is, we assume that η η 0 < γ. ˆ A (ã, p, η, σ α ) = σ α α pσ α exp [pσ α ( α + ã)] α + ã [1 + exp [pσ α ( α + ã)]] 2 φ ( α η) d α σ α For large p, A (ã, p, η, σ α ) σ α = O ( (pσ α ) 1) Furthermore, recall that ˆ s = A (ã, p, η, σ α ) = α exp (pσ α ( α + ã)) φ ( α + η) d α [1 + exp (p ( α + ã))] If we let α = α + η, then, = ˆ exp (pσ α ( α + ã)) A (ã, p, η, σ α ) = φ ( α + η) d α α [1 + exp (p ( α + ã))] ˆ exp (pσ α ( α + ã η)) [1 + exp (p ( α + ã η))] φ ( α + η) d α = A (ã η, p, η + η, σ α ) α Thus, for large p s = A (ã, p, η, σ α ) = A (ã η, p, η + η, σ α + σ α ) + O ( (pσ α ) 1) s = A (ã, p, η, σ α ) = A (ã µ, η + η, p, σ α + σ α ) + O ((p σ α ) 1) ˆ B (ã, p, η, σ α ) = ˆ B (ã η, p, η + η, σ α ) = α α ( ) p exp [p (α + ãσ α )] α α [1 + exp [p (α + ãσ α )]] 2 φ η dα σ α σ α p exp [p (α + (ã η) σ α )] α [1 + exp [p (α + (ã η) σ α )]] 2 φ σ α ( ) α η η dα σ α 33

34 Now, by change of variables, let α = α ησ α. Then, ˆ ( ) p exp [p (α + ãσ α )] α + ησ α α B (ã η, p, η + η, σ α ) = α [1 + exp [p (α + ãσ α )]] 2 φ η dα σ α σ α ˆ ( ) p exp [p (α + ãσ α )] α = B (ã, p, η, σ α ) + η α [1 + exp [p (α + ãσ α )]] 2 φ η dα σ α B (ã, p, η, σ α ) (ã + η) +ã2 (ã + η) φ (ã + η) = ãφ +O ( p 1) = σ α σ α σ α Then, for let ã = η/2 η 2 /4 + 1 < 0. ãφ (ã + η) σ α ( 1 + ã (ã + η))+o ( p 1) B (ã, p, σ α ) σ α = 0 MR (η + η, σ α + σ) = p [1 + [ B (ã, p, η, σ α ) + η ( φ (ã + η) + O ( p 1) + O ( ( η) 2))] ] 1 s = p [1 + [ [ B (ã, p, η, σ α ) + η + [ 1 + ã (ã + η)] σ ] φ (ã + η) + O ( p 1) + O ( ] ( η) 2)] 1 s σ MR (η + η, σ α + σ) = p [ 1 + B (ã, p, η, σ α ) 1 s + B 2 [ η ( φ (ã + η) + O ( p 1) + O ( ( η) 2))] s ] [ [[ = p 1 + B (ã, p, η, σ α ) 1 s + B 2 η + [ 1 + ã (ã + η)] σ ] ( φ (ã + η) + O ( p 1) + O ( ( η) 2))] s ] σ α Then, = pb [( 2 φ (ã + η) + O ( p 1) + O ( ( η) 2))] s [ [ã (ã + η) 1] σ ] p B [( 2 φ (ã + η) + O ( p 1) + O ( ( η) 2))] s σ α η We already know that [ p 1 + ] [ s = p 1 + ãφ (ã + η) ] s ã φ (ã + η) 34