hotelling model assumptions

J Econ Lit 36(4):2065–2107, Ledoit O, Wolf M (2004) A well-conditioned estimator for large dimensional covariance matrices. In that case, the Wald statistic does not have a limiting \(\chi ^{2}\) distribution (see Andrews 2001). Give the assumptions of the Hotelling linear or "main street" model. In the case where the parameters of the Hotelling model vary between sub-periods, the linearity in the resource stock of the extraction is accepted if there is no evidence to reject the null hypothesis (29) for all sub-periods. Cambridge University Press, New York, Slade ME, Thille H (2009) Whither Hotelling: tests of the theory of exhaustible resources. I will assume that most readers are familiar with Hotelling’s game/the median voter theorem game. As the country effect \(e_{i}=(z_{0i},\theta _{0i})\) may vary between sub-periods, we restrict the analysis of the structural break of the model to the fixed part \((\alpha , \beta , \gamma , \eta , \mu , \delta )\) of the vector of parameters \(\omega \). Since the model has many parameters (6 + two times the number of countries producing the resource), the number of instruments used to compute the GMM estimator is very large. The discrete form associated to the dynamic efficiency condition (9) is given by: Substituting from static efficiency condition (8) into the dynamic efficiency condition (52) and rearranging, we obtain the following dynamic of the market price: where i is the country or firm index, t the time index and \(AC_{it}\) is the average extraction cost. ". The assumptions mentioned above may yield good approximations in many applications, but for transport scheduling, they are oversimpliﬁcations. These indications include the choice of instrumental variables, the test of structural stability, the computation of the GMM estimator, and the test of null assumptions \(H_{01}\), \(H_{02}\), and \(H_{03}.\). Trans-port demand is usually price-sensitive, since people can choose not to travel, Highly abstract model: does not refer to anything concrete 2. 7.2.6 - Model Assumptions and Diagnostics Assumptions ... One should be aware that, even though Hotelling's T-square test is robust to violations of assumptions of multivariate normality, the results of Bartlett's test are not robust to normality violations. For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Springer Nature Abstracting and Indexing). To test the null assumption \(H_{01}\), we use the t-statistic. NOTATIONS AND ASSUMPTIONS The notations and assumptions in this study are as follows: 1) 1) Homogeneous consumers are uniformly distributed on Some are distributional assumptions about the residuals. where \(W_{it}\) is the \(it^{th}\) row of W and \(d_{t}(R_{1}) \) is a dummy variable, which is equal to one when \(t \in R_{1}\). These results depend on whether firms use different extractive technologies or whether the structural break observed on resource prices is taken into account. Testing the Assumptions and Predictions of the Hotelling Model. Learn more about Institutional subscriptions. Hotelling’s linear city model was developed by Harold Hotelling in his article “ Stability in Competition ”, in 1929. Create and merge two data vectors ... model using nlmer() function for nonlinear mixed models and obtained the same results. As a result, the stock of resource will be fully depleted. The Hotelling model is extended to include the production technology and labor input. We consider nonlinear functional forms for the extraction cost and resource demand to develop an empirical Hotelling model with technological progress and stock dependent extraction costs. Although game theory as such allows for relaxing these assumptions, doing so often leads to intractable results or the nonexistence of equilibria (Halpern and Pass 2015). Furthermore, customers choose the shop without considering competing shops, while in daily life, it I suspect that there was no significant departure from linearity. a space–time Hotelling model that introduces a unit vertical time axis in the classical Hotelling unit interval model. The significance level is printed as .000 (i.e., p < .0005). \end{aligned}$$, $$\begin{aligned} S(0)= & {} \frac{q(0)}{g}(1-e^{-gT})\nonumber \\= & {} S(0)(1-e^{-gT}). An important advantage of agent-based models is that they allow for Denote by \(w_{it}\) the instrumental variable and \(X_{it}\) the cumulative resource stock extracted at time t for country i, respectively. See Davidson and Mackinnon (2003) for further details on GMM estimators. \end{aligned}$$, $$\begin{aligned} e^{-gT}=0 \end{aligned}$$, $$\begin{aligned} \lambda _{t+1}-\lambda _{t}=\delta \lambda _{t}-\beta \frac{q_{t+1}}{S_{t+1}}AC_{t+1}. Because users accessing a given energy source pay di erent costs, Hotelling’s assumption of a single demand curve misses important features of world energy markets and leads to predictions that are easily refuted. Article. 1. Rev Econ Stud 55:615–640, Atewamba C (2013) Managment of nonrenewable natural resources under the hotelling rule. Hotelling’s linear city model was developed by Harold Hotelling in his article “Stability in Competition” in 1929 [ 2 ]. Indeed, the GMM estimator residuals are used to calculate a new estimate of the covariance matrix \(\Sigma \), which is then used to obtain a second GMM estimator, which in turn is used to derive another GMM estimator, until the procedure converges relative to a given criterion. In carrying out any statistical analysis it is always important to consider the assumptions for the analysis and confirm that all assumptions are satisfied. 1. Correspondence to Yet none of these have ever considered the effect of multiple agents controlling multiple locations. The Hotelling Model with Multiple Demands1 G erard Gaudet Stephen W. Salant2 July, 2014 1Forthcoming in Handbook on the Economics of Natural Resources, eds Robert Halvorsen and Dave Layton, Cheltenham, U.K.: Edward Elgar Publ. Furthermore, the Hotelling model may sustain a zero long-run growth rate in resource prices. Downloadable (with restrictions)! But these costs must be small, because the people at the end of the beach continue to buy the same amount no matter how far they are from the nearest vendor. Hence, It follows that the constant g is given by, as desired. the various RePEc services. \end{aligned}$$, $$\begin{aligned} z_{0}\theta _{0}=\frac{g[\alpha (\beta -\alpha +1)-\beta ]-\alpha (\gamma +\delta ) }{(\mu -\delta +\eta g)g^{\beta }} q_{0}^{\eta -\beta +\alpha -1}, \end{aligned}$$, $$\begin{aligned} {\textit{Left}}(37)=\left( \mu +\eta g -\delta \right) p(t)-\left( g(\beta -\alpha +1)-\frac{\beta }{\alpha }g -(\delta +\gamma )\right) C_{q}(z(t),q(t),S(t)) \end{aligned}$$, $$\begin{aligned} p(t)= & {} p(0)e^{(\mu +\eta g)t}=\theta _{0}q_{0}^{-\eta }e^{(\mu +\eta g)t} \end{aligned}$$, $$\begin{aligned} C_{q}(t)= & {} C_{q}(z(0),q(0),S(0))e^{(-\gamma -(\alpha -1)g + \beta g)t}= \alpha z_{0}^{-1}q_{0}^{\alpha -1}S_{0}^{\beta }e^{(-\gamma -(\alpha -1)g + \beta g)t}. The data from population i is sampled from a population with mean vector \(\boldsymbol{\mu}_{i}\). It was first developed in 1931 by Harold Hotelling. See Davidson and Mackinnon (2003) for more details on the construction of the objective function \(Q_{T}(\omega )\) and the Newton method of optimization. Instead, they can simply refer the document which explains the basis for various calculations which are taking place in the model. \end{aligned}$$, $$\begin{aligned} \dot{\lambda }(t)=\left( \frac{\dot{\theta }(t)}{\theta (t)} -\eta \frac{\dot{q}(t)}{q(t)}\right) p(t)-\left( -\frac{\dot{z}(t)}{z(t)} +(\alpha -1)\frac{\dot{q}(t)}{q(t)}-\beta \frac{\dot{S}(t)}{S(t)}\right) C_{q}(z(t),q(t),S(t)). (17) and (18). 7.2.6 - Model Assumptions and Diagnostics Assumptions. Using panel data on fourteen nonrenewable natural … These results depend on whether firms use different extractive technologies or whether the structural break observed on resource prices is taken into account. To illustrate the Hotelling rule, let us consider as basic model where inSo denotes an economy’s total stock of resource and Rt denotes the total extraction at time t (Gaitan et al. $$\begin{aligned} \frac{dC_{q}(z(t),q(t),S(t))}{dt}= & {} \left( -\frac{\dot{z}(t)}{z(t)} +(\alpha -1)\frac{\dot{q}(t)}{q(t)} -\beta \frac{\dot{S}(t)}{S(t)}\right) C_{q}(z(t),q(t),S(t)), \end{aligned}$$, $$\begin{aligned} \frac{dp(t)}{dt}= & {} \left( \frac{\dot{\theta }(t)}{\theta (t)} -\eta \frac{\dot{q}(t)}{q(t)}\right) p(t). GMM estimators are known to be consistent, asymptotically normal, and efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions. Therefore, there is no evidence against the GMM specification used in this paper. \end{aligned}$$, \(\partial h_{j} / \partial \omega _{i}\), https://doi.org/10.1007/s10640-015-9922-0. If the assumptions are attached to the model itself, the user need not be trained about the assumptions that have been changed. (ii) The distribution of customers is uniform on the segment (with unit J Multivar Anal 88(2):365–411, Lin C, Wagner G (2007) Steady state growth in a Hotelling model of resource extraction. As data for market price \(p_{t}\), extraction rate \(q_{it}\) and average extraction cost \(AC_{it}\) are available, this equation can be used to estimate the primitives \(\alpha \), \(\beta \), \(\gamma \) and \(z_{0i}\) of the Hotelling model. We find evidence of stock-dependent extraction costs for most resources. In this paper we explore the classic Hotelling model and some of its implications. 2G erard Gaudet is professor emeritus in the Department of Economics, University of Montreal and research fellow at CIREQ (gerard.gaudet@umontreal.ca); … The original Hotelling-Downs model su ers from some problematic assumptions: customers always choose the near-est shop without considering the distance, contradicting to the fact that a shop is no more attractive to a customer if it is too far away. May 2015; Environmental and Resource Economics 66(1) Some would suggest that if your model is a standard Multilevel Model (i.e. \end{aligned}$$, $$\begin{aligned} \mu +\eta g - \delta = \left( -\gamma -(\alpha -1)g +\beta g -\delta -\frac{\beta }{\alpha }g\right) \Phi (0); \qquad \Phi (0)=\frac{ C_{q}(z(0),q(0),S(0))}{p(0)}. However, since they each have two locations, the outcome Let's recall the four assumptions underlying the Hotelling's T-square test. J Environ Econ Manag 54(1):68–83, Livernois J (2009) On the empirical significance of the Hotelling rule. Testing the Assumptions and Predictions of the Hotelling Model. Atewamba, C., Nkuiya, B. Before running the Newton’s method to minimize the objective function \(Q_{T}(\omega )\), we replace the covariance matrix \(\Sigma \) in \(Q_{T}(\omega )\) by the Ledoit and Wolf (2004) HAC estimator obtained from an initial estimation of the model by the NLS method. The first model of product differentiation is due to Hotelling (1929). ", Margaret E. Slade & Henry Thille, 2009. Give the assumptions of the Hotelling linear or main street model. In Hotelling’s Location Model, firms do not exercise variations in product characteristics; firms compete and price their products in only one dimension, geographic location. Hotelling's Model. Their estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. Therefore, at the terminal date T, \(S(T)=0\). If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form . I too have considered a multi-factor ANOVA and a Hotelling T^2-test – in fact I had initially done the analysis using the Hotelling test. If a linear market is 10 kilometers long, there are two suppliers, firm A located 2 kilometers from time left end of the market and firm B located at the right end of the market, and transportation cost t is 1 per unit distance, what are the equilibrium prices and profits per firm? Unlike the chickens’, the model’s question has an easy answer. Before explaining the model, I will start by making several (unrealistic) assumptions that will help simplify the analysis: For each sub-period a GMM estimator is obtained by minimizing an objective function \(Q_{T}(\omega )\) obtained from the moment condition (55).Footnote 12 To compute the GMM estimator \(\hat{\omega }\) for each sub-period, we use the Newton’s method for constrained nonlinear minimization. Hotelling made following assumptions while suggesting his theory a) the cost of exploring and producing oil is small compared with the price of the oil. We can, for example, let money sway matters, allowing candidates to “buy” votes. With many instruments, the estimate of the covariance matrix with the usual procedure, the Newey–West HAC estimator of the covariance matrix \(\hat{\Sigma }\) is generally not well conditioned. The Hotelling's Trace for DEFAULT is printed in the "Multivariate Tests" table in the General Linear Model output. Finally, they allow … PubMed Google Scholar. The initial model is able to approximate the actual charges below 17,000 USD, but as the actual charges go above 20,000 USD, the gap between actual charges and fitted values keeps increasing. Environ Resource Econ 66, 169–203 (2017). Article. This paper empirically examines whether the assumptions and predictions of the Hotelling model are consistent with patterns observed in data. In this paper, we empirically examine whether the assumptions and predictions of the Hotelling model are consistent with patterns observed in data. 3 Overview As with any statistical manipulation, there are a specific set of assumptions under which we operate when conducting multilevel models … Because profits are equivalent in the two models, the results on equilibrium content choice correspond to those in quadratic Hotelling models (see, e.g., d’Aspremont et al., 1979).In particular, if α and β are restricted to be positive, firms in a two-stage location-cum-price game choose maximal differentiation in equilibrium. Subscription will auto renew annually. For this analysis, based on 700 cases and two values for DEFAULT, the Hotelling's Trace is .209, which is converted to an F of 48.537 with 3 and 696 degrees of freedom. In fact, a sufficient condition for the moment conditions (55) to be verified is that they cannot be rejected in each regime. Now lots of assumptions are hidden in this model. Testing the Assumptions and Predictions of the Hotelling Model. These results depend on whether firms use different extractive technologies or whether the structural break observed on resource prices is taken into account. http://link.springer.com/10.1007/s10640-015-9922-0, Testing the Assumptions and Predictions of the Hotelling Model, A well-conditioned estimator for large-dimensional covariance matrices, A well conditioned estimator for large dimensional covariance matrices, DES - Working Papers. Hereafter, OIL oil, NG natural gas, GOL gold, HC hard coal, SC brown coal, PHO phosphate, BAU bauxite, COP copper, IRO iron, LEA lead, NIC nickel, SIL silver, TIN tin, ZIN zinc. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation. ", Roberto Ferreira da Cunha & Antoine Missemer, 2020. These assumptions are similar to those for Hotelling’s T-square test (see Hotelling’s T-square for Two samples). Furthermore, the Hotelling model may sustain a zero long-run growth rate in resource prices. It has spawned numerous papers on the extrapolation of its concepts. In this paper, we empirically examine whether the assumptions and predictions of the Hotelling model are consistent with patterns observed in data. 1 https://doi.org/10.1007/s10640-015-9922-0, DOI: https://doi.org/10.1007/s10640-015-9922-0, Over 10 million scientific documents at your fingertips, Not logged in General contact details of provider: http://www.springer.com . We consider nonlinear functional forms for the extraction cost and resource demand to develop an empirical Hotelling model with technological progress and stock dependent extraction costs. Public profiles for Economics researchers, Various rankings of research in Economics & related fields, Curated articles & papers on various economics topics, Upload your paper to be listed on RePEc and IDEAS, RePEc working paper series dedicated to the job market, Pretend you are at the helm of an economics department, Data, research, apps & more from the St. Louis Fed, Initiative for open bibliographies in Economics, Have your institution's/publisher's output listed on RePEc. T. 2: A Two-Group Multivariate Analysis # 1. Both shop owners want their shops to be where they will get most market share of customers. A theoretical model of resource extraction 2.1. In the current article, we continue the series by describing methods to evaluate the validity of the Cox model assumptions.. Now, substituting the stock growth rate (14) into (54), we get a tractable form of the Hotelling rule. \end{aligned}$$, $$\begin{aligned} p_{t} - p_{t-1}=\delta (p_{t-1}-\alpha AC_{it-1}) +\alpha (AC_{it}- AC_{it-1}) -\beta \frac{q_{it}}{S_{it}}AC_{it}. Previously, we described the basic methods for analyzing survival data, as well as, the Cox proportional hazards methods to deal with the situation where several factors impact on the survival process.. In this model he introduced the notions of locational equilibrium in a duopoly in which two firms have to choose their location taking into consideration consumers’ distribution and transportation costs. (37) and rearranging, we obtain. Can J Econ 40(4):1033–1059, Hansen LP, Heaton J, Yaron A (1996) Finite-sample properties of some alternatives GMM estimators. We conﬁrm the model’s validity for 8 of 14 minerals. Google Scholar, Andrews D, Fair R (1988) Inference in econometric models with structural change. Some applications to retail competition in a duopoly are also discussed. Tax calculation will be finalised during checkout. \end{aligned}$$, $$\begin{aligned} \dot{H}=(-\alpha g-\gamma +\beta g)H. \end{aligned}$$, $$\begin{aligned} H(t,q(t),S(t),\lambda (t))=H(0,q(0),S(0),\lambda (0))e^{(-\gamma +g(\beta -\alpha ))t}. Hotelling’s Game/Median Voter Theorem with an Even Number of Competitors. Differentiating with respect to time, we get, If there is extraction at time 0, then \(H(0,q(0),S(0),\lambda (0))>0\) and there will be extraction at all dates, so that. Likewise, allowing for alternative underlying distributions (e.g., of consumers over space in the Hotelling model) might yield a model without equilibria in pure strategies (Caplin and Nalebuff 1991). & Wagner, Gernot, 2007. In this model he introduced the notions of locational equilibrium in a duopoly in which two firms have to choose their location considering consumers’ distribution and transportation costs. It also allows you to accept potential citations to this item that we are uncertain about. If a linear market is 10 kilometers long, there are two suppliers, firm A located 2 kilometers from time left end of the market and firm B located at the right end of the market, and transportation cost t is 1 per unit distance, what are the equilibrium prices and profits per firm? Like the Hotelling model we have only two players. It is a very useful model in that it enables us to prove in a simple way such claims as: “the larger the … As in Andrews and Fair (1988), we derive the Wald statistic. The paper presents a model of the Hotelling rule and examines its applicability to real life phenomena. The results of the Wald statistic reported in this table should be interpreted with caution because some of the parameter estimates fall on the boundary of the parameter space (for example \(\alpha =1\) ). \(R_{r}\) is a subset of \(\mathbb {N}\) and \(R_{1} \cap R_{2}= \varnothing \). We find evidence of stock-dependent extraction costs for most resources. The basic Hotelling model In this section, we present a theoretical model of optimal … \end{aligned}$$, $$\begin{aligned} {\textit{Left}}(37)= & {} \left( \mu -\delta -\eta g \right) \theta _{0}q_{0}^{-\eta }e^{(\mu +\eta g)t}\nonumber \\&- \left( g(\beta -\alpha +1)-\frac{\beta }{\alpha }g -(\delta +\gamma )\right) \alpha z_{0}^{-1}q_{0}^{\alpha -1}S_{0}^{\beta }e^{(-\gamma -(\alpha -1)g + \beta g)t}\nonumber \\= & {} z_{0}^{-1}\left( \mu -\eta g-\delta \right) (z_{0}\theta _{0})q_{0}^{-\eta }e^{(\mu +\eta g)t}\nonumber \\&-\left\{ g[\alpha (\beta -\alpha +1)-\beta ] -\alpha (\delta +\gamma )\right\} z_{0}^{-1}q_{0}^{\alpha -1}S_{0}^{\beta }e^{(-\gamma -(\alpha -1)g + \beta g)t}\nonumber \\ \end{aligned}$$, $$\begin{aligned} {\textit{Left}}(37)= & {} \frac{q_{0}^{\alpha -1}X_{0}^{\beta }}{z_{0}}\left\{ g[\alpha (\beta -\alpha +1)-\beta ] -\alpha (\delta +\gamma )\right\} \left\{ e^{(\mu +\eta g)t}-e^{(-\gamma -(\alpha -1)g + \beta g)t} \right\} \nonumber \\ \end{aligned}$$, $$\begin{aligned} \mu +\eta g=-\gamma -(\alpha -1)g + \beta g. \end{aligned}$$, $$\begin{aligned} H= & {} pq-C(z,q,S)-\lambda q\nonumber \\= & {} pq-z^{-1}q^{\alpha }S^{-\beta }-(p-C_{q})q\nonumber \\= & {} -z^{-1}q^{\alpha }S^{-\beta }+ \alpha z^{-1}q^{\alpha -1}S^{-\beta }q\nonumber \\= & {} - (1-\alpha )z^{-1}q^{\alpha }S^{-\beta }. Calvin Atewamba. II. Using panel data on fourteen nonrenewable natural resources to estimate this empirical Hotelling model, we get qualitatively different results as compared to the related literature. b) … A problem with the Hotelling model when applied to commerce is that the results are very sensitive to the cost assumption. We consider nonlinear functional forms for the extraction cost and resource demand to develop an empirical Hotelling model with technological progress and stock dependent extraction costs. In this generalized model, both close substitutes and bidders are hypothetically distributed at the interval [0;1], types of bidders are continuous, and each bidder's valuations for close substitutes are not independent. not mixed designs) to then just use the lme package to streamline the model building process. The Hotelling model suggests that the price of oil in future should rise at rate equal to interest rate. volume 66, pages169–203(2017)Cite this article. Now, evaluating (38) at date \(t=0\), we get, Solving Eq. the models used for processing a multi-dimensional continuous type in his paper inspired our studies. Hotelling Model We first take the locations of the sellers as given (afterwards we are going to determine them endogenously) and assume firms compete in prices. We consider nonlinear functional forms for the extraction cost and resource demand to develop an empirical Hotelling model with technological progress and stock dependent extraction costs. In this paper, we empirically examine whether the assumptions and predictions of the Hotelling model are consistent with patterns observed in data. The utility of consumption would be denoted by U(Rt).The objective is to maximize the marginal net revenue of extraction of the non-renewable resource.