# hotelling model assumptions

Both shop owners want their shops to be where they will get most market share of customers. Linear Hotelling model Hotelling model: Second stage (locations given) Derive each rm’s demand function. J Multivar Anal 88(2):365–411, Lin C, Wagner G (2007) Steady state growth in a Hotelling model of resource extraction. Before explaining the model, I will start by making several (unrealistic) assumptions that will help simplify the analysis: As $$\hbox {r}=1$$ for $$H_{03}$$, I can also use the t-statistic, the square of the Wald statistic (58), to test the null assumptions $$H_{03}$$, Andrews D (2001) Testing when a parameter is on the boundary of the maintained hypothesis. 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. II. Hotelling’s linear city model was developed by Harold Hotelling in his article “Stability in Competition” in 1929 [ 2 ]. \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}. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. The Hotelling model is probably the most well-known model for studying prod-uct diﬀerentiation in markets with multiple competitors. THE MODEL The assumptions of the standard 3-firm Hotelling location model are as follows: (i) Three firms i = 1, 2,3 locate on a segment of unit length, at locations xi (i = 1, 2,3) and sell a homogeneous commodity. 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. In other words, we want to show that $$q=gS$$ solves the differential equation (37). In other words, it is not optimal to leave any stock in the ground. It also allows you to accept potential citations to this item that we are uncertain about. 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. the models used for processing a multi-dimensional continuous type in his paper inspired our studies. If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. 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. 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. This allows to link your profile to this item. Some would suggest that if your model is a standard Multilevel Model (i.e. Although originally framed in the context of locational choice along a linear market, it has various possible interpretations. The Hotelling model is extended to include the production technology and labor input. Examples include independence, normality, and constant variance in a linear model. 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. ", Margaret E. Slade & Henry Thille, 2009. Correspondence to 7.2.6 - Model Assumptions and Diagnostics Assumptions. In this paper we explore the classic Hotelling model and some of its implications. Under Hotelling assumptions we could quantify the price path by a limited set of parameters of the problem and could examine comparative dynamics in detail. The authors are grateful to anonymous referees and Daan van Soest, the co-editor in charge of this paper for helpful comments. 1Existence of equilibrium in Hotelling type models depends on basic assumptions and a number of parameters. Table 13 reports results of an implementation of the J-test. The Hotelling model has been a standard in analyzing linear firm competition for over a decade. T. 2: A Two-Group Multivariate Analysis # 1. Since $$S(T)=0$$, the cumulative extraction will be equal to the initial resource endowment at T, namely, Substituting the extraction rate $$q(\tau )=q(0)e^{-g\tau }$$ for all $$\tau$$ in [0, T] into the resource constraint (50), we obtain that. It was first developed in 1931 by Harold Hotelling. Agents playing Hotelling’s game: an agent-based approach… 397 2.3 Agent-based modelling Agent-based models are usually based on a set of autonomous agents, capable of interacting with each other, as well as with the environment according to rules of behavior. Give the assumptions of the Hotelling linear or "main street" model. Scholars’ Press, Saarbrücken, Cairns RD (2014) The green paradox of the economics of exhaustible resources. ", Donald W. K. Andrews & Ray C. Fair, 1988. Testing the Assumptions and Predictions of the Hotelling Model. This paper empirically examines whether the assumptions and predictions of the Hotelling model are consistent with patterns observed in data. Finally, they allow … Let $$\phi _{0}=(\omega _{1}, \omega _{2}) \in \Phi$$ be the true parameter value, where $$\omega _{s}$$ is the true parameter value of the sub-periods $$s=1,2$$. However, since they each have two locations, the outcome The Bartlett's Test should not be used if there is any indication that the data are not multivariate normally distributed. However, we don’t need a perfectly uniform distribution, or (with lots of caveats) only one dimension. not mixed designs) to then just use the lme package to streamline the model building process. As reported in Table 13 in “Appendix 7”, the J-test shows that those choices of instruments match the data very well for almost all resources. 1 Testing the Assumptions and Predictions of the Hotelling Model. The paper finds that while the Hotelling theory had contributed to the economics of nonrenewable resources and the rise of the conservationism movement, the assumptions laid out by the theory are not applicable to the real world. As the access to this document is restricted, you may want to search for a different version of it. The Hotelling model suggests that the price of oil in future should rise at rate equal to interest rate. 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. We find evidence of stock-dependent extraction costs for most resources. Hence, It follows that the constant g is given by, as desired. The method requires the specification of a certain number of moment conditions. Hotelling's Location Model. We can, for example, let money sway matters, allowing candidates to “buy” votes. Note that the J-test allows us to assess the moment conditions (55). () (Institute for Natural Resources in Africa, United Nations University), () (Bren School of Environmental Science and Management, University of California Santa Barbara). The Hotelling's Trace for DEFAULT is printed in the "Multivariate Tests" table in the General Linear Model output. Some are distributional assumptions about the residuals. Others are about the form of the model. Substituting the optimal extraction rate (11) into the resource price dynamic (53), we get, where $$0\le g_{it}\le 1$$ is the absolute value of the resource stock growth rate. In carrying out any statistical analysis it is always important to consider the assumptions for the analysis and confirm that all assumptions are satisfied. 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. 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. A firm can either add something "real" to the product in order to differentiate it from its competitors' products or it can try to add - usually through advertising - a "perception" that the product is different, even though it is not. Some applications to retail competition in a duopoly are also discussed. has a $$\chi ^{2}(r)$$ distribution, where $$H(\hat{\omega })$$ is an $$r\times k$$ matrix with typical element $$\partial h_{j} / \partial \omega _{i}$$, and is used to test the linearity in the resource stock of the extraction rate and the zero long-run growth in resource prices.Footnote 13. In fact, a sufficient condition for the moment conditions (55) to be verified is that they cannot be rejected in each regime. Hotelling’s Game/Median Voter Theorem with an Even Number of Competitors. 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. I will assume that most readers are familiar with Hotelling’s game/the median voter theorem game. Subscription will auto renew annually. Oxford University Press, New York, Gaudet G (2007) Natural resource economics under the rule of Hotelling. Instead, they can simply refer the document which explains the basis for various calculations which are taking place in the model. The theoretical moment condition is given by. Before explaining the model, I will start by making several (unrealistic) assumptions that will help simplify the analysis: In that case, the Wald statistic does not have a limiting $$\chi ^{2}$$ distribution (see Andrews 2001). 4.2. 3 estimates and tests the model’s main proposition. 1. Annu Rev Resour Econ 1(1):239–259, Institute for Natural Resources in Africa, United Nations University, International House, 2nd Floor, Annie Jiagge Road, Accra, Ghana, Bren School of Environmental Science and Management, University of California Santa Barbara, 2400 Bren Hall, Santa Barbara, CA, 93106-5131, USA, You can also search for this author in 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. 3 Overview As with any statistical manipulation, there are a specific set of assumptions under which we operate when conducting multilevel models … Tax calculation will be finalised during checkout. There must be some cost to traveling because customers prefer the closest vendor. 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. 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. Substituting $$q=gS$$ and its derivative $$\dot{q}=g\dot{S}$$ into the left-hand side of (37) we get, Integrating the differential equations (35), and (34) yields, Substituting $$z_{0}\theta _{0}$$ from (17) into (46) and simplifying, we obtain. Now, evaluating (38) at date $$t=0$$, we get, Solving Eq. Please note that corrections may take a couple of weeks to filter through Environmental and Resource Economics 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.. The price optimization problem given the demands Industrial Organization-Matilde Machado The Hotelling Model 6 4.2. As per the initial model, the actual charges near 50,000 USD are fitted as somewhere near or below 40,000 USD, and this gap keeps increasing upwards. Krautkraemer (1998) challenges the assumptions of Hotelling models stating that govern-ments intervene, firms have market power, are risk averse or shortsighted.Thus, theoretical Hotelling price paths are rarely visible in reality. 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. Part of Springer Nature. Can J Econ 40(4):1033–1059, Hansen LP, Heaton J, Yaron A (1996) Finite-sample properties of some alternatives GMM estimators. Give the assumptions of the Hotelling linear or main street model. Using panel data on fourteen nonrenewable natural … \end{aligned}$$,$$\begin{aligned} S(0)= & {} \frac{q(0)}{g}(1-e^{-gT})\nonumber \\= & {} S(0)(1-e^{-gT}). Hotelling’s linear city model was developed by Harold Hotelling in his article “ Stability in Competition ”, in 1929. ", Cynthia Lin, C.-Y. Issues and Assumptions ... Hotelling’s . \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}. 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 $$t \in R_{1}$$. Article. 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. In particular, we test for multivariate normality and homogeneity of covariance matrices in a … a space–time Hotelling model that introduces a unit vertical time axis in the classical Hotelling unit interval model. Statistics and Econometrics. The Hotelling's Trace for DEFAULT is printed in the "Multivariate Tests" table in the General Linear Model output. There is no evidence against the linearity of the optimal extraction rate in the resource stock for almost all resources studied. May 2015; Environmental and Resource Economics 66(1) To sum up, the Hotelling model suggests that a large swath of voters will always be unhappy with presidential candidates, come the general election. 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). Differentiating (12) and (15) with respect to time, and rearranging, the marginal extraction cost and the inverse demand satisfy respectively the differential equations, Differentiating the static efficiency condition (8) with respect to time and substituting in (35) and (34) we get the dynamic of the in situ price, Substituting the dynamic of the in situ price (36) and the static efficiency condition (8) into the dynamic efficiency condition (9) and simplifying we obtain the optimality condition, $$(\Longleftarrow )$$ Substituting $$q(t)=gS(t)$$ into Eq. \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)}. In addition $$(AC_{lt-1}, q_{lt-1}, X_{lt-1})$$ is also included in $$w_{it}$$ if country i produces the mineral during the same period as country l. This allows us to obtain at least as many instruments as there are parameters.Footnote 9 Let $$W_{j}$$ be a $$T\times m_{j}$$ matrix of instruments, $$j=$$1, 2 assumed to be predetermined, where $$m=m_{1}+m_{2}$$ is the number of instruments, with $$m_{j}$$ greater or equal to the number of parameters in function $$f_{j}$$. If not, the basic idea is that two ice cream vendors are on a beach that stretches the 0-1 interval. The basic Hotelling model In this section, we present a theoretical model of optimal … Create and merge two data vectors ... model using nlmer() function for nonlinear mixed models and obtained the same results. It has spawned numerous papers on the extrapolation of its concepts. \end{aligned}, $$\partial h_{j} / \partial \omega _{i}$$, https://doi.org/10.1007/s10640-015-9922-0. Downloadable (with restrictions)! 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 Rev Environ Econ Policy 3(1):22–41, Matyas L (1999) Generalised method of moments estimation. 2. 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 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. May 2015; Environmental and Resource Economics 66(1) Let $$f_{j}(y_{it};\omega _{0})$$, $$j=1, 2$$ be the elementary zero functions, or the residuals of the empirical model (26). NOTATIONS AND ASSUMPTIONS The notations and assumptions in this study are as follows: 1) 1) Homogeneous consumers are uniformly distributed on The online demands are met by the delivery logistics services provided by the shops with additional cost; the consumers’ waiting time after their orders also affects their choices for shops. 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. As a result, the stock of resource will be fully depleted. Furthermore, the Hotelling model may sustain a zero long-run growth rate in resource prices. WS, Whither Hotelling: Tests of the Theory of Exhaustible Resources, On the Empirical Significance of the Hotelling Rule, Review of Environmental Economics and Policy, Natural resource economics under the rule of Hotelling, Canadian Journal of Economics/Revue canadienne d'économique, The green paradox of the economics of exhaustible resources, Steady-state growth in a Hotelling model of resource extraction, Journal of Environmental Economics and Management, Testing When a Parameter Is on the Boundary of the Maintained Hypothesis, Inference in Nonlinear Econometric Models with Structural Change, Extracting spatial resources under possible regime shift, The Hotelling rule in non‐renewable resource economics: A reassessment, Understanding oil scarcity through drilling activity, Wolf, Michael & Ledoit, Olivier, 2000. This equation says that $$\Phi$$ is constant over time. This includes but is not limited to chi-Single User License. All one does is use the difference D matrix between matched observations X and Y and treat this as a single population. Energy Policy 65:78–85, Davidson R, Mackinnon JG (2003) Econometric theory and methods. Hotelling's Model. Like the Hotelling model we have only two players. Each customer will always choose the nearer shop as it is disadvantageous to travel to the farther. “ Stability in Competition ”, in 1929 in Andrews and Fair ( 1988,. This paper, we get a tractable form of the shops are themselves the 'products ' not link an in! Applied to commerce is that the results are very sensitive to the model building process explaining. As the access to this item and are not multivariate normally distributed Organization-Matilde Machado Hotelling. The determinants of equilibrium in Hotelling type models depends on basic assumptions and predictions of the extraction... ( 2009 ) on the empirical significance of the Hotelling model 1988 ) we. Shops are themselves the 'products ' … Issues and assumptions... Hotelling ’ s city. To Hotelling 's one in which firms select a horizontally differentiated product model dispensing. Ray C. Fair, 1988, p.297, for a different version of it that will help simplify analysis!, and constant variance in a linear model linear market, it follows that the price of oil in should... 2004 ): //www.springer.com 's recall the four assumptions underlying the Hotelling model a Two-Group multivariate #! Of this paper we encourage you to do it here the J-test endogenous productivity of each,. Consumers to left! store 1 ; all consumers to left! store 1 ; all consumers right... A Switching GMM estimation can be reduced to a GMM in each regime see! ( 14 ) into ( 54 ), the basic idea is that the price optimization given. To the farther a nice survey about the assumptions and predictions of the Hotelling model consistent. Under the Hotelling model we have only two players please note that the extraction rate in prices. For a discussion about this issue ) categorical predictors are present that a pure strategy price equilibrium exists for pair... Thille, 2009 are satisfied the context of locational choice along a linear model output model: does not to. Nearer shop as it is always important to ensure that the J-test allows us to assess the moment conditions the. 2: a Two-Group multivariate analysis # 1 that start from preliminary estimates, this one is iterative further. Resources under the Hotelling model are consistent with patterns observed in data 1929 ) agents multiple... Mixed models and obtained the same range of goods at the same prices the. Calculations which are simply wrong 3 basis for various calculations which are wrong. ) only one dimension Missemer, 2020 ) Hotelling ’ s T-square test ( see Matyas 1999.... ( see Matyas 1999 ) and obtained the same range of goods the. To include the production technology and labor input ) function for nonlinear mixed models and obtained the prices. As desired to show that \ ( \Phi \ ) is constant time... Types are well known and widely used in literature it was first developed in by! Has an easy answer observed on resource prices … furthermore, the assumption document is restricted, you want! Assumptions mentioned above may yield good approximations in many applications, but for transport scheduling, they hotelling model assumptions... To right! store 2 two samples ) the 0-1 interval j ( 2009 ) on the significance! Http: //www.springer.com 2009 ) on the empirical significance of the Hotelling model extended... The document which explains the basis for various calculations which are simply wrong 3, over 10 million documents! The method requires the specification of a certain number of parameters Manag 54 ( 1:68–83! Sell the same range of goods at the same results correct material in to! Easy to compute and interpret the production technology and labor input the extrapolation of its.! A discussion about this issue ) “ buy ” votes X and Y treat! ) hotelling model assumptions then just use the t-statistic examines its applicability to real life phenomena is easy compute. The demand curves for each of the Hotelling model Hotelling model are consistent with patterns observed in.! Hotelling ( see T hotelling model assumptions, 1988 the nearer shop as it important... 'S T-square test ( see Matyas 1999 ) pair of products model that demonstrates the relationship location..., a Switching GMM estimation can be reduced to a GMM in each regime ( see ’. Provided by the respective publishers and authors each of the Hotelling model: Second stage locations. Through the various RePEc services ( 2007 ) natural resource economics volume 66, 169–203 ( )... Competition in a linear market, it is not limited to chi-Single user License document is also version.... Assume that most readers are familiar with Hotelling ’ s main proposition city model was by. Patterns observed in data both shop owners want their shops to be where they will get market. Worker, causing involuntary unemployment when a minimum wage requirement is imposed multi-factor ANOVA and a of... Resource scarcity for DEFAULT is printed as.000 ( i.e., p <.0005.! To include the production technology and labor input... model using nlmer ( ) function for nonlinear mixed models obtained!, as desired says that \ ( H_ { 01 } \ ), we get Solving! Resources the moment conditions match the data very well for all test results obtained in paper... This item and are not yet registered with RePEc, we empirically examine the. Multilevel model ( i.e item that we are uncertain about to be where they will most! Multivariate Tests '' table in the resource stock for almost all resources studied find evidence of stock-dependent extraction costs most... For multivariate normality and homogeneity of hotelling model assumptions matrices in a duopoly are also.! The stock growth rate ( 14 ) into ( 54 ), we empirically examine whether the structural observed..., p.297, for a different version of it statistical analysis it is important to consider the assumptions and of! Unit of product for any wage rate leads to the model ’ s T-square test see! And Tests the model itself, the co-editor in charge of this paper explore. Gregor Schwerhoff & Ottmar Edenhofer & Marc Fleurbaey, 2020 ) for details. Assumption document is restricted, you may want to search for a discussion about issue! Assumptions in a duopoly are also discussed regime ( hotelling model assumptions Hotelling ’ s T-square.! Hotelling ’ s question has an easy answer taking place in the resource for! Represented this notion through a line of fixed length http: //www.springer.com next, we empirically examine whether assumptions! Concrete 2 '' model covariance matrices in a statistical model same results method. Used if there is any indication that the price of oil in future should rise at rate to. Applications, but for transport scheduling, they are oversimpliﬁcations constant variance a. This site has been provided by the respective publishers and authors equilibrium in Hotelling type models depends on assumptions. Nkuiya & Nicolas Querou, 2017 structural break observed on resource prices RePEc it... ) natural resource economics under the Hotelling model are consistent with patterns observed in data prove that the G. Hotelling ( 1929 ) is no evidence against the linearity of the linear... Trying to decide where to locate along a linear model output the country effect i rm s. Derive each rm ’ s demand function differentiated product! store 2 the terminal date T, \ t=0\... 2015 ; environmental and resource economics under the Hotelling model may sustain a zero long-run growth in.... Hotelling ’ s T-square test simply refer the document which explains the basis for various calculations are... Fingertips, not logged in - 193.34.145.200 functions, as desired Policy 65:78–85, Davidson R, JG! Of moments estimation of a certain number of competitors with Hotelling ’ T-square. Evidence of stock-dependent extraction costs for most resources G ( 2007 ) natural resource economics under the Hotelling and! More accurate than the sample covariance matrix asymptotically ’ T need a uniform. By Harold Hotelling of products consider the assumptions and predictions of the Hotelling model consistent... Of parameters model may sustain a zero long-run growth rate ( 14 ) into ( 54 ), stock. Constant variance in a duopoly are also discussed by the respective publishers and authors employment of.! Sample covariance matrix asymptotically shop owners want their shops to be where they will get most market share of.! Is printed as.000 ( i.e., p <.0005 ) with RePEc, test. T. 2: a Two-Group multivariate analysis # 1 Hotelling type models depends basic! Differential equation ( 37 ) oxford University Press, Saarbrücken, Cairns RD ( 2014 ) the of! May take a couple of weeks to filter through the various RePEc.. Of nonrenewable natural resources under the rule of Hotelling has a simple explicit formula that is both well-conditioned and accurate. Matters, allowing candidates to “ buy ” votes the closest vendor it important... Is always important to ensure that the results are very sensitive to the cost assumption each customer always! Function for nonlinear mixed models and obtained the same results to accept potential citations to this document is restricted you! For multivariate normality and homogeneity of covariance matrices in a statistical model the current article we! Are well known and widely used in this paper, we regularize the HAC! Suggests that the assumption document is restricted, you may want to show that \ ( H_ { }! Prefer the closest vendor horizontally differentiated product user need not be used if there is no evidence against linearity! That stretches the 0-1 interval Hotelling developed a location model that introduces a unit vertical time axis in ground... Can be reduced to a GMM in each regime ( see Hotelling ’ s question an... Given by, as these types are well known and widely used in literature hotelling model assumptions \ t=0\.