University of Economics, Prague Faculty of Economics
Major specialisation: Economic Analysis
Analysis of Apartment Rental Price Adequateness Across Prague Districts
Diploma thesis
Author: Bc. Eliška Pulcová
Supervisor: PhDr. Ing. Martin Janíčko, Ph.D.
Year: 2021
I hereby declare on my word of honour that I have written the diploma thesis independently with using the listed literature.
Bc. Eliška Pulcová
I would like to express my sincere gratitude to my supervisor, PhDr. Ing. Martin Janíčko, Ph.D. and my friend Ing. Karel Šafr, Ph.D. for beneficial advice and comments provided while writing the diploma thesis.
Abstract
The aim of the thesis is to develop a pricing model from where the author can derive whether the price per square meter of rental apartments is below or above the average market price per square meter of rental apartments in Prague per cadastral municipality, where possible. Author expects that price per month will be higher the closer to the center of Prague is the rental apartment located. This hypothesis is confirmed by all of the performed regression analysis that takes into account dummy variables controlling for location.
Key Words: Industrial Organization, Firm Behavior, Product Differentiation, Statistical Methods, Estimation
Contents
Abstract ... 6
Introduction ... 8
Literature review ... 10
Conclusion of theoretical part ... 23
Comparison of the rental markets in Brussels, Berlin, and Prague ... 25
The Model ... 28
Dataset ... 32
Descriptive Statistics ... 33
Empirical analysis ... 34
Conclusion of the practical part ... 52
Appendix ... 54
References ... 65
Introduction
The aim of the thesis is to develop a pricing model from where the author can derive whether the price per square meter of rental apartments is below or above the average market price per square meter of rental apartments in Prague per cadastral municipality, where possible. Data used for analysis were fetched from one real estate portal www.sreality.cz. Data were scrapped during three days in June 2020 and are maximum five months old.
The thesis is observing determinants affecting the price per square meter of rental apartments in Prague per cadastral municipality during the development of the price model. The bid price is considered as equal to the realized price for there is unfortunately no way to know the actual realized price of apartment lease. I am focusing on the following explanatory variables:
cadastral municipality, floor level, flat area, distance to a bus/tram/train stop measured in meters, distance to the underground station measured in meters, civic amenities, proximity to nature and parking slot for a personal car. Variables are selected based on Salop’s (1979), Perloff’s (1985), and partially also Hotelling’s (1929) contributions in the field of spatial differentiation which stresses importance of non-monetary benefits such as location and walking distance for an efficiency-seeking customer.
The first part of the thesis is a theoretical part and is composed of two sections. The first section is the literature and theory review.
The next section is an overall description of current rental situation in Prague compared to the current rental situation in Berlin and Brussels.
The practical part of the thesis is composed of three sections. The first section introduces the dataset and used statistical methods, the second section describes the dataset and the third section is the regression analysis. Regression analysis models relationship between dependent variable, independent variables and various quantity of dummy variables that control for locations. Regression analysis consists of four main and four reduced models. The first analysis estimates the relationship between dependent variable price per month of rental apartment and 18 independent variables. The next three regression analysis models relationship between dependent variable, 18 independent variables and dummy variables. First type of location is as follows. I divide 112 cadastral municipalities of Prague into 22 areas with similar
municipalities and the last type of location is division of Prague into municipal areas. Each regression analysis is estimated by OLS, robust regression is also performed.
Literature review
Literature touching upon the contributions in the field of spatial differentiation includes Salop (1979) dealing with monopolistic competition with outside goods and his circle model, that is a famous variation of Hotelling’s (1929) theory of location model and Perloff (1985) searching for equilibrium with product differentiation.
In formalization of Hotelling (1929) spatial competition model, heterogeneous consumers have different preferences among the available brands. As analyzed by Hotelling and generalized and extended by Lancaster (1979) and Salop (1979), competition is treated as a localized phenomenon. Every consumer purchases number of brands (often one) which is also limited from a small subset of most preferred brands . As such, this model might better be classified as a location model rather than as monopolistic competition. In Hotelling‘s model, brands mayor may not be relocated if entry occurs. In contrast to this formalization is the representative consumer model associated with Chamberlin (1933). As analyzed by Dixit and Stiglitz (1976) and Spence (1976), a representative consumer purchases many brands, varying the proportions of each according to their prices and utility weights that are exogenously given. These models involve competition by all brands for each representative consumer compared to the localized competition of the linked oligopoly model.
None of the models is clearly superior to the other. The representative consumer model has the property of multibrand competition. Compared to the spatial model focuses clearly on brand attributes and brand reformulation that face the entry competition.
In this section, we deal with individual articles. Salop in his paper analyzes a model of spatial competition in which a second commodity is explicitly treated. He assumes two-industry economy with a zero-profit equilibrium with symmetrically located firms that may exhibit rather strange properties. Demand curves are kinked, although firms make Nash conjectures.
Suppose equilibrium is a tangency solution outside of the kink. In that case, the short run and long run responses to changes in parameter are conventional, but if equilibrium lies at the kink, the effects of changes in parameter are perverse. Price is rigid in the short run faces small cost changes. Increases in costs lower equilibrium prices in the long run. If market size increases, then prices rise. The welfare properties are also perverse at a kinked equilibrium. Decreases in cost and increases in market size lower both consumer and aggregate welfare.
Two industries are as follows. The one is a monopolistically competitive industry with differentiated brands and has decreasing average costs; the other is a competitive market
specification, l is different brand and his evaluation is inferior according to preferences. Product space is an infinite line of a circle. The model is a benchmark for subsequent analyses with non-uniform preferences across empirically validated product spaces. By eliminating technical problems, this model allows a focus on the essential interactions of firms in an industry.
If there are n brands of the differentiated product at prices p and locations l, a consumer whose preferences are described as l* will purchase one unit from brand. If the maximum surplus of utility lower price across brands outweighs the surplus from the other homogeneous goods.
Denoting surplus by s, we have this decision rule according to which purchasing one unit of the brand has to satisfy the following conditions:
[𝑈(𝑙!, 𝑙∗) − 𝑝!] ≥ 𝑠 (1)
If we assume constant transportation costs, we have decision rule given by:
𝑈(𝑙!, 𝑙∗) = 𝑢 − 𝑐|𝑙!− 𝑙∗| (2)
Where the distance |𝑙! − 𝑙∗| is the shortest arc length between l and l*. The effective reservation price is given by
𝑣 = 𝑢 − 𝑠 > 0 (3)
Now the author explores the existence and properties of a symmetric zero profit Nash equilibrium (SZPE). Symmetric means an equilibrium where brands are equally spaced around the circular product space and charge identical prices. Zero profits refer to free entry leading to a situation that every brand is earning zero profits. Nash equilibrium in this case means that each brand chooses a best price, given the perception that all of other brands hold their prices constant. The last requirement is that the number of brands is an integer. The goal is to derive the perceived demand curve for a single representative brand as a function of other brand’s prices and locations and then finding a tangency between the average cost and the demand curve. Demand curve can be divided between three parts: the monopoly, competitive and supercompetitive part. The monopoly region consists of prices where the brand’s market consists of consumers for whom no other brand exceeds the surplus of homogeneous outside goods . The competitive region consists of prices where some customers would purchase goods
from some other differentiated brand. The supercompetitive region consists of prices in which all of the customers of the closest neighboring brand are captured.
Figure 1. Typical demand curve:
Presume the representative brand charges price p and its nearest competitors located at distance 1/n charge p. First, we define the potential monopoly market of representative brand. The representative brand captures all consumers living within a distance where the net surplus given in (1) is nonnegative. Denoting the maximum distance by x and substituting into
[𝑣 − 𝑐|𝑙! − 𝑙∗| − 𝑝!] ≥ 0 (4)
we have:
𝑥 5 = (𝑣 − 𝑝)/𝑐 (5) If there are L customers around the circle its monopoly demand qm is given by:
𝑞# = $%& (𝑣 − 𝑝) (6)
Figure 2. The circular market:
Further, we define competitive region, namely the region in which consumers reside in the potential monopoly market of two brands and purchase from the one offering higher net surplus. If the brands are located a distance apart of 1/n and the neighbouring brand on one side charges a price p, then from (3) the representative brand captures all of the consumers within a distance x given by
𝑣 − 𝑐𝑥 − 𝑝 ≤ 𝑣 − 𝑐 9'(− 𝑥: − 𝑝 (7)
Denoting by x we have:
𝑥 =$&' (𝑝 +(&− 𝑝) (8)
And a firm’s competitive demand 𝑞& = 2𝐿) is given by
𝑞& =%&(𝑝 +(&− 𝑝) (9)
If we differentiate (6) and (9) we get the slopes 𝑠𝑙(𝐷) of the demand curve in these two regions:
𝑠𝑙(𝐷#) = −$%& (10) 𝑠𝑙(𝐷&) = −&% (11)
The result is as follows. The demand is more elastic in the monopoly region than in the competitive region. The monopoly region exhibits higher prices.
Assume neighbour from the right has a potential monopoly market illustrated in Figure 3. At prices above v, the representative brand gains no customers. Since reducing prices below v, it captures demand from the homogeneous good according to the monopoly slope c/2L.
As illustrated in Figure 3 Market segments.
Figure 3. Market segments
Figure 4. Market overlap
As the price becomes low enough its monopoly market overlaps the monopoly market of its neighbour as we can see in Figure 4 Market overlap.
As it diminishes price further, it begins to capture customers from its neighbour according to competitive slope c/L that is steeper compared to monopoly slope.
The price where the customers are indifferent between the representative firm at p and the neighbour at p is given by
𝑝* = 𝑝 −(& (12) The representative firm captures the entire market of its neighbor if the price is below pz. Hence the representative brand’s demand has a discontinuity at pz from this predatory pricing.
Figure 2 illustrates the typical shape of these three regions. It shifts according to the prices and locations of the neighbouring brands. Since demand can never exceed the monopoly demand, the kink is always on monopoly curve. Market may be so competitive as to make the kink nonexistent. Nonexistent kink occurs when the neighbour’s potential monopoly market includes the location of the representative brand.
The author further discusses existence of a symmetric zero-profit equilibrium (SZPE).
Definition of symmetric zero profit equilibrium is as follows. The price p and a number of brands n such that every equally spaced Nash price setter’s maximum profit price earns zero profit. If there is equilibrium the representative brand’s demand curve and average cost curve will be tangent, for then the zero-profit point is surely also one of maximum profits.
Figure 5 illustrates three equilibrium configurations that are possible.
Figure 5. Equilibrium configuration
Author now discusses each of the equilibria. At the monopoly equilibrium, some of the consumers lying between two neighbouring brands may not purchase the differentiated good.
Hence, neighbours' markets may not overlap and each of them can act as a monopolist that is constrained only by the outside product.
At kinked equilibrium as displayed in Figure 5 markets touch. As the extension of the monopoly demand curve lies above the average cost curve, the monopoly price pm lies lower than kinked equilibrium price pk. As we can see from the last graph in Figure 5 the monopoly markets completely overlap at the competitive equilibrium. However, pc may be lower or higher than pm, depending on demand and technologies.
Figure 6:
Figure 6 illustrates the existence of the equilibrium. Author puts together each demand curve, drew the average cost curve and then can refer that existence of an equilibrium configuration requires maximum profit so point E is zero-point. In comparison to zero-point G that does not satisfy maximum profit because it is dominated by point F.
Given those we can point out the two conditions which have to SZPE satisfy. Marginal revenue has to be less than or equal to marginal cost and price has to be equal to average cost.
If we assume constant marginal cost m and fixed cost F, the SZPE is given by
𝑝 + 𝑞,-,.≤ 𝑚 (12) 𝑝 = 𝑚 +/. (13)
From the monopoly equilibrium we can derive the monopoly price and number of brands, which is:
𝑝# = 𝑚 +$(&
! (14) 𝑛# = '
√$A&%/ (15) And from the competitive equilibrium we gain price and number of brands given by
𝑝& = 𝑚 +(&
" (16) 𝑛& = B𝑐𝐿/𝐹 (17) The values for kinked equilibria lie between monopoly and competitive prices and numbers of brands. As there is no tangency at the kinked equilibrium
𝑝 + 𝑞,-,.≤ 𝑚 (18)
holds as inequality. Price is given by the monopolistic demand function:
𝑝1 = 𝑣 − 9%&: 𝑞 = 𝑣 − 𝑐/𝑛 (19) If we set price equal to average cost we get
/
%𝑛1+(&
# = 𝑣 − 𝑚 (20)
The demand discontinuity implies the nonexistence of any SZPE. Representative firm can capture the neighbor’s entire market at prices lower than predatory price p-c/n, but given the conditions for existence of a SZPE is such pricing behaviour non profitable . A sufficient condition for this is that predatory price does not exceed MC m, for price equal to or lower than marginal cost, which is a losing strategy because fixed costs are still present.
Supercompetitive behaviour is not profitable as the equilibrium price never exceeds m + c/n.
Similarly, if MC is increasing, then the price that captures entire market lies below the minimum AC price. But if MC is decreasing, then such price cuts can generate profit and cause SZPE to be nonexistent.
Another author dealing with product differentiation is Perloff (1985). Perloff in his article synthesizes two diverse approaches namely spatial competition model, that had been published by Hotelling (1929) and extended by Salop (1979), and Chamberlin (1933) with his representative consumer model that has been extended by Stiglitz (1976) and Spence (1976) and in which a representative consumer purchases brands where proportions of each vary according to their prices and utility weights that is given exogenously.
In the beginning Perloff evaluates consumer’s preference of all possible brands and then focuses on brand attributes. At the same time he assumes that competition is not localized. It means that every brand competes with every other brand.
Further, the author analyzes a model of industry equilibrium where differentiated brands exist in a product class. Assume that there are an unlimited possible number of different brands, each consumer assigns relative values to these brands according to his preferences, further he assumes that there exist n brands and there is a finite number of consumers L. Each of the consumers has no monopsony power. Initially, the author assumes that there are n brands of whom have no monopsony power. Further every consumer purchases the brand that maximizes his net surplus. Pi is the price of the ith brand, si is its surplus, and ai is a single element of the consumer's preference vector. The term best buy implies to the brand with the highest net surplus for a particular consumer. There can be situations that for some prices even the best buy may give negative surplus or surplus less than some threshold opportunity value, v.
Demand would be zero in these cases. However, this possibility adds complication to the current model so the author this possibility further ignores.
Author supposes that preferences are symmetric. That means the aggregate preferences for every brand i are independent and identically distributed. A consumer will choose a brand which maximizes his surplus and therefore is his best buy. He examines the case where each consumer purchases exactly one unit of his best buy. The expected demand for brand i, in this case, equal to the proportion of consumers who buy that brand times the number of consumers, L:𝑄!(𝑝', 𝑝$, … , 𝑝() =𝑃𝑟 𝑃𝑟 H𝑠! ≥ 𝑠2 I 𝐿 (21)
He assumes constant marginal cost c for every firm, then its expected profits are given by that profit equation:
𝜋!(𝑝', 𝑝$, … , 𝑝() = (𝑝!− 𝑐)𝑄!(𝑝', 𝑝$, … , 𝑝() − 𝐾 (22)
,where K are fixed (sunk) costs of each firm.
Each firm is risk-neutral, maximizes expected profits and takes the prices of the other firms as given. So he basically analyzes the "Bertrand-Nash in price" equilibrium.
The marginal revenue equals marginal cost condition is hence as follows:
𝑝! = 𝑐! −3$93(-%,….,-&)
$/9-$ (23)
First he considers the existence of a single-price equilibrium.
While a single-price equilibrium may seem plausible given our symmetry assumption, it may not be necessary. Thus, he assumes for now that equilibrium entails identical prices for all firms. Assuming that all firms except firm i charge an identical price p, then we have
𝑝(𝑛) = 𝑐 +;(()' (24) The equation above determines a single-price equilibrium that lies strictly above the competitive price until 𝑀(𝑛) > 0 . This condition prevails for all finite number of firms n until the preference density is differentiable.
Proposition 1. An increase in preference intensity raises the equilibrium price. If brands are perfect substitutes the equilibrium price approaches the competitive price c and we have the usual Bertrand price model for which p = c for all n greater than or equal to 2.
So assume that the number of firms is not fixed. Entry competition, where can increase the number of firm n will affect the single-price equilibrium. Even in traditional Cournot models of imperfect competition, entry may not lower the equilibrium price (Seade, 1980). Entry shifts each firm's demand curve inward, the elasticity of demand may not rise and consequently the equilibrium price may not fall. On the other hand, a complete characterization gains for the case of unbounded entry. If each firm has strictly positive fixed costs K, the market can support only a finite number of firms. Instead, if we ignore the fact that number of firms has to be integer then a zero-profit equilibrium is characterized by the usual Chamberlinian tangency of demand with average cost. Zero profits condition (price equals average cost) is:
𝑝 = 𝑐 +(<% (25)
where L/n is the quantity every firm sells in a symmetric equilibrium. Only if the level of fixed costs approaches zero, that is the situation where entry is perfectly free, may the number of competitors become infinite. Author further presents two propositions that have conditions for the perfectly free entry namely the price to be equal to the perfectly competitive price even when consumers have different brand preferences and brands are not perfect substitutes.
The Nash equilibrium price approaches the competitive price if all firm’s Nash demand curves become perfectly elastic, for that case even a small increase in price causes a firm to lose all of its customers. There are two cases for consideration. First case is that, if preferences are bounded from above (that is Proposition 2), a brand only gains those customers for whom the brand is the best buy; then, as the number of firms increase infinitely large, a brand only gains those customers who value the brand at the level equal to the upper bound of the consumer preference density. Similarly, as the number of brands is infinitely large, any consumer's preference for his next highest-valued brand also approaches the upper bound of the consumer preference density. There exist many close ties among consumer preferences for the available brands.
In other words, all of the firms have close substitutes for the firm's brand. As a result, if the firm increases its price even slightly, every consumer will choose another brand instead. As a small increase in price causes a loss of all customers this situation implies demand as perfectly elastic, and price is reduced to the competitive level. Further consider the case of an unbounded preference density. In that case consumer's valuations of his most preferred brand and the next- best substitute may not cluster together resulting to not perfectly elastic demand. Instead, the elasticity of demand depends on the rate at which the preference density approaches zero as measured by the condition where preferences are unbounded from above and goes to infinity (that is Proposition 3). The preference density conditions coming from Propositions 2 and 3 have intuitive interpretations. Each consumer place a finite maximum valuation on entire brand supply implies that each brand's demand curve cuts the price axis at some finite price. Hence we can be sure that there is some sufficiently high but finite price premium at which any brand is not purchased by any of its consumers. Proposition 3 covers cases in which the demand curve does not cut the price axis. Even at an infinitely high price premium, some consumers prefer a brand. This condition thus concerns the elasticity of demand at high prices.
Previously the author has discussed the effect of reductions in the level of fixed costs K. As K goes to 0, we showed that the number of firms the market can support becomes unbounded.
But there have to be fulfilled the conditions of Propositions 2 or 3 in order to obtain perfect competition. In both cases the firm-consumer ratio becomes zero. If the conditions of neither of the propositions hold, in the perfectly free entry case, an unbounded number of firms is in accordance to price greater than marginal cost. In the cases in which Propositions 2 or 3 hold, rises in the size of the market (as measured by L) rises the number of firms n which in turn increases M(n). Thus, in the limit, as L approaches infinity, then n approaches infinity, M(n) approaches infinity, and the equilibrium price approaches the perfectly competitive level. In these cases, the firm-consumer ratio (n/L) becomes zero. If the assumptions of Propositions 2 or 3 do not hold, the economy may not approach perfect competition as the size of the market rises and the number of firms does become unbounded. Overall entry competition (due to decrease in fixed costs or rise in the number of consumers) does not guarantee that the price reduces to marginal cost. Price approaches marginal cost if (i) the demand curve cuts the price axis (Proposition 2) or (ii) the speed of convergence toward the axis is fast enough (Proposition 3). However, cases do exist in which the market is very large, there is an unbounded number of firms, yet each firm has a market power. However, for a firm to keep market power under these conditions, preferences must be unbounded. If consumers' willingness to pay is bounded (as their assets are), we may refuse these possibilities and concentrate on the case where are preference bounded.
Further author examine the uniqueness of single price equilibria and the possibility of multiprice equilibria. At the beginning we examines uniqueness issue by proving that a multiplicity of single price equilibria is impossible if we have a fixed number of firms. So the Proposition 4 is as follows.
𝑝 −&-=-;(()' (26)
The left hand side of equation above is monotonically increasing in p and the right hand side is monotonically decreasing. As the left hand side is equal to zero at p=c and the right hand side is positive for all positive finite p, a single equality must occur at a price above c. But this result does not rule out the additional possibility of multiprice equilibria.
While we have not ruled out the possibility of existence of multiprice equilibria in general, possible equilibrium can be refused in the case of a duopoly where number of firms is equal to
In duopoly, the probability of obtaining representative customer with preferences
𝜃 = (𝜃', 𝜃$) (27) is for firm 1 given by equation
𝑃𝑟 𝑃𝑟 (𝑠' ≥ 𝑠$) = 𝑃𝑟 (𝜃'− 𝜃$ ≥ 𝑝$− 𝑝') (28)
The distribution H(𝜇), where
𝜇 ≡ 𝜃'− 𝜃$ (29)
𝜇 is symmetric, it has mean equal to zero, so that H(0) is equal to 1/2.
If we substitute the definition of 𝜇 into equation above and normalize the number of consumers to one (L=1) so that expected sales is equal to the representative probability, so we obtain these two equations that define the quantity
𝑄'(𝑝', 𝑝$) = 𝐻(𝑝$− 𝑝') (30) 𝑄$(𝑝', 𝑝$) = 1 − 𝐻(𝑝$− 𝑝') (31)
Further we can calculate from that the equations that define the price, which is 𝑝' = 𝑐 +=(-?(-'>-%)
'>-%) (32)
𝑝$ = 𝑐 +'> =(-?(- '>-%)
'>-%) (33)
When we subtract two equations above, we obtain 𝑝$ − 𝑝' = ?(-'
'>-%)[1 − 2𝐻(𝑝$− 𝑝')] (34)
And as H (0) is equal to ½, the equation above is only satisfied at the single price p=p1=p2. This single price equilibrium is unique and is given by:
𝑝 = 𝑐 +=(@)?(@) (35) This means that two-price equilibria can be easily ruled out. The derived model can be also reinterpreted for examination of the interaction of imperfect information and product differentiation as follows. Assume that consumers are not perfectly informed about the availability of competing brands in the market. For instance, consider a market in which there exist three brands but every single consumer is aware of only two of them. If each of the three
possible pairs of brands have equal likelihood to be known by any consumer, then on average one-third of all consumers know and thus choose between brands one and two, other third of all consumers choose between brands one and three, and last third choose between brands two and three. These subsets therefore can be defined as three duopoly submarkets. Calculating demands, the slope of the demand curve faced by the first brand for each of its two submarkets is given by equation
->&
- = -;(()' (36)
for n is equal to 2.
As it has other two submarkets, the slope of brand one's demand curve is twice the slope given by equation above. However, its elasticity of price is unchanged because the quantity sold in the two duopoly submarkets is also twice that of one submarket. Therefore the price in equilibrium for the aggregate market that consists of two duopoly submarkets equals the duopoly equilibrium price. Generalizing this argument to an n brand industry, suppose each consumer is aware of only k < n brands. Then there exist
𝑚 = ( 𝑛 𝑘 ) (37)
equal-sized submarkets, each of them consists of k brands competing for L/m consumers. Each submarket is identical and equilibrium is achieved at the k-firm equilibrium price p(k). A similar analysis can be made for the type of localized competition that characterizes Hotelling- style models of spatial competition. Consumers strictly prefer the k stores located nearby to the other n - k more distant stores in that model.
Conclusion of theoretical part
Conclusion from Salop (1979) is as follows. Explicit attention has been paid to the role the other commodity (second industry) plays in determining the properties of monopolistically competitive equilibrium. The major contribution of the approach is providing a rationale of the kinked demand curve in terms of symmetric Nash conjectural variations. The industry equilibrium model illustrates properties when equilibrium occurs at a Chamberlinian tangency away from the kink. The properties of equilibria occurring at the kink are perverse. Industry prices do not conform to small cost changes in the short run.
The process of entry and exit is the only way the industry conforms to a cost changes. Only through the process of entry and exit that the industry conform to a cost changes. Above that, at the kinked equilibrium, the long run response to increasing in a cost is exit by some brands followed by a decline in industry prices as remaining brands better utilize economies of scale.
In a deterrence equilibrium sequential entrants locate in such a way that no new entrant is located in the interval between two firms. As a result, the deterrence equilibrium has half the number of brands than the SZPE. If the SZPE is competitive, the deterrence equilibrium may be competitive, kinked or monopolistic. Which of those equilibrium prevails depends on the particular parameter values. As Hay (1976) showed the deterrence equilibrium in competitive market has its prices higher than at the SZPE. The kinked or monopoly deterrence equilibria may lead to lower prices than equilibria in the competitive SZPE. If the SZPE is kinked, the deterrence equilibrium lies at the monopoly point and causes lower prices and unserved sections of the circular market. Finally, the kink appears robust as the number of dimensions of products space increases, but as Archibald-Rosenbluth (1975) and Weiss (1977) stress equilibrium may not exist with higher dimensional product spaces.
Summary from the article from Perloff is as follows. First, as preferences become stronger, equilibrium price increases. Second, if the value that consumers obtain from each possible brand is bounded, perfectly free entry eliminates monopoly schema. Third, only one single- price equilibrium exists. Fourth point is that if a significant number of consumers have identical tastes, then a single-price equilibrium may not exist. The model I analyzed in detail is Chamberlinian in nature; every brand competes equally with every other available brand.
Although the model considers differences in the preferences of individual consumers, there could be obtained a representative consumer model of the kind analyzed by Spence (1976) and Dixit and Stiglitz (1977) by treating the joint preference density as the aggregation of the preferences of a representative consumer. Results from localized (spatial) competition is as
follows. From additional brands in the market consumers get additional utility when more brands are available.
Although every consumer has some most-preferred brand (best buy) in this approach, the concept of "localized" (spatial) competition is only imperfectly captured by treating consumers as considering only a subset of the available brands. In this case, entry does not entirely eliminate market power. The alternative approach would be for every consumer to have exactly one brand valuation equal to some preferences and all other brands are valued less according to some compensation (transport cost) function.
According to theory, larger supply of differentiated rental apartments leads to the point, where consumer has greater probability to get better combination of his preferences and then to gain greater utility. That combination can be for instance preference for nature or park close to rental apartment but at the same time he wants to satisfy the preference of possibility to be in the centre of the city up to 45 minutes. If this combination exists on the market of rental apartment, consumer is willing to pay extra amount of money if consumer has budget which will cover that extra amount of money. Another preference combination can be school close to the rental apartment but at the same time consumer want to have rental apartment located in silent part of the city away from the centre.
Theoretical part discusses the market structure and equilibrium in the Salop model with no free entry that best represents the real estate market, i.e. the focus is on the supply side optimal behavior, but empirical part studies rather the relationship between product characteristics and equilibrium price, which may be qualified as a demand side issue. Furthermore, we have clear uniform preference on some of the dimensions, which implies vertical differentiation based on quality and not on individual preferences (location).
Comparison of the rental markets in Brussels, Berlin, and Prague
In the beginning of this section we describe the single markets with their specification and then in the last paragraph we compare them.
We start with rental situation in Brussels. Brussels is the capital city of Belgium and an international administration center for NATO. This is why the city fetches high prices for the private property and rental market. Brussels is divided into two areas that are made up of 19 districts within the Municipality area. Woluwe is the residential district where the NATO headquarters is situated. This is a pricey district that attracts high income earners. Although property prices in the capital are the highest in the country, they’re still significantly cheaper than in London or Paris. Unusually, the tenant is responsible for most repairs and improvements required during their tenancy. City centre rents are typically around €1,000–1,500 per month for a two-bedroom apartment, and lower in less popular areas. The Belgian government provides a tool to calculate an estimated rental price of rented apartment, which can also help tenant determine if he is paying too much or too low rent. Adverts must include an accurate rent, otherwise the landlord can be fined. Negotiating the rent is rare. Expect an annual rent increase (or occasionally decrease) tied to the cost of living index. This will be applied retrospectively for the preceding three months. Base rent may be increased either at contract renewal (short term lets) or every three years (standard and long term leases). The tenancy agreement should list the details. In Brussels, utilities are the responsibility of the renter and are usually paid separately. The rent may include water and sewage, but any others are rare. In addition, there are often fees owed to the housing association (particularly for apartment blocks) or the commune (e.g., for garbage collection). The deposit will be up to three month’s rent. It must be handled separately from the landlord’s other finances and the rent, and should be placed in a special account. There is the standard of a nine-year contract (often known as a 3-6-9 contract as the landlord can only increase the base rent every three years). In Belgium, a short-term contract is three years or less, however, the standard contract of nine years can actually be more flexible. Short-term contracts impose a penalty for giving notice before the end of the contract; in many cases, tenant will be charged for the full duration of the contract if she leaves early. Longer contracts – from nine years to the long-term contract of up to 25 years – impose penalties (up to three months’ rent) for giving notice in the first three years;
after four years, no penalty applies for breaking a contract. Even where penalties apply, the tenant can give three months’ notice at any time. Still, landlords will also have to pay a penalty
of several months’ rent to the tenant if they give the tenant notice to leave. Another alternative for renting in Brussels is co-living spaces. Locations like Morton Place offer individual bedrooms and communal living spaces, such as kitchens and social areas. Although Brussels is home to over a million people, it’s only about 20km in diameter. The city center is primarily apartment blocks and old townhouses split into apartments, but this quickly gives way to suburbs. As a result, it’s easy to find a house with a garden.
Further we consider Berlin. Berlin has 12 official districts. Since the fall of the Berlin Wall in 1989, the division between the city has been eroded by new-builds and standardized services.
It has reinvented itself as an exciting, innovative, modern capital and is an appealing destination for students, entrepreneurs and expats from around the world. In 2013, the city welcomed 50,000 new residents, swelling its numbers to around 3.5 million people. It’s both walkable and easy to cycle, so owning a car is a matter of choice, not necessity. Moreover, the city’s history is still visible on its streets, but it is not mired in the past. With a high demand for property, finding a place can be challenging. Rent is around 1 200 EUR or more for luxurious properties, small properties, and apartments in particularly desirable areas. House prices have risen dramatically in Germany in the last 10 years, with a knock-on effect on rents. Rents have increased by over 28% since 2007, with properties in desirable areas increasing by up to 10%
annually. This is partly due to the German housing market being a sound investment in the wake of the 2008 economic crisis, and partly due to an influx of new residents. Popular expat and arty districts, such as Prenzlauer Berg, are particularly hit. Locals are even campaigning against the increases, partly on the basis that the city is losing its artists and musicians. On the other hand, it’s difficult for a landlord to increase the rent significantly while a tenant is in situ.
Berlin has strict regulations about rental increases: they should not happen more than once per year, and not total more than 20% in three years. The property description should indicate whether the rent includes any utilities. If it doesn’t, it’s a ‘cold rent’ (Kaltmiete) compared to
‘warm rent’ (Warmmiete) that will include heating bills and may include other costs. A deposit (Kaution) equal to three months’ rent is standard. The sum should not be greater than three months’ rent. A deposit must be kept in an escrow account, a type of savings account that is separate from the landlord’s or estate agent’s business accounts. Most leases are unlimited or have a two-year initial contract before becoming unlimited. This means there are no contract amendment or renewal periods. A tenant may therefore stay as long as they choose, giving three months’ notice when they quit. To remove a tenant, the landlord must either serve an eviction notice (by going through the courts) or give three months’ notice. Most Berliners
(85%) rent their home, and as a result there is a constant supply of apartments and houses coming up for rent. The best deals are to in buildings where the landlord has owned the property for a significant period. The worst deals are from profit-hungry land investors.
Now I focus on Prague rental market. Prague is divided into 55 municipal areas and into 112 cadastral areas. Apartment rental typically costs between 10,000 CZK and 30,000 CZK per month according to real estate websites such as www.sreality.cz or reality.idnes.cz, depending on how tenant wants to live in terms of area and location of the apartment. The closer to the centre the more expensive per sqm. Utilities are not usually included in the rent and can cost an additional 1000-3000 CZK per month and per person depending on the area of the apartment. Tenant can either have utilities billed to him or can have it billed to the owner of the flat. If utilities are billed to landlord tenant ends up paying a monthly deposit and at the end of the year his landlord calculates the total cost for the year. For a short-term lease the tenant does not need a green card but for a long-term lease a residency permit is required.
A short-term lease is more expensive (30-40% more per month), but if an expat does not have a green card it can be his only option. As far as contracts go, a lease is typically for one year and tenant need to give between one and three months’ notice before leaving the flat. Tenants typically have to pay a deposit equivalent to one month's rent on any contract. There is flat sharing as an option as well. Some of the nicer and less touristy neighborhoods include Dejvice, Pohořelec, Střešovice, Břevnov, Letná and Vinohrady but there are plenty of other great places to live. Average net salary in Berlin is 2 486.56 EUR per month and average cost for 3 bedrooms apartment is 1 819.41 EUR in city centre and 1 231.11 EUR outside of centre Average net salary in Brussels is 2,422.97 EUR and average cost for 3 bedrooms apartment is 1,536.36 EUR in city centre and 1,274.78 EUR outside of centre. Average net salary in Prague is 32 397 CZK and average cost for 3 bedrooms apartment is 32 000 CZK in city centre and 22 845 CZK outside of centre. As we can see, person in Berlin spend on average from around 49 % to 73 % of net salary on rent, person in Brussles spend on average from around 52 % to 63 % of net salary on rent and person in Prague spend on average from around 70,5 % to 98 % of net salary on rent. Rent costs and net salaries are according to www.numbeo.com.
The Model
The price per month of rental housing as dependent variable is measured in CZK, independent variables are categorical and numerical. Categorical data, such as equipment or balcony acquire binary values zero, if condition is not satisfied, and one, if condition is satisfied. Continuous variables such as restaurant, train or tram acquire values in meters. Dummy variables control for location affiliation and represent where the rental apartment is located. Author uses regression analysis for inferring causal relationships between the independent and dependent variables and this relationship is estimated by OLS. In other words how independent variables and dummy variables affect level of independent variable. Data sample is not random, because they were scrapped from www.sreality.cz web server during short period of time and therefore are not representative. It means that statistical inference presented in the thesis can only be applied for advertisement of rental apartment on web server www.sreality.cz, because data sample was selected from this web page. Data was scraped during June 2020 and the oldest advertisement was from February 2020, the youngest advertisement was from June 2020. I developed Java Spring Boot application using Selenium Framework for scraping the data sample. I model four main relationships between dependent variable and various quantity of independent and dummy variables.
I expect that price per month will be the higher the closer to the centre of Prague is the rental apartment located. Centre of Prague is defined as Prague 1. I also expect that the dependent variable will be the higher the closer to bus, tram, train or underground station the rental apartment is located, but only if a rental apartment is located in outskirts of Prague. Outskirts of Prague is defined as part of Prague 5, part of Prague 4, part of Prague 10, part of Prague 9, part of Prague 8, part of Prague 6.
Prague City districts
I assume that relationship between independent variables and dependent variable is not significant for bus, train, tram or underground station and also for shops and restaurants if the rental apartment is located closer to the Prague centre due to high density of these variables in the city centre. Compared to that author assumes, that floor number, if cellar, garage or lift is present in the house where rental apartment is located and school, post, pharmacy and sport field will have significant relationship on level of the dependent variable price per month.
Regression analysis 1:
First regression analysis estimates relationship between dependent variable price per month and independent variables. In this regression analysis data are not separated into locations.
Author estimated two models, first model uses all of independent variables and the second model uses only independent variables that came out significant in the first model.
Theoretical regression equation for the first model is as follows: (38) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎!+ 𝑏A𝑎𝑡𝑚!+ 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C
𝑏𝑢𝑠!+ 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟!+ 𝑏E𝑓𝑙𝑜𝑜𝑟!+ 𝑏F𝑔𝑎𝑟𝑎𝑔𝑒! + 𝑏𝑠𝑝𝑜𝑟𝑡𝑓𝑖𝑒𝑙𝑑! + 𝑏'@𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦! + 𝑏'' 𝑝𝑜𝑠𝑡!+ 𝑏'$𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'A𝑠𝑐ℎ𝑜𝑜𝑙!+ 𝑏'B𝑠ℎ𝑜𝑝!+ 𝑏'C𝑙𝑖𝑓𝑡!+ 𝑏'D𝑡𝑟𝑎𝑖𝑛! + 𝑏'E𝑡𝑟𝑎𝑚!
+𝑏'F𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑! + 𝑒!
Theoretical regression equation for a reduced model is as follows: (39) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎!+ 𝑏A𝑎𝑡𝑚!+ 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C
𝑏𝑢𝑠!+ 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟!+ 𝑏E𝑔𝑎𝑟𝑎𝑔𝑒! + 𝑏F𝑠𝑝𝑜𝑟𝑡𝑓𝑖𝑒𝑙𝑑! + 𝑏G𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦!+ 𝑏'@𝑝𝑜𝑠𝑡! + 𝑏''𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'$𝑠𝑐ℎ𝑜𝑜𝑙! + 𝑏'A𝑠ℎ𝑜𝑝! + 𝑏'B𝑙𝑖𝑓𝑡! + 𝑏'C𝑡𝑟𝑎𝑖𝑛! + 𝑏'D𝑡𝑟𝑎𝑚! + 𝑏'E𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑!+ 𝑒!
Regression analysis 2:
Second regression analysis estimates relationship between dependent variable price per month, every independent variables and dummy variables controlling for first type of location. Author divided cadastral municipalities of Prague into 22 areas, the quantity of dummy variables is 20, author removed oblast 22 because of singularity problem. Oblast 1 consists only of Josefov and therefore oblast 1 is reference area for this model.
I estimated two models, first model uses all of independent variables and all of dummy variables and the second model uses only independent and dummy variables that was significant in the first model.
Theoretical regression equation for the second model is as follows: (40)
𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎! + 𝑏A𝑎𝑡𝑚! + 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C𝑏𝑢𝑠! + 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟! + 𝑏E𝑓𝑙𝑜𝑜𝑟! + 𝑏F𝑔𝑎𝑟𝑎𝑔𝑒!+ 𝑏G𝑙𝑖𝑓𝑡! + 𝑏'@𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦!+ 𝑏''𝑝𝑜𝑠𝑡!
+ 𝑏'$𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'A𝑠𝑐ℎ𝑜𝑜𝑙! + 𝑏'B𝑠ℎ𝑜𝑝!+ 𝑏'C𝑠𝑝𝑜𝑟𝑡𝑓𝑖𝑒𝑙𝑑 + 𝑏'D𝑡𝑟𝑎𝑖𝑛! + 𝑏'E𝑡𝑟𝑎𝑚! + 𝑏'F𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑!+ 𝑗'𝑜𝑏𝑙𝑎𝑠𝑡2! + 𝑗$𝑜𝑏𝑙𝑎𝑠𝑡3! + 𝑗A𝑜𝑏𝑙𝑎𝑠𝑡4! + 𝑗B𝑜𝑏𝑙𝑎𝑠𝑡5! + 𝑗C𝑜𝑏𝑙𝑎𝑠𝑡6! + 𝑗D𝑜𝑏𝑙𝑎𝑠𝑡7! + 𝑗E𝑜𝑏𝑙𝑎𝑠𝑡8! + 𝑗F𝑜𝑏𝑙𝑎𝑠𝑡9! + 𝑗G𝑜𝑏𝑙𝑎𝑠𝑡10! + 𝑗'@𝑜𝑏𝑙𝑎𝑠𝑡11! + 𝑗''𝑜𝑏𝑙𝑎𝑠𝑡12! + 𝑗'$𝑜𝑏𝑙𝑎𝑠𝑡13! + 𝑗'A𝑜𝑏𝑙𝑎𝑠𝑡14!+ 𝑗'B𝑜𝑏𝑙𝑎𝑠𝑡15! + 𝑗'C𝑜𝑏𝑙𝑎𝑠𝑡16! + 𝑗'D𝑜𝑏𝑙𝑎𝑠𝑡17! + 𝑗'E𝑜𝑏𝑙𝑎𝑠𝑡18!+ 𝑗'F𝑜𝑏𝑙𝑎𝑠𝑡19! + 𝑗'G𝑜𝑏𝑙𝑎𝑠𝑡20! + 𝑗$@𝑜𝑏𝑙𝑎𝑠𝑡21! + 𝑒!
Theoretical regression equation for the reduced model is as follows: (41) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎! + 𝑏A𝑎𝑡𝑚! + 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C𝑏𝑢𝑠! + 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟! + 𝑏E𝑓𝑙𝑜𝑜𝑟! + 𝑏F𝑔𝑎𝑟𝑎𝑔𝑒! + 𝑏G𝑙𝑖𝑓𝑡! + 𝑏'@𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦!
+ +𝑏''𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'$𝑠ℎ𝑜𝑝! + 𝑏'A𝑡𝑟𝑎𝑚!+ 𝑏'B𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑! + 𝑗'𝑜𝑏𝑙𝑎𝑠𝑡3! + 𝑗$𝑜𝑏𝑙𝑎𝑠𝑡4! + 𝑗A𝑜𝑏𝑙𝑎𝑠𝑡5! + 𝑗B𝑜𝑏𝑙𝑎𝑠𝑡6!+ 𝑗C𝑜𝑏𝑙𝑎𝑠𝑡7! + 𝑗D𝑜𝑏𝑙𝑎𝑠𝑡8!+ 𝑗E𝑜𝑏𝑙𝑎𝑠𝑡9!+ 𝑗F𝑜𝑏𝑙𝑎𝑠𝑡10! + 𝑗G𝑜𝑏𝑙𝑎𝑠𝑡11! + 𝑗'@𝑜𝑏𝑙𝑎𝑠𝑡12! + 𝑗''𝑜𝑏𝑙𝑎𝑠𝑡13!+ 𝑗'$𝑜𝑏𝑙𝑎𝑠𝑡14! + 𝑗'A𝑜𝑏𝑙𝑎𝑠𝑡15! + 𝑗'B𝑜𝑏𝑙𝑎𝑠𝑡16!
+ 𝑗'C𝑜𝑏𝑙𝑎𝑠𝑡17!+ 𝑗'D𝑜𝑏𝑙𝑎𝑠𝑡18! + 𝑗'E𝑜𝑏𝑙𝑎𝑠𝑡19! + 𝑗'F𝑜𝑏𝑙𝑎𝑠𝑡20! + 𝑗'G𝑜𝑏𝑙𝑎𝑠𝑡21! + 𝑒!
Regression analysis 3:
Third regression analysis estimates relationship between dependent variable price per month, independent variables and dummy variables controlling for second type of location. Second type of location is all cadastral municipalities that are included in the dataset. Prague has 112 cadastral municipalities and there are 100 cadastral municipalities in dataset present.
I estimated two models, first model uses all independent variables and all dummy variables and the second model uses only independent and dummy variables that were significant in the first model. The first theoretical regression equation contains all of the independent variables and 99 dummy variables.
Theoretical regression equation for reduced model is as follows: (42) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎!+ 𝑏A𝑎𝑡𝑚!+ 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C
𝑏𝑢𝑠!+ 𝑏D𝑓𝑙𝑜𝑜𝑟!+ 𝑏E𝑔𝑎𝑟𝑎𝑔𝑒! + 𝑏F𝑙𝑖𝑓𝑡! + 𝑏G𝑡𝑟𝑎𝑚!+ 𝑗'𝐷𝑒𝑗𝑣𝑖𝑐𝑒!+ 𝑗$𝐻𝑟𝑎𝑑č𝑎𝑛𝑦! +𝑗A𝐽𝑜𝑠𝑒𝑓𝑜𝑣!+ 𝑗B𝐾𝑜šíř𝑒! + 𝑗C𝑀𝑎𝑙á 𝑆𝑡𝑟𝑎𝑛𝑎!+ 𝑗D𝑁𝑜𝑣é 𝑀ě𝑠𝑡𝑜! + 𝑗E𝑃𝑜𝑑𝑜𝑙í!
+ 𝑗F𝑆𝑡𝑎𝑟é 𝑀ě𝑠𝑡𝑜! + 𝑗G𝑆𝑡ř𝑒š𝑜𝑣𝑖𝑐𝑒! + 𝑗'@𝑇𝑟𝑜𝑗𝑎! + 𝑒! Regression analysis 4:
The last regression analysis estimates relationship between dependent variable price per month, every independent variable and dummy variables controlling for third type of location. Third type of location is all of Municipal areas of Prague that are included in the dataset. Prague has 10 Municipal areas and there are all of them in dataset present. Prague 1 is the reference area.
Author estimated two models, first model uses all of independent variables and all of dummy variables and the second model uses only independent and dummy variables that were significant in the first model.
Theoretical regression equation for the last model is as follows: (43) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎!+ 𝑏A𝑎𝑡𝑚!+ 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C 𝑏𝑢𝑠! + 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟! + 𝑏E𝑓𝑙𝑜𝑜𝑟! + 𝑏F𝑔𝑎𝑟𝑎𝑔𝑒!+ 𝑏G𝑠𝑝𝑜𝑟𝑡𝑓𝑖𝑒𝑙𝑑!+ 𝑏'@𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦! + 𝑏'' 𝑝𝑜𝑠𝑡!+ 𝑏'$𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'A𝑠𝑐ℎ𝑜𝑜𝑙!+ 𝑏'B𝑠ℎ𝑜𝑝!+ 𝑏'C𝑙𝑖𝑓𝑡!+ 𝑏'D𝑡𝑟𝑎𝑖𝑛! + 𝑏'E𝑡𝑟𝑎𝑚!
+𝑏'F𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑! + 𝑗'𝑜𝑏𝑙𝑎𝑠𝑡2! + 𝑗$𝑜𝑏𝑙𝑎𝑠𝑡3!+ 𝑗A𝑜𝑏𝑙𝑎𝑠𝑡4!
+𝑗B𝑜𝑏𝑙𝑎𝑠𝑡5! + 𝑗C𝑜𝑏𝑙𝑎𝑠𝑡6! + 𝑗D𝑜𝑏𝑙𝑎𝑠𝑡7! + 𝑗E𝑜𝑏𝑙𝑎𝑠𝑡8! + 𝑗F𝑜𝑏𝑙𝑎𝑠𝑡9! + 𝑗G𝑜𝑏𝑙𝑎𝑠𝑡10! + 𝑒!
Theoretical regression equation for the reduced model is as follows: (44) 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ! = 𝑏@+ 𝑏'𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡! + 𝑏$𝑎𝑟𝑒𝑎!+ 𝑏A𝑎𝑡𝑚!+ 𝑏B𝑏𝑎𝑙𝑐𝑜𝑛𝑦! + 𝑏C
𝑏𝑢𝑠!+ 𝑏D𝑐𝑒𝑙𝑙𝑎𝑟!+ 𝑏E𝑓𝑙𝑜𝑜𝑟!+ 𝑏F𝑔𝑎𝑟𝑎𝑔𝑒! + 𝑏G𝑙𝑖𝑓𝑡! + 𝑏'@𝑝ℎ𝑎𝑟𝑚𝑎𝑐𝑦!+ 𝑏''𝑝𝑜𝑠𝑡! + 𝑏'$𝑟𝑒𝑠𝑡𝑎𝑢𝑟𝑎𝑛𝑡! + 𝑏'A𝑡𝑟𝑎𝑖𝑛! + 𝑏'B𝑡𝑟𝑎𝑚! + 𝑏'C𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑! + 𝑗'𝑜𝑏𝑙𝑎𝑠𝑡2! + 𝑗$𝑜𝑏𝑙𝑎𝑠𝑡3! + 𝑗A𝑜𝑏𝑙𝑎𝑠𝑡4!
+𝑗B𝑜𝑏𝑙𝑎𝑠𝑡5! + 𝑗C𝑜𝑏𝑙𝑎𝑠𝑡6! + 𝑗D𝑜𝑏𝑙𝑎𝑠𝑡7! + 𝑗E𝑜𝑏𝑙𝑎𝑠𝑡8! + 𝑗F𝑜𝑏𝑙𝑎𝑠𝑡9! + 𝑗G𝑜𝑏𝑙𝑎𝑠𝑡10! + 𝑒!
The following tests are performed: t-test, F-test, test for multicollinearity, test for normality of residuals and test for robustness. Robustness refers to the strength of a statistical procedures according to the specific conditions of the statistical analysis. I chose random sample from the dataset and performed regression analysis again. If the direction of parameters compared to original model changed it means there is a problem with robustness in the dataset.
Dataset
The thesis exploits cross-sectional data for Prague. Data was scraped during three days in June 2020 from www.sreality.cz website using script, which author of the thesis wrote in Java programming language using Selenium framework and Chromedriver library. Data was stored in Postgres database running in the Docker virtual environment. Data from the website are not older than 5 months. The analysis uses one dependent variable, 18 independent variables and different quantities of dummy variables dependent on character of regression analysis. The dependent variable is the price per month of rental housing measured in CZK. Independent variables are as follows.
Table 1: Independent variables
equipment area atm balcony bus cellar floor garage sport field pharmacy post
office
restaurant school shop lift train tram underground
Following independent variables are categorical: accommodation, balcony, cellar, garage and lift. Meaning of these independent variables is as follows. For instance the independent variable equipment means that the rental apartment is rented equipped or is rented unequipped.
The rest of the independent variables are numerical variables and express the distance from the rented apartment. They are measured in meters and were listed in the rental offer. Around 5,7 percent of the data was missing so the author chose the strategy to add the mean of particular independent variable in municipal district instead of removing them. Author removed 60 observation due to too high (for example 500 000 CZK per month) or too low (for example 4500 CZK per month) rent per month.
