# Testing Cost Inefficiency under Free Entry in the Real Estate Brokerage Industry

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1 Web Appendix to Testing Cost Inefficiency under Free Entry in the Real Estate Brokerage Industry Lu Han University of Toronto Seung-Hyun Hong University of Illinois December 23, 2010 A Proof for Proposition 1 Proposition 1 Suppose that Assumptions 1-3 hold. Then, N e > N. That is, the equilibrium number of real estate agents is socially excessive. Proof: Differentiating (2) with respect to N yields V (N) = C(q, N) N ( C q q N + C ) F N = C(q, N) + q C q N C N F (16) where (16) follows from the symmetric assumption q = Q/N. Adding and subtracting τp q and rearranging the terms yields ( V (N) = π(n) τp C ) q N C q N The socially optimal number of agents, N, satisfies V (N ) = 0. At the free-entry equilibrium number of agents, we have ( V (N e ) = π(n e ) q e τp C ) N e C q N = π(n e ) w q e ( C(q, N e ) + F q C q ) N e C N = π(n e ) w q (AC(q e, N e ) MC(q, N e )) N e C } {{ } } {{ N} Loss of Economies of Scale Wasteful Competition (17) where the second equality follows from (1), and the last equality follows from definitions that AC C(q,N)+F q and MC C q. Note that the first term, π(n e ) w, is equal to zero under free entry. The second term is negative under Assumption 3, as declining average costs mean that 1

2 marginal cost is necessarily below average cost. The third term is negative under Assumption 2. Thus, V (N e ) < V (N ) = 0. We complete the proof by establishing the global concavity of V (N). From (16), we have V (N) = 2 C N N 2 C N 2 + 2q 2 C q N q2 2 C N q 2, which is negative under Assumptions 1-2. It follows that N e > N. B Simple Model with Product Differentiation In this section, we explicitly take into account product differentiation and examine how this would affect the direction of the entry bias identified in Proposition 1. Following Mankiw and Whinston (1986), we define the total consumer surplus in the following way: G(Nf(q), P, τ) Note that total consumer surplus now depends not only the aggregate level of output, but also on the number of real estate agents. For given P and τ, the function G( ) satisfies the following assumptions. Assumption 4. f(0) = 0, f (q) > 0, and f (q) 0 for all q 0. Assumption 5. G (z) > 0 and G (z) 0 for all z Nf(q) 0. Assumptions 4-5 imply that households prefer increasing variety of real estate brokerage service and that the types of brokerage service provided by different agents are substitutes. In equilibrium, consumer maximization implies that the commission that a household pays to a real estate agent is exactly equal to its marginal utility, G (Nf(q))f (q). Thus, the equilibrium level profit per agent is π G (Nf(q))f (q)q C(q, N) F The objective of the social planner is to select N to solve: max W (N) G(Nf(q)) NC(q, N) NF N To examine whether entry is socially efficient, we begin by differentiating W (N) as follows ( W (N) = G f(q) + Nf q ) ( C q N N q N + C ) C(q, N) N Adding and subtracting G f q and rearranging terms give W (N) = π + N q ( G f C ) N C N q N + ( G f f q ) The socially optimal number of agents, N, satisfies W (N ) = 0. 2

3 The market equilibrium requires that G f = τp (consumer optimality) and that τp = AC(q, N e ) (free entry condition). Thus the market equilibrium number of agents, N e, satisfies W (N e ) = π(q, N e ) + N e q N (AC(q, N e ) MC(q, N e )) } {{ } Loss of Economies of Scale N e C N } {{ } +G ( f f q ) } {{ } Wasteful Competition Product Differentiation The equation above is a generalization of (17). The first three terms are identical to those in (17), and the sum of these components is negative under Assumptions 1-3, as shown earlier. The fourth term represents the effect of product differentiation, and Assumptions 4-5 imply that it is positive. More specifically, if households have a preference for variety of brokerage service, a marginal real estate agent increases consumer surplus by providing more variety, but does not capture this gain in profits. If this effect dominates the loss of economies of scale and wasteful competition, then entry will be socially insufficient. Nevertheless, it is plausible to conclude that entry is still socially excessive, to the extent that the product differentiation effect is limited to a certain degree. (18) C Evidence on Price Competition and Product Differentiation The main empirical findings in this article have established evidence for cost inefficiency under free entry in the real estate brokerage market. As discussed in Section 2, entry could potentially benefit consumers either through price competition or through product differentiation. Given the lack of information on the demand side for the real estate brokerage service in the PUMS, we cannot recover the magnitude of these potential benefits. In this appendix, using separate datasets, we show that consumers benefits from possible price competition and product differentiation are quite limited, thereby providing evidence for the two simplifying assumptions described at the end of Section 2. C.1 Price Competition It is well known that real estate brokerage is characterized by relatively fixed commission rates (e.g., FTC report, 1983; Hsieh and Moretti, 2003). One observation from the American Bankers Association is that one would expect to see variations in brokerage commissions across geographical regions as the supply and demand varies dramatically across the United States, but little if any variation exists. In light of the motivation of this paper, it is reasonable to ask whether entry has helped bring down commission rates in recent years. Tables 8 and 9 indirectly show the relationship between entry and commission rates. Table 8 shows a positive correlation between changes in log of housing prices and changes in log of the number (or fraction) of real estate brokers/agents in MSA. We compute the correlation coefficients using the MSAs whose geographical boundaries did not change between 1990 and This positive relationship is consistent with evidence presented by Hsieh and Moretti (2003). To examine the relationship between housing prices and commission rates, we then use the Consumer Expenditure Surveys (CEX), because the PUMS does not contain any information on commission fees. We follow Hsieh and Moretti (2003), and estimate the commission rate by dividing total selling expenses (including commission fees) by the price of housing. To reduce measurement errors, we drop observations with implausibly large or small estimated commission rate (less than 1 percent or more than 10 percent), again following Hsieh and Moretti (2003). The 3

4 total number of observations then becomes 520, since only a handful of observations in the CEX reported selling their houses during the survey period. Table 9 reports the relationship between housing prices and commission rates. Though commission rates vary slightly across different price ranges, there is no clear positive or negative relationship between the increase in housing price and commission rates. Consequently, we find that the number of real estate agents is positively correlated with housing prices, but housing prices do not seem to be correlated with commission rates. Therefore, entry is less likely to put downward pressure on commission rates. To further explore the relationship between entry and commission rates, Table 10 resents the changes in real estate agent growth and the changes in average commission rates between 2002 and 2003 across 6 regions of the United States. The aggregate statistics are derived from the data based on the 2004 Real Trends Brokerage Performance Report. Using the percentage changes from 2002 to 2003, we control for region-specific fixed effects that might affect both entry and commission rates. The percentage changes in the average commission rates were respectively 1.63%, 0%, and -1.15% in FW, MA, and NE, which are the regions with high growth in the number of real estate agents. Given that the average commission rate was around 5.1% during that period, the magnitude of the changes in commission rates was negligible. Though lack of more detailed information prevents us from further investigation, the descriptive evidence here suggests that entry does not have noticeable effects on commission rates. C.2 Product Differentiation Just like price competition, product differentiation in the real estate brokerage industry is impeded by implicit agreement or state legislation. Many states in the U.S. have imposed so-called minimum-service requirements on real estate agents. As the name suggests, these laws and regulations enumerate specific tasks that a real estate agent must perform for a client. According to the joint statement from the FTC and DOJ, minimum-service requirements limit the degree of differentiation in the real estate brokerage service and reduce the options available to consumers by prohibiting agents from offering a la carte services. While agents are required to provide the standardized package of service to their clients, they may differentiate themselves by trying to improve the quality of their service. Even in this case, entry could still be socially excessive (e.g., Berry and Waldfogel, 1999). The reason is that such improvements result in stolen business. In equilibrium, agents do not take into account the fact that the increase in own-quality reduces the demand for other agents, and therefore over-invest in quality improvement. In this section, we empirically examine whether entry benefits consumers by improving the quality of the brokerage service. Because the PUMS does not provide valid information on the quality of home matching process, we use the 2001 Home Buyer and Seller Survey conducted by the NAR. After excluding the MSAs with fewer than 10 observations and excluding transactions without the assistance of a real estate agent or broker, the sample consists of 3, 145 observations on home buyers and 1, 785 observations on home sellers. We merge this sample with the PUMS 2000 to obtain the MSA-level information on the number of real estate agents and brokers. To measure the quality of real estate brokerage service, we use the household s response to the following question in the NAR survey: Would you use the same agent again for the future home transaction? We define an indicator variable REA satisfaction to be equal to 1 if the answer is yes. We then estimate the probit model in which the dependent variable is REA satisfaction and the independent variables include the number of real estate agents out of total labor force in the 4

5 local market, time on the market, buyers discount (sellers concessions for home sellers sample), housing characteristics, neighborhood characteristics, market conditions, search/selling process characteristics and home buyer/seller characteristics. As shown in Table 11, the key coefficients on the number of real estate agents in both the buyer and the seller samples are insignificant. This suggests that increasing number of real estate agents has a negligible impact on consumers evaluation of the quality of the brokerage service. Therefore, entry by traditional real estate agents does not lead to obvious improvement in home matching quality. D Implementation of NPL with Finite Mixture To implement a finite mixture version of the NPL algorithm, we assume that ξ has a discrete distribution with finite points of support. We then consider the following log-likelihood function at the h-th iteration of the NPL algorithm for given R i and Ŵi ( M K ) L h h 1 (θ, ϕ) = ln ϕ k f k (X m ; θ, N m,k, ω k), (19) m=1 k=1 h 1 N m,k where θ = {α, β, δ R, δ W, δ C }; X m is a vector of market-level observables; is the number of real estate agents in market m for type k, estimated from the (h 1)-th NPL iteration; ξ takes values from {ω 1,..., ω K }, and ξ is i.i.d across markets with probability mass function ϕ k = Pr(ξ = ω k ). Because we allow for market-specific unobserved heterogeneity, we consider the market-level density function for type k given by f k (X; θ, N h 1 k, ω k ) = I Pr(d i = 1 X i ; θ, i=1 h 1 N k, ω k ) d i (1 Pr(d i = 1 X i ; θ, N h 1 k, ω k )) 1 d i. where we suppress market subscripts m. The maximum likelihood estimation of the finite mixture model is usually difficult to estimate, because the log-likelihood function as in (19) does not allow for any additive separability. However, Arcidiacono and Jones (2003) develop an approach to reintroduce additive separability in the log-likelihood function. The key insight of their approach is that maximizing the log of the unconditionally-type-avearged likelihood (ln[ k ϕ kf k (X m ; θ)]) is equivalent to maximizing the conditionally-type-averaged log-likelihood ( k Pr(k X m) ln[f k (X m )]), where Pr(k X m ) is the probability that market m is of type k, conditional on having observed X m. We thus incorporate the Expectation-Maximization (EM) algorithm proposed by Arcidiacono and Jones (2003) as follows. At the beginning of iteration l, we use θ l 1 and ϕ l 1 to update h 1 Pr(k X m ; θ, ϕ), where we suppress N to simplify the notation, and from Bayes theorem, we have Pr(k X m ; θ, ϕ) = m,k ϕ k f k (X m;θ,ω k ) P K k=1 ϕ kf k (X m;θ,ω k ). This completes the E step. In the M step, we obtain the ML estimates ϕ l by ϕ l k = 1 M M m=1 Pr(k X m; θ l 1, ϕ l 1 ), and the ML estimates θ l by θ l = arg max θ M m=1 k=1 K Pr(k X m ; θ l 1, ϕ l 1 ) ln[f k (X m ; θ, ω k )]. We repeat the E step and the M step until θ and ϕ converge. 5

7 Table 8: Correlation Between Entry and Housing Price a changes in log(average price of houses) changes in log(n) changes in log(n/m) a The table reports the correlation coefficients between the changes in log of the number of real estate agents and the changes in the average housing prices between 1990 and 2000 in the PUMS. In the table, N denotes the number of real estate agents, and M denotes total labor force in a market. Table 9: Commission Rate from the CEX a Year Rate House Price Rate House Price Rate a Table reports the average commission rates by year, and by housing prices (at \$10,000 intervals). The commission rate is computed from the Consumer Expenditure Surveys. 7

8 Table 10: Agent Growth and Commission Rate Growth Across Regions b Region Agent Growth Commission Rate Growth SW/MTN 8.50% 1.94% MW 8.70% -4.27% SE 9.50% -2.18% FW 11.10% 1.63% MA 11.10% 0.00% NE 12.00% -1.15% a Sources: This table is derived from the statistics based on the 2004 Real Trends 500 Brokerage Performance Report. The agent growth and commission rate growth are computed as the percentage changes in real estate agents and percentage changes in average commission rates from 2002 to NE indicates Northeast region; MA indicates Mid-Atlantic region; SE indicates Southeast region; MW indicates Midwest region; SW/MTN indicates SW/Mountain region and FW indicates Far West region. b Sources: This table is derived from the statistics based on the 2004 Real Trends 500 Brokerage Performance Report. The agent growth and commission rate growth are computed as the percentage changes in real estate agents and percentage changes in average commission rates from 2002 to NE indicates Northeast region; MA indicates Mid-Atlantic region; SE indicates Southeast region; MW indicates Midwest region; SW/MTN indicates SW/Mountain region and FW indicates Far West region. Table 11: Effects of Competition on Consumer Satisfaction a Home Buyer Sample Home Seller Sample Dependent Variable REA Satisfaction REA Satisfaction Agents/Labor Force (12.83) (16.95) Number of Observations a Standard errors in parentheses. The data source is the 2001 National Association of Realtors Surveys on Home Buyers and Sellers. Additional control variables include housing characteristics, neighborhood characteristics, market conditions, search/selling process characteristics and home buyer/seller characteristics. 8

9 Table 12: Counterfactual Results for Changes in Internet Search Intensity a Variable Mean S.D. P10 P50 P90 Increase Internet adoption rate by 10% and full search intensity N initial equilibrium 3,427 3, ,969 8,387 N counterfactual 3,503 3, ,021 8,654 % change in N mean of R i initial eqm. 42,632 8,060 33,579 42,144 52,657 mean of R i counterfactual 38,801 8,015 29,212 37,907 48,528 % change in mean of R i mean of Ĉi initial eqm. 13,951 5,183 8,615 12,559 20,306 mean of Ĉi counterfactual 11,993 5,188 6,857 10,755 17,802 % change in mean of Ĉi T C initial equilibrium 46,288,585 64,559,029 1,958,049 18,607, ,664,544 T C counterfactual 41,427,847 59,505,869 1,611,195 14,994, ,628,656 % change in T C #MSAs 160 a The counterfactual results are computed using new equilibrium entry probabilities. In the table, the mean reports the weighted mean, using total labor force as weights. P10 is the 10th percentile, and similarly for P50 and P90. T C is MSA-level total cost. % change reports the percentage change between the initial equilibrium value and the counterfactual equilibrium value 9

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