Housing Markets with Competitive Search∗ Charla Ka Yui Leung, The City University of Hong Kong Jun Zhang, The Chinese University of Hong Kong† June 2007 Abstract: Three striking empirical regularities have been repeatedly reported: the positive correlation between housing prices and trading volume, between housing price and the time-on-themarket, and the existence of price dispersion. This paper provides perhaps the first unifying framework for these phenomena based on a simple competitive search framework. With price bargaining and endogenously determined market tightness, some heterogeneity on the buyer’s and/or the seller’s side is included in the pricing formula, which helps explain several empirical finding in the housing markets, including the price dispersion, positive correlation between housing prices and the time-on-the-market. Moreover, when the costly search effort is introduced, we are able to explain the well-documented price-volume correlation. Intuitively, the buyers with higher waiting costs are more eager to purchase a house faster, and are more likely to accept a relatively higher price. They will also provide more search efforts, and thus raise the trading volume. To the sellers, the free entry assumption implies a tradeoff between the housing prices and the speed they sell the house in a decentralized market. As a result, if they ask for higher housing prices, the expected time on the market would be longer. JEL Classification Codes: D830, E300, R210, R310 Keywords: housing market, competitive search, price dispersion, trading volume, time on the market. ∗
Acknowledgment: The authors are grateful to Paul Anglin, Morris Davis, Qian Gong, John Quigley, Ping
Wang, William Wheaton, Abullah Yavas, as well as participants of the Midwest Macroeconomis Meetings at the Federal Reserve Bank of Cleveland, The Fourth Biennial Conference of Hong Kong Economic Association at the Chinese University of Hong Kong, the International Conference on Real Estates and the Macroeconomy at the China Center for Economic Research in Peking University, for many helpful comments and suggestions; and Chinese University of Hong Kong, City University of Hong Kong for financial support. All errors are ours. †
Correspondence: Jun Zhang, Department of Economics, Chinese University of Hong Kong, Shatin, Hong
Kong. (Phone) (852) 2609 8005; (Fax) (852) 2603 5805; (Email)
[email protected].
1
Introduction
Both casual observations and serious empirical research agree that the housing market is characterized by a strong decentralized pattern of exchange with severe search frictions. In contrast to most commodities and assets that are traded in the centralized markets, empirical "anomalies" such as price dispersion, a nontrivial time on the market (TOM) positively associated with housing prices, the positive correlation between housing prices and trading volumes, etc., are repeatedly reported, which are in sharp contrast to the prediction from the traditional Walrasian settings. This paper provides perhaps the first effort to develop a unified competitive search framework that can explain most of the anomalies. The modelling choice is indeed intuitive. The existence of price dispersion widely observed in the real estate market is difficult to be reconciled with the Walrasian framework, and naturally leads one to a search-theoretic setting (for instance, see Gabriel et al (1992), Leung et al (2006)). Our competitive search framework focuses on two key determinant of the price dispersion: the decentralized pattern of exchange and the heterogeneities on the seller’s and/or the buyer’s side. For simplicity, we assume that buyers have different waiting costs, which can be interpreted literally or viewed as a reduced form representation of other form of heterogeneity. The "waiting costs" we employ in this paper can represent the additional costs the buyers have to pay if the purchase of the house is delayed. For instance, some buyers may have "safer" alternatives such as living with parents, while some buyers may be facing soonexpiring rental contracts. For simplicity, we abstract from these complications and attempt to capture the buyer heterogeneity through differential waiting costs. Or, some buyers may need to pay higher transportation cost if the purchase is delayed. The expectation of a rising nominal price in real estates can also be regarded as an important component as the waiting costs, since they have to pay more if they purchase later. In this sense, several business-cycle related macroeconomic variables may affect the waiting costs as well. The competitive search framework is chosen with a reason. In earlier search-theoretic frameworks explaining price dispersion, such as Axell (1974), Butters (1977), Reinganum (1979), von zur Muehlen (1980), Burdett and Judd (1983), Diamond (1985), Rob (1985), Salop and Stiglitz (1985), Benabou (1988, 1992a,b, 1993), Wheaton (1990), and Rauh (2001), both the sellers and buyers are unable to differentiate themselves before the beginning of the
1
search process. One of the consequences is the difficulty in address the time-on-the-market issues along with the price dispersion. In contrast, with the help of advisement via various channels, a modern real estate market is usually separated into several submarkets targeting some specific features of the buyers and/or the sellers, in order to improve the resource allocation. Hence, we adopt the competitive search framework based on the seminal work in Moen (1997),1 where the seller are free to enter either of the submarkets and the tightness of the submarkets is endogenized. The price dispersion is mainly driven by the heterogeneity in the buyers and/or the sellers incorporated in the formula of the negotiated prices. Meanwhile, the free-entry assumption implies the trade-off between the housing price and the speed of sales for the sellers. As a consequence, there exists a positive correlation between price and the time on the market. Thus, the current setup captures the search frictions commonly observed with a flavor of competition, and it is broadly consistent with the empirical findings. For example, Merlo and Ortalo-Magné (2004) find that sellers post different prices to target various types of consumers, while the submarket with a higher listing price has a lower matching rate and a longer time on the market. Harding et al (2003) demonstrate that the buyer characteristics would affect the bargaining power and thus the negotiated price. The positive correlation between TOM and transaction price are found in the work of Kang and Gardner (1989), Forgey et al (1996), Leung, Leong and Chan (2002), Anglin et al (2003), among others. With the introduction of endogenous search effort, we can also illustrate, within the competitive search framework, the positive relationship between housing prices and trading volume, as pointed out in empirical works of Fisher et al (2003), Leung, Lau and Leong (2002). Intuitively, the buyers with higher waiting costs are more eager to purchase a house, and hence provide more search efforts. The increase in search intensity would lead to higher trading volumes. It also holds in the case with a positive shock in the waiting costs. As a consequence, this paper provides a new search-based explanation about the positive correlation between housing prices and trading volume, other than the down-payment explanation in Stein (1995), Ortalo-Magne and Rady (2006). More specifically, Ortalo-Magne and Rady (2006) seems to capture the short-run dynamics while this paper focuses on the steady state relationship. It 1
It is close to directed search, or directed matching framework originated in Peters (1991) and Montgomery
(1991).
2
is consistent with the empirical finding of Leung, Lau and Leong (2002), which suggests that the short-run dynamics of the housing market is driven by down-payment effect, where the longer-run relationship between housing price and trading volume is due to search friction. The remaining of the paper is structured as follows: The baseline model with competitive search and heterogeneous waiting costs will be introduced in Section 2, while several extensions, including costly search efforts, the introduction of some supply/demand scheme, and the waiting costs for the sellers, will be discussed in Section 3. Section 4 concludes.
2
A Baseline Model of Housing Price Dispersion
2.1
A Tale of Two Submarkets
Time is continuous. There is a continuum of buyers who have different waiting costs. To simplify the exposition, we focus on the case with only two types, high and low. And without loss of generality, the waiting time for high class H is higher than that of low class L, cH > cL . Let B i be the measure for the buyers with a flow waiting cost of ci , i = H, L. Similarly, we define S H and S L as the measure of sellers focusing on the two "submarkets" respectively. Notice that the "submarkets" may or may not be separated on the basis of locations. Submarket i simply represents the trading which involve type i buyers, i = H, L. The sellers may post a relatively higher posting price or choose some agencies targeting high-end buyers, etc., in an effort to differentiate them from those who are urgent to sell the house. For simplicity, we assume complete and perfect information between the agents. In each submarket i, i = H, L, the number of successful matching in an infinitesimal period is governed by a random matching function, M(B i , S i ), which exhibits constant return to scale in B and S, positive but diminishing marginal returns in each argument. We define the market tightness θi = B i /S i in the sense that it is more difficult for a buyer to find a seller in a tighter market. For each submarket i, we can define η i as the flow matching rate for a buyer to find a seller in submarket i such that ηi =
M(B i , S i ) = M(1, 1/θi ) Bi
3
(1)
Similarly, the flow matching rate for a seller to find a buyer, µi , satisfies µi =
2.2
M(B i , S i ) = M(θi , 1) = θi η i Si
(2)
Housing Prices and Bellman Equations
Recognizing the buyers’ heterogeneity in waiting costs, sellers would post different prices in the two submarkets. The actual price P i in the submarket i would be determined by a Nash bargaining solution, which will be discussed in the next subsection. Since the sellers are free to enter each of the two submarkets, the entry value, ΠE , is the same for all the sellers. Let V i denote the value for type-i buyers, while Ω as the value of a house owner, which is independent of the waiting cost level. For simplicity, we assume that both ΠE and Ω are exogenously determined. We are now well-equipped to obtain the following Bellman equations rΠE = µi (P i − ΠE )
(3)
rV i = −ci + η i (Ω − P i − V i )
(4)
From equation (4) we can obtain Vi =
−ci + η i (Ω − P i ) −ci /η i + (Ω − P i ) = r + ηi 1 + r/η i
(5)
Note that 1/η i is actually the mean waiting time for the buyers. Hence, equation (5) means that the buyer’s value equals the discounted net gain from purchasing a house, while the net gain is the consumer’s surplus net of the waiting cost during the waiting period. Similarly, equation (3) yields ΠE =
µi P i Pi = r + µi 1 + r/µi
(6)
Note that, in the Poisson process with an arrival rate of µi , the expected time-on-the-market is 1/µi . As a consequence, the equation (6) just states that the entry value of the house seller is just the housing price times the discount factor associated with the waiting period. In the baseline model, we, for simplicity, regard the buyers reservation values (ΠE ), the value for owning a house (Ω), and the numbers of buyers (B H and B L ) as exogenous variables. As shown in Section 3, the relaxation of the these restrictions would not affect the main qualitative results. 4
2.3
Seller’s Free Entry and the Price-TOM Relation
While the sellers are free to choose either of the submarkets, their entry values would be the same for both submarkets. As a consequence, the expected time-on-the-market must be longer (shorter) for the submarket with a higher (shorter) price, which in line with the proposal of the trade-off between the price and selling rate in Anglin (2004). Proposition 1 summarizes the above finding. Proposition 1 (Price-TOM Relation): In a competitive search framework where sellers are free to enter either submarket, the submarket with a higher (lower) price must have a longer (shorter) expected time-on-the-market. The empirical evidence in Kang and Gardner (1989), Forgey et al (1996), Leung, Leong and Chan (2002), Anglin et al (2003), among others, support our claim.2
2.4
The Bargaining Process and the Price Dispersion
The housing price is determined by the Nash bargaining, which means that it will solve the following joint surplus maximization problem © i ª E 1/2 i i 1/2 max − Π ) (Ω − P − V ) (P i P
The solution is
ΠE + Ω − V i . 2 Based on equations (1), (2), (5), (6), and (7), we can derive Pi =
rΩ + ci 2r + η i + µi 2r 1 = = + i + 1. i E i Π r µ M(θ , 1) θ
(7)
(8)
Observe that the right hand side is decreasing in θi . Hence, θH < θL , i.e., it is easier for the buyers to find a house in the type-H submarket. While the price in a tighter market must be lower, as suggested by equation (6) and Proposition 1, we have P H > P L . The results can be summarized as follows:
2
Actually, the relationship between the price and the actual time-on-the-market may not be strictly linear,
since we only find the linear relationship between the price and the expected time-on-the-market.
5
(i = H, L)
ci ↑
Ω↑
ΠE ↑
r↑
Pi
+
+
+
?
µi
−
−
+
+
−
−
+
+
fixed
fixed
θi Bi
fixed fixed
Si
+
+
M(B i , S i )
−
−
+
+
−
−
Table 1: Comparative Statics for the Baseline Model Proposition 2 (Price Dispersion): In a competitive search framework with heterogeneous waiting costs, housing prices would be different even for the houses and sellers with the same traits. Specifically, the buyer with higher waiting costs would pay a higher housing price, in an effort to reduce the waiting costs by making the purchase faster. The above results are in line with the empirical findings. For example, Merlo and OrtaloMagné (2004) find that sellers post different prices to target various types of consumers, while the submarket with a higher listing price has a lower matching rate and a longer time on the market. Leung et al (2006) claim that the price dispersion cannot be only attribute to randomness or econometric mis-specification since the degree of price dispersion systematically varies with some macroeconomic variables.3
2.5
Comparative Statics and the Price-Volume Correlation
On the basis of equations (6) and (8), we can obtain the results of comparative statics as shown in Table 1. Since the trading volume for a given period is proportional to the matching rate M(B i , S i ), we can claim the price-volume comovements feature of our baseline 3
Actually, housing price dispersion exists as long as the heterogeneous waiting costs enter the pricing
formula, no matter how the prices are determined. The employment of the Nash bargaining scheme is just for simpliciy. The use of other price determination would not affect the qualitative results. Empirically, Harding et al (2003) do find that the household characteristics would affect the bargaining power and thus the negotiated price.
6
in Proposition 3. Proposition 3 (Price-Volume Correlation): In the baseline model with fixed entry value for the sellers and fixed number of buyers, housing prices and the trading volumes would move in the same directions given the demand shocks in the waiting costs, or the values for owning a house. On the other hand, housing prices and the trading volumes would move in the opposite directions given the supply shocks in the entry value of house sellers. The effects from the shocks in the time preference is ambiguous. The intuition is quite straightforward. When the capitalized waiting cost (ci /r) or the value for owning a house (Ω) is higher, buyers are more eager to buy a house, and would accept a higher housing price, making the housing market more lucrative than before. While the seller’s reservation value are fixed, more sellers would enter the housing market until the congestion effect on the selling rates draw the sellers’ entry value back to their reservation values. With more sellers, the trading volume would also rise. On the otherhand, if the seller’s reservation value increases, less houses are available in the market. Hence the market is tighter, and the buyers have to pay a higher price. Actually, the results on the price-volume comovements are close to those from the basic supply/demand analysis.4 However, the rationale is a bit different. In our baseline model, the price-volume comovements are mainly due to the trade-off between the prices and the selling rate in the case of fixed entry value for the sellers, as well as the positive relationship between selling rate and the trading volume given the fixed number of buyers. The lead-lag pattern for the price-volume comovement depends on how fast the sellers can react to the shocks. The comovement would be contemporary when the sellers are responsive to the market changes, for instance, in the secondary market. In contrast, the supply in the primary market would be subject to a long lag due to the long construction period for the real estates. Empirically, Fisher et al (2003), Leung, Lau and Leong (2002), among others, find strong contemporary comovments between housing prices and the trading volumes, while Leung, Lau and Leong (2002) also find that price would lead the trading volume by 24-48 months in the monthly data. In Section 3, we show that the contemporary price-volume 4
As a comparison, this flavor is lost in the conventional asset pricing model due to the employment of the
representative agent framework.
7
comovements would be strengthened given costly search efforts.
2.6
Determinants of the Waiting Costs
In order to understand the relevance of the waiting costs in the backdrop of the macroeconomic changes, a discussion about the determinants of the waiting costs may be helpful. To buyers, the waiting costs are usually the costs they have to pay if the purchase of the house is delayed, for instance, the rents or hotel costs, the transportation costs, etc. The expectation of a rising nominal price in real estates can also be regarded as an important component as the waiting costs, since they have to pay more if they purchase later. In this sense, several business-cycle related macroeconomic variables may affect the waiting costs as well. To sellers, the waiting costs include the expectation of a declining housing prices, the cost of monitoring the house when they are abroad, etc.
2.7
Detailed Solutions in a Simplified Case
To obtain a closed-form solution, a widely used random matching function is adopted M(B i , S i ) = m0
BiS i Bi + Si
(9)
which depicts the case that the agents would meet each other with equal likelihood at an meeting rate of m0 . For simplicity, we assume m0 = 1. Note that this matching technology implies µi + η i = 1. As a result, equation (8) turns out to be µi =
ΠE r(2r + 1) rΩ + ci
Using equation (2) and (9), we can obtain θi =
µi ΠE r(2r + 1) = 1 − µi rΩ + ci − ΠE r(2r + 1)
and η i = 1 − µi = 1 −
8
ΠE r(2r + 1) rΩ + ci
Now we can derive the values of the buyers Vi =
Ω − 2ci − ΠE 2r + 1
and the housing price P i = ΠE +
3
rΩ + ci 2r + 1
Possible Extensions
We keep the above model setup as simple as possible to show that the heterogeneity in waiting costs can result in price dispersion in housing market. There are also some possible extensions to be explored in the future as an effort to make the theory closer to the reallife situations. Note that the price dispersions and the positive price-TOM relation exist in any competitive search framework with heterogeneous waiting costs (or other dimensions of heterogeneity that enter the pricing formula), the following extensions would affect the results in Proposition 1 and 2. However, the pattern of the price-volume comovements, as shown in Proposition 3, would change in the following extensions.
3.1
Costly Search Efforts
Proposition 3 claims a positive price-volume comovement if the sellers can adjust the supply of new houses quickly to the market changes. However, in reality, there is usually a significant lag in the housing supplies, especially in the first-hand market.5 We now introduce costly search efforts into the baseline to investigate the short-run price-volume comovements. In the short run, the market tightness, θi , is assumed to be fixed. However, a type-i buyers can provide a search effort, ei , at the unit cost of κi , i = H, L. The random matching function now becomes M(ei B i , S i ), where ei B i indicates the measure of effective type-i buyers, and the effective market tightness turns out to be ei B i /S i = ei θi , i = H, L. Now we have the 5
It usually takes years to construct a building with hundreds of flats. The exception may be the pre-
manufatured houses in the US. However, it is not common in most big cities due to the limited supply of land.
9
selling rate M(ei B i , S i ) M(ei θi , 1) = η = Bi θi i
and the buying rate M(ei B i , S i ) = M(ei θi , 1) = θi η i Si The Bellman equation for the sellers would be the same as before, while the value function µi =
for the buyers becomes ª © i i i i i i − κ e + η (Ω − P − V ) , rV i = max −c i e
i
or V = max i e
The first order condition implies
½
−ci − κi ei + η i (Ω − P i ) r + ηi
¾
.
¤ £ i c + κi ei + r(Ω − P i ) M1 (ei θi , 1) = κi (r + η i ),
(10)
where Mj (.) is the derivative of the matching function M with respect to the j-th argument. While the housing price is given as P i = ΠE (1 + r/µi ), and the Nash bargaining solution of price suggests rΩ + ci + κi ei 2r + η i + µi , = ΠE r µi
(11)
the equation (11) can be rewritten as M1 (ei θi , 1)(rΩ + ci + κi ei ) − κi (2r + η i + µi ) = 0.
(12)
The derivations in the Appendix claim that, at the short-run equilibrium, the optimal search effort is increasing in ci and Ω, decreasing in κi , while the effects from r and θi are ambiguous. Pi =
κi (µi + r) (rΩ + ci + κi ei )(µi + r) = . (2r + η i + µi )r M1 (ei θi , 1)r
Hence the housing price is increasing in ci and Ω, while the effects from κi , r and θi are ambiguous. Meanwhile, P i and ei would move to the same direction, given a shock in θi . The results of the comparative statics analysis are summarized in Table 2. 10
ci ↑ Ω ↑ θi
(i = H, L)
r ↑ κi ↑
ei
+
+
?
?
µi
−
+
+
?
?
Pi
−
+
+
?
?
?
M(ei B i , S i )
+
+
?
?
−
Price-Volume Comovements
+
+
+
?
?
Table 2: Comparative Statics for the Model with Costly Search Effort
Intuitively, when the waiting cost or the value of owning a house is higher, the buyers are more eager to buy a house, and thus provide more search efforts. As a result, both prices and trading volumes increase. When the search cost is higher, search effort would be lower, but the total effect on the price is ambiguous. Meanwhile, in a tighter market, the longer waiting period implies a higher total waiting cost and a larger total search cost. While the waiting cost effect and the search cost effect point to different directions, the total effect is ambiguous in general. Nonetheless, the market tightness effects turn out to have the same sign for the trading volume and the price. Proposition 4 summarizes the finding about the short-run (contemporary) price-volume comovements. Proposition 4 (Contemporary Price-Volume Comovements): With the introduction of costly search efforts into the baseline model, there exist positive contemporary price-volume comovements provided the shocks either in the waiting cost, or in the value for owning a house, or in market tightness. Proposition 4 implies a positive relationship between housing prices and trading volume even in a short run, which supports the empirical evidence by Fisher et al (2003), Leung, Lau and Leong (2002).
11
3.2
A More Flexible Supply/Demand Scheme
So far, we stick to the assumption of fixed free entry values for the sellers and fixed number of buyers. In reality, it is usually not the case. In this subsection, we employ a more flexible supply/demand scheme. Now the number of sellers is given by an increasing function S = S(ΠE ), while the number of buyers satisfies the increasing functions B i = B i (V i ), for i = H, L. Denote h as the proportion of sellers who enter the type-H submarket. The remaining is the same as the baseline model. Hence, we still have the following equations: Vi =
−ci + η i (Ω − P i ) r + ηi
(13)
µi P i r + µi
(14)
ΠE =
ΠE + Ω − V i 2 i i M(B , S ) = M(1, 1/θi ) ηi = i B i M(B , S i ) = M(θi , 1) = θi η i µi = Si Pi =
Now the measures of market tightness satisfies θH =
B H (V H ) B L (V L ) L = , and θ hS(ΠE ) (1 − h)S(ΠE )
After substituting the prices, the selling rate and the buying rate into equations (13) and (14), we have four equations (since i = H, L) and four unknown variables, V H , V L , ΠE , and h. Theoretically, we can solve the system, but the general case seems too complicated for close-form analysis. In general, we may expect that a higher price would lead to a larger number of sellers and a smaller number of buyers, leaving the changes in the trading volume ambiguous. However, in the case with higher expected housing price in the future, the value of owning a house rises, which has a positive effect on the number of buyers. As a result, the trading volumes are more likely to increase with the housing prices.
12
3.3
Waiting and Search Costs for the Sellers
We may also introduce the waiting and search costs from the sellers’ side. Some sellers may be more urgent to sell the house than the others. Hence, they may post more advertisement and list the house in more agencies, which implies higher search costs for them. As a consequence, their bargaining power would be relative lower, which will also contribute to the degree of price dispersions.
3.4
Access to Both Submarkets for the Type-H Buyers
In the baseline model, the type-H buyers can only focus on the submarket H. In reality, the urgent buyers may look at both submarkets, while the type-L buyers would only focus on the cheap houses. We can modify the baseline model accordingly, but we expect that the qualitative results would not change.
4
Conclusion
To a certain extent, the idea behind this paper is analogous to a theme park visit. In those famous theme parks, visitors do not know ex ante whether it will be very crowded and they would need to wait in a long queue before they can enjoy some machines, or a ghost house. They can choose to buy a more expensive "quick-pass" and they will save the time of waiting, or buy a cheaper normal-pass and hope that they may not need to wait that long ex post. Normally, people with a higher "waiting cost" such as those foreign visitors would prefer the more expensive options. For some others, they prefer to wait. Thus, the time-on-the-queue (TOQ) will be negatively related to the price of the "pass". Similarly, some buyers in this paper have higher waiting cost than the others, and they prefer to search their houses in a higher-priced submarket. Unlike the theme park visit, however, the supply side of the housing market is endogenous. The sellers will take the buyers’ strategies as given and then self-select into different submarkets. Moreover, while pass-purchasing is certain, house-purchasing is not. Even within each sub-market, there is a random matching process among potential buyers and sellers. And while the price of the pass is given, the housing price in each submarket will be determined through a Nash bargaining 13
process, which will in turn depend on the market-tightness of the corresponding submarket. Perhaps more importantly, this paper differs from the theme park visit example in that there are three stylized facts for this paper to mimic, namely, the existence of price dispersion, the positive correlation between the market price and trading volume, and that between the transaction price and the time-on-the-market (TOM). The empirical "anomalies" found against the Walrasian predictions can be explained within our competitive search framework. The free-entry assumption implies the positive correlation between housing prices and the time on the market. With the introduction of costly search efforts, the buyers with higher waiting costs are more eager to purchase a house, and hence provide more search efforts. The increase in search intensity would lead to higher trading volumes. It also holds in the case with a positive shock in the waiting costs. In addition, we show that price dispersion can exist easily even with perfect information and perfect competition in the ex ante sense, as long as the trades are decentralized. Future work can be extended in several directions. For instance, "middlemen" are missing in this analysis. Previous partial equilibrium analysis, such as Yavas (1994, 1995), shows that the introduction of intermediary may affect the equilibrium configuration, and efficiency under some conditions. Future research can study the implications of introducing "broker" in this competitive search framework. Second, while this paper focuses on the heterogeneity in waiting costs, there is still needs to do some empirical study to justify the role of waiting costs directly. Moreover, we may consider other dimensions in the heterogeneity whenever the empirical evidence suggests so. Perhaps a promising alternative is to explicitly model the coexistence of financially constrained and unconstrained agents. Another candidate for future research is to merge the current housing market model with the conventional neoclassical framework, as in Lagos and Wright (2005). These directions are indeed being pursued.
14
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Appendix Proof for the comparative statics for Subsection 3.1 (Short Run) The short-run equilibrium is governed by the equation M1 (ei θi , 1)(rΩ + ci + κi ei ) − κi (2r + η i + µi ) = 0
Let LHS denote the left-hand side of the above equation, and M as a short-hand notation for M(ei θi , 1). Note that ¢ dLHS ¡ i i i = c + κ e + rΩ M11 + κi M1 θi − κi (1 + θi )M1 < 0 dei dLHS = M1 (ei θi , 1) > 0 dci dLHS = rM1 (ei θi , 1) > 0 dΩ
dLHS = ei M1 (ei θi , 1) − (2r + η i + µi ) dκi (ci + rΩ) M1 θi
