You get what you give: theory and evidence of reciprocity in the sharing economy

Abstract

We develop an analytical framework of peer interaction in the sharing economy that incorporates reciprocity, the tendency to increase (decrease) effort in response to others’ increased (decreased) effort. In our model, buyers (sellers) can induce sellers (buyers) to exert more effort by behaving well themselves. We demonstrate that this joint increased effort can improve the utility of both parties and influence the market equilibrium. We also show that bilateral reputation systems, which allow both buyers and sellers to review each other, are more responsive to reciprocity than unilateral reputation systems. By rewarding reciprocal behavior, bilateral reputation systems generate trust among strangers and informally regulate their behavior. We test the predictions of our model using data from Airbnb, a popular peer-to-peer accommodation platform. We show that Airbnb hosts that are more reciprocal receive higher ratings and that higher rated hosts can increase their prices. Therefore, reciprocity affects equilibrium prices on Airbnb through its impact on ratings, as predicted by our analytical framework.

Appendices

Appendix A: Proofs

1.1 A.1 Proof of proposition 1

Proposition 1

The host’s average rating on Airbnb is positively related to her reciprocity weight, i.e.,∀α_i ≥ 0 and α_i≠ 0,

$$ \frac{\partial R_{airbnb}}{\partial\alpha_{h}}>0 $$

(26)

where

$$ R_{airbnb}\equiv{\int}_{i}r_{h,i}(e_{h},e_{i}) di $$

(27)

is the average rating of the host.

Proof

We solve the subgame equilibrium at Period 2. Note that at Period 2, both players choose their optimal effort level as the best response to the other’s effort. Then each player reports their utility of accommodation in the rating.

Plugging r_{h, i}(e_i, e_h) = v_h + α_iu(e_i, e_h) and r_i(e_i, e_h) = v_i + α_hu(e_i, e_h) into the ex-post utility of host and guest i, we have:

$$\begin{array}{@{}rcl@{}} U_{i}(e_{i}|e_{h})&=&\max\limits_{e_{i}}\left\{v_{h}+\alpha_{i} u(e_{i},e_{h})-P-C(e_{i})+\beta_{i}\left[v_{i}+\alpha_{h} u(e_{i},e_{h})\right]\right\} \end{array} $$

(28)

$$\begin{array}{@{}rcl@{}} U_{h}(e_{h}|e_{i})&=&\max\limits_{e_{h}}\left\{v_{i}+\alpha_{h} u(e_{i},e_{h})+\beta_{h}\left[v_{h}+\alpha_{i} u(e_{i},e_{h})\right]\right\}. \end{array} $$

(29)

Combining the first order conditions of the two optimality problems, we have:

$$\begin{array}{@{}rcl@{}} e_{i}^{*} &=& A(\alpha_{h}+\beta_{h}\alpha_{i})^{\frac{1-k}{2}}(\alpha_{i}+\beta_{i}\alpha_{h})^{\frac{1+k}{2}}\\ e_{h}^{*} &=& B(\alpha_{h}+\beta_{h}\alpha_{i})^{1-\frac{k}{2}}(\alpha_{i}+\beta_{i}\alpha_{h})^{\frac{k}{2}} \end{array} $$

where $A\equiv \left (\frac {k}{c_{i}}\right )^{\frac {1-k}{2}}\left (\frac {1-k}{c_{h}}\right )^{\frac {1+k}{2}}$ and $B\equiv \left (\frac {k}{c_{i}}\right )^{1-\frac {k}{2}}\left (\frac {1-k}{c_{h}}\right )^{\frac {k}{2}}$.

Thus,

$$\begin{array}{@{}rcl@{}} r_{h,i}(e_{i,}e_{h})&=&v_{h}+\alpha_{i}{e_{i}^{k}}e_{h}^{1-k} \\ &=&v_{h}+\alpha_{i}\left[A\left( \alpha_{h}+\beta_{h}\alpha_{i}\right)^{\frac{1-k}{2}}\right.\\ &&\left.\left( \alpha_{i}+\beta_{i}\alpha_{h}\right)^{\frac{1+k}{2}}\right]^{k}\\ &&\left[B\left( \alpha_{h}+\beta_{h}\alpha_{i}\right)^{1-\frac{k}{2}}\left( \alpha_{i}+\beta_{i}\alpha_{h}\right)^{\frac{k}{2}}\right]^{1-k}. \end{array} $$

(30)

From the above, we have:

$$ \frac{\partial r_{h,i}}{\partial\alpha_{h}}>0 $$

(31)

and since $R_{airbnb}\equiv \int r_{h,i}di$,

$$ \frac{\partial R_{airbnb}}{\partial\alpha_{h}}>0. $$

(32)

□

1.2 A.2 Proof of proposition 2

Proposition 4.2

The ratings on both review systems depend positively on the weights of reciprocity and reputation, α_h and β_h.However, given the same pool of guests, the host’s average rating on Airbnb responds more to the reciprocity weight, while the rating on the unilateral review system responds more to reputation weight, i.e.:

$$ \frac{\partial R_{airbnb}}{\partial\alpha_{h}}>\frac{\partial R_{uni}}{\partial\alpha_{h}}>0 ~\text{and}~ \frac{\partial R_{uni}}{\partial\beta_{h}}>\frac{\partial R_{airbnb}}{\partial\beta_{h}}>0. $$

(33)

where $R_{airbnb}\equiv {\int }_{i}r_{h,i}^{air}(e_{h},e_{i}) di$ is the host’s average rating on Airbnb, and $R_{uni}\equiv {\int }_{i}r_{h,i}^{uni}(e_{h},e_{i}) di$ is the host’s average rating on an unilateral review system.

Proof

From previous results, we have:

$$\begin{array}{@{}rcl@{}} e_{i}^{*} &=& A(\alpha_{h}+\beta_{h}\alpha_{i})^{\frac{1-k}{2}}(\alpha_{i}+\beta_{i}\alpha_{h})^{\frac{1+k}{2}}\\ e_{h}^{*} &=& B(\alpha_{h}+\beta_{h}\alpha_{i})^{1-\frac{k}{2}}(\alpha_{i}+\beta_{i}\alpha_{h})^{\frac{k}{2}}. \end{array} $$

For the same host and the same guest pool on both the unilateral review system platform and Airbnb, and given identical values of the parameters, except β_i = 0 under the unilateral review system, we have:

$$ \frac{\partial r_{h,i}^{air}}{\partial\alpha_{h}}>\frac{\partial r_{h,i}^{uni}}{\partial\alpha_{h}}>0 $$

(34)

$$ 0<\frac{\partial r_{h,i}^{air}}{\partial\beta_{h}}<\frac{\partial r_{h,i}^{uni}}{\partial\beta_{h}}. $$

(35)

Since $R_{airbnb}\equiv {\int }_{i}r_{h,i}^{air}(e_{h},e_{i}) di$ and $R_{uni}\equiv {\int }_{i}r_{h,i}^{uni}(e_{h},e_{i}) di$, we have:

$$ \frac{\partial R_{airbnb}}{\partial\alpha_{h}}>\frac{\partial R_{uni}}{\partial\alpha_{h}}>0 ~\text{and}~ \frac{\partial R_{uni}}{\partial\beta_{h}}>\frac{\partial R_{airbnb}}{\partial\beta_{h}}>0. $$

(36)

Thus, under a unilateral reputation system, hosts that care more about reputation are ranked higher than hosts that care more about the shared experience utility. The opposite is true on bilateral reputation systems like Airbnb.¹⁰□

1.3 A.3 Proof of proposition 3

Proposition 3

On Airbnb, prices increase after a positive shock on rating and decrease after a negative shock on ratings. Given the same price P ₁ at period 1, and ratings R _{a i r b n b} and $R^{\prime }_{airbnb}$ disclosed at period 2, we have the following relationship for prices posted at period 3:

$$ if ~R_{airbnb}> R^{\prime}_{airbnb} ~ then ~ P_{3}(R_{airbnb}) > P_{3}(R^{\prime}_{airbnb}). $$

(37)

Proof

Ex-ante, guests determine whether to enter the market according to V _i(P). Guest i enters the market if and only if V _i(P) ≥ 0, given P. Therefore, the marginal guest is guest i^∗ where $V_{i^{*}}(P)= 0$. For each guest i at period 3, the expected utility V _i(P) is positively determined by her inference of α_h, β_h.

From previous results, we have $\frac {\partial u(e_{i},e_{h})}{\partial \alpha _{h}}>0$ and $\frac {\partial u(e_{i},e_{h})}{\partial \beta _{h}}>0$. Together with r_i = v_i + α_hu(e_i, e_h), we have:

$$ \frac{\partial r_{i}}{\partial\alpha_{h}}>0 $$

(38)

$$ \frac{\partial r_{i}}{\partial\beta_{h}}>0. $$

(39)

Then, since U_i(e_i, e_h, r_i, v_h) = v_h + α_iu(e_i, e_h) + β_ir_i, we have that ∀(α_i, β_i) > 0,

$$ \frac{\partial U_{i}(e_{i},e_{h},r_{i},v_{h})}{\partial\alpha_{h}}>0 $$

(40)

$$ \frac{\partial U_{i}(e_{i},e_{h},r_{i},v_{h})}{\partial\beta_{h}}>0, $$

(41)

where u(e_i, e_h) is the shared experience utility and U_i(e_i, e_h, r_i) is ex-post utility of guest i during the accommodation stay.

At period 3, V _i(P₃) is the ex-ante utility of the guest i if she decides to transact with the host. The guest i forms her expectation based on the publicly observed price, P₃, and the average rating of the host, $R_{h}\equiv \int r_{h,j} dj$, i.e.:

$$ V_{i}(P)=\int U_{i}(e_{i},e_{h},r_{i},v_{h}) dF^{updated}({\Psi}_{h}), $$

(42)

where F^updated(Ψ_h) is the posterior distribution of the host’s characteristic parameter vector (α_h, β_h, v_h). The Bayesian-updated guests update the prior distribution F(Ψ_h) upon the signal R_h to derive F^updated(Ψ_h) .

From Eqs. 40, 41, and 42, we have that ∀(α_h, β_h, v_h) ∈Ψ_h, the expected value of U_i(e_i, e_h, r_i, v_h) are positively related to α_h and β_h, i.e.,

$$ \frac{\partial V_{i}(P_{3})}{\partial\alpha_{h}}>0 $$

(43)

$$ \frac{\partial V_{i}(P_{3})}{\partial\beta_{h}}>0. $$

(44)

The two conditions above imply that the ex-ante utility to transact is higher if the host is perceived to have higher weight on shared experience or reputation, respectively.

Meanwhile, from previous results, we have:

$$ \frac{\partial r_{h,i}}{\partial\alpha_{h}}>0 $$

(45)

$$ \frac{\partial r_{h,i}}{\partial\beta_{h}}>0. $$

(46)

Conditions (45) and Eq. 46 show that the hosts with higher value of α_h and β_h have higher rating r_{h, i} given the same guest i. Then, from $R_{h}\equiv \int r_{h,i} di$, we have that, given the same guest pool, the average rating, R_h, reveals a higher value of (α_h, β_h).

Suppose that two hosts are identical, except for their ratings, i.e., they have the same price P₁, same location, and similar property, but $R_{airbnb}> R^{\prime }_{airbnb}$. Then the higher rating R_airbnb is a positive signal relative to $R^{\prime }_{airbnb}$, i.e., the expected value of the host’s characteristics is better for the host with R_airbnb. Thus, if the two hosts post identical P₃, the expected demand toward the host with higher average ratings would be higher, i.e., $Q(P_{3}, R_{airbnb})>Q(P_{3},R^{\prime }_{airbnb})$.

Since each period-3 guest is more willing to transact with a host having R_airbnb than with a host having $R^{\prime }_{airbnb}$, the host with the higher rating posts higher price P₃ in this monopolistic pricing setting (assuming that the hosts are faced with the same pool of guests).

Formally, let’s suppose P₁ and Q₁ are the same for Host A and Host B, and, without loss of generality, assume the mass of guests enter the market have the same set of parameters, (α_i, β_i). Let Host A have an average rating R_airbnb and Host B an average rating $R^{\prime }_{airbnb}$. If $R_{airbnb}>R^{\prime }_{airbnb}$, then from Eqs. 38 and 39, at least one of the following conditions holds:

$$\begin{array}{@{}rcl@{}} \alpha_{A}>\alpha_{B} \\ \beta_{A}>\beta_{B} \end{array} $$

Then, for any positive P₃,

$$ V_{i}(transact with A |P_{3}) > V_{i}(transact with B |P_{3})a $$

(47)

The marginal guest is defined as the one with V _i(transact|P₃, R_airbnb) = 0. Then, from Eq. 47, we know that the marginal guest transacting with Host A can bear a higher P₃ compared to that trading with Host B, since Host A is expected to have higher value of α or β or both.

Since P₃ is determined by:

$$ \max\limits_{P_{3}} \left\{P_{3}Q(P_{3})+\int U_{h}\left( e_{i},e_{h},v_{h}^{*},r_{h,i}\right) dF\left( {\Psi}_{g}\right)\right\}, $$

(48)

we have that, given that Ψ_g is the same for Host A and B, $P_{3}(R_{airbnb})>P_{3}(R^{\prime }_{airbnb})$. □

1.4 A.4 Relaxing the truth-telling assumption

As defined in Section 3.3, we have that:

$$ r_{h,i}\equiv \phi\left( v_{h}+u\left( e_{i},e_{h}\right)\right). $$

(49)

Plugging the equation above into $R_{airbnb}\equiv \int r_{h,i}di$, we have:

$$ R_{airbnb}=\int \phi (v_{h}+u(e_{i},e_{h}))di. $$

(50)

Now, assume all hosts are faced with an identical pool of guests. Let ω_i ≡ v_h + u(e_i, e_h). From previous results, we have:

$$ \frac{\partial \omega_{i}}{\partial \alpha_{h}}>0. $$

(51)

Since ϕ is a weakly increasing mapping, we have

$$ \frac{\partial \phi(\omega_{i})}{\partial \omega_{i}}\geq 0 $$

(52)

The inequality is strict for some ω_i.

From Eqs. 51 and 52, we have $\frac {\partial \phi (\omega _{i})}{\partial \alpha _{h}}\geq 0$ and the inequality is strict for some ω_i, i.e.:

$$ \frac{\partial r_{h,i}}{\partial \alpha_{h}}>0 $$

(53)

Appendix B: Alternative models

1.1 B.1 A Simple model

We start with a simple model where we assume the host to be a risk-neutral profit maximizer. The service quality offered by the host is exogenously given and fixed across transactions. Further, the service quality is private information of the host and only revealed to the guests during their stay in the host’s property. The distribution of service quality is common knowledge. We assume a monopolistic host and a continuum of guests with heterogeneous tastes. The guests are located on a line and the host is located at the center. The heterogeneous taste of guest i is modeled as x_i, which denotes the distance between the host and the guest i, and it follows a uniform distribution over [0, 1]. Without loss of generality, we assume the effort cost of publishing a review is zero. The timing of the game is as follows:

• in Period 1, the host posts price P₁ and a continuum of guests – the early guests – enter the market. The early guests can only observe P₁. As stated above, the expected value of service quality, E[v_h], is common knowledge. Guests decide whether to request accommodation. The transaction volume for this set of guests, Q₁, is realized.

• in Period 2, the accommodation stay takes place. Service quality is now revealed to the guest who requests the accommodation. The utility of guest i is v_h − x_i. At the end of this period, the guest i publishes a rating r_{h, i} for the host.

• in Period 3, the host observes the ratings received in period 2 and post a new price P₃. A continuum of guests – the late guests – enter the market. Their heterogeneous taste parameter x_i follows the same distribution of the early guests. They observe the average rating for host h disclosed in Period 2 and the price P₃. They make their accommodation decision accordingly.

We assume that guests truthfully report their utility in the ratings, i.e., r_{h, i} = v_h − x_i, where v_h denotes the service quality and x_i denotes the heterogeneous taste of guest i. We don’t consider this truthtelling assumption is a strong one here, since under this scenario, an agent does not have an incentive to collude with the host in inflating ratings. Meanwhile, if agents truthfully report service quality in the ratings, informative ratings have value to other users on the platform. Thus, the truthtelling assumption is justified by the phenomenon of “warm glow” discussed in Andreoni (1990)¹¹

Moreover, we assume that the distribution of v_h and x_i is common knowledge and, hence, E[v_h] is known ex-ante to the guests and E[x_i] is known ex-ante to the host.

In this setting, the only choice variable of the host is price. The objective function of the host is given by:

$$ V_{h}(P) =\max\limits_{P}\{E[PQ-C(Q)|P]\} $$

(54)

where P denotes price and Q denotes the transaction volume. V _h(P) denotes the ex-ante utility contingent on choosing price P. Since the host is risk-neutral and only interested in expected profits, the utility function coincides with expected profits.

Similarly, the only choice variable of a guest is whether to request accommodation. A guest requests accommodation from the host if and only if her expected utility from the accommodation is non-negative. The guest’s ex-ante utility prior to the accommodation is given by:

$$ V_{i}(transact|P)=E[v_{h}]-x_{i}-P $$

(55)

where V _i(transact|P) is the ex-ante utility of the guest i if she books the accommodation. P denotes the price set by the host, v_h denotes the service quality, and x_i denotes the heterogeneous taste of guest i. we assume x_i to have uniform distribution over [0, 1].

After staying in the host’s property, the guest i publishes a rating r_{h, i} = v_h − x_i. The host’s rating $R\equiv \int r_{h,i} di$ depends only on v_h and E[v_h] as E[v_h] determines the pool of guests requesting accommodation from the host.

In the solution below, we solve backwards for Perfect Bayesian Equilibrium.

Proof

In the first period, early guests make a decision based on E[v_h], which is common knowledge. When the host chooses P₁, the marginal guest i^∗ who is indifferent about reserving the accommodation or not, is given by:

$$ E[v_{h}]-x_{i^{*}}-P_{1}= 0. $$

(56)

Guests with $x\in [0,x_{i^{*}}]$ book the accommodation. Hence, the first period transaction volume is:

$$ Q_{1}=x_{i^{*}}=E[v_{h}]-P_{1}. $$

(57)

Now let’s consider the rating a host receives. The average rating, denoted by R, is given by:

$$ R\equiv\frac{{\int}_{0}^{Q_{1}}v_{h}-x_{i}dx_{i}}{Q_{1}}=v_{h}-\frac{Q_{1}}{2}. $$

(58)

Since both R and Q₁ are common knowledge in Period 3, late guests can infer the value of v_h from $v_{h}=R+\frac {Q_{1}}{2}$. Therefore, the marginal guest in Period 3 is given by $v_{h}-x_{j^{*}}-P_{3}= 0$, i.e.,

$$ Q_{3}=x_{j^{*}}=v_{h}-P_{3}. $$

(59)

Hence, the host maximizes profits in Period 3 by solving the following optimality problem:

$$ \max\limits_{P_{3}} P_{3}Q_{3}=\max\limits_{P_{3}}\left\{P_{3}(v_{h}-P_{3})\right\}. $$

(60)

From the first order condition (FOC), we have $P_{3}^{*}=\frac {v_{h}}{2}$. Then in Period 1, assuming hosts discount future revenue at rate β, a host solves:

$$ \max\limits_{P_{1}}\left\{P_{1}(E[v_{h}]-P_{1})+\beta P_{3}^{*}\left( v_{h}-P_{3}^{*}\right)\right\}. $$

(61)

From the FOC, we have:

$$ P_{1}=\frac{E[v_{h}]}{2}. $$

(62)

Then, from Eqs. 57 and 62, we have $Q_{1}=\frac {E[v_{h}]}{2}$. Thus, we have:

$$ R=v_{h}-\frac{Q_{1}}{2}=v_{h}-\frac{E[v_{h}]}{4}. $$

(63)

For a pool of hosts with $v_{h}\in [\underline {v}_{h},\bar {v}_{h}]$, the average value of v_h is E[v_h]; hence, the average ratings of hosts is $M=\int R_{j}dj=\frac {3}{4}E[v_{h}]$.

Let R_c denote the rating casual hosts and R_p denote the rating of professional hosts. For R_c to be systematically higher than R_p, we have to assume that the average service quality of casual hosts is higher than that of professional hosts, i.e., $E\left [{v_{h}^{c}}\right ]>E\left ({v_{h}^{p}}\right )$, where $E\left [{v_{h}^{c}}\right ]$ and $E\left [{v_{h}^{p}}\right ]$ denote the average service quality of casual hosts and of professional hosts, respectively. Note that casual and professional hosts differ in their market participation frequency. In this simple model, the service quality of the host is not endogeneously chosen by the host, hence we lack a mechanism to link a host’s market participation frequency with their ratings. Thus, in order for $E\left [{v_{h}^{c}}\right ]>E\left [{v_{h}^{p}}\right ]$, we need to assume that market participation negatively correlates with the exogenously given service quality. However, this does not seem to be a natural assumption to make. As higher service quality can translate into higher ratings that attracts future business, hosts choosing to participate more frequently should not have less incentive to work hard toward achieving high ratings. If anything, this rationale suggests that market participation frequency should be positively correlated with service quality. We conclude that this simple model cannot easily explain why professional hosts have lower ratings. □

1.2 B.2 Relaxing the truth-telling assumption

Next, motivated by the observation that the guests who do not report ratings are likely to have had a worse experience (Fradkin et al. 2017), we relax the truth-telling assumption. In doing so, we allow selection bias in ratings. To allow for selection bias, we assume that guests provide a rating to a host only if the rating is above a threshold 𝜃_i, i.e.:

$$ \textbf{1}\{rating\}= 1 \text{ iff } r_{h,i}>\theta_{i}. $$

(64)

Then the ex-ante utility of the guest i in period 1 is given by:

$$ V_{i}(transact|P)= E[v_{h}]-x_{i}-P+ E[r_{i}-\textbf{1}\{rating\}\alpha_{i} |(v_{h}-x_{i})-r_{h,i}|], $$

(65)

where r_{h, i} denotes the rating guest i gives to the host and r_i denotes the rating the host gives to the guest i. The term α_i denotes the weight of the host’s reputation in guest i’s utility.

Compared with the guest’s ex-post utility in the simple model (55), the current utility function has two new parts. The first part includes 1{rating}|(v_h − x_i) − r_{h, i}| and the rating threshold 𝜃_i. The difference between the disclosed rating and the true level of the guest’s utility captures the guest’s disutility derived from reporting a rating different from the true value of service quality and deteriorating rating informativeness. The term 𝜃_i captures the cost (e.g., psychological cost) for a guest to give a low rating. Together, these terms capture the trade-off faced by the guest when choosing whether to rate a host and what rating to disclose. While psychological costs may encourage guests to inflate ratings, guests also have an incentive to provide informative ratings as a contribution to other users on the platform. The two forces work in opposite direction and together determine the rating guests report.

The second part is r_i, the rating guest i receives from the host, which captures the fact that the guest i has reputation concerns. The reason for r_i to be part of the guest’s utility is to match the Airbnb setting, where a host can rate the guest, and this rating affects whether future hosts will accept the guest’s accommodation request.

In all our analyses, we only consider ratings produced under the new simultaneous Airbnb reputation mechanism. Therefore, strategic rating manipulation of ratings is not a concern, i.e., hosts cannot strategically collude with guests to exchange high ratings, which implies that r_i is not a function of r_{h, i}.

In the equilibrium of this model, a host receives a higher rating than in the simple model, and the inflated ratings depend on the average level of 𝜃_i associated with the pool of the guests. While 𝜃_i may be affected by the interaction between guests and hosts, it seems unlikely that it is also correlated with market participation, since this information is not revealed to guests. Unless we are willing to assume such a correlation, we cannot easily explain the difference in ratings observed in Why do casual hosts have higher ratings than professionals?. The formal proof of this statement is as follows.

Proof

Under this model, the only choice variable of the guest during Period 2 is r_{h, i}. If guest i decides to request the accommodation, her ex-ante utility in Period 1 is given by:

$$\begin{array}{@{}rcl@{}} V_{i}(transact|P) &=& \max\limits_{r_{h,i}} E[v_{h}]-x_{i}-P+ E[r_{i}-\alpha_{i} \textbf{1}\{rating\}|r_{h,i}\\ &&-~(v_{h}-x_{i})|\} \\ &\text{where }& ~\textbf{1}\{rating\}= 1 iff r_{h,i}>\theta_{i}. \end{array} $$

(66)

The ex-post utility of guest i at the end of Period 2 is given by:

$$\begin{array}{@{}rcl@{}} u_{i}(r_{h,i})&=& \max\limits_{r_{h,i}} \{v_{h}-x_{i}+r_{i}-\alpha_{i}\textbf{1}\{rating\}|r_{h,i}-(v_{h}-x_{i})|\}\\ &\text{where}& ~\textbf{1}\{rating\}= 1 iff r_{h,i}>\theta_{i}. \end{array} $$

(67)

With respect to the best response of guest i, two options exist:

• 1. If r_{h, i} ≥ 𝜃_i, then:

  $$ V_{i}(transact|P)=E[v_{h}]-x_{i}-P+E[r_{i}-\alpha_{i}|r_{h,i}-(v_{h}-x_{i})|] $$

  (68)
  $$ u_{i}(r_{h,i})= [v_{h}-x_{i}+r_{i}-\alpha_{i}|r_{h,i}-(v_{h}-x_{i})|] $$

  (69)

  where u_i(r_{h, i}) is the ex-post utility after the accommodation stay. Then, from Eq. 69 and α_i > 0, we have:

  $$ r_{h,i}=v_{h}-x_{i}. $$

  (70)

• 2. If r_{h, i} < 𝜃_i, then, the guest i does not publish a rating.

Then, the ex-ante utility for a guest to enter the market is given by:

$$\begin{array}{@{}rcl@{}} V_{i}(transact|P) & = & E[v_{h}]-x_{i}-P+E[r_{i}]+\alpha_{i}[Prob(v_{h}-x_{i}>\theta_{i})\\ &&*[v_{h}-x_{i}-(v_{h}-x_{i})] \\ &&+~(1-Prob(v_{h}-x_{i}>\theta_{i}))*0]. \end{array} $$

(71)

Note that under simultaneous ratings, guest i cannot directly influence r_i by choosing the rating she gives to the host. Thus, the only choice of guest i in Period 1 is to transact if and only if E[v_h] − x_i − P ≥ 0. Therefore, the following conditions still hold: $Q_{1}=x_{i}^{*}=E[v_{h}]-P_{1}$ and $P_{1}=\frac {E[v_{h}]}{2}$.

Then, the average rating of a host under this model, denoted as R^biased, is:

$$\begin{array}{@{}rcl@{}} R^{biased} & \equiv & \frac{{\int}_{i rate}r_{h,i}di}{{\int}_{i rate}1di}=\frac{{\int}_{i rate}(v_{h}-x_{i})di}{{\int}_{i rate}1di}\\ & = & v_{h}-\frac{{\int}_{i rate}x_{i}df(x_{i})}{{\int}_{i rate}di}\\ & = & v_{h}-\frac{{\int}_{0}^{x^{**}}x_{i}df(x_{i})}{{\int}_{0}^{x^{**}}df(x_{i})} \end{array} $$

where x^∗∗≡ min{v_h − 𝜃_i, E[v_h] − P₁}, and f is the density function of x_i.

Then, denoting the average rating in the simple model as $R^{unbiased}\equiv \frac {{\int }_{i}r_{h,i}di}{{\int }_{i}idi}$, we have that:

$$ R^{biased}-R^{unbiased}=g(\theta)>0 $$

(72)

i.e., the average rating under selection bias is higher than the average rating without selection bias, and their difference is a function of 𝜃. Under this condition, to observe systematically lower ratings for professional hosts, we need to assume that ${v_{h}^{p}}$ is systematically lower than ${v_{h}^{c}}$ or that the distribution of ${v_{h}^{c}}$ first order stochastically dominates (FOSD) the distribution of ${v_{h}^{p}}$.

Alternatively, to explain the rating patterns observed, we could assume that guests of professional and casual hosts differ systematically in their 𝜃, the psychological cost of leaving a bad review. Specifically, guests of professional hosts must have a systematically lower psychological threshold than guests of casual hosts. However, guests cannot observe hosts market participation and, thus, they cannot discern between professional or casual hosts; and hosts cannot infer the guests’ 𝜃, so they cannot select a specific type of guest. Thus, this self-selection of guests depending on the host type (and vice versa) is unlikely to occur in practice.¹²□

1.3 B.3 Relaxing the risk neutral assumption

In this section, we relax the risk-neutral assumption and allow the behavior of the guests to enter into the utility function of the host. We propose this modification because of the nature of Airbnb transactions. Since Airbnb hosts share their own properties, it is natural to expect them to be risk-averse toward guest misconduct.

Formally speaking, we assume the effort of guest i, denoted as e_i, to enter into the utility of the host, and the host’s utility is assumed to be concave with respect to e_i. The utility function of the host is given by:

$$ V_{h}(P)=\max\limits_{P}\{E[PQ-C(Q)|P]+E[(r_{h,i}+u_{h}(e_{i}))|P]\} $$

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where, as before, P denotes price, Q denotes the transaction volume, and v_h(P) denotes the ex-ante utility contingent on choosing price P. r_{h, i} is the rating the host receives from the guest i. The term u_h(e_i) shows that the guest’s effort e_i affects the host’s welfare. The concavity of u_h captures the risk aversion of the host. Then, hosts trade off expected profits against the possibility of guest misconduct in accepting guests. While this assumption reduces the number of guests a host will accept, the host’s rating does not affect service quality since it is still exogenously given and fixed. Formally, the only choice variable is still P and the optimality problem of the host is reduced to:

$$ \max\limits_{P}\{E[PQ-C(Q)|P]+E[r_{h,i}]\}. $$

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Independently of which assumptions are invoked on r_{h, i}, the absence of strategic manipulation of r_{h, i} makes it impossible to alter the optimality problem, which therefore is analogue to the simple model discussed above.

Moreover, since service quality is exogenously given in this scenario, to explain the systematically lower ratings to professional hosts, we would need to assume that the service quality of casual and professional hosts follow different distributions and that the distribution of ${v_{h}^{c}}$ FOSD the distribution of ${v_{h}^{p}}$.

1.4 B.4 Endogenous service quality

Next, we relax the exogenous service quality assumption. We allow service quality to vary between transactions. This assumption is consistent with the high heterogeneity of service quality on Airbnb. In this scenario, we consider three alternative models.

1.4.1 B.4.1 Model I: Hosts only care about profit and reputation

First, we endogenize service quality without changing the assumption that hosts care only about profit and reputation. The optimality problem of the host in Period 1 is given by:

$$ U_{h,1}(P)=\max\limits_{P}\{E[PQ-C(Q)|P]+E[U_{h,2}(e_{h},r_{h,i})]\}. $$

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Demand is realized in Period 1 while only reputation concerns and effort cost determine effort levels in Period 2. Then, the host’s optimality problem in Period 2 is:

$$ U_{h,2}(e_{h},r_{h,i})=\max\limits_{e_{h}}\{\beta_{h}r_{h,i}-C_{h}(e_{h})\}. $$

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Because of their higher market participation, professional hosts have a higher weight on reputation concerns, i.e., professional hosts have systematically higher β_h. With the assumption that guests report hosts’ effort levels in ratings r_{h, i}, i.e., r_{h, i} = e_h.

From the FOC, we have:

$$ C^{\prime}_{h}(e_{h})^{*}=\beta_{h}. $$

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The above condition solves for the optimal level of $e_{h}^{*}$.

If the effort cost function is identical across hosts, i.e., the function C_p(e_h) = C_c(e_h) ≡ C(e_h) for all e_h, and effort costs increase with effort level, i.e., C^′(e_h) > 0,∀e_h, then professional hosts exert higher effort levels due to their higher β_h, i.e., from β_p > β_c and $C^{\prime }_{h}>0$, we have $e_{p}^{*}>e_{c}^{*}$.

Even if the effort cost function of professional hosts is systematically different from that of casual hosts, because of economies of scale, it is unlikely that the latter have systematically lower marginal effort cost than professional hosts.That is, C^′(e_p) ≤ C^′(e_c) for each e_h. Even if we assume C^′(e_p) > C^′(e_c) for some e_h, the difference in marginal effort cost has to be large enough to offset the effect of β_h so that casual hosts can exert systematically higher effort under this model. Therefore, this model cannot easily explain why professional hosts have lower ratings.

1.4.2 B.4.2 Model II: Hosts also care about guest behavior

In this model, hosts also care about guest behavior. Hosts on Airbnb are relatively “small” service providers compared to firms in the accommodation industry, such as Hilton or Marriott. Therefore, a natural assumption is that hosts are risk- averse with respect to possible guest misconduct. We model this by letting guest conduct enters hosts’ utility function. However, as we show, simply introducing guest conduct in the utility function of a risk- averse host does not suffice to explain why professional hosts have lower ratings than casual hosts. Formally speaking, if we assume e_i and e_h are separable in the host’s utility, guest behavior cannot alter the host’s effort. Thus, we have to further assume a non-separable function of e_h and e_i as we do in our main model.

As U_h takes a seperable functional form of e_i and e_h, without loss of generality, we assume U_h is given by:

$$ U_{h}(e_{h},r_{h,i})=\max\limits_{e_{h}}{g(e_{i})-C_{h}(e_{h})+\beta_{h}r_{h,i}} $$

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where g(e_i) captures the effect of guests’ conduct on the host’s utility, and the concavity of g(⋅) models the host’s risk averse attitude toward the guest misconduct.

Assuming r_{h, i} = e_h, we have the same FOC as the previous model, i.e., $C^{\prime }_{h}(e_{h})^{*}=\beta _{h}$. This means that e_i does not determine the optimal level of e_h and therefore it cannot explain the difference in ratings between casual and professional hosts.

1.4.3 B.4.3 Model III: Hosts have interdependent preference

Next, we assume that the host has interdependent preference (without assuming the reciprocity feature discussed in our main theoretical framework), i.e., the utility of guest i, denoted as U_i, enters into the host’s utility function.

Under this assumption, the optimality problem of the host is given by:

$$ U_{h}(e_{h},e_{i},r_{h,i})=\max\limits_{e_{h}} {g(e_{i})+\alpha_{h} U_{i}(e_{i})-C_{h}(e_{h})+\beta_{h}r_{h,i}} $$

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where U_i is the ex-post utility of guest i given by U_i(e_i, e_h, r_i) = f(e_h) − C_i(e_i) + β_ir_i. Also, e_i and e_h are the efforts of guest i and the host h, respectively. The function g(⋅) denotes how much hosts care about a guest’s behavior, and f(⋅) denotes guest utility derived from host’s effort.

To derive a closed-from solution of the optimal effort level, we assume $f(e_{h})=e_{h}^{1-k}$ and $g(e_{i})={e_{i}^{k}}$, and the effort cost function to be quadratic, i.e., $C(e_{h})=\frac {c_{h}}{2}{e_{h}^{2}}$.

After invoking r_i = g(e_i), r_{h, i} = f(e_h), we have:

$$ (\alpha_{h}+\beta_{h})f^{\prime}(e_{h})=C^{\prime}_{h}(e_{h}). $$

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Then, we have $e_{h}^{*}=\left (\frac {(1-k)(\alpha _{h}+\beta _{h})}{c_{h}}\right )^{\frac {1}{k + 1}}$ and $\frac {\partial e_{h}^{*}}{\partial \beta _{h}}>0$.

If casual hosts differ from professional hosts only in β_h, and professional hosts have a higher level of β_h, we have $e_{p}^{*} > e_{c}^{*}$. Therefore, $r_{h,i}^{c}<r_{h,i}^{p}$, ∀i. Thus, the average rating of professional hosts is expected to be higher than casual hosts, contradicting what we observe in the Airbnb data.

In order to explain the systematically lower ratings of professional hosts, we need an additional assumption. Specifically, we need to assume that casual hosts are systematically more altruistic than professionals. This assumption can be implemented by adding the parameters α^g in front of the interdependent utility term in the gross utility of the host and assuming that the difference on the altruism weights are larger than the difference between reputation weights. Formally stated:

$$\begin{array}{@{}rcl@{}} \alpha_{c} - \alpha_{p}> \beta_{p} - \beta_{c}>0, \end{array} $$

where α_c and α_p are the weights on guests’ utility of the non-professional hosts and the professional hosts, respectively.

By invoking this assumption, we allow casual hosts to have systematically higher intrinsic altruism, and the difference between altruistic attitude is large enough to offset the difference between reputation concerns. However, we consider altruistic behavior to be a larger departure from standard assumptions and our explanation based on reciprocity to be more natural for the setting of Airbnb.