Replication code

Parameters

	name		value
demand (theta)	price	$\alpha$	-0.0068
	type 1	$\beta_1$	-12.5906
	type 2	$\beta_2$	-12.1095
	type 3	$\beta_3$	-11.7011
	type 4	$\beta_4$	-11.3012
	quality	$\gamma$	4.8860
	prior	$a$	12.2260
		$b$	2.1134
supply (c)	mean entry cost (type 1)	$\bar \kappa_1$	55,496
	mean entry cost (type 2)	$\bar \kappa_2$	96,673
	mean entry cost (type 3)	$\bar \kappa_3$	161,946
	mean entry cost (type 4)	$\bar \kappa_4$	270,233
	mean fixed cost (type 1)	$\bar \phi_1$	2,580
	mean fixed cost (type 2)	$\bar \phi_2$	3,577
	mean fixed cost (type 3)	$\bar \phi_3$	4,562
	mean fixed cost (type 4)	$\bar \phi_4$	5,751
other (params)	discount factor	$\delta$	0.995
	revenue fee	$f$	0.142
	review prob.	$\upsilon_r$	0.7041
	max. no. of reviews	$\bar N$	20
	arrival rate	$\mu$	10,000
	max. no. of listings	$J$	10,000

Demand

Function U_s(p,theta,t,params) characterizes a guests's indirect utility of renting a property in state $x=(N,K,j)$, where $j = 1,2,3,4$ is the property's (observed) type.

$$U_x = \gamma\frac{a + K(x)}{a + b + N(x)} + \beta(x) + \alpha ((1+f)p- t) + \epsilon = u(p,x) + \epsilon$$

$p$ is the daily rental rate of the listing; $t$ is the counterfactual per-unit subsidy. For the moment, we set $t$ equal to zero.

The unobserved quality $\omega$ is unknown to guests and hosts. However, $\omega$ is known to be iid $Beta(a,b)$ distributed. After observing the number of good reviews $K$ and bad reviews $N-K$ agents form an expectation about the unobserved quality, $E[\omega|N,K]$.

$\epsilon$ is iid T1EV extreme value distributed.

$\mathbf{s}$ is the state distribution. $s(x)$ pins down the number of properties in each state. Function ccp_s(p,P,s,theta,t,params) characterizes the probability that a guest intends to book the property at rate $p$ provided that all remaining hosts set their prices according to $P(x)$.

$$ccp(p,x) = \frac{\exp(u(p,x))}{1+\sum_xs(x)\exp(u(P(x),x))}$$ For later use, we also work out the first-order (dccp_s(p,P,s,theta,t,params)) and second-order (d2ccp_s(p,P,s,theta,t,params)) derivatives of $ccp(p,x)$ with respect to $p$.

$$ccp'(p,x) = ccp(p,x)(1 - ccp(p,x))\alpha(1+f) $$

$$ccp''(p,x) = ccp(p,x)(ccp(p,x)^2 - ccp(p,x))\alpha^2(1+f)^2 $$

The number of arriving guests is $Poisson(\mu)$ distributed. Function q_s(p,P,s,theta,t,params) characterizes the probability that at least one of these consumers books the property, again assuming its rental rate is $p$ while everyone else follows the pricing rule $P(x)$.

$$q(p,x) = 1 - \exp(-\mu \cdot ccp(p,x))$$

Function dq_s(p,P,s,theta,t,params) and function d2q_s(p,P,s,theta,t,params) describe the first- and second-order derivatives of $q(p,x)$ with respect to $p$.

$$q'(p,x) = \mu\exp(-\mu \cdot ccp(p,x))ccp'(p,x)$$

$$q''(p,x) = \mu\exp(-\mu \cdot ccp(p,x))(ccp''(p,x)-\mu\cdot ccp'(p,x))$$

Strictly speaking, $q_s$ is the daily booking probability. As a time period in the model is a 4-week interval ("month"), we interpret $q_s$ as the monthly occupancy rate.

State Transitions

If a property is booked ($q(p,x) = 1$), $x$ changes with probability $\upsilon_r = 70.41$% between periods. Conditional on being booked, it receives a good review ($\Delta N = 1, \Delta K = 1$) with probability $\frac{a+K(x)}{a+b+N(x)}$. Conditional on being booked, it receives a bad review ($\Delta N = 1, \Delta K = 0$) with probability $\left(1-\frac{a + K(x)}{a+b+N(x)}\right)$. The probability of getting a good review and the probability of getting a bad review are $\rho^g(p,x)$ and $\rho^b(p,x)$ respectively. States where $N=20$ are terminal and the probability of getting a review is zero.

$$\rho^g(p,x) = \upsilon_rq(p,x)\frac{a+K(x)}{a+b+N(x)}$$

$$\rho^b(p,x) = \upsilon_rq(p,x)\left(1-\frac{a + K(x)}{a+b+N(x)}\right)$$

Accordingly, the probability $\rho^0(p,x)$ of getting no review is $1-\rho^g(p,x)-\rho^b(p,x)$. States are arranged in increasing order of type $j$ and, for a given type, in increasing order of $N$ and, for a given $N$, in increasing order of $K$. $S$ is the state space. Note: $S$ is in params.

$$ S = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 & 0 \\ 2 & 2 & 1 & 0 & 0 & 0 \\ ... & ... & ... & ... & ... & ... \\ 11 & 17 & 0 & 0 & 1 & 0 \\ 12 & 17 & 0 & 0 & 1 & 0 \\ 13 & 17 & 0 & 0 & 1 & 0 \\ ... & ... & ... & ... & ... & ... \\ 19 & 20 & 0 & 0 & 0 & 1 \\ 20 & 20 & 0 & 0 & 0 & 1 \end{bmatrix} $$

Function dT_s(dq,theta,params) stores the transition matrix $\mathbf{T}(p)$. It turns out that the way states are ordered the number of zeros between $\rho^0(p,x)$ and $\rho^g(p,x)$ is $N$.

	$(0,0,1)$	$(0,1,1)$	$(1,1,1)$	$(0,2,1)$	$(1,2,1)$	$(2,2,1)$	...	$(20,20,4)$
$(0,0,1)$	$\rho^0_{(0,0,1)}$	$\rho^b_{(0,0,1)}$	$\rho^g_{(0,0,1)}$	0	0	0	...	0
$(0,1,1)$	0	$\rho^0_{(0,1,1)}$	0	$\rho^b_{(0,1,1)}$	$\rho^g_{(0,1,1)}$	0	...	0
$(1,1,1)$	0	0	$\rho^0_{(1,1,1)}$	0	$\rho^b_{(1,1,1)}$	$\rho^g_{(1,1,1)}$	...	0
$(0,2,1)$	0	0	0	$\rho^0_{(0,2,1)}$	0	0	...	0
$(1,2,1)$	0	0	0	0	$\rho^0_{(1,2,1)}$	0	...	0
$(2,2,1)$	0	0	0	0	0	$\rho^0_{(2,2,1)}$	...	0
...	...	...	...	...	...	...	...	0
$(20,20,4)$	0	0	0	0	0	0	...	1

Function dT_s(q,theta,params) and d2T_s(q,theta,params) store the first-order and second-order derivatives of $\mathbf{T}(p)$ respectively. Notice:

$$\rho^{0\prime}(p,x) = -\upsilon_rq'(p,x)$$

$$\rho^{0\prime\prime}(p,x) = -\upsilon_rq''(p,x)$$

$$\rho^{g\prime}(p,x) = \upsilon_rq'(p,x)\left(\frac{a+K(x)}{a+b+N(x)}\right)$$

$$\rho^{g\prime\prime}(p,x) = \upsilon_rq''(p,x)\left(\frac{a+K(x)}{a+b+N(x)}\right)$$

$$\rho^{b\prime}(p,x) = \upsilon_rq'(p,x)\left(1-\frac{a+K(x)}{a+b+N(x)}\right)$$

$$\rho^{b\prime\prime}(p,x) = \upsilon_rq''(p,x)\left(1-\frac{a+K(x)}{a+b+N(x)}\right)$$

Market Entry & Exit

Types are equally distributed in the host population, meaning 2,500 properties have a certain type. If a host is inactive and has not yet entered the market, they can do so at the start of the following month at entry cost $\kappa_j$ which is iid drawn from $Exponential(\bar \kappa_j)$, $j=1,2,3,4$. Let $\lambda_j$ denote the entry rate.

$$ \lambda_j = 1-\exp(-\delta V((0,0,j))]\bar\kappa_j^{-1} ) $$

Denote the number of properties of type $j$ by $s_j$. The expected, total entry costs of type $j$ hosts in a given month is the number of inactive hosts $(J/4 - s_j)$ times $\mathbb{E}[\kappa_j|\phi_j\geq \delta V(0,0,j)]$.

$$ \text{Total entry costs} = \sum_{j}\left(J/4 - \sum_xs_j(x)\right)\left(\lambda_j\bar \kappa_j - (1-\lambda_j)\delta V((0,0,j))\right) $$

If a host is active they have entered the market. At the end of each month they have to pay the operating cost $\phi_j$ for the following month, regardless of whether the property is booked or not. $\phi_j$ is iid $Exponential(\bar \phi_j)$ distributed. Let $\chi(p,x)$ denote the exit rate.

$$ \chi(p,x) = \exp(-\delta \mathbb{E}_{\tilde x}[V(\tilde x)|p,x]\bar\phi_j^{-1} ). $$

$\tilde x$ denotes the state in the next month. Note that the host's expectation depends on $p$ because the property is likely to transition to a new state if it is booked.

The expected, total operating costs of properties in a certain state in a given month are the number of active hosts $s(x)$ times $\mathbb{E}[\phi(x)|\phi(x)\leq \delta \mathbb{E}_{\tilde x}[V(\tilde x)|p,x]]$

$$ \text{Total operating costs} = \sum_{x}s(x)\left((1-\chi(p,x))\bar \phi(x) - \chi(p,x)\delta \mathbb{E}_{\tilde x}[V(\tilde x)|p,x]\right) $$

F_s(p,P,s,q,chi,lamb,theta,t,params) contains the expanded transition matrix $\mathbf{F}(p)$. It accommodate transitions from and to inactivity by expanding $\mathbf{T}(p)$ by an additional state.

	$(0,0,1)$	$(0,1,1)$	$(1,1,1)$	$(0,2,1)$	$(1,2,1)$	$(2,2,1)$	...	$(0,0,2)$	...	$(20,20,4)$	$(20,20,4)$
$(0,0,1)$	$(1-\chi_{(0,0,1)})\rho^0_{(0,0,1)}$	$(1-\chi_{(0,0,1)})\rho^b_{(0,0,1)}$	$(1-\chi_{(0,0,1)})\rho^g_{(0,0,1)}$	0	0	0	...	...	...	0	$\chi_{(0,0,1)}$
$(0,1,1)$	0	$(1-\chi_{(0,1,1)})\rho^0_{(0,1,1)}$	0	$(1-\chi_{(0,1,1)})\rho^b_{(0,1,1)}$	$(1-\chi_{(0,1,1)})\rho^g_{(0,1,1)}$	0	...	...	...	0	$\chi_{(0,1,1)}$
$(1,1,1)$	0	0	$(1-\chi_{(1,1,1)})\rho^0_{(1,1,1)}$	0	$(1-\chi_{(1,1,1)})\rho^b_{(1,1,1)}$	$(1-\chi_{(1,1,1)})\rho^g_{(1,1,1)}$	...	...	...	0	$\chi_{(1,1,1)}$
$(0,2,1)$	0	0	0	$(1-\chi_{(1,2,1)})\rho^0_{(1,2,1)}$	0	0	...	...	...	0	$\chi_{(1,2,1)}$
$(1,2,1)$	0	0	0	0	$(1-\chi_{(1,2,1)})\rho^0_{(1,2,1)}$	0	...	...	...	0	$\chi_{(1,2,1)}$
$(2,2,1)$	0	0	0	0	0	$(1-\chi_{(2,2,1)})\rho^0_{(2,2,1)}$	...	...	...	0	$\chi_{(2,2,1)}$
...	...	...	...	...	...	...	...	...	...	...	...
$(0,0,2)$	0	0	0	0	0	0	...	$(1-\chi_{(0,0,2)})\rho^0_{(0,0,2)}$	...	0	$\chi_{(0,0,2)}$
...	...	...	...	...	...	...	...	...	...	...	...
$(20,20,4)$	0	0	0	0	0	0	...	...	...	$1 - \chi_{(20,20,4)}$	$\chi_{(20,20,4)}$
$\varnothing_1$	$\lambda_1$	0	0	0	0	0	...	...	...	0	$1-\lambda_1$
$\varnothing_2$	0	0	0	0	0	0	...	$\lambda_2$	...	0	$1-\lambda_2$
$\varnothing_3$	0	0	0	0	0	0	...	...	...	0	$1-\lambda_3$
$\varnothing_4$	0	0	0	0	0	0	...	...	...	0	$1-\lambda_4$

Solving The Model

solver(theta,c,guess,t,tol,params) finds an oblivious equilibrium of the model. guess contains starting values for the prices $\mathbf{\hat P}$, the state distribution $\mathbf{\hat s}$ and the value function $\mathbf{\hat V}$.

Price Update

Conditional on guess $\mathbf{\hat V}$ and assuming that there are $\hat s(x)$ competitors in state $x$ who set their prices according to $\hat P(x)$, a host operating a property in state $x$ maximizes $V(x)$ over $p$.

$$ V(p,x) = 30q(p,x)p - (1-\chi(p,x))\phi(x) + \delta \mathbf{T}\mathbf{\hat V} $$

The FOC requires that $V'(p,x) = 0$. The first-order Taylor series approximation around $p_0$ is $V'(p,x) = V'(p_0,x) + V''(p_0,x)(p-p_0)$. We find $p$ by iterating

$$p = p_0 - \frac{V'(p_0,x)}{V''(p_0,x)}$$

until $|p-p_0| \leq 0.1$.

dV_s(p,P,s,V,theta,phi_bar,t,params) and d2V_s(p,P,s,V,theta,phi_bar,t,params) store the first- and second-order derivative of $V(p,x)$ with respect to p respectively.

$$ V'(p,x) = 30(q(p,x) + q'(p,x)p) + (1 - \chi(p,x))\delta \mathbf{T}'(p)\mathbf{\hat V} $$

$$ V''(p,x) = 30(2q'(p,x) + q''(p,x)) + (1 - \chi(p,x))\delta \mathbf{T}''(p)\mathbf{\hat V} - \chi(p,x)\frac{(\delta \mathbf{T}(p)\mathbf{\hat V})^2}{\phi(x)} $$

In code:

while dP>.1:        P1 = P0 - dV_s(P0,P_old,s_old,V_old,theta,phi_bar,t,params)/d2V_s(P0,P_old,s_old,V_old,theta,phi_bar,t,params)        P1 = np.where(np.isnan(P1) == True,P_old,np.where((P1<0),0,np.where((P1>1000),1000,P1)))        dP = np.max(np.abs(P1 - P0))        P0 = P1

Value Function Update

Having found $p$ that solves the host's pricing problem, we let $\mathbf{P}=p$ and update the value function.

$$ V(x) = 30q(P(x),x)P(x) - (1-\chi(P(x),x))\phi(x) + \delta \mathbf{T}(p)\mathbf{\hat V} $$

In code:

q_new = q_s(P_new,P_new,s_old,theta,t,params) T = T_s(P_new,P_new,s_old,q_new,theta,t,params) eV = T @ V_old V_new = 30 * (q_new * P_new.T) + delta * eV - (1 - np.exp(-delta * eV/phi_bar)) * phi_bar

Entry & Exit Rate Updates

We use $\mathbf{V}$ and $\mathbf{P}$ to compute $\lambda(x)$, $\chi(P(x),x)$ and, ultimately, $\mathbf{F}(P)$.

In code:

eV = T @ V_new chi = np.exp(-delta * eV/phi_bar).flatten() lamb = (1-np.exp(-delta * V_new.reshape((231,4),order='F')[0,:]/[kappa1,kappa2,kappa3,kappa4])) F = F_s(q_new,chi,lamb,theta,params)

State Distribution Update

We use $\mathbf{F}(P)$ to compute the stationary state distribution. Specifically, we iterate $\mathbf{s}$ until $|\mathbf{s} - \mathbf{s}_0|\leq 0.01$.

$$ \left[\mathbf{s},J/4-s_1,J/4-s_2,J/4-s_3,J/4-s_4\right] = \left[\mathbf{s}_0,J/4-s_1,J/4-s_2,J/4-s_3,J/4-s_4\right]\mathbf{F}(P) $$

In code:

while np.max(np.abs(s_new - s_old))>10e-3:        s_old = s_new        s_new = (np.array([np.append(s_old,np.array([J/4-s_old[0,:231].sum(),J/4-s_old[0,231:462].sum(),J/4-s_old[0,462:693].sum(),J/4-        s_old[0,693:].sum()]))])@ F)[:1,:-4]

Solution

We update $\mathbf{\hat P} = \mathbf{P}$, $\mathbf{\hat s}=\mathbf{s}$ and $\mathbf{\hat V}=\mathbf{V}$ and repeat the algorithm until convergence, i.e.,

$$|\mathbf{P}-\mathbf{\hat P}|\leq \text{tol} \ \text{ and } \ |\mathbf{s}-\mathbf{\hat s}|\leq \text{tol} \ \text{ and } \ |\mathbf{V}-\mathbf{\hat V}|\leq \text{tol}.$$

tol is set to 0.000001. To save time, we solve the host's pricing problem only if $\mathbf{V}$ changes substantially, i.e., by more than 10% since the last time we solved for $p$.

Our initial guess of $P(x)$ is $300 for all $x$. The initial guess of the state distribution is that half of the properties are in the market, while half are not. Those that are in the market are uniformely distributed across states. The initial guess for the value function is the PDV of the revenue earned by the host if they as well as all competitors set a price of $300.

In code:

P_init = np.array([[300] * len(S)]) s_init = np.array([[J/(2 * len(S))] * len(S)]) V_init = (30 * q_s(300,P_init,s_init,theta,0,params) * P_init.T)/(1-delta) s_star = np.where(s_star<0,0,s_star)

The solution to the model is $\mathbf{V}^\ast, \mathbf{s}^\ast, \mathbf{P}^\ast, \mathbf{\chi}^\ast, \mathbf{\lambda}^\ast$. We use the solution to compute $q({P}^*(x),x)$.

In code:

V_star,s_star,P_star,chi_star,lamb_star = solver(theta,c,[P_init,s_init,V_init],tol,params)

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Data Generating Process

We generate 4 years worth of mock data. We draw the average number of properties from the equilibrium state distribution. We associate each property with the number of reviews, equilibrium price (after adding some noise), and demand (after adding some noise) of the corresponding state. We repeat this for $13\times 4$ times.

In code:

for t in range(1,13 * 4+1): if (t==1): index = np.repeat(range(0,924),states p = (P_star.T + np.random.normal(loc = 0, scale = 25.0, size = (P_star.T).shape)) q = q_s(p,p,s_star,theta,0,params) data = np.hstack((np.zeros((len(index),1)) + t, S[index,:], p[index,:], q[index,:]+ np.random.normal(loc = 0, scale = 0.15, size = q[index,:].shape))) else: p = (P_star.T + np.random.normal(loc = 0, scale = 25.0, size = (P_star.T).shape)) q = q_s(p,p,s_star,theta,0,params) data = np.vstack((data, np.hstack((np.zeros((len(index),1)) + t, S[index,:], p[index,:], q[index,:] + np.random.normal(loc = 0, scale = 0.15, size = q[index,:].shape))))) data = pd.DataFrame(data,columns=['period','K','N','type 1','type 2','type 3','type 4','p','q']) data.to_pickle('data.pkl')

Demand Estimation

We estimate the demand parameters using GMM. xi(omicron,adata,params) stores the structural error term $\xi_{it}$ of property $i$ at time $\tau$.

$$\xi_{i\tau}(\omicron) = \ln(ccp_{i\tau}) - \ln(ccp_{0\tau}) - u_{i\tau}(\omicron)$$

Inversion

We retrieve $ccp_{it}$ from the data by inverting $q_{i\tau}$.

$$ ccp_{i\tau} = -\ln(1 - q_{i\tau})/\mu $$

$ccp_{0\tau}$ is then $1-\sum_i ccp_{i\tau}$. Notice that, order to arrive at the regression equation, we must take the logarithm twice. This introduces additional bias from measurement error and complicates the estimation.

Rather than estimating $\theta = (a,b,\alpha,\boldsymbol{\beta},\gamma)$ directly, we estimate $\omicron = (\psi,\iota,\alpha,\boldsymbol{\beta},\gamma)$ to facilitate the estimation.

$$ \frac{a}{a+b} = \frac{1}{1+\exp(-\psi)} $$

$$ (a+b) = \exp(\iota) $$

Objective Function

The objective function is stored in O(omicron,adata,W,params). Let $I$ denote the total number of observations of the dataset.

$$\left(\frac{1}{I}\mathbf{Z}^T\boldsymbol{\xi}(\omicron)\right)^TW\left(\frac{1}{I}\mathbf{Z}^T\boldsymbol{\xi}(\omicron)\right)$$

$\mathbf{Z}$ is the set of instruments. Here, we simply use the rental rate $p$, number of reviews $N$ and $K$ and the average rating $r=1+4(K/N)$, as the prices in the mock data are simply a function of $s$.

We minimize $O(\omicron)$ using the analytical gradient.

$$ \nabla O(\omicron) = 2\left(-\frac{1}{I}\mathbf{Z}^T\nabla u(\omicron)\right)^T W\left(\frac{1}{I}\mathbf{Z}^T\boldsymbol{\xi}(\omicron)\right) $$

dO(omicron,adata,W,params) contains $\boldsymbol{\xi}'(\omicron)$. It requires $\nabla u(\omicron)$ (dU(omicron,adata,params)).

Minimization

To initiate the minimization, we choose appropriate starting values. For the first step of two-step GMM we set the weighting matrix equal to the inverse of the variance-covariance matrix of the instruments.

$$ W_1 = I\left(\mathbf{Z}^T\mathbf{Z}\right)^{-1} $$

In code:

start_values = [0,0,0,-10,-10,-10,-10,0] W1 = np.linalg.inv( ((Z(start_values,data,params)).T @ (Z(start_values,data,params)))/len(data)) res_demand = minimize(O, start_values, args=(data,W1,params), method='BFGS',jac=dO)

In the second step, we choose the efficient weighting matrix. Let $\hat \omicron$ be our estimation result from the first stage.

$$ W_2 = \left(\left(-\frac{1}{I}\mathbf{Z}^T\nabla u(\hat \omicron)\right)\left(-\frac{1}{I}\mathbf{Z}^T\nabla u(\hat \omicron)\right)^T\right)^{-1} $$

In code:

xi_hat = xi(res_demand.x,data,params) W2 = np.linalg.inv( ((xi_hat * Z(res_demand.x,data,params)).T @ (xi_hat * Z(res_demand.x,data,params)))/len(data) ) res_demand = minimize(O, start_values, args=(data,W2,params), method='BFGS',jac=dO)

Standard Errors

As we have chosen the efficient weighting matrix in the second step, the (heteroscedasticity robust) standard errors simplify to

$$ \frac{1}{I}\left( diag\left( \left( \left(\frac{1}{N}\mathbf{Z}^T\nabla u(\hat \omicron)\right)^TW_2\left(\frac{1}{I}\mathbf{Z}^T\nabla u(\hat \omicron)\right) \right)^{-1} \right) \right)^\frac{1}{2} $$

In code:

G_bar = ( (Z(res_demand.x,data,params).T @ (-dU(res_demand.x,data,params))) )/len(data) W2 = np.linalg.inv( ((xi_hat * Z(res_demand.x,data,params)).T @ (xi_hat * Z(res_demand.x,data,params)))/len(data) ) S_hat = np.diag(np.linalg.inv((G_bar.T @ W2) @ G_bar))**.5/len(data)

Estimation Results

parameter	estimate	standard error
$\phi$	1.8388	(0.0002)
$\iota$	2.6677	(0.0004)
$\alpha$	-0.0068	(0.0000)
$\beta_1$	-12.9853	(0.0019)
$\beta_2$	-12.4921	(0.0019)
$\beta_3$	-12.0770	(0.0019)
$\beta_4$	-11.6858	(0.0019)
$\gamma$	5.2759	(0.0020)

We convert $\hat \omicron$ to $\hat \theta$.

In code:

theta_hat = [expit(res_demand.x[0])*np.exp(res_demand.x[1]), (1-expit(res_demand.x[0]))*np.exp(res_demand.x[1]), res_demand.x[2], res_demand.x[3:7], res_demand.x[7]]

Our estimates of $\psi$ and $\iota$ correspond to $a=12.3102$ and $b=1.8890$. Notice that our estimates are slightly biased as the demand inversion is non-linear and the measurement error is not fully captured by the structural error term.

Supply Estimation

Objective Function

We estimate $\mathbf{c} = (\phi_1,\phi_2\phi_3,\phi_4,\kappa_1,\kappa_2,\kappa_3,\kappa_4)$ by maximizing the logarithm of the likelihood of the equilibrium state distribution $\mathbf{s}$ over $\mathbf{c}$. This requires that we infer the average number of listings $\mathbf{s}^d$ from the (mock) data.

In code:

s_d = np.array([(data.groupby(['x'])['period'].count()/data.groupby(['period']).mean().shape[0]).reindex(np.arange(0,len(S)), fill_value=0)])

l(k,theta,guess,tol,s_d,params) stores the log-likelihood function (times -1).

$$ \text{Log-likelihood} = \sum_{x} s^d(x) \ln \left(s^\ast(x|\mathbf{c}) \right) + \sum_j\left(\frac{J}{4}-\sum_{x}s_j^d(x)\right)\ln\left(\frac{J}{4}-\sum_{x}s_j^\ast(x|\mathbf{c})\right) $$

We exclude states for which we do not observe any observations in the mock data or for which the model predicts that there are no observations as otherwise the log-likelihood is undefined.

Maximization

tol is set to 1. Each candidate for $\mathbf{c}$ requires us to solve the model. We use the same guess as in the 'Solving the Model' section to initiate the solution algorithm. After that, we use the model solution for the previous set of candidates as the starting values to find the model solution for the next set of candidates. Furthermore, we use the demand estimates $\hat \theta$. To facilitate the search of a maximum, we search over $\ln(\mathbf{c})$, thereby excluding negative values. k0 contains the starting values.

In code:

k0 = np.log([100000,100000,100000,100000,3000,3000,3000,3000]) res_supply = minimize(l, k0, args=(theta,[P_init,s_init,V_init],tol,s_d,params), method='BFGS')

Standard Errors

We use the the numerical approximation of the inverse Hessian $(H(\mathbf{\hat c}))^{-1}$ (res_supply.hess_inv) to compute the standard errors of the estimates.

$$ \sqrt{\frac{diag\left((H(\mathbf{\hat c}))^{-1}\right)}{I}} $$

(np.diag(res_supply.hess_inv)/len(data))**0.5

We use the delta method to compute the standard errors of $c$.

$$ \sqrt{\left(\frac{\partial f(\mathbf{c})}{\partial \mathbf{c}}\right)^2\frac{diag\left((H(\mathbf{\hat c}))^{-1}\right)}{I}} $$

In code:

((np.exp(res_supply.x) * np.diag(res_supply.hess_inv) * np.exp(res_supply.x))/len(data))**0.5

Estimation Results

parameter	estimate	standard error	parameter	estimate	standard error
$\ln(\bar \kappa_1)$	10.8625	(0.00016)	$\bar \kappa_1$	52183	(9.0024)
$\ln(\bar \kappa_2)$	11.4392	(0.00016)	$\bar \kappa_2$	92888	(15.7957)
$\ln(\bar \kappa_3)$	11.9692	(0.00017)	$\bar \kappa_3$	157812	(27.2976)
$\ln(\bar \kappa_4)$	12.4895	(0.00020)	$\bar \kappa_4$	265529	(53.6090)
$\ln(\bar \phi_1)$	7.84021	(0.00001)	$\bar \phi_1$	2541	(0.0216)
$\ln(\bar \phi_2)$	8.17893	(0.00001)	$\bar \phi_2$	3565	(0.0268)
$\ln(\bar \phi_3)$	8.42697	(0.00001)	$\bar \phi_3$	4569	(0.0312)
$\ln(\bar \phi_4)$	8.66163	(0.00001)	$\bar \phi_4$	5777	(0.0671)

Counterfactual Analysis

We simulate the model forward for 10 years - starting at the stationary equilibrium ($\mathbf{V}^\ast, \mathbf{s}^\ast, \mathbf{P}^\ast$) - (1) if every $j$ type host in the market receives a monthly lump-sum subsidy of $ $Sub_j$ (Sub) and/or (2) if consumers receive a $ $t$ (t) subsidy for each day they book a property that has not been reviewed before. The corresponding function is stored in simulation1(theta,c,sol,t,Sub,It,params).

We calculate the sum of host profits, the consumer surplus and the cost of the subsidy per month.

Subsidy Cost

The expected cost of the lump-sum subsidy is $Sub\sum_xs(x)$ per month. The cost of the per-unit subsidy is $30t\sum_xs(x)q(x)$.

$$ \text{Subsidy cost} = \sum_jSub_j\sum_xs(x) + 30t\sum_xs(x)q(x) $$

In code:

((30 * s_new * q_new.T) @ t) + (s_new @ Sub)

Consumer Surplus

We calculate the expected consumer surplus per month. As each property can only be booked once, we focus on the consumer surplus from the inside good, i.e., the consumer surplus from Airbnb bookings.

$$ \text{Consumer surplus} = -\frac{30}{\alpha}\left(\sum_x s(x)q(x)\right)\ln\left(1 + \sum_{x} s(x)\exp(u(x))\right) + \text{constant} $$

In code:

-(s_new @ q_new) * 30 * np.log(1 + (s_new @ np.array([np.diagonal(np.exp(U(P_new,theta,t,params)))]).T))/alpha

Note that our consumer surplus measure likely understates the true consumer surplus.

Aggregate Profit

We distinguish hosts who are in the market and hosts who are outside the market. Hosts in the market receive the expected monthly revenue of renting out the property as well as the lump-sum subsidy. They also pay the operating cost. See the 'Market Entry & Exit' section for the total expected operation costs per month.

$$ \text{Profit (inside)} = \sum_{x}s(x)\left(30q(x)P(x) + Sub_j - \left((1-\chi(p,x))\bar \phi(x) - \chi(p,x)\delta \mathbb{E}_{\tilde x}[V(\tilde x)|p,x]\right)\right) $$

In code:

s_new @ ((1+f)*np.array([np.diagonal(30 * q_s(P_new,P_new,s_new,theta,t,params)*P_new)]).T + Sub - (np.array([np.repeat(c[4:],231)]).T - np.array([chi_new]).T * (delta * eV_in + np.array([np.repeat(c[4:],231)]).T)))

Hosts who are currently outside the market do not earn revenue but pay the entry cost if they decide to enter that month.

$$ \text{Profit (outside)} = -\sum_{j} (J/4 - \sum_{x}s_j(x))\left(\lambda_j\bar \kappa_j - (1-\lambda_j)\delta V((0,0,j))\right) $$

-(J/4-s_new[0,:231].sum()) * (c[0] - (1-lamb_new[0]) * (delta * eV_out[0] + c[0]))

-(J/4-s_new[0,231:462].sum()) * (c[1] - (1-lamb_new[231])*(delta * eV_out[1] + c[1]))

-(J/4-s_new[0,462:693].sum()) * (c[2] - (1-lamb_new[462])*(delta * eV_out[2] + c[2]))

-(J/4-s_new[0,693:].sum()) * (c[3] - (1-lamb_new[693])*(delta * eV_out[3] + c[3]))

Social Welfare

We calculate social welfare as the present discounted value of the sum of host profits and the consumer surplus less the cost of the subsidy over the 10 year time horizon.

$$ \text{Welfare} = \sum_{\tau=1}^{130} \delta^{\tau-1}(\text{Consumer surplus} + \text{Profit (inside)} + \text{Profit (outside)} - \text{Subsidy cost}) $$

Welfare Maximization

For counterfactual 1, we maximize social welfare over $Sub_j, j=1,2,3,4$ by repeatedly simulating the model forward. The function that maximizes welfare over $Sub_j$ is Sub_prim(Sub,theta,c,sol,It,params). Initially, we set the the lump-sum subsidy to zero.

In code:

minimize(Sub_prim, [0,0,0,0], args=(theta_hat,c_hat,[P_star,s_star,V_star],'constrained',130,params), method='BFGS')

We find that a lump-sum subsidy corresponding to more or less 20-30% (depending on property type) of producer surplus (i.e., revenue) maximizes welfare.

type	$Sub^*$ in $	$Sub^*$ in % of revenue	$\Delta$ # properties
1	$409.51	21.32%	79.61
2	$646.95	23.68%	77.04
3	$945.56	26.47%	65.74
4	$1309.57	28.99%	86.71

Subsidizing market entry raises social welfare by a bit below $2,000 per day. The cost of the subsidy amount to roughly $25,000 per day. Each Airbnb guest is on average better off by about $15.29 per day. Each host gains about $8.38 per day.

For counterfactual 2, we search for the welfare-maximizing subsidy $t$ if entry/exit is efficient (lest we conflate two distinct effects of $t$, on consumer booking decisions and hosts' decisions to enter or exit the market). As hosts raise their prices in response to $t$ being paid to consumers, they will enter the market more frequently and exit the market less often. We adjust the lump-sum subsidy downward to keep the revenues of hosts at their efficient level. The function that maximizes welfare over $t$ is t_prim(t,theta,c,sol,Sub,It,params). The forward simulation that keeps host revenues at their optimal level is simulation2(theta,c,sol,t,Sub,It,params).

In code:

W1_c,CS1_c,PS1_c,GS1_c,P1_c,s1_c,V1_c = simulation1(theta_hat,c_hat,[P_star,s_star,V_star],np.zeros((S.shape[0],1)),Sub_c,1000,params) minimize(t_prim, [0,0,0,0], args=(theta_hat,c_hat,[P1_c,s1_c,V1_c],[409.51, 646.95, 945.56, 1309.57],130,params), method='BFGS')

We find that a per-day subsidy corresponding to about 20-21% of the rental rate maximizes welfare. From a welfare perspective, rental rates should be 11-16% lower. All changes are relative to counterfactual 1.

type	$t^\ast$	$t^\ast$ in % of price	price	$\Delta$ price in $	$\Delta$ price in %	$\Delta$ demand	$\Delta$ # properties
1	$41.60	20.40%	$212.94	-$32.58	-15.98%	17.63%	-18.00
2	$47.28	20.61%	$243.63	-$33.06	-14.41%	16.13%	-15.67
3	$53.98	20.96%	$277.82	-$33.66	-13.07%	14.90%	-13.92
4	$59.47	20.57%	$316.06	-$32.59	-11.27%	12.97%	-11.33

Subsidizing social learning raises social welfare by a bit less than $2,000 per day. This compares to about $24,000 paid in subsidies each day. Each guest is better off by about $9.93 per day. By design, host profit does not change in any meaningful way.