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\begin_body

\begin_layout Title
Code Guide
\end_layout

\begin_layout Author
Anthony Lee Zhang
\end_layout

\begin_layout Section
Overview
\end_layout

\begin_layout Standard
The basis of the calibration is to iterate the operator:
\begin_inset Formula 
\[
\mathcal{T}\left[\hat{V}\right]=\max_{q}\left(q-\tau\right)p_{\hat{V}\left(\cdot\right),F\left(\cdot\right)}\left(q\right)+\left(1-q\right)\left[\gamma+\delta\mathbb{E}_{G\left(\cdot\mid\cdot\right)}\left[\hat{V}\left(\gamma^{\prime}\right)\mid\gamma\right]\right]
\]

\end_inset


\end_layout

\begin_layout Standard
on candidate 
\begin_inset Formula $\hat{V}\left(\cdot\right)$
\end_inset

 functions until convergence, and then characterize the steady state of
 this.
 We will computationally do this by operating on quantile grids, with a
 total of 
\begin_inset Formula $qres$
\end_inset

 gridpoints.
 Any functions of 
\begin_inset Formula $\gamma$
\end_inset

, such as 
\begin_inset Formula $V\left(\gamma\right),WTP\left(\gamma\right)$
\end_inset

, etc.
 will be represented as length 
\begin_inset Formula $qres$
\end_inset

 vectors, which for code clarity we'll keep together in a data.table.
 So, for example, we will have vectors:
\end_layout

\begin_layout Itemize
\begin_inset Formula $Fgam=\left[\frac{1}{qres},\frac{2}{qres},\ldots\frac{qres-1}{qres},1\right]$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\gamma=\left[F^{-1}\left(0\right),F^{-1}\left(\frac{1}{qres}\right),\ldots F^{-1}\left(\frac{qres-1}{qres}\right)\right]$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $V\left(\gamma\right)=\left[V\left(F^{-1}\left(0\right)\right),V\left(F^{-1}\left(\frac{1}{qres}\right)\right)\right]$
\end_inset


\end_layout

\begin_layout Section
solve_value_function
\end_layout

\begin_layout Subsection
Overview
\end_layout

\begin_layout Standard
solve_value_function takes as input a data table specifying the quantile
 vector 
\begin_inset Formula $\gamma$
\end_inset

, and an initial value function guess 
\begin_inset Formula $V\left[\cdot\right]$
\end_inset

, as well as parameters such as discount rate/decay rate, and iterates the
 
\begin_inset Formula $\mathcal{T}$
\end_inset

 operator until convergence.
 It outputs the equilibrium value function, and some auxiliary information
 like equilibrium sale prices and probabilities, 
\begin_inset Formula $WTP$
\end_inset

's, etc.
\end_layout

\begin_layout Subsection
Inputs
\end_layout

\begin_layout Standard
solve_value_function begins with a uniformly spaced quantile grid vector.
 Supposing we use 
\begin_inset Formula $qres$
\end_inset

, this will be:
\end_layout

\begin_layout Standard
The user specifies a value distribution 
\begin_inset Formula $F\left(\cdot\right)$
\end_inset

 over use values 
\begin_inset Formula $\gamma$
\end_inset

 by inputting its quantile vector, as:
\end_layout

\begin_layout Standard
We will think of everything in terms of 
\begin_inset Quotes eld
\end_inset

quantile vectors.
\begin_inset Quotes erd
\end_inset

 So, for example, 
\begin_inset Formula $\gamma\left[q\right]$
\end_inset

 refers to the value of 
\begin_inset Formula $\gamma$
\end_inset

 at the 
\begin_inset Formula $q$
\end_inset

'th quantile.
 We will also need to specify a candidate value function 
\begin_inset Formula $\hat{V}\left[q\right]$
\end_inset

 to start the Bellman iteration.
 These vectors will be given to solve_value_function in a 
\begin_inset Formula $qres$
\end_inset

-row data table, with the following columns:
\end_layout

\begin_layout Itemize
\begin_inset Formula $Fgam$
\end_inset

: Quantile vector 
\begin_inset Formula $\left[\frac{1}{qres},\frac{2}{qres},\ldots\frac{qres-1}{qres},1\right]$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $fgam$
\end_inset

: Density at 
\begin_inset Formula $q$
\end_inset

, by construction equal to 
\begin_inset Formula $\frac{1}{qres}$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\gamma$
\end_inset

: Use value at 
\begin_inset Formula $q$
\end_inset

th quantile of 
\begin_inset Formula $F$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $V$
\end_inset

: Candidate value function at 
\begin_inset Formula $q$
\end_inset

th quantile
\end_layout

\begin_layout Standard
In addition, we specify the following parameters:
\end_layout

\begin_layout Itemize
\begin_inset Formula $\delta$
\end_inset

: discount rate
\end_layout

\begin_layout Itemize
\begin_inset Formula $decay\_rate$
\end_inset

: beta decay parameter
\end_layout

\begin_layout Itemize
\begin_inset Formula $beta\_shape$
\end_inset

: beta shape parameter
\end_layout

\begin_layout Itemize
\begin_inset Formula $max\_runs$
\end_inset

: max iterations of Bellman operator (never reached in practice)
\end_layout

\begin_layout Itemize
\begin_inset Formula $Vtol$
\end_inset

: sup norm tol of Bellman operator
\end_layout

\begin_layout Itemize
\begin_inset Formula $quiet$
\end_inset

: whether to print output
\end_layout

\begin_layout Itemize
\begin_inset Formula $tau\_try$
\end_inset

: value of tax rate 
\begin_inset Formula $\tau$
\end_inset


\end_layout

\begin_layout Subsection
Outputs
\end_layout

\begin_layout Standard
The output of solve_value_function appends the following vectors to the
 data table:
\end_layout

\begin_layout Itemize
\begin_inset Formula $EV$
\end_inset

: Period 
\begin_inset Formula $t+1$
\end_inset

 expected value
\end_layout

\begin_layout Itemize
\begin_inset Formula $WTP$
\end_inset

: Willingness to pay, where we have 
\begin_inset Formula $WTP\left[q\right]=\gamma\left[q\right]+\delta EV\left[q\right]$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $V$
\end_inset

: Value function of the 
\begin_inset Formula $q$
\end_inset

th quantile asset owner
\end_layout

\begin_layout Itemize
\begin_inset Formula $best\_saleprob$
\end_inset

: optimal sale probability
\end_layout

\begin_layout Itemize
\begin_inset Formula $best\_p$
\end_inset

: optimal price, equal to 
\begin_inset Formula $WTP\left[1-best\_saleprob\left[q\right]\right]$
\end_inset

 (i.e.
 willingness to pay of the marginal buyer)
\end_layout

\begin_layout Subsection
Details
\end_layout

\begin_layout Subsubsection
Continuation value calculation
\end_layout

\begin_layout Standard
We can implement the decay operator 
\begin_inset Formula $\mathbb{E}_{G\left(\cdot\mid\cdot\right)}\left[\hat{V}\left(\gamma^{\prime}\right)\mid\gamma\right]$
\end_inset

 as a matrix; since 
\begin_inset Formula $G$
\end_inset

 is defined in quantile space, this interfaces well with the quantile grid.
 In particular, let 
\begin_inset Formula $P_{\beta}\left(x\right)$
\end_inset

 be the Beta CDF respectively, shape parameters 
\begin_inset Formula $C\beta,\ C\left(1-\beta\right)$
\end_inset

.
 For given quantile 
\begin_inset Formula $q\in\left\{ 1,\ldots,qres\right\} $
\end_inset

, we derive the decay vector: 
\begin_inset Formula 
\[
y_{\beta,q}=\left[P_{\beta}\left(\frac{1}{q}\right),\ P_{\beta}\left(\frac{2}{q}\right)-P_{\beta}\left(\frac{1}{q}\right),\ldots1-P_{\beta}\left(\frac{q-1}{q}\right),0,0,\ldots0\right]
\]

\end_inset


\end_layout

\begin_layout Standard
i.e.
 a grid approximation to the decay distribution.
 Then, conditional on 
\begin_inset Formula $q$
\end_inset

, we can implement the expectation operator numerically as: 
\begin_inset Formula 
\[
\mathbb{E}_{G\left(\cdot\mid\cdot\right)}\left[\hat{V}\left(F^{-1}\left(q^{\prime}\right)\right)\mid q\right]=y_{\beta,q}\cdot\hat{V}
\]

\end_inset


\end_layout

\begin_layout Standard
In particular we can stack these decay vectors into a matrix: 
\begin_inset Formula 
\[
decay\_matrix=\left[\begin{array}{c}
y_{\beta,1}\\
y_{\beta,2}\\
\vdots\\
y_{\beta,1000}
\end{array}\right]=\left[\begin{array}{cccc}
1 & 0 & 0 & \ldots\\
P_{\beta}\left(\frac{1}{2}\right) & 1-P_{\beta}\left(\frac{1}{2}\right) & 0 & \ldots\\
P_{\beta}\left(\frac{1}{3}\right) & P_{\beta}\left(\frac{2}{3}\right)-P_{\beta}\left(\frac{1}{3}\right) & 1-P_{\beta}\left(\frac{2}{3}\right) & \ldots\\
\vdots & \vdots & \vdots & \ddots
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Standard
And this lets us express the 
\begin_inset Quotes eld
\end_inset

continuation utility vector
\begin_inset Quotes erd
\end_inset

 as a matrix operator: 
\begin_inset Formula 
\[
E\hat{V}=decay\_matrix\cdot\hat{V}
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Maximization
\end_layout

\begin_layout Standard
Note that we can also derive a WTP vector, which we'll call 
\begin_inset Formula $WTP_{\hat{V}}$
\end_inset

, naturally, and an associated inverse demand distribution...
\end_layout

\begin_layout Standard
For each quantile 
\begin_inset Formula $q$
\end_inset

 we can thus evaluate the objective function, as:
\begin_inset Formula 
\[
\mathcal{T}\left[\hat{V}\right]\left[q\right]=\max_{q^{\prime}}\left(q^{\prime}-\tau\right)p_{\hat{V}\left(\cdot\right),F\left(\cdot\right)}\left(q^{\prime}\right)+\left(1-q^{\prime}\right)\left[\gamma+\delta\mathbb{E}_{G\left(\cdot\mid\cdot\right)}\left[\hat{V}\left(F^{-1}\left(q^{\prime}\right)\right)\mid q\right]\right]
\]

\end_inset

 
\begin_inset Formula 
\[
=\max_{q^{\prime}}\left(q^{\prime}-\tau\right)WTP_{\hat{V}}\left[q^{\prime}\right]+\left(1-q^{\prime}\right)F^{-1}\left(q\right)+\left(1-q^{\prime}\right)\delta\left(M_{\beta}\hat{V}\right)\left[q\right]
\]

\end_inset


\end_layout

\begin_layout Standard
Since we can calculate 
\begin_inset Formula $M_{\beta}\hat{V}$
\end_inset

, and we know 
\begin_inset Formula $WTP_{\hat{V}}$
\end_inset

, this is now a fairly straightforwards maximization problem, so to evaluate
 
\begin_inset Formula $\mathcal{T}\left[\hat{V}\right]$
\end_inset

 we can just loop over all 1000 
\begin_inset Formula $q$
\end_inset

 values in each iteration.
 Somewhat counterintuitively, it turned out, in my numerical experiments,
 that rather than looping, it is actually faster to generate a 1000 x 1000
 grid and use a group-by operation with data.table to do all the maximizations
 together, so this is the approach I adopt in the code, although it is somewhat
 awkward.
\end_layout

\begin_layout Section
solve_steadystate
\end_layout

\begin_layout Standard
For a given 
\begin_inset Formula $\tau$
\end_inset

, the stationary equilibrium defines a Markov process over quantiles 
\begin_inset Formula $q$
\end_inset

.
 By solving for the stationary equilibrium of the Markov process, we can
 characterize steady-state values, etc.
\end_layout

\begin_layout Subsection
Overview
\end_layout

\begin_layout Standard
The Markov transition process can be decomposed into two steps:
\end_layout

\begin_layout Enumerate
Trade: Quantile 
\begin_inset Formula $q$
\end_inset

 sets a saleprob 
\begin_inset Formula $q^{*}$
\end_inset

, and sells to all agents with higher values
\end_layout

\begin_layout Enumerate
Decay: Owner's value decays by the Beta decay process
\end_layout

\begin_layout Subsection
Inputs
\end_layout

\begin_layout Standard
Takes as input a data_table output from solve_value_function, that is, with
 the columns: 
\end_layout

\begin_layout Itemize
\begin_inset Formula $Fgam$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $fgam$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\gamma$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $EV$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $WTP$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $V$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $best\_saleprob$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $best\_p$
\end_inset


\end_layout

\begin_layout Standard
In addition we specify the incidental parameters:
\end_layout

\begin_layout Itemize
\begin_inset Formula $decay\_rate$
\end_inset

: Beta decay parameter
\end_layout

\begin_layout Itemize
\begin_inset Formula $beta\_shape$
\end_inset

: Beta shape parameter
\end_layout

\begin_layout Itemize
\begin_inset Formula $quiet$
\end_inset

: Whether to print output
\end_layout

\begin_layout Itemize
\begin_inset Formula $efficient$
\end_inset

: Whether we want equilibrium behavior, or socially efficient behavior (hacky,
 always set to 0 for equilibrium)
\end_layout

\begin_layout Subsection
Outputs
\end_layout

\begin_layout Standard
Appends the following rows to the data table:
\end_layout

\begin_layout Itemize
\begin_inset Formula $ss$
\end_inset

: Stationary density over 
\begin_inset Formula $\gamma$
\end_inset

 values of owners
\end_layout

\begin_layout Itemize
\begin_inset Formula $val\_ss$
\end_inset

: Stationary density over 
\begin_inset Formula $\gamma$
\end_inset

 values of users -- slightly different from owners, because if the owner
 sells in period 
\begin_inset Formula $t$
\end_inset

, we count the 
\begin_inset Formula $\gamma$
\end_inset

 value of the buyer here.
\end_layout

\begin_layout Itemize
\begin_inset Formula $buyer\_dist$
\end_inset

: In steady state, distribution over buyers' quantiles
\end_layout

\begin_layout Itemize
\begin_inset Formula $seller\_dist$
\end_inset

: In steady state, distribution over sellers' quantiles
\end_layout

\begin_layout Subsection
Details
\end_layout

\begin_layout Subsubsection
Trade
\end_layout

\begin_layout Standard
Given a quantile 
\begin_inset Formula $q$
\end_inset

 and her optimal choice 
\begin_inset Formula $q_{q}^{*}$
\end_inset

, transition is uniform from 
\begin_inset Formula $q_{q}^{*}$
\end_inset

 upwards, so, 
\begin_inset Formula 
\[
T_{q}=\left[0,\ldots0,\frac{1}{1-q_{q}^{*}},\frac{1}{1-q_{q}^{*}},\ldots\frac{1}{1-q_{q}^{*}}\right]
\]

\end_inset


\end_layout

\begin_layout Standard
This can be stacked into a matrix:
\begin_inset Formula 
\[
tradeprob\_matrix=\left[\begin{array}{ccccc}
0 & \frac{1}{1-q_{1}^{*}} & \frac{1}{1-q_{1}^{*}} & \frac{1}{1-q_{1}^{*}} & \ldots\\
0 & 0 & \frac{1}{1-q_{2}^{*}} & \frac{1}{1-q_{2}^{*}} & \ldots\\
0 & 0 & \frac{1}{1-q_{3}^{*}} & \frac{1}{1-q_{3}^{*}} & \ldots\\
0 & 0 & 0 & 0 & \ldots\\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Decay
\end_layout

\begin_layout Standard
Decay matrix is just 
\begin_inset Formula $M_{\beta}$
\end_inset

 from above:
\begin_inset Formula 
\[
decay\_matrix=\left[\begin{array}{c}
y_{\beta,1}\\
y_{\beta,2}\\
\vdots\\
y_{\beta,1000}
\end{array}\right]=\left[\begin{array}{cccc}
1 & 0 & 0 & \ldots\\
P_{\beta}\left(\frac{1}{2}\right) & 1-P_{\beta}\left(\frac{1}{2}\right) & 0 & \ldots\\
P_{\beta}\left(\frac{1}{3}\right) & P_{\beta}\left(\frac{2}{3}\right)-P_{\beta}\left(\frac{1}{3}\right) & 1-P_{\beta}\left(\frac{2}{3}\right) & \ldots\\
\vdots & \vdots & \vdots & \ddots
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Stationary distribution
\end_layout

\begin_layout Standard
Now we can get the full transition matrix as: 
\begin_inset Formula 
\[
M=decay\_matrix\cdot tradeprob\_matrix
\]

\end_inset


\end_layout

\begin_layout Standard
This is always ergodic.
 So we can use a formula from Resnick (Adventures in Stochastic Processes,
 pg 138) to get the unique stationary distribution:
\begin_inset Formula 
\[
\pi^{\prime}=\left(1,\ldots,1\right)\left(I-M+ONE\right)^{-1}
\]

\end_inset


\end_layout

\begin_layout Standard
The stationary distribution over 
\begin_inset Formula $q$
\end_inset

 values then allows us to compute all the values of interest.
\end_layout

\end_body
\end_document