randomFirms.m

%% randomFirms.m
%{
The function |randomFirms| randomly draws a realization of the surviving number
of active firms $N'$ given the current period's initial number of active firms $n$,
demand state $c$ and cost shock $w$. The computation of the number of firms
follows the construction of the unique symmetric Nash equilibrium of the
one-shot survival game, which takes advantage of the monotonicity of $v_S(n,c)$.
Given $n$, $c$, and $w$, the possible realizations of $N'$ can be classified
using the following cases:

\begin{itemize}

\item $v_S(1,c) \leq \exp (w)$: In this case, even a monopolist would not
be willing to remain active. Thus, all firms leave the market for sure and
there is no entry. This case is only relevant when $n>0$, i.e. when there is a
positive number of incumbents. If the number of incumbents is equal to
zero, and the above condition holds, then there will not be any entry and
the number of firms remains at zero.

\item $v_S(n,c) \leq \exp (w) < v_S(1,c)$: In this case, a monopolist finds
survival profitable, but the current number of firms $n$ does not. Thus,
some firms will be leaving the market and firms' survival strategies
satisfy $a_S(n,c, w) \in (0,1)$. $N'$ follows a binomial
distribution with $n$ trials and success probability $a_S(n,c,w)$.
This case is only well defined when $n>1$.

\item $v_S(n,c) > \exp (w) $: All $n$ incumbents find survival profitable.
In addition, there may be some entry. We will use the entry strategies to
compute $n'$. In fact, we will count for how many $n'$ the entry condition
$v_S(n',c) > (1+\varphi) \exp (w) $ is satisfied. This case is only
relevant if $n<\check n$, i.e. the current number of incumbents is
strictly less than $\check n$. If $n=\check n$, there will not be any
entry by construction.

\end{itemize}

The function takes as inputs last period's post-survival number of active firms
|N|, the current number of consumers |C|, the realized cost shock |W|, and
the equilibrium post-survival value function, |vS|. |N|, |C|, and |W|
are column vectors of length |rCheck| containing one entry per
market. The output is the column vector |Nprime| of length |rCheck|,
which contains the post-survival number of active firms in each market.
%}

function Nprime = randomFirms(N, C, W, Settings, Param, vS)

Nprime = NaN(size(N));

for nX = 1:Settings.rCheck

    switch (true)
        %\% All firms leave (only relevant if N(nX)>0)
        case N(nX) > 0 && (vS(1, C(nX)) <= exp(W(nX)))
            Nprime(nX) = 0;

        %\% Some firms leave (only relevant if N(nX) > 1)
        case N(nX) > 1 && (vS(max(1,N(nX)), C(nX)) <= exp(W(nX)))
            aS = mixingProbabilities(N(nX), C(nX), W(nX), vS);
            Nprime(nX) = binornd(N(nX), aS);

        %\% All incumbents stay; there may be entry.
        case N(nX) < Settings.nCheck && (vS(max(1, N(nX)), C(nX)) > exp(W(nX)))
            Nprime(nX) = N(nX) + sum(vS((N(nX) + 1):Settings.nCheck, C(nX)) ...
                - (1 + Param.phi) .* exp(W(nX)) > 0);

        %\% Remaining cases include N(nX)=0 and no entry, or N(nX)=Settings.nCheck and no exit).
        otherwise
            Nprime(nX) = N(nX);
    end
end
% This concludes |randomFirms|.
end