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plotRegion.m
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plotRegion.m
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function [ ] = plotRegion(parameters, target, points, markers, ...
error_tol, inBW)
% =========================================================================
% PLOTREGIONBW plots the survival region in black and white by splitting up
% the survival region into a given number of points and then drawing
% colored or b/w lines to denote decisions; the start and end of each line
% is determined by doing a binary search for the borders of the survival
% region at a given beta
% =========================================================================
% INPUT ARGUMENTS:
% parameters (struct) generated by loadParameters
% target (axes) target a subplot of the current figure
% by calling subplot(x,y,plot_num)
% points (int) number of points to draw lines over,
% more points increases visual accuracy
% but takes more time
% markers (int) number of markers for x and beta to
% place on the plot, eg setting to 10
% will result in 100 points
% error_tol (numeric) error of approximation of survival
% region borders
% inBW (boolean) set to true for black and white plots
% =========================================================================
% OUTPUT: Updates targetted subplot
% =========================================================================
%% Prepare plot
cla(target);
hold(target, 'on');
%% Unpack parameters
% Exogenous
beta_hat = parameters.beta_hat;
price = parameters.price;
emp = parameters.emp;
I_1 = parameters.I_1;
I_2 = parameters.I_2;
gamma = parameters.gamma;
rho = parameters.rho;
delta = parameters.delta;
alpha = parameters.alpha;
xi = parameters.xi;
psi = parameters.psi;
elas_D = parameters.elas_D;
a1 = parameters.a1;
a2 = parameters.a2;
% Endogenous
t_1a = parameters.taxes_1a;
t_2a = parameters.taxes_2a;
p_1a = parameters.prices_1a;
p_2a = parameters.prices_2a;
p_2b = parameters.prices_2b;
%% Define integration range
x_range = linspace(0, max([p_1a, p_2a, p_2b]), points);
beta_range = linspace(0, beta_hat, points);
%% Main
for beta_ind = 1:points
beta = beta_range(beta_ind);
% get optimal decision at current beta and x = 0
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(0, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector( ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b );
x_max = max(x_range);
x_upper = 0;
% while producing at current (beta, x) still proftiable
while decisionvector(1) ~= 1
% check higher x values until a different decision is found
x_lower = x_upper;
x_upper = x_max;
% binary search
[ x_upper, decisionvector, ~ ] = binarySearchDecision(x_lower, ...
x_upper, beta, I_1, I_2, gamma, rho, delta, alpha, t_1a, t_2a, ...
p_1a, p_2a, p_2b, error_tol);
% set line color
if all(decisionvector == [1,1,1]) % black = no invest
mcolor = [1 1 1]/1e6;
elseif all(decisionvector == [2,1,2]) % cyan = produce(1), drop second(a), produce second(b)
mcolor = [0 1 1];
elseif all(decisionvector == [2,3,2]) % blue = produce(1), invest second(a), produce second(b)
mcolor = [0 0 1];
elseif all(decisionvector == [2,2,2]) % green = produce(1), produce second (both)
mcolor = [0 1 0];
elseif all(decisionvector == [3,2,2]) % red = invest(1), produce second (both)
mcolor = [1 0 0];
elseif all(decisionvector == [3,1,2]) % magenta = invest(1), drop second (a), produce second (b)
mcolor = [1 0 1];
elseif all(decisionvector == [2,2,1]) % orange = produce(1), produce second (a), drop second (b)
mcolor = [1 0.5 0];
else % yellow = ??
mcolor = [1 1 0];
end
if inBW
mcolor = brighten(rgb2gray(mcolor), 0.9);
end
%plot(beta, x_upper, 'Marker', '.', 'Color', [0.5 0.5 0.5])
if (mod(beta_ind, 1) == 0 || points < 1000)
plot([beta, beta], [x_lower, x_upper], 'Color', mcolor, ...
'LineWidth',2)
end
end
end
%% Add points
beta_range2 = linspace(beta_hat*0.02, beta_hat*0.98, markers);
x_range2 = linspace(max(x_range)*0.02, max(x_range)*0.98, markers);
for i = 1:length(beta_range2)
beta = beta_range2(i);
for j = 1:length(x_range2)
x = x_range2(j);
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b);
if all(decisionvector == [1,1,1]) % idle
mtype = '.';
elseif all(decisionvector == [2,1,2]) % produce(1), drop second(a), produce second(b)
mtype = '*';
elseif all(decisionvector == [2,3,2]) % produce(1), invest second(a), produce second(b)
mtype = 'x';
elseif all(decisionvector == [2,2,2]) % produce(1), produce second (both)
mtype = 'o';
elseif all(decisionvector == [3,2,2]) % invest(1), produce second (both)
mtype = '+';
elseif all(decisionvector == [3,1,2]) % invest(1), drop second (a), produce second (b)
mtype = '^';
elseif all(decisionvector == [2,2,1]) % produce(1), produce second (a), drop second (b)
mtype = 'v';
else % something else
mtype = 's';
end
plot(beta, x, 'Marker', mtype, 'Color', 'k', 'MarkerSize', 4)
end
end
end
%% Auxiliary functions
function [ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b)
% =====================================================================
% FINDOPTIMALDECISION gets a firms optimal decision given a set of
% params, x, and beta
% =====================================================================
% Max profit in period 2a for firms that have invested in period 1
[pi_I_2a, ind_I_2a] = max([0, p_2a - (1+rho)*x - ...
t_2a*(1-gamma)*beta]);
% Noninvesting max profit in period 2a
[pi_N_2a, ind_N_2a] = max([0, p_2a - x - t_2a*beta, p_2a - x - ...
t_2a*(1-gamma)*beta - I_2]);
% Max profit in period 2b for firms that have invested in period 1
[pi_I_2b, ind_I_2b] = max([0, p_2b - (1+rho)*x]);
% Noninvesting max profit in period 2b
[pi_N_2b, ind_N_2b] = max([0, p_2b - x]);
% Present value expected profit for a firm investing in period 1
pi_I_1a = p_1a - (1+rho)*x - t_1a*(1-gamma)*beta - I_1 + delta* ...
(alpha*(pi_I_2a) + (1-alpha)*(pi_I_2b));
% Present value expected profit for a firm producing in period 1
pi_N_1a = p_1a - x - t_1a*beta + delta*(alpha*(pi_N_2a) + (1-alpha)*(pi_N_2b));
% Firms may exit, produce in period 1, or invest in period 1
profit_mat = [0, pi_N_1a, pi_I_1a];
[~, ind] = max(profit_mat);
end
function [ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b)
% =====================================================================
% CREATEDECISIONVECTOR converts a set of optimal decisions into a
% vector representation of the optimal decision for a firm
% =====================================================================
% in each period, 1 = drop, 2 = produce, 3 = invest
% by default 1 => not producing first period
decisionvector = ones(1,3);
if ind == 2 % firms not investing first period
decisionvector = [ind, ind_N_2a, ind_N_2b];
elseif ind == 3 % firms investing first period
decisionvector = [ind, ind_I_2a, ind_I_2b];
end
end
function [ x_upper, decisionvector, debug ] = binarySearchDecision( ...
x_lower, x_upper, beta, I_1, I_2, gamma, rho, delta, alpha, t_1a, ...
t_2a, p_1a, p_2a, p_2b, error_tol)
% =====================================================================
% BINARYSEARCHDECISION searches the range between x_lower and x_upper
% for the boundary where the decision at some output x is different
% then the inital decision at x_lower; essentially it scans upwards
% from a given x for the next survival region boundary
% =====================================================================
debug = [x_lower, (x_lower + x_upper)/2, x_upper];
% get base decision
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x_lower, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b);
decision_base = decisionvector;
while true
% get decision at x maximum
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x_upper, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b);
decision_max = decisionvector;
% get decision at x minimum
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x_lower, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b);
decision_min = decisionvector;
% get decision at x mean
x_mid = mean([x_lower, x_upper]);
[ ind, ind_I_2a, ind_N_2a, ind_I_2b, ind_N_2b ] = ...
findOptimalDecision(x_mid, beta, I_1, I_2, gamma, rho, ...
delta, alpha, t_1a, t_2a, p_1a, p_2a, p_2b);
[ decisionvector ] = createDecisionVector(ind, ind_I_2a, ...
ind_N_2a, ind_I_2b, ind_N_2b);
decision_mid = decisionvector;
if x_upper - x_lower < error_tol
decisionvector = decision_min;
break;
elseif all(decision_mid == decision_base)
x_lower = x_mid;
x_mid = mean([x_lower, x_upper]);
else
x_upper = x_mid;
x_mid = mean([x_lower, x_upper]);
end
debug = [debug; x_lower, x_mid, x_upper];
end
end