second_derivative.m

%% Higher Order Derivatives--Cobb Douglas
% *back to* <https://fanwangecon.github.io *Fan*>*'s* <https://fanwangecon.github.io/Math4Econ/ 
% *Intro Math for Econ*>*,*  <https://fanwangecon.github.io/M4Econ/ *Matlab Examples*>*, 
% or* <https://fanwangecon.github.io/MEconTools/ *MEconTools*> *Repositories*
%% 
% We have the following general form for the Cobb-Douglas Production Function
% 
% $$Y(K,L) = K^{\alpha} \cdot L^{\beta}$$
% 
% The first order condition is
% 
% $$\frac{dY(K,L)}{dL} = (\beta) \cdot K^{\alpha} \cdot L^{\beta-1}$$
% 
% The derivative we have obtained is just another function. We can take additional 
% derivatives with respect to this function. 
% 
% $$\frac{\text{d}^2 Y(K,L)}{dL^2} = (\beta)\cdot(\beta-1)\cdot K^{\alpha} \cdot 
% L^{\beta-2}$$
% 
% Matlab symbolic toolbox gives us the same answer:

syms L K0 alpha beta
f(L, K0, alpha) = K0^(alpha)*L^(beta);
frsDeri = diff(f, L)
secDeri = diff(diff(f, L),L)
%% 
% You can specify an additional parameter for the matlab _diff_ function, if 
% we want to take multiple derivatives:

syms L K0 alpha beta
f(L, K0, alpha) = K0^(alpha)*L^(beta);
% 5 for 5th derivative
fifthDeri = diff(f, L, 5)
%% Curvature and Second Derivative, Concave Function
% Let's graph out the second derivative when $\beta=0.5$. The production function 
% is concave (concave down). For a function that is twice continuously differentiable, 
% the function is concave if and only if its second derivative is non-positive 
% (never accelerating). 

alpha = 0.5;
beta = 0.5;
K0 = 1;
% Note that we have 1 symbolic variable now, the others are numbers
syms L
f(L) = K0^(alpha)*L^(beta);
% note fDiff1L >= 0 always
fDiff1L = diff(f, L)
% note fDiff2L <= 0 always
fDiff2L = diff(f, L, 2)
% Start figure
figure();
hold on;
% fplot plots a function with one symbolic variable
fplot(f, [0.2, 3])
fplot(fDiff1L, [0.2, 3])
fplot(fDiff2L, [0.2, 3])
title({'Concave f(x), with K=1, beta=0.5 (decreasing return to scale for L)' 'First and Second Derivatives'})
ylabel({'First derivative is slope=f increase or decrease' '2nd deri is curvature=f increase faster or slower'})
xlabel('Current level of Labor')
legend(['f(x)'], ['First Derivative'], ['Second Derivative'], 'Location','SE');
grid on
%% Curvature and Second Derivative, Convex Function
% Let's graph out the second derivative when $\beta=1.2$. The production function 
% is convex (concave up). For a function that is twice continuously differentiable, 
% the function is convex if and only if its second derivative is non-negative 
% (never decelerating). 

alpha = 0.5;
beta = 1.2;
K0 = 1;
% Note that we have 1 symbolic variable now, the others are numbers
syms L
f(L) = K0^(alpha)*L^(beta);
% Note here fDiff1L >= 0
fDiff1L = diff(f, L)
% Note here fDiff2L >= 0
fDiff2L = diff(f, L, 2)
% Start figure
figure();
hold on;
% fplot plots a function with one symbolic variable
fplot(f, [0.1, 3])
fplot(fDiff1L, [0.1, 3])
fplot(fDiff2L, [0.1, 3])
title({'Convex f(x), with K=1, beta=1.2 (increasing return to scale for L)' 'First and Second Derivatives'})
ylabel({'First derivative is slope=f increase or decrease' '2nd deri is curvature=f increase faster or slower'})
xlabel('Current level of Labor')
legend(['f(x)'], ['First Derivative'], ['Second Derivative'], 'Location','NW');
grid on
%% 
%