Higher Order Derivatives Cobb Douglas

1 Higher Order Derivatives–Cobb Douglas

Go to the MLX, M, PDF, or HTML version of this file. Go back to fan’s MEconTools Package, Matlab Code Examples Repository (bookdown site), or Math for Econ with Matlab Repository (bookdown site).

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.

\[\frac{{\textrm{d}}^2 Y(K,L)}{dL^2 }=(\beta )\cdot (\beta -1)\cdot K^{\alpha } \cdot L^{\beta -2}\]

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)

frsDeri(L, K0, alpha) =

1em \(\displaystyle {K_0 }^{\alpha } \,L^{\beta -1} \,\beta\)

secDeri = diff(diff(f, L),L)

secDeri(L, K0, alpha) =

1em \(\displaystyle {K_0 }^{\alpha } \,L^{\beta -2} \,\beta \,{\left(\beta -1\right)}\)

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)

tenthDeri(L, K0, alpha) =

1em \(\displaystyle {K_0 }^{\alpha } \,L^{\beta -5} \,\beta \,{\left(\beta -1\right)}\,{\left(\beta -2\right)}\,{\left(\beta -3\right)}\,{\left(\beta -4\right)}\)

1.1 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)

fDiff1L(L) =

1em \(\displaystyle \frac{1}{2\,\sqrt{L}}\)

% note fDiff2L <= 0 always
fDiff2L = diff(f, L, 2)

fDiff2L(L) =

1em \(\displaystyle -\frac{1}{4\,L^{3/2} }\)

% 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

1.2 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)

fDiff1L(L) =

1em \(\displaystyle \frac{6\,L^{1/5} }{5}\)

% Note here fDiff2L >= 0
fDiff2L = diff(f, L, 2)

fDiff2L(L) =

1em \(\displaystyle \frac{6}{25\,L^{4/5} }\)

% 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