Higher Order Derivatives Cobb Douglas

1 Higher Order Derivatives–Cobb Douglas

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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) =

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

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

secDeri(L, K0, alpha) =

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

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) =

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

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) =

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

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

fDiff2L(L) =

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

% 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) =

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

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

fDiff2L(L) =

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

% 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