1.10 Euler’s Theorem on Homogeneous Functions
Definition 1.10.1. A function \(F(x_1,x_2,\dots , x_n)\) is homogeneous of degree \(p\) if \[F(\lambda x_1,\lambda x_2,\dots , \lambda x_n) = \lambda ^p F(x_1,x_2,\dots ,x_n)\] for all \(\lambda >0\) and all points \((x_1,\dots ,x_n)\) in its domain.
Example 1.10.2. For \(F(x,y) = x^4 + 2xy^3 - 5y^4\), \[F(\lambda x, \lambda y) = \lambda ^4 x^4 + 2\lambda ^4xy^3 - 5\lambda ^4 y^4 = \lambda ^4 (x^4 +2xy^3 - 5y^4),\] so \(F\) is homogeneous of degree \(4\). In general a polynomial is homogeneous of degree \(p\) exactly when every term has total degree \(p\).
Theorem 1.10.3 (Euler). If \(F(x_1,\dots , x_n)\) is differentiable and homogeneous of degree \(p\), then \[ x_1 \frac {\partial F}{\partial x_1} + x_2 \frac {\partial F}{\partial x_2} + \cdots +x_n \frac {\partial F}{\partial x_n} = p F .\]
Proof. Fix the point \((x_1,\dots ,x_n)\) and differentiate the identity \(F(\lambda x_1,\dots ,\lambda x_n) = \lambda ^p F(x_1,\dots ,x_n)\) with respect to \(\lambda \). By the chain rule the left side gives \(\sum _i x_i\,\frac {\partial F}{\partial x_i}(\lambda x_1,\dots ,\lambda x_n)\), and the right side gives \(p\lambda ^{p-1}F(x_1,\dots ,x_n)\). Setting \(\lambda = 1\) gives the result. □
For the function of Example 1.10.2, \[x F_x + y F_y = x(4x^3 + 2y^3) + y(6xy^2 -20y^3) = 4x^4 + 8xy^3 - 20y^4 = 4F,\] as the theorem predicts.
Example 1.10.4. If \(F\) is homogeneous of degree \(p\) with continuous second partial derivatives, show that \[x^2F_{xx} + 2xyF_{xy} + y^2F_{yy} = p(p-1)F,\] and verify it for \(F = x^3 + y^3\).
Solution. Differentiate \(xF_x + yF_y = pF\) with respect to \(x\) and to \(y\): \[F_x + xF_{xx} + yF_{xy} = pF_x,\qquad xF_{xy} + F_y + yF_{yy} = pF_y .\] Multiply the first by \(x\), the second by \(y\), and add: \[x^2F_{xx} + 2xyF_{xy} + y^2F_{yy} = (p-1)(xF_x + yF_y) = p(p-1)F .\] For \(F = x^3 + y^3\), the left side is \(x^2(6x) + 0 + y^2(6y) = 6(x^3 + y^3)\), and \(p(p-1) = 6\).
Example 1.10.5. Show that \(u = \sin ^{-1}\frac {x^2 + y^2}{x + y}\) satisfies \(xu_x + yu_y = \tan u\).
Solution. The function \(u\) itself is not homogeneous, but \(\sin u = \frac {x^2+y^2}{x+y}\) is homogeneous of degree \(1\). Applying Theorem 1.10.3 to \(\sin u\), with the chain rule giving \((\sin u)_x = \cos u\;u_x\), \[x\cos u\;u_x + y\cos u\;u_y = \sin u,\qquad \text {so}\qquad xu_x + yu_y = \frac {\sin u}{\cos u} = \tan u .\]
Example 1.10.6. A firm’s output is \(P(L,K) = 4L^{0.7}K^{0.3}\), where \(L\) is labour and \(K\) capital. Show that \(LP_L + KP_K = P\), and interpret the result.
Solution. \(P(\lambda L,\lambda K) = 4\lambda ^{0.7 + 0.3}L^{0.7}K^{0.3} = \lambda P\), so \(P\) is homogeneous of degree \(1\), and Euler’s theorem gives \(LP_L + KP_K = P\). Directly, \(LP_L = 0.7P\) and \(KP_K = 0.3P\). The partial derivatives \(P_L\) and \(P_K\) are the marginal products of labour and capital; the identity says that if each input were paid its marginal product, the payments would exactly use up the output, with labour receiving \(70\%\) and capital \(30\%\). Degree \(1\) means constant returns to scale: doubling both inputs doubles the output.
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