Introduction to Modern Economic Growth The other important assumption is that of constant returns to scale Recall that F exhibits constant returns to scale in K and L if it is linearly homogeneous (homogeneous of degree 1) in these two variables More specifically: Definition 2.1 Let z ∈ RK for some K ≥ The function g (x, y, z) is homo- geneous of degree m in x ∈ R and y ∈ R if and only if g (λx, λy, z) = λm g (x, y, z) for all λ ∈ R+ and z ∈ RK It can be easily verified that linear homogeneity implies that the production function F is concave, though not strictly so (see Exercise 2.1) Linearly homogeneous (constant returns to scale) production functions are particularly useful because of the following theorem: Theorem 2.1 (Euler’s Theorem) Suppose that g : RK+2 → R is continuously differentiable in x ∈ R and y ∈ R, with partial derivatives denoted by gx and gy and is homogeneous of degree m in x and y Then mg (x, y, z) = gx (x, y, z) x + gy (x, y, z) y for all x ∈ R, y ∈ R and z ∈ RK Moreover, gx (x, y, z) and gy (x, y, z) are themselves homogeneous of degree m − in x and y Proof We have that g is continuously differentiable and (2.2) λm g (x, y, z) = g (λx, λy, z) Differentiate both sides of equation (2.2) with respect to λ, which gives mλm−1 g (x, y, z) = gx (λx, λy, z) x + gy (λx, λy, z) y for any λ Setting λ = yields the first result To obtain the second result, differentiate both sides of equation (2.2) with respect to x: λgx (λx, λy, z) = λm gx (x, y, z) Dividing both sides by establishes the desired result 43 Ô