2. (12 marks) Consider the properties of homogeneity and homotheticity. a. Define homogeneity of degree r as it relates to f(x1, ... , xn).

b. Let Q(L, K) be a linearly homogenous (i.e. constant-returns-to-scale) production function where L and K are labour and capital inputs, respectively, and MPPi is the marginal physical product of input i {L,K}. Prove that MPPK = (k) and MPPL = (k) k (k), where k K/L and (k) Q(L,K)/L = Q(1, k), verifying that the marginal physical products depend on the capital-labour ratio only.

c. For x = (x1, ... , xn), prove that f(x) C 1 is homogenous of degree r if and only if fixi n i=1 = rf(x). For the sufficiency direction of the proof, consider using the Fundamental Theorem of Calculus and the monotonicity of the natural logarithm.

d. Define homotheticity as it relates to a composite function h(x1, ... , xn) = f(z) where z = g(x1, ... , xn).

e. Define for a function h(x1, ... , xn) the -intensity expansion path in the xi-xj plane.

f. Prove that every expansion path of a homothetic function h(x1, ... , xn) is a ray.