Introduction to Modern Economic Growth Assumption (Continuity, Differentiability, Positive and Diminishing Marginal Products, and Constant Returns to Scale) The production function F : R3+ → R+ is twice continuously differentiable in K and L, and satisfies ∂F (K, L, A) > 0, ∂K ∂ F (K, L, A) < 0, FKK (K, L, A) ≡ ∂K FK (K, L, A) ≡ ∂F (K, L, A) > 0, ∂L ∂ F (K, L, A) FLL (K, L, A) ≡ < ∂L2 FL (K, L, A) ≡ Moreover, F exhibits constant returns to scale in K and L All of the components of Assumption are important First, the notation F : R3+ → R+ implies that the production function takes nonnegative arguments (i.e., K, L ∈ R+ ) and maps to nonnegative levels of output (Y ∈ R+ ) It is natural that the level of capital and the level of employment should be positive Since A has no natural units, it could have been negative But there is no loss of generality in restricting it to be positive The second important aspect of Assumption is that F is a continuous function in its arguments and is also differentiable There are many interesting production functions which are not differentiable and some interesting ones that are not even continuous But working with continuously differentiable functions makes it possible for us to use differential calculus, and the loss of some generality is a small price to pay for this convenience Assumption also specifies that marginal products are positive (so that the level of production increases with the amount of inputs); this also rules out some potential production functions and can be relaxed without much complication (see Exercise 2.4) More importantly, Assumption imposes that the marginal product of both capital and labor are diminishing, i.e., FKK < and FLL < 0, so that more capital, holding everything else constant, increases output by less and less, and the same applies to labor This property is sometimes also referred to as “diminishing returns” to capital and labor We will see below that the degree of diminishing returns to capital will play a very important role in many of the results of the basic growth model In fact, these features distinguish the Solow growth model from its antecedent, the Harrod-Domar model (see Exercise 2.13) 42