Final answer:
The false statement among the given options regarding the production function is B: 'It does not show diminishing marginal product of labor.' The given function actually does indicate diminishing marginal product of labor, which contradicts the claim in statement B.
Step-by-step explanation:
The question revolves around a given production function Q=L³K¹/³, and it examines the truth of a set of statements relating to marginal products and returns to scale. Given the marginal product of labor (MPL) as 3L²K¹/³ and the marginal product of capital (MPK) as 1/3L³K²/³, we can analyze each statement.
Statement A expresses diminishing marginal product of capital, which aligns with the given MPK because as K increases, MPK decreases due to the negative exponent on K. Statement B is false since in the production function the power on L is greater than one, indicating that as more labor is added, the output increases at a decreasing rate which shows a diminishing marginal product of labor.
Statement C describes constant returns to scale, and it is true for this function because if we were to scale both input factors L and K by any given factor, the output Q would be scaled by the same factor which is characteristic of constant returns to scale. Lastly, Statement D claims there is diminishing marginal rate of technical substitution (MRTS), which is reflected in the changing ratio of MPL to MPK as either L or K is varied, showing a decreasing MRTS. Therefore, the answer to which statement is false would be B: 'It does not show diminishing marginal product of labor'.