Review session for Midterm #1

Q = min(aK,bL)= min(λaK,λbL)= λmin(aK,bL)= λ[Q]Therefore the production function has constant returns to scale.6.19 A firm produces a quantity Q of breakfast cereal using labor L and material M with theproduction function Q = 50 ML + M + L . The marginal product functions for this production functionareMP L= 25 M L +1MP M= 25 L .M +1a) Are the returns to scale increasing, constant, or decreasing for this production function?To determine the nature of returns to scale, increase all inputs by some factor λand determine if output goes up by a factor more than, less than, or the same as λQ = 50 λMλL + λM + λLλQλ= 50λ ML + λM + λLQλ= λ ⎡50ML M L⎤⎣+ +⎦Q = λQλBy increasing the inputs by a factor of λ output goes up by a factor of λ . Since output goes up bythe same factor as the inputs, this production function exhibits constant returns to scale.b) Is the marginal product of labor ever diminishing for this production function? If so, when? Is it evernegative, and if so, when?The marginal product of labor isMMPL= 25 + 1LSuppose M > 0 . Holding M fixed, increasing L will have the effect of decreasing MPL. Themarginal product of labor is decreasing for all levels of L . The MPL, however, will never benegative since both components of the equation above will always be greater than or equal tozero. In fact, for this production function, MPL≥ 1.23