Question

1. Consider a firm that has the following CES production function: Q = f(L,K) = [aL^ρ + bK^ρ]^1/ρ

   where ρ ≤ 1. Please clearly show each STEP and make sure your handwriting is LEGABLE. Thank you

   1. Derive the MRTS for this production function. Does this production function exhibit a diminishing MRTS? Justify using derivatives and in words. What does this imply about the shape of the corresponding isoquants? (10 points)

   2. What are the returns to scale for this production function? Show and explain. Explain what will happen to cost if the firm doubles its output. (8 points)

   3. Explain the difference between the concepts of diminishing marginal product and diminishing marginal rate of technical substitution. (5 points)

   4. Explain the difference between the concepts of diminishing marginal product and diminishing returns to scale. (5 points)

Answer 1

1) bKP-1 I Q = f(4K)= [aLP + b k pre MRTS = MPL/MPIS ve fall+ b k e le-1 . ap Let - aLl-1 Ye [alf+ 6K e] vet.beket MRTS = a (KLL) I-P Now & MATS = (a) kt. - (1-P) (LJ-P as PLI 80 AMRTS TO SO MRTS is diminishing, Thus Iso Quants are down ward sloping & convex to origin Ang AL #al, f al, ak)= falanie + bakit le = [a LP+ b k 1] 746 - z ł [all + b Kelle =a fckily Hence CRS Constant Retums to Scale And of fiom doubles olf, Cost will also double And diminishing MP when as any Input 1. Its Marginal productivity rises at decreasing rate diminishing MRTs when one Input use 1, then More & More of other Good have to be given up, keeping op Level unchanged.

D) returns to scale is a long run concept

When all Production inputs are doubled , then output rises by less than double

( When no factor of Production is fixed )

While diminishing MP is short run concept, where keeping at least one factor fixed , when more of one input is used, it's Marginal Productivity falls