Question

Q#02 Check whether the following production function exhibits (10 Marks) Constant Returns to Scale Increasing Returns to scale Decreasing Returns to scale . i. Y = Kal1-a ii. Y = (KL-ay iii. Y = KOLB iv. Y = (K 1/4L 1/8), v. Y = KL

Answer 1

on (1) Y = kal-a Constant returns to seate ~: Y=tik,) = kal'a (cobb-Dougler poaduetion fumettong If all inputs increased by t. → Y- f (k+, L+) I-a (Ktja (Lt)!-a Ka+a. L'-a t' = t.t-a .ka, L-a t+1-4 Kaz-a = (xa Ling) - () By increasing the inputs by it, the output (y) also increased exactly by t. This shows the production function, Y= kalla Constant returns to scale. is (ii) y = (ka 41-9) → Deereasing returns to scate. by If all inputs increase by t y = f(kt, Lt) ** (ka Lwag [from solution (i)] ta y By increasing the inputs by t the output y decreases by ts. This shows the production function Y= (ka L-aya is decreasing returns to seale. (iii) Y= ka Lo → increasing returns to seale If all inputs increase by t y = f(kt, LA) kt) L+) = K + L +

= +*+8, ka La tate Y by increasing the inputs by it, the output y increases by This shows the production function , Y = Kale is +&+B increasing returns to scale. live Y= ( KL) increasing returns to seale * If all inputs increase by t, y = f(KA, L# = [[k +)2. (L+))' (kb. t'a. L'*. t'a) '= cka. L'a)? *****) #**+ *). Y So, by increasing the inputs by t, the output y increases by t(a + /6). This shows that the production fonetion, Y= (K 2) > Es încreasing returns to seale. Y = 1 & KL² Increasing returns to scale. If all inputs Increase by t. = kt ²t² 5.: = f(ku, Lt) & (KA) (LA) ² =t² (1k2² 22) = ť?(0 by increasing the inputs by t, the output, I, increases by t2 Hence this in increasing returns returns to scale.