Question

Briefly show whether the following production functions exhibit increasing, decreasing, or constant returns to scale:

Y = K2/3 + L2/3

Y = min {2L+K, 2K+L}

Y = 20*L1/5*K4/5

Answer 1

Answer

(a)

(i) A function exhibit constant returns to scale if F(tK,tL) = t*F(K,L),
(ii) A function exhibit increasing returns to scale if F(tK,tL) > t*F(K,L)
(iii) A function exhibit decreasing returns to scale if F(tK,tL) < t*F(K,L).

Here Y = F(K,L) = (tK)^2/3 + (tL)^2/3

=> F(tK,tL) = (tK)^2/3 + (tL)^2/3 = t^2/3(K^2/3 + L^2/3) < t(K^2/3 + L^2/3) = tF(K,L)

Note ; (for all t > 1, t^a > t^b if a > b)

Hence F(tK,tL) < t*F(K,L) for all t > 1

Hence this function exhibit decreasing returns to scale.

(b)

(i) A function exhibit constant returns to scale if F(tK,tL) = t*F(K,L),
(ii) A function exhibit increasing returns to scale if F(tK,tL) > t*F(K,L)
(iii) A function exhibit decreasing returns to scale if F(tK,tL) < t*F(K,L).

Here Y = F(K,L) = min {2L+K, 2K+L}

=> F(tK,tL) = min {2tL + tK, 2tK + tL}  = min {t(2L+K), t(2K+L)} = t*min {2L+K, 2K+L} = tF(K,L)

Hence F(tK,tL) = t*F(K,L) for all t > 1

Hence this function exhibit Constant returns to scale.

(c)

(i) A function exhibit constant returns to scale if F(tK,tL) = t*F(K,L),
(ii) A function exhibit increasing returns to scale if F(tK,tL) > t*F(K,L)
(iii) A function exhibit decreasing returns to scale if F(tK,tL) < t*F(K,L).

Here Y = F(K,L) = L^1/5K^4/5

=> F(tK,tL) = (tL)^1/5(tK)^4/5 = t^{1/5 + 4/5}(L^1/5K^4/5) < t*L^1/5K^4/5 = tF(K,L)

Hence F(tK,tL) = t*F(K,L) for all t > 1

Hence this function exhibit Constant returns to scale.