Answer to Question #147354 in Economics of Enterprise for Inaya Singh

Question

53. For each of the following production functions, determine whether returns to scale are decreasing, constant, or increasing.

i) Q = 2k + 3L + KL

ii) Q = 20 K0.6 L0.5

iii) Q = 100 + 3K + 2L

iv) Q = 5Ka Lb, Where a+ b = 1

v) Q = K/L

Answer 1

Production function: Y = F(K,L)

We augment the T to the factor t

$f(tL, tK) = t^{α + β} + f(L,K)$

if α + β = 1, the f(L,K) exhibit Constant Returns to Scale

if α + β > 1, the f(L,K) exhibit Increasing Returns to Scale

if α + β < 1, the f(L,K) exhibit Decreasing Returns to Scale

i) Q = 2K + 3L + KL

Let K = 3 and L = 3

Q = 6 + 9 + 9 = 24

Let’s double the input, L = 6 and K = 6

Q = 12 + 18 +36 = 66

We observe that output got more than double by doubling the input values.

So, Q exhibits Increasing Returns to Scale.

ii) Q = 20K^0.6L^0.5

Augmenting Q by factor λ

Q(λK, λL) = 30 (λL)0.6(λK)0.5 = λ^1.1 \times Q(K,L)

Since, α + β = 1.1 > 1

So, Q exhibits Increasing Returns to Scale.

iii) Q = 100 + 3K + 2L

Let K = 2 and L = 2

Q = 100 + 6 + 4 = 110

Let's double the inputs, L = 4 and K = 4

Q = 100 + 12 + 8 = 120

We observe that output got less than double by doubling the input values.

So, Q exhibits Decreasing Returns to Scale.

iv) Q = 5K^aL^b, Where a+ b = 1

Augmenting Q by factor λ

Q(λK, λL) = 5(λL)^a(λK)^b = $λ^1\times Q(K,L)$

Since, α + β = 1

So, Q exhibits Constant Returns to Scale.

v) Q = K/L

Augmenting Q by factor λ

Q(λK, λL) = (λK / λL) = (K/L)

So, the production function exhibits Constant Returns to Scale