Special Production Functions and Cobb-Douglas Production Function

Special Production Functions:

Linear Production Function (Perfect Substitutes, σ = ∞)

Q = cL + dK , c and d are positive constants.

Example: natural gas or fuel oil in manufacturing process.

Company data storage between high-capacity and low-capacity computers.

Fixed-proportions Production Function (Perfect Complements, σ = 0, Leontief Function)

Q = min(aL,bK), a and b are positive constants.

Example: fixed portions of oxygen and hydrogen atoms to make water molecules one frame with two tires for bicycle one chassis with four tires for a car.

Cobb-Douglas Production Function:

Q = ALα Kβ, A, α, β are positive constants.

Homogeneous Function of degree r:

y = f (x, z)

If we k-fold all the independent variables x and z, f (kx, kz) ≡ kr f (x, z) = kr y

If r = 1, the function is also known as linear homogeneous function.

Example: Identify the following functions.

a) y = 3x2 + xz − 2z2
b) y= (x/3z) + 5
c) y = xa z1−a

Returns to scale for a Cobb-Douglas Production Function:

Let L1 and K1denote the initial quantities of labor and capital, and let Q1 denote the initial output, so Q1 AL1α K1β . Now let’s increase all input quantities by the same proportional amount λ , where λ > 1, and let Q2 denote the resulting volume of output:

Q2 A (λ L1)α (λ K1)β = λα+β AL1α K1β = λα +β Q1. From this, we can see that if:

a) α + β > 1, then λα +β > λ , and so Q2 > λ Q1 (increasing returns to scale IRS)
b) α + β = 1 , then λα+β = λ , and so Q2 = λ Q1 (constant returns to scale CRS)
c) α + β < 1, then λα+β < λ , and so Q2 < λQ1 (decreasing returns to scale DRS)

Constant Elasticity of Substitution (CES) Production Function:

σ is independent of MRTSL,K or input ratio (K/L) or even output Q

Q = A[aL-p + (1− a)K-p]-r/p, where A > 0, 0 < a < 1, ρ ≥ −1.

r is the degree of homogeneity. σ = (1/p)

a) If ρ = −1 (σ = ∞) , Q = A[aL + (1− a)K]r (isoquant is a straight-line).

b) If ρ = 0 (σ = 1), we need a trick because we can’t define 1∞

Taking log on both sides, we can get logQ = log A – (r/p) log[aL-p + (1− a)K-p]

But the second term of r.h.s. is indeterminate because of 0/0. The best way to solve this problem is to use L’Hospital rule.

2172_L hospital rule.jpg

(You can check out why Cobb-Douglas has σ = 1)

c) If ρ →∞ (σ = 0 ). Leontief Production Function Q = min[aL, bK]

1904_leontief production.jpg

Returns to Scale: revisited

f (kL, kK) = kf (L, K) : CRS
f (kL, kK) > kf (L, K) : IRS
f (kL, kK) < kf (L, K) : DRS

1827_return to scale.jpg

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