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 = AL^α K^β, 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) ≡ k^r f (x, z) = k^r 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 = x^a z^1−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 AL₁^α K₁^β . 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 = λ^α+β AL₁^α K₁^β = λ^{α +β} Q₁. From this, we can see that if:

a) α + β > 1, then λ^{α +β} > λ , and so Q₂ > λ Q₁ (increasing returns to scale IRS)
b) α + β = 1 , then λ^α+β = λ , and so Q₂ = λ Q₁ (constant returns to scale CRS)
c) α + β < 1, then λ^α+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.

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

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

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

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