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## 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 = AL

^{α}K^{β}, A, α, β are positive constants.:Homogeneous Function of degree ry = f (x, z)

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

^{r}f (x, z) = k^{r}yIf r = 1, the function is also known as linear homogeneous function.

Example: Identify the following functions.

a) y = 3x

^{2}+ xz − 2z^{2}b) y= (x/3z) + 5

c) y = x

^{a}z^{1−a}:Returns to scale for a Cobb-Douglas Production FunctionLet L

_{1}and K_{1}denote the initial quantities of labor and capital, and let Q_{1}denote the initial output, so Q_{1}AL_{1}^{α}K_{1}^{β}. Now let’s increase all input quantities by the same proportional amount λ , where λ > 1, and let Q_{2}denote the resulting volume of output:Q

_{2}A (λ L_{1})^{α}(λ K_{1})^{β}= λ^{α+β}AL_{1}^{α}K_{1}^{β}= λ^{α +β}Q_{1}. From this, we can see that if:a) α + β > 1, then λ

^{α +β}> λ , and so Q_{2}> λ Q_{1}(increasing returns to scale IRS)b) α + β = 1 , then λ

^{α+β}= λ , and so Q_{2}= λ Q_{1}(constant returns to scale CRS)c) α + β < 1, then λ

^{α+β}< λ , and so Q_{2}< λQ_{1}(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: revisitedf (kL, kK) = kf (L, K) : CRS

f (kL, kK) > kf (L, K) : IRS

f (kL, kK) < kf (L, K) : DRS

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