Production Theory and Firm Efficiency

Production theory studies the technological capabilities of firms, the optimal choice of production factors, and efficiency analysis. Key tools are production functions, cost functions, and quantitative methods for efficiency assessment (DEA, SFA). These tools are used by regulators, banks, and consultants to evaluate the performance of companies and state institutions.

Production Functions

The production function f: ℝᴸ₊ → ℝ₊ maps vectors of factors (xₗ) to maximum output. Key characteristics: returns to scale and elasticity of substitution.

Returns to scale: f(tx) = tˢ f(x) — increasing (s > 1, IRS), constant (s = 1, CRS), decreasing (s < 1, DRS).

Cobb-Douglas: y = AK^α L^β. With competitive markets (p·MPₖ = wₖ): capital share in revenue = α = wₖK/pY, labor share = β. Elasticity of substitution between K and L: σ = 1 (unitary — at any factor prices, their shares in expenses are constant). α + β > 1: IRS (e.g., a large plant). α + β = 1: CRS (replication). α + β < 1: DRS (limited managerial resource).

CES: y = [αK^{−ρ} + (1−α)L^{−ρ}]^{−1/ρ}. Elasticity of substitution σ = 1/(1+ρ). ρ → 0: σ → 1 (Cobb-Douglas). ρ → ∞: σ → 0 (Leontief, fixed proportions). ρ → −1: σ → ∞ (linear, perfect substitutes).

Cost Function and Shephard's Lemma

Cost minimization problem: c(w, y) = min_{x≥0} w·x subject to f(x) ≥ y. Optimality condition: MRTS_{lk} = wₗ/wₖ — isoquant is tangent to the isocost at the optimal x*. Geometrically: find the tangency point of the isoquant with the cheapest isocost.

Properties of the cost function c(w,y): Non-decreasing in w (more expensive resources → higher costs). Homogeneous of degree one in w: c(tw, y) = t·c(w, y) — doubling all prices doubles costs. Concave in w (second derivatives ≤ 0). Increasing in y.

Shephard's Lemma: ∂c(w,y)/∂wₗ = xₗ*(w,y) — conditional demand for the factor. From concavity in w: ∂²c/∂wₗ² ≤ 0 → ∂xₗ/∂wₗ ≤ 0 — law of demand for factors (when the price of a factor rises, its conditional demand decreases or remains unchanged).

Translogarithmic cost function: ln c = α₀ + Σₖ αₖ ln wₖ + β ln y + (1/2)Σₖ Σⱼ γₖⱼ ln wₖ ln wⱼ + ... Flexible form, approximates any regular function. Econometric estimates of substitution elasticities in industry.

DEA — Data Envelopment Analysis

DEA method (Charnes-Cooper-Rhodes, 1978): for n production units (DMU) with inputs xᵢ and outputs yᵢ, a “best practice frontier” is constructed — the convex hull of the observations.

Efficiency of unit j (CCR model with CRS): solve LP:

θⱼ = min θ subject to: Σᵢ λᵢ yᵢ ≥ yⱼ, Σᵢ λᵢ xᵢ ≤ θ xⱼ, λᵢ ≥ 0

Decoding: seek a virtual “reference” DMU (weighted mix of efficient ones) that produces at least yⱼ, using no more than θ·xⱼ resources. θⱼ = 1 → DMU j is efficient (lies on the frontier). θⱼ = 0.8 → uses 25% more resources than its efficient counterparts.

VRS vs CRS: CCR model (CRS): no restriction Σλᵢ = 1. BCC model (VRS): add Σλᵢ = 1 — only allow analogs scalable to the close size. Difference θ_{CRS}/θ_{VRS} — losses from suboptimal scale.

Applications of DEA: Banks (efficiency assessment of branches: inputs — staff, assets; outputs — loans, profit). Hospitals (staff, equipment → discharged patients, quality assessment). Universities (expenses, number of faculty → publications, graduates). Government programs (taxpayer costs → social outcomes).

Stochastic Frontier (SFA)

SFA Model: ln yᵢ = f(xᵢ; β) + vᵢ − uᵢ, where vᵢ ~ N(0, σᵥ²) — random noise (measurement errors, luck), uᵢ ≥ 0 — “inefficiency” (nonnegative, u ~ N⁺(0, σᵤ²)).

Technical efficiency: TEᵢ = exp(−uᵢ) ∈ (0,1]. Estimation via MLE — simultaneously evaluate parameters β, σᵥ, σᵤ. SFA’s usefulness: separates noise and inefficiency (DEA cannot).

Numerical Example

4 hospitals (Doctors, Beds → Treated): A(20, 100 → 500), B(15, 80 → 380), C(25, 110 → 520), D(18, 90 → 450).

DEA analysis: θ_A = 1.00 (efficient, on the frontier), θ_B = 0.97 (uses 3% excess resources), θ_C = 1.00 (efficient), θ_D = 1.02 → 1.00 after normalization. Hospital B could produce 380 discharges with 14.6 doctors and 77.6 beds (saving 3%).

Assignment: 4 hospitals: A(20,100→500), B(15,80→380), C(25,110→520), D(18,90→450). Solve DEA (CCR) for each manually or via Python (pulp). Who is “best practice”? For each inefficient one, find “target” input values. Compare with SFA estimation (suppose ln y = β₀ + β₁ ln Doctors + β₂ ln Beds + v − u).