🏭 Production Theory, Cost Functions & Firm Behavior

Firms transform productive inputs (labor $L$, capital $K$, intermediate materials $M$) into marketable outputs $q$ according to technological constraints defined by the production function $q=f(K,L)$.

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1. ⚙️ Technology & Production Sets

The production set $Y\subset {\mathbb{R}}^n$ comprises all feasible input-output plans. For a two-input production function $q=f(K,L)$:

1. Marginal Product of Labor ($M{P}_L$): $M{P}_L\equiv \frac{\partial f(K,L)}{\partial L}>0$
2. Marginal Product of Capital ($M{P}_K$): $M{P}_K\equiv \frac{\partial f(K,L)}{\partial K}>0$
3. Law of Diminishing Marginal Returns: $\frac{{\partial }^{2}f}{\partial L^2}<0,\frac{{\partial }^{2}f}{\partial K^2}<0$

1.1 Marginal Rate of Technical Substitution ($MRTS$)

The slope of an isoquant ($dq=0$) represents the rate at which capital can be substituted for labor while preserving constant output:

$$MRT{S}_{L,K}\equiv -{\frac{dK}{dL}|}_{dq=0}=\frac{M{P}_L}{M{P}_K}$$

1.2 Elasticity of Substitution ($\sigma$)

Measures the percentage change in the capital-labor ratio relative to the percentage change in $MRTS$:

$$\sigma \equiv \frac{\mathrm{\%}\mathrm{\Delta }(K/L)}{\mathrm{\%}\mathrm{\Delta }MRT{S}_{L,K}}=\frac{d\ln (K/L)}{d\ln (M{P}_L/M{P}_K)}$$

• Leontief ($\sigma =0$): Fixed proportions $q=min(aK,bL)$.
• Cobb-Douglas ($\sigma =1$): $q=A{K}^{\alpha }{L}^{\beta }$.
• Linear Substitutes ($\sigma =\mathrm{\infty }$): $q=aK+bL$.
• CES (Constant Elasticity of Substitution): $q=A{[\delta K^{-\rho }+(1-\delta )L^{-\rho }]}^{-1/\rho }$ where $\sigma =\frac{1}{1+\rho }$.

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2. 📉 Returns to Scale

Let $f(\lambda K,\lambda L)={\lambda }^{r}f(K,L)$ for $\lambda >1$:

• Constant Returns to Scale (CRS) ($r=1$): Doubling inputs doubles output ($AC$ is constant).
• Increasing Returns to Scale (IRS) ($r>1$): Doubling inputs more than doubles output ($AC$ falls $⟹$ economies of scale).
• Decreasing Returns to Scale (DRS) ($r<1$): Doubling inputs less than doubles output ($AC$ rises $⟹$ diseconomies of scale).

For Cobb-Douglas $q=A{K}^{\alpha }{L}^{\beta }$, the degree of homogeneity is $r=\alpha +\beta$.

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3. 💰 Cost Minimization & Duality

To produce a targeted output level $q_0$ at minimum cost given wage $w$ and rental rate of capital $r$:

$$\underset{K,L}{min}wL+rK\text{s.t.}f(K,L)\ge q_0$$

Lagrangian: $\mathcal{L}=wL+rK-\lambda (f(K,L)-q_0)$. First-Order Conditions:

$$w=\lambda M{P}_L,r=\lambda M{P}_K⟹MRT{S}_{L,K}=\frac{M{P}_L}{M{P}_K}=\frac{w}{r}$$

The Lagrange multiplier $\lambda$ represents Marginal Cost: $\lambda =MC(q)=\frac{\partial C(w,r,q)}{\partial q}$.

3.1 Cost Curves Taxonomy

Costs ($)
  ▲
  │                     MC
  │                     /  ATC
  │         \          /  /   AVC
  │          \  ___   /  /   /
  │           \/   \_/__/__/
  │           /\___ /  /
  │          /     /  /
  │         /     /  /
  │        /     /  /        AFC
  │       /     /  /        __---___
  └──────┴─────┴──┴───────────────────► Output (q)
        q_shutdown  q_breakeven

1. Total Cost: $TC(q)=FC+VC(q)$
2. Average Total Cost: $ATC(q)=\frac{TC(q)}{q}=AFC(q)+AVC(q)$
3. Marginal Cost: $MC(q)=\frac{dTC}{dq}=\frac{dVC}{dq}$
4. Intersections: $MC$ crosses $AVC$ and $ATC$ exactly at their respective global minimum points.

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4. 📐 The Envelope Theorem & Long-Run Cost

In the short run, capital is fixed ($K=\overline{K}$), giving short-run average cost $SRAC(q,\overline{K})$. In the long run, all inputs are variable. The long-run average cost ($LRAC$) is the lower envelope of all $SRAC$ curves:

$$LRAC(q)=\underset{K}{min}SRAC(q,K)$$

By the Envelope Theorem:

$$\frac{d{C}_{LR}(w,r,q)}{dq}={\frac{\partial C_{SR}(w,r,q,K^{\ast })}{\partial q}|}_{K=K^{\ast }(q)}$$

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5. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Cobb-Douglas Cost Function Derivation

Problem: A firm operates with production function $q=K^{1/3}{L}^{2/3}$. Input prices are wage $w$ and rental rate $r$.

1. Derive the conditional factor demands $L^{\ast }(w,r,q)$ and $K^{\ast }(w,r,q)$.
2. Derive the total cost function $C(w,r,q)$, average cost $AC(q)$, and marginal cost $MC(q)$.
3. Verify Shephard's Lemma for labor input: $\frac{\partial C}{\partial w}=L^{\ast }$.

Step-by-Step Rigorous Derivation:

1. Cost-minimization tangency condition:

   $$\frac{M{P}_L}{M{P}_K}=\frac{\frac{2}{3}{K}^{1/3}{L}^{-1/3}}{\frac{1}{3}{K}^{-2/3}{L}^{2/3}}=\frac{2K}{L}=\frac{w}{r}⟹K=\frac{w}{2r}L$$

2. Substitute into production constraint:

   $$q={\left(\frac{w}{2r}L\right)}^{1/3}{L}^{2/3}={\left(\frac{w}{2r}\right)}^{1/3}L⟹L^{\ast }(w,r,q)={\left(\frac{2r}{w}\right)}^{1/3}q$$$$K^{\ast }(w,r,q)=\frac{w}{2r}{\left(\frac{2r}{w}\right)}^{1/3}q={\left(\frac{w}{2r}\right)}^{2/3}q$$

3. Total Cost Function:

   $$C(w,r,q)=w{L}^{\ast }+r{K}^{\ast }=w{\left(\frac{2r}{w}\right)}^{1/3}q+r{\left(\frac{w}{2r}\right)}^{2/3}q=[2^{1/3}+2^{-2/3}]w^{2/3}{r}^{1/3}q=\frac{3}{2^{2/3}}{w}^{2/3}{r}^{1/3}q$$

4. Marginal & Average Cost:

   $$MC(q)=AC(q)=\frac{3}{2^{2/3}}{w}^{2/3}{r}^{1/3}(\text{Constant Returns to Scale: }\alpha +\beta =1)$$

5. Shephard's Lemma Check:

   $$\frac{\partial C}{\partial w}=\frac{3}{2^{2/3}}\cdot \frac{2}{3}{w}^{-1/3}{r}^{1/3}q=2^{1/3}{\left(\frac{r}{w}\right)}^{1/3}q=L^{\ast }(w,r,q)\text{Q.E.D.}$$