Advanced Microeconomics/Production

Properties of Production Sets

The production vector $Y=(y_1,y_2,\dots y_n)$ where $y_i>0$ represents an output, and $y_i<0$ an input

• Y is non empty
• Y is closed (includes its boundary)
• No free lunch - $y\ge 0\to Y=0$ (no inputs, no outputs)
• possibility of inaction $(0\in Y)$
• Free disposal
• Irreversability - can't make output into inputs
• Returns to scale:
  • Non-increasing: $\mathrm{\forall }y\in Y,\alpha y\in Y\mathrm{\forall }\alpha \in [0,1]$
  • Non-decreasing: $\mathrm{\forall }y\in Y,\alpha y\in Y\mathrm{\forall }\alpha >1$
  • Constant: $\mathrm{\forall }y\in Y,\alpha y\in Y\mathrm{\forall }\alpha \ge 0$
• Additivity: $y\in Y\text{ and }{y}^{\prime }\in Y\to y+y^{\prime }\in Y$
• Convexity: $y,y^{\prime }\in Y\text{ and }a\in [0,1]\to ay+(1-a)y^{\prime }\in Y$

Profit maximization

Example

$\begin{array}{cc}\max & p_2{y}_2-p_1{y}_1\\ \text{s.t. }& [y_1,y_2]\in Y\\ & f(y_1,y_2)\le k\\ ℒ(y_1,y_2,\lambda )& =p_2{y}_2-p_1{y}_1+\lambda [k-f(y_1,y_2)]\\ {ℒ}_1& =-p_1-\lambda f_1=0\\ {ℒ}_2& =p_1-\lambda f_2=0\\ {ℒ}_{\lambda }& =k-f(y_1,y_2)=0\\ \end{array}$

Single Output

$y=f(Z)$ where $Z=(z_1,z_2,\dots ,z_n)$
$\begin{array}{cc}& \underset{y,Z}{\max }py-w\\ \text{subject to }& y=f(z)\\ & \text{ -or- }\\ & \underset{Z}{\max }pf(Z)-wZ\\ \frac{\partial ℒ}{\partial z_i}& =p{f}_i-w_i\le 0\\ \end{array}$

marginal revenue product

Marginal revenue product is the price of output times the marginal product of input $MRP=p\cdot f_i$
The first order conditions for profit maximization require the marginal revenue product to equal input cost for all inputs (actually) used in production,
$p{f}_i=w_i\mathrm{\forall }{z}_i>0$

marginal rate of techical substitution

$\begin{array}{cc}p{f}_1& =w_1\\ p{f}_2& =w_2\\ & \to \frac{f_1}{f_2}=\frac{w_1}{w_2}f(z_1,z_2)& =\overline{y}\\ f_{1}d{z}_1& +f_{2}d{z}_2=0\\ \frac{d{z}_2}{d{z}_1}& =-\frac{f_1}{f_2}\end{array}$

Properties of profit functions and supply

• Profit functions exhibit homogeneity of degree 1 $\pi =pf(z)-wz$ doubling all prices doubles nominal profit
• supply functions exhibit homogeneity of degree 0

Cost Minimization

The optimal CMP gives cost function <align>\funcd{c}{w,q}</align>

Example

$\begin{array}{cc}\underset{z_1,z_2}{\min }& w_1{z}_1+w_2{z}_2\text{ s.t. }f(z_1,z_2)\le q\\ ℒ(z_1,z_2,\lambda )& =w_1{z}_1+w_2{z}_2-\lambda [f(z_1,z_2)-q]\\ {ℒ}_1& =w_1-\lambda f_1=0\\ {ℒ}_2& =w_2-\lambda f_2=0\\ {ℒ}_{\lambda }& =f(z_1,z_2)-q=0\\ \end{array}$

$\frac{w_1}{w_2}=\frac{f_1}{f_2}\frac{m{p}_1}{m{p}_2}$ The ratio of input prices equals the ratio of marginal products

$\frac{w_1}{f_1}=\frac{w_2}{f_2}$ The marginal cost of expansion through $z_1$ equals the marginal cost of expansion through $z_2$

$\begin{array}{cc}C(w_1,w_2,q)& =w_1{z}_1^{*}+w_2{z}_2^{*}\\ & =w_1{z}_1^{*}+w_2{z}_2^{*}-\lambda [f(z_1^{*},z_2^{*})-q]\\ \frac{\partial C}{\partial w_1}& =z_1^{*}\\ \frac{\partial C}{\partial w_2}& =z_2^{*}\\ \frac{\partial C}{\partial q}& =\lambda \text{- marginal cost}\\ \end{array}$

The solution to the CMP gives factor demands, $z_i^{*}=z_i(w,q)$ and the cost function $\sum w_i{z}_i=c(w,q)$

Cost Functions

• $P>ATC$ gives positive economic profit, short run and long run
• In short run, fixed costs are irrelevant. Shut down if $p<AVC$
• minimum efficient scale: $mc=ATC=P$

No economic profits in the long run, given free entry for any $P>ATC$ firms enter, in the long run $\pi \to 0$ until $p=ATC\to \pi =0$