### Properties of Production Sets

The production vector ${\displaystyle Y=(y_{1},y_{2},\ldots y_{n})}$ where ${\displaystyle y_{i}>0}$ represents an output, and ${\displaystyle y_{i}<0}$ an input

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

### Profit maximization

#### Example

{\displaystyle {\begin{aligned}\max &\;p_{2}y_{2}-p_{1}y_{1}\\{\mbox{s.t. }}&[y_{1},y_{2}]\in Y\\&f(y_{1},y_{2})\leq k\\{\mathcal {L}}(y_{1},y_{2},\lambda )&=p_{2}y_{2}-p_{1}y_{1}+\lambda [k-f(y_{1},y_{2})]\\{\mathcal {L}}_{1}&=-p_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=p_{1}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=k-f(y_{1},y_{2})=0\\\end{aligned}}}

##### Single Output

${\displaystyle y=f(Z)}$ where ${\displaystyle Z=(z_{1},z_{2},\ldots ,z_{n})}$
{\displaystyle {\begin{aligned}&\max _{y,Z}py-w\\{\mbox{subject to }}&y=f(z)\\&{\mbox{ -or- }}\\&\max _{Z}pf(Z)-wZ\\{\frac {\partial {\mathcal {L}}}{\partial z_{i}}}&=pf_{i}-w_{i}\leq 0\\\end{aligned}}}

### marginal revenue product

Marginal revenue product is the price of output times the marginal product of input ${\displaystyle MRP=p\cdot f_{i}}$
The first order conditions for profit maximization require the marginal revenue product to equal input cost for all inputs (actually) used in production,
${\displaystyle pf_{i}=w_{i}\;\forall z_{i}>0}$

### marginal rate of techical substitution

{\displaystyle {\begin{aligned}pf_{1}&=w_{1}\\pf_{2}&=w_{2}\\&\rightarrow {\frac {f_{1}}{f_{2}}}={\frac {w_{1}}{w_{2}}}f(z_{1},z_{2})&={\bar {y}}\\f_{1}dz_{1}&+f_{2}dz_{2}=0\\{\frac {dz_{2}}{dz_{1}}}&=-{\frac {f_{1}}{f_{2}}}\end{aligned}}}

### Properties of profit functions and supply

• Profit functions exhibit homogeneity of degree 1 ${\displaystyle \pi =pf(z)-w{z}}$ 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

{\displaystyle {\begin{aligned}\min _{z_{1},z_{2}}&\,w_{1}z_{1}+w_{2}z_{2}{\mbox{ s.t. }}f(z_{1},z_{2})\leq q\\{\mathcal {L}}(z_{1},z_{2},\lambda )&=w_{1}z_{1}+w_{2}z_{2}-\lambda [f(z_{1},z_{2})-q]\\{\mathcal {L}}_{1}&=w_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=w_{2}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=f(z_{1},z_{2})-q=0\\\end{aligned}}}

${\displaystyle {\frac {w_{1}}{w_{2}}}={\frac {f_{1}}{f_{2}}}\;{\frac {mp_{1}}{mp_{2}}}}$  The ratio of input prices equals the ratio of marginal products

${\displaystyle {\frac {w_{1}}{f_{1}}}={\frac {w_{2}}{f_{2}}}}$  The marginal cost of expansion through $z_1$ equals the marginal cost of expansion through ${\displaystyle z_{2}}$

{\displaystyle {\begin{aligned}{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 {\mbox{- marginal cost}}\\\end{aligned}}}

The solution to the CMP gives factor demands, ${\displaystyle z_{i}^{*}=z_{i}({w},q)}$  and the cost function ${\displaystyle \sum w_{i}z_{i}=c(w,q)}$

## Cost Functions

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

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