# Advanced Microeconomics/Production
### Properties of Production Sets

The production vector $Y=(y_{1},y_{2},\ldots y_{n})$ 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 - $y\geq 0\rightarrow Y=0$ (no inputs, no outputs)
* possibility of inaction $(0\in Y)$
* Free disposal
* Irreversability - can't make output into inputs
* Returns to scale:
** Non-increasing: $\forall y\in Y,\,\alpha y\in Y\forall \alpha \in [0,1]$
** Non-decreasing: $\forall y\in Y,\,\alpha y\in Y\forall \alpha >1$
** Constant: $\forall y\in Y,\,\alpha y\in Y\forall \alpha \geq 0$
* Additivity: $y\in Y{\mbox{ and }}{y}^{\prime }\in Y\rightarrow y+{y}^{\prime }\in Y$
* Convexity: $y,{y}^{\prime }\in Y{\mbox{ and }}a\in [0,1]\rightarrow ay+(1-a){y}^{\prime }\in Y$

### Profit maximization
#### Example

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

##### Single Output

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

### marginal revenue product

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

### marginal rate of techical substitution

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

### Properties of profit functions and supply

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

${\begin{aligned}\min _{z_{1},z_{2}}&\,w_{1}z_{1}+w_{2}z_{2}{\mbox{ s.t. }}f(z_{1},z_{2})\leq q\\{\mathcal {L}}(z_{1},z_{2},\lambda )&=w_{1}z_{1}+w_{2}z_{2}-\lambda [f(z_{1},z_{2})-q]\\{\mathcal {L}}_{1}&=w_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=w_{2}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=f(z_{1},z_{2})-q=0\\\end{aligned}}$
${\frac {w_{1}}{w_{2}}}={\frac {f_{1}}{f_{2}}}\;{\frac {mp_{1}}{mp_{2}}}$ The ratio of input prices equals the ratio of marginal products
${\frac {w_{1}}{f_{1}}}={\frac {w_{2}}{f_{2}}}$ The marginal cost of expansion through $z_1$ equals the marginal cost of expansion through $z_{2}$
${\begin{aligned}{C}(w_{1},w_{2},q)&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}\\&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}-\lambda [f(z_{1}^{*},z_{2}^{*})-q]\\{\frac {\partial C}{\partial w_{1}}}&=z_{1}^{*}\\{\frac {\partial C}{\partial w_{2}}}&=z_{2}^{*}\\{\frac {\partial C}{\partial q}}&=\lambda {\mbox{- marginal cost}}\\\end{aligned}}$
The solution to the CMP gives factor demands, $z_{i}^{*}=z_{i}({w},q)$ 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 $p=ATC\rightarrow \pi =0$