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: $$y,{y}^{\prime} \in Y \mbox{ and } a\in[0,1] \rightarrow ay+(1-a){y}^{\prime}\in Y $$

Example
$$ \begin{align} \max&\;p_2y_2 - p_1y_1 \\ \mbox{s.t. } &[y_1,y_2] \in Y \\ &f(y_1,y_2) \leq k \\ \mathcal{L}(y_1,y_2,\lambda) &= p_2y_2-p_1y_1 + \lambda [k-f(y_1,y_2)]\\ \mathcal{L}_1 &= -p_1 -\lambda f_1 = 0\\ \mathcal{L}_2 &= p_1 -\lambda  f_2 = 0\\ \mathcal{L}_\lambda &= k -f(y_1,y_2) = 0\\ \end{align} $$

Single Output
$$y=f(Z)$$ where $$Z=(z_1,z_2,\ldots,z_n)$$ $$ \begin{align} &\max_{y,Z} py - w \\ \mbox{subject to } &y=f(z)\\ &\mbox{ -or- }\\ &\max_{Z} pf(Z) - wZ \\ \frac{\partial \mathcal{L}}{\partial z_i} &= pf_i - w_i \leq 0\\ \end{align} $$

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
$$ \begin{align} pf_1 &= w_1 \\ pf_2 &= w_2 \\ &\rightarrow \frac{f_1}{f_2} = \frac{w_1}{w_2} f(z_1,z_2) &= \bar{y}\\ f_1dz_1 &+ f_2dz_2 = 0\\ \frac{dz_2}{dz_1} &= -\frac{f_1}{f_2} \end{align} $$

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 \funcd{c}{w,q}

Example
$$ \begin{align} \min_{z_1,z_2} &\,w_1z_1+w_2z_2 \mbox{ s.t. } f(z_1,z_2) \leq q\\ \mathcal{L}(z_1,z_2,\lambda) &= w_1z_1+w_2z_2 - \lambda [f(z_1,z_2) - q]\\ \mathcal{L}_1 &= w_1-\lambda f_1 = 0 \\ \mathcal{L}_2 &= w_2-\lambda f_2 = 0 \\ \mathcal{L}_\lambda &= f(z_1,z_2) - q=0\\ \end{align} $$

$$ \frac{w_1}{w_2} = \frac{f_1}{f_2} \; \frac{mp_1}{mp_2} $$ 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 $$z_2$$

$$ \begin{align} {C}(w_1,w_2,q)&=w_1z_1^*+w_2z_2^*\\ &=w_1z_1^*+w_2z_2^* -\lambda [f(z_1^*,z_2^*)-q]\\ \frac{\partial C}{\partial w_1} &= z_1^*\\ \frac{\partial C}{\partial w_2} &= z_2^*\\ \frac{\partial C}{\partial q} &= \lambda \mbox{- marginal cost}\\ \end{align} $$

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

Cost Functions

 * $$P>ATC$$ gives positive economic profit, short run and long run
 * In short run, fixed costs are irrelevant. Shut down if $$pATC$$ firms enter, in the long run $$\pi\to 0$$ until $$p=ATC\rightarrow \pi=0$$