Advanced Microeconomics/Production

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Properties of Production Sets[edit]

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[edit]


\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\\

Single Output:[edit]

y=f(Z) where Z=(z_1,z_2,\ldots,z_n)

&\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\\

marginal revenue product[edit]

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[edit]

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}

Properties of profit functions and supply[edit]

  • 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[edit]

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

\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

&=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}\\

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[edit]

  • 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