• Chapter 10
  • Cost:
  • Assumption:
  • Cost minimizing input:
    • Compensated Factor demands:
    • Example 2
    • Opitimal point $(k^* , l^* )$
    • Lagrangian multiplier $\lambda$:
    • Expansion Path:
  • Total Cost Function:
  • Average Cost Function:
  • Marginal Cost Function:
    • Example 2 continued:
  • Graphical analysis of Total Costs
    • Constant return to scale production function
    • Properties of the total cost function
  • Short run/ Long run Distinction
    • Short run

Chapter 10

Economic profits:

Cost minimization (for a fixed level of output):

Cost:

• Explicit cost:
  Wages paid to employees (financial payment, rent, buying equipment etc.), and the cost of raw materials.

• Implicit cost:
  opportunity cost of the entrepreneur not related to financial payment(time spent on the firm), capital used for production.

• Opportunity Cost:
  remuneration the input would receive in its best alternative employment.

Assumption:

• two inputs $k$ and $l$:
  entrepreneurial costs are included in capital costs.

• inputs are hires in perfectly competitive markets, firms are f=price taker in input markets, meaning that price is given.

Cost minimizing input:

Compensated Factor demands:

$TC = wl+vk$,

→ $k=-\frac{w}{v}l+\frac{TC}{v}$

Given $min (wl+vk)$, such that $q^-=f(k,l)$

→ $L = wl+vk + \lambda(q^- -f(k,l))$
solve for F.O.C:

$\frac{\partial L}{\partial l} = w-\lambda \frac{\partial f}{\partial l}= 0$

$\frac{\partial L}{\partial k} = v-\lambda \frac{\partial f}{\partial l}= 0$
…

→ $\frac{w}{v} = \frac{\partial f}{\partial l} / \frac{\partial f}{\partial k}= RTS$

Compensated Factor demands:
$l^* = l^c (v, w, q^-)$,
$k^* = k^c (v, w, q^-)$

Example 2

$q=f(k,l) = k^{\frac{1}{3}}l^{\frac{2}{3}}$
Given $min (wl+vk)$, such that $q^-=k^{\frac{1}{3}}l^{\frac{2}{3}}$ → $q^- -k^{\frac{1}{3}}l^{\frac{2}{3}}=0$

→ $L = wl+vk + \lambda(q^- -k^{\frac{1}{3}} l^{\frac{2}{3}})$
Solve for F.O.C: $\frac{\partial L}{\partial l} = w - \frac{1}{3} \lambda k^{\frac{1}{3}} l^{-\frac{1}{3}} = 0$
…

→ $\frac{w}{v}= \frac{2}{3} k^{\frac{1}{3}} l^{-\frac{1}{3}} / \frac{1}{3} k^{-\frac{2}{3}} l^{\frac{2}{3}}$

∴ $l^* = (\frac{2v}{w})^{\frac{1}{3}} q^-$, $k^* = (\frac{2}{2v})^{\frac{2}{3}} q^-$.

Opitimal point $(k^* , l^* )$

• Given output $q_0$ , we want to find the least point in the isoquant.
  $k=-\frac{w}{v}l+\frac{Total ~ Cost}{v}$
  
  
  
  
  
  
  
  

Lagrangian multiplier $\lambda$:

how much the total cost would increase if output $q_0$ would increase by a small amount.

Expansion Path:

• Firm can determine the cost minimizing combinations of $k$ and $l$ for every level of output.

• Set of combination of optimal amount of $k$ and $l$ is called expansion path.
  
  
  
  
  
  

  • not have to be a straight line;
    use of some input may increase faster than others, depends on the shape of the isoquants).

  • not have to be upward sloping;
    use of an input falls as output expands, that input is an inferior input.

Total Cost Function:

• Substituting the compensated factor demand function in the total cost:
  $C=wl^c(w,vq)+ vk^c(w,v,q)$,
  → total cost function: $C=C(w,v,q)$

• minimum cost to produce the given quantity $q$ with given $k$ and $l$.

Average Cost Function:

Total cost function per unit of output: $AC(v,w,q)=\frac{C(w,v,q)}{q}$

Marginal Cost Function:

compute the change in the total cost function for a small change in output priduced.
$MC(w,v,q)= \frac{\partial C(w,v,q)}{\partial q}$

Example 2 continued:

since we calculated that $l^* = (\frac{2v}{w})^{\frac{1}{3}} q^-$, $k^* = (\frac{2}{2v})^{\frac{2}{3}} q^-$.

$C(w,v,q)= wl^* +vk^* = w(\frac{2v}{w})^{\frac{1}{3}} q^- + v(\frac{2}{2v})^{\frac{2}{3}} q^-$

$= (2^{\frac{1}{3}}+2^{-\frac{2}{3}}) w^{\frac{2}{3}} v^{\frac{1}{3}} q$

→ AC $= (2^{\frac{1}{3}}+2^{-\frac{2}{3}}) w^{\frac{2}{3}} v^{\frac{1}{3}}$

→ MC $= (2^{\frac{1}{3}}+2^{-\frac{2}{3}}) w^{\frac{2}{3}} v^{\frac{1}{3}}$

Graphical analysis of Total Costs

Constant return to scale production function

• $k_1$ unit of capital and $l_1$ units of labor are the optimal choice to produce one unit of output, $C(q=1, v,w)=vk_1 + wl_1$

• $k_m=mk_1, l_m=ml_1$ when product $m$ units.

• optimal choice $\frac{w}{v} =RTS$, price did not change, RTS depends only on $k/l$

  • ratio $k_m/l_m = k_1/l_1$
    
    
    
    
    
    
    
    

small business, just stared: total cost rises slowly
constant return to scale, compensated amount of labor, specialization firm becomes larger, total cost rises very fast, lack of management, too many poople to control.

increasing productibility
loss of efficiency

AC: U shape,
when MC $<$ AC, AC goes down
when MC $>$ AC, AC goes up
cross at the bottom at cost function ($min ~ AC$)

Example
$q=f(k,l)=min(ak,bl)$ → $ak=bl=q$

$C(w,v,q)=vk+vl$
$C(w,v,q)=q(\frac{v}{a}+\frac{w}{b})$

Properties of the total cost function

• Homogeneity, cost function is homogenious of degree one

• Increasing in $q$

• Non-decreasing in $v$ and $w$

• Concave in input prices,

  • price of labor goes up, firm’s optimal: firm will replace workers with capital.
    (consequence. before everyone get money, after some get more, some get zero.)

Short run/ Long run Distinction

Short run

• Easier to hire (and fire )workers
• Difficult to change the level of the capital stock

1. Increase Production in Short run:

   • Difficult to acquire more capital
   • hire more workers and use existing capital more intensively
   • not optimal level of amount of capital, cost of production is higher than in the long run.

2. Fixed Capital $k_1$, firm is free to vary its labor only, plug in $q=f(k,l)$

$q=f(k,l)=k^{\frac{1}{2}} l^{\frac{1}{2}}$, with $min wl+vk$
$k^{\frac{1}{2}} l^{\frac{1}{2}} = q^-$
since $k=k^-$ constant,
$q^- = k_-^{\frac{1}{2}} l^{\frac{1}{2}}$
$l^* =(w,v,q^-, k^-)= \frac{q^2}{k^-}$

3. Short run total cost $=(w,v,q^-,k^-) = wl^* +vk^- = w\frac{q^2}{k^-} +vk^-$

• fixed cost: $vk^-$, cannot be change in short run
• $wl^* ~$ variable costs

4. Long run total cost $=(w,v,q^-)=2w^{\frac{1}{2}} v^{\frac{1}{2}} q^-$
   (choose capital as Zero, all costs varies)
   
   

5. $k^-$ is fixed, so RTS can not be the ratio of input changing price.

• tangent cannot hold.
  
  
  
  
  
  

6. Short Run average total cost (SAC)

SAC $= \frac {short ~ run ~ total ~ costs}{total ~ output} = \frac {SC}{q}$