• chapter 9 Production Functions
  • Production Function:
  • Marginal Physical Product
  • Diminishing Marginal Productivity
  • Average Physical Product
  • Isoquant Maps:
    • Marginal rate of technical substitution (RTS)
  • Return to Scale:
    • Two factors:
    • Constant return-to-scale production function
  • Elasticity of substitution:
    • Linear (Perfect Substitutes):
    • Fix proportional, (perfect complements):
    • Cobb Douglas production function:
    • CES production function:

chapter 9 Production Functions

Production Function:

• A firm’s Production Functions for a good $q$ is the maximum amount of the good that can be produced using alternative combinations of inputs

• $q=f(x_a, ...,x_n)$

• Consider capital $k$, labor $l$, then $q=f(k,l)$

Marginal Physical Product

Additional output that can be produced by increasing by a small amount one of the input while holding other inputs constant.

• Marginal Physical Product of capital $MP_k = \frac{\partial q}{\partial k}= \frac{\partial f}{\partial k}=f_k$

• Marginal Physical Product of labor $MP_l = \frac{\partial q}{\partial l}= \frac{\partial f}{\partial l}=f_l$
  
  

Diminishing Marginal Productivity

• An input depends on how much of that input is used.

• Marginal Product decreases

$\frac{\partial MP_k}{\partial k}=\frac{\partial^2 f}{\partial k^2}= f_{kk}=f_{11}<0$, $\frac{\partial MP_l}{\partial l}=\frac{\partial^2 f}{\partial l^2}= f_{ll}=f_{22}<0$,

Average Physical Product

An input also depends on how much of the other input is used.

$AP_l=\frac{output}{labor input} = \frac{q}{l}=\frac{f(k,l)}{l}$

• $AP_l$ depends on the amount of capital employed.
• $AP_k$ depends on the amount of labor employed.

Example:

$q = f(k,l)=600k^2l^2-k^3l^3$

Marginal Product of labor $= \frac{\partial q}{\partial l}= \frac{\partial f}{\partial l} = 1200k^2l-3k^3l^2$

$k=10$

$MP_l=120000l-3000l^2$

$\frac{\partial MP_l}{\partial l} = \frac{\partial^2 f}{\partial l^2} = 120000-6000l<0$

$l>20$

when $k=10$, $max f(k,l)= max (600k^2l^2-k^3l^3) = max (60000l^2-1000l^3)$

f.o.c $MP_l=0, 120000l-3000l^2=0$

$l=40$ (max profits)

$f(k,l)= 600k^2l^2-k^3l^3$
$AP_l = \frac{f(k,l)}{l}= 600k^2l-k^3l^2$
$k=10, 60000l-1000l^2$

Isoquant Maps:

• shows the combinations of $k$ and $l$ that can be used to produce a given level of output $(q_0)$, meaning all $(k,l)$ such that $f(k,l) = q_0$

Marginal rate of technical substitution (RTS)

• RTS is diminishing for increasing inputs of labor.

• $RTS(l ~ for ~ k) = - \frac{dk}{dl}, when ~(q=q_0)$
  
  
  
  
  

• Marginal rate of technical substitution rate (RTS):
  the rate at which capital can be substituted for labor while holding output constant along an isoquant.

Example:
$dq = qf(k,l) = \frac{\partial f}{\partial k} dk + \frac{\partial f}{\partial l} dl = MP_k dk + MP_l dl$
$dq=0$
$MP_k dk + MP_l dl = 0$, if stays on same isoquant map,

$-\frac{dk}{dl}=\frac{MP_l}{MP_k}$

Return to Scale:

• The effect of a proportional change in input and output.

• $q=f(k,l)$, all inputs are multiplied by the same positive constant $t>1$, then

Two factors:

• greater division os labor and specialization of function(divide complex jobs in simple tasks)

• because of monitoring and communication, loss in efficiency because management may become more difficult given the larger scale of the firm.

Constant return-to-scale production function

• Homogeneous of degree one in inputs. $f(tk,tl)=t^1f(k,l)=tq$

• Marginal Productivity functions are homogeneous of degree zero.(Derivative is one degree less than the function).

  • Marginal Productivity of any input depends only on the ratio of capital and labor.

  • RTS between $k,l$ depends only on the ratio of $k$ to , not the scale of operation.

• the RTS will be the same on all isoquants (same shape and same space).
  
  
  
  
  
  

• for $n$ inputs, $q=f(x_1,x_2,..., x_n)$, RTS is same as two inputs.
  
  

Example, marginal product for constant RTS

$MP_e=\frac{\partial f(k,l)}{\partial l} = g(k,l)$

$g(\frac{k}{l}, \frac{l}{l})=(\frac{1}{e})^0 g(k,l)=g(k,l) = MP_l$

$g(\frac{k}{l})=MP_l$

Elasticity of substitution:

• Elasticity of substitution $\sigma$, measures how easy it is to substitute one input with a different one.

• proportional change in $k/l$ relative to the proportional change in the RTS along an isoquant.

• $\sigma$ is always positive because $k/l$ and RTS moves to same direction.

• $\sigma = \frac{d(k/l)}{dRTS}\frac{RTS}{k/l} = \frac{d ln(k/l)}{d ln RTS}$

• RTS and $k/l$ will change as we move from point A to B.
  $\sigma$ is the ratio of these proportional changes.
  
  
  
  
  
  
  
  

• if define the elasticity of substitution between two input to be proportional change in RTS, we need to hold output and the levels of other inputs constant.

Linear (Perfect Substitutes):

Given $q=f(k,l) = ak+bl$,

1. $f(tk,tl) = atk+btl = t(ak+bl)= tq$ , → linear.

2. • isoquant constant.
   • $-\sigma =$ ∞.
     
     
     
     
     
     

Fix proportional, (perfect complements):

Given $q= min(ak,bl), a,b >0$

meaning capital and labor must always be used in a fixed ration.

→ $k/l = b/a$ is constant, $\sigma =0$

→ No substitution between labor and capital is possible, constant RTS.

Cobb Douglas production function:

Given $q= f(k,l) = Ak^al^b, where A, a, b >0$
different any RTS,

$f(tk,tl) = Atk^a tl^b = A t^{a+b} k^a l^b = t^{a+b} f(k, l)$

CES production function:

Given $q=f(k,l)=[k^p+l^p]$, where $p≤1, p≠0, \gamma >0$

when $\gamma = 1$, $\sigma = \frac{1}{1- \rho}$