### Review Chapter 6: Firms and Production

```Chapter 6: Firms and Production
• Firms’ goal is to maximize their profit.
• Profit function: π= R – C = P*Q – C(Q)
where R is revenue, C is cost, P is price, and
Q is quantity
• Production function: the relationship between
the quantities of inputs used and the maximum
quantity of output that can be produced. It
summarizes the technology of transforming
inputs into outputs. e.g.) q = f(L,K)
• Fixed input vs. variable input
1
Short-Run: At least one factor
of production is fixed
For production function: q = f(L,K)
• Average product of labor (AP) = q/L
• Marginal product of labor (MP) = △q/△L
• AP increases when MP exceeds AP and
decreases when MP is exceeded by AP.
2
3
Diminishing Marginal Returns
(or diminishing marginal product)
• If a firm keeps adding one more unit of input, holding all
other inputs and technology constant, the extra output it
obtains will become smaller eventually.
• Why?
– Too many workers per machine
– Increases the cost of managing labors, etc.
4
Example: A Cobb-Douglas Function
• Production Function:
q  10L0.5 K 0.5
• Capital (K) is fixed. Only labor (L) is variable.
• The marginal product of Labor is
q
 10  0.5L( 0.51) K 0.5  5 L0.5 K 0.5
L
• The second derivative of q w.r.t. L is
 2q
( 0.51)
0.5
1.5
0.5

5

(

0
.
5
)
L
K


0
.
25
L
K
0
L2
which is negative: Concave function.
Assume that K is fixed at 100. Draw the production function and the
marginal product of Labor.
5
Long-Run: All inputs are variable
• Firms can vary input mix to achieve the most efficient
production.
• Isoquant: a curve that shows the efficient combinations
of labor and capital that can produce a single level of
output (similar to indifference curve)
q  q( L, K )
• Marginal Rate of Technical Substitution (MRTS):
– the extra units of one input needed to replace one
unit of another input that allows a firm to produce
the same level of output
– slope of an isoquant (i.e., dK )
dL
6
Diminishing marginal rate
of technical substitution
7
Unique Isoquants
8
MRTS and Marginal Products
• By definition of isoquant: q  q( L, K )
• To see the small change in q, totally
differentiate an isoquant:
q
q
dq 
dL 
dK  0
L
K
Marginal increase in Change
output from increasing L
in L
Total increase in output
from increasing L by dL
MPL dK
 

 MRTS
MPK dL
9
Example: A Cobb-Douglas Function
q  10L0.5 K 0.5
•
Production Function:
•
Capital (K) is not fixed (long-run).
•
The marginal product of Labor is
q
 10  0.5L( 0.51) K 0.5  5 L0.5 K 0.5
L
•
The marginal product of Capital is
q
 10  0.5L0.5 K ( 0.51)  5 L0.5 K 0.5
K
The marginal rate of technical substitution (MRTS) is
dK
MPL
0.5L0.5 K 0.5
K
MRTS 




dL
MPK
0.5L0.5 K 0.5
L
Draw the isoquant curve.
10
Returns to Scale
• How much output changes if a firm increases all
its inputs proportionately.
• Long-run concept
• Constant Returns to Scale (CRS):
t * f(x1, x2) = f(tx1, tx2)
• Increasing Returns to Scale (IRS):
t * f(x1, x2) < f(tx1, tx2)
• Decreasing Returns to Scale (DRS):
t * f(x1, x2) > f(tx1, tx2)
11
Reasons for increasing or decreasing returns to scale
- Increasing Returns to Scale (IRS):
- A larger plant may allow for greater specializations of
inputs.
- Decreasing Returns to Scale (DRS):
- Management problems may arise when the
production scale is increased, e.g., cheating by
workers.
- Large teams of workers may not function as well as
small teams.
12
13
For a Cobb-Douglas production function:

q  AL K

• If we double all inputs,


 
A(2L) (2K )  2
• CRS if
• IRS if
• DRS if


 
AL K  2
q
   1
   1
   1
14
Productivity and Technical Change
• Technical change:
– Neutral technical change
• q = A*f(L,K)
– Non-neutral technical change
• e.g. from labor-using to labor-saving
15
Illustration of Neutral Technical Change
K
Isoquants
q = 30 → q = 45
q = 20 → q = 30
q = 10 → q = 15
L
16
Illustration of Non-neutral or Biased
Technical Change
K
K-using or
L-saving
L-using or
K-saving
Original
isoquants
17
L
```