Production & Costs Continued…
Agenda:
I.
Consumer and Producer Theory:
similarities and differences
II. Isoquants & The Marginal Rate of Technical Substitution
III. Diminishing vs. Decreasing Returns
IV. Isocosts
V. Putting it together: Optimal Production &
Examples
Consumer Utility Maximization
Indifference curves
Utility curves
U x px

U y p y
Different notation
Same meaning!
Y
px

X
py
The Production Mountain
Q  f ( K , L; time)
A  f ( p, l ; time)
Quantity per unit of labor
holding capital constant
Quantity per unit of capital
holding labor constant
Isoquants:
Combinations of capital and
labor that produce a given
quantity
Long Term we can vary
both capital and labor
The Marginal Rate of Technical Substitution
If we change the amount of capital we use, how much do we need
to change the amount of labor to make the same quantity?
Q  MPK K  MPL L  0
MPK K  MPL L
Isoquants
K MPL


L MPK
Airplane Game Isoquants
paper
Y
X
1
1
1
labor
Isoquants vs. Indifference Curves
Isoquants
Convexity from diminishing marginal
rate of technical substitution
More is better
Quantity is a cardinal measure
Can only change both capital and labor
in the long run.
Indifference Curves
Convexity from preference assumption
More is better
Utility is an ordinal measure
Individuals make trade-offs both at one
time and over time
Diminishing vs. Decreasing returns
All Isoquants are convex and slope down: diminishing MRTS
Quantity increases at a decreasing rate as all inputs increase: decreasing returns
+20
+40
+60
+60
+60
+90
+60
Isocost Lines
What is an equation to represent the total cost of production?
C=PKK + PLL
Can we re-arrange this to fit the equation for a line in (L,K) space?
C PL
K

L
PK PK
What is the optimal input combination GIVEN cost or quantity?
“No matter what the structure of industry may be… (for profit or not for profit) … the
objective of most producers is to produce any given level and quality of output at the
lowest possible cost. Equivalently, the producer wants to produce as much output as
possible from a given expenditure on inputs.” (Frank p. 233)
Duality
Maximize Q given C
w = cost of labor
K MPL* PL w



L MPK * PK r
r = cost of capital
MPK * MPL*

r
w
Marginal products per dollar
Minimize C given Q
Example:
If the MRTS between capital and labor is 1/2, the interest rate is
5% (use 5) and the wage rate is $10 per hour, is the firm
maximizing production?
1 K w 10

 
2 L r
5
How should the firm adjust its mix of capital and labor?
The firm could be making more for the same cost!
Use LESS labor,
MORE capital
2
The firm is spending more than it has to!
1/2
Example:
If the marginal product of labor is 5 and the marginal product of
capital is 2, the price of labor is $20 and the cost of capital is 4%,
is the firm optimizing production?
MPL 5 20 w
 

MPK 2 4
r
or 
To increase the
marginal product
of labor, reduce
labor.
MPL
5 MPK 2 To decrease the
marginal product



w
20
r
4 of capital,
increase capital.
You can NOT control interest rates or reduce wages in perfect
capital or labor markets.
(that said… change term structure, reduce benefits, training, perks…)