This document discusses the economics of input combinations for production. It introduces isoquant curves which show combinations of multiple inputs that produce the same output level. Iso-cost lines show combinations of inputs that can be purchased with a given budget, based on input prices. The least-cost combination of inputs to produce a certain output occurs where the iso-cost line is tangent to the isoquant curve, implying equal marginal rates of productivity per dollar spent on each input.

1. Economics of Input
input combinations
Prepared By: Milan Padariya

2. Topics of Discussion
Concept of isoquant curve
Concept of an iso-cost line
Least-cost use of inputs
2

3. Physical Relationships
3

4. Use of Multiple Inputs
This lecture will refer to situations where
we have multiple variable inputs
Labor, machinery rental, fertilizer application,
pesticide application, etc.
4

5. Use of Multiple Inputs
Our general single input production function
looked like the following:
 Output = f(labor | capital, land, energy, etc)
Variable Input
Fixed Inputs
Lets extend this to a two input production
function
 Output = f(labor, capital | land, energy, etc)
Variable Inputs
5
Fixed Inputs

6. Use of Multiple Inputs
Output (i.e. Corn Yield)
Phos. Fert.
250
Nitrogen Fert.
6

7. Use of Multiple Inputs
If we take a slice at a level of
250
output we obtain what is
referred as an isoquant
Similar to the indifference
curve we covered when we
reviewed consumer theory
Shows collection of multiple
inputs that generates a
particular output level
There is one isoquant for each
output level

8. Isoquant means “equal quantity”
Output is
identical along
an isoquant and
different across
isoquants
Two inputs

9. Slope of an Isoquant
The slope of an isoquant is referred to as
the Marginal Rate of Technical Substitution
(MRTS)
 The value of the MRTS in our example is given
by: MRTS = Capital
Labor
 Provides a quantitative measure of the
changes in input use as one moves along a
particular isoquant

10. Slope of an Isoquant
The slope of an isoquant is the
Capital
Q=Q*
Marginal Rate of Technical
Substitution (MRTS)
 Output remains unchanged along an
isoquant
 The ↓ in output from decreasing labor
must be identical to the ↑ in output from
adding capital as you move along an
isoquant
K*
L*
Labor

11. MRTS here is
–4 1=–4

12. What is the slope over
range B?
MRTS here is
–1 1 = –1

13. What is the slope over
range C?
MRTS here is
–.5 1 = –.5

14. Slope of an Isoquant
Since the MRTS is the slope of the
isoquant, the MRTS typically changes
as you move along a particular
isoquant
MRTS becomes less negative as shown
above as you move down an isoquant

15. Introducing Input Prices

16. Plotting the Iso-Cost Line
Lets assume we have the following
 Wage Rate is $10/hour
 Capital Rental Rate is $100/hour
What are the combinations of Labor and
Capital that can be purchased for $1000
 Lets introduce the Iso-Cost Line

17. Plotting the Iso-Cost Line
Capital
Firm can afford 10 hours of
capital at a rental rate of $100/hr with a
budget of $1,000
10
Firm can afford 100 hour of labor at a wage
rate of $10/hour for a budget of $1,000
 Combination of Capital and Labor
costing $1,000
 Referred to as the $1,000 Iso-Cost
Line
100
Labor

18. Plotting the Iso-Cost Line
 How can we define the equation of this isocost line?
Given a $1000 total cost we have:
$1000 = PK x Capital + PL x Labor
→ Capital =
(1000 PK) – (PL PK) x Labor
 →The slope of an iso-cost in our example is
given by:
Slope = –PL ÷ PK
(i.e., the negative of the ratio of the price
of the two inputs)

19. Plotting the Iso-Cost Line
Capital
2,000 PK
20
Doubling of Cost
Original Cost Line
Note: Parallel cost lines
given constant prices
10
500
PK
5
Halving of Cost
Labor
50
500
PL
200
100
2000
PL

20. Plotting the Iso-Cost Line
Capital
$1,000 Iso-Cost Line
Iso-Cost Slope = – PK
10
PL = $10
PL = $20
50
100
PL
PL = $5
200
Labor

21. Plotting the Iso-Cost Line
Capital
20
$1,000 Iso-Cost Line
Iso-Cost Slope = – PK
PK = $50
PL
10
PK = $100
5
PK = $200
50
100
200
Labor

22. Least Cost Combination
of Inputs

23. Least Cost Input Combination
TVC are predefined Iso-Cost Lines
Capital
TVC*** > TVC** > TVC*
Q*
TVC***
Pt. C: Combination of inputs that cannot produce
Q*
Pt. A: Combination of inputs that have the
highest of the two costs of producing Q*
Pt. B: Least cost combination of inputs to
produce Q*
A
TVC**
B
TVC*
C
Labor

24. Least Cost Decision Rule
The least cost combination of two inputs
(i.e., labor and capital) to produce a certain
output level
 Occurs where the iso-cost line is tangent to
the isoquant
 Lowest possible cost for producing that level
of output represented by that isoquant
 This tangency point implies the slope of the
isoquant = the slope of that iso-cost curve at
that combination of inputs

25. Least Cost Decision Rule
When the slope of the iso-cost = slope of the
isoquant and the iso-cost is just tangent to the
isoquant
–MPPK
MPPL
=
Isoquant
Slope
– (PK
PL)
Iso-cost
Line Slope
We can rearrange this equality to the
following

26. Least Cost Decision Rule
MPPL
PL
MPP per dollar
spent on labor
MPPK
Pk
=
MPP per dollar
spent on capital

27. Least Cost Decision Rule
The above decision rule holds for all variable
inputs
• For example, with 5 inputs we would have the
following
MPP1
P1
MPP1 per $ spent
on Input 1
MPP2
P2
=
MPP3
P3
MPP2 per $ spent
on Input 2
MPP4
P4
=…
…
=
MPP5
P5
MPP5 per $ spent
on=
Input 5

28. Least Cost Input Choice for 100 Units of Output

Point G represents 7 hrs of capital and
60 hrs of labor
 Wage rate is $10/hr and rental rate is
$100/hr
 → at G cost is
$1,300 = (100 7) + (10 60)
7
60

29. Least Cost Input Choice for 100 Units of Output


G represents a total cost of $1,300 every
input combination on the iso-cost line costs
$1,300
With $10 wage rate → B* represent 130
units of labor: $1,300 $10 = 130
7
60
130

30. Least Cost Input Choice for 100 Units of Output
 Capital rental rate is $100/hr
13
 → A* represents 13 hrs of
capital, $1,300 $100 = 13
130