Economics of Inputinput combinations

Prepared By: Milan Padariya

Topics of DiscussionConcept of isoquant curveConcept of an iso-cost lineLeast-cost use of inputs

2

Physical Relationships

3

Use of Multiple Inputs

This lecture will refer to situations where we have multiple variable inputsLabor, machinery rental, fertilizer application,

pesticide application, etc.

4

Use of Multiple InputsOur general single input production function

looked like the following: Output = f(labor | capital, land, energy, etc)

Lets extend this to a two input production function Output = f(labor, capital | land, energy, etc)

Variable Input Fixed Inputs

Fixed InputsVariable Inputs

5

Use of Multiple InputsOutput (i.e. Corn Yield)

250

Nitrogen Fert.

Phos. Fert.

6

Use of Multiple InputsIf we take a slice at a level of

output we obtain what is referred as an isoquantSimilar 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

250

Output isidentical alongan isoquant and different across isoquants

Isoquant means “equal quantity”Isoquant means “equal quantity”

Two inputs

Slope of an IsoquantThe 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

Slope of an IsoquantThe slope of an isoquant is the

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

Labor

CapitalQ=Q*

L*

K*

MRTS here is– 4 ÷ 1 = – 4

MRTS here is– 4 ÷ 1 = – 4

What is the slope overrange B?

MRTS here is–1 ÷ 1 = –1

MRTS here is–1 ÷ 1 = –1

What is the slope overrange C?What is the slope overrange C?

MRTS here is–.5 ÷ 1 = –.5

MRTS here is–.5 ÷ 1 = –.5

Slope of an IsoquantSince the MRTS is the slope of the

isoquant, the MRTS typically changes as you move along a particular isoquantMRTS becomes less negative as shown

above as you move down an isoquant

Introducing Input Prices

Plotting the Iso-Cost LineLets 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

Plotting the Iso-Cost Line

Labor

Capital

10

100

Firm can afford 100 hour of labor at a wage rate of $10/hour for a budget of $1,000

Firm can afford 100 hour of labor at a wage rate of $10/hour for a budget of $1,000

Firm can afford 10 hours ofcapital at a rental rate of $100/hr with a budget of $1,000

Firm can afford 10 hours ofcapital at a rental rate of $100/hr with a budget of $1,000

Combination of Capital and Labor costing $1,000 Referred to as the $1,000 Iso-Cost

Line

Plotting the Iso-Cost LineHow can we define the equation of this iso-

cost 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)

Plotting the Iso-Cost Line

Labor

Capital

20

200

10

5

10050

Doubling of Cost

Note: Parallel cost linesgiven constant prices

Original Cost Line

2,000÷PK

500 ÷ PK

Halving of Cost

500 ÷ PL 2000 ÷ PL

Plotting the Iso-Cost Line

Labor

Capital

10

10050 200

$1,000 Iso-Cost Line

Iso-Cost Slope = – PK ÷ PL

PL = $5PL = $10

PL = $20

Plotting the Iso-Cost Line

Labor

Capital

10

10050 200

$1,000 Iso-Cost Line

Iso-Cost Slope = – PK ÷ PL

20

5

PK = $200

PK = $100

PK = $50

Least Cost Combinationof Inputs

Least Cost Input Combination

Labor

CapitalTVC are predefined Iso-Cost Lines

TVC*

TVC**

TVC***

TVC*** > TVC** > TVC*

A

B

C

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*

Q*

Least Cost Decision RuleThe 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

Least Cost Decision RuleWhen the slope of the iso-cost = slope of the

isoquant and the iso-cost is just tangent to the isoquant

–MPPK ÷ MPPL = – (PK ÷ PL)

We can rearrange this equality to the following

Isoquant Slope

Isoquant Slope

Iso-cost Line SlopeIso-cost

Line Slope

Least Cost Decision Rule

=

L K

L k

MPP MPP

P P

MPP per dollar spent on labor

MPP per dollar spent on labor

MPP per dollar spent on capitalMPP per dollar spent on capital

Least Cost Decision Rule

3 51 2 4

1 2 3 4 5

MPP MPPMPP MPP MPP

P P P P P

The above decision rule holds for all variable inputs• For example, with 5 inputs we would have the

following

MPP1 per $ spent on Input 1

MPP1 per $ spent on Input 1 = MPP2 per $ spent

on Input 2

MPP2 per $ spent on Input 2 = …… =MPP5 per $ spent

on Input 5

MPP5 per $ spent on Input 5=

Least Cost Input Choice for 100 Units of OutputLeast Cost Input Choice for 100 Units of Output

7

60

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

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

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

130

Least Cost Input Choice for 100 Units of OutputLeast Cost Input Choice for 100 Units of Output

130

Capital rental rate is $100/hr → A* represents 13 hrs of capital,

$1,300 $100 = 13

Capital rental rate is $100/hr → A* represents 13 hrs of capital,

$1,300 $100 = 1313

Least Cost Input Choice for 100 Units of OutputLeast Cost Input Choice for 100 Units of Output