ME Production & Cost Theory

8/10/2019 ME Production & Cost Theory

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Production Theory and Analysis


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Production

Productionrefers to the transformation of inputs or

resources into outputs of goods and services

Creation of utility


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Characteristics of goods & services to be

classified as Production are:

Created by human labor and capital

Satisfy human wants directly or indirectly Are comparatively scarce and have economic

value

Have a definite monetary price/cost


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EnterpriseCapitalLaborLand

Immobile

Passive

Heterogeneous

Active

Mobile

Variableproductivity

Structures

Equipment

Cap.goodsMoney

Innovative function

Risk

Decision making

Factors of Production

INPUTS


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INPUTS

CAPITAL

EntrepreneurWorkersLand &

Structures

LABOR

Machinery

plant &

equipment

Natural

Resources


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Factors of Production

Inputs

Fixed Inputs

Variable Inputs

Short Run- At least one input is fixed

Long Run- All inputs are variable

The length of long run depends on industry.


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Level & Scale of Production

Level of productioncan be altered changing the

proportion of variable inputs

Output = Fixed inputs + Variable inputs

Scale of productioncan be altered by changing the

supply of all the inputs only in the long run

Output = Total inputs(variable inputs)


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Concept of Product

Total Product- total volume of goods producedduring a specific period of time

Average Product- the per unit product of avariable factor

Marginal Product- the rate at which totalproduct increases / addition to total productresulting from a unit increase in the quantity of thevariable factor


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Production Function With Two Inputs

Input & output are measured in physical units

Assumption- Technology is constant the during analysis period

- All units of L & K are homogenous

K Q6 10 24 31 36 40 395 12 28 36 40 42 40

4 12 28 36 40 40 363 10 23 33 36 36 332 7 18 28 30 30 281 3 8 12 14 14 12

1 2 3 4 5 6 L

Q = f(L, K)


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Production Function with Two Inputs

Discrete Production Surface


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Production Function with Two Inputs

Continuous Production Surface


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Production Function with

One Variable Input

Total Product TP = Q = f(L)

Marginal Product MPL=

TPL

Average Product APL=TP

LProduction orOutput Elasticity

Q/QL/L

Q/ LQ/L

== =ELMPL

APL


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One Variable Input

L Q

0 0

1 3

2 8

3 12

4 14

5 14

6 12

Total, Marginal, and Average Product of Labor, and Output Elasticity


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One Variable Input

L Q MPL APL EL

0 0 - - -1 3 3 3 1

2 8 5 4 1.25

3 12 4 4 14 14 2 3.5 0.57

5 14 0 2.8 0

6 12 -2 2 -1

Total, Marginal, and Average Product of Labor, and Output Elasticity


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Production Function

with One Variable Input


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-2

-1

0

1

2

3

4

5

0 1 2 3 4 5 6 7

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7

6

A

B

C

D E

F

A

B CD

EF

TotalProduct

Marginal

& Average

Product

Labor

Labor

Production Function with One Variable Input

TP

MP

AP


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-2

-1

0

1

2

3

4

5

0 1 2 3 4 5 6 7

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7

6

A

B

C

D E

F

B C

AD

EF

I

TotalProduct

Marginal

& Average

Product

Labor

Labor

The Law of Diminishing Returns &

Stages of Production

Stage I of Labor Stage II of Labor Stage III of Labor

TP

MP

AP

G


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Optimal Use of the Variable Input

Marginal Revenue

Product of Labor

MRPL= (MPL)(MR)

Marginal ResourceCost of Labor

MRCL=TCL

Optimal Use of Labor MRPL= MRCL


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L MPL MR = P

2.50 4 $10

3.00 3 10

3.50 2 10

4.00 1 10

4.50 0 10


Assumption : Firm hires additional units of labor at constantwage rate = $20


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L MPL MR = P MRPL MRCL

2.50 4 $10 $40 $20

3.00 3 10 30 20

3.50 2 10 20 20

4.00 1 10 10 20

4.50 0 10 0 20

Use of Labor is Optimal When L = 3.50


Assumption : Firm hires additional units of labor at constantwage rate


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2.5 3.0 3.5 4.0 4.5

40

30

20

10

0

MRCL= w = $20

dL= MRPL

Units of Labor Used

$


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Exercise

The marginal product of labor equation for a firm

is given by: MPL= 10(K/L)0.5

Currently the firm is using 49 units of capital and

100 units of labor. Capital usage is fixed, but labor

can be varied. If the price of labor is $20 per unit

and the firms' output sells for $4, is the firm

producing efficiently in the short run? If not,explain and determine the optimal rate of labor

input.


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MRPL = MRCL = w

28 20 Not efficient

L = 196


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Production With Two Variable Inputs

Isoquants show combinations of two inputsthat can produce the same level of output.

K

Q

6 10 24 31 36 40 395 12 28 36 40 42 404 12 28 36 40 40 36

3 10 23 33 36 36 332 7 18 28 30 30 281 3 8 12 14 14 12

1 2 3 4 5 6 L


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Isoquants


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Economic Region of ProductionFirms will only use combinations of two inputs that are in

the economic region of production.

W


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MRTS = -(-2.5/1) = 2.5

Marginal Rate of Technical Substitution

Absolute value of the slope of isoquant is called the MRTS

K


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Production With Two Variable Inputs

Perfect Substitutes Perfect Complements

6

4

2

2 4 6 8 10 12

6

4

2

2 4 6

CapitalCapital

Labor Labor0 0

-1K

2L

2K

1L

C

A

B


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10

8

6

4

2

2 4 6 8 10

Capital

Labor

1K

1L

AB C = $100, w = r = $10A

B

Isocost Lines

slope = -w/r = -1

vertical intercept = 10


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4 8 10 12 14 16 20

14

108

4

A

A

A

BB B B*

0 Labor

Capital

Isocost Lines

Isocost Lines

AB C = $100, w = r = $10

AB C = $140, w = r = $10

AB C = $80, w = r = $10

AB* C = $100, w = $5, r = $10


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Optimal Combination of Inputs for

Minimizing costs or Maximizing output

MRTS = w/r

Isocost Lines

AB C = $100, w = r = $10

AB C = $140, w = r = $10

AB C = $80, w = r = $10


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Product lines

A product line shows the movementfrom one isoquant to another as we

change both factors or a single factor.

The product line describes thetechnically possible alternative paths of

expanding output and what path

actually be chosen by the firm willdepend on the prices of factors.


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The product curve passes through the origin if

all factors are variable. If only one factor isvariable (the other being kept constant) the

product line is a straight line parallel to the

axis of variable factor. The K/L ratio

diminishes along the product line.


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Optimal input combinations:

Slope of isoquant = Slope of isocost line

(absolute)Slope of isoquant = (absolute) Slope of isocost line

MRTS = w

r

Since MRTS = MPL/ MPK

MPL = w

MPK r

MPL = MPKw r

If MPL = 5, MPK =4, and w = r

MPL > MPK

w r


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Profit Maximization

MRP(input) = MRC(input)

with constant input prices

MRP(input) = input price

To maximize Profits:

MRPL= w = (MPL)(MR)

MRPK= r = (MPK)(MR)

MPL = MPKw r

MPL = w

MPK r


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Optimal Combination of Inputs

Effect of a Change in Input Prices


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Returns to Scale

Production Function Q = f(L, K)

Q = f(hL, hK)

If = h, constant returns to scale.

If > h, increasing returns to scale.

If < h, decreasing returns to scale.


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Returns to Scale

ConstantReturns to

Scale

IncreasingReturns to

Scale

DecreasingReturns to

Scale


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Empirical Production Functions

Cobb-Douglas Production Function

Q = AKaLb

If a + b = 1, constant returns to scale.

If a + b > 1, increasing returns to scale.If a + b


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Q=ALa Kb

Where Q= Manufacturing output

L= quantity of labour employed

K= quantity of capital employed

A, a, b are constant or parameter of funcions.

1. the sum of exponents of factors i.e. a+b measures returns toscale. If

If a + b = 1, constant returns to scale.

If a + b > 1, increasing returns to scale. If a + b


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2. Average and marginal product of factors

depend upon the ratios in which factors are

combined for the production of a

commodity.

APL=Q/L=ALaKb/L=A(K/L)b

Thus ,average product of labour depends on

the ratios of the factor(K/L) and does not

depend upon the absolute quantities of thefactor used.


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CobbDouglas Production Function

and Output Elasticities of Factors

The exponents of labour and capital in

cobb-douglas production function measure

output elasticities of labour and capital.

Output elasticity of a factor refers to the

relative or percentage change in output

caused by a given percentage change in a

variable factor, other factors and inputs

remaining constant.


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Elasticity of Technical Substitution (between

factors) =

Proportionate change in factor proportions

K/L

Proportionate change in MRTS between

labour and capital


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Exercise

Do the following production functions have constant,

increasing or decreasing returns of scale? ( K, L, M

are inputs)

a. Q = 0.5X + 2Y + 40Z

b. Q = 3L + 10K + 500

c. Q = K0.3L0.5

d. Q = 4A2+ 6B2+ 8AB

e. Q = 10L 0.5K 0.6


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A. constant returns to scale.

B. constant returns to scale.

C. Decreasing returns to scale

D. increasing returns to scale.

E.increasing returns to scale


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Exercise

Medical Testing Labs, Inc., provides routine testing servicesfor blood banks in the Los Angeles area. Tests are supervisedby skilled technicians using equipment produced by twoleading competitors in the medical equipment industry.Records for the current year show an average of 27 tests per

hour being performed on the Testlogic-1 and 48 tests perhour on a new machine, the Accutest-3. The Testlogic-1 isleased for $18,000 per month, and the Accutest-3 is leased at$32,000 per month. On average, each machine is operated 25eight-hour days per month.

a. Does Medical Testing Lab usage reflect an optimal mix oftesting equipment?

b. If tests are conducted at a price of $6 each while labor and allother costs are fixed, should the company lease moremachines?


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a) (27*25*8)/ 18000 = (48*25*8) / 32000 = 0.3

In both instances, the last dollar spent on each machineincreased output by the same 0.3 units, indicating anoptimal mix of testing machines.

b) For each machine hour, the relevant question isTestlogic-1

27 (25 8) $6 > $18,000 or $32,400 > $18,000.

Accutest-3

48 (25 8) $6 > $32,000 or $57,600 > $32,000.In both cases, each machine returns more than its

marginal cost (price) of employment, and expansionwould be profitable.


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The marginal product of labor for international

trading is given by the equation

MPL = 10K0.5/L0.5Currently the firm is using 100 units of capital and

121 units of labor. The capital stock is constant but

the labor can be varied. If the price of labor is 10/-

and price of output is Rs. 2/- per unit, is the firm

operating efficiently in the short run? If not,

determine the optimal rate of labor input.

Exercise


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Answer: not optimally, L = 400


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The production function is : Q = 20K0.5L0.5

With marginal product functions

MPK= 10L0.5/K0.5MPL= 10K

0.5/L0.5

If the price of capital is Rs. 5/- and price of labor is Rs.4/- per unit, determine the expansion path for the firm.

The firm currently is producing 200 units of output perperiod using input rates of L = 4 and K =25. is this anefficient input combination? Why or why not? If not,determine the efficient input combination for producingan output rate of 200.

If the price of labor increases from Rs 4 to Rs 8 per unit,determine the efficient input combination for an outputrate of 200. What is the capital labor ratio now?


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Answer: K= 0.8L,

L= 11.18 and K= 8.94,

L = 7.905 and K = 12.65


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Suppose the price of one unit of labor is $10

and price of one unit of capital is $2.50. Use this information to determine the isocost

equations corresponding to a total cost of $200 and$500.

Plot these two isocost lines on a graph

If the price of labor falls from $10 per unit to $8 perunit, determine the new $500 isocost line and plot it

on the same diagram used in part (b) Answer: K = 2004L and K = 80- 4L, K =

2003.2L


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4. Given the production function Q = 30K0.7L0.5

and input prices r = 20 and w = 30.

Determine an equation for the expansion path

What is the efficient input combination for an

output rate of Q = 200? For 500?

Answer: K = 2.1L, for 200: L = 3.15 and K = 6.62,for 500: L = 6.765 and K = 14.207

Th d t f t t t l tifi d bli


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The revenue dept. of a state govt. employs certified publicaccountants (CPAs) to audit corporate tax returns and

book keepers to audit individual returns. CPAs are paid

$31200 per yr, while the annual salary of a bookkeeper is$18200. Given the current staff of CPAs and bookkeepers,a study made by the depts economist shows that addingone year of a CPAs time to audit corporate returns resultsin an additional tax collection of $52000. In contrast, an

additional bookkeeper adds $41600 per year in additionaltax revenue. If the depts objective is to maximize tax revenue collected, is

the present mix of CPAs and bookkeepers optimal? Explain

If the present mix of CPAs and bookkeepers is not optimal,

explain what re-allocation should be made. That is, should thedepartment hire more CPAs and fewer bookkeepers or viceversa.

Answer: CPAs1.67 and bookkeepers2.29

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