Lecture 4 MGMT 7730 - © 2011 Houman Younessi Supply: Production What is the optimal level of...

Lecture 4

MGMT 7730 - © 2011 Houman Younessi

Supply: Production

What is the optimal level of output?

How should we decide among alternative production processes?

How would investment in production equipment affect labor costs?

Will building extra production capacity increase or decrease costs?

Lecture 4


Supply: Production

1500

Quantity(widgets)

Labor

1200

900

600

300

3 6 9 12 15

Total Output

Lecture 4


LaborCapital (# of

Machines)Output

Average Output(Average Product)

(Q/L)

Marginal Output (Marginal Product) (ΔQ/ΔL)

Marginal Output (Marginal Product) (dQ/dL)

0 5 0 -- -- --

1 5 49 49 49 67

2 5 132 66 83 98

3 5 243 81 111 123

4 5 376 94 133 142

5 5 525 105 149 155

6 5 684 114 159 162

6.67 5 792.6 118.9 162.9 163.34

7 5 847 121 163 163

8 5 1008 126 161 158

9 5 1161 129 153 147

10 5 1300 130 139 130

11 5 1419 129 119 107

12 5 1512 126 93 78

13 5 1573 121 61 43

14 5 1596 114 23 2

15 5 1575 105 -19 -45

Lecture 4


1500

Quantity(widgets)

Labor

1200

900

600

300

3 6 9 12 15

Total Output

A

B

Lecture 4


Average and Marginal output (product) curves

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

-10

-30

-50

10

30

50

70

90

110

130

150

170

A

B

Marginal OutPut (Marginal Product)

dQ/dL

Average Output (Average Product)

Q/L

C

Lecture 4


Average output (product)

L

QQ or in general

X

QPo

Where X is a given input into production

Average output is either zero or positive

Maximum Average output Point A is precisely that: the point where average output is at its highest.

Lecture 4


Marginal output (product)

dX

dQ

X

QMP

X

0lim

Where X is a given input into production

A measure of productivity, it measures the rate of change of output (production) as an extra unit of input is added.

A production reaches maximum productivity with respect to a given input (assuming all other inputs are constant) when marginal output for that particular input is maximized (Point B) .

Lecture 4



X

Q

dX

dQThe difference between and

1500

1200

900

600

300

3 6 9 12 15

Assumes continuous units (e.g. labor in labor-hours 2.73 units is allowed)

Assumes discrete units (e.g. labor in persons n or n+1 but not 1.5n)

Lecture 4



As expected, output is maximized when marginal output (marginal product) is zero (point C).

0

0)(max @

dL

dQ

MPLQ

dL

dQMP

In general Output = Q(L,M,N,X,Y,Z,…)

In our simple case Output=Q(L)

As such:

Lecture 4


Marginal output (product)Exercise:

Show that in all cases, average output is maximum when marginal output is equal to it.

e.g. point A

Average product (average output) is

Maximum average product (average output) is when its derivative is equal to zero

Average product is zero when:

X

QXP )(

0)(0)(1

)/()/()/()(2

X

Q

dX

dQ

X

Q

dX

dQ

X

X

dXdXQdXdQX

dX

XQd

dX

XPd

)()(

0

XPXMPX

Q

dX

dQX

Q

dX

dQ

Lecture 4


The Law of Diminishing Marginal Returns

If equal increments of an input are added whilst at least one other input is held constant, there is a point beyond which the resulting product output would start to decrease.

Note:1. This is an empirical generalization and not a law of nature

2. It is assumed that production technology does not change

Example:

Adding staff but not workstations

Lecture 4


Optimal Resource UtilizationHow much of a resource (input variable) should a firm use?

What do want to maximize:

A. Output

B. Revenue

C. Profit ?Correct answer is: C

Lecture 4


Optimal Resource Utilization

Marginal Revenue Product (MRP) is the amount that an additional unit of the variable input adds to the firm’s total revenue (i.e. if MRPY is the marginal revenue product of input Y):

Y

TRMRPY

However note that :Y

Q

Q

TR

Q

Q

Y

TR

Y

TRMRPY

As such: )( YY MPMRMRP

Therefore: The marginal revenue product of an input equals to that input’s marginal product times the firm’s marginal revenue

Lecture 4


Marginal Product Expenditure (MPE) is the amount that an additional unit of the variable input adds to the firm’s total costs (i.e. if MPEY is the marginal product expenditure of input Y):

Y

TCMPEY

However note that :Y

Q

Q

TC

Q

Q

Y

TC

Y

TCMPEY

As such: )( YY MPMCMPE

Therefore: The marginal product expenditure of an input equals to that input’s marginal product times the firm’s marginal cost

Similarly:Optimal Resource Utilization

Lecture 4


Optimal Resource UtilizationTo maximize profit (globally): MCMR

or:

Q

TC

Q

TR

Multiplying both sides by:Y

Q

Y

Q

Q

TC

Y

Q

Q

TR

Rearranging:

YY MPEMRP

orQ

Q

Y

TC

Q

Q

Y

TR

Lecture 4


Optimal Resource UtilizationExample:

A brewery has fixed plant and equipment. The output of this brewery is shown to relate to its labor usage according to the following equation where L is the number of workers hired per day and Q is the quantity of beer produced.

2398 LLQ

This is a thirsty country and the brewery can sell all the beer it can produce for $20 a gallon.

This is also a developing country so the brewery can hire as many workers as it needs for $40 a day.

How many workers should the brewery hire?

Lecture 4


Optimal Resource Utilization

The brewery's marginal revenue is MR= $20

The brewery's marginal product expenditure is MPEL= $40

Marginal product for labor is: LdL

dQMPL 698

Marginal revenue product for labor is: )( LL MPMRMRP

To maximize, we must have:

16

40)698(20

L

L

MPEMRP LL

Lecture 4


Multi-variable Production Functions

A production function is very rarely a function of only one variable.

For example Q may be a function of: ),,( TMLfQ

Where L stands for cost of Labor, M stands for cost of Machinery and T stands for cost of Transportation

In most cases – certainly all cases covered in this course - the mathematics and formulations are exactly the same as for the case of a single variable instance except that we assume – in turn – that all other variables except the one of our interest are constants. We also use the partial differential notation to indicate multi-variability.

Lecture 4


Multi-variable Production FunctionsExample:

Our brewery has the production function:

How much should they spend on transportation if it costs $200 to transport one gallon of beer?

20

200)2(20

)(2

20

T

T

MPEMPMRMRP

T

T

QMP

MR

TTT

T

2412 2510120 TMLQ

Lecture 4


Isoquants: Choosing Combinations of InputsWhat are the possible combinations of various inputs that are capable of

producing a certain quantity of output?

50

Capital

Labor6

40

30

20

10

1 2 3 4 5

100

200

A

B

Capital/Labor Isoquants

C300

7

Lecture 4


Marginal Rate of Substitution

The rate at which one input can be substituted for another in such a way that the output remains constant

2

1

X

X

Change in one input

Change in another input

1

2

1

2

dX

dXMRS

X

XMRS

Rate of change of one

over the other input

At the limit

Lecture 4


Marginal Rate of Substitution

Exercise:

Show that the marginal rate of substitution is equal to the ratio of the marginal output (marginal product) of the inputs concerned.

€

MRS =−dX2

dX1

−dX2

dX1

=dX2

dX1

×∂Q∂Q

dX2

dX 1

= −(∂Q / ∂X 1)

(∂Q / ∂X 2)= −

MP1

MP2

By definition:

Therefore:

Lecture 4


Isoquants: Choosing Combinations of InputsIsoquants may have regions of positive slope!

These are regions where the quantities of BOTH inputs must increase in order to maintain a fixed level of production.

For the slope of an isoquant to be positive, it means that the marginal product of one or the other of the inputs must be negative. This is not efficient!

As such we restrict ourselves to operating within the range where the slope of the isoquants are negative.

Such region is called the Economic Region of Production

Lecture 4


50

Capital

6

40

30

20

10

1 2 3 4 5 7

200

300

100

Labor

Economic Region of Production

Economic Region of Production

Lecture 4


Isocosts: Choosing Combinations of Inputs

What combinations of inputs can we obtain for the same expenditure?

In a two input situation, this is very simple, for a fixed outlay (e.g. of money), we will have:

KYPXP YX

Where X and Y represent the amount of each input respectively, and PX and PY represent the unit cost for each unit of each input.

Rewriting this we get: XP

P

P

KY

Y

X

Y

Which is the equation for a straight line called the isocost line of X and Y

Lecture 4


Isocosts: Choosing Combinations of Inputs

Y

X

K/PY

K/PX

Slope = -PX/PY

Lecture 4


Choosing Combinations of InputsWhat point on the isocost line would you pick?

Pick the point that lies on both the isocost curve and the highest isoquant

Y

X

K/PY

K/PX

Q

Lecture 4


Choosing Combinations of Inputs

Analytically:

The Q point that optimizes the input combinations is the point where the isocost line is a tangent to the isoquant.

We know that the slope of the isocost line is:

We also know that the slope of the isoquant curve (at a given point) is:

Therefore the Q point is where the slope of the isocost curve equals the ratio of the marginal products:

In fact in general when more than two inputs are involved:

Y

X

P

P

Y

X

MP

MP

Y

X

Y

X

MP

MP

P

P

n

n

c

c

b

b

a

a

P

MP

P

MP

P

MP

P

MP ...

Lecture 4


Example:

An automobile manufacturer has determined that the quantity of automobiles they manufacture relates to the number (in thousands) of factory workers (F) and the number (in thousands) of office workers (O):

22 5.01200020000 FFOOQ

The monthly salary for an office worker is $4000 and the monthly wage for a factory worker is $2000.

how many office workers and how many factory workers should the company hire if they have allotted $2,600,000 a month to wages?

Lecture 4


F

F

O

O

P

MP

P

MPWe know that:

FF

QMP

OO

QMP

F

O

12000

220000We calculate the marginal products:

2000

4000

F

O

P

Pand that

2000

2000

12000

4000

220000

OF

FO

26000000)2000(20004000

2600000020004000

OO

FO

Substituting:

Therefore:

And as such:60002000

4000600020002600000

OF

O

Lecture 4


Example:

ABCO has a production equation as follows:

Q is the output, L is the number of workers and K is the number of machines.

The wages of a worker is $8 per hour and the price of a machine is $2 per hour.

If ABCO produces 80 units of output per hour, how many workers and machines should it use?

Again:

K

K

L

L

P

MP

P

MP

21

)(10 LKQ

and

€

MPL =∂Q∂L

=5(KL)

12

MPK =∂Q

∂K= 5( L

K)

12

Lecture 4


so if:K

K

L

L

P

MP

P

MP then:

2

)(5

8

)(5 21

21

KL

LK

and therefore: LKLK 42

585

as Q=80

164

4

))4((1080 21

LK

L

LL

Lecture 4


Optimum Lot SizeIt is often more economical to produce items in lots. This is because doing so would reduce set up time for equipment and project start-up costs.

For example if it costs S dollars to set up a machine to produce widgets and each widget manufacture cost is W, then if we produce 100 widgets we will have a total cost of manufacture per widget of:

(S+100W)/100

If we produce 1000,000 widgets, then the manufacturing cost per unit will be

(S+1000,000W)/1000000

So, it would pay to have as large a run as possible.

However, things are not as easy. There is a force acting against us

Lecture 4


Optimum Lot Size

We have to carry all those widgets in inventory until we need them, which will cost money. Let us say the cost of carrying a widget in inventory is B.

What should be our production lot size to minimize costs?

We know that total set up cost equals SQ/L where S is the set up cost, Q is the quantity required, and L is the number of widgets produced in a given run.

We wish to find the optimum size of L that minimizes the total cost of

We also know that the cost of holding an item in inventory is B, as such the average cost of holding inventory is BL/2.

Lecture 4


Optimum Lot Size

We need to find a formula for cost.

MI CCC

Therefore:L

SQBLC

2

To minimize:

B

SQL

L

SQB

dL

dC

2

02 2

Lecture 4


Output Elasticity and Scalability

What would happen to output quantity if we doubled all inputs?

A. It would double

B. It would less than double

C. It would more than double

Answer:

IT DEPENDS

Lecture 4


Output Elasticity and Scalability

It depends on a measure called output elasticity.

Output elasticity is the measure of percent change in output resulting from a one percent increase in ALL inputs.

Given a quantity Q say of X and Y, that is Q(X,Y), calculate Q’(1.01X+1.01Y)

If :

1'

1'

1'

Q

QQ

QQ

Q Then the production has constant return to scale

Then the production has increasing return to scale

Then the production has decreasing return to scale

Lecture 4 MGMT 7730 - © 2011 Houman Younessi Supply: Production What is the optimal level of...

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