CHAPTER SIX THE THEORY AND ESTIMATION OF...

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CHAPTER SIX THE THEORY AND ESTIMATION OF PRODUCTION

For the manager of a profit-maximizing firm, seeking the maximum total

revenue may not ensure attaining maximum profits. Profit is the difference

between total revenues and total costs.

Total revenue depends on price and quantity.

For a given price, under competition, a successful manager should aim at

two intermediate targets leading to the final target of profit maximization:

o First, in the short run, to choose the optimal quantity of each input

o Second, in the long run, to choose the optimal size of the firm or the

production capacity that minimizes the cost

The manager will be able to tackle these targets only if he has enough and

accurate knowledge concerning the production function or the input output

relationships, and that is the focus of this chapter.

The first part of the chapter introduces the production function in short run

and long run, while the second part of the chapter is concerned with the

estimation of the production function.

The Production Function:

What level of output should the firm produce depends on the production

function it has, among other things.

The production function is the technical relationship, which shows the

maximum attainable production levels using different combinations of input

at a given state of technology within a given period of time.

Mathematically, the production function can be expressed as

Q = f (L, K, N, T, …)

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Where Q is level of output produced; L is the number workers (including

entrepreneurship); K is the capital; N is land; T is the state of technology;

and … refers to other inputs used in the production process

The relationship between inputs and output assumes:

a. Fixed state of technology

b. Efficient use of input combinations

c. Given time period

For simplicity we will often consider a production function of two inputs:

labor and capital. Labor and capital are both composite inputs that include

all other factors of production, so, the production function is normally

written in the follows implicit form:

Q = f (L, K)

The above equation tells us that the first partial derivatives of output, with

respect to each of the inputs, are positive, i.e., Q has a positive relationship

with L and K. The second partial derivatives of output with respect to each

of the inputs are negative, indicating that the production function has some

maximum point.

Production depends on the time-frame in which the firm is operating (short

run and long run)

The short-run production function shows the maximum quantity of good

or service that can be produced by a set of inputs when at least one of the

inputs used remains unchanged.

The long-run production function shows the maximum quantity of good

or service that can be produced by a set of inputs, assuming the firm is free

to vary the amount of all the inputs being used.

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Short-Run Analysis of Total, Average, and Marginal Product We use alternative terms in reference to inputs such as factors, factors of

production, and resources

Also alternative terms are usually used in reference to outputs such as

quantity (Q), total product (TP), and product

The relationship between output and the quantity of labor employed,

assuming fixed capital, can be described using the following three concepts:

o Total product

o Marginal product

o Average product

Total Product:

Total product is the total output produced in a given period.

In the short run, the total production curve shows the maximum output

produced using a certain set of variable inputs (as labor and row

materials…) in addition to one or more of fixed inputs (as the size of the

plant, equipments, area of land).

However, because of our assumption for simplicity, the production function

depends only on labor and capital. Hence, to increase output in the short

run, a firm must increase the amount of labor employed.

The total product curve shows how total product changes with the quantity

of labor employed

The area below TP curve is attainable at every level of labor, while the area

are above TP is unattainable.

Example:

To understand the nature of the relation between the different measures

of production in the short run, let us take a simple example of a small

farm where capital (area of the farm, water well, and equipments) is

fixed, and the number of workers is the only variable input.

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Under these set of assumptions, total production (TP), marginal Product

(MP), and average product (AP), may be represented by the following

hypothetical values:

Number of Workers

(L)

Total Product

(TP)

Marginal Product of labor

(MPL)

Average product of labor (APL)

0 0 --------- --------- 1 8 8 8 2 18 10 9 3 29 11 9.67 4 39 10 9.75 5 47 8 9.4 6 52 5 8.67 7 56 4 8 8 52 -4 6.5

From the table above, you may notice that total product increases as more

workers are hired. First, total product increases at high speed making big

jumps, then it slows down but still in creasing as it approach its maximum,

then total product starts to decrease as more labor are employed.

In scientific terms we may describe the behavior of the total product in the

short run by saying that, TP increases first at an increasing rate to reach an

infliction (turning) point, after that it keeps increasing but at a decreasing

rate until it reaches its maximum level, then it starts to fall.

The following graph reflects these phases of production:

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The Marginal Product:

The marginal product of labor (MPL) is the change in output (or Total

Product) that results from a one-unit change in the variable input (quantity

of labor employed), or one additional hour of work, with all other inputs

remaining the same.

LQ

LTP

labor ofquantity the in Changeoutput total in ChangeMPL ∆

∆=

∆∆

==

Graphically, MPL measures the slope of the total product curve.

As you see in the graph above, the slope of the TP curve starts very close

to zero at the origin, then increases (production speeds up) to reach its

maximum at the infliction (turning) point (point a) on the TP curve, where

MPL curve reaches its maximum. The slope of TP curve decreases

gradually (production slows down) to reach zero at L2 as MPL falls to

TP , AP , MP , MP < AP MP > 0

TP , AP , MP then , MP > AP

AP

Stage III Stage II

Stage I

a

MPL

L1 L2 0

TP

TP

L

TP , AP , MP , MP < AP MP < 0

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intersects the L axis. Total product starts to fall from there on as the slope

of the MPL curve become negative.

From the above table notice how the Marginal product of labor increases to

reach its maximum of 11, then falls after that to take a negative value

where total product starts to fall.

Almost every production process has two features: initially, increasing

marginal returns (IMR), and eventually diminishing marginal returns (DMR).

Initially increasing marginal returns

o When the marginal product of a worker exceeds the marginal

product of the previous worker, the marginal product of labor

increases and the firm experiences increasing marginal returns.

o Increasing marginal returns arise from increased specialization

and division of labor.

Then, diminishing marginal returns o Diminishing marginal returns arises from the fact that employing

additional units of labor means each worker has less access to

capital and less space in which to work.

o Diminishing marginal returns will start to take effect, when MP

starts to decrease.

o Diminishing marginal returns are so pervasive that they are

elevated to the status of a “law.”

o The law of diminishing returns states that as a firm uses more

of a variable input with a given quantity of fixed inputs, the

marginal product of the variable input at some point will eventually

falls, where each additional unit contribute less production than

the preceded unit”.

o The law of diminishing returns is a short run concept that

describes the change in the marginal product of the variable input.

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It is important to keep in mind that all units of the variable input are

assumed of equal productivity, and the only reason for variations in its

marginal product (productivity) can be attributed to its order in utilization in

the production process.

Referring to our previous example, at some early stage you may notice

increasing returns, where the MP curve has a positive slope. The reason

here is clear because the larger the number of workers is the higher the

productivity of individual workers due to specialization and teamwork

privileges.

As more workers are added to the same quantity of the fixed input, at some

point diminishing returns will be in effect due to crowdness in the work

place, and inadequacy of the fixed input, as the number of workers

increase each worker will have less fixed input.

In general:

o When MP is , firm is experiencing increasing marginal returns.

o When MP is but positive, firm is experiencing diminishing

(decreasing) marginal returns.

o When MP is and negative, firm is experiencing negative

marginal returns.

Example:

Given a production function:

Q = 15X – 5X2,

Find the level of input at which diminishing marginal returns (DMRs)

starts

MP = dQ/dX = 15 – 10X

Set MP = 0 and solve for X

15 – 10X = 0 ⇒ X = 15/10 = 1.5

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The Average Product: The average product of labor (APL) refers to the share of each worker in

the total production.

It is equal to total product divided by the quantity of labor employed. It is

the total Product per unit of input used.

LQ

LTPAP L

L ==

Example:

Given a production function:

Q = 15X – 5X2, then X515XX5

XX15

XQAP

2

−=−== = 0

⇒ X = 15/5 = 3

(Notice that AP reach 0 at twice L as MP)

The relationship between TP and MP

Because of IMR, TP increases initially at increasing rate. Then, because of

DMR, TP increases at decreasing rate.

The point that TP change its pace from increasing at increasing rate to

increasing at decreasing rate (i.e., the point where DMR start its course) is

called the turning point. When TPL is at its turning point MPL is at its

maximum,

1. If MPL is increasing and positive, TPL is increasing at increasing rate

2. If MPL is decreasing and positive, TPL is increasing at decreasing rate

3. If MPL = 0, TPL is at its maximum

4. If MPL < 0, TPL is decreasing

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The relationship between AP and MP When MP > AP (MP Curve above AP Curve), then AP is rising

o If your marginal grade in this class is higher than your grade point

average, then your GPA is rising

When MP < AP (MP Curve below AP Curve), then AP is falling

o If your marginal grade in this class is lower than your grade point

average, then your GPA is falling

When MP = AP, (MP Curve intersect AP Curve) then AP is at its maximum

o If your marginal grade in this class is similar to your grade point

average, then your GPA is unchanged

The Three Stages of Production in the Short Run:

The relationship between TP, MP, and AP can be used to divide the SR

production function into three stages of production.

Stage I: o Starts from zero units of the variable input to where AP is maximized

(where APL = MPL).

o At this stage, AP is rising.

o In our example, stage I ends where four workers are being hired at

L1 on the figure above.

o For a profit-maximizing firm, it is irrational to limit production to any

level within the first stage. The rising AP throughout this stage

causes the average cost to fall and profits to rise as production

expands.

Stage II: o From the maximum AP to where MP=0 (or TP is maximized).

o At this stage, AP declining but MP is positive.

o In our example, Stage II is between L1 and L2 levels of employment.

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o This is the rational stage of production, over which the firm manager

has to figure out the optimal number of workers that maximizes

profits.

Stage III: o From where MP=0 (or TP is maximum) onwards.

o At this stage, MP is negative or TP is declining.

o In our example, Stage III starts at L2 (MPL = 0) and above.

o Production in this stage is also irrational because the firm incurs

higher costs to hire more workers; while the total revenue is falling as

TP decreases.

At which stage should the firm operate? According to economic theory, in the short run, firms should operate in

Stage II. Specialization and team work continue to add more output when

additional variable input is used. Fixed input is being properly utilized ⇒

efficient use of resources

Why not Stage I?

o Stage I indicates that the fixed input is underutilized since AP as

more and more of the variable input is used i.e., can increase output

per unit by increasing the amount of the variable input ⇒ inefficient

use of resources

Why not Stage III?

o Firm uses more of its variable inputs to produce less output. Fixed

input is over utilized or overused ⇒ inefficient use of resources

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The Partial Elasticity of Production:

In the short run, it is possible to measure the elasticity of production with

respect to each of the variable inputs.

In a similar way to the concept of demand elasticity, production elasticity measures the responsiveness of production to changes in one variable

input when other inputs are held constant.

It measures the percentage change in the quantity produced due to one

percent change in labor input.

It is known as the partial elasticity of production.

The elasticity of production with respect to labor may be presented in the

following formula:

L

LL AP

MP

LQLQ

QL

LQ

LL

QQ

LL

QQ

L%Q%E =∆

∆

=×∆∆

=∆

×∆

=∆

∆

=∆∆

=

By rearranging the basic elasticity formula, the elasticity of production

turned to be equal to the ratio of the marginal product to the average

product of the variable input.

Look at the graphical illustration of the three stages of production; compare

the location of the marginal and the average product curves. You will easily

notice how the elasticity of production changes across the three stages of

production.

In the first stage, when MPL > APL, then the labor elasticity, EL > 1.

o A one percent increase in labor will increase output by more than 1

percent.

In the second stage, when MPL < APL, the labor elasticity 0< EL <1.

o A one percent increase in labor will increase output by less than 1

percent.

In the third stage of production, when MPL is negative, EL< 0.

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Using the above form of partial elasticity of production, the firm manager,

my easily find out the stage of his firm production. All he needs is

estimating the production functions and the quantity of the major variable

input used in production.

The production elasticity of capital has the identical in form, except K

appears in place of L.

Example:

Consider the production function, Q = 100 L – L2, find the stage of

production if the firm uses 20 workers.

Solution

MPL = 100 – 2L = 100 – 2 (20) = 60

APL = 100 – L = 100 – 20 = 80

EL = MPL / APL = 60 /80.

Since EL is positive and less than one, the firm is producing in stage II.

Example:

If Q = 9L2 – L3

(Not that power 3 determines 3 stages)

a. Find the ranges of the 3 stages of production

b. Find at which stage the firm is operating if L= 5, and L = 3

c. Find L value at the starting of DMRs

Solution

a. Stage I: Between L= 0 and L where AP is maximized

232

LL9LL

LL9

LQAP −=−==

dAP/dL = 9 – 2L = 0 ⇒ L = 9/2 = 4.5

Stage I from 0 to 4.5

Stage II: From L= 4.5 to L where MP = 0

MP = dQ/dL = 18L – 3L2 = 0 divide by 3L

6 – L = 0 ⇒ L = 6

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Stage II from 4.5 to 6

Stage III: 6 onward

b. Suppose L = 5, at which stage the firm is operating?

2015

5)5(9)5(3)5(18

LL9L3L18

APMP

2

2

2

2

=−−

=−−

= < 1 ⇒ at stage II

Suppose L = 3, at which stage the firm is operating?

1827

3)3(9)3(3)3(18

LL9L3L18

APMP

2

2

2

2

=−−

=−−

= > 1 ⇒ at stage I

c. Diminishing returns starts when MP reaches its maximum; i.e., when

dMP/dL = 0

MP = 18L – 3L2

dMP/dL=18 – 6L = 0 ⇒ L=18/6=3

Example:

Q = 8L – 0.5L2

(Not that power 2 determines only 2 stages of production: II and III)

Law of diminishing returns starts immediately when production begins

Stage II: AP = 8 -0.5L

dAP/dL = -0.5 ⇒ L= 0

Stage II starts where L = 0

Stage III: MP = dQ/dL = 8 – L = 0

⇒ L = 8

Stage III starts at 8

To find AP = 0

8 – 0.5L = 0 ⇒ L = 8/0.5 = 16

Notice that AP = 0 is at twice level

of L than when MP = 0

Exercise:

If Q = 50L + 6L2 - 0.5L3, find the 3 stages of production.

AP

L

MP

MP, AP

II III 0

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Optimal level of the Variable input: Given that Stage II is the best for profit-maximizing firm, then what is the

optimal level of variable input should the firm use? In other words, at which

point on production function should the firm operate?

The answer depends upon: how many units of output the firm can sell, the

price/ of the product, and the monetary costs of employing the variable

input.

The Optimal quantity of the variable input is the quantity that allows the

firm to maximize its profits.

The question facing the manager is: what is the optimal number of workers?

To make things easy, let us assume that the firm buys its inputs and sell its

products in competitive markets, which means, it can hire any number of

workers at the market going wage (W) and sell any quantity of its product

at the market going price (P).

Now we may present the firm profit function in the following form:

π = TR –TC = PXQ – (FC + WL)

Where: P is the Price of the good which is assumed constant, Q is the total

product, FC is the fixed input cost, W is the labor wage which is assumed

constant, and L is the number of workers.

Now, the question is: how many workers the firm should hire in order to

maximize its profits? To find out the answer, we should take the first

derivative of the profit function with respect to L, set it equal to zero and

solve for the value of L as follows:

WMP*Por0WLQ*P

L L ==−∂∂

=∂π∂

Which says that, the optimal number of workers is that number at which the

value of the marginal product of labor is equal to the market wage rate; or

at which the marginal revenue product of labor (MRPL) equals the

marginal labor cost (MLC).

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Total revenue product (TRP) refers to the market value of the firm’s

output, computed by multiplying the total product by the market price (Q * P)

Marginal revenue product of labor (MRPL) is the change in the market

value of the firm’s output resulting from one unit change in the number of

workers used ( TRP/ L). It can be also computed by multiplying the

marginal product by the product price (MP *P)

Total labor cost (TLC) refers to the total cost of using the variable input,

labor, computed by multiplying the wage rate (which assumed to be fixed)

by the number of variable input employed (W * L)

Marginal labor cost (MLC) is the change in total labor cost resulting from

one unit change in the number of workers used

Because the wage rate is assumed to be constant regardless of number of

inputs used, MLC is the same as the wage rate (w).

o If MRPL > MLC ⇒ π as L

o If MRPL < MLC ⇒ π as L

o If MRPL = MLC ⇒ π is maximum ⇒optimal input level

In summary, A profit-maximizing firm operating in perfectly competitive

output and input markets will be using the optimal amount of an input at the

point in which the monetary value of the input’s marginal product is equal to

the additional cost of using that input (MRP = MLC) or (MRP = wage rate)

As you may notice here again we are comparing marginal revenues and

marginal cost. Therefore, the decision rule is to hire more workers as long

as the value of the production contributed by the additional worker (P*MPL)

exceeds the cost of hiring an additional unit of labor which is equal to the

wage (W) paid to worker, under competition in the labor market.

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Example:

Suppose the firm short run production function has the quadratic form: Q =

100L - L2,

The firm hires any number of workers at a wage of $80, and sells any

quantity of its output at a price of $2. Find how many workers should this

firm hire to maximize its profits.

Solution:

The optimal number of workers is reached when P*MPL = W

MPL = 100 - 2L

P*MPL = 2 * (100 – 2L) = 200 – 4L = 80

P*MPL = W ⇒ 200 – 4L =80 ⇒ 120 = 4L ⇒ L = 30 workers

If the wage rate falls to $ 60, the optimal number of workers increases to 35

workers, which represents a movement along the MRPL curve.

The MRPL curve thus shows the negative relationship between the wage

rate and the number of labor demanded.

At a constant wage rate, the higher the price of the output is the greater the

number of workers demanded would be, for this reason, the demand for

labor (MRPL) is considered a derived demand from the demand for the

output.

Exercise:

Q = 50L + 6L2 - 0.5L3

If the price is of the good is BD10 and the wage of each worker is BD400,

what is the level of production where profit will be at maximum level.

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The Optimal Mix of Variable Inputs: Suppose the firm production function has more than one variable input in

the short run. In order to maximize profits, the manager has to choose the

quantities of each input that will minimize cost.

Let the production function be as follows:

Q = f (X, Y, K),

Where X and Y are the two variable inputs

Let CX is the cost of X and CY be the cost of Y, while PQ is the price of the

firm output.

For simplicity let us assume that the firm buys its inputs and sell its product

in competitive markets, where the firm can buy or sell any quantities at a

constant price.

Now by taking the first derivative of the profit function with respect to each

of the variable inputs and equate it to zero we get:

(7) C

MP C

MPas written-re can equation This

(5) CC

MPMP

(4)by (3) Divide(4) C MP *P(3) C MP *P

(2) and (1) From

(2) 0 C YQ*P

y

(1) 0 C XQ*P

X

)Y*C X*(C - Q*P TC - TR

Y

Y

X

X

Y

X

Y

X

YYQ

XXQ

YQ

XQ

YX

=

=

=

=

=−∂∂

=∂π∂

=−∂∂

=∂π∂

+==π

Thus, the optimal condition for a profit-maximizing firm employing two

variable inputs X and Y is attained where the ratio of the marginal products

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of the variable inputs is equal the ratio of their costs. Or, the marginal

product of X to its cost must be equal to the marginal product of Y to its

cost.

In other words, the marginal product per the amount of money spent on X

must be equal to the marginal product per the amount of money spent on Y.

Political and economic risk factors may outweigh this relationship

To get a better understanding of this condition, let us solve the following

simple example:

Example:

o Suppose you are the production manager of a company that makes

computer parts in Malaysia and Algeria.

o At the current production levels and inputs utilization you found that:

Malaysian marginal product of labor (MPM ) =18 Units

Algerian marginal product of labor (MPA ) = 6 Units

Wage rate in Malaysia Wm = $ 6/hr

Wage rate in Algeria WA = $ 3/hr

o In which country should the firm hire more workers?

Solution

o Looking at the wage rates you might be tempted to hire more workers

and expand production in Algeria, where wages are relatively lower.

o However, production theory suggests that the firm should not only look

at input’s cost but also to the MP of each input relative to the cost.

o By examining the marginal product per dollar in each country, you will

find that:

23$6

WMP3

6$18

WMP

A

A

M

M ==>== ;

Which means that: an additional dollar spent on labor in Malaysia would

yield 3 units, but would yield only 2 units if spent in Algeria.

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o Therefore, the firm’s profit will rise as it shifts reallocate its labor budget

away from Algeria toward Malaysia. Shifting one dollar from hiring labor

in Algeria would reduce production by 2 units, if this dollar is spent on

hiring more labor in Malaysia it would generate 3 units.

o By this move, the firm would make a net gain of one unit. The firm would

continue to cut down its number of workers in Algeria and expand its

employment in Malaysia until equality is restored.

o As the firm increases employment in Malaysia, MPM would start to fall

following the law of diminishing returns. At the same time as the firm

reduces employment in Algeria, the law of diminishing returns will start

to raise MPA until eventually equilibrium is reached, and profit is

maximized.

Example:

o Suppose labor and capital are both variable inputs and some other input

such as land is fixed, and suppose that

MPL = 12 units, MPk =24, w =$6 and r = $8,

Solution

o MPL /w = 12/6 =2 ⇒ spending one additional dollar on labor gives two

units of output

o MPK /r = 24/8 = 3 ⇒ spending one additional dollar on capital gives

three units of output

o So use more capital and less labor since capital is cheaper per dollar

spent than labor (capital is more productive)

o Bur, as more capital is used its MP , and as less labor is used its MP .

o This will continue until the two ratios are equal.

o Suppose MPL to 15 and MPK to 20, then

MPL/w = 15/6 = 2.5 and MPK/r = 20/8 = 2.5

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Example:

o A firm has the following production function:

Q = 20 E – E2 + 12 T – 0.5 T2;

o Where E is the number of engineers and T is the number of technicians.

o The average annual salary for engineers is B.D. 9600; and for

technicians is B.D. 4800.

o The firm budget for hiring engineers and technicians is B.D. 336000 per

year.

o Calculate the optimal number of engineers and technicians.

Solution

o The budget constraint: 336000 = 9600E + 4800T (1)

o The optimization condition:

4800

T129600

E220C

MPC

MP

T

T

E

E −=

−⇒= (2)

o So, 10 – E = 12 - T or E = (T - 2) (3)

o By substituting (3) in (1) then;

3360 = 96E + 48T

3360 = 96 (T-2) + 48T

3360 = 96T – 192 +48T

144T = 3552

T = 3552/144 = 24.7

E = 24.7 – 2 = 22.7

Budget Constraint = 9600 (22.7) + 4800 (24.7) =336000

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The Long-Run Production Function As you know by now, the long run is a period of time long enough to allow

the firm to change all its inputs. Effectively, all inputs are variable.

As the firm increases all its inputs in the long run, it actually changes the

scale of its production activity.

The total elasticity of production, the increase in production in response to

the firm proportional increase in the scale of the production process is

called returns to scale.

If all inputs into the production process are doubled, three things can

happen:

1. Output can be more than double, increasing returns to scale. A larger scale of production allows the firm to divide tasks into more

specialized activities, thereby increasing labor productivity. It also

enables the firm to justify the purchase of more sophisticated (hence,

more productive) machinery. These factors help in explaining why

the firm can experience increasing returns to scale.

2. Output can exactly double, constant returns to scale. 3. Output can be less than double, decreasing returns to scale.

Operating on a larger scale might create certain managerial

inefficiency (e.g., communication problems, bureaucratic red tape)

and hence cause decreasing returns to scale.

Graphically, the returns to scale concept can be illustrated using the

following graphs.

Q

X,Y

IRTS Q

X,Y

CRTSQ

X,Y

DRTS

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One way to measure returns to scale is to use a coefficient of output

elasticity:

inputs all in %Q %EQ ∆

∆=

o If EQ > 1, production function shows increasing returns to scale (IRTS)

o If EQ = 1, production function shows constant returns to scale (CRTS)

o If EQ < 1, production function shows decreasing returns to scale (DRTS)

Example:

If Q = 5L + 7K; and L = 10 & K = 10

Q1 = 5(10) + 7 (10) = 120 units

Now if each input increases by 25%, then L = 12.5 & K = 12.5

Q2 = 5(12.5) + 7 (12.5) = 150 units

% Q = (150 - 120)/120= 25%

A 25% increase in L & K led to a 25% increase in Q ⇒ CRTS

Example:

Q = 50X + 50Y +100

X = 1, Y = 1 ⇒ Q 50(1) + 50 (1) +100 = 200

If X =2, Y = 2 ⇒ Q 50(2) + 50 (2) +100 = 300

% Q = (300 - 200)/200 = 50%

% in all inputs = 100%

⇒ EQ = 50%/100% = 0.5 < 1 ⇒ DRTS

Example:

Q = 50X2 + 50Y2

X=1, Y=1 ⇒Q = 50+50 =100

X=2, Y=2 ⇒Q = 200+200 =400

% Q = (400 - 100)/100 = 300%

⇒ EQ = 300%/100% = 3 > 1 ⇒ IRTS

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Forms of Production Functions 1. Short run: existence of a fixed factor to which is added a variable factor

o One variable, one fixed factor: Q = f(L)K

2. Increasing marginal returns followed by decreasing marginal returns

o Cubic function: Q = a + bL + cL2 – dL3

3. Diminishing marginal returns, but no Stage I

o Quadratic function: Q = a + bL - cL2

4. Power function: Q = aLb

If b > 1, MP increasing

If b = 1, MP constant

If b < 1, MP decreasing

o Can be transformed into a linear equation when expressed in

logarithmic terms

logQ = log a + bLog L

5. Cobb-Douglas Production Function:

o Cobb and Douglas have introduced their production function in

1928.

o The Cobb-Douglas production function is a non-linear power or

exponential function in the following form: Q = a Lb K1-b

o Both capital and labor inputs must exist for Q to be a positive

number

o b and 1- b are the elasticities of production with respect o labor and

capital.

o b and 1- b are constants. b + b -1 = 1 ⇒ CRTS

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o In later version, Cobb-Douglas relaxed this requirement and

rewrote the equation as follows Q = aLbKc

o Can be increasing, decreasing, or constant returns to scale

b + c > 1, IRTS

b + c = 1, CRTS

b + c < 1, DRTS

o Permits us to investigate MP for any factor while holding all others

constant

o Each of the coefficients is usually less than one showing that the

production takes place in stage two.

o Each exhibits diminishing marginal returns

o b and c represents the partial elasticity of production:

b = MPL/APL, c = MPK/APK

So b can be found using MPL and APL. Same thing for C.

o Can be estimated by linear regression analysis

log Q = log a +b log L + c log K

o Can accommodate any number of independent variables

Q = a X1b X2

c X3d …. Xk

n

o Does not require that technology be held constant

o Shortcomings:

Cannot show MP going through all three stages of production.

Cubic function is necessary

Cannot show a firm or industry passing through increasing,

constant, and decreasing returns to scale

Specification of data to be used in empirical estimates`

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Example:

Q = 75 X0.25 Y0.75

X=1, Y=1 ⇒Q = 75*1*1 =75

X=2, Y=2 ⇒Q = 75*1.19*1.68 = 149.94 = 150

% Q = (150 - 75)/75 = 100%

⇒ EQ = 100%/100% = 1 ⇒ CRTS

For Cobb-Douglas production function

o Sum of Exponents = 1 ⇒ CRTS

o Sum of Exponents > 1 ⇒ IRTS

o Sum of Exponents < 1 ⇒ DRTS

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Choosing the Optimal Capacity: (Reading)

Production levels do not depend on how much a company wants to

produce, but on how much its customers want to buy.

Although the manager may be well informed about the three stages of

production for his firm, in the short rum, he might realize that the market

demand is consistently lower than expected.

Capacity Planning: refers to the planning the amount of fixed inputs that

will be used along with the variable inputs. Good capacity planning

requires:

o Accurate forecasts of demand

o Effective communication between the production and marketing

functions

Suppose the firm is forced to produce in the first stage as a result of the

inadequate demand in the short run. In this case the firm is underutilizing

its capacity or its fixed inputs, in other words, the firm would have excess or

idle capacity. Fixed costs are the cost of fixed inputs, that part of the firm

total cost, which is independent of the level of production. A firm producing

in stage one with some idle capacity, would have high average cost (per

unit cost), and therefore, will not maximize profits.

In the long run, such firm should consider reducing its production capacity

to the optimal size that enables it to maximize profits

By the same token, a firm facing a growing demand beyond its production

capacity should consider expanding its capacity in the long rune to meet

the market demand while producing in the rational range that would allow

maximum profits.

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Statistical Estimation of Production Functions (Reading) Model specification;

Specifying the model means to choose the exact explicit equation that

would best illustrate the production function of the firm. To make the

appropriate choice, the researcher may plot output against each of the

independent variables to find out in which stage the firm is producing. It is

possible to estimate the production function in the short run when at least

one input is constant, if it takes a form that shows all the three stages of

production, as in the case of Q = 200 + 0.2L + 0.25 L2 – 0.1L3,

or if it shows only two stages as in the case of

Q = 200 +0.2L - 0.25 L2

which shows only the second and the third stages of the production.

The most popular form of the production function is the power function. In

its implicit form, the power function with n variables is:

Q = (AX1bX2

c……..Xnm),

which has the following appealing properties:

1. It can be transformed into a log-linear form that suits the use of LS

method for linear regression analysis.

2. In the short run, it allows the estimation of the marginal product of one

variable holding the others constant.

3. In the long run, it can be used to estimate returns to scale, as all inputs

are variable.

4. The powers measure the partial elasticity of production with respect to

each variable input. If b>1, then marginal product exceeds the average

product and the later is rising in the first stage. If 0<b<1, the marginal

product is less than the average product, and the later is falling in the

second stage. If b<0, the marginal product is negative and the

production takes place in the third stage.

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Data measurement problems; 1. In studies at the micro levels, i. e., at the firm or a group of firms’ level,

researchers depend on firm’s records as the main source of data.

However, in studies at the macro level, sectoral or aggregate economy

level, data will be obtained from published national accounts statistics

(GDP, GDP by sector, Employment, Investment, Inflation rate, the stock

of real capital….).

Gathering data for aggregate functions can be difficult.

2. For a firm producing one output, the output Q is normally measured in

physical units (tons, barrels, ..). But for a firm producing more than one

product with different measurement units, or in the case of macro

studies that use aggregate data of output, the output in physical units

should be converted into values using per unit costs or market retail or

whole sale products prices. However, changes in these values should

reflect the changes in the quantities demanded of the products.

Therefore, such values must be deflated annually to eliminate any

inflation distortion from the output data.

3. Inputs should be measured as “flow” rather than “stock” variables, which

is not always possible.

o Usually, the most important input is labor.

o Most difficult input variable is capital.

Inputs should be measured as “flow variables” (for instance, Gallons

per month) to mach the way in which output is measured. However,

sometimes input data may not be available as flows, particularly in

the case of capital, where data on the flow of capital used in the

production process might not be readily available. Instead, annual

depreciation of capital may be used. But again depreciation taken

from the firm books is not an accurate measure of the capital input

used in production; it is always based on some accounting

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techniques, which are mostly affected by the tax law. Furthermore,

some fixed assets are not depreciated at all like land. As a way out

of these problems, researchers use gross capital stock (the historical

value of plant and equipments) or net capital stock (gross capital

minus accumulated depreciation).

Time Series versus Cross Section Data:

If the researcher uses time series data, it is recommended to include a time

variable or a dummy variable to capture the effect of technological

improvements on output. It is also important to adjust data measured in

monetary units to the rate of inflation.

In cross section studies, researchers should take into consideration the

variations between firms using levels of technology, and the should adjust

variables measured in monetary terms to price and wage differentials

between various geographical regions.

CHAPTER SIX THE THEORY AND ESTIMATION OF...

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