cost of production / Chapter 6(pindyck)

TOPICS •Production and firm •The production function •Short run versus Long run •Production with one variable input(Labour) •Average product •Marginal product •The slopes of the production curve •Law of diminishing marginal returns •Production with two variable inputs •Isoquant •Isoquant Maps •Diminishing marginal returns •Substitution among inputs •Returns to scale •Describing returns to scale

Production and firm • The theory of the firm that describes

how a firm makes cost-minimizing production decisions and how the firm’s resulting cost varies with its output.

• The production decisions of firms are analogous to the purchasing decisions of consumers, and can likewise be understood in three steps:

1. Production technology

2. Cost constraints

3. Input choices

The production function • Factors of production: inputs that are used into the

production(Ex-labour, capital, and materials)

• The production function:

Q=F(K,L)

• A production function indicates the highest output q

that a firm can produce for every specified

combination of inputs.

• This equation relates the quantity of output to the

quantities of the two inputs, capital (K)and

labour(L).

• inputs and outputs are flows.

• Note that the above equation applies to a given

technology.

• As the technology becomes more advanced and the

production function changes, a firm can obtain more

output for a given set of inputs.

• Production functions describe what is technically

feasible when the firm operates efficiently

Short run versus Long run • Short run: Period of time in which quantities of one

or more production factors cannot be changed.

• Fixed input: Production factor that cannot be varied.

• Long run: Amount of time needed to make all production inputs variable.

Production with one variable input(Labour)

Average product

• The first column shows the amount of labour, the

second the fixed amount of capital, and the third

total output. When labour input is zero, output is also

zero. Output then increases as labour is increased

up to an input of 8 units.

• The fourth column in Table shows the average

product of labour (APL) which is the output per unit

of labour input.

• The average product is calculated by dividing the

total output q by the total input of labour L.

• The average product of labour measures the

productivity of the firm's workforce in terms of how

much output each worker produces on average.

Marginal product

• The fifth column of Table shows the marginal

product of labour (MPL).

• This is the additional output produced as the labour

input is increased by 1 unit.

• For example, with capital fixed at 10 units, when the

labour input increases from 2 to 3, total output

increases from 30 to 60, creating an additional

output of 30 (i.e., 60 - 30) units. The marginal

product of labour can be written as ΔQ / ΔL - in

other words, the change in output ΔQ resulting from

a one-unit increase in labour input ΔL.

The slopes of the production curve

•The total product curve in (a)

shows the output produced for

different amounts of labour input.

•The average and marginal

products in (b) can be obtained

(using the data in Table) from the

total product curve.

•At point A in (a), the marginal

product is 20 because the tangent

to the total product curve has a

slope of 20.

•At point B in (a) the average

product of labour is 20, which is

the slope of the line from the

origin to B.

•The average product of labour at

point C in (a) is given by the slope

of the line OC.

•To the left of point E in (b),

the marginal product is

above the average product

and the average is

increasing; to the right of E,

the marginal product is

below the average product

and the average is

decreasing.

•As a result, E represents

the point at which the

average and marginal

products are equal, when

the average product

reaches its maximum.

•At D, when total output is

maximized, the slope of the

tangent to the total product

curve is zero, as is the

marginal product.

Law of diminishing marginal returns

• The law of diminishing marginal returns states that

as the use of an input increases in equal increments

(with other inputs fixed), a point will eventually be

reached at which the resulting additions to output

decrease.

• The law of diminishing marginal returns applies to a

given production technology.

• Over time, however, inventions and other

improvements in technology may allow the entire total

product curve in to shift upward, so that more output

can be produced with the same inputs.

• Figure in the next slide illustrates this principle.

Initially the output curve is given by O1 but

improvements in technology may allow the curve to

shift upward, first to O2 and later to O3.

•Labour productivity

(output per unit of

labour) can increase if

there are

improvements in

technology, even

though any given

production process

exhibits diminishing

returns to labour.

•As we move from point

A on curve O1 to B on

curve O2 to C on curve

O3 over time, labour

productivity increases.

Production with two variable inputs

• We have completed our analysis of the short-run

production function in which one input, labour, is

variable, and the other, capital, is fixed. Now we turn

to the long run, for which both labour and capital are

variable. The firm can now produce its output in a

variety of ways by combining different amounts of

labour and capital.

Isoquant

Isoquant : the term

may be defined as

Curve showing all

possible

combinations of

inputs that yield the

same output.

Isoquant map •Isoquant map: When a

number of isoquants are

combined in a single

graph, we call the graph an

isoquant map.

•A set of isoquants, or

isoquant map, describes

the firm’s production

function.

•Output increases as we

move from isoquant q1 (at

which 55 units per year are

produced at points such as

A and D), to isoquant q2 (75

units per year at points

such as B) and to isoquant

q3 (90 units per year at

points such as C and E).

•Holding the amount of capital

fixed at a particular level—say

3, we can see that each

additional unit of labour

generates less and less

additional output.

• For example, when labour is

increased from 1 unit to 2

(from A to B), output increases

by 20 (from 55 to 75). However,

when labour is increased by an

additional unit (from B to C),

output increases by only 15

(from 75 to 90).Thus there are

diminishing marginal returns to

labour both in the long and

short run. •There are also diminishing marginal returns to capital. With labour

fixed, the marginal product of capital decreases as capital is increased.

For example, when capital is increased from 1 to 2 and labour is held

constant at 3, the marginal product of capital is initially 20 (75 - 55) but

falls to 15 (90 - 75) when capital is increased from 2 to 3.

Diminishing marginal returns

Substitution among inputs • The slope of each isoquant indicates how

the quantity of one input can be traded off against the quantity of the other, while output is held constant.

• When the negative sign is removed, we call the slope the marginal rate of technical substitution (MRTS).

• marginal rate of technical substitution (MRTS): Amount by which the quantity of one input can be reduced when one extra unit of another input is used, so that output remains constant.

• The marginal rate of technical substitution of labour for capital is the amount by which the input of capital can be reduced when one extra unit of labour is used, so that output remains constant.

•Keeping q= 75 (constant)

• MRTS is equal to 2 when

labour increases from 1

unit to 2 and output is

fixed at 75.However, the

MRTS falls to 1 when

labour is increased from 2

units to 3, and then

declines to 2/3 and to 1/3.

• Clearly, as more and

more labour replaces

capital, labour becomes

less productive and

capital becomes relatively

more productive.

• MRTS is closely related to the marginal products of labour

MPL and capital MPK. To see how, imagine adding some

labour and reducing the amount of capital sufficient to

keep output constant.

• The additional output resulting from the increased labour

input is equal to the additional output per unit of

additional labour (the marginal product of labour) times

the number of units of additional labour:

• Additional output from increased use of labour = (MPL)(ΔL)

• Similarly, the decrease in output resulting from the

reduction in capital is the loss of output per unit reduction

in capital (the marginal product of capital) times the

number of units of capital reduction:

• Reduction in output from decreased use of capital =

(MPK)(ΔK)

• Because we are keeping output constant by moving along

an isoquant, the total change in output must be zero. Thus,

(MPL)(ΔL)+ (MPK)(ΔK)=0

Two special cases • In the first case,

shown in Figure

inputs to production

are perfect

substitutes for one

another.

• Here the MRTS is

constant at all points

on an isoquant. As a

result, the same

output (say q3) can be

produced with mostly

capital (at A), with

mostly labour (at C),

or with a balanced

combination of both

(at B).

Case 1

Case 2 • fixed-proportions production

function: Production function

with L-shaped isoquants, so

that only one combination of

labour and capital can be

used to produce each level

of output.

• The fixed-proportions

production function

describes situations in which

methods of production are

limited.

• When the isoquants are L-

shaped, only one

combination of labor and

capital can be used to

produce a given output (as at

point A on isoquant q1, point

B on isoquant q2, and point C

on isoquant q3). Adding more

labor alone does not

increase output, nor does

adding more capital alone.

Returns to scale

• Returns to scale: Rate at which output

increases as inputs are increased

proportionately.

• Incresing retuns to scale: Situation in which

output more than doubles when all inputs

are doubled.

• Constant retuns to scale: Situation in which

output doubles when all inputs are doubled.

• Decreasing returns to scale: Situation in

which output less than doubles when all

inputs are doubled.

Describing returns to scale When a firm’s production

process exhibits constant

returns to scale as shown by a

movement along line OA in part

(a), the isoquants are equally

spaced as output increases

proportionally.

However, when there are

increasing returns to scale as

shown in (b), the isoquants

move closer together as

inputs are increased along

the line.

Empirical production function

• Cobb-Douglas Production Function

Q=ALaKb

• A , a and b is to be estimated empirically

• a and b are the output elasticity of labor and capital, respectively. These values are constants determined by available technology.

• a + b = 1, the production function has constant returns to scale.

• a + b < 1, decreasing returns to scale.

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

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