production. an activity that creates value inputs for production raw materials, labor, land,...

Production

Production

an activity that creates value

Inputs for Production

raw materials, labor, land, capital, & entrepreneurial or managerial talent.

Capital includes tools, machinery, equipment, & physical facilities.

Production Function

Q = f(X1,X2,X3,X4,…,Xn)

where Q is the quantity of output that can

be produced with amounts of inputs,

X1,X2,X3,X4,…,Xn.

Short run

time period so short that the amounts of some inputs can not be changed

For example, the quantity of plant & heavy equipment can not be changed in a short time period.

Long run

time period long enough for all inputs to be changed

Fixed input

an input whose quantity can not be changed in the short run

Variable input

an input whose quantity can be changed in a short period of time

Examples: labor, raw materials

The scale of a firm’s operation is determined by its fixed inputs.

We can look at the productivity of a variable input given a fixed level of fixed input.

Marginal Product (MP) of the variable input X

Discrete MP = ΔQ/ ΔX = ΔTP/ ΔX

Change in output resulting from a one-unit change in the quantity of input

Continuous MP = dQ/dX = dTP/dX

Rate of change in total output as the usage of the variable input increases by very small amounts.

Graphical Interpretation of MP

The continuous MP is the slope of the total product curve at a particular point.

The discrete MP is the slope of the line segment connecting 2 points on the total product curve.

Q = TP

X

Example: Q = 21X + 9X2 – X3

X Q = TP

0 0

1 29

2 70


X Q = TPDiscrete MP

ΔQ/ΔX

0 0 –

1 29 29

2 70 41


X Q = TPDiscrete MP

ΔQ/ΔXContinuous MP

dQ/dX =21+ 18 X – 3X2

0 0 – –

1 29 29 36

2 70 41 45

Average Product (AP)

AP = Q / X = TP / X

Amount of product per unit of input

Can be calculated for variable or fixed inputs


AP = Q / X = (21X + 9X2 – X3) / X

= 21 + 9X – X2

X = 1: AP = 29

X = 2: AP = 35

Graphical Interpretation of AP = Q / X

X

The AP of a particular value of X1 can be interpreted as the slope of the line from the origin to the corresponding point on the curve.

Q = TP

X1 →

Q1

0

In this graph, we see that initially, AP is increasing

X

Q = TP

0

and then decreasing

X

Q = TP

0

Principle of Diminishing Marginal Returns

As the amount of a variable input is increased and combined with a specified amount of fixed inputs, a point is eventually reached where the resulting increases in the quantity of output get smaller & smaller.

In other words, as the amount of variable input increases, eventually the MP of the variable input falls.

Total Product, Marginal Product, &

Average Product Curves

Q = TP

XMP AP

X

AP

MP

incr marg returns

pt of dim marg returns

Diminishing marginal returns set in when MP starts to fall (but is still positive).

The TP curve gets flatter as the slope of TP falls.

Q = TP

XMP AP

X

AP

MP

pt of dim avg returns

Diminishing average returns set in when AP starts to fall. (Remember that AP is the slope of the line from the origin to the point on the TP curve.)

Q = TP

XMP AP

X

AP

MP

marginal returns become negative

dim total returns

Diminishing total returns set in when the TP curve turns downward and MP becomes negative.

Isoquant

a curve showing all possible efficient combinations of input that are capable of producing a certain quantity of output

(Note: iso means same, so isoquant means same quantity)

Isoquant for 100 units of output

100

Quantity of capital used per unit of time

Quantity of labor used per unit of time

K1

K2

K3

K4

L1 L2 L3 L4

100 units of output can be produced in many different ways including L1 units of labor & K1 units of capital, L2 units of labor & K2 units of capital, L3 units of labor & K3 units of capital, & L4 units of labor & K4 units of capital.

Isoquants for different output levels

50

100

125



As you move in a northeasterly direction, the amount of output produced increases, along with the amount of inputs used.

If you move out from the origin along a ray with constant slope, the input combinations have a

constant capital-labor ratio.

50100

125



140

15 24 36 45

15

12

8

5

Each of the indicated points uses one-third as much capital as labor.

It is possible for an isoquant to have positively sloped sections.



In these sections, you’re increasing the amounts of both inputs, but output is not increasing, because the marginal product of one the inputs is negative.

The lines connecting the points where the isoquants begin to slope upward are called ridge lines.



ridge lines

No profit-maximizing firm will operate at a point outside the ridge lines, since it can produce the

same output with less of both outputs.



L1 L2

K2

K1

Notice, for example, that since points A & B are on the same isoquant, they produce the same amount of output.

However, point B is a more expensive way to produce since it uses more capital & more labor.

B

A

Marginal rate of technical substitution (MRTS)

The slope of the isoquant

The rate at which you can trade off inputs and still produce the same amount of output.

For example, if you can decrease the amount of capital by 1 unit while increasing the amount of labor by 3 units, & still produce the same amount of output, the marginal rate of technical substitution is 1/3.

What is the MRTS or slope of the isoquant?

Q2



KA

KB

LA LB

A

B

Q1

C

Consider 2 points A & B on the same isoquant.

Let’s divide the movement between A & B into 2 parts, from A to C, & from C to B.Moving from A to C, ΔQ = (ΔQ/ΔK) ΔK .Moving from C to B, ΔQ = (ΔQ/ΔL) ΔL .Moving from A to B, ΔQ = (ΔQ/ΔK) ΔK + (ΔQ/ΔL) ΔL

= MPK ΔK + MPL ΔL .

Since A & B are on the same isoquant, ΔQ = 0.

So, MPK ΔK + MPL ΔL = 0 .

MPK ΔK = - MPL ΔL .

ΔK/ΔL = - MPL/MPK

slope of isoquant

Marginal Rate of Technical Substitution (MRTS)

or slope of an isoquant

ΔK/ΔL = - MPL/MPK

the negative of the ratio of the marginal products of the inputs, with the input on the

horizontal axis in the numerator.

How does output respond to changes in scale in the long run?

Three possibilities:

1. Constant returns to scale

2. Increasing returns to scale

3. Decreasing returns to scale

Constant returns to scale

Doubling inputs results in double the output.

Increasing returns to scale

Doubling inputs results in more than double the output.One reason this may occur is that a firm may be able to use production techniques that it could not use in a smaller operation.

Decreasing returns to scale

Doubling inputs results in less than double the output.One reason this may occur is the difficulty in coordinating large organizations (more paper work, red tape, etc.)

Graphs of Constant, Increasing, & Decreasing Returns to Scale

Constant Returns to Scale: isoquants for output levels 50, 100, 150, etc. are evenly spaced.

Capital

Labor

150

100

50

150

Capital

Labor

50100

150

Capital

Labor

10050

Increasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. get closer & closer together.

Decreasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. become more widely spaced.

Methods of estimating production functions

1. using statistical analysis of time series or cross-sectional data.

2. based on experimentation or experience with day-to-day operations.

A commonly used production function is the Cobb-Douglas function

Q = AL1 K2 M3

where K is the quantity of capital, L is the quantity of labor, & M is the quantity of raw materials. A, 1, 2, & 3 are parameters that depend on the specific case. Also, 1, 2, & 3 are between 0 & 1.

If 1+ 2 + 3 = 1, we have constant returns to scale.

If 1+ 2 + 3 > 1, we have increasing returns to scale.

If 1+ 2 + 3 < 1, we have decreasing returns to scale.

Suppose Q = AL.5 K.2 M.5

Show that with 1+ 2 + 3 = .5 + .2 + .5 = 1.2 > 1, this production function does have increasing returns to scale, by showing that

doubling inputs results in more than double the output.

Let Q’ be the output resulting from doubling the inputs. Then Q’ = (A)(2L).5 (2K).2 (2M).5 = (A) (2.5 L.5) (2.2 K.2) (2.5 M.5) = (A) (2.5) (2.2) (2.5)(L.5 K.2 M.5) = (A) (2 .5 + .2 + .5)(L.5 K.2 M.5) = 21.2 (A L.5 K.2 M.5) > 2 (A L.5 K.2 M.5) 2QSo doubling the inputs more than doubles the output.

production. an activity that creates value inputs for production raw materials, labor, land,...

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