Production Theory and Estimation FALL 20 14 - 15 by Dr Loizos Christou.

Production Theory and Estimation

FALL 2014-15

by

Dr Loizos Christou

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The Production Function

Production refers to the transformation of inputs or resources into outputs of goods and services. In other words, production refers to all of the activities involved in the production of goods and services, from borrowing to set up or expand production facilities, to hiring workers, purchasing row materials, running quality control, cost accounting, and so on, rather than referring merely to the physical transformation of inputs into outputs of goods and services.

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For example

A computer company hires workers to use machinery, parts, and raw materials in factories to produce personal computers.

The output of a firm can either be a final commodity or an intermediate product such as computer and semiconductor respectively.

The output can also be a service rather than a good such as education, medicine, banking etc.

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The Organization of Production

Inputs Labor, Capital, Land

Fixed Inputs Variable Inputs Short Run

At least one input is fixed Long Run

All inputs are variable

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Inputs: are the sources used in the production of goods and services and can be broadly classified into labour, capital, land, natural resources, and entrepreneurial talent.

Fixed input: are those that cannot be readily changed during the time period under consideration such as a firm’s plant and specialized equipment.

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Variables Inputs: are those can be varied easily and on very short notice such as raw materials and unskilled labour.

The time period during which at least one input is fixed called the short-run and if all inputs are variable, we are in the long-run.

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A production function is an equation, tables, or graph showing the maximum output of a commodity that a firm can produce per period of time with each set of inputs.

Both inputs and outputs are measured in physical rather than in monetary units. Here technology is assumed to remain constant during the period of the analysis.

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The general equation of the production function of a firm using labour (L) and capital (K) to produce a good or service (Q) or shows the maximum amount of output (Q) that can be produced within a given time period with each combination of (L) and (K). This can be defined as follows:

Q= f (L,K)

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Production Function With Two Inputs

K Q6 10 24 31 36 40 395 12 28 36 40 42 404 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) The table shows

that by using 1 unit of labour (1L) and 1 unit of capital (1K), the firm would produce 3 units of o/p (3Q).

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Discrete Production Surface

The previous table are shown graphically in this figure. The height of bars refers to the max o/p that can be produced with each combination of labour and capital shown on the axes.

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Continuous Production Surface

In this figure, If we assume that i/p’s and o/p’s are continuously divisibly, we would have the continuous production surface.

This indicates that by increasing L2 with K1 of capital, the firm produces the o/p by height of cross section K1AB. Increasing L1 with K2, we have cross section K2CD.

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Production Function With One Variable Input

When discussing production in the short run, three definitions are important:

Total product Marginal product Average product

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Total Product

Marginal Product

Average Product

Production orOutput Elasticity

TP = Q = f(L)

MPL =TP L

APL =TP L

EL =MPL

APL

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Total ProductTotal Product

Total product (TP) is another name for output in the short run.

TP = Q = f (L)

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Marginal Product

The marginal product (MP) of a variable input is the change in output (or TP) resulting from a one unit change in the input.

MP tells us how output changes as we change the level of the input by one unit.

Consider the two input production function Q=f (L,K) in which input L is variable and input K is fixed at some level.

The marginal product of input L is defined as holding input K constant.

MPL =TP L

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Average Product

The average product (AP) of an input is the total product divided by the level of the input.

AP tells us, on average, how many units of output are produced per unit of input used.

The average product of input L is defined as holding input K constant.

APL =TP L

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Production Function With One Variable Input-Example

L Q MPL APL EL

0 0 - - -1 3 3 3 12 8 5 4 1.253 12 4 4 14 14 2 3.5 0.575 14 0 2.8 06 12 -2 2 -1

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

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The Law of Diminishing Returns

• As additional units of a variable input are combined with a fixed input, after a point the additional output (marginal product) starts to diminish. This is the principle that after a point, the marginal product of a variable input declines.

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The Law of Diminishing Returns

X

MP

Increasing Returns

Diminishing Returns Begins

MP

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The Three Stages of Production

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Stage I: The range of increasing average product of the variable input. From zero units of the variable input to

where AP is maximized Stage II: The range from the point of

maximum AP of the variable i/p to the point at which the MP of i/p is zero. From the maximum AP to where MP=0

Stage III: The range of negative marginal product of the variable input. From where MP=0 and MP is negative.

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In the short run, rational firms should only be operating in Stage II.

Why Stage II? Why not Stage I and III? In Stage III- MPLis negative

In Stage I- MPK is negative

In Stage II- MPL and MPK are both positive but decline

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The Three Stages of Production-Example

Labor Unit (L)

Total Product

(Q or TP)

Average Product

(AP)

Marginal Product

(MP)0 01 10,000 10,000 10,0002 25,000 12,500 15,0003 45,000 15,000 20,0004 60,000 15,000 15,0005 70,000 14,000 10,0006 75,000 12,500 5,0007 78,000 11,143 3,0008 80,000 10,000 2,000

Stage II

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The Three Stages of Production-Example

What level of input usage within Stage II is best for the firm? Is there a precise point.

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.

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Optimal Use of the Variable Input

How much labor or the variable input should the firm use in order to maximize profit.

The firm should employ an additional unit of labor as long as the extra revenue genereted until the extra revenue equals the extra cost.

Where MRP=MLC.

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Marginal RevenueProduct of Labor

MRPL = (MPL)(MR)

Marginal ResourceCost of Labor MRCL =

TC L

Optimal Use of Labor MRPL = MRCL

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Optimal Use of the Variable Input-Example

L MPL MR = P MRPL MRCL

2.50 4 $10 $40 $203.00 3 10 30 203.50 2 10 20 204.00 1 10 10 204.50 0 10 0 20

Use of Labor is Optimal When L = 3.50

MRPL=MRxMPL--------MRC=W

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Production With Two Variable Inputs

--In the long run, all inputs are variable.

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

-In other words, Production isoquant shows the various combination of two inputs that the firm can use to produce a specific level of output.

-Firms will only use combinations of two inputs that are in the economic region of production, which is defined by the portion of each isoquant that is negatively sloped.-A higher isoquant refers to a larger output, while a lower isoquant refers to a smaller output.

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IsoquantsK Q6 10 24 31 36 40 395 12 28 36 40 42 404 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

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Production Isoquant

Economic region of production: Negatively sloped portions of the isoquants within the ridge lines represents the relevant economic region of production.

Ridge lines: The lines that separate the relevant (i.e., negatively sloped) from the irrelevant ( or positively sloped) portions of the isoquant.

This refers to stage II where the MPLand MPK are both positive but declining and producers never want to operate outside this region.

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Economic Region of Production

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Marginal Rate of Technical Substitution: The absolute value of the slope of the isoquant. It equals the ratio the marginal products of the two inputs. Slope of isoquant indicates the quantity of one input that can be traded for another input, while keeping output constant.

MRTS = -K/L = MPL/MPK

Substitution among inputsSubstitution among inputs

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MRTS = -(-2.5/1) = 2.5

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Perfect Substitutes Perfect Complements

When an isoquant is straight line or MRTS is constant, inputs are perfect substitutes whilst an isoquant is right angled, inputs are perfect complements.

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Optimal Combination of Inputs

To determine the optimal combination of labor and capital, we also need an isocost line.

Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

C wL rK

C wK L

r r

C Total Cost

( )w WageRateof Labor L

( )r Cost of Capital K

Slope of isocost

Vertical intercept of isocost

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Example: Isocost Lines

AB Total Cost = c = $100w=r=$10c/r = $100/$10 = $10k (vertical intercept)-w/r = -$10/$10 = -1(slope)

A’B’Total Cost = c = $140w=r=$10c/r = $140/$10 = $14k -w/r = -$10/$10 = -1

A’’B’’ Total Cost = c = $80w=r=$10c/r = $80/$10 = $8k -w/r = -$10/$10 = -1

AB*

C = $100,

w = $5,

r = $10

c/r = $100/$10 =$10k

-w/r = -$10/$5 = -1/2

MRTS = w/r;

since MRTS = MPL/ MPK, condition for optimal combination of inputs as MPL/ MPK= w/r

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Expansion Path

Expansion path: joinning points of tangency of isoquants and isocost of optimal input combination. The optimal input combination required to minimize the cost of producing a given level of maximum output that the firm can produce at the tangency of an isoquant and an isocost.

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If the price of an input declines, the firm will substitute the cheaper input for another inputs in production in order to reach a new optimal input combination.

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Returns to ScaleHow does output vary with the scale of production?

Production Function Q = f(L, K)

Q = f(hL, hK)

If = h, then f has constant returns to scale.

If > h, then f has increasing returns to scale.

If < h, the f has decreasing returns to scale.

Returns to scale describes what happens to total output as all of the inputs are changed by the same proportion.

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Returns to Scale

Graphically, the returns to scale concept can be illustrated using the following graphs.

The long run production process is described by the concept of returns to scale.

Q

X,Y

IRTS Q

X,Y

CRTSQ

X,Y

DRTS

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

output can more than double increasing returns to scale (IRTS)

output can exactly double constant returns to scale (CRTS)

output can less than double decreasing returns to scale (DRTS)

Returns to Scale

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Constant Returns to

Scale

Increasing Returns to

Scale

Decreasing Returns to

Scale

Returns to Scale

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Empirical Production Functions

Several Useful Properties :1. The Marginal Product of capital and the

marginal Product of labor depend on both the quantity of capital and the quantity of labor used in production, as is often the case in the real world.

2. K and L are represents the output elasticity of labor and capital and the sum of these exponents gives the returns on scale. a + b = 1 Constant return to scale a + b > 1 Increasing return to scale a + b <1 Decreasing return to scale

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Empirical Production Functions

Cobb-Douglas Production Function

Q = AKaLb

Estimated using Natural Logarithms

ln Q = ln A + a ln K + b ln L

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Empirical Production Functions-Example

A bus ltd in a district has estimated the following Cobb-Douglas production function using monthly observations for the past four years:

ln Q = ln A + a ln K + b ln L+ c Ln G

Ln Q = 2.303+ 0.40 ln K + 0.60 Ln L+ 0.20 ln G

(3.40) (4.15) (3.05)

R2=0.94 DW=2.20 F= 25.6

Q is the number of bus miles driven, K is the number of buses the firm operates, L is the number of bus drives it employes each day, and G is the gallons of gasoline it uses.

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Innovations and Global Competitiveness

Product Innovation Process Innovation Product Cycle Model Just-In-Time Production System Competitive Benchmarking Computer-Aided Design (CAD) Computer-Aided Manufacturing (CAM)

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The EndThe End

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Production Theory and Estimation FALL 20 14 - 15 by Dr Loizos Christou.

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Transcript of Production Theory and Estimation FALL 20 14 - 15 by Dr Loizos Christou.