Production Theory and Estimation Department of Business Administration FALL 2007-08 by Asst. Prof....

Production Theory and Estimation

Department of Business Administration

FALL 2007-08

by

Asst. Prof. Sami Fethi

2 Managerial Economics © 2006/07, Sami Fethi, EMU, All Right Reserved.

Production Theory

The Production FunctionThe 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.


Production Theory

For exampleFor 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.


Production Theory The Organization of ProductionThe Organization of Production

Inputs– Labor, Capital, Land

Fixed InputsVariable InputsShort Run

– At least one input is fixedLong Run

– All inputs are variable


Production Theory

The Organization of ProductionThe Organization of Production

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.


Production Theory

The Organization of ProductionThe Organization of Production

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.


Production Theory


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.


Production Theory


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)


Production Theory

Production Function With Two InputsProduction 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).


Production Theory

Production Function With Two InputsProduction Function With Two Inputs

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.


Production Theory Production Function With Two InputsProduction Function With Two Inputs

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.


Production Theory

Production Function With One Variable InputProduction Function With One Variable Input

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

Total product Managerial product Average product


Production Theory


Total Product

Marginal Product

Average Product

Production orOutput Elasticity

TP = Q = f(L)

MPL =TP L

APL =TP L

EL =MPL

APL


Production Theory

Total ProductTotal Product

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

TP = Q = f (L)


Production Theory

Marginal ProductMarginal 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


Production Theory Average ProductAverage 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 asholding input K constant.

APL =TP L


Production Theory

Production Function With One Variable Input Production Function With One Variable Input ExampleExample

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


Production Theory



Production Theory

The Law of Diminishing Returns 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.


Production Theory

The Law of Diminishing ReturnsThe Law of Diminishing Returns

X

MP

Increasing Returns

Diminishing Returns Begins

MP


Production Theory

The Three Stages of ProductionThe Three Stages of Production


Production Theory The Three Stages of ProductionThe Three Stages of Production

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.


Production Theory



Production Theory


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

Why Stage II? Why not Stage I and III?


Production Theory The Three Stages of ProductionThe Three Stages of Production

ExampleExample

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


Production Theory

The Three Stages of ProductionThe Three Stages of ProductionExampleExample

What level of input usage within Stage II is best for the firm?

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.


Production Theory

Optimal Use of the Variable InputOptimal Use of the Variable Input

Marginal RevenueProduct of Labor MRPL = (MPL)(MR)

Marginal ResourceCost of Labor MRCL =

TC L

Optimal Use of Labor MRPL = MRCL


Production Theory


A profit-maximizing firm operating in perfectly competitive output and input markets will be using the optimal amount of an input at the point at which the monetary value of the input’s marginal product is equal to the additional cost of using that input.

Where MRP=MLC.


Production Theory Optimal Use of the Variable InputOptimal Use of the Variable Input

ExampleExample

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


Production Theory



Production Theory Production With Two Variable InputsProduction 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.


Production Theory

Production With Two Variable InputsProduction With Two Variable Inputs

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


Production Theory

Production IsoquantProduction 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.


Production Theory Production With Two Variable InputsProduction With Two Variable Inputs

Economic Region of Production


Production Theory


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


Production Theory


MRTS = -(-2.5/1) = 2.5


Production Theory


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 comlements.


Production Theory

Optimal Combination of InputsOptimal 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 Wage Rateof Labor L

( )r Cost of Capital K

Slope of isocost

Vertical intercept of isocost


Production Theory



Production Theory Example: Example: Isocost Lines

AB Total Cost = c = $100w=r=$10c/r = $100/$10 = $10k-w/r = -$10/$10 = -1

A’B’Total Cost = c = $140w=r=$10c/r = $140/$10 = $14k (vertical intercept)-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


Production Theory OptOptimal Input Combination for imal Input Combination for

Minimizing Cost or Maximizing OutputMinimizing Cost or Maximizing Output

Cost-Minimization

Simply by changing the proportions of factors K and L, it may be able to decrease total costs without affecting total revenue - and thus increase profits. Thus, we first begin with analyzing the cost-minimizing choice of technique for a firm which seeks to produce a particular level of output.


Production Theory

Con’t

It is important to note that this story is not the whole story behind the producer's profit-maximizing decision. In cost-minimization, we leave the decision on the level of output out of the picture. The full, profit-maximization story would require that output level enter as a variable and not as a given. We shall consider this later.


Production Theory

Con’t

Where C is total factor costs and w and r are the rental rates for labor and capital respectively. Notice that this cost equation defines a function in L-K space of the following linear form,

L = C/w - (r/w)K. This is known as an isocost curve and is depicted as a downward-sloping straight line, as we see in Figure 1.


Production Theory Con’t

Any factor input combination on a particular isocost curve has the same total costs, C. The vertical intercept of the isocost curve is C/w and the slope of the isocost curve is -(r/w). The horizontal intercept is obviously C/r. Thus, for different levels of costs, C, there will be different (but parallel) isocost curves. The isocost curves closest to the origin represent relatively low total costs, those furthest away represent relatively high total costs. Thus, referring to the isocost curves, C < C* < C’.

Note that a change in the factor prices, r or w, will change the slopes of all the isocost curves.



In order for the cost minimizing firm, is then to minimize cost given that a certain level of output must be reached. This is stated as follows:

min C = wL + rK

Y* = F(K, L)

In other words, as the output level Y* is given, then the consraint is the isoquant Y* depicted in previous Figure. What the firm seeks to do is thus find the factor combination, (K, L), which has the lowest total cost and yet still produces output Y*.


Production Theory

Con’t

Diagramatically, this is represented as the tangency of the given isoquant Y* and the lowest İsocost curve, C*. This will be at point e* in Figure 1, which represents factor input combination K*, L*.

Notice that point e is unattainable: although the factor combination corresponding to e yields a lower cost (as it lies on isocost curve C which is below isocost curve C*), it does not produce Y* and thus will not be considered.


Production Theory

Con’t

In contrast, point e` is attainable as it lies on the isoquant Y*, but obviously, as it also lies on the higher isocost curve C` , it is more costly than the combination e*. Thus, point e* - and thus capital employment K* and labor employment L* - will be the optimal choice of inputs for a firm which seeks to minimize the costs of producing Y*.

We can obtain this tangency solution from the cost-minimization problem via simple mathematical programming. Setting up in Lagrangian form, we have:

E = rK + wL + λ(Y* - f(K, L))



Where λ is the Lagrangian multiplier. This yields the following first order conditions for a maximum are (we are assuming an interior solution):

E/ dK = r - λfK = 0

E/dL = w - λfL = 0

E/dλ = Y* - λf(K, L) = 0

Combining the first two yield:

r/w = fK/fL


Production Theory

Con’t

Thus, for a minimum, the (negative of the) slope of the lowest isocost curve must equal the ratio of marginal products, or MRTS, which, as we know, is the (negative of the) slope of the isoquant Y*. So, minimum cost is achieved by finding the factor combination which yields a tangency between the isoquant and the lowest isocost curve.


Production Theory Output-Maximization

In our cost-minimization problem, we assumed that the producer was trying to find the input combinations that would produce a given level of output at minimum cost. An alternative way of thinking about the producer's decision on factor inputs is to ask him to find the highest amount of output he can produce for a given total cost. Such an exercise could be expressed as the following:

max Y = f (K, L)

C* = wL + rK


Production Theory

Output-Maximization

Notice that now the constraint is now a given level of costs. This is analogous to a consumer case: the entrepreneur is now given a "budget" (the maximum amount of costs he is allowed to incur, C*) and will thus try to achieve as much output as he can out of this by choosing the appropriate factor input combinations.


Production Theory

Output-Maximization Diagramatically, we impose

on the producer a particular isocost curve and then ask to choose factor inputs such that output is maximized. In the Figure, we see that the maximimum output is represented by the isoquant Y* which is tangent to the given isocost curve, C*, at point e*. Point e’ is not output-maximizing as that factor combination produces a lower level of output Y’< Y*), while point e’’ is not available, as costs at e’’ would be greater than C*, thus our constraint would be violated.

LKMRTC /MPMP


Production Theory


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


Production Theory

Optimal Combination of Inputs

Effect of a Change in Input Prices


Production Theory Returns to Scale-How does output vary with Returns to Scale-How does output vary with

the scale of production?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.


Production Theory Returns to ScaleReturns 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

IRTSQ

X,Y

CRTSQ

X,Y

DRTS


Production Theory If all inputs into the production process

are doubled, three things can happen:

output can more than doubleincreasing returns to scale (IRTS)

output can exactly doubleconstant returns to scale (CRTS)

output can less than doubledecreasing returns to scale (DRTS)


Production Theory

Returns to Scale

Constant Returns to

Scale

Increasing Returns to

Scale

Decreasing Returns to

Scale


Production Theory

Empirical Production Functions

Cobb-Douglas Production Function

Q = AKaLb

Estimated using Natural Logarithms

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


Production Theory Empirical Production FunctionsEmpirical 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


Production Theory

Innovations and Global CompetitivenessInnovations and Global Competitiveness

Product InnovationProcess InnovationProduct Cycle ModelJust-In-Time Production SystemCompetitive BenchmarkingComputer-Aided Design (CAD)Computer-Aided Manufacturing (CAM)


Production Theory

The EndThe End

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Production Theory and Estimation Department of Business Administration FALL 2007-08 by Asst. Prof....

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