production theory.ppt

Production Theory

Production Theory 



The production theory basically addresses itself to the question: If you have fixed amount of inputs, how much output can you get ? The state of technology and engineering knowledge is assumed to remain constant. The production function specifies the maximum output that can be produced with a given quantity of inputs. It is defined for a given state of engineering and technical knowledge.

Total, Avera Average ge and Margi Marginal nal Product 



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Total product is the total amount of output produced in physical units like bushels of wheat or quintals of rice. Average Average product equals the total output divided by total units of input. Marginal (or extra) product of an input is the extra output produced by one additional unit of that input while other inputs are held constant.

Total, Average Average and Marginal Produc Productt 

The following diagram shows the total product curve rising as additional inputs of labor are added, holding other things constant. However, However, total total product product rises ri ses by smaller and smaller increments as additional units of labor are added.

Total Product TP 4,000

3,000

t

c u d or P l ta o T

2,000

1,000

0

1

2

3

4

5

Marginal Product MP

3,000 )r

o b al f

o ti

n u r

e p(

2,000

t

P l a ni gr a M

c u d or

1,000

0

1

2

3

4

5

Marginal Product 

The preceding diagram shows the declining steps of marginal product. It must be understood that each dark rectangle is equal to the equivalent dark rectangle in the total product curve.

Total, Average Average and Marginal Produc Productt (1)

(2)

(3)

(4)

Units of

Total

Labor input

Product

Marginal Product

Average Product

0

0 2000

1

2000

2000 1000

2

1500

3000 500

3

1167

3500 300

4

950

3800 100

5

3900

780

The Law of Diminishing Returns 



The Law of Diminishing Returns holds that we will get less and less extra output when we add additional doses of an input while holding other inputs fixed. In other words, the marginal product of each unit of input will decline as the amount of that input increases. The Law of Diminishing Returns expresses a very basic relationship. As more of an output such as labor is added to a fixed amount of land, machinery and other inputs, the labor will have less and less of the other factors to work with. The land gets more crowded, the machinery is overworked, and the marginal product of labor declines.

The Law of Diminishing Returns 

You might find that the first hour of studying economics on a given day is productive – you learn new laws and facts, insights and history. The second hour might find your attention wandering a bit, with less learned. The third hour might show that diminishing returns have h ave set in with a vengeance, and by the next day the third hour is a blank in your memory. Hence, hours devoted to studying should be spread out rather than crammed into the day before exams.

Returns to Scale 



While marginal products refer to infusion of a single input, we are sometimes interested in the effect of increasing all inputs. Returns to scale is the study of the effect on production if all inputs or the scale of inputs is altered. Returns to scale are of three types: Constant Returns to scale denote a case where a change in all inputs leads to a proportional change in output. For example, if labor, land, capital and other inputs are doubled, then under constant returns to scale output would also double.


Increasing returns to scale (also referred to as economies of scale) scale) arise when an increase in all inputs leads to a more-than-proportionate increase in the level of output. For example, an engineer planning a small-scale chemical plant will generally find that increasing the inputs of labor, capital and materials by 10% will increase the total output by more than 10%. Engineering studies have determined that many manufacturing processes enjoy modestly increasing returns to scale.


Diminishing returns to scale occurs when a balanced increase of all inputs leads to a less-thanproportionate increase in total output. Many productive activities involving natural resources like agriculture, mining, vineyards etc show decreasing returns to scale.

Returns to Scale

Short Run and Long Run 

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To account for the role of time in i n production and costs, we distinguish between two different time periods. We define the short run as a period in which firms can adjust production by changing variable factors such as materials and labor but cannot change fixed factors such as land and capital. The long run is a period sufficiently long that all factors including fixed factors can be adjusted.

Production Function 

Assume that all factors or inputs of production can be grouped into two broad categories: labor (L) and capital (K). The general equation for the production function is Q = f(K,L) The above function defines the maximum quantity of output that could be obtained from a given rate of labor and capital inputs. Output may be in physical terms or intangible terms like in case of services.


Production function is a purely engineering concept which is devoid of economic econom ic content. conten t. However However,, if the production function were to be more meaningful it has to be integrated to economic theory. Such integration will give answers to many managerial issues like least-cost capital-labor combination or the most profitable output rate.


Production implies the maximum output rate which is related to efficient management of resources. Firms that do not manage resources efficiently (as defined by the production production function) will be rendered uncompetitive uncompetitive and unprofitable. Hence, they go out of business.


Economists use a variety variety of functional forms to describe production. The multiplicative form, generally referred to as Cobb-Douglas production is widely used in economics because it has properties representative of many production processes. It is represented by the equation: Q = AKα Lβ Consider a production function with parameters A=100, α =0.5 and β =0.5. That is Q = AK0.5L0.5

Production Function and Returns to Scale 

From the preceding Cobb-Douglas production function, we may conclude on the returns to scale with the following chart: Sum of exponents ( Returns to Scale + ) Less than one

Decreasing

Equal to one

Constant

More than one

Increasing


A production table shows the maximum output associated with each of a number of input combinations. The following table shows production rates for various input combinations applied to the production function.

Productio Prod uction n Table

K

8

283 40 400 49 490 56 565 63 632 69 693 74 748 80 800

7

265 37 374 45 458 52 529 59 592 64 648 70 700 74 748

6

245 34 346 42 424 49 490 54 548 60 600 64 648 69 693

5

224 31 316 38 387 44 447 50 500 54 548 59 592 63 632

4

200 28 283 34 346 40 400 44 447 49 490 52 529 56 565

3

173 24 245 30 300 34 346 38 387 42 424 45 458 49 490

2

141 20 200 24 245 28 283 31 316 34 346 37 374 40 400

1

100 14 141 17 173 20 200 22 224 24 245 26 265 28 283 1

2

3

4 L

5

6

7

8


Two important relationships emerge from the preceding production tables. Firstly, the table indicates that there are a variety of ways to produce a particular rate of output. For example, 245 units of output can be produced with any of the following input combinations :

Production Function Combination

K

L

a

6

1

b

3

2

c

2

3

d

1

6


This implies that there is substitutability substitutability between the factors of production. The firm can use a capital intensive production process characterized characterized by combination a, a labor intensive process such as d , or a process that uses a resource combination somewhere between these extremes, such as b or c .


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The concept of substitution is important to us because it means that managers can change the mix of capital and labor in response to changes in the relative prices of these inputs. Secondly, the preceding table also suggests a relationship between the input deployment and output generation. gene ration.


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For example, maximum production with 1 unit of capital and 4 units of labor is 200. Doubling the input rates to K=2, L=8 results in the rate of output doubling to Q-400. The relationship between output change and proportionate changes in both inputs is referred to as returns to scale which in this case is constant returns to scale.


Although the production table provides considerable information on production possibilities, it does not allow for the determination of the profit-maximizing profit-maximizing rate of output or even the best way to produce some specified rate of output. Hence, the production function needs to be integrated with economic theory to determine optimum allocation of resources.


For a two-input production process, the total product of labor (TP L) is defined as the maximum rate of output forthcoming from combining various rates of labor input with a fixed capital output. Denoting the fixed capital input as K, the total product of labor function is: TPL = f(K,L)


Similarly, the total product of capital function is: TPK = f(K,L)

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Two other product relations may be examined. Firstly, marginal product (MP) is defined as the change in output per one-unit change in the variable input. Thereby, MP of labor is: MPL = Q/ L


The MP of capital is:

MPK =

Q/

K

For infinitesimal changes in the variable variable output, the MP function is the first derivative derivative of the production function with respect to the t he variable input.


For the general Cobb-Douglas function Q = AKα Lβ



The marginal products are MPK = dQ/dK =

AK

-1

L

and MPL = dQ/dL =

AK L

-1


Average product (AP) is total product per unit of the variable input and is found by dividing the rate of output by the rate of the variable input. The average product of labor function is APL = TPL/L

and the average product of capital is APK = TPK/K

Production Function Consider a hypothetical production function. If capital is fixed at two units, the rates of output generated by combining various levels of labor with two units of capital (ie. The total product of labor) are shown in the following table. The average and marginal products of labor are also shown. The total product function can be thought of as a cross section or vertical slice of a three-dimension production surface as shown in the next figure.

Production Function Rate of labour input (L)

TPL

APL

MPL

0

0

-

-

1

20

20

20

2

50

25

30

3

90

30

40

4

120

30

30

5

140

28

20

6

150

25

10

7

155

22

5

8

150

19

-5

Production Function

Production Function

Diminishing Marginal Returns 

This law states that when increasing amounts of the variable input are combined with a fixed level of another input, a point will be reached where the marginal product of the variable input will decline. This is called the law of diminishing marginal returns. This law is not a theoretical argument but is based on actual observation of many production processes.


Suppose the capital stock is fixed at K1. The total product of labor function f(K1,L) is shown as the line starting at K1 and extending through point A. Similarly, if the labor input is fixed at L2, the total product of capital function is shown as the line beginning at L 2 and going through points A and C.

Relationships among the Product Functions 

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A set of typical total, average and marginal product functions for labor is shown in the following figure. Total product begins at the origin, increases at a increasing rate over the range O to G and then increases at a decreasing rate. Beyond I, total product actually declines. Explanation – Initially, the input proportions are inefficient – there is too much of the fixed factor, capital. As the labor input is increased from 0 to G, output rises more than in proportion to the increase in the labor input.

Production Function With One Variable Input

Relationships among the Product Functions 



1.

That is, marginal product per unit of labor increases as a better balance of labor and capital inputs is achieved. As the labor input is increased beyond G, diminishing marginal returns set in and marginal product declines. The following relationships between total, average and marginal products are worth noting: Marginal product reaches a maximum at G’ which corresponds to an inflection point

Relationships among the Product Functions

2.

G on the total product function. At the inflection point, total product changes from increasing at an increasing rate to increasing at a decreasing rate. Marginal product intersects average product at the maximum point on the average product curve. This occurs at labor input rate H’. Note that whenever the MR curve is rising it is above the AR curve and whenever the MR curve is faling it is below the AR curve.

Relationships among the Product Functions

3.

This implies that the intersection must happen only at the highest value of AR. Marginal product becomes negative at the point I’ which corresponds to the point where the total product curve is at the maximum.

Optimal Employment of a Factor of Production 

The General Motors Corporation has a worldwide physical capital stock valued at about $70 billion. Consider this to be the fixed input for the firm. About 760,000 workers are employed to use this capital stock. What principles guide the decisions about the level of employment? In general, to maximize profit, the firm should hire labor as long as the additional revenue associated with hiring of another unit of labor exceeds the cost of employing that unit. For example, suppose that the marginal product of an additional worker

Optimal Employment of a Factor of Production 

is two units of output (ie. Automobiles) and each unit of output is worth $20,000. Thus the additional revenue to the firm will be $40,000 if the worker is hired. If the additional cost of a worker (wage rate) is $30,000, that worker will be hired because $10,000, the difference between additional revenue and additional cost, will be added to profit. However, if the wage rate is $45,000, the worker should not be hired because profit would be reduced by $5,000.

Optimal Use of the Variable Input 



How much labor should the firm use to maximize profits? The answer is that the firm should employ an additional unit of labor as long as the extra revenue generated from the sale of the output produced exceeds the extra cost of hiring the unit of labor. In the following graph, it pays the firm to hire more labor as long as the marginal revenue product (MRPL) exceeds the marginal resource cost of hiring labor (MRCL) and until MRPL=MRCL. At MRCL = $20 the firm maximizes total profit.

Optimal Use of the Variable Input

Optimal Use of the Variable Input •

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

Properties: 1.

Isoquants are downward sloping because, as one factor is removed (and we move down one axis), more of another factor must be added to maintain the old level of output (moving up the other axis).

2.

They are convex to the origin, because increasing amounts of a second factor are required to compensate for unit decreases in the first (MRTS) (MRTS)

Optimal Use of the Variable Input Isoquants

Optimal Use of the Variable Input 

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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. Ridge lines separate the relevant (negatively sloped) from the irrelevant (positively sloped) portions of the isoquants. In the following figure, ridge lines join points on the various isoquants where the isoquants have zero slope.

Production With Two Variable Inputs Economic Region of Production

Production With Two Variable Inputs 

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Marginal Rate of Technical Substitution is the absolute value of the slope of the isoquant. MRTS = -∆ K/∆ L = MPL/MPK

Between points N and R on the isoquant 12Q, MRTS = 2.5.

Production With Two Variable Inputs MRTS MRTS = -(-2.5/1) = 2.5

Perfect Substitutes and Complementary inputs 

The shape of an isoquant reflects the degree to which one input can be substituted for another in production. The smaller the curvature of an isoquant, the greater is the degree of substitutability of inputs in production. On the other hand, the greater the curvature of an isoquant, the smaller is the degree of substitutability.

Perfect Substitutes and Complementary inputs 

In the following figure, when an isoquant is a straight line (so that its absolute slope or MRTS MRTS is constant), inputs are perfect substitutes. In the left panel, 2L can be substituted for 1K regardless of the point of production on the isoquant. With the right angled isoquants in the right panel, efficient production can take place only with 2K/1L. Thus, labor and capital are perfect complements. Using only more labor or only more capital does not increase output (ie. MPL=MPK=0) MPL=MPK=0)

Production With Two Variable Inputs Perfect Substitutes Perfect Complements

Optimal Combination of Inputs The isoquant is a physical relationship that denotes different ways to produce a given rate of output. The next step toward determining the optimal combination of capital and labor is to add information on the cost of those inputs. i nputs. This cost information is introduced by a function called a production isocost. Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

Optimal Combination of Inputs 

Given the per unit prices of capital (r) and labor (w), the total expenditure (c) on capital and labor input is C = wL + rK w Wage Rate of Labor (L ) =

r

=

Cost of Capital (K )




For Eg., if r=3 and w = 2, the combination of 10 units of capital and 5 units of labour will cost $40 i.e. 40 = 3(10) + 2(5). For any given cost C, the isocost line defines all combinations of capital and labour that can be purchased for C. Solving for K as a function of L, K

C =

r

w −

r

L


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Changes in the budget amount cause the isocost line to shift in a parallel manner. Changes in either the price of capital or labour cause both the slope and one intercept of the isocost function to change.

Optimal Combination of Inputs

AB

C = $100, w = r = $10

A’’B’’ C = $80, w = r = $10

A’B’

C = $140, w = r = $10

AB*

C = $100, w = $5, r = $10


When both capital and labor are variable, variable, determining the optimal input rates of capital and labor requires that the technical information from the production function (i.e., the isoquants) be combined with the market data on input prices (i.e., the isocost functions).


At the tangency of the isoquant and isocost, the slopes of the two functions fun ctions are equal. Thus, the marginal rate of technical substitution (i.e., (i.e., the slope of the isoquants) equals the price of labour divided by the price of capital. That is, MRTS = w/r The above identity is a necessary condition for efficient production.

Optimal Combination of Inputs MRTS = w/r


These principles can be used to test for efficient resource allocation in production. The slope of the isocost is the negative of the ratio of the wage rate and price of capital (i.e., (i.e., -w/r) and that the slope of the isoquant is the negative n egative of the ratio of the marginal product of labour to that of capital (i.e., -MP L/MPK). Further, at the point of tangency, the slopes of both isocost and isoquant are equal.


Thus, -MPL / MPK = -w/r MPL / MPK = w/r Or MPL/w = MPK/r

Optimal Combination of Inputs Effect of a Change in Input Prices

Sources of Economies of Scale 

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There are several reasons why increasing i ncreasing returns to scale occur. Firstly, technologies that are cost effective at high levels of production pr oduction generally have higher unit costs at lower levels of output. Secondly, increasing returns accrue due to specialization of labor. As a firm becomes larger, the demand for employee expertise in specific areas grows. Instead of being generalists, workers can concentrate on learning all the aspects of particular segments of the production process. Thirdly, Thirdly, increasing returns are a re also the result of inventory economies. economies. Large firms f irms may not need to increase inventories or replacement parts proportionately with size.

Economies of Scope 

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Firms often find that per unit costs are lower l ower when two or more products are produced. Sometimes, the firm will have excess excess capacity that can be used to produce other products with little or no increase in its capital costs. For e.g. airlines can rearrange their seating system to convert a passenger plane into a cargo plane. Certain firms have taken advantage of their unique skills or comparative advantage in marketing to develop products that are complementary with the firms existing products. For e.g. Proctor and Gamble sells all kinds of cleaning products

Discussion Questions 

1) Explain the concept of a production function. Why is having only qualitative information about the production function inadequate for making decisions about efficient input combinations and the profitmaximizing rate of output?


Ans: A production function is a relationship that shows the maximum rate of output forthcoming from any specified rate of input for capital and labor. The production function provides information only about the maximum rate of output associated with given input rates. It says nothing about the costs of those input combinations or which is preferred.


2) Explain the law of diminishing marginal returns and provide an example of this phenomenon.


Ans: The law of diminishing marginal returns states that when one input is held constant while the other is increased, a point will be reached where the marginal product of the variable variable input will diminish. Examples of diminishing marginal returns in higher education might include: a) additional hours of study each day may result in progressively smaller increases in average grades attained; b) increasing the number of hours of computer time in a research project project may result in progressively progressively smaller increases in research output.


3) What is the difference between short run and the long run? What are examples of a firm where the short run would be quite short (eg. A few days weeks) and where would it be quite long (eg. Several months or years). Explain.


Ans: The short run is that period of time during which the input rate of at least one factor of production is fixed. In the long run, all factors of production can be varied. An example of firms where the short run would be quite “short” would be a lawn-care business. Such a business could very quickly increase or decrease its stock of trucks, lawnmowers and labor. In contrast, electrical generating plants typically require several years to change the stock of generators. An airline may require a number of months to increase the stock of airplanes.


4) What is meant by the statement that “firms operate in the short run and plan in the long run”. Relate this statement to the operation operation of the college or university that you are attending.


Ans: This statement means that at any point in time, virtually all firms have one input that is fixed in amount. That is, they must make operating decisions based on that fixed input. Hence, it is said that they operate in the short run. However, most firms are planning or at least considering changes in the scale of their operations. Because this takes some time, it is said the firm is planning in the long run.


5) Legislation in the USA requires that most firms pay workers at least a specified minimum wage per hour. Use principles of marginal productivity to explain how such laws might affect the quantity of labor employed.


Ans: It has been shown that the profitmaximizing firm will hire labor until the marginal revenue product of that labor is equal to the wage rate. In general, to the extent that a legislated minimum wage would increase the wage rate above the wage rate that would have prevailed in the absence of such legislation, less labor l abor will be employed.


6) What would the isoquants look like if all inputs were nearly perfect substitutes in a production process? What if there was near zero substitutability substitutability between inputs?


Ans: If two inputs are perfect substitutes the isoquant is a straight line uniform slope or MRTS.


7) Explain why the isocost function will shift in a parallel fashion if the cost level changes, but the isocost will pivot about one of the intercepts if the price of either input changes.


Ans: The slope of the isocost is determined by the ratio of the input i nput prices. If the cost level changes but the input prices are constant, then the slope does not change – the function shifts parallel to the original function. However, if either input price changes, the ratio of the input prices change implying that the slope of the isocost function will change,.


8) Suppose wage rates at a firm are raised 10%. Use theoretical principles of production to show how the relative substitution of one input for another occurs as a result of the increased price of labor. Provide an example of how input substitution has been made in higher education.


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Ans: If wage rates increase by 10%, the slope of the isocost function will change. The original equilibrium point must occur at a tangency of the isoquant and isocost functions. When the slope of the isocost changes, the equilibrium point occurs where the capital-labor is higher. In general, the input rate for a factor that has increased in relative cost would fall relative to the input rate for the other input. Examples of input substitution in higher education include: a) substituting larger classrooms for additional professors and b) the substitution of movies and video tape lectures fotr personal lectures.


  

9) When estimating production functions, what would be some of the problems of measuring output and inputs for each of the following? A) A multiproduct firm B) A construction company C) An entire economy


Ans: A) For any firm, there may be a problem providing adequate measures of the capital and labor inputs. For example, both inputs come in varying kinds and qualities, and it is difficult to reduce these to a common unit. Labor comes in the form of different kinds of skills (ie. Carpenters, bricklayers, etc) and within those skills, the productivity of labor may vary significantly from person to person. The same problem applies to capital inputs. How does one adequately compare one hour of time on a drill press to an hour of time on a lathe?


B) Coming up with an aggregate measure of output also is difficult in some firms. For example, a multiproduct firm often produces a variety of disparate products. It may be difficult to reduce these outputs to a common unit. The same problems apply in a construction company where a variety of different labor skills and capital equipment are used to build houses and other structures.



C) For the economy as a whole, the basic problem is one of having to combine very disparate types of capital, labor and output into aggregate measures of these three variables.


10) The following table shows the relation ship between hours of study and final examination grades in each of three classes for a particular student, who has a total of 15 hours to prepare for these tests. If the objective is to maximize the average average grade in the three classes, how many hours should this student allocate to preparation for each of these classes? Explain your approach to this problem

Discussion Questions Manage rial Econom ics Hours

History

Grade

Hours

Chemist ry

Grade

Hours

Grade

Discussion Questions Managerial Economics Hours

Grade

History istory Hours

Grade

Chemist hemistry ry Hours

Grade

0

40

0.

50

0.

30

1.

50

1.

60

1.

50

2.

59

2.

69

2.

60

3.

67

3.

77

3.

66

4.

74

4.

84

4.

71

5.

79

5.

90

5.

74

6.

83

6.

95

6.

76

7.

86

7.

96

7.

77

8.

88

8.

97

8.

77

9.

89

9.

97

9.

77

10.

89

10. 10.

97

10. 10.

77

production theory.ppt

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