production theory.ppt

Production TheoryProduction 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, Average and Marginal Product

Total product is the total amount of output produced in physical units like bushels of wheat or quintals of rice.

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.


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

Total Product

4,000

3,000

2,000

1,000

0

To

tal P

rod

uct

1 2 3 4 5

Labor

TP

Marginal Product

3,000

2,000

1,000

0

Ma

rgin

al P

rod

uct

(p

er

un

it o

f la

bo

r)

1 2 3 4 5

Labor

MP

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.


(1)Units of

Labor input

0

1

2

3

4

5

(2)Total

Product0

2000

3000

3500

3800

3900

(3)Marginal Product

2000

1000

500

300

100

(4)Average Product

2000

1500

1167

950

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 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.

Returns to Scale

Increasing returns to scale (also referred to as economies of 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.

Returns to Scale

Diminishing returns to scale occurs when a balanced increase of all inputs leads to a less-than-proportionate 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

To account for the role of time in 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

Production function is a purely engineering concept which is devoid of economic content. 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 Function

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 function) will be rendered uncompetitive and unprofitable. Hence, they go out of business.

Production Function

Economists use a 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 = AKL

Consider a production function with parameters A=100, =0.5 and =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

Production Function

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.

Production Table

8 283 400 490 565 632 693 748 800

7 265 374 458 529 592 648 700 748

6 245 346 424 490 548 600 648 693

5 224 316 387 447 500 548 592 632

4 200 283 346 400 447 490 529 565

3 173 245 300 346 387 424 458 490

2 141 200 245 283 316 346 374 400

1 100 141 173 200 224 245 265 283

1 2 3 4 5 6 7 8

K

L

Production Function

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

Production Function

This implies that there is substitutability between the factors of production. The firm can use a capital intensive production process 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.

Production Function

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.

Production Function

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.

Production Function

Although the production table provides considerable information on production possibilities, it does not allow for the determination of the 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.

Production Function

For a two-input production process, the total product of labor (TPL) 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)

Production Function

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

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

Production Function

The MP of capital is:

MPK = Q/ K

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

Production Function

For the general Cobb-Douglas function

Q = AKL

The marginal products areMPK = dQ/dK = AK -1L

andMPL = dQ/dL = AK L-1

Production Function

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 isAPK = 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

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.

Production Function

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 L2 and going through points A and C.

Relationships among the Product Functions

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 FunctionWith One Variable Input


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:

1. Marginal product reaches a maximum at G’ which corresponds to an inflection point


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.

2. 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.


This implies that the intersection must happen only at the highest value of AR.

3. 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 theVariable 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.


• 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)


Isoquants


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 TwoVariable Inputs

Economic Region of Production


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.


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 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)


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. 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 WageRateof 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,

C wK L

r r


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.


AB C = $100, w = r = $10 A’B’ C = $140, w = r = $10

A’’B’’ C = $80, w = r = $10 AB* C = $100, w = $5, r = $10


When both capital and labor are 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 are equal. Thus, the marginal rate of technical substitution (i.e., the slope of the isoquants) equals the price of labour divided by the price of capital. That is,

MRTS = w/rThe above identity is a necessary condition for efficient production.


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., -w/r) and that the slope of the isoquant is the negative of the ratio of the marginal product of labour to that of capital (i.e., -MPL/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

OrMPL/w = MPK/r


Effect of a Change in Input Prices

Sources of Economies of Scale

There are several reasons why increasing returns to scale occur. Firstly, technologies that are cost effective at high levels of production 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, increasing returns are also the result of inventory economies. Large firms may not need to increase inventories or replacement parts proportionately with size.

Economies of Scope

Firms often find that per unit costs are lower when two or more products are produced. Sometimes, the firm will have 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 profit-maximizing 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 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 may result in 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 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 profit-maximizing 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 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 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 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.


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 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


Managerial Economics

History Chemistry

Hours Grade Hours Grade Hours Grade


Managerial Economics

History Chemistry

Hours Grade Hours Grade 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. 97 10. 77

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