Chapter 9 page 1

Chapter 9: Production.Goal:

* Production Function –The relationship

between input and output.

+ Isoquant.

+ Average Product, Marginal Product.

Short run Production Function and the idea

of Diminishing Return.

Long run Production Function and the idea

of Return to Scale.

Topic 4 Extra page2

Objective: To examine how firm and industry supply curves are derived.

Introduction: Up to this point we have examined how the market demand function is derived. Next, we will examine the supply side of the market. We will explore how firms minimize costs and maximize productive efficiency in order to produce goods and services. By effectively combining labour and capital, the firm develops a production process with the objective of efficient resource allocation and cost minimization. The firm is assumed to produce a given output at minimum cost.

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Topic 4 Extra page3

The Production Function Definitions: Factors of Production: are factors used to produce output. Example: Labour Capital - machines -buildings Land Natural resources State of technology: consists of existing knowledge about method of production.

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The quantity that a firm can produce with its factors of production depends on the state of technology. The relationship between factors of production and the output that is created is referred to as the production function.

“The production function describes the maximum quantity of output that can be

produced with each combination of factors of production given the state of technology.”

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Topic 4 Extra page5

For any product, the production function is a table, a graph or an equation showing the maximum output rate of the product that can be achieved from any specified set of usage rates of inputs. The production function summarizes the characteristics of existing technology at a given time; it shows the technological constraints that the firm must deal with.

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Mathematically,

Let x1 and x2 be inputs then output is produced by the following:

Y = F(x1,x2).

Frequently encountered inputs: Capital (K) and Labor (L)

4Ch 9: Production

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The Production“Mountain”

5Ch 9: Production

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Important production functions

1. The Leontief technology (fixed-proportions

technology)

Y(x1,x2) = min(ax1,bx2)

Example: Y(K,L) = min(1/6L,K)

2. The Cobb-Douglas technology (More realistic)

Example Cobb-Douglas p.f.:

3. Perfectly substitutable inputs p.f.:

Y(K,L)=aK+bL

2/12/1 LKY

6Ch 9: Production

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Production Function in General.

Production function describes the

relationship between inputs and output.

The production function Frequently encountered inputs: Capital (K) and

Labour (L)production function = Y(K,L).

◦ Idea of fixed and variable inputs.

Example: Land is fixed input and Farmers are

variable input. In the above case, capital is fixed and

labour is variable.

Production Function in General

Important production functions

1. The Leontief technology (fixed-proportions technology)

◦ Y(x1,x2) = min(ax1,bx2)

◦ Example: Y(K,L) = min(1/6L,K)

2. The Cobb-Douglas technology

◦ Found to be realistic

◦ Example Cobb-Douglas p.f.:

3. Perfectly substitutable inputs p.f.: Y(K,L)=aK+bL

◦ Example: Y(K,L)=2.5K+L

Production Function in General.

Total Product: Represents the relationship between a

variable input and total output given other inputs.

Marginal Product: The change in the total

product that occurs in response to a unit

change in the variable input (all other inputs

held fixed).

Average Product (of a variable input): is defined

as the total product divided by the quantity of

that input.

◦ Geometrically, it is the slope of the line joining the

origin to the corresponding point on the total

product curve.

Short run Production Function.

Short run: the longest period of time during which at

least one of the inputs used in a production process

cannot be varied.

Assuming K is fixed at K0, the short run production

function is then Y(K,L)= K0L β

Law of diminishing return: if other inputs are fixed, the

increase in output from an increase in the variable input

must eventually decline. In other word, the marginal

product will increase then decrease as more variable

input is added. This is a short run phenomenon.

Short run Production Function.

Short run Production Function.

Short run Production Function

Exercise:

◦ A firm/s short-run production function is

given by: for

for

Sketch the production function.

Find the maximum attainable production. How much

labour is used at that level?

Identify the ranges of L utilization over which the MPL is

increasing and decreasing.

Identify the range over which the MPL is negative.

25

4Q L 0 2L

213

4Q L L 2 7L

Short run Production Function.

Key things to remember:

◦ When marginal product curve lies above the

average product curve, the average product

curve must be rising and vice versa.

◦ If there are more than one production

processes, allocate the resource so that the

marginal products are the same across

different production processes.

◦ Always allocate resource to the activity that

yields higher marginal product.

Long run Production Function.

Long run: The shortest period of time required

to alter the amounts of all inputs used in a

production process.

Y(K,L)= KαLβ . Now both K and L are variable

inputs.

Graphical expression of the production function

– the isoquant

◦ Isoquant is the set of all input combinations that yield

a given level of output. Similar to the indifference

curve.

Long run Production Function.

Marginal Rate of Technical Substitution.

Returns to scale:

◦ What happens if we doubled all inputs?

Output doubles too constant returns to scale.

Y(cK,cL) = cY(K,L)

Output less than doubles decreasing returns to scale.

Y(cK,cL) < cY(K,L)

Output more than doubles increasing returns to scale.

Y(cK,cL) > cY(K,L)

Production Function

Exercise:

◦ Consider the following production function

Determine whether they exhibit

diminishing/constant or increasing marginal return

of labour.

Determine whether they exhibit

decreasing/constant/increasing return to scale.

0.7 0.2Q K L

2Q K L

The Marginal Rate of Technical Substitution

The marginal rate of technical substitution (MRTS) measures the rate of substitution of one factor for another along an isoquant.

“The marginal rate of technical substitution is the rate at which a firm can substitute capital

and labour for one another such that the output is constant.”

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Note: An isoquant cannot have a positive slope.

An increase in one factor of production causes output to increase. Hence this increase in one factor must be offset by a decrease in other factor in order to keep output at the same level.

As the firm moves along the isoquant from left to right, the slope decreases. The firm substitutes labour for capital, but at a diminishing rate. When this occurs there is a diminishing marginal rate of technical substitution.

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Returns to Scale and the Cobb-Douglas Production Function A common production function is the Cobb-Douglas production function. Algebraically: q=ALaKb where A, a and b are constant and greater than zero.

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To determine the returns to scale for this function, we could change labour and capital by a factor ‘m’ and then determine if output changes by more than, equal to or less than ‘m’ times. q=A (mL)a(mK)b q=AmaLam b K b q=ma+b[ALaK b] Since originally output was q=ALaKb, we can determine if output will increase by either less than m times if a+b<1 (because ma+b<m), by exactly m times if a+b=1 or by more then m times if a+b>1.

Chapter 9 page 50