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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.
Chapter 9 page 4
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.
Chapter 9 page 5
Topic 4 Extra page4
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.”
Chapter 9 page 6
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.
Chapter 9 page 7
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
Chapter 9 page 9
The Production“Mountain”
5Ch 9: Production
Chapter 9 page 10
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
Chapter 9 page 11
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.”
Chapter 9 page 41
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.
Chapter 9 page 42
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.
Chapter 9 page 49
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