CDAE 254 - Class 17 Oct. 23

Last class: Result of the midterm exam 5. Production functions

Today: 5. Production functions

Next class:5. Production functions6. Costs

Important date:Problem set 5: due Thursday, Nov. 1

Problem set 5-- Due at the beginning of class on Thursday, Nov. 1

-- Please use graph paper to draw graphs

-- Please staple all pages together before you turn them in

-- Scores on problem sets that do not meet the above requirements will be discounted.

Problems 5.1., 5.2., 5.4., 5.6. and 5.8.

5. Productions5. Productions

5.1. Production decisions

5.2. Production functions

5.3. Marginal physical productivity

5.4. Isoquant and isoquant map

5.5. Return to scale

5.6. Input substitution

5.7. Changes in technology

5.8. An example

5.9. Applications

5.1. Production decisions 5.1.1. An overview of an economy

5.1.2. Definition of a firm

5.1.3. Production decisions of a firm

5.1.4. Decision making process

5.2. Production functions 5.2.1. What is a production function?

5.2.2. General notation

5.2.3. A simplified notation:

q = f (K, L)

5.2.4. An example

5.2.5. Limitations of production functions

5.3. Marginal physical productivity 5.3.1. What is marginal physical product?

of an input?

The change in output associated with a one-unit change in the input while

holding all other factors constant.

An example:

5.3. Marginal physical productivity 5.3.2. How to derive MP of an input?

Example 1:

q = 20 + 0.5 F + 10 L + 0.2 M

MPF = 0.5 MPM= 0.2

Example 2: q = 10 + 0.4 F - 0.01 F2

MPF = 0.4 - 0.02 F

5.3. Marginal physical productivity 5.3.3. Diminishing marginal physical

productivity As an input continues to increase,

the MP of the input will eventually decrease.

5.3. Marginal physical productivity 5.3.4. Relationship between total output

and MP:

-- A graphical analysis (Fig. 5.1)

-- Summary: When MP > 0, q is increasing

When MP = 0, q is at the highest

When MP < 0, q is decreasing

5.3. Marginal physical productivity 5.3.5. Marginal physical productivity

and average physical productivity -- What is AP?

-- Relationship between MP and AP:

when MP > AP, AP is increasing

when MP < AP, AP is decreasing

when AP = MP, AP is at the highest

5.4. Isoquant and isoquant map 5.4.1. A graphical analysis (Fig. 5.2)

5.4.2. What is an isoquant? A curve representing various

combinations of inputs that will produce the same amount of output.

Note: It is similar to an indifference curve

5.4.3. What is an isoquant map?

Note: It is similar to an indifference curve map

5.4. Isoquant and isoquant map 5.4.4. Rate of technical substitution (RTS)

RTS = - (Change in K)/(Chang in L) - slope of the isoquant

Note that RTS is a positive number and this is similar to the marginal rate of substitution (MRS)

5.4.5. How to calculate & interpret RTS?

5.5. Returns to scale 5.5.1. Definition: The rate at which output

increases in response to proportional increases in all inputs

5.5.2. Graphical analysis (Fig. 5.3):

(1) Constant returns to scale

(2) Decreasing returns to scale

(3) Increasing returns to scale

5.6. Input substitution 5.6.1. General situations (Fig. 5.2.)

5.6.2. Fixed-proportions (Fig. 5.4.)

5.6.3. Perfect-substitution

5.7. Changes in technology 5.7.1. A graphical analysis

(1) The curve labeled by q0 = 100

represents the isoquant of the old

technology:

100 units of the output can be produced by different combinations of L and K.

e.g., Point B: L= 20 and K= 20

Point E: L= 10 and K= 40

Point F: L= 30 and K= 14

5.7. Changes in technology 5.7.1. A graphical analysis (2) The curve labeled by q0* = 100

represents the isoquant of the new

technology:

100 units of the output can be produced by different combinations of L and K.

e.g., Point A: L= 15 and K= 14

Point C: L= 20 and K= 9

Point D: L= 10 and K= 20

5.7. Changes in technology 5.7.1. A graphical analysis

(3) Comparison of the two technologies in producing 100 units of the output:

From B to A:

From B to D:

From B to C:

From E to D:

From F to A:

5.7. Changes in technology 5.7.2. Technical progress vs. input

substitution (1) Input substitution (move along q0 = 100)

e.g., from Point B to Point E:

L reduced from ( ) to ( )

K increased from ( ) to ( )

APL increased from ( ) to ( )

APK reduced from ( ) to ( )

5.7. Changes in technology 5.7.2. Technical progress vs. input substitution

(2) Technical progress (move from q0 = 100 to q0* = 100)

e.g., from Point B to Point D:

L reduced from ( ) to ( )

K has no change

APL increased from ( ) to ( )

APK has no change

5.8. An example 5.8.1. Production function:

where q = hamburgers per hour

L = number of workers

K = the number of grills

5.8.2. What is the returns to scale of this function?

When L = 1 and K = 1, q =

when L = 2 and K = 2, q =

when L = 3 and K = 3, q =

LKq 10

5.8. An example 5.8.3. How to construct (graph) an isoquant?

-- For example q = 40

-- Simplify this function:

4010 LKq

4010 LK

4LK

16LK

5.8. An example 5.8.4. How to construct (graph) an isoquant?

-- Calculate K for each value of L (Table 5.3):

when L=1, K= ( )

when L=2, K= ( )

……

when L=10, K= ( )

-- Draw the isoquant of q=40

16LK LK 16

5.8. An example 5.8.5. Technical progress

-- A new production function:

-- Construct the new isoquant of q=40

when L=1, K= ( )

when L=2, K= ( )

when L=3, K= ( ) ……

-- Draw the new isoquant of q=40

LKq 20

LK2040 LK4