1. CDAE 254 - Class 17 Oct. 23 Last class: Result of the midterm exam 5. Production functions Today: 5. Production functions Next class: 5.Production functions 6. Costs Important date: Problem set 5: due Thursday, Nov. 1

2. 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.

3. 5. Productions 5. 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

4. 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. 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

6. 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:

7. 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 MP F = 0.5 MP M = 0.2 Example 2: q = 10 + 0.4 F - 0.01 F 2 MP F = 0.4 - 0.02 F

8. 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.

9. 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

10. 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

11. 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

12. 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?

13. 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

14. 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

15. 5.7. Changes in technology 5.7.1. A graphical analysis (1) The curve labeled by q 0 = 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

16. 5.7. Changes in technology 5.7.1. A graphical analysis (2) The curve labeled by q 0 * = 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

17. 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:

18. 5.7. Changes in technology 5.7.2. Technical progress vs. input substitution (1) Input substitution (move along q 0 = 100) e.g., from Point B to Point E: L reduced from ( ) to ( ) K increased from ( ) to ( ) AP L increased from ( ) to ( ) AP K reduced from ( ) to ( )

19. 5.7. Changes in technology 5.7.2. Technical progress vs. input substitution (2) Technical progress (move from q 0 = 100 to q 0 * = 100) e.g., from Point B to Point D: L reduced from ( ) to ( ) K has no change AP L increased from ( ) to ( ) AP K has no change

20. 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 =

21. 5.8. An example 5.8.3. How to construct (graph) an isoquant? -- For example q = 40 -- Simplify this function:

22. 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

23. 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