Lecture # 10 Inputs and Production Functions (cont.) Lecturer: Martin Paredes

1. Lecture # 10 Inputs and Production Functions (cont.) Lecturer: Martin Paredes

2. Outline • The Production Function (conclusion) • Elasticity of Substitution • Some Special Functional Forms • Returns to Scale • Technological Progress

3. Elasticity Of Substitution Definition: The elasticity of substitution measures how the capital-labor ratio, K/L, changes relative to the change in the MRTSL,K.  = % (K/L)= d (K/L) . MRTSL,K% MRTSL,K d MRTSL,K (K/L) • In other words, it measures how quickly the MRTSL,K changes as we move along an isoquant.

4. Elasticity Of Substitution Notes: • In other words, the elasticity of substitution measures how quickly the MRTSL,K changes as we move along an isoquant. • The capital-labor ratio (K/L) is the slope of any ray from the origin to the isoquant.

5. Example: Elasticity of Substitution • Suppose that… • At point A: MRTSAL,K = 4 KA/LA = 4 • At point B: MRTSBL,K = 1 KB/LB = 1 • What is the elasticity of substitution?

6. K Example: The Elasticity of Substitution MRTSA = 4 KA /LA = 4 • A Q L 0

7. K Example: The Elasticity of Substitution MRTSA KA /LA • A KB/LB = 1 • B Q MRTSB = 1 L 0

8. Example: Elasticity of Substitution • % (K/L) = -3 / 4 = - 75% • % MRTSL,K = -3 / 4 = - 75% •  = % (K/L)= - 75% = 1 • % MRTSL,K - 75%

9. Special Functional Forms • Linear Production Function Q = aL + bK where a,b are positive constants • Properties: • MRTSL,K = MPL = a (constant) MPK b • Constant returns to scale •  = 

10. K Example: Linear Production Function Q0 L 0

11. K Example: Linear Production Function Slope = -a/b Q1 Q0 L 0

12. Special Functional Forms • Fixed Proportions Production Function Q = min(aL, bK) where a,b are positive constants • Also called the Leontief Production Function • L-shaped isoquants • Properties: • MRTSL,K = 0 or  or undefined •  = 0

13. Frames Example: Fixed Proportion Production Function Q = 1 (bicycles) 1 0 Tires 2

14. Frames Example: Fixed Proportion Production Function 2 Q = 2 (bicycles) Q = 1 (bicycles) 1 0 Tires 2 4

15. Special Functional Forms • Cobb-Douglas Production Function Q = ALK where A, ,  are all positive constants • Properties: • MRTSL,K = MPL = AL-1K = K MPK ALK-1L •  = 1

16. K Example: Cobb-Douglas Production Function Q = Q0 0 L

17. K Example: Cobb-Douglas Production Function Q = Q1 Q = Q0 0 L

18. Special Functional Forms • Constant Elasticity of Substitution Production Function Q = (aL + bK)1/ where , ,  are all positive constants • In particular,  = (-1)/ • Properties: • If  = 0 => Leontieff case • If  = 1 => Cobb-Douglas case • If  =  => Linear case

19. K Example: The Elasticity of Substitution  = 0 L 0

20. K Example: The Elasticity of Substitution  = 0  =  L 0

21. K Example: The Elasticity of Substitution  = 0  = 1  =  L 0

22. K Example: The Elasticity of Substitution  = 0  = 0.5  = 1  =  L 0

23. K Example: The Elasticity of Substitution  = 0  = 0.5  = 1  = 5  =  L 0

24. K Example: The Elasticity of Substitution "The shape of the isoquant indicates the degree of substitutability of the inputs…"  = 0  = 0.5  = 1  = 5  =  L 0

25. Returns to Scale Definition: Returns to scale is the concept that tells us the percentage increase in output when all inputs are increased by a given percentage. Returns to scale = % Output . % ALL Inputs

26. Returns to Scale • Suppose we increase ALL inputs by a factor  • Suppose that, as a result, output increases by a factor . • Then: • If  >  ==> Increasing returns to scale • If  =  ==> Constant returns to scale • If  <  ==> Decreasing returns to scale.