Cobb – Douglas Production Function

1. Cobb – DouglasProduction Function

2. Production Function A production function describes a mapping from quantities of inputs to quantities of an output as generated by a production process. Definition: A standard production function which is applied to describe much output two inputs into a production process make. It is used commonly in both macro and micro examples.

3. Cobb – DouglasProduction Function • is an econometric model that shows relation between scale of production and scale of inputs essential to the production • These essential inputs are: labour and capital(in agricultural research the area of farming is considered to be the third input).

4. Cobb – Douglas Production Function For capital K, labor input L, and constants b0, b1, and b2, the Cobb-Douglas production function is: where: Qt – production Kt – capital Lt – labour b0, b1, b2– parameters of production function Et– error term

5. Cobb – Douglas Production Function Log-linearization simplifies the function,meaning just that taking logs of both sides of a Cobb-Douglass function gives one better separation of the components.

6. Cobb – Douglas Production Function • To obtain the estimators, OLS in matrix notation should be used. • To interpret your result and to make futher analysis of characteristics of production function, antilog for constant term should be calculated.

7. Cobb – Douglas Production Function where

8. Cobb – Douglas Production Function where

9. Cobb – Douglas Production Function where: Qt – production, in thousand zloty Kt – capital, in million zloty Lt – labour, the number of employees b0, b1, b2– parameters to be estimated Et– error term Log-linearization

10. Cobb – Douglas Production Function

11. Cobb – Douglas Production Function

12. Cobb – Douglas Production Function

13. Cobb – Douglas Production Function

14. Cobb – Douglas Production Function

15. Cobb – Douglas Production Function

16. Cobb – Douglas Production Function

17. Cobb – Douglas Production Function The analysis of production function consists of examinating such characteristics as: • Elasticity of production • Returns to scale • Total product • Average product • Marginal product • Marginal rate of substitution

18. Cobb – Douglas Production Function Elasticity of production • Capital elasticity of production – average change in production (in %) associated with a 1%increament in capital, with the labour held constant; • Labour elasticity of production– average change in production(in %) associated with a 1%increament in labour, with the capital held constant

19. Cobb – Douglas Production Function Capital elasticity of production (b1) 0,583%. Labour elasticity of production (b2) 0,131%,

20. Cobb – Douglas Production Function • If capital increases 1 % and labour will be unchanged, the production will increase 0,583%. • If labour increases 1%, the production will increase 0,131%, holding capital constant.

21. Cobb – Douglas Production Function Returns to scale • Fixed income in relation to production scale – constant returns to scale, outputs increase as fast as inputs • Increasing returns to scale – outputs increase faster than inputs • Decreasing returns to scale – outputs increase slower than inputs

22. Cobb – Douglas Production Function • If labour increases 4% and capital increases 2,5%, the production will increase 1,98% • If labour decreases 3% and capital decreases 1,8%, the production will decrease 1,44%

23. Cobb – Douglas Production Function Total product – total scale of production calculated on given quantity of each input Assume K=45 mln zl, L=900 employees If the company employes 900 people and its capital is 45 mln zl, the total production is equal 2 043 280 zloty

24. Cobb – Douglas Production Function Average product – production calculated on 1 unit of input: • Average product related to capital • Average product related to labour

25. Cobb – Douglas Production Function Average product – production calculated on 1 unit of input: • Average product related to capital If capital is equal 45 mln zland labour 900 people, production of45406 zł can be reached from one unit of input (1 mln zl).

26. Cobb – Douglas Production Function Average product – production calculated on 1 unit of input: • Average product related to labour If capital is equal 45 mln zland labour 900 people, production of2 270 zł can be reached from one unit of input (1 person).

27. Cobb – Douglas Production Function Marginal product – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal product related to capital • Marginal product related to labour

28. Cobb – Douglas Production Function Marginal product – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal product related to capital If capital increases 1 unit (1 mln zl over 45 mln zl) and labour is constant the production will increase 26472 zł

29. Cobb – Douglas Production Function Marginal product – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal product related to labour If labour increases 1 person (over 900) and capital is constant the production will increase 297 zł

30. Cobb – Douglas Production Function Marginal rate of substitution – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal rate of substitution related to capital • Marginal rate of substitution related to labour

31. Cobb – Douglas Production Function Marginal rate of substitution – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal rate of substitution related to capital (substitute capital for labour) 1 unit of capital (1 mln of capital) can be substituted for the employment level increased by 89 employees without changing the production scale.

32. Cobb – Douglas Production Function Marginal rate of substitution – shows how will respond the scale of production if we change quantity of one input and others stay unchanged • Marginal rate of substitution related to labour (substitute labour for capital) 1 unit of labour (1 person) can be substituted for the capital level increased by 0,011 mln zł without changing the production scale.