Cobb-Douglas Production Function presentation.pptx

cobb doudlas production function descrioption

1. Cobb-Douglas Production Function Presented by: Sital Wagle Pardip Dhungana Nirmal Bastola Asmita kshettri Bijaya Bhurtel Rabindra Poudel Himal Thapa

2. Cobb-Douglas Production Function The Cobb-Douglas production function is based on empirical study of the American manufacturing industry made by two economists Paul H. Douglas and Charles W. Cobb. This production function basically described technical relationship between amount of input and amount of output. It takes only 2 inputs labor and capital for whole production.

3. Mathematically, Q = AL α K β Where, Q = Output K = Capital L = Labor α = Output elasticity for labor β = Output elasticity for capital A = Constant parameter.

4. This production function is based on Linear Homogeneous Production Function. It means constant return to scale

5. Properties Factor Intensity The factor intensity can be measured by taking the ratio between α and β . If > 1, there is a operation of labor intensive production technique. If , there is a operation of capital intensive production  

6. 2. Efficiency of production The efficiency of production can be measured by the coefficient A. If the value of A is higher, there is higher of efficiency of production. If the value of A is lower, there is lower degree of efficiency of production .

7. 3. Returns to scale R eturns to scale are defined as the change in output as factor inputs change in the same proportion. It is a long run concept. The various degrees of returns to scale can be measured by taking the sum of α and β. Let α +β= V If V >1, there is operation of increasing returns to scale. If V = 1, there is operation of constant returns to scale. If V < 1, there is operation of decreasing returns to scale.

8.   Types of Returns to Scale Decreasing Returns to Scale Constant Returns to Scale Increasing Returns to Scale Type equation here.  

9. 4. Average productivities of inputs a). The average productivity of labor (AP_L) is defined as the output (Q) per unit of labor (L). It can be calculated using the following formula: AP_L = Q / L = AL^αK^β / L Where, • Q is the total output. • L is the quantity of labor. • A is a constant representing total factor productivity. • α is the output elasticity of labor, which shows how much output responds to a change in labor. • K is the quantity of capital. • β is the output elasticity of capital, which shows how much output responds to a change in capital.

10. By simplifying the formula, We get, AP_L = AL^(α - 1) K^β This equation shows that the average productivity of labor depends on the levels of labor and capital and their respective elasticities.

11. b. Average Productivity of Capital (AP_K) The average productivity of capital (AP_K) is defined as the output (Q) per unit of capital (K). The formula is: AP_K = Q / K = AL^αK^β / K Simplifying the formula, we get: AP_K = AL^α K^(β - 1) This equation shows that the average productivity of capital depends on the levels of labor and capital and their respective elasticities.

12. 5. Marginal Productivities of inputs The concept of marginal products of inputs is crucial in understanding how additional units of inputs (like labor or capital) affect the output in a production process. In the context of the Cobb-Douglas production function, the marginal products are derived from taking the partial derivatives of the production function with respect to each input . Cobb-Douglas production function: Y=AlαKβ

13. A . Marginal Product of Labor (MP L ) The Marginal Product of Labor is the additional output produced as a result of adding one more unit of labor while keeping other inputs constant. Mathematically, it is the partial derivative of the production function with respect to labor (L). MPL = ∂Q/∂L= ∂ (AL α K β ) /∂L​= AK β .αL α-1 = α (AL α K β ) / L =α (AP L ) · A scales the MPL. · α the output elasticity of labor, indicates how responsive output is to changes in labor input. · Lα−1 shows the effect of the current level of labor on the MPL. · Kβ indicates that the level of capital affects the MPL.

14. B. The Marginal Product of Capital (MP K ) It is the additional output produced as a result of adding one more unit of capital while keeping other inputs constant. Mathematically, it is the partial derivative of the production function with respect to capital (K) MPK = =∂Q/∂K = ∂ (AL α K β )/ ∂K​=AL α .βK β-1 = β (AL α K β ) / K =β (AP K ) A scales the MPK. β, the output elasticity of capital, indicates how responsive output is to changes in capital input. Lα shows the effect of the current level of labor on the MPK. Kβ−1 shows the effect of the current level of capital on the MPK.

15. 6. The marginal rate of technical substitution The marginal rate of technical substitution (MRTS) is the measure with which one input factor is reduced while the next factor is increased without changing the output. It is an economic illustration that explains the level at which one factor of input must decline. While maintaining the same level of production, another factor of production is increased. It shows how you can replace one input with another input without altering the resulting output . NOTATIONS: MRTS(LK)​: Marginal Rate of Technical Substitution of Labor for Capital. MRTS(KL):Marginal Rate of Technical Substitution of Capital for LaboR . MP(L)​: Marginal Product of Labor. MP(K)​: Marginal Product of Capital. α : Output elasticity of labor. β: Output elasticity of capital. L: Labor input. K : Capital input. Q: Quantity of outpu t

16. A. MRTS(LK)​ ( MRTS of Labour for Capital ) MRTS(LK)=MP(L)/MP(K) = α K/ β ​L B. MRTS(KL) (MRTS of Capital for Labour ) MRTSKL​=MP(L)/​MP(K) ​​= α (K)/ β (L)​

17. In the fig, IQ is the isoquant representing the combinations A, B, C, D & E containing different units of two factors of production, capital & labour which yield the same level of output to the producer equal to 100 units.

18. 7. The elasticity of technical substitution The elasticity of technical substitution measures the responsiveness of the ratio of two inputs to changes in their marginal rate of technical substitution (MRTS). In simpler terms, it tells us how easily one input can be substituted for another in the production process when the relative efficiency of those inputs changes. Mathematical Representation; The elasticity of technical substitution is defined as: (σ) = = 1  

19. Where, ( K ) is the quantity of capital. ( L ) is the quantity of labor. ( MRTS ) is the marginal rate of technical substitution. This formula expresses the percentage change in the ratio of capital to labor (K/L) relative to the percentage change in the MRTS.

20. Importance of Cobb-Douglas 1 . It has been used widely in empirical studies of manufacturing industries and in inter-industry comparisons. 2. It is used to determine the relative shares of labor and capital in total output. 3. It is used to prove Euler's Theorem. 4. Its parameters a and b represent elasticity coefficients that are used for inter-sectorial comparisons. 5. This production function is linear homogenous of degree one which shows constant returns to scale if α+β, there are increasing returns to scale and if a +β<1, there are diminishing returns to scale. 6. Economists have extended this production function to more than two variables.

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