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Cobb – Douglas Production Function. Production Function. A production function describes a mapping from quantities of inputs to quantities of an output as generated by a production process.
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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.
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).
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
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.
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.
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
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
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
Cobb – Douglas Production Function Capital elasticity of production (b1) 0,583%. Labour elasticity of production (b2) 0,131%,
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.
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
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%
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
Cobb – Douglas Production Function Average product – production calculated on 1 unit of input: • Average product related to capital • Average product related to labour
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).
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).
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
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
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
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
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.
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.
