Basic Reminders about Production Function

Production Function With Complementary Factors

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(isoquant $Y=1$).

Production Function With Substituable Factors

$Y=F(K,L)$, $F$ continuous and twice derivable, $F_{K}^{\prime }>0$, $F_{L}^{\prime }>0$,MATH

and the Inada conditions:

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Marginal Rate of Substitution

The differential of the production function $Y=F(K,L)$:

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along an isoquant $dY=0$ and the Marginal Rate of Substitution (MRS) of Capital to Labor can be defined by

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Elasticity of substitution

Elasticity of the substitution of Capital to Labor:

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Hypothesis of the remuneration of factors to their marginal productivity,

$u$ nominal cost of capital,

$w$ nominal rate of wage,

$p$ level of prices

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Returns to Scale

. Constant if $\forall \lambda ,$ MATH

. Decreasing if MATH MATH

. Increasing if MATH MATH

Constant Returns to Scale

Euler theorem:

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that means

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Product Exhaustion Theorem

Zero-Profit

Reasoning per capita:

$y=Y/L$ per capita product (or mean productivity of labor)

$k=K/L$ per capita capital

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Frontier of the prices of factors

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Cobb-Douglas Function

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with $0<\alpha ,$ $\beta <1$

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If $\beta =1-\alpha $, constant returns to scale. Then

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and frontier of the prices of factors:

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C.E.S Function

Function of Constant Elasticity of Substitution (CES)

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with $0<a<1$, $\gamma >-1$, $\gamma \neq 0$

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Constant returns to scale if $\gamma \mu =1$, increasing if $\gamma \mu >1$, decreasing if $\gamma \mu <1$. $\gamma \mu $ is the scale elasticity. The per capita formulation of the CES function with constant returns to scale:MATH