### Basic Reminders about Production Function

#### Production Function With Complementary Factors

(isoquant
).

#### Production Function With Substituable Factors

,
continuous and twice derivable,
,
,

and the Inada conditions:

#### Marginal Rate of Substitution

The differential of the production function
:

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

#### Elasticity of substitution

Elasticity of the substitution of Capital to Labor:

Hypothesis of the remuneration of factors to their
marginal productivity,

nominal cost of capital,

nominal rate of wage,

level of prices

#### Returns to Scale

. Constant if

. Decreasing if

. Increasing if

**Constant Returns to Scale**

Euler theorem:

that means

**Product Exhaustion Theorem**

Zero-Profit

Reasoning per capita:

per capita product (or mean productivity of labor)

per capita capital

Frontier of the prices of factors

#### Cobb-Douglas Function

with

If
,
constant returns to scale. Then

and frontier of the prices of factors:

#### C.E.S Function

Function of Constant Elasticity of Substitution (CES)

with
,
,

Constant returns to scale if
,
increasing if
,
decreasing if
.
is the scale elasticity. The per capita formulation of
the CES function with constant returns to
scale: