### Basic Reminders about Production Function

(isoquant ).

#### Production Function With Substituable Factors

, continuous and twice derivable, , ,

#### 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:
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9336621761322021, "perplexity": 15024.33894493833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588242.22/warc/CC-MAIN-20211027181907-20211027211907-00023.warc.gz"}