Abstract
Suppose that a firm has n goods to serve as inputs and/or outputs. If the firm uses \( y_{i}^{ - } \) units of good i as an input and produces \( y_{i}^{ + } \) units of the good, where \( y_{i}^{ - } \), \( y_{i}^{ + } \in {\mathbb{R}}_{ + } , \) then \( y_{i} \equiv y_{i}^{ + } - y_{i}^{ - } \) is the net output of good i.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
A weak version of technological efficiency is as follows: a production plan \( y \in {\mathbb{Y}} \) is technologically efficient if there is no \( \hat{y} \in {\mathbb{Y}} \) such that \( \hat{y} \gg y. \) See the definition of inequalities \( > \) and \( \gg \) in Notation and Terminology.
- 2.
This is from Arrow and Enthoven (1961, p. 796). For sufficient conditions of quasi-convexity when \( n > 2 \), see Wang (2008, Chap. 3).
- 3.
Denote \( F\left( {y_{1} ,y_{2} } \right) \equiv - \frac{{G_{{y_{1} }} \left( {y_{1} ,y_{2} } \right)}}{{G_{{y_{1} }} \left( {y_{1} ,y_{2} } \right)}}. \) Then, \( \frac{d}{{dy_{1} }}F\left[ {y_{1} ,y_{2} \left( {y_{1} } \right)} \right] = F_{{y_{1} }} + F_{{y_{2} }} \frac{{dy_{2} }}{{dy_{1} }}. \)
- 4.
The equation \( f^{{\prime }} \left( x \right) = f\left( x \right)/x \) gives the general solution \( f\left( x \right) = Ax, \) where \( A \) is an arbitrary constant, implying \( f^{{\prime }} \left( x \right) = A. \)
- 5.
See the Appendix for the definition of homogenous functions.
- 6.
This works since, after changes \( w_{1} \leftrightarrow w_{2} , \) \( x_{1} \leftrightarrow x_{2} , \) and \( a \leftrightarrow b, \) the original problem is the same.
- 7.
A function \( c\left( {w,y} \right) \) is a cost function if there exists an increasing and concave function f such that
\( c\left( {w,y} \right) \equiv \mathop { \hbox{min} }\limits_{x} \left\{ {w \cdot x|y = f\left( x \right)\begin{array}{*{20}c} \\ \\ \end{array} } \right\},\quad\forall \left( {w,y} \right) \in {\mathbb{R}}_{ + }^{n + 1} . \)
- 8.
See Fig. 10 for a graphic explanation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Wang, S. (2018). Producer Theory. In: Microeconomic Theory. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0041-7_1
Download citation
DOI: https://doi.org/10.1007/978-981-13-0041-7_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-0040-0
Online ISBN: 978-981-13-0041-7
eBook Packages: Economics and FinanceEconomics and Finance (R0)