A Holistic Visual "Wheel of Relationships"
in Production Theory
Alexandra Naumenko*, [email protected]
Seyyed Ali Zeytoon Nejad Moosavian*, [email protected]
*. Ph.D. Students in Economics, Department of Economics, North Carolina State University
Abstract: Production theory in advanced microeconomics
Symbols and Notations:
comprises two intricate optimization problems: cost
minimization and profit maximization. The cost minimization
problem essentially involves deriving conditional factor demand
functions and the cost function. The profit maximization problem
involves deriving the output supply function, the profit function,
and unconditional factor demand functions. Additionally, there are
numerous mathematical equations in the setting of production
theory which aid us in deriving each of the mentioned functions
from another. These include Shephard’s lemma, Hotelling’s
lemmas, direct and indirect mathematical relations, and other
mathematical elements. All of these elements contribute to
forming a neat wheel of relationships in production theory.
However, these complexities and unseen, underlying linkages
can bring about difficulties for both instructors to teach as well as
students to learn the material. Much of this complexity has its root
in the intricacies inherent in the mathematical theory of duality.
The primary purpose of this poster is to illuminate these
theoretical complexities through a holistic visual wheel of
relationships in production theory in order to ease the teaching
and learning of advanced production theory. More specifically,
this poster introduces two instructional visuals which can be used
as complementary teaching tools to demonstrate the connections
between the profit-maximization and cost-minimization problems
as well as other components of production theory in modern
microeconomics. This visual clarification also allows for easier
elaboration of three notions of optimality which must exist in any
economic production system, i.e. technical optimality,
allocative optimality, and scale optimality. The focus of the
present poster is on the two extreme market structures: perfect
competition and monopoly.
max: Maximize
min: Minimize
s.t.: Subject to
x: Vector of Input Quantities
W: Vector of Input Prices
y: Firm’s Output Quantity (in general and also for a price-taking firm)
Y: Market Output Quantity (in general sense and also for a monopolistic firm)
C(W,x): Cost Relation (OR, the Amount of Cost)
y=f(x): Production Function
xC (W,y): Vector of Conditional Factor Demands (conditional on output)
C(W,y): “The” Cost Function (OR, total cost function, i.e. TC)
AC: Average Cost Function
AVC: Average Variable Cost Function
MC: Marginal Cost Function
Zero-Profit Point: The point at which AC=MC
Shutdown Point: The point at which AVC=MC
PZero-Profit: The price corresponding to the Zero-Profit point
PShutdown: The price corresponding to the Shutdown point
PPC: The optimal price for a perfectly competitive firm
PMON: The optimal price for a monopolistic firm
yZP: The output level corresponding to the Zero-Profit point
ySD: The output level corresponding to the Shutdown point
y*PC: The optimal quantity for a perfectly competitive firm
y*MON: The optimal quantity for a monopolistic firm
π(P,W,y): Profit Relation for a perfectly competitive firm
π(P(Y),W,Y): Profit Relation for a monopolistic firm, which finally reduces to
π(W,Y)
TR: Total Revenue which equals P.Y for a perfectly competitive firm and P(Y).Y
for a monopolistic firm
AR: Average Revenue
MR: Marginal Revenue
y*(P,W): Output Schedule or Supply Function for a perfectly competitive firm
P: Output Price for a perfectly competitive firm
P(Y): Inverse Demand Function for the output produced by a monopolistic firm
Y*(W): Output quantity schedule in terms of input prices for a monopolistic firm
P*(W): Output price schedule in terms of input prices for a monopolistic firm
XU (W,y): Unconditional Factor Demand for a perfectly competitive firm
XU (W): Unconditional Factor Demand for a monopolistic firm
Π(P,W): “The” Profit Function for a perfectly competitive firm, in which
everything is in terms of prices
Π(W): “The” Profit Function for a monopolistic firm
Figure 1) Spectrum of Firm Types and Market Structures
Number of
Firms in the
Market
1
2
∞
…
Towards More Market Power
Towards More Competition
Monopolistic
Competition
Imperfect
Competition
Duopoly
Monopoly
Perfect
Competition
Intermediate Cases
(which involve strategic interactions):
Studied Based on the Conventional Microeconomic Analysis
as Well as Game-Theoretic Analyses
Two Polar Cases (Extreme Cases):
Studied Based on the Conventional Microeconomic Analysis
Figure 3) Wheel of Relationships in Production Theory
(Case of Perfect Competition)
Figure 2) Wheel of Relationships in Production Theory
(Case of Monopoly)
Cost Minimization:
Profit Maximization:
Cost Minimization:
Profit Maximization:
A General Story Applicable to Any Type
of Firms in Any Markets
A Specific Story Applicable to
a Monopolistic Firm
A General Story Applicable to Any Type
of Firms in Any Markets
A Specific Story Applicable to a Price-Taking, ProfitMaximizing Firm in a Perfectly Competitive Market
s.t. y=f(x)
 Invert x (W , y )
c
Constrained
Minimization
Through
Lagrangian
𝒙
𝒎𝒂𝒙 π(P(Y),W,Y) = P(Y).Y − C(W,Y)
𝒀
Cost,
Rev. &
Price
𝒙
𝒎𝒊𝒏C(W,x)=W.x
PMON
AC
to get W ( x c , y )
 C (W , x) 
C (W ( x c , y ), y ))
xC (W,y)
 Invert Y*(W) to get W(Y)
 Work with Π(W)=TR(W)-TC(W)
 Substitute W(Y) into TR(W) to get the TR(Y)
MC
D=AR=P
Unconstrained
Maximization
term of π(W,Y)
 Substitute W(Y) into TC(W) to get TC(Y),
MR
YMON
P*(W)
𝒚
 Invert x c (W , y )
Constrained
Minimization
Through
Lagrangian
to get W ( x c , y )
 C (W , x) 
P*(W)=P(Y*(W))
AC
PPC
Unconstrained
Maximization
D=AR=P=MR
y*PC

Invert y*(P,W) for
each of its arguments

Substitute W(y.P)
into the TR term of Π(P,W)

C (W ( x , y ), y ))
Substitute P(y.W)
into the TC term of Π(P,W)
Output (y)
xC (W,y)
xU (W)
Y*(W)
MC
c
then using W(Y) write TC(W) as W.h(Y), which
is the cost function, to get the complete π(W,Y)
Output (Y)
𝒎𝒂𝒙 π(P,W,y) = P.y − C(W,y)
s.t. y=f(x)
Cost,
Rev. &
Price
𝒎𝒊𝒏C(W,x)=W.x
y*(P,W)
xU (P,W)
xc (W,y*(P,W)) = xU (P,W)
xc (W,Y*(W)) = xU (W)
C (W , y )  C (W , x c (W , y ))
C (W , y )

Shephard’s

W
i
Lemma
xic (W , y )
OR
C (W , y )  W .x (W , y )
c
AC
AVC
MC
 x (W )
U
i
Point
MC
AC
PZero-Profit
AVC
PShutdown
Shutdown
 ( P, W ) 
 ( P,W , y * ( P,W ))
Hotelling’s
Lemma
 ( P, W )

P
y * ( P, W )
 ( P, W )

Hotelling’s
Wi
Lemma
 xU ( P , W )
i
Π(P,W) = P.y*(P,W) - C(W,y*(P,W))
Zero-Profit
Point
MC
AC
AVC
Shutdown
Point
Created By:
Alexandra Naumenko ([email protected])
Ali Zeytoon Nejad ([email protected])
ySD yZP Output (y)
Figure 4) The Process of Determining the Potential Market Outcomes
Firm and Market Types
Perfect Competition:
A Price-Taking, ProfitMaximizing Firm in a Perfectly
Competitive Market
Identify
• the setting of
the markets
• and the type of
the firm
xic (W , y )
C(W,y)
PZero-Profit
Point
Problem and
Context Identification
Lemma
C (W , y )  W .x c (W , y )
AC
AVC
MC
Zero-Profit
C (W , y )

Shephard’s
Wi
OR
Cost &
Price
Cost &
Price
Hotelling’s
Lemma for
Monopoly
 (W )

Wi
Π(W) = P*(W).Y*(W) - C(W,Y*(W))
C(W,y)
PShutdown
 (W ) 
 ( P * (W ),W , Y * (W ))
C (W , y )  C (W , x c (W , y ))
Monopoly:
A Price-Determining, ProfitMaximizing Firm in a
Monopolistic Output Market
Inputs to
the Theory
• Production function form
“y=f(x)”
• Input prices “W”
Our Black Box
(Cost and
Production
Theory)
Technical
Optimality
Cost
and
Production
Theory
• Production function form
“y=f(x)”
• Input prices “W”
• Inverse demand function
for output “P(Y)”
Figure 5) Three Notions of Optimality in Production Theory
(Physical)
Outputs of
the Theory
• When to shutdown in SR/LR
• Optimal Output OR Optimal
Scale of Production (y*)
• Optimal input (x*)
• Maximized amount of profit
• Output market price “P”
ySD yZP Output (y)
Technical
Optimality
(Input)
Allocative
Optimality
(Π*)
Technical
Optimality
Cost
and
Production
Theory
• Optimal Output OR Optimal
Scale of Production (Y*)
• Optimal input (x*)
• Optimal output price (P*)
• Maximized amount of profit
(Π*)
(Output)
Scale/Size
Optimality
Created By:
Alexandra Naumenko ([email protected])
Ali Zeytoon Nejad ([email protected])