Unit 6.
Analyses of SR Costs & Profits
as Functions of Output Q
‘Liquid Gold’ Economics?

Recent increases in crude oil prices
have prompted much interest in trying
to figure out their likely consequences
on fuel prices ‘at the pump’. Politicians
and others often wonder if pump price
increases are ‘out of line’ with actual
increases in the cost of the crude oil
input. Are their concerns warranted?
No More Babies?

The CEO of Memorial Hospital recently conducted
financial reviews of all departments in the hospital.
During the review process, the head of the obstetrics
unit proposed trying to increase the number of
babies delivered in the department to make it more
profitable. After reviewing the department’s financial
statements for the previous month, the CEO
discovered the unit delivered 540 babies that
generated total costs of $3.132 million and total
revenues of $2.754 million. The CEO raised the
question as to why they would want to increase the
number of deliveries when the unit was already
losing $700 per delivery? How should the unit’s head
respond to this concern?
New Product Launch Advice

Assume Compaq is scheduled to launch next
month a new server at a cost of $5,500. This
server will be competing against Dell’s version
that was just introduced to the market. Dell’s
server has basically the same features (and
even a few more) for a cost of $4,500. To
date, Compaq has invested more than $2.5
million in the development of its new server.
What advice would you give Compaq on
launching its new server, keeping in mind all
the development money the company has
already invested into the product?
How to Produce?

Several years ago, John Deere was about to begin building
a capital-intensive factory to produce large, four-wheeldrive farm tractors. Then, grain prices dropped
dramatically which reduced tractor demand. Deere
management considered 1) stopping the construction of
its own factory and, instead, 2) purchasing a Canadian
company that could add to their tractor assembly capacity.
Management recognized the company would have higher
fixed costs, but lower marginal costs if it were to go ahead
with construction of its own plant. Which course of action
would you have recommended be pursued by Deere
management?
Lower Price to Sell More?

Joe is the district sales manager for Agri Green. He
has five sales representatives, each with their own
geographical territory, reporting directly to him. One
of his reps has noted it has become increasingly
difficult to compete against other products with the
company’s current stance on maintaining relatively
high prices by industry standards. The rep has
proposed permission to cut price by 10%. With
current prices, the company’s profit margin is 25%.
The sales rep is confident he/she could sell 50%
more product with the 10% price reduction. Should
Agri Green give the sales rep permission to sell at a
10% lower price?
“Gentlemen, Stop Your Engines”

Decker Truck Lines owns and operates about
600 semi tractor-trailers. Rising diesel fuel
prices have been cutting into the company’s
profit. Management is looking at alternative
ways of reducing diesel fuel expenses. One
strategy being considered is to offer drivers
incentives (bonuses) to reduce idling time while
out on the road. What specific information
would be needed to implement such a plan and
when would this plan result in increased profits
for the company?
Revenue/Cost Analysis

Cy Shops’ manager has provided you with the
following information for the business (q =
units of product sold)
TR = 44q – q2
TVC = 4q
TFC = 120
The manager wants you to calculate the level
of quantity sales that will result in the
company a) breaking even, b) maximizing its
profit, and c) maximizing its sales. What do
you tell the manager?
Costs of Production
(Overview of Reality)

Production costs are determined by
1) the productivity of inputs used by a
business firm and
2) the prices paid for inputs used. The more
productive the inputs are (i.e. the more
efficient the production process is), the
lower the costs of production will be.
Likewise, lower input prices also result in
lower costs of production.
Cost Concepts

SR and LR




Cash and Noncash

SR  fixed & variable
LR  variable only
Fixed and Variable




Fixed  don’t change
w/output
Variable  vary w/output
Cash
= ‘explicit’
Noncash = ‘implicit’
= ‘opportunity’
foregone
= ‘lost’ income
Total & Average & Marginal
TFC, TVC, TC
AFC, AVC, ATC
MC
Opportunity Cost Examples
Activity
Opportunity Cost
Work at home
Lost wages
Own & operate a business
Lost wages
Lost interest
Own & operate farm land
Lost rent
Lost interest
Own & operate machinery
Lost interest
Lost rent
Attend college
Lost wages
Skip class/party
Lost knowledge
Lower grade
Go to class/study
Lost work
Lost sleep
Cost Concepts

1. Total
=

2. Average
=

3. Marginal
=
=
total dollar cost associated
with a given q of output
dollar cost PER UNIT
OF OUTPUT
ADDITIONAL COST
per unit of ADDITIONAL
OUTPUT
added cost of producing
one more unit of output
Cost Graphs
Graphical Derivation of TVC from
TP (L = variable input)
1.
2.
3.
Multiply L by W to get TVC
Rotate graph 90° counter clockwise
Flip graph 180°
Cost Graphs (cont’d)
Cost Graphs (cont’d)
General Cost Equations
Cost Concept
Average
Total
Fixed
AFC = TFC/Q
TFC = AFC · Q
Variable
AVC = TVC/Q
TVC = AVC · Q
Total
ATC = TC/Q
TC = ATC · Q
TFC in Avg Cost Graph
Total
TVC in Avg Cost Graph
TC in Avg Cost Graph
Solving for TVC as function of q of output
given production function equation:

Step #1:


Solve for L as a function of q given the
production function equation (i.e. solve for
the inverse equation)
Step #2:

In the TVC equation, TVC = wL, substitute
the L as a function of q equation for L
Calculating Cost Equations
from Production Info


Assume
AFC
= TFC/q = 1000/q
q = 50L [L=(1/50)q = .02q]
w = $20,000
TFC = $1,000
AVC
= TVC/q = 400q/q = 400
ATC
= AFC + AVC
= 1000/q + 400
MC
=
Calculations
TVC = w•L(q)
= w (.02q)
= (20,000)(.02q)
= 400q
TC
= TFC + TVC
= 1000 + 400q
 T C = 400
q
Oil Production & Cost Questions
1.
2.
If there are 44 gallons (output = Q) of oil
per barrel (input = B), what is the
corresponding production function
equation?
Given the price of a barrel of oil, what is the
TVC and AVC equations for producing
gallons of oil, and how do these changes
with changes in the price of a barrel of oil?
Review of some cost & production fn
concept relationships
MC 
 TC
Q
AVC 

TVC
WL
Q

WL
Q
Q

W
M PL
W

APL
q  f ( k , L )  SR production fn
L = f(q)
TVC=WL=wf(q)
SR Profit Max (output)

 = TR – TC
max
 /q = 0
 TR/q - TC/q = 0
 MR – MC = 0
 MR = MC
Optimal Output Level
Profit-maximizing level of output
 A manager should keep producing
additional output up to the point where
the additional income equals the
additional cost from the last unit
 MR = MC

NOTE: Optimal Input Level
(e.g. labor)

 MRP = MFC

 MPL • MR = w
w

 MR =

 MR = MC
M PL
MRP vs MR

MRP
=
additional revenue per
additional unit of input

MR
=
additional revenue per
additional unit of output
MFC vs MC

MFC
=
additional cost per
additional unit of input
(= marginal factor cost)

MC
=
additional cost per
additional unit of output
Profit Max Input Side = Profit Max Output Side
Profit Max-Output Side
(Alternative Cases)
Case
TR
TC
1
Linear
Linear
2
Linear
Nonlinear
3
Nonlinear
Linear
4
Nonlinear
Nonlinear
Profit Max Level of Output


Nonlinear TR & Nonlinear TC 
Decreasing MR, Increasing MC
The ‘Profit’ Equation

=
=
=
=
=
=
=
TR – TC
TR – TVC - TFC
PQ – (AVC)Q – TFC
(P-AVC)Q – TFC
(P-AVC)Q – (AFC)Q
(P-AVC-AFC)Q
(P-ATC)Q
P Setter π
P Taker π
Four Math Cases/Examples of
Profit Maximization
Case
TR
MR
TFC
TVC
TC
MC
1
10q
10
120
4q
120+4q
4
2
10q
10
120
.1q2
120+.1q2
.2q
3
44q-q2
44-2q
120
4q
120+4q
4
4
44q-q2
44-2q
120
.1q2
120+.1q2
.2q
Breakeven (B.E.) Analysis
=0
  TR – TVC – TFC = 0
  PQ – (AVC)Q – TFC = 0
 equation w/4 variables
(P, Q, AVC, TFC)
 given any 3, solve for 4th
 B.E. Q = TFC/(P-AVC)
 B.E. P = AVC + AFC
Note:
 analysis   = desired amt

Case #1 – Breakeven Q
 TR = TC
  10q = 120 + 4q
  6q = 120
  q = 20
TR
Check:

= 10q
= 10(20)
= 200
- TFC
-120
-120
-120
- TVC
-4q
-4 (20)
-80
=
=
=
=0
 B.E. Q due to  P to $8
(From $0)
TR = TC
 8q = 120 + 4q
 4q = 120  q = 30
Case #1  Max
MR = 10 > MC = 4
  Keep increasing q to increase profit

TR for a P Setting Firm (sets P but
Q sold is variable) e.g. P2 < P1
$
TR1 (P = P1)
TR2 (P = P2)
Q
Case #2 -  Max



MR = MC
 10 = .2q
 Max  =
 q* = 50
TR
= 10q
= 10(50)
= 500
- TFC
-120
-120
-120
- TVC
-.1q2
-.1(50)2
-250
= 130
Quadratic Formula

= formula that finds values of X that
result in a quadratic equation’s value
=0

Equation: aX2 + bX + c = 0

Formula: X =
 b  and 
2a
( b  4 ac )
2
Case #2 Breakeven Q



TR = TC
 10q = 120 + .1q2
 .1q2 – 10q + 120 = 0
 a=.1, b=-10, c=120
 q 
 (  10 ) 
2
2 (.1)

10 
100  48
.2

10 
52
.2
Check:
q=86.06TR-TC=10(86.06)-120-.1(86.06)2
= 860.6-120-740.6=0
Q=13.95TR-TC=10(13.95)-120-.1(13.95)2
= 139.50-120-19.50=0
(  10 )  4 (.1)(120 )

10  7 .211
.2
 86 .06 and 13.95
Stay-even Analysis
=> Determining the volume required to offset
a change in costs, prices, or other factors.
=> Set profit equations equal and solve for
unknown.
=> Π1 = Π2
=> P1Q1 – AVC1Q1 – TFC = P2Q2 – AVC2 Q2 TFC
For which of the following situations would the
farmer produce corn in the SR?

A. Price of corn = $2.00, AVC = $1, TFC = $100

B. Price of corn = $1.75, TVC = $1.50Q, TFC = $100

C. Price of corn = $2.50, TVC = $2.75Q, TFC = $100

D. Price of corn = $2.00, AVC = $1.00, TFC = $400

E.
Price of corn = $1.75, TVC = $1.50Q, TFC = $275
Produce or Shut Down in SR?


Let p = max  by producing
= TR – TVC – TFC
SD = Max  if shut down
= - TFC
Produce if p > SD





TR - TVC – TFC > -TFC
TR – TVC > 0
TR > TVC
TR/q > TVC/q
P > AVC
Shut Down Profits
Output q produced by a Ptaking firm
Shut Down Graph
Derivation of Market S (Qs) from
Firm S (qf )
P = MC
  P = 5 + 10qf
  10q* = -5 + P
  qf = -1/2 + .1P
  qs = 100qf = -50 + 10P
  10P = 50 + Qs
  P = 5 + .1Qs

LR Output P Disequilibrium
LR Output P Equilibrium
LR Break Even ?

While firms may stay in business in the
SR even though they are losing money,
in the LR firms need to make money.
 In LR, firms need to cover all costs
and have a NPV > 0.
Real-World Cost Analysis
Complexities
1.
2.
3.
4.
Need to calculate costs of multiple variable
and multiple fixed inputs.
Some individual inputs may have fixed and
variable components.
Some inputs are use to produce multiple
outputs, so need to allocate or assign input
costs across products.
Calculating input costs often more difficult
than a simple input price x input quantity
calculation (e.g. how to value/cost
depreciable inputs).