1. Cost Functions
Dr. Andrew McGee
Simon Fraser University

2. Where we stand…

3. What we know…
• Production function: (K,L) → Q
• Suppose we know
– w = wage rate of L
– R = rental rate of K
• Then we can determine how much it costs for a
firm to produce an amount Q=f(K,L) assuming
that the firm endeavors to minimize its costs
• That is, we can figure out TC(Q), so all that
remains to solve the firm’s profit maximization
problem is to determine, P(Q), which depends on
market structure

4. Types of costs
• Accounting costs
– Things for which you pay money
– E.g., bills, labor costs, rent
– Other things like depreciation (i.e., lost value of
assets that lose value over time)
• Economic costs
– Opportunity costs of doing business
• Values of highest foregone options of inputs
– Workers (how much they could earn elsewhere)
– Machines (how much they could earn if put to some other use)
– Entrepreneur (how much he/she could earn as a salaried
employee at some other firm)

5. Overlap between Accounting &
Economic Costs
• There is significant overlap between accounting &
economic costs:
– E.g., labor costs=wages=opportunity cost of labor in a
competitive labor market
• Economic costs > accounting costs
– Entrepreneur’s salary < opportunity costs because
entrepreneur receives a large share of firm’s
accounting profits as owner of firm
– Investors’ opportunity costs are not reflected in firm’s
costs (compensated through accounting profits by
way of dividends, capital gains)

6. Economic & Accounting Profit

7. Economic Costs (forget about
accounting costs for now)
Price-takers in input
markets
w r
L K

8. Deriving Total Cost Function

9. Cost Minimization Problem

10. Cost Minimization

11. Example

12. Cost minimization

13. Depicting Cost Minimization
K
TC3=rK3 All input combinations on the iso-expenditure
TC2=rK2 (iso-cost) lines are input combinations that
result in the same total cost of production
w/r=MRTS
TC1=rK1 Total cost is declining in the southwesterly
direction
TC0=rK0 A
Isoquant
q0
-w/r This firm can produce q0 for less than
L TC3, but it cannot produce q0 for TC1
or TC2.
Notice that for this and every output
level and given input prices (w & r)
Isocost or isoexpenditure lines there is a unique minimum cost of
production.
Our goal is to derive TC(w,r,q)

14. Duality: Output maximization
• For every constrained maximization problem
there exists a constrained minimization
problem yielding the same solution for
appropriate parameter values (“duality”)
• Dual problem of cost minimization: maximize
output subject to expenditure constraint

15. Output Maximization
K
w/r=MRTS Solving the Lagrangian will yield the
E=rK same optimal condition: w/r=MRTS
A
q3
q2
q1
q0 L
E=wL

16. Demand for inputs?
K
A
q3
q2
q1
q0 L
Demand for L at different wage rates? NO!

17. Demand for inputs?
• Can we derive firm’s demand for inputs (K,L)
using solutions to cost minimization problem
(much like we derived demand for goods using
the solution to the consumer choice
problem)?
• No!
• Cost minimization holds output constant, but
firm’s demand for K & L obviously depends on
how much output it chooses to produce.

18. Deriving Total Cost Function
K
q E
TC3=rK3 Cost expansion path: from
this expansion path we q0 TC0
TC2=rK2
can obtain the TC q1 TC1
associated with each TC(w,r,q)
TC1=rK1 q2
output level q TC2
TC0=rK0 q3 TC3
q3
q2
q0 q1 L

19. Example: Cobb-Douglass production
function

20. Example: Cobb-Douglass production
function
• →wL=rK
• Suppose w=r=$4. Then L=K. If L=K & q=40, we
have
• This implies that TC=$4*4+$4*4=$32 (lowest
possible cost of producing 40 units)
• &
• (extra output for last $1 spent on inputs)

21. Example: Cobb-Douglass production
function

22. Example: Deriving the Cost Function for
a Cobb-Douglass production function

23. Other important cost functions

24. Graphing the Cost Functions
$
TC(q) Observations:
1. Ceteris paribus invoked (w & r held
constant)
2. MC initially declining through q0
(may not always be true)
-Results from IRS (benefits to
specialization)
-Shape of the cost functions
q depends on the production
q0 q1
function
$/unit MC(q)=dTC/dq 3. MC intersects AC at minimum value
of AC
-MC<AC for q<q1
-MC>AC for q>q1
4. For q>q0, MC is increasing. This is
AC(q)=TC(q)/q the region of DRS in the production
function.
q
q0 q1

25. Example: Deriving MC and AC
functions

26. CRS production functions & cost
functions
CRS production functions have
$ linear TC functions because MC
TC(q)
is constant:
$
TC(q)
IRS CRS DRS
q q

27. CRS production functions & cost
functions
• What is AC when production function exhibits
CRS everywhere? In previous example and in
all such examples, AC(q)=TC(q)/q=c, a
constant.
When f(K,L) exhibits CRS everywhere,
AC(q)=MC(q)=constant.
$/unit
When you see constant marginal costs, you
immediately know something about the
AC=MC productions function. Likewise, when you see
CRS, you immediately know something about
the MC>
q

28. Effects of input price increase
• When the price of one of the two inputs goes
up, what happens if producers wish to
maintain the same level of output?
– Producers will substitute the now relatively
cheaper input for some of the input whose price
increased
– Total costs definitely do not decrease following an
input price increase. If they do, the producer
could not have been minimizing costs to start with

29. Partial elasticity of substitution
between inputs

30. Partial elasticity of substitution
between inputs
• Partial elasticity of substitution (s) measures
how firms change input mix in response to
price changes
• High s → firms change input mix (K/L ratio)
substantially in response to small changes in
relative input prices
• Low s → firms change input mix (K/L ratio)
little in response to changes in relative input
prices

31. Long vs. Short Run Costs
• Short run = period of time during which at least
one input cannot be changed
• Long run = any time horizon during which all
input can be changed
• Variable input = input that can be changed in the
short run
• Fixed input = input that cannot be changed in the
short run
– Some inputs take time to be delivered or made; this
time-to-delivery defines the short run
• All inputs are variable in the long run

32. Short run cost functions

33. Short run cost functions

34. Enveloping the Short-run cost curves
• Envelope of the STC: The set of lowest costs of
production on any STC (i.e., for any fixed level
of K) for every possible output level
• The envelope of the STC curves defines the TC
curve
• Similarly, the envelope of the SATC curves
defines the AC curve

35. Enveloping the Short-run cost curves
TC
TC(w,r,q)
Imagine that each STC curve
corresponds to the firm’s cost
schedule with a different
number of plants. How many
plants should the firm build to
produce q1, q2, q3, and q4 units
of output?
q
q1 q2 q3 q4
This TC curve is linear. What does that tell us?

36. Enveloping the Short-run cost curves
TC
TC The envelope of the
STC curves need not
be linear.
q
q1 q2 q3 q4

37. Enveloping the Short-run cost curves
Cost per
unit
AC=MC
q
This is a CRS production function
How are we deriving the MC curve from the SMC curves?

38. Enveloping the Short-run cost curves
Not CRS
Cost per
unit
MC
AC
At MES, AC reaches
its minimum values
and
AC=MC=SATC=SMC
q
MES=minimum efficient scale
For each output level, find the SMC curve corresponding to the SATC just tangent
to the AC curve. From this SMC curve we derive the MC (in the LR) of producing
that output level.

39. Example: Short run Cobb Douglass
Costs
Fixed input STC
=1
=4
=9

40. Example: Short run Cobb Douglass
Costs
• If in the short run you happen to be using the capital level
that minimizes the costs of producing output level q in the
long run (K1*), then you must also be minimizing costs in
the short run, meaning that the derivative of the STC at this
capital level is zero. Use this fact to solve for K1*:
• Plug this back into the STC. You are now effectively using
the K level that minimizes LR costs while using the L-level
that minimizes costs for any given K-level. Thus you must be
minimizing costs in the LR, so this is the TC function:
• You can check that this is indeed the same as the TC we
derived earlier for this Cobb Douglass production function