W5-1
Producer Theory-2
Topic 2 (cont.)
Econ20002 summer
Intermediate Microeconomics
Svetlana Danilkina
Lectures, week 5
The harder you work,
the luckier you get.
Plato
Overview
2. Cost minimisation
a. What is cost?
b. Long Run
 isocosts,
 optimal input bundle, tangency condition, how to
calculate; example
 comparative statics;
 LR output expansion path;
 average costs; economy of scale;
 returns of scale and economy of scale
 technological progress
c. Short Run
 total, marginal and average costs;
 relationship between MC and AC;
 relationship between MC and MP;
 lump-sum and unit tax.
d. LR and SR compared
 output expansion paths;
 total, average and marginal costs. W5-2ne
xt
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ct
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re
W5-3
2a. What is cost?
• opportunity cost = value of an item in the
next best alternative use
• economic cost = total opportunity cost of
all inputs used in production
• sunk cost = expenditure on goods and
services that have no resale value or
alternative use
W5-4
2b. Cost minimisation in long run
how to produce:
• what is the optimal combination of inputs (with smallest
cost) to produce a given amount of output and prices of
inputs
• The optimal inputs to use will depend on the quantity of
output and prices of inputs – capital and labour.
• optimal inputs are called factor demands:
K(r, w, q) and L(r, w, q)
• Then cost function:
C(r, w, q) = r*K(r, w, q) + w*L(r, w, q)
minimum cost necessary to produce output q
W5-5
What $3,600 can buy you?
K
L
Assume that capital costs r=$40 per hour and
labour w=$20 per hour.
Suppose we have enough money to afford
L=100 and K=40 (input bundle from previous
slide), i.e. we have $20*100+$40*40=$3,600
What other combinations of inputs can we
afford?
slope = - w/r = -0.5
wL + rK = cost
20L + 40K = $3,600
180100
40
90
Slope = - w/r = -0.5
W5-6
K
L
All the bundles of labour and capital on this line cost exactly
$3,600.
We will call it an isocost line – all the bundles of labour and
capital that cost the same, given prices of labour and capital.
It shows you the trade-off between capital and labour on the
factor markets (markets for inputs): if you buy one less hour
of labour, you can afford to buy 0.5 more hours of capital.
wL + rK = cost
20L + 40K = $3,600
180100
40
90
Slope = - w/r = -0.5
What $3,600 can buy you?
slope = - w/r = -0.5
W5-7
K
L
What if I prepared to spend more than $3,600 on labour
and capital?
For example, $4,200?
wL + rK = cost
$3,600 isocost:
20L + 40K = $3,600
180100
40
90
Slope = - w/r = -0.5
210
105 $4,200 isocost:
20L + 40K = $4,200
What $4,200 can buy you?
slope = - w/r = -0.5
W5-8
K
L
What if I prepared to spend less than $3,600 on labour
and capital?
For example, $2,600?
- w/r = -0.5
wL + rK = cost
$3,600 isocost:
20L + 40K = $3,600
180100
40
90
Slope = - w/r = -0.5
210
105 $2,600 isocost:
20L + 40K = $2,600
130
65
What $2,600 can buy you?
W5-9
Three isocosts
K
L
Three different isocost lines
$4,200 isocost
180
90
210
105 $2,600 isocost
130
65 $3,600 isocost
W5-10
Isocosts map
K
L
We can draw an isocost line for any value of cost.
That gives us isocosts map.
$4,200 isocost
180
90
210
105
$2,600 isocost
130
65 $3,600 isocost
Cost decreases that way
- w/r
wL + rK = cost
Slope = - w/r
W5-11
What input bundle should I buy
if I want to produce output 500?
K
L
r = $40 and w = $20.
If I buy L=100 and K=40 it will cost me
$20*100+$40*40=$3,600.
Is it a good idea?
180100
40
90 q = 500
isoquant
slope = - w/r = -0.5
W5-12
K
L
Not if I want to keep costs down!
I can buy a bit more labour and a bit less
capital to produce the same 500 output.
That would cost me less money.
The new input bundle is on the same
isoquant, but on a lower isocost => produce
the same output at lower cost.
- w/r = -0.5
180100
40
90
q = 500
isoquant
What input bundle should I buy
if I want to produce output 500?
W5-13
K
L
Can I do even better?
I want to find an input bundle on the 500
isoquant which is cheapest, i.e.
I want to minimise cost to produce
specific amount of output.
q = 500
isoquant
What input bundle should I buy
if I want to produce output 500?
Cost decreases that way
wL + rK = cost
W5-14
K
L
To minimise cost to produce 500
units of output, you should choose
the bundle of inputs on the 500
isoquant, such that at this point the
isoquant is tangent to an isocost
line.
• slope of the isoquant = – MRTSLK
• slope of an isocost = -w/r
q = 500
isoquant
What input bundle should I buy
if I want to produce output 500?
MRTSLK = w/r
slope = - w/r = -0.5
L*
K*
isocost line
W5-15
K
L
isoquant for
the output q
f(K, L) = q
isocost line
The optimal input bundle
is on the isoquant for the
output q where it is tangent
to an isocost line.
MRTSLK = w/r
slope = - w/r
should remind you
Consumer Theory
Cost minimization in LR: optimal
bundle of inputs – general case
What is the minimum cost the firm must incur to produce
a given level of output if factor prices are r and w?
L*
K*
 The slope of the isoquant is (minus) MRTS
(marginal rate of technical substitution)
= the ratio of the marginal products =
= trade-off between inputs in production
 The slope of the isocost line is the ratio of the
wage to rental cost of capital = trade-off between
inputs on the factor markets
 So, at the cost minimising choice: trade-off in
production = trade-off on the market
L
K
MP w
MP r
=
The intuition of choosing cost
minimising inputs
= , or
=
W5-16
Rearrange the last equation to get a common
sense rule for minimising cost!
Suppose I have an extra $1. I can
hire more labour or capital.
The left hand side is output per
extra $1 spent on labour.
The right hand side is output per
extra $1 spent on capital.
You can ONLY be cost minimising if
each input gives you the same
output per extra $1 spent.
L KMP MP
w r
=
Otherwise, you need to change your composition of inputs
and get more of input that brings more output per extra $1
and less of the other input.
W5-17
Balance inputs
to get the
same ‘bang for
a buck’
W5-18
K
L
isoquant for
the output q
f(K, L) = q
slope = - w/r
How to calculate cost minimising
bundle of inputs and the cost.
L*
K*
Use two conditions to find L* and K*:
• the input bundle is on the isoquant
• at the tangency pointisoquant: = (, )tangency: =
Note that optimal labour and
capital, and, therefore, cost
will depend on output q and
input prices r and w.
* *Cost: C wL rK= +
W5-19
The isoquants for this production function look like the ones on the previous
slide, so we need to find tangency points.
The marginal products of factors of production are: = 2, = 2.
Then MRTS is given by = = 2.
The tangency condition to find optimal bundle of inputs is
= ⇒ 2 = ⇒ = 2 .
Substituting it in the isoquant: = 2 = 2


2 = 2

2
3
gives us the demand for labour: ∗ =
2
⁄2 3
1
3.
Substituting it back into the tangency condition we obtain the demand for
capital: ∗ = 2

∗ = 2


2
2
3
1
3 = 2

1
3
1
3.
The corresponding long run cost function is
= ∗ + ∗ = w 2 2313 + 2 1313 = �3 223 132313
Example: production function q(K,L) = LK2
W5-20
Comparative statics: change in factor prices leads to
the substitution of one factor for the other.
K
L
isoquant for
the output q
isocost
rK+20*L=constant
Assume that we move production from the
country with high wage $20 to the country
with low wage $10, but r is the same.
If wage decreases, the firm will buy more
labour and less capital to produce the
same output.
- 20/r - 10/r
isocost
rK+10*L=constant
optimal input bundle for wage = $20
optimal input bundle for
wage = $10
increase in demand for labour
de
cr
ea
se
in
d
em
an
d
fo
r c
ap
ita
l
Jobs are going overseas
W5-21
Long run output expansion path: output
changes, input prices are fixed.
K q3
Long run output
expansion path
L
q2
q1
isocost
line C2
q
Cost CLR
q1 q2 q3
C3
C2
C1
A
B
C
- w/r
* *Cost: C ( ) ( ) ( )LR q wL q rK q= +
isocost
line C3
isocost
line C1
W5-22
Long run output expansion path: output
changes, input prices are fixed.
q
Cost CLR
K
q3
Long run output
expansion path
L
q2
q1
C3
C2
C1
A
B
C
- w/r
q3q2q1
isocost
line C2
isocost
line C3
isocost
line C1
* *Cost: C ( ) ( ) ( )LR q wL q rK q= +
W5-23
Average cost = slope of the line from origin.
Average cost could increase/decrease/stay the same
when output increases.
It depends on the shape of cost function (which itself
depends on production technology).
Average cost: ACLR(q) = CLR(q)/q
q
Cost CLR
q1 q2 q3
C3
C2
C1
q
ACLR
q1 q2 q3
Here AC is decreasing
W5-24
Average cost: ACLR(q) = CLR(q)/q
ACLR
AC is
increasing
q
Cost CLR
C3
C2
C1
q3q2q1
qq3q2q1
ACLR AC is
constant
q
Cost CLR
C3
C2
C1
q3q2q1
qq3q2q1
W5-25
The long run output expansion path connects all the tangency points and is
given by the tangency condition: = 2

.
It is a straight line from the origin, the slop of which depend on the wage
and rent and is equal to 2

.
The cost function is = �3
2
2
3

1
3
2
3
1
3.
The average cost AC q =

= �3
2
2
3

1
3
2
3

2
3
is decreasing with output q.
The marginal cost MC q =

= �1
2
2
3

1
3
2
3

2
3
is decreasing with output q.
Example: production function q(K,L) = LK2
(continued)
q
Cost CLR
q
ACLR
Long run output
expansion path
L
K
W5-26
q
Most typical production processes have U-shaped LR average cost.
q1 q2 q3
U-shaped LR average cost
ACLREconomy of scale (AC decreasing).
Diseconomy of scale
(AC increasing).
Economy of scale: double output => cost less than doubled
Diseconomy of scale: double output => cost more than doubled
W5-27
Returns to scale: what happens to output when you, for
example, double all inputs?
IRS: output more than doubles
CRS: output doubles
DRS: output less than doubles
If the production exhibits IRS, then AC is decreasing (to
produce double output, we need less than double inputs, so
cost is less than double) => economy of scale.
Note that the optimal proportion of inputs can change as output
increases. As a result, a firm could have economies of scale
without IRS in production.
What is the relationship between returns
to scale and economies of scale?
it’s complicated...
W5-28
example: economy of scale without IRS
q = 1
10
q = 2
20
8
16
K
L
Double inputs => double output
(CRS at this point);
but the firm substitutes: uses less
than double labour and more
than double capital => produces
q=2 at less than double the cost
of q=1.
=> Economy of scale
optimal input
bundle for
q=2
W5-29
example: CRS
If production exhibits CRS everywhere
and input prices are fixed (i.e. a firm can’t
save by buying in bulk), then average
cost is always constant: to double output
=> you need to double inputs => cost
doubles.
q = 1
10
q = 2
20
8
16
K
L
ACLR
qq3q2q1
W5-30
Technological progress
• “Are Robots Taking Our Jobs?” by Borland
and Coelli (Australian Economic Review,
2017): Is it true that computer based
technologies cause a substantial decrease
in the amount of work available? Only
about 9% of Australian workers are at high
risk of their jobs being automated (and not
40% as predicted by some); technology
also creates new jobs.
Robots are coming to take our jobs !?
q=1 (old)
q=1
(new)
K
L
improvement in
technology
• Technological progress might be biased
in favour of capital, or in favour of highly
skilled but not unskilled labour…
• “The Global Decline in the Labour Share” by Karabarbounis and Neiman
(QJE, 2014): the decline in the cost of capital (i.e., lower prices of the
computers) explains about a half of the decline in the share of income
earned by labour.
Most of the time a firm has limited ability to make
adjustments to its capital due to contracts with suppliers or
due to lag times in constructing new manufacturing plants.
In these cases, the only thing that they can do is to adjust
other inputs. In the “short-run” the firm is stuck with a fixed
amount of capital.
Managers of the firm can only adjust production on the
labour margin. They can run shifts around the clock, but
they cannot adjust capital.
2c. Cost minimisation in short run
W5-31
W5-32
Cost minimisation in short run
what is the optimal amount of labour (with smallest cost) to
produce a given amount of output if the available amount of
capital is fixed?
find optimal amount of labour => SR factor demand:
SR cost function:
SR total cost SR fixed cost SR variable cost
minimum cost necessary to produce output q given that capital is fixed
SRL (r, w, q; K)
SR SRC (r, w, q; K) = rK + wL (r, w, q; K)
SRC (q) = F + VC(q)
33 of 55Copyright © 2013 Pearson Education, Inc. • Microeconomics • Pindyck/Rubinfeld, 8e.
The Shape of a typical SR Cost Curve
W5-33
W5-34
Total, marginal, and average costs
Total cost of production
TC = F + VC
F – fixed cost (to set up business; capital)
VC – variable cost
Average cost for most technologies, AC is U-shaped
Marginal cost ( ) dTC d F VC dF dVC dVCMC
dq dq dq dq dq
+
= = = + =

;
TC F VC F VCAC
q q q q
VC FAVC AC AVC
q q
+
= = = +
= = +
W5-35
The relationship between
marginal and average costs
• MC < AC→ AC decreases
the cost of the last unit is smaller than the cost of an average unit
• MC > AC → AC increases
the cost of the last unit is larger than the cost of an average unit
The same for MC and AVC
q
AC
MC
AVC
For U-shaped AC: the
marginal cost passes
through the minimum
point of AC.
The same for U-
shaped AVC.
5-36
The relationship between
marginal and average costs
= ()

()

= ′ − ()
2
=
= ′ − �()

= − ()

• MC < AC→ ACʹ(q) < 0, AC(q) decreases
• MC > AC → ACʹ(q) > 0, AC(q) increases
• MC = AC → ACʹ(q) = 0, AC(q) is at its minimum
The quotient
rule of
differentiation
The slope of
AC curve
W5-37
Relationship between MCSR and MPL
( )MC = VC wL w L
q q q
∆ ∆ ∆
= =
∆ ∆ ∆
it’s simple...
MP = L
q
L
∆
∆
MC =
L
w w
q MP
L
=
∆
∆
MCSR
L
w
MP
=
 MPL is usually increasing for small outputs => MC is
decreasing
 When diminishing marginal returns kick in, MPL is
decreasing => MC increases
 MC is smallest when MP is largest.
W5-38
Lump-sum tax
Lump-sum tax on drinking establishments = licence
fee of T: TCT = TC + T increases by T
ACT
MC
AC
q
; increases by
( ) ( ) ; same
T
T
TC T T TAC AC
q q q
d TC T dTC d T dTCMC MC
dq dq dq dq
+
= = +
+
= = + = =
W5-39
per-unit tax
per unit tax on drinking establishments = t dollars per
each drink: TCt = TC + tq
ACtMC
AC
q
MCt
Both MC and AC shift
up by t.
; increases by
( ) ( ) ; increases by
T
T
TC tqAC AC t t
q
d TC tq dTC d tqMC MC t t
dq dq dq
+
= = +
+
= = + = +
W5-40
The harder you work,
the luckier you get.
Plato
Work is not a wolf – it won’t
run away into the forest.
Russian proverb
motivational / antimotivational?
W5-41
Do you think things are better or
worse than you think?
Test yourself:
1.Where does the majority of the world
population live?
a) Low-income countries
b) Middle-income countries
c) High-income countries
2. In the last 20 years, the proportion of the
world population living in extreme poverty
has…
a) Almost doubled
b) Remained more or less the same
c) Almost halved
3.How many people in the world have some
access to electricity?
a) 20%
b) 50%
c) 80%