and Supply MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Firm Optimisation Prerequisites October 2015 1 Moving on from the optimum We derive the firms reactions to changes in its environment ID: 260008
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Slide1
The Firm: Demand and Supply
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Firm: Optimisation
Prerequisites
April 2018
1Slide2
Moving on from the optimum…
We derive the firm's reactions to changes in its environment
These are the response functionsWe will examine three types of themResponses to different types of market events In effect we treat the firm as a black box
market
prices
the firm
output level;
input demands
April 2018
2Slide3
The firm as a “black box”
Behaviour can be predicted by necessary and sufficient conditions for optimumThe FOC can be solved to yield behavioural response functionsTheir properties derive from the solution functionwe need the solution function’s propertiesapply them again and again
April 2018
3Slide4
Overview
April 2018
4
Conditional
Input Demand
Output
Supply
Ordinary
Input Demand
Short-run problem
Firm: Comparative Statics
Response function for stage 1 optimisation Slide5
The first response function
m
S
wi
zi subject to
q
f
(
z
),
z ≥ 0
i = 1Review
the cost-minimisation problem and its solutionChoose z to minimise
Cost-minimising value for each input:
zi* = Hi
(w,
q
),
i
=
1,2,…,
m
The firm’s cost function
:
C
(
w
,
q
) := min
S
w
i
z
i
vector of
input prices
Specified output level
{
f
(
z
)
³
q
}
H
i
:
conditional
input demand function Demand for input i, conditional on given output level q
The “stage 1” problem
The solution function
could be a well-defined function or a correspondence
A graphical approach
April 2018
5Slide6
Mapping into (
z
1
,w1)-space
z1
z
2
z
1
w
1
Left-hand panel: conventional
case of
Z
the slope of the
tangent:
value of
w
1
Repeat for a lower value of
w
1
…and again to get…
Green curve:
conditional demand curve
H
1
(
w
,
q
)
Constraint set is convex, with smooth boundary
Response function is a continuous map:
Now try a different case
April 2018
6Slide7
Another map into (
z
1
,w1)-space
z
1
z
2
z
1
w
1
Left-hand panel:
nonconvex
Z
Start with a high value of
w
1
Repeat for a very low value of
w
1
Points “nearby” `work the same way
But what happens in between?
A
demand correspondence
Constraint set is nonconvex
Response
is
discontinuous
:
jumps in
z*
Map
multivalued
at
discontinuity
Multiple inputs at this price
no price yields
a solution here
April 2018
7Slide8
Conditional input demand function
Assume that single-valued input-demand functions exist How are they related to the cost function C? What are their properties?How are their properties related to those of C
?tip if you’re not sure about the cost function:check the presentation “Firm Optimisation”revise the five main properties of the function C
April 2018
8Slide9
Use the cost function
Yes, it's Shephard's
lemma
Recall this relationship?Ci (w
, q) =
z
i
*
So we have:
C
i
(w, q) = Hi(w
, q) Link between conditional input demand and cost functions
Differentiate this with respect to wj
Cij (w,
q) = H
j
i
(
w
,
q
)
conditional input demand function
Second derivative
Slope of input conditional demand function: effect of
D
w
j
on
z
i
*
for given
q
The slope:
¶
C
(
w
,
q
)
————
¶
w
i
Optimal demand for input
i
Two simple results:
April 2018
9Slide10
Simple result 1
Use a standard property
¶ 2()
¶2(
)
——— = ———
¶
w
i
¶w
j ¶wj ¶wi
second derivatives of a function “commute”
So in this caseC
ij (w, q) =
Cji
(
w
,
q
)
The order of differentiation is irrelevant
Therefore we have:
H
j
i
(
w
,
q
) =
H
i
j
(
w
,
q
)
The effect of the price of input
i
on conditional demand for input j equals the effect of the price of input j on conditional demand for input
i
April 2018
10Slide11
Simple result 2
Use the standard relationship:
Cij (
w, q) = H
ji(
w
,
q
)
Slope of conditional input demand function derived from second derivative of cost function
We can get the special case:
C
ii (w, q
) = Hii(w, q)
We've just put j =
i
Because cost function is concave:C
ii
(
w
,
q
)
0
A general property
Therefore:
H
i
i
(
w
,
q
)
0
The relationship of conditional demand for an input with its own price cannot be positive
and so…
April 2018
11Slide12
Conditional input demand curve
H
1
(w,q)
z
1
w
1
Consider the demand for input 1
Consequence of result 2?
H
1
1
(
w
,
q
) < 0
“Downward-sloping” conditional demand
In some cases it is
possible
that
H
i
i
= 0
Corresponds to
case where isoquant is kinked: multiple
w
values consistent with same
z
*
April 2018
12Slide13
Conditional demand function: summary
Nonconvex Z yields discontinuous HCross-price effects are symmetricOwn-price demand slopes downward (exceptional case: own-price demand could be constant)
April 2018
13Slide14
Overview
April 2018
14
Conditional
Input Demand
Output
Supply
Ordinary
Input Demand
Short-run problem
Firm: Comparative Statics
Response function for stage 2 optimisationSlide15
The second response function
April 2018
15
From the FOC:
p
=
C
q
(
w, q*
), if q* > 0pq* ³
C(w, q*)
“Price equals marginal cost”“Price covers average cost”
Review the profit-maximisation problem and its solutionChoose q
to maximise:
pq
–
C
(
w
,
q
)
The “stage 2” problem
q
*
= S
(
w
,
p
)
S
is the
supply
function
(again it
may
be a correspondence)
input prices
output price
profit-maximising value for output:Slide16
Supply of output and output price
Use the FOC:
Cq
(w, q
*) = p
“marginal cost equals price”
Use the supply function for
q
:
C
q
(w, S
(w, p) ) = pGives an equation in w and p
Differentiate with respect to
p Cqq
(w,
S
(
w
,
p
) )
S
p
(
w
,
p
) = 1
Use the “function of a function” rule
Rearrange:
1
S
p
(
w
,
p
)
=
————
C
qq
(
w
, q*
)Gives slope of supply functionPositive if MC is increasing
Differential of
S with respect to
pApril 2018
16Slide17
The firm’s supply curve
C/q
C
q
p
q
AC (green)
and MC
(red) curves
For given p read off optimal
q
*
Continues
down to
p
Check what
happens below
p
p
_
–
q
_
|
Case illustrated is for
f
with first decreasing AC, then increasing AC, Response is a
discontinuous
map: jumps in
q*
Multivalued
at the discontinuity
Multiple
q*
at this price
no price yields
a solution here
Supply response
given
by
q
=
S
(
w
,
p
)
April 2018
17Slide18
Supply of output and price of input j
Use the FOC:
Cq
(w, S
(w,
p
) ) =
p
Differentiate with respect to
w
j
Cqj (w, q*
) + Cqq (w, q*
) Sj (
w, p) = 0
Use the “function of a function” rule again
Same as before: “price equals marginal cost”
Rearrange:
C
qj
(
w
,
q
*
)
S
j
(
w
,
p
) = – ————
C
qq
(
w
,
q
*
)
Supply of output must fall with
w
j
if MC increases with
w
j
Remember, this is positive
April 2018
18Slide19
Supply function: summary
Supply curve slopes upwardSupply decreases with the price of an input, if MC increases with the price of that inputNonconcave f yields discontinuous SIRTS means
f is nonconcave and so S is discontinuous
April 201819Slide20
Overview
Conditional
Input Demand
Output
Supply
Ordinary
Input Demand
Short-run problem
Firm: Comparative Statics
Response function for combined optimisation problem
April 2018
20Slide21
The third response function
Demand for input
i, conditional on output
q
zi*
=
H
i
(
w
,q)
q* = S (w, p)
Supply of outputzi
* = H
i(w, S(w
, p) )
D
i
(
w
,
p
)
:=
H
i
(
w
,
S
(
w
,
p
) )
Now substitute for
q
*
:
Demand for input
i
(unconditional )
input prices
output price
Stages 1 & 2 combined…
Recall the first two response functions:
Use this relationship to analyse
firm’s response to price changes
Use this to define a new function:
April 2018
21Slide22
Demand for i and the price of output
Take the relationship D
i(w, p
) = Hi
(
w
,
S
(
w
, p))
Di increases with p iff Hi increases with
q. Reason? Supply increases with price ( Sp > 0 )
But we also have, for any q:Shephard’s
Lemma again
Substitute in the above:
D
p
i
(
w
,
p
) =
C
qi
(
w
,
q
*
)
S
p
(
w
,
p
)
Demand for input
i
(D
i
)
increases with p iff
marginal cost (Cq) increases with wi
Differentiate with respect to
p:
Dpi(w,
p) = Hqi(w, q*) Sp(
w, p
)
“function of a function” rule again
H
i
(
w
,
q
) =
C
i
(w, q)
Hqi (w
, q) = C
iq (w,
q)April 201822Slide23
Demand for i and the price of
jApril 2018
23
Differentiate with respect to w
j:
D
j
i
(
w
, p) = H
ji(w, q*) + H
qi(w, q*)S
j (w,
p)Again take the relationship
Di(
w
,
p
) =
H
i
(
w
,
S
(
w
,
p
))
Use Shephard’s Lemma again:
H
q
i
(
w
,
q
)
=
C
iq
(
w
,
q
)
Use this and the previous result on Sj (
w, p) to give a decomposition into a “substitution effect” and an “output effect”:
“substitution effect”
“output effect”
Cjq(w, q*)
Dj
i
(
w
,
p
) =
H
j
i(
w,
q*)
Ciq(w,
q*)
Cqq(w,
q*) .
Substitution effect is just slope of conditional input demand curve
Output effect is [effect of Dwj on q][effect of Dq on demand for i]Slide24
Results from decomposition formula
The effect
wi on demand for input j equals the effect of wj on demand for input i
Take the general relationship:
Now take the special case where
j
=
i
:
We
know
this is symmetric in
i
and j
We
know this is negative or zero
If wi increases, the demand for input i cannot rise
Symmetric
in
i
and
j
C
iq
(
w
,
q
*
)
C
jq
(
w
,
q
*
)
D
j
i
(
w
,
p
) =
H
j
i
(
w
,
q*) Cqq
(w, q
*) .
cannot be
positiveCiq(w,
q*)
2
D
i
i
(
w
,
p
) =
H
ii(w,
q*)
Cqq(w,
q*).
April 2018
24Slide25
Input-price fall: substitution effect
Change in cost
conditional demand curve
price fall
z
1
w
1
H
1
(
w
,
q
)
*
z
1
z
1
*
: initial
equilibrium
grey arrow: fall in
w
1
shaded area: value of
price
fall
initial price level
original
output level
Notional increase in factor input if output target is held constant
April 2018
25Slide26
z
1
Input-price fall: total effect
price fall
z
1
w
1
*
z
1
z
1
*
:
initial equilibrium
green line: substitution
effect
z
1
**
:
new equilibrium
**
initial price level
Conditional demand at original output
ordinary demand curve
Conditional demand at new output
April 2018
26Slide27
Ordinary demand function: summary
Nonconvex Z may yield a discontinuous DCross-price effects are symmetricOwn-price demand slopes downwardSame basic properties as for
H functionApril 2018
27Slide28
Overview
Conditional
Input Demand
Output
Supply
Ordinary
Input Demand
Short-run problem
Firm: Comparative Statics
Optimisation subject to side-constraint
April 2018
28Slide29
The short run: concept
This is not a moment in time It is defined by additional constraints within the modelCounterparts in other economic applications where one may need to introduce side constraints
April 2018
29Slide30
The short-run problem
April 2018
30
subject to the standard constraints:
We build on the firm’s standard optimisation problem
Choose
q
and
z
to maximise
q £ f (
z)q ³ 0, z ³
0 P := pq –
m
S wi zi
i
=1
z
m
=
`
z
m
But we add a
side condition
to this problem:
Let
`
q
be the value of
q
for which
z
m
=
`
z
m
would have been freely chosen in the unrestricted cost-min problem…Slide31
The short-run cost function
{zm
=`zm }
C(w,
q, zm ) := min S w
i
z
i
C
(
w, q) £
C(w, q, zm ) ~ _
The solution function with the side constraintBy definition of the cost function. We have “=” if q =`
q
Short-run AC ≥ long-run AC.SRAC = LRAC at q =`q
So, dividing by
q
:
Supply curves
~ _
~ _
Compare with the ordinary cost function
Short-run demand for input
i
:
H
i
(
w
,
q
,
z
m
) =
C
i
(
w
,
q, z
m
)
~ _ ~ _
Follows from Shephard’s Lemma
C
(
w
,
q
)
C
(
w
, q,
zm )______ _________ £ q qApril 2018
31Slide32
MC, AC and supply in the short and long run
C/q
C
q
q
p
q
C/q
C
q
~
~
green curve: AC if
all inputs
variable
`
q
: given
output level
red
curve:
MC if
all inputs variable
black
curve:
AC if input
m
kept
fixed
brown
curve:
MC
if input m kept fixed
LR supply
curve
follows LRMC
SR supply
curve
follows
SRMC
Supply
curve
steeper
in the short run
SRAC touches LRAC at
given
output
SRMC cuts LRMC at
given
output
April 2018
32Slide33
Conditional input demand
H
1
(w,q)
z
1
w
1
Brown curve: demand for
input 1
“Downward-sloping” conditional demand
Conditional demand curve is steeper in the short run
H
1
(
w
,
q
,
z
m
)
~ _
Purple curve:
demand for input 1 in
problem
with
the side constraint
April 2018
33Slide34
Key concepts
Basic functional relationsprice signals firm input/output responses
April 2018
34
Hi(
w
,
q
)
S
(w,p
) Di(w,p)
Hi(w, S
(w,p)) =
Di(w,p)
demand for input
i,
conditional on output
supply of output
demand for input
i
(unconditional )
And they all hook together like this:Slide35
What next?
Analyse the firm under a variety of market conditionsApply the analysis to the consumer’s optimisation problem
April 2018
35