Prerequisites
Almost essential
Firm: Optimisation
Frank Cowell: Microeconomics
October 2011
The Firm: Demand and Supply
MICROECONOMICS
Principles and Analysis
Frank Cowell
Moving on from the optimum...
Frank Cowell: Microeconomics
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We derive the firm's reactions to changes in its
environment.
These are the response functions.
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We will examine three types of them
Responses to different types of market events.
In effect we treat the firm as a Black Box.
the firm
The firm as a “black box”
Frank Cowell: Microeconomics
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Behaviour can be predicted by necessary and
sufficient conditions for optimum
The FOC can be solved to yield behavioural
response functions
Their properties derive from the solution function
We need the solution function’s properties…
…again and again
Overview...
Frank Cowell: Microeconomics
Firm: Comparative
Statics
Conditional
Input Demand
Response
function for stage
1 optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
The first response function
Frank Cowell: Microeconomics
Review the cost-minimisation
problem and its solution


Choose z to minimise
m
The “stage 1” problem
S wi zi subject to q  f(z), z≥0
i=1

The firm’s cost function:
C(w, q) := min S wizi
The solution function
{f(z) q}

Cost-minimising value for each input:
zi* = Hi(w, q), i=1,2,…,m
may be a well-defined
function or may be a
correspondence
Specified
vector of output level
input prices
 Hi is the conditional input
demand function.
Demand for input i, conditional
on given output level q
A graphical
approach
Mapping into (z1,w1)-space
Frank Cowell: Microeconomics
Conventional case of Z.
Start with any value of w1 (the
slope of the tangent to Z).
Repeat for a lower value of w1.
...and again to get...
z2
w1
...the conditional demand curve
 Constraint set is
convex, with smooth
boundary
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
 Response function is
a continuous map:

z1
H1(w,q)
z1
Now try a
different case
Another map into (z1,w1)-space
Frank Cowell: Microeconomics
Now take case of nonconvex Z.
Start with a high value of w1.
Repeat for a very low value of w1.
Points “nearby” work the same
way.
z2
 But what happens in between?
w1
A demand correspondence

 Constraint set is
nonconvex.
 Response is a
discontinuous map:
jumps in z*

Multiple inputs
at this price




z1
no price yields
a solution here
z1
Map is multivalued at
the discontinuity
Conditional input demand function
Frank Cowell: Microeconomics
Link to
cost
function
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Assume that single-valued inputdemand functions exist.
How are they related to the cost
function?
What are their properties?
How are they related to properties of the
cost function?
Do you
remember
these...?
slope: cost function
UseThethe
Frank Cowell: Microeconomics
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C(w, q)
————
wi
Optimal demand
for input i
Recall this relationship?
Ci(w, q) = zi*
conditional input
demand function

So we have:
Ci(w, q) = Hi(w, q)

Second
derivative
Differentiate
this with
respect to wj
Cij(w, q) = Hji(w, q)
...yes, it's Shephard's
lemma
Link between conditional input
demand and cost functions
Slope of input demand
function
Two simple
results:
Simple result 1
Frank Cowell: Microeconomics
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Use a standard property

So in this case
2(2(
——— = ———
wi wj
wj wi
Cij(w, q) = Cji(w, q)

Therefore we have:
Hji(w, q) = Hij(w, q)
second derivatives of a function
“commute”
The order of differentiation is
irrelevant
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.
Simple result 2
Frank Cowell: Microeconomics
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Use the standard relationship:
Cij(w, q) = Hji(w, q)
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We can get the special case:
Slope of conditional input
demand function derived from
second derivative of cost function
We've just put j=i
Cii(w, q) = Hii(w, q)

Because cost function is concave:
A general property
Therefore:
The relationship of conditional
demand for an input with its own
price cannot be positive.
Cii(w, q)  0

Hii(w, q)  0
and so...
Conditional input demand curve
Frank Cowell: Microeconomics
Link to
“kink”
figure
 Consider the demand for
input 1
 Consequence of result 2?
w1
H1(w,q)
 “Downward-sloping”
conditional demand
 In some cases it is
also possible that Hii=0
H11(w, q) < 0
z1
 Corresponds to the
case where isoquant is
kinked: multiple w
values consistent with
same z*.
Frank Cowell: Microeconomics
For the conditional demand
function...
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Nonconvex Z yields discontinuous H
Cross-price effects are symmetric
Own-price demand slopes downward
(exceptional case: own-price demand could be
constant)
Overview...
Frank Cowell: Microeconomics
Firm: Comparative
Statics
Conditional
Input Demand
Response
function for stage
2 optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
The second response function
Frank Cowell: Microeconomics
Review the profit-maximisation
problem and its solution
Choose q to maximise:

pq – C (w, q)

From the FOC:
p  Cq (w, q*)
pq*  C(w, q*)

The “stage 2” problem
profit-maximising value for output:
q*
= S (w, p)
input
prices
output
price
“Price equals marginal cost”
“Price covers average cost”
 S is the supply function
(again it may actually be a
correspondence)
Supply of output and output price
Frank Cowell: Microeconomics
Use the FOC:
Cq (w, q) = p

“marginal cost equals price”
Use the supply function for q:
Cq (w, S(w, p) ) = p
Gives an equation in w and p

Differential of S
with respect to p
Differentiate with respect to p
Cqq (w, S(w, p) ) Sp (w, p) = 1


Use the “function of a
function” rule
Positive if MC is
Gives the slope
increasing.
Rearrange:
1 . function.
Sp (w, p) = ————
Cqq (w, q)
of the supply
The firm’s supply curve
Frank Cowell: Microeconomics
p
 The firm’s AC and MC curves.
 For given p read off optimal q*
 Continue down to p
 What happens below p
Cq
C/q
 Case illustrated is
for f with first IRTS,
then DRTS.
Response is a
discontinuous map:
jumps in q*
Multiple q* at
this price
_p –
no price yields
a solution here
Supply response is
given by q=S(w,p)
|
_q
q Map is multivalued
at the discontinuity
Frank Cowell: Microeconomics
Supply of output and price of
input j
Use the FOC:
Cq (w, S(w, p) ) = p
Same as before: “price
equals marginal cost”
Differentiate with respect to wj
Cqj(w, q*) + Cqq (w, q*) Sj(w, p) = 0
Use the “function of a
function” rule again
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Rearrange:
Cqj(w, q*)
Sj(w, p) = – ————
Cqq(w, q*)
Remember, this is
positive
Supply of output must fall
with wj if marginal cost
increases with wj.
For the supply function...
Frank Cowell: Microeconomics
Supply curve slopes upward
 Supply decreases with the price of an input,
if MC increases with the price of that input
 Nonconcave f yields discontinuous S
 IRTS means f is nonconcave and so S is
discontinuous

Overview...
Frank Cowell: Microeconomics
Firm: Comparative
Statics
Conditional
Input Demand
Response
function for
combined
optimisation
problem
Output
Supply
Ordinary
Input Demand
Short-run
problem
The third response function
Frank Cowell: Microeconomics
input
prices
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Recall the first two response functions:
zi* = Hi(w,q)
Demand for input i,
conditional on output q
q* = S (w, p)
Supply of output
Now substitute for q* :
zi* = Hi(w, S(w, p) )

Stages 1 & 2 combined…
Use this to define a new function:
Di(w,p) := Hi(w, S(w, p) )
output
price
Demand for input i
(unconditional )
Use this relationship to
analyse further the firm’s
response to price changes
Demand for i and the price of output
Frank Cowell: Microeconomics
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Take the relationship

Differentiate with respect to p:
“function of a
Di(w, p) = Hi(w, S(w,
p)).
function” rule again
Dpi(w, p) = Hqi(w, q*) Sp(w, p)

But we also have, for any q:
Hi(w, q) = Ci(w, q)
Hqi (w, q) = Ciq(w, q)

Di increases with p iff Hi
increases with q. Reason?
Supply increases with price
( Sp>0).
Shephard’s Lemma again
Substitute in the above:
Dpi(w, p) = Cqi(w, q*)Sp(w, p)
Demand for input i (Di)
increases with p iff
marginal cost (Cq)
increases with wi .
Demand for i and the price of j
Frank Cowell: Microeconomics

Again take the relationship
Di(w, p) = Hi(w, S(w, p)).

Differentiate with respect to wj:
Dji(w, p) = Hji(w, q*) + Hqi(w, q*)Sj(w, p)

Use Shephard’s Lemma again:
Hqi(w, q) = Ciq(w, q) = Cqi(w, q)
“substitution
Use this and the previous
result on Sj(w, p) to give a
effect”
decomposition into a “substitution effect” and an “output effect”:

Dji(w, p) = Hji(w, q*) 
Cjq(w, q*)
 Ciq(w, q*)
Cqq(w, q*)
.
“output
effect”
Results from decomposition formula
Frank Cowell: Microeconomics

Take the general relationship:
The effect wi on demand
for input j equals the
effect of wj on demand
for input i.
Ciq(w, q*)Cjq(w, q*)
Dji(w, p) = Hji(w, q*)  
Cqq(w, q*)
.
We already know
this is symmetric in i
and j.

Obviously
symmetric in i
and j.
Now take the special case where j = i:
Ciq(w, q*)2
Dii(w, p) = Hii(w, q*)  
Cqq(w, q*)
.
We already know this
is negative or zero.
cannot be
positive.
If wi increases, the
demand for input i
cannot rise.
Frank Cowell: Microeconomics
Input-price fall: substitution
effect
 The initial equilibrium
w1
 price of input falls
conditional
demand curve
original
 value to firm of price fall,
given a fixed output level
output level
H1(w,q)
price
fall
initial price
level
Change in cost
Notional increase in
factor input if output
target is held constant
z1*
z1
Input-price fall: total effect
Frank Cowell: Microeconomics
Conditional
demand at
original output
w1
 The initial equilibrium
 Substitution effect of inputprice of fall.
 Total effect of input-price fall
Conditional
demand at new
output
price
fall
initial price
level
Change in cost
ordinary demand
curve
z1*
z**
1
z1
The ordinary demand function...
Frank Cowell: Microeconomics
Nonconvex Z may yield a discontinuous D
 Cross-price effects are symmetric
 Own-price demand slopes downward
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Same basic properties as for H function
Overview...
Frank Cowell: Microeconomics
Firm: Comparative
Statics
Conditional
Input Demand
Optimisation
subject to sideconstraint
Output
Supply
Ordinary
Input Demand
Short-run
problem
The short run...
Frank Cowell: Microeconomics
This is not a moment in time but…
 … is defined by additional constraints
within the model
 Counterparts in other economic applications
where we sometimes need to introduce side
constraints

The short-run problem
Frank Cowell: Microeconomics
We build on the firm’s standard optimisation problem
 Choose q and z to maximise

P := pq –
m
S wizi
i=1

subject to the standard constraints:
q  f(z)
q  0, z  0

But we add a side condition to this problem:
zm = `zm
Let `q be the value of q for which zm =`zm would have
been freely chosen in the unrestricted cost-min problem…

The short-run cost function
Frank Cowell: Microeconomics
~
_
C(w, q, zm ) := min S wi zi
{zm =`zm }
Short-run demand for input i:

~
_
~
_
i
H (w, q, zm) =Ci(w, q, zm )
Compare with the ordinary cost
function
~
_
The solution function with
the side constraint.
Follows from Shephard’s
Lemma

C(w, q)  C(w, q, zm )

So, dividing by q:
~
_
C(w,
q) C(w,
q, zm )
_______
_________

q
q
By definition of the cost
function. We have “=” if q =`q.
Short-run AC ≥ long-run AC.
SRAC = LRAC at q =`q
Supply curves
Frank Cowell: Microeconomics
MC, AC and supply in the short and
long run
 AC if all inputs are variable
 MC if all inputs are variable
 Fix an output level.
p
 AC if input m is now kept fixed
~
Cq
 MC if input m is now kept fixed
 Supply curve in long run
Cq
~
C/q
 Supply curve in short run
C/q
SRAC touches LRAC at
the given output

SRMC cuts LRMC at
the given output
q
q
The supply curve is
steeper in the short run
Conditional input demand
Frank Cowell: Microeconomics
 The original demand curve
for input 1
 The demand curve from the
problem with the side
constraint.
w1
H1(w,q)
 “Downward-sloping”
conditional demand
 Conditional demand
curve is steeper in the
short run.
~
_
H1(w, q, zm)
z1
Key concepts
Frank Cowell: Microeconomics
Basic functional relations
 price signals  firm  input/output responses

Review

Hi(w,q)
demand for input i,
conditional on output
Review

S (w,p)
supply of output
Di(w,p)
demand for input i
(unconditional )
Review 
And they all hook together like this:

Hi(w, S(w,p)) = Di(w,p)
What next?
Frank Cowell: Microeconomics
Analyse the firm under a variety of market
conditions.
 Apply the analysis to the consumer’s
optimisation problem.
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