The Firm: Demand and Supply

Report
Prerequisites
Almost essential
Firm: Optimisation
THE FIRM: DEMAND AND
SUPPLY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Firm- Demand & Supply
1
Moving on from the optimum…
 We derive the firm's reactions to changes in its
environment
 These are the response functions
• 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
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Frank Cowell: Firm- Demand & Supply
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The firm as a “black box”
 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
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Frank Cowell: Firm- Demand & Supply
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Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response
function for stage
1 optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
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The first response function

Review the cost-minimisation
problem and its solution

Choose z to minimise
The “stage 1” problem
m
S wi zi subject to q  f (z), z ≥ 0
i=1

The firm’s cost function:
The solution function
C(w, q) := min S wizi
{f(z)  q}

Cost-minimising value for each input:
zi* = Hi(w, q), i=1,2,…,m
could be a welldefined function or a
correspondence
March 2012
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
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Mapping into (z1,w1)-space
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


 Response function is
a continuous map:

z1
H1(w,q)
z1
Now try a
different case
March 2012
Frank Cowell: Firm- Demand & Supply
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Another map into (z1,w1)-space
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 discontinuous
map: jumps in z*
Multiple inputs
at this price




z1
Map is multivalued at the
discontinuity
z1
no price yields
a solution here
March 2012
Frank Cowell: Firm- Demand & Supply
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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
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Use the cost function

Recall this relationship?
…yes, it's Shephard's lemma
Ci(w, q) = zi*
The slope:
 C(w, q)
————
 wi

Optimal demand
for input i
So we have:
Ci(w, q) =
Hi(w,
q)
Link between conditional input
demand and cost functions
conditional input
demand function
 Differentiate this with respect to wj
Cij(w, q) = Hji(w, q)
Slope of input demand function
Second
derivative
March 2012
Two simple
results:
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Simple result 1
 Use a standard property
second derivatives of a function
“commute”
 So in this case
The order of differentiation is irrelevant
 2()
2()
——— = ———
 wi wj
wj wi
Cij(w, q) = Cji(w,q)
 Therefore we have:
Hji(w, q) = Hij(w, q)
March 2012
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
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Simple result 2


Cij(w, q) = Hji(w, q)
Slope of conditional input
demand function derived from
second derivative of cost function
We can get the special case:
We've just put j = i
Use the standard relationship:
Cii(w, q) = Hii(w, q)

Because cost function is concave: A general property

Therefore:
Cii(w, q)  0
Hii(w, q)  0
The relationship of conditional
demand for an input with its own
price cannot be positive
and so…
March 2012
Frank Cowell: Firm- Demand & Supply
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Conditional input demand curve
w1
 Consider the demand for input 1
 Consequence of result 2?
H1(w,q)
 “Downward-sloping”
conditional demand
 In some cases it is also
possible that Hii = 0
H11(w, q) < 0
z1
March 2012
 Corresponds to the
case where isoquant is
kinked: multiple w values
consistent with same z*
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For the conditional demand function…
 Nonconvex Z yields discontinuous H
 Cross-price effects are symmetric
 Own-price demand slopes downward
 (exceptional case: own-price demand could be constant)
March 2012
Frank Cowell: Firm- Demand & Supply
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Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response function
for stage 2
optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
Frank Cowell: Firm- Demand & Supply
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The second response function

Review the profit-maximisation
problem and its solution
 Choose q to maximise:
The “stage 2” problem
pq – C (w, q)

From the FOC:
p  Cq (w, q*)
pq*  C(w, q*)

“Price equals marginal cost”
“Price covers average cost”
profit-maximising value for output:
 S is the supply function
q* = S (w, p)
input
prices
March 2012
output
price
(again it may actually be a
correspondence)
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Supply of output and output price
 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 Use the “function of a function” rule
Cqq (w, S(w, p) ) Sp (w, p) = 1
Positive if MC is
increasing
 Rearrange:
1
.
Sp (w, p) = ———— Gives slope of supply function
Cqq (w, q)
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The firm’s supply curve
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
March 2012
Supply response is
given by q=S(w,p)
|
_q
q Map is multivalued
at the discontinuity
Frank Cowell: Firm- Demand & Supply
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Supply of output and price of input j
 Use the FOC:
Cq (w, S(w, p) ) = p
Same as before: “price
equals marginal cost”
Use the “function of a function”
 Differentiate with respect to wj
Cqj(w, q*) + Cqq (w, q*) Sj(w, p) = 0 rule again
 Rearrange:
Cqj(w, q*)
Sj(w, p) = – ————
Cqq(w, q*)
Supply of output must fall
with wj if marginal cost
increases with wj
Remember, this is
positive
March 2012
Frank Cowell: Firm- Demand & Supply
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For the supply function…
 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
March 2012
Frank Cowell: Firm- Demand & Supply
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Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response function for
combined
optimisation problem
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
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20
The third response function
 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) )

Use this to define a new function:
Di(w,p) := Hi(w, S(w, p) )
input
prices
March 2012
Stages 1 & 2 combined…
output
price
Demand for input i (unconditional )
Use this relationship to
analyse further the firm’s
response to price changes
Frank Cowell: Firm- Demand & Supply
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Demand for i and the price of output
 Take the relationship
Di(w, p) = Hi(w, S(w, p))
 Differentiate with respect to p:
Dpi(w, p) = Hqi(w, q*) Sp(w, p)
Di increases with p iff Hi
increases with q. Reason? Supply
increases with price ( Sp>0)
“function of a
function” rule again
 But we also have, for any q:
Hi(w, q) = Ci(w, q)
Hqi (w, q) = Ciq(w, q)
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
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Demand for i and the price of j
 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)
 Use this and the previous result on Sj(w, p) to give a
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”
.
“substitution
effect”
March 2012
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Results from decomposition formula
 Take the general relationship:
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.
The effect wi on demand for
input j equals the effect of wj
on demand for input i
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.
March 2012
If wi increases, the demand for
input i cannot rise
cannot be
positive.
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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*
March 2012
z1
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Input-price fall: total effect
w1
Conditional
demand at
original output
 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
ordinary demand
curve
z1*
March 2012
z**
1
z1
Frank Cowell: Firm- Demand & Supply
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The ordinary demand function…
 Nonconvex Z may yield a discontinuous D
 Cross-price effects are symmetric
 Own-price demand slopes downward
 Same basic properties as for H function
March 2012
Frank Cowell: Firm- Demand & Supply
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Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Optimisation subject
to side-constraint
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
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28
The short run…
 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
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The short-run problem
 We build on the firm’s standard optimisation
problem
 Choose q and z to maximise
m
P := pq – 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…
March 2012
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The short-run cost function
~
_
 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
~
_
C(w, q)  C(w, q, zm )
 So, dividing by q:
~
_
C(w,
q) C(w,
q, zm )
_______
_________

q
q
March 2012
The solution function with
the side constraint
Follows from Shephard’s Lemma
By definition of the cost
function. We have “=” if q =`q
Short-run AC ≥ long-run AC.
SRAC = LRAC at q =`q
Supply curves
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31
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
March 2012
q
The supply curve is
steeper in the short run
Frank Cowell: Firm- Demand & Supply
32
Conditional input demand
 The original demand curve for input 1
w1
H1(w,q)
 The demand curve from the problem
with the side constraint
 “Downward-sloping” conditional
demand
 Conditional demand curve is
steeper in the short run
~
_
H1(w, q, zm)
z1
March 2012
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33
Key concepts
 Basic functional relations
 price signals  firm  input/output responses

Hi(w,q)
demand for input i,
conditional on output
Review

S (w,p)
supply of output
Review

Di(w,p)
demand for input i
(unconditional )
Review
And they all hook together like this:
 Hi(w, S(w,p)) = Di(w,p)
March 2012
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34
What next?
 Analyse the firm under a variety of market conditions
 Apply the analysis to the consumer’s optimisation problem
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