The Firm: Demand and Supply
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Transcript The Firm: Demand and Supply
Prerequisites
Almost essential
Firm: Optimisation
THE FIRM: DEMAND AND
SUPPLY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Firm- Demand & Supply
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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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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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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
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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
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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
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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
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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)
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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…
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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
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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)
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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
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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
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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
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Supply response is
given by q=S(w,p)
|
_q
q Map is multivalued
at the discontinuity
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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
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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
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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
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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
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Stages 1 & 2 combined…
output
price
Demand for input i (unconditional )
Use this relationship to
analyse further the firm’s
response to price changes
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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”
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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.
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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*
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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*
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z**
1
z1
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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
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Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Optimisation subject
to side-constraint
Output
Supply
Ordinary
Input Demand
Short-run
problem
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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…
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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
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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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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
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q
The supply curve is
steeper in the short run
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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
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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)
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What next?
Analyse the firm under a variety of market conditions
Apply the analysis to the consumer’s optimisation problem
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