The Firm: Demand and Supply

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The Firm Demand and Supply

• Microeconomia III (Lecture 3)
• Tratto da Cowell F. (2004),
• Principles of Microeoconomics

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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.

market prices
output level input demands
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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 functions properties
• again and again.

4
Overview...
Firm Comparative Statics
Conditional Input Demand
Response function for stage 1 optimisation
Output Supply
Ordinary Input Demand
Short-run problem
5
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), z0 i1

• The firms cost function

• The solution function

C(w, q) min S wizi
f(z) ³q

• Cost-minimising value for each input

• Hi is the conditional input demand function.
• Demand for input i, conditional on given output
  level q

zi Hi(w, q), i1,2,,m
A graphical approach
vector of input prices
may be a well-defined function or may be a
correspondence
Specified output level
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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

H1(w,q)
Now try a different case
z1
z1
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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.

• But what happens in between?

z2
w1

• A demand correspondence

• Constraint set is nonconvex.

Multiple inputs at this price

• Response is a discontinuous map jumps in z

• Map is multivalued at the discontinuity

z1
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?
• What are their properties?
• How are they related to properties of the cost
  function?

Do you remember these...?
Link to cost function
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Use the cost function

• The slope
• C(w, q)
• wi

Optimal demand for input i

• Recall this relationship?
• Ci(w, q) zi

• ...yes, it's Shephard's lemma

conditional input demand function

• So we have
• Ci(w, q) Hi(w, q)

• Link between conditional input demand and cost
  functions

Second derivative

• Differentiate this with respect to wj
• Cij(w, q) Hji(w, q)

• Slope of input demand function

Two simple results
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Simple result 1

• Use a standard property
• 2(?) 2(?)
• wi wj wj wi

• second derivatives of a function commute

• So in this case
• Cij(w, q) Cji(w, q)

• The order of differentiation is irrelevant

• Therefore we have
• Hji(w, q) Hij(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.

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Simple result 2

• Slope of conditional input demand function
  derived from second derivative of cost function

• Use the standard relationship
• Cij(w, q) Hji(w, q)

• We can get the special case
• Cii(w, q) Hii(w, q)

• We've just put ji

• Because cost function is concave
• Cii(w, q) ? 0

• A general property

• The relationship of conditional demand for an
  input with its own price cannot be positive.

• Therefore
• Hii(w, q) ? 0

and so...
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Conditional input demand curve

• 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 Hii0

H11(w, q) lt 0

• Corresponds to the case where isoquant is
  kinked multiple w values consistent with same z.

Link to kink figure
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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 ³ 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)

• (again it may actually be a correspondence)

output price
input prices
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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
• Cqq (w, S(w, p) ) Sp (w, p) 1

• Use the function of a function rule

Positive if MC is increasing.

• Rearrange
• 1 .
• Sp (w, p)
• Cqq (w, q)

• Gives the slope of the supply function.

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The firms supply curve
p

• The firms AC and MC curves.

• For given p read off optimal q

• Continue down to p

• What happens below p

Cq

• Supply response is given by qS(w,p)

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

• Map is multivalued at the discontinuity

q
no price yields a solution here
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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

• Differentiate with respect to wj
• Cqj(w, q) Cqq (w, q) Sj(w, p) 0

• Use the function of a function 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

• Demand for input i, conditional on output q

zi Hi(w,q)

• Supply of output

q S (w, p)

• Now substitute for q

• Stages 1 2 combined

zi Hi(w, S(w, p) )

• Use this to define a new function

• Demand for input i (unconditional )

Di(w,p) Hi(w, S(w, p) )
input prices
output price

• Use this relationship to analyse further the
  firms 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)).

function of a function rule again

• 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 ( Spgt0).

• But we also have, for any q

Hi(w, q) Ci(w, q) Hqi (w, q) Ciq(w, q)

• Shephards 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)).

function of a function rule yet again

• Differentiate with respect to wj
• Dji(w, p) Hji(w, q) Hqi(w, q)Sj(w, p)

• Use Shephards 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

Ciq(w, q)Cjq(w, q) Dji(w, p) Hji(w, q) ?
???????? Cqq(w, q) .
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Results from decomposition formula

• 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) .

• Now take the special case where j i

Ciq(w, q)2 Dii(w, p) Hii(w, q) ? ????
Cqq(w, q).

• If wi increases, the demand for input i cannot
  rise.

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Input-price fall substitution effect
w1

• The initial equilibrium

• price of input falls

conditional demand curve

• value to firm of price fall, given a fixed
  output level

H1(w,q)
price fall
Notional increase in factor input if output
target is held constant
Change in cost
z1

z1
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Input-price fall total effect
w1

• The initial equilibrium

• Substitution effect of input-price of fall.

• Total effect of input-price fall

price fall
Change in cost
z1
z1


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 firms standard optimisation
  problem
• Choose q and z to maximise

m S wizi i1
P pq

• subject to the standard constraints

q f (z)
q ³ 0, z ³ 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
_

• The solution function with the side constraint.

C(w, q, zm ) min S wi zi
zm zm

• Short-run demand for input i

• Follows from Shephards Lemma

_ _
Hi(w, q, zm) Ci(w, q, zm )

• Compare with the ordinary cost function

_

• By definition of the cost function. We have
  if q q.

C(w, q) C(w, q, zm )

• Short-run AC long-run AC.SRAC LRAC at q q

• So, dividing by q

_
Supply curves
C(w, q) C(w, q, zm ) _______ _________
q q
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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

• Supply curve in short run

C/q
C/q

• SRAC touches LRAC at the given output

?

• SRMC cuts LRMC at the given output

• The supply curve is steeper in the short run

q
?
q
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Conditional input demand

• The original demand curve for input 1

• The demand curve from the problem with the side
  constraint.

H1(w,q)

• Downward-sloping conditional demand

• Conditional demand curve is steeper in the short
  run.

_
H1(w, q, zm)
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Key concepts

• Basic functional relations
• price signals ? firm ? input/output responses

• Hi(w,q)
• S (w,p)
• Di(w,p)

demand for input i, conditional on output supply
of output demand for input i (unconditional )
Review
Review
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 consumers optimisation
  problem.