Prerequisites. Almost essential Firm: Optimisation. The Firm: Demand and Supply. MICROECONOMICS Principles and Analysis Frank Cowell . October 2011. the firm. Moving on from the optimum. We derive the firm's reactions to changes in its environment.
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Almost essential
Firm: Optimisation
MICROECONOMICS
Principles and Analysis
Frank Cowell
October 2011
market
prices
output level;
input demands
Firm: Comparative Statics
Conditional
Input Demand
Response function for stage 1 optimisation
Output
Supply
Ordinary
Input Demand
Shortrun problem
m
Swi zisubject toq f(z), z≥0
i=1
C(w, q) := minSwizi
{f(z) ³q}
zi* = Hi(w, q), i=1,2,…,m
A graphical approach
may be a welldefined function or may be a correspondence
Specified output level
vector of
input prices
z2
w1
H1(w,q)
Now try a different case
z1
z1
z2
w1
Multiple inputs at this price
z1
z1
no price yields
a solution here
Link to cost function
Optimal demand for input i
conditional input demand function
Second derivative
Two simple results:
¶wi ¶wj¶wj ¶wi
Cii(w, q) = Hii(w, q)
and so...
z1
Conditional input demand curveH1(w,q)
H11(w,q) < 0
Link to “kink” figure
Firm: Comparative Statics
Conditional
Input Demand
Response function for stage 2 optimisation
Output
Supply
Ordinary
Input Demand
Shortrun problem
pq –C (w, q)
p £Cq(w, q*)
pq*³ C(w, q*)
q* = S (w, p)
output price
input prices
Differential of S with respect to p
Positive if MC is increasing.
–
_

q
_
The firm’s supply curvep
Cq
C/q
Multiple q* at this price
q
no price yields
a solution here
Remember, this is positive
Firm: Comparative Statics
Conditional
Input Demand
Response function for combined optimisation problem
Output
Supply
Ordinary
Input Demand
Shortrun problem
zi*= Hi(w,q)
q* = S (w, p)
zi*= Hi(w, S(w, p) )
Di(w,p):= Hi(w, S(w, p) )
input prices
output price
Di(w, p) = Hi(w, S(w, p)).
“function of a function” rule again
Dpi(w, p) = Hqi(w, q*) Sp(w, p)
Hi(w, q) = Ci(w, q)
Hqi (w, q) = Ciq(w, q)
Dpi(w, p) = Cqi(w, q*)Sp(w, p)
“output effect”
Demand for i and the price of jDi(w, p) = Hi(w, S(w, p)).
Dji(w, p) = Hji(w, q*)+ Hqi(w, q*)Sj(w, p)
Hqi(w, q)= Ciq(w, q) = Cqi(w, q)
Cjq(w, q*)
Dji(w, p) = Hji(w, q*) Ciq(w, q*)
Cqq(w, q*) .
We already know this is symmetric in i and j.
Obviously symmetric in i and j.
We already know this is negative or zero.
cannot be positive.
Results from decomposition formulaCiq(w, q*)Cjq(w, q*)
Dji(w, p) = Hji(w, q*)
Cqq(w, q*) .
Ciq(w, q*)2
Dii(w, p) = Hii(w, q*)
Cqq(w, q*).
output level
initial price level
Inputprice fall: substitution effectw1
conditional demand curve
H1(w,q)
price fall
Notional increase in factor input if output target is held constant
Change in cost
z1
*
z1
Conditional demand at original output
Conditional demand at new output
initial price level
ordinary demand curve
Inputprice fall: total effectw1
price fall
Change in cost
z1
z1
*
**
z1
Firm: Comparative Statics
Conditional
Input Demand
Optimisation subject to sideconstraint
Output
Supply
Ordinary
Input Demand
Shortrun problem
m
Swizi
i=1
P := pq –
q £f (z)
q³0, z³ 0
zm = `zm
~ _
C(w, q, zm ) := min S wi zi
{zm=`zm }
~ _ ~ _
Hi(w, q, zm) =Ci(w, q, zm )
~ _
C(w, q) £ C(w, q, zm)
~ _
Supply curves
C(w, q) C(w, q, zm)
_______ _________
£
q q
p
~
Cq
Cq
~
C/q
C/q
q
q
z1
Conditional input demandH1(w,q)
~ _
H1(w, q, zm)
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: