Transcript Profit

Profit Maximization
Profit Maximizing Assumptions
• Firm: Technical unit that produces goods or
services.
• Entrepreneur (owner and manager)
– Gains the firm’s profits and suffers losses and has the
goal of maximizing profit.
– Transforms inputs (aka factors) into outputs through
the technology of the production function.
– Decides how much of each input to use and what
quantity to produce.
Realistic?
• For this class, we assume corporation’s with
shareholders and boards and executives function
like the entrepreneur.
– Obviously, managers and executives have other
incentives besides maximizing profit.
– But there is a more developed theory of the firm for
corporations that we will not get into.
• Profit maximization?
– Libraries and water utilities are strictly non-profit.
– Most hospitals claim to be non-profit, but they act
like for-profit hospitals.
Why Firms?
• You could ask why even have firms?
• Why don’t entrepreneurs outsource
EVERYTHING?
• Transactions costs make that infeasible. Or not.
– In the 1870s-1950s vertical integration was the norm
(River Rouge plant included a steel mill and processed
rubber).
– Since then, outsourcing has been growing.
– New communication technology has driven this more
towards the entrepreneur-only model.
Profit
• Profit maximization could easily be 1/2 of the
text as we assume firms maximize profit under a
host of situations: perfectly competitive,
monopoly, price discrimination, oligopoly,
monopsony, etc.
• However, the basics are perfect competition
(price taker) and monopoly (price setter).
Price Taker vs Price Setter
• Profit is TR-TC
• Price Taker (competitive firm)
– Treat the price they face as given when choosing quantity
• Price Setter (single price, no strategic behavior)
– Price is chosen along with quantity.
• Short and long run options are the same
– Short Run, quantity decision includes the shut down option
– Long Run, consideration of returns to scale and entry and exit
Profit
• Profit = TR-TC
– Total costs include all implicit and explicit costs (unlike
accounting cost that would only include explicit costs).
• In our model, we assume the firm rents capital at a rate of v.
But that is exactly the same as if the firm owned the capital
but could rent it out to another firm at a rate of v.
• Accounting profit using the firm’s owned resources in the
next best alternative use
– Includes Value of the entrepreneur’s time
– Selling off owned factors and investing elsewhere
– For us, SC = VC + FC = wL + vK
Price Takers
• The rest of this lecture focuses on price takers.
– Homogeneous output
– No barriers to entry/exit in long run
– Many sellers
– Perfect price information
Revenue: Price Taker
• Price Taker
P
Market Demand
However, price taker
assumption is that no
firm is big enough to
be able to affect the
market price.
p = Pm
1,000 2,000 3,000
3,000,000
q
• If a firm charges p > Pm , they will sell q = 0
• Demand for firm’s output is p = Pm, the firm can sell as
many as it wants, until q = 3,000,000, and then need to
lower the price to sell more.
Revenue: Price Taker
• So for price taker, we assume decreasing returns
to scale precludes getting large enough to have
production influence price:
P
Demand for firm:
Pm = MR = AR
p = Pm
1,000
3,000
5,000
q
• So R = p·q, where p = Pm
MR 
d p  q 
dq
 p,
AR 
p  q 
q
p
Cost and Short Run Supply
• Let’s for the moment assume production exhibits IRS and then DRS.
• Firms will, in the LR, choose a level of K commensurate with getting the lowest
possible SAC.
• The price will be driven to the low point of AC, the break even price.
• At this starting point, the firm’s SAC, AVC, and SMC are relevant in the short run
• Other returns to scale options considered in the long run.
Exhibits
IMR, DMR
C
C
SRC
SRC
Exhibits
IMR, DMR
AC
SAC
SMC
Exhibits
IMR, DMR
MC
SMC
AC
SAC
Ehhibits
IRS, DRS
q
pbe
AVC
q
Cost and Short Run Supply
• But for the moment, we don’t care which level of K the firm has or what the AC
curve looks like.
• In the SR, here is what we have to work with.
• To determine the profit maximizing level of q = q*, and whether shut down and
produce q = 0, MC and SAVC are most important.
C
SC
SC
Exhibits
IMR, DMR
AC
SRAC
SRMC
Exhibits
IMR, DMR
SAC
AVC
SMC
q
q
Profit Max
• Maximize π = R(q) - C(q)
• FOC for this yields q where slope of π function = 0
SC
R=p·q
SC
π=R-SC
MR=P
slope of R
π maximized at q where MR=SMC
FOC, derivative of π function is zero
SMC =
slope of SC
q
q
Profit Max
• Checking the SOC too.
R
SC
SC
π=R-SC
MR=P
slope of R
π also minimized at q where
MR=SMC (FOC satisfied here too)
Which is why we check SOC, to
make sure profit is falling where
MR = SMC
(i.e. MR is falling relative to SMC)
SMC =
slope of SC
q
q
Profit Max
•
•
•
•
The more common graph
Maximize π = R(q) - C(q)
FOC for this yields MR=SMC, which it does twice.
SOC ensures MC is rising relative to MR
R
C
SC
SC
AC
SAC
SMC
Exhibits
IMR, DMR
SMC
q
q
Price Taker Profit Max
• So long as you are better off producing than
not,
• As you increase q, the change in profit = MRSMC.
• Produce until MR = SMC and marginal profit
(change in profit as q increases) is falling.
To Maximize Profit
Q
MR
99
TR
12
SMC
1188
SC
Change in
Profit =
(MR-MC)
1000
I pulled these starting values out of the air
Profit
188
To Maximize Profit
Q
MR
TR
SMC
SC
Change in
Profit =
(MR-MC)
1000
Profit
99
12
1188
100
12
1200
7
1007
+5
193
101
12
1212
8
1015
+4
197
102
12
1224
9.10
1024.10
+2.90
199.90
103
12
1236
10.40
1034.50
+1.60
201.50
104
12
1248
12
1046.50
0
201.50
105
12
1260
13.80
1060.30
-1.80
199.70
106
12
1272
16
1076.30
-3
195.70
Note, at π max, AR > ATC. When AR = ATC, π = 0
188
Graphically
MR = P, SMC
MC
16
13.80
MR = P
12
10.40
9.10
8
7
100 101 102 103 104 105 106 q
MC and qs: If price was $16, then the firm would
produce 106.
MR = P, SMC
MC
MR = P
16
13.80
12
10.40
9.10
8
7
100 101 102 103 104 105 106 q
MC and qs: If price was $10.40, then the firm
would produce 103.
MR = P, SMC
MC
16
13.80
12
MR = P
10.40
9.10
8
7
100 101 102 103 104 105 106 q
MC and qs: If price was $8.00, then the firm
would produce 101.
MR = P, SMC
MC
16
13.80
12
10.40
9.10
MR = P
8
7
100 101 102 103 104 105 106 q
MC and Supply
• Price Takers
– The MC curve tells us the profit maximizing qs by the
firm at any price.
– Since it is the MC curve that determines the
relationship between p and the quantity to supply,
the SMC curve IS the firm’s short run supply curve.
– Important caveat, if suffering a loss, firm might want
to shut down if the loss is larger than FC.
• Side note: Price Setters
– set the price, they do not respond to it, so they have
no supply curve.
Shut Down Option
(price takers and price setters)
• Shut down: Short run situation where the firm
produces a quantity of 0 while remaining in
the industry. It is still considered to be in the
industry as long as it cannot rid itself from its
fixed inputs.
• The firm could start producing very easily by
employing some of the variable input.
Intuition
• A firm bearing a loss can produce qs = q*, (where
MR=MC) or can shut down, produce qs = 0.
– If qs = q*: π = R – VC – FC
– If the firm shuts down: π = -FC (that is, has a loss =
FC)
• If FC is greater than the loss from producing, qs =
q*. If FC is less than the loss from producing qs =
q*, better to shut down and produce qs=0.
• Decision Rule: Shut down if
R = 10,000
R = 7,000
– FC > R – VC – FC
Profit from
shut down
Profit from
q=q*
VC = 8,000
FC = 4,000
π = -2,000
-4,000 < -2,000
qs = q *
VC = 8,000
FC = 4,000
π = -5,000
-4,000 > -5,000
qs =0
Decision Rule
• Shut down if:
–FC > R – VC – FC
0 > R – VC
R > VC
• So long as revenue covers all variable cost, the
loss will be less than FC so q = q* .
Side note: Which can change the firm’s
output decision, a change in Variable
Cost and/or a change in Fixed Cost?
• Shut down if: – FC > 
• – FC > R – VC – FC
• FC is on both sides, so a change in FC does not affect
the relationship or the decision.
• Ok, yes, fixed costs are fixed (don’t vary with output)
• health insurance premiums rise
• Tony Romo signs a $108m extension.
• But a change in either R or VC could change the
decision.
Price Taker in the Short Run
• Simple, just MR = MC
• Maximize profit w.r.t. q
• Maximize profit w.r.t. L
Profit Max 1
• Simple, supply is SMC, find q where SMC = p.





SC  SC v , w , q;K as fro m last ch ap ter SC  w  L w , q;K  v  K
d SC
dq

 SM C w , q;K

s


Set P  SM C w , q;K an d so lve fo r q  q  p 
s
Ch eck to en su re th at  > -FC, if n o t, th en sh u t d o w n

Profit Max 2, MR = MC
• Optimize by choosing q.



m ax   p  q  SC v , w , q;K 1 , w h ere SC  SC v , w , q;K 1
q
*
FO C


 q  p  SM C w , q;K 1  0
or


p  SM C w , q;K 1 , ch o o se q su ch th at th e M R = S M C
So lve fo r q to get th e firm su p p ly fu n ct io n , q  q  p 
*
s
SO C
 qq  
d(SM C)
dq
 0 , w h ich it w ill b e if D M R
Ch eck to en su re th at  > -FC, if n o t, th e n sh u t d o w n

Profit Max 3, MRPL=w
• Optimize by choosing L.
m a x   p  f  K ,L   vK  w L
1
1
L
FO C
 L  p  fL  w  0
p  fL  w , ch o o se L su ch th a t th e M R PL = w

So lve fo r L to get th e la b o r d em a n d fu n c tio n , L  L w ,p ,K 1
*
*



Plu g in to q  f K 1 ,L to get p ro fit m a xim izin g q  q  p 

*
s

Plu g in to   p  f K 1 ,L  vK 1  w L to get m a xim a l p ro fit fu n ctio n     w ,p 
SO C
 LL  p  fLL  0 , w h ich it w ill b e if D M R
C h eck to en su re th a t  > -FC , if n o t, th e n sh u t d o w n
Producer Surplus
• Producer surplus is the amount by which a
firm is better off than shut down (q=0)
• If shut down, loss is = FC
• By producing, the firm covers this potential loss,
plus gains profit.
• If profit = 0, then better off by the amount of FC
• PS = π + FC
• PS = R-VC-FC+FC
• PS = R-VC
Producer Surplus
• If π = 0, then producer surplus = FC
• If π < 0, but -π < FC, producer surplus > 0
• If π < 0, and -π = FC, producer surplus = 0
• Shut down point
• If π < 0, and -π > FC, producer surplus < 0
• Will shut down, so PS = 0 and profit = -FC.
Profit Maximization
Price Taker, Long Run
• Returns to Scale Matter
– IRS, LMC falling
– CRS, LMC constant
– DRS, LMC rising
Increasing Returns to Scale
• IRS only: incompatible with competition as the
biggest firm has the lowest average cost…
natural monopoly results
C
AC
MC
C
AC
MC
q
q
Decreasing Returns to Scale
• DRS only: an infinite number of infinitely small
firms.
C
AC
MC
C
MC
AC
q
q
Constant Returns to Scale
• CRS only: any size firm can produce at the same
AC. AC = AVC = MC (Firm LRS is horizontal at MC).
C
AC
MC
C
AC=MC
q
q
IRS, DRS
• IRS, DRS: MC rising. Firm LRS = MC above AC (exit
otherwise.
C
C
AC
MC
MC
AC
Exhibits
IRS, DRS
q
q
IRS, CRS, DRS
• IRS, CRS, DRS: MC rising, but flat spot while CRS.
• Perhaps most realistic, but not easy to solve – or find a
production function that creates this.
C
C
AC
MC
DRS
Firm LRS = MC for p ≥ pBE
MC
CRS
AC
IRS
q
q
Profit Max. vs. Perfect Comp.
• We will eventually assume that in the long run K will be
fixed to yield this SAC and the market price will be pbe (so
in the LR q* will be at the low point of SAC)
• But in this chapter, we want to explore the possibility that
price will exceed pbe for a while. So we need a firm LRS
curve. AC
MC
SAC
SMC
MC
SMC
SAC
AC
Firm exit if p < pBE
q
Production and Exit
• Essentially, a firm’s long run supply curve will
be its long run MC curve…
• While shut down is a viable option in the short
run, in the long run all costs can be avoided by
exiting the market.
• If p < pbe, (minimum value of AC curve), the
firm should exit the industry.
• So firm long run supply is MC above pbe.
Price Taker in the Long Run
• Simple, just MR = MC
• Maximize profit w.r.t. q
• Maximize profit w.r.t. K, L
Profit Max 1
• Simple, set MC = P, find q.
C  C  v , w , q  a s fro m la st ch a p te r  C  w  L  v , w , q   v  K  v , w , q  
*
dC
*
*
 M C  v ,w ,q 
dq
R e ve n u e : p  q
dR
dq
 M R q
Se t M R = M C a n d so lve fo r q  q  p 
*
SO C :
d M C 
dq
s
0
C h e ck to e n su re th a t  > -FC , if n o t, th e n sh u t d o w n
Profit Max , MR=MC
• Optimize by choosing q
m a x   p  q  C  v , w , q  , w h e re C  C  v , w , q  co m e s fro m
*
q
co st m in im iza tio n .
FO C
q  p  M C  v ,w ,q   0
or
p  M C  v , w , q  , ch o o se q su ch th a t th e M R = M C
So lve fo r q to ge t th e firm su p p ly fu n ctio n , q  q  p 
*
SO C
d 
2
 qq 
dq
2

d(M C)
dq
 0, or
d(M C)
dq
 0 w h ich it w ill b e if D R S
C h e ck to e n su re th a t  > 0, if n o t, th e n e xit
Profit Max, MRPL=w; MRPK=v
• Optimize by choosing inputs
m a x   p  f  K ,L   vK  w L
L
FO C
 L  p  fL  w  0  p  fL  w , M R PL = w
 K  p  fK  v  0  p  fK  v , M R PK = v
*
*
So lve fo r L , K to ge t th e fa cto r d e m a n d fu n ctio n s
*
L = L(w ,v,p )
*
K = K(w ,v,p )
P lu g in to q  f  K ,L  to ge t p ro fit m a xim izin g su p p l y fu n ctio n
*
*
q  q  K(v , w ,p),L(v , w ,p)   q(p)
*
Profit Max, choose K and L
• Ratio of FOC
 L : p  fL  w
 K : p  fK  v
fL
fK

w
v
So th e p ro fit m axim izin g in p u t ch o ice also m in im izes
th e co st o f p ro d u cin g th at level o f q .
Profit Max, choose K and L
• SOC
Th e  fu n ctio n is strictly co n cave at L , K
*
H 
 LL
 LK
 KL
 KK
*
 0 , n egative d efin ite
 LL  p  fLL ,  LK  p  fLK
 KL  p  fKL ,  KK  p  fKK

H  p  fLL  fKK   fLK 
2
2
0
So lo n g as fLL  0 an d fKK  0 , an d fLK is sm all.
2
Profit Max, choose K and L
• Profit function, maximal profits for a given w,
v, and p.
Plu g K =K(v , w ,p) an d L  L(v , w ,p) in to
*
*
  p  f  K ,L   vK  w L
to get th e p ro fit o p tim izin g p ro fit fu n c tio n :
  p  f  K(v , w ,p),L(v , w ,p)   vK(v , w ,p)  w L(v , w ,p)
Properties of the Profit Function
• Homogeneous of degree one in all prices
– with inflation, K*, L*, and q* are the same profit will keep up
with that inflation
• Nondecreasing in output price
– Δ profit ≥ 0 with Δ p > 0
• Nonincreasing in input prices
– Δ profit ≤ 0 with Δ w or Δ v > 0
• Convex in output prices
– profits from averaging those from two different output prices
will be at least as large as those obtainable from the average of
the two prices
 (p 1 , v , w )   (p 2 , v , w )
2
 p1  p2


,v,w 
 2

Envelope Results
• Long run supply
d
dp
 f  K*,L * 
  p  f  K(v , w ,p),L(v , w ,p)   vK(v , w ,p)  w L(v , w ,p)
d
dp
d
dp
d
dp
d
dp
 f  K(v , w ,p),L(v , w ,p)   p  fK K p  fL L p  d p  vK p d p  w L p d p
 f  K(v , w ,p),L(v , w ,p)   p fKK p d p  p fLL p d p  vK p d p  w L p d p
 f  K(v , w ,p),L(v , w ,p)   p fKK p d p  vK p d p  p fLL p d p  w L p d p
 f  K(v , w ,p),L(v , w ,p)    p fK  v  K p d p   p fL  w  L p d p
B y FO C
d
dp
d
dp
d
dp
 f  K(v , w ,p),L(v , w ,p)    0  K p d p   0  L p d p
 f  K(v , w ,p),L(v , w ,p)  w h ich is f(K* ,L* )  q *
give s u s th e lo n g ru n su p p ly e q u a tio n : q  q(v , w ,p)
*
To maximize profit
when there is a change
in price , q=q*= f(K*,
L*), continue producing
such that
L* = L(w, v, p)
K* = K(w, v, p)
Envelope Results
• Profit maximizing factor demand functions
  p  f  K(v , w ,p),L(v , w ,p)   vK(v , w ,p)  w L(v , w ,p)
d
*
 p  fK K w  fL L w  d w  vK w d w   L  w L w d w 
dw
d
dw
d
dw
d
dw
 p fK K w d w  p fL L w d w  vK w d w  L  w L w d w
*
 p fK K w d w  vK w d w  p fL L w d w  w L w d w  L
*
  p fK  v  K w d w   p fL  w  L w d w  L
*
dw
dw

*
*
 L
dw
dv
 K(v , w ,p)
To m axim ize p ro fit w h en
  0 K w d w   0 L w d w  L
d
d
K  K*  K(w , v , p)
B y FO C
d
K 
th ere is a ch an ge in v,
B y FO C
d
Sim ilarly,
*
To maximize profit when there is
a change in w, choose L such that
L = L*=L(w, v, p)
gives u s th e d em a n d eq u a tio n , L*  L(v , w ,p )
Comparative Statics
• Price taker
• Long run
Comparative Statics (∂L/∂w, ∂K/∂w)
• K* and L * back into the FOC to create the
following identities
p  fL  L(w , v ,p),K(w , v ,p)   w  0
p  fK  L(w , v ,p),K(w , v ,p)   v  0
d iffe re n tia te w .r.t. w
L
*
p  fLL
w
L
 p  fLK
*
p  fKL
w
 p  fKK
K
*
w
K
1 0
*
w
0
Comparative Statics (∂L/∂w, ∂K/∂w)
L
*
p fLL
p fLK
w
p fKL
p fKK
K
*
1

0
w
L
*
w
K

p fKK
p (fLL fKK  fLK )
2
 p fKL
*
w

2

p (fLL fKK  fLK )
2
0
2

0
Comparative Statics
(∂L/∂w, ∂K/∂w)
• Increase in wage, increases MC, q* falls
K
L falls, K rises
L
Comparative Statics
(∂L/∂w, ∂K/∂w)
• Increase in wage, increases MC, q* falls
K
L falls, K falls
L
Comparative Statics (∂L/∂v, ∂K/∂v)
L
*
p fLL
p fKL
p fLK
v
p fKK
K
*
0

1
v
L
 p fLK
*

v
K
v

p (fLL fKK  fLK )
2
*

N o te th at:
2
p fLL
p (fLL fKK  fLK )
2
2
L
*

0
v

K
w
b ecau se
0
*
fLK  fKL
Comparative Statics (∂L/∂p, ∂K/∂p)
• K* and L * back into the FOC to create the
following identities

 L (w , v ,P),K

(w , v ,P)   v  0
P  fL L (w , v ,P),K (w , v ,P)  w  0
P  fK
*
*
*
*
d ifferen tiate w .r.t. P
L
*
fL  P  fLL
p
L
 P  fLK
*
fK  P  fKL
p
 P  fKK
K
*
0
p
K
*
p
0
Comparative Statics (∂L/∂p , ∂K/∂p)
L
*
p fLL
p fLK
w
p fKL
p fKK
K
*

fL
fK
w
 fL fKK  fK fLK 

0
2
 p p(fLL fKK  fLK ) 
L
*
 fK fLL  fL fLK 

0
2
p
p(fLL fKK  fLK ) 
K
*
So long as fKL is positive or
small, these will both be >
0. Since an increase in P
causes MR to rise, at least
one of these must be > 0
Comparative Statics
(∂L/∂p, ∂K/∂p)
• Increase in p, increases MR, q* rises
K
L rises, K rises
L
Comparative Statics
(∂L/∂p, ∂K/∂p)
• Increase in p, increases MR, q* rises
K
L rises, K falls
K inferior
Obviously, L could be
inferior instead
Expansion path
L
Comparative Statics (∂q/∂p)
• q = f(L,K)
• q*=f(L*=L(w,v,p),K*=K(w,v,p))
• How does this respond to a change in p?
q
*
p
L
*
 fL
p
 fK
K
*
p
an d w e n o w kn o w
L
*
p
K

*
p

 fL fKK  fK fLK
p(fLL fKK  fLK )
2
 fK fLL  fL fLK
p(fLL fKK  fLK )
2
Comparative Statics (∂q/∂p)
• And so we can substitute to get:
q
*
p
 fL
 fL fKK  fK fLK
p(fLL fKK  fLK )
2
 fK
 fK fLL  fL fLK
p(fLL fKK  fLK )
2
a n d fin a lly
2
2


f
f

2
f
f
f

f
f 
q
L K LK
K LL 
 L KK

0
2
p
p(fLL fKK  fLK )
*
w h e re fL fKK  2 fL fK fLK  fK fLL  0 if iso q u a n ts a re co n ve x to o rigin
2
2
(re q u ire d fo r co st m in im iza tio n )
w h e re fLL fKK  fLK  0 if p ro d u ctio n fu n ctio n is co n ca ve a t K a n d L
2
*
*
(re q u ire d fo r risin g m a rgin a l co st a t q , p ro fit d e cre a sin g in q )
*
The Short Run and the Long Run
Le Châteliar Principle
• How does the short run demand for L differ
from the long run demand?
K
Increase in the wage rate
K2
K0
q*(w1)
q*(w2)
LL*
Ls*
L*
L
Short Run Profit Max
 
FO C :   p  f  L ,K   w  0
SO C :   p  f  L ,K   0
D e m a n d : L  L  w ,p ,K  , o n ce K is fixe d , v d o e s n o t a ffe ct L d e cisio n .
m a x   p  f L ,K  w L  vK
L
L
LL
LL
*
L
S
S
To ge t:

*
w
, su b stitu te d e m a n d (L* ) in to F O C
 
 
 p  fL L w ,p ,K ,K  w  0
S
L
d iffe re n tia te w .r.t. w
L
S
p  fLL
L
w
S
w

1 0
1
p  fLL
 0 , a s fLL  0
Long Run vs. Short Run
L
S
w

1
p  fLL
0
R em em b er th e co m p arative statics resu lt:
L
L
w
L

L
w
L
L

L
w
L

S

L
w
p(fLL fKK  fLK )
2
S
L
w
fKK
w
L

S

w

0
fKK
p(fLL fKK  fLK )
2

1
p  fLL
(fLL fKK  fLK )
2
fKK fLL
p(fLL fKK  fLK )fLL
2

fKK fLL  fLL fKK  fLK
p(fLL fKK  fLK )fLL
2
p  fLL (fLL fKK  fLK )
2
2

fLK
And we can deduce
2
p fLL (fLL fKK  fLK )
2
> 0, by SOC
0
Short Run Profit Max
• Input demand in the long run is more elastic.
L
L
w
L
S

L
w
L
L
w
L
L

p fLL (fLL fKK  fLK )
2
0
0
w
L
S

L
w
2
S
L
w

fLK
, b u t b o th n e ga tive , so m u lt b y -1
w
L
S

w
, LR ch a n ge in L is > th e sh o rt ru n ch a n ge .
“These types of relations are …referred to as Le Châtelier effects, after the similar
tendency of thermodynamic systems to exhibit the same types of responses.” –
Silberberg, 3rd ed. (p. 85)
Cobb-Douglas Examples
• Three cases:
•
qK
.25 .25
L
.5 .5
•
qK L
•
q  KL
qK
.25
.25
L
• Cost Min
o
.25 .25
m inL  w L  vK   (q  L K
)
FO C
LL  w 
K
.25
.75
L
.25
 0; L K  v 
Exp a n sio n p a th : K 
wL
v
L
K
.75
o
.25 .25
 0; L   q  L K
0
qK
.25
.25
L
• Cost Min
SO C
.25
3 K
L LL 
7
4L
.25
 0; L KK 
4K
.25
L
.75
4K
4L
.25
3 K
.75
7
K
4L
4L
4K
.25
4
.25

.75
4L K
L
5
4
32L K
5
4
0
4
.25
.75

.75 .75
4L K


5
4
5
64L K
.25
3 L
.75 .75

7
4
K
0
3 L
7
4K

3

5
5
 32L 4 K 4
4


 5 5 0
 8L 4 K 4

4

5
4
64L K
5
4

3
3


5
5
5
5
 64L 4 K 4 64L 4 K 4



qK
• L * , K*
.5
2  v 
2  w 
L*  q 
 ; K*  q 

w 
 v 
.25
.25
L
.5
• C*
.5
*
2  v 
2  w 
C  wq 
  vq 

w 
 v 
C  2q
*
2
 wv 
.5
.5
M C  4q  w v  ; P  4q  w v  ; q 
*
AC  2q  w v 
*
.5
.5
.5
*
p
4  wv 
.5
qK
.25 .25
L
• MC, AC
MC,AC
M C  4qwv 
*
A C  2q  w v 
*
q
.5
.5
qK
.25 .25
L
• Profit Max

.25 .25
m ax   p L K
  w L  vK
FO C
L 
pK
.25
.75
4L
.25
 w  0;  K 
Exp a n sio n p a th : K 
wL
v
pL
4K
.75
v0
qK
.25
.25
L
• Profit Max
SO C
 3p K
 LL 
.25
7
16L
 3p K
7
16L
4
16L 4 K
8p
3
.25
 3p L
3
4
7
16K
2
32  LK 
0
4
p
4
4
7
16K
3
16L K

 0;  KK 
.25
p
3
.25
 3p L
1.5
0
4
4

9p
2
 KL 
256  LK 
.25
7

4
p
2
 KL 
.25
256  LK 
7
4
qK
• L * , K*
p
L* 
16 v
• Π*
2
1.5
w
.5
; K* 
p
16 w
.25 .25
L
2
1.5 .5
v


p
q 
1.5 .5 
 16 w v 
.25

p
*
  p
 4  w v  .5

2
2





p
p
  v
w
1.5 .5 
1.5 .5 

 16 w v 
 16 v w 

2
*
*
 
p
2
8wv 
.5


p

1.5 .5 
 16 v w 
2
.25

p
4wv 
.5
qK
.25 .25
• Envelope Results
*
q 


p
*
L 
*
*
K 

p
4wv 
*

w

v
p
16 v
*

.5
2
1.5
p
16 w
w
.5
2
1.5 .5
v
L
qK
.25 .25
L
• Profit Max
MC,AC
M C  4qwv 
*
A C  2q  w v 
*
MR
L* 
p
16 v
2
1.5
w
.5
; K* 
2


p
q 
1.5 .5 
 16 w v 
*
p
16 w
.25
q*
2
1.5 .5
v
2


p

1.5 .5 
 16 v w 
.25

p
4wv 
.5
q
.5
.5
.5 .5
qK L
• Cost Min
o
.5 .5
m inL  w L  vK   (q  L K )
FO C
LL  w 
K
.5
.5
2L
.5
 0; L K  v 
Exp a n sio n p a th : K 
wL
v
L
2K
.5
o
.5 .5
 0; L   q  L K
0
.5 .5
qK L
• Cost Min
SO C
L LL 
K
.5
.5
 0; L KK 
1.5
4L
.5
L
.5
2K
K
0
2L
K
.5
K
.5
1.5
2L
4L
.5

.5
4L K
L
2K

.5

.5 .5
8L K
.5 .5

L
4K
1.5
0
.5
.5

.5 .5
4L K


.5 .5
16L K
.5
L
4K
1.5

  

0
 8L.5K .5  4L.5K .5




.5 .5
16L K






 16L.5K .5 16L.5K .5 


.5 .5
qK L
• L * , K*
.5
• C*
 v 
L*  q   ; K* 
w
.5
w
q 
 v 
 v 
w
C  w q    vq  
w
 v 
.5
.5
*
C  2q  w v 
*
.5
M C  4  w v  ; P  4  w v  ; q  any
*
AC  2 wv 
*
.5
.5
.5
*
.5 .5
qK L
• MC, AC
MC,AC
MC  2wv 
*
AC  2 wv 
*
q
.5
.5
.5 .5
qK L
• Profit Max

.5 .5
m ax   p L K
  w L  vK
FO C
L 
pK
.5
.5
2L
.5
 w  0;  K 
Exp a n sio n p a th : K 
wL
v
pL
2K
.5
v0
.5 .5
qK L
• Profit Max
SO C
p K
 LL 
p K
.5
1.5
4L
.5
1.5
4L
.5
 0;  KK 
4K
1.5
0
p
.5 .5
4L K
.5
p
p L
.5 .5
4K
4L K
p L
1.5

p
2
16  LK 

p
2
16  LK 
0
.5 .5
qK L
• L * , K*
L*  u n d e fin e d ; K*  u n d e fin e d
• Π*
*
q  any
*
  u n d e fin e d
• Envelope Results? No.
.5 .5
qK L
• MC, AC
MC,AC
At MR1 > MC, π-max at q = ∞
At MR2 < MC, π-max at q = 0
At MR = MC, π-max at any q (π=0)
MR1
M C  AC  2  w v 
*
*
MR2
q
.5
q  KL
• Cost Min
o
m inL  w L  vK   (q  LK)
FO C
o
L L  w   K  0; L K  v   L  0; L   q  LK  0
Exp a n sio n p a th : K 
wL
v
q  KL
• Cost Min
SO C
L LL  0; L KK  0
0
K
K
0
L

L
    KL    KL    2KL   0
0
q  KL
• L * , K*
• C*
.5
 v 
 w
L*   q  ; K*   q 
 w
 v 
.5
 v 
 w
*
C  wq   vq 
 w
 v 
C  2 qw v 
*
.5
.5
.5
.5
.5
wv
 wv 
 wv 
*
MC  
 ; p
 ; q  2
p
 q 
 q 
*
 wv 
AC  2 

q


*
.5
q  KL
• MC, AC
MC,AC
 wv 
*
AC  2 

q


 wv 
*
MC  

q


.5
.5
q
q  KL
• Profit Max
m a x   p  LK   w L  vK
FO C
 L  p K  w  0;  K  p L  v  0
Exp a n sio n p a th : K 
wL
v
Cost minimizing tangency
q  KL
• Profit Max
SO C
 LL  0;  KK  0
0
p
p
0
2
 0p  0
• SOC indicate we have a profit min, not a max.
q  KL
• L * , K*
L* 
• Π*
v
p
; K* 
w
p
*
q  SO C n o t sa tisfie d
*
  SO C n o t sa tisfie d
q  KL
• MC, AC
MC,AC
 wv 
*
AC  2 

 q 
 wv 
*
MC  

 q 
.5
.5
Maximizing profit
means q=∞, where
MR ≠ MC
MR
Only produce
units where
MR < MC!
q*
q
Appendix
Short Run vs. Long Run Labor demand
• Demand for labor can be written:

L (w , v ,p)  L w ,p ,K  K (w , v ,p)
L
S
L
• Differentiate w.r.t. w:
L
L
S
w

w
L
S

K

K
L
w
• But let’s try to sign
L
S
K

K
L
w

The slopes of the SR and LR factor
demand functions differ by a term that
is the product of two effects: change in
K from a change in w and the change
in L that WOULD be caused by a
change in the fixed amount of K.
– Differentiate original equation w.r.t. v
L
L
v
L
S

K

K
L
v
Now what?
• From the differential w.r.t. v:
L
L
v
L
S

K

K
L
sign-wise, all we
know is this is < 0
v
Which tells us these two must have opposite signs.
• Rearrange for this:
L
L
L
S
  vL
K
K
v
Because of the
opposite signs, this
is < 0
• And substitute into:
L
w
L
S

w
L
S

K

K
L
w
Now what?
• Yields:
L
L
L
w
L
K
S

L
  vL 
w K w
v
• And from reciprocity
L
L
v
• Yielding:

K
L
w
 KL 


S

w
L
L




L
K
w w
v
2
Finally
• And we can say:
 K 


S

w
L
L




L
K
w w
L
2
>0
v
<0
• Since all terms are < 0, it is clear that the short
run effect is smaller than the long run effect.