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 from last chapter SC  w  Ls w,q;K  v  K

dSC
 SMC w,q;K
dq



Set P  SMC w,q;K and solve for q  qs p 
Check to ensure that  > -FC, if not, then shut down

Profit Max 2, MR = MC
• Optimize by choosing q.



max   p  q  SC v,w,q;K1 , where SC*  SC v,w,q;K1
q
FOC


q  p  SMC w,q;K1  0
or


p  SMC w,q;K1 , choose q such that the MR = SMC
Solve for q to get the firm supply function, q*  qs p 
SOC
d(SMC)
qq  
 0, which it will be if DMR
dq
Check to ensure that  > -FC, if not, then shut down

Profit Max 3, MRPL=w
• Optimize by choosing L.
 
max   p  f K1 ,L  vK1  wL
L
FOC
L  p  fL  w  0
p  fL  w, choose L such that the MRPL = w

Solve for L* to get the labor demand function, L*  L w,p,K1
 
Plug into   p  f K ,L   vK

Plug into q  f K1 ,L to get profit maximizing q*  qs p 
1
1
 wL to get maximal profit function     w,p 
SOC
LL  p  fLL  0, which it will be if DMR
Check to ensure that  > -FC, if not, then shut down
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 as from last chapter  C*  w  L  v,w,q   v  K  v,w,q  
dC*
 MC  v,w,q
dq
Revenue: p  q
dR
 MR  q
dq
Set MR=MC and solve for q*  qs p 
d MC 
SOC :
0
dq
Check to ensure that  > -FC, if not, then shut down
Profit Max , MR=MC
• Optimize by choosing q
max   p  q  C  v,w,q  , where C*  C  v,w,q  comes from
q
cost minimization.
FOC
q  p  MC  v,w,q   0
or
p  MC  v,w,q  , choose q such that the MR = MC
Solve for q to get the firm supply function, q*  q p 
SOC
d2 
d(MC)
d(MC)
qq  2  
 0, or
 0 which it will be if DRS
dq
dq
dq
Check to ensure that  > 0, if not, then exit
Profit Max, MRPL=w; MRPK=v
• Optimize by choosing inputs
max   p  f K,L   vK  wL
L
FOC
L  p  fL  w  0  p  fL  w, MRPL = w
K  p  fK  v  0  p  fK  v, MRPK = v
Solve for L* , K* to get the factor demand functions
L* =L(w,v,p)
K* =K(w,v,p)
Plug into q  f K* ,L*  to get profit maximizing supply function
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 w

fK v
So the profit maximizing input choice also minimizes
the cost of producing that level of q.
Profit Max, choose K and L
• SOC
The  function is strictly concave at L* , K*
H
LL
LK
KL
KK
 0, negative definite
LL  p  fLL , LK  p  fLK
KL  p  fKL , KK  p  fKK


H  p  fLL  fKK   fLK   0
2
2
So long as fLL  0 and fKK  0, and fLK2 is small.
Profit Max, choose K and L
• Profit function, maximal profits for a given w,
v, and p.
Plug K* =K(v,w,p) and L*  L(v,w,p) into
  p  f K,L   vK  wL
to get the profit optimizing profit function:
  p  f K(v,w,p),L(v,w,p)   vK(v,w,p)  wL(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
(p1 ,v,w)  (p2 ,v,w)
 p1  p2


,v,w 
2
 2

Envelope Results
• Long run supply
d
 f K*,L *
dp
  p  f K(v,w,p),L(v,w,p)   vK(v,w,p)  wL(v,w,p)
d
 f K(v,w,p),L(v,w,p)   p  fKKp  fLL p  dp  vKpdp  wL pdp
dp
d
 f K(v,w,p),L(v,w,p)   pfKKpdp  pfLL pdp  vKpdp  wL pdp
dp
d
 f K(v,w,p),L(v,w,p)   pfKKpdp  vKpdp  pfLL pdp  wL pdp
dp
d
 f K(v,w,p),L(v,w,p)   pfK  v Kpdp  pfL  w L pdp
dp
By FOC
d
 f K(v,w,p),L(v,w,p)    0 Kpdp   0 L pdp
dp
d
 f K(v,w,p),L(v,w,p)  which is f(K*,L*)  q*
dp
d
gives us the long run supply equation: q*  q(v,w,p)
dp
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)  wL(v,w,p)
d
 p  fKKw  fLL w dw  vK wdw  L*  wL wdw 
dw
d
 pfKKwdw  pfLL wdw  vK wdw  L*  wL wdw
dw
d
 pfKKwdw  vK wdw  pfLL wdw  wL wdw  L*
dw
d
 pfK  v Kwdw  pfL  w L wdw  L*
dw
By FOC
d
  0Kwdw   0 L wdw  L*
dw
By FOC
Similarly,
d
 K(v,w,p)
dv
To maximize profit when
K*  
there is a change in v,
K  K*  K(w, v, p)
To maximize profit when there is
a change in w, choose L such that
L = L*=L(w, v, p)
d
 L*
dw
d

gives us the demand equation, L*  L(v,w,p)
dw
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
differentiate w.r.t. w
L*
K*
p  fLL
 p  fLK
1  0
w
w
L*
K*
p  fKL
 p  fKK
0
w
w
Comparative Statics (∂L/∂w, ∂K/∂w)
L*
pfLL pfLK w 1

pfKL pfKK K* 0
w
pfKK
L*
 2
0
2
w p (fLL fKK  fLK )
pfKL
K*
 0
 2
2
w p (fLL fKK  fLK ) 
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
pfLL pfLK v
0

*
pfKL pfKK K
1
v
*
pfLK
L
 0
 2
2
v p (fLL fKK  fLK ) 
*
pfLL
K
 2
0
2
v p (fLL fKK  fLK )
*
Note that:
L* K*

v w
because
fLK  fKL
Comparative Statics (∂L/∂p, ∂K/∂p)
• 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
differentiate w.r.t. P
L*
K*
fL  P  fLL
 P  fLK
0
p
p
L*
K*
fK  P  fKL
 P  fKK
0
p
p
Comparative Statics (∂L/∂p , ∂K/∂p)
L*
pfLL pfLK w fL

pfKL pfKK K* fK
w
L*  fL fKK  fK fLK 

0
2
p p(fLL fKK  fLK ) 
K*  fK fLL  fL fLK 

0
2
p p(fLL fKK  fLK ) 
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*
L*
K*
 fL
 fK
p
p
p
and we now know
L*  fL fKK  fK fLK

p p(fLL fKK  fLK2 )
K*  fK fLL  fL fLK

p p(fLL fKK  fLK2 )
Comparative Statics (∂q/∂p)
• And so we can substitute to get:
f f  f f
f f  f f
q*
 fL L KK K LK2  fK K LL L LK2
p
p(fLL fKK  fLK )
p(fLL fKK  fLK )
and finally
q

p
*
  fL 2 fKK  2fL fK fLK  fK2 fLL 
p(fLL fKK  fLK )
2
0
where fL 2 fKK  2fL fK fLK  fK2 fLL  0 if isoquants are convex to origin
(required for cost minimization)
where fLL fKK  fLK2  0 if production function is concave at K* and L*
(required for rising marginal cost at q* , profit decreasing 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
 
FOC:   p  f L,K   w  0
SOC:   p  f L,K   0
Demand: L  L  w,p,K  , once K is fixed, v does not affect L decision.
max   p  f L,K  wL  vK
L
L
LL
LL
*
S
LS
To get:
, substitute demand (L*) into FOC
w
 
 
*L  p  fL LS w,p,K ,K  w  0
differentiate w.r.t. w
LS
p  fLL
1  0
w
LS
1

 0, as fLL  0
w p  fLL
Long Run vs. Short Run
LS
1

0
w p  fLL
Remember the comparative statics result:
fKK
LL

0
2
w p(fLL fKK  fLK )
fKK
LL LS
1



w w p(fLL fKK  fLK2 ) p  fLL
fKK fLL
(fLL fKK  fLK2 )
LL LS



2
w w p(fLL fKK  fLK )fLL p  fLL (fLL fKK  fLK2 )
fLK2
LL LS fKK fLL  fLL fKK  fLK2



0
2
2
w w p(fLL fKK  fLK )fLL
pfLL (fLL fKK  fLK )
> 0, by SOC
And we can deduce
Short Run Profit Max
• Input demand in the long run is more elastic.
fLK2
LL LS


0
2
w w pfLL (fLL fKK  fLK )
LL LS

0
w w
LL LS

, but both negative, so mult by -1
w w
LL
LS

, LR change in L is > the short run change.
w w
“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 L
•
qK L
•
q  KL
.25 .25
.5 .5
qK L
.25 .25
• Cost Min
minL  wL  vK  (qo  L.25K.25 )
FOC
K.25
L.25
LL  w  .75  0; LK  v  .75  0; L   qo  L.25K.25  0
L
K
wL
Expansion path: K 
v
qK L
.25 .25
• Cost Min
SOC
LLL 
3K.25
7
4L
L.25
4K.75
32L
7
4L 4

4L.75K.75

5
4K
K.25
4L.75
3K.25
0
K.25
4L.75
4
 0; LKK 
5
4K 4
3L.25
7
0
4
L.25
4K.75
 3



3 


 5 5 
.75 .75
5
5
5
5
5
5 
4L K
64L 4K 4 64L 4K 4  64L 4K 4 64L 4K 4 
3L.25
7
4K 4
 3 

 5 5   5 5 0
 32L 4K 4  8L 4K 4
qK L
.25 .25
• L *, K*
.5
.5
2 v 
2w
L*  q   ; K*  q  
w
v
• C*
.5
.5
*
2 v 
2w
C  wq    vq  
w
v
C  2q
*
2
 wv 
.5
MC  4q  wv  ; P  4q  wv  ; q 
.5
*
AC*  2q  wv 
.5
.5
*
p
4  wv 
.5
qK L
.25 .25
• MC, AC
MC,AC
MC*  4q wv 
.5
AC  2q wv 
.5
*
q
qK L
.25 .25
• Profit Max


max   p L.25K.25  wL  vK
FOC
pK.25
pL.25
L  .75  w  0; K  .75  v  0
4L
4K
wL
Expansion path: K 
v
qK L
.25 .25
• Profit Max
SOC
LL 
3pK.25
7
16L
4
 0; KK 
16K
3pK.25
7
16L
p
3
16L

3pL.25
7
0
4
p
p KL 
16L 4K 4 9p KL 


7
7
.25
4
3pL
256 LK 
256 LK  4
3
4
3
4K 4
8p2
32 LK 
1.5
16K
0
3
7
4
2
.25
2
.25
qK L
.25 .25
• L *, K*
p2
p2
L* 
; K* 
1.5 .5
16v w
16w1.5v.5
• Π*
.25
.25

 

p
p
q 
1.5 .5  
1.5 .5 
 16w v   16v w 
2
*
2

p
4  wv 
.5
2
2
 p
 



p
p
*
  v
  p
 w

.5
1.5
.5
1.5 .5 
 4  wv    16w v 
 16v w 


* 
p2
8  wv 
.5
qK L
.25 .25
• Envelope Results
*

p
*
q 

p 4  wv .5

p
L 

w 16v1.5w.5
*
2

p
*
K 

v 16w1.5 v.5
*
*
2
qK L
.25 .25
• Profit Max
MC,AC
MC  4q wv 
.5
*
AC*  2q wv 
.5
MR
q*
p2
p2
L* 
;
K*

16v1.5w.5
16w1.5v.5
.25
.25

 

p2
p2
q 
1.5 .5  
1.5 .5 
 16w v   16v w 
*

p
4  wv 
.5
q
qK L
.5 .5
• Cost Min
minL  wL  vK  (qo  L.5K.5 )
FOC
K.5
L.5
LL  w  .5  0; LK  v  .5  0; L   qo  L.5K.5  0
2L
2K
wL
Expansion path: K 
v
qK L
.5 .5
• Cost Min
SOC
K.5
L.5
LLL  1.5  0; LKK  1.5  0
4L
4K
0
K.5
2L.5
L.5
2K.5

K.5
2L.5
K.5
4L1.5

4L.5K.5
L.5
2K.5



 
 




4L.5K.5 16L.5K.5 16L.5K.5  16L.5K.5 16L.5K.5 
L.5
4K1.5


  


0
.5 .5
.5 .5 
.5 .5

8L K
 8L K  4L K
qK L
.5 .5
• L *, K*
.5
• C*
.5
v
w
L*  q  ; K*  q 
w
v
.5
.5
v
w
C  wq   vq 
w
v
*
C  2q wv 
.5
*
MC*  4  wv  ; P  4  wv  ; q*  any
.5
AC  2  wv 
*
.5
.5
qK L
.5 .5
• MC, AC
MC,AC
MC*  2  wv 
.5
AC  2  wv 
.5
*
q
qK L
.5 .5
• Profit Max


max   p L.5K.5  wL  vK
FOC
pK.5
pL.5
L  .5  w  0; K  .5  v  0
2L
2K
wL
Expansion path: K 
v
qK L
.5 .5
• Profit Max
SOC
pK.5
pL.5
LL  1.5  0; KK  1.5  0
4L
4K
pK.5
4L1.5
p
4L.5K.5
p
2
p2
4L.5K.5  p

0
.5
16 LK  16 LK 
pL
4K1.5
qK L
.5 .5
• L *, K*
L*  undefined; K*  undefined
• Π*
q*  any
*  undefined
• Envelope Results? No.
qK L
.5 .5
• 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
MC*  AC*  2  wv 
.5
MR2
q
q  KL
• Cost Min
minL  wL  vK  (qo  LK)
FOC
LL  w  K  0; LK  v  L  0; L   qo  LK  0
wL
Expansion path: K 
v
q  KL
• Cost Min
SOC
LLL  0; LKK  0
0
K
L
K
0
  KL  KL  2KL  0
L

0
q  KL
• L *, K*
• C*
.5
.5
 v
 w
L*   q  ; K*   q 
 w
 v
.5
.5
 v
 w
*
C  w q   v q 
 w
 v
C*  2  qwv 
.5
.5
.5
wv
 wv 
 wv 
*
MC  
; p
; q  2


p
 q 
 q 
*
.5
 wv 
AC  2 

q


*
q  KL
• MC, AC
MC,AC
.5
 wv 
AC*  2 

q


.5
 wv 
MC*  

q


q
q  KL
• Profit Max
max   p LK   wL  vK
FOC
L  pK  w  0; K  pL  v  0
wL
Expansion path: K 
v
Cost minimizing tangency
q  KL
• Profit Max
SOC
LL  0; KK  0
0 p
p 0
 0  p2  0
• SOC indicate we have a profit min, not a max.
q  KL
• L *, K*
• Π*
v
w
L*  ; K* 
p
p
q*  SOC not satisfied
*  SOC not satisfied
q  KL
• MC, AC
MC,AC
.5
 wv 
AC*  2 

 q 
.5
 wv 
MC*  

 q 
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:

LL (w,v,p)  LS w,p,K  KL (w,v,p)
• Differentiate w.r.t. w:
L LS LS KL



w w K w
• But let’s try to sign
LS KL

K 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
LL LS KL


v K v
Now what?
• From the differential w.r.t. v:
LL LS KL


v K v
sign-wise, all we
know is this is < 0
Which tells us these two must have opposite signs.
• Rearrange for this:
LL
LS v
 L
K K
v
Because of the
opposite signs, this
is < 0
• And substitute into:
L LS LS KL



w w K w
Now what?
• Yields:
LL
L LS v KL

 L
w w K w
v
• And from reciprocity
LL KL

v w
• Yielding:
 KL 


L LS  w 


KL
w w
v
2
Finally
• And we can say:
 K 


L LS  w 


KL
w w
v
L
2
>0
<0
• Since all terms are < 0, it is clear that the short
run effect is smaller than the long run effect.