Elasticities
Own-elasticity:
 AA
(1.1)
dq A
 dq  p 
q
  A    A  A 
dp A
 dp A  q A 
pA
Note that a perfect competitor—that is, firm that loses all of its output for an infinitesimal
increase in price (i.e., dPA is arbitrarily small)—has a own-elasticity of minus infinity. As a
matter of convention, demand elasticities are usually written as positive numbers. Since q is
negative, this requires a negative sign on the right-hand side.
Note that Equation 1.1 is written in terms of differentials. Over finite ranges, an arc elasticity
may be calculated:
 AA
(1.2)
q
q p
q


p
p q
p
As p  0, Equation 1.2 becomes Equation 1.1.
  1 elastic (MR  0)
  1 unit elastic (MR  0)
  1 inelastic (MR  0)
Some special cases of demand functions:
Constant elasticity of demand:
(1.3)
since
(1.4)
q  apb has a constant elasticity of demand b
b
dq p d  ap  p
p apb


 abpb1 
bb
dx q
dq q
q
q
Cross-elasticity:
1
 BA
(1.5)
dqB
 dq  p 
q
 B   B  A 
dp A  dp A  qB 
pA
Relationship between own- and cross-elasticities.
n
Consider
px
i 1
(1.6)
i i
 B, where xi  xi ( p). Then
n
x
x
dB
 0  x1  p1 1   pi i
dp1
p1 i 2 p1
Rearranging
n
x
x1
 p1
 x1   pi i
p1
p1
i 2
(1.7)
Dividing by x1 and multiply the last term by 1:
(1.8)

x 1 xi p1
p1 x1 x1 n
   pi i
x1 p1 x1 i 2 p1 x1 xi p1
Rearranging
(1.9)

n
p1 x1
p x x p
 1  1 i i i
x1 p1
i  2 xi p1 x1 p1
or
n
(1.10)
r
r1
11  1    i1 i ,
i 2
where ri  pi xi . Let si  ri / B be the revenue share of the expenditure on the product of firm i.
Then
(1.11)
11  1 
1 n
  i1si .
s1 i 2
2
3
Diversion Ratios
Diversion ratio:
DBA 
(1.12)
dqB
dq A
Relationship:
(1.13)
DBA
 dqB
 q
dq
 B  B
dq A  dp A
 p
 A
  dp A
 p
 A
  dq A
 q
 A

 q     q 
  B    BA   B 
  q A    AA   q A 


Diversion as a measure of upward pricing pressure. Look at lost profits from total diversion and
recoupment from diversion to other merging firm.
Value of total diversion for Firm A:
(1.14)
 p A  cA  d A
Where dA is the total units diverted from Firm A (say because of a relative price increase).
Recapture by Firm B
(1.15)
 pB  cB  d AB
Where dAB is the unit quantity of dA capture by Firm B.
The diversion ratio is then
(1.16)
DAB 
d AB
dA
The merged firm’s incentive to increase prices postmerger presumably is a function of the
proportion of profits it can recapture through the merger:
(1.17)
 pB  cB  d AB   pB  cB  D
 pA  cA  d A  pA  cA  AB
[SO WHAT?]
4
Residual Demand
Residual demand for firm f:
(1.18)
Df  p   D  p   So  p 
where D(p) is the market demand function and So(p) is the supply function of the other firms.
Differentiating with respect to p yields:
(1.19)
dD f
dp

dD dSo

dp dp
Multiplying both sides by p/Df and by 1:
p dD f
p dD D p dSo S o


D f dp
D f dp D D f dp S o
(1.20)
p dD D
p dSo So

D dp D f So dp D f

Translating into elasticities and noting that D f / D is the market share msf of firm f and So / D is
the aggregate market share of the other firms 1 ms f  :
(1.21)
 Df   D
1  ms f
1
  oS
ms f
ms f

where  Df is the elasticity of the residual demand function of firm f and  oS is the elasticity of
supply of the remaining firms in the market. This is sometimes written as:
(1.22)
 Df 
 D   oS 1  ms f 
ms f
5
Relationship of Firm and Market Elasticities
To show:
In a market of n identical firms, the firm own-price elasticity is equal to n times
the market own-elasticity.
Assume that there are n identical firms in the market, each producing quantity q (for an
aggregate market output of nq  Q ) at price p. Each firm has an own-price elasticity of i, and
the market has a market own-price elasticity of ..
Market own-price elasticity . is defined as:
Q
Q

p
p
Rearranging:
Q
p

Q
p
In this partial equilibrium analysis, assume that one firm changes its production by an amount
q and that all other firms remain at their original production levels q, so that q  Q .
Assume further that the one firm’s quantity change is so small that we can ignore any effect on
price (i.e., the changes are really differentials). Then:
Q q q
p



Q
Q nq
p
where the last equality comes from the previous equation. The definition of own price-elasticity
for a firm is:
q
q
i 
p
p
or
q
Q
Q
q
nq
Q
i 
n
n
 n
p
p
p
p
p
p
6
Profit Maximization, Margins, and Elasticities
Profits:
(1.23)
  x  R  x  C  x
 p  x x  C  x
First order condition:
d
 MR  MC  0
dx
(1.24)
 p
dp
dC
x
0
dx
dx
Although identifying the price p is usually straightforward, determining marginal cost mc can be
much more difficult because of difficulties in identifying which costs are variable (which
depends on the time period) and where common variable costs must be allocated.
Second order condition:
(1.25)
d 2 dMR dMC


0
dx 2
dx
dx
In order words, for the critical point to be a (local) profit maximum, the profit function must be
convex. This also implies that the marginal revenue curve must cut the marginal cost curve from
above.
Note that:
(1.26)
MR  p 
dp
x
dx
(1.27)
 dp x 
 p 1 

 dx p 
(1.28)
1 

 p 1  D 
  
At a profit-maximizing equilibrium, MR  MC , so that:
(1.29)
1 

MC  p 1  D 
  
or rearranging:
7
(1.30)
p  mc
1
1
 D  D
p


This is known as the inverse elasticity rule, which tells us that the Lerner index becomes smaller
the more elastic the firm’s residual demand.
We can also ask how MR changes with price:
(1.31)

d
pq( p) 
 pq( p)   q( p)  pq( p)  q( p) 1 
  q( p)(1   )
dx
q( p) 

8
Lerner Index
Lerner index:
L
(1.32)
p  mc
p
where p is the current price and mc is the current marginal cost. Expressed as a percentage, the
Lerner index gives the percentage of price p that constitutes the markup. From Eq. __, the Lerner
index is equal to the absolute value of the reciprocal of the own-elasticity of the firm’s residual
demand function.
[CHECK THIS:] The value of the Lerner index declines with output:
(1.33)
dp
 dp dmc 

p   p  mc 


dL  dx dx 
dx

2
dx
p
Relation of the Lerner index to elasticities through Equations __ and __:
(1.34)
L
ms f
p  mc
1
 D  D
p
f
   oS 1  ms f
9

HHI and Cournot Oligopoly
First order condition for profit maximization:
0
(1.35)
 i
 p(Q)  p(Q)qi  ci(qi )
qi
where Q is total quantity and qi is the quantity produced by firm i. Rearranging, dividing both
sides by p(Q) and multiplying by Q/Q:
p(Q)  ci(qi )
p(Q)qi
Qp(Q) qi si


 .
p(Q)
p(Q)
p(Q) Q 
(1.36)
qi
and  is the industry elasticity of demand. Multiplying by the market share si and
Q
summing over all firms i = 1, … , n yields:
where si 
n

(1.37)
i 1
p(Q)  ci(qi )
1 n
HHI
si   si2 
p(Q)
 i 1

Note that, in the symmetric case, si  1 n , so that:
(1.38)
1

n
 si2 
i 1
2
1 n 1 1 HHI
1



  
 i 1  n   n 2  n

1
n
or
(1.39)
1
 HHI
n
10
Bertrand Duopoly Model
Homogeneous products
Two suppliers: firm 1 and firm 2. Firms simultaneously compete in prices and offer homogenous
products so that consumers always buy from the firm that offers the lowest price.
Production costs: Ci  qi ci where c2  c1
Market demand: q  D( p ) where p  min  p1 , p2 
Individual demand:
pi  p j
0

 D ( pi )
qi  
pi  p j
(1.40)
 2
 D  pi  pi  p j
Profit functions:  i   pi  ci  qi  pi , p j 
Differentiated products
Same basic model, only qi is determined by residual demand function.
For a linear residual demand function
11
Critical Loss
General idea: A firm will increase price as long as the profits it earns with the price increase is
greater than the profits it earns without the price increase:
(1.41)
 p  p  q  q   c  q  q   pq  cq
Assuming constant average costs. Note that when the demand curve is downward sloping, a
positive Δp implies a negative Δq. Rearranging:
p  q  q     p  c  q
(1.42)
The left-hand side provides the gross revenue increase resulting from the increase in margin from
the price increase on the sales that the firm continues to make, while the right-hand side provides
the gross revenue loss resulting from the loss of margin on the sales that the firm no longer
makes at the higher price.
So firms price where the benefit of a price increase is just offset by the cost of the price increase:
p  q  q     p  c  q
(1.43)
The critical loss is the Δq that creates the equality in Equation 1.32. We can also express the
critical loss in terms of a fraction Δp/p by dividing the above equation by pq and rearranging:
 p  c  q
p  q 
1 
  

p 
q 
 p  q
(1.44)
Collecting terms and rearranging
(1.45)
p
q
%p
p



q  p p  c  % p  m
 p  p 


Where %p is the percentage price increase and m is the percentage gross margin. This implies
that the percentage critical loss declines as the gross margin increases.
12
Unilateral Effects
Standard Cournot model
For each firm i
 i  p( x) xi  Ci ( xi )
(1.46)
where x is the output vector and xi is the scalar output of firm I, so that x   xi .
n
First order condition for a profit maximum:
(1.47)
d i
p Ci
 p  xi

0
dxi
xi xi
Profit-maximizing function for the merged firm:
(1.48)
  1   2  p( x) x1  C1 ( x1 )  p( x) x2  C2 ( x2 )
First order condition for merged firm:
(1.49)
d
p
p C1
 p  x1
 x2

0
dx1
x1
x1 xi
p
 0, so that the right-hand side of Eq. 1.33 evaluated at the premerger level of
x1
output is less than zero. If the profit function is convex (from the second order condition—see
Eq. ), this implies that the premerger level of x1 is too large to maximize profits postmerger and
should be reduced. A reduction in the output of x1 then reduces total output and hence increases
the market price p.
Note that x2
13
HMG Example 5
Example 5: Products A and B are being tested as a candidate market. Each sells for $100,
has an incremental cost of $60, and sells 1200 units. For every dollar increase in the price
of Product A, for any given price of Product B, Product A loses twenty units of sales to
products outside the candidate market and ten units of sales to Product B, and likewise
for Product B. Under these conditions, economic analysis shows that a hypothetical
profit-maximizing monopolist controlling Products A and B would raise both of their
prices by ten percent, to $110. Therefore, Products A and B satisfy the hypothetical
monopolist test using a five percent SSNIP, and indeed for any SSNIP size up to ten
percent. This is true even though two-thirds of the sales lost by one product when it raises
its price are diverted to products outside the relevant market.
In the perfectly symmetric case,
(1.50)
p* p
DM

p
2 1  D 
where p* is the postmerger price, p the premerger price, D the diversion ration and M the
percentage gross margin. See Carl Shapiro, Mergers with Differentiated Products, 10 Antitrust
23 (1996).
Solving for p*:
(1.51)
p* 
DM
p p 
2 1  D 

DM 
p 1 

 2 1  D  
In Example 5, p1  p2  100, D12  D21  10 / 10  20 , and M1  M 2  100  60 /100  0.4. So
(1.52)
 1/ 3 0.4  
p*  100 1 
  100  10  110
2 1  1/ 3 

Note that D is only a ratio—it depends only on the relative magnitudes of total units lost and
units lost to the other firm. So one might surmise that the same price increase would result if the
for a dollar price increase in Product A, firm 1 loses only 3 units in total, of which 1 unit goes to
firm 2, because D remains at 1/3. But Equation 1.30 holds only when p is the profit-maximizing
premerger price. Under our new diversion facts, a premerger price of 100 (as in the example) is
much too low. Consider:
(1.53)
p
pc
1
p x
p
 

p
 x x p
x
14
Substituting the values of c, Δp, and Δx yields:
p  60 1 1200

p
3 p
(1.54)
so that p  460. That is, in the new hypothetical where there is a diversion of only 3 units for
every dollar price increase, the profit-maximizing price is 460 rather than 100. Using Equation
1.30, the merger would increase prices to
(1.55)
Δp

 1/ 3 0.4  
DM 
p*  p 1 
  460 1 
  460  92  552
2 1  1/ 3 
 2 1  D  

Diversion
Recapture
Total
Price 1 Cost 1 Quantity 1 Profits 1 Firm 2 Other Total Price 2 Cost 2 Profits 2 Profits
100
60
1200 48000
48000
10
110
60
900 45000
100
200
300
110
60
5000 50000
Diversion
Recapture
Δp
Price 1 Cost 1 Quantity 1 Profits 1 Firm 2 Other Total Price 2 Cost 2 Profits 2
460
60
1200 480000
92
552
60
924 454608
92
184
276
552
60
45264
Total
Profits
480000
499872
Another way to see this is through the residual demand curve. Note that Example 5 specifies that
each firm sells 1200 units, even through this fact is not a variable in the postmerger price
equation. Total quantity produced is important, since this fact, along with the lost of 30 units in
demand for every dollar increase in price (which tells us that the residual demand function is
linear), allows us to determine the each firm’s demand function and calculate the firm’s profitmaximizing price.
Linear demand function:
qi  a  bpi
(1.56)
Since b is the (negative) slope of the demand function and we know two points on the demand
curve, we can solve for b.
(1.57)
b 
1200  1170
 30
100  101
Taking one point and solving for a, 1200  a  30(100) or a  4200. So the demand function
behind Example 5 is
15
(1.58)
qi  4200  30 pi
for each firm i.
Rearranging to find the inverse demand curve:
(1.59)
p
4200  q
30
Profits:
(1.60)
First order condition:
(1.61)
q 
 4200  q


 60   q  80  
30 
 30


  q  p  c  q 
d
2q
 80 
0
dq
30
so that q  1200 and p  100 as Equation 5 states.
16
Margins, Elasticities, Entry and Repositioning
From Eq. __
(1.62)
pc 1

p

Under the usual assumptions, a firm will enter if it can make profits, that is, if its revenues
exceed its fixed and variable costs:
(1.63)
p2 q2  F  c2 q2  0
or
(1.64)
 p2  c2  q2  F
If a firm has not entered premerger, then presumably this condition is not fulfilled. Under what
circumstances will the condition be fulfilled postmerger? [NEED A DEMAND CURVE THAT
IS A FUNCTION OF P1 AND P2, AND A CROSS-ELASTICITY OF DEMAND]
17
Envelope Theorem
V (a)  max f  x, a 
(1.65)
x
where a is a parameter. Let x*(a) be the optimal control variable, so that x* solves the maximum
equation:
V  a   f  x *(a), a 
(1.66)
Then
dV f
  x *(a), a 
da a
(1.67)
That is, only the direct effect of the change in a should be taken into account. The indirect effect
of a change in a through x has no first-order effect.
Proof:
From Eq. 1.66, V  a   f  x *(a), a  . Differentiate both sides of this identity with respect to a:
dV  a 
(1.68)
But
da
f  x *(a), a 
(1.69)
x

f  x *(a), a  x(a)
x
a

f  x *(a), a 
a
 0, since f  x *(a), a  is a maximum but the definition of x*. So
dV  a 
da

f  x *(a), a 
a
.
18
Appendix: Basic Derivative Forms
Chain rule: If y  y (u ( x)), then
dy dy du
 ·
dx du dx
(1.70)
Product rule:
d
dv
du
(u·v)  u·  v· .
dx
dx
dx
(1.71)
Reciprocal rule:
d  1   g ( x)


dx  g ( x)  ( g ( x)) 2
(1.72)
Quotient rule: If f ( x) 
(1.73)
g ( x)
, then
h( x )
f ( x) 
g ( x)h( x)  g ( x)h( x)
.
[h( x)]2
Multivariate
Chain rule for several variables: Let z = f(x, y), where x = g(t) and y = h(t), and g(t) and h(t) are
differentiable with respect to t. Then:
(1.74)
dz z dx z dy


.
dt x dt y dt
Young’s theorem: Let f be a differentiable function of n variables. If each of the cross partials fij
and fji exists and is continuous at all points in some open set S of values of (x1, ..., xn) then
fij (x1, ..., xn) = fji (x1, ..., xn) for all (x1, ..., xn) in S.
19
Total Derivatives and Total Differentials
Total derivative
Let f ( p)  f ( x( p), y ( p)). Suppose that p increases from p to p   p. The change of x and y
dx
dy
f
f
will be  x   p and  y   p. The increase of f will be  f   x   y. So
dp
dp
p
p
f dx
f dy
f 
 p
 p.
x dp
y dp
Dividing by  p and let  p  0 (so that  becomes = ) yields
(1.75)
df f dx f dy


.
dp x dp y dp
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