Univariate Calculus
Professor Peter Cramton
Economics 300

Outline
• Rules of differentiation
– Sum rule
– Scale rule
– Product rule
– Power rule
– Exponential rule
– Chain rule
– Quotient rule
– Logarithmic rule

Outline
• Second derivatives and convexity
• Economic applications
– Risk aversion and utility functions
– Elasticities
– Total revenue
– Average, marginal, and total cost

Rules of differentiation
• Used to evaluate derivative of any function
• Much easier than working with basic definition dy dx
 lim
0
( x ) f x
 x

Sum rule y

( )

( )

( ) dy dx
 
( )
 
( )
 
( )

Scale rule y

( )

( ) dy dx
 
( )
 
( )

Product rule y

( )

( ) ( ) dy dx
 
( )
 
( ) ( )
 

Power rule y

( )
 kx n dy dx
 
( )
 nkx n

1

Exponential rule y

( )
 e kx dy dx
 
( )
 ke kx

Chain rule y

( )

( ( )) dy dx
 
( )
 
( ( ))

( ) y

( ) u

( ) dy
 dy du dx du dx

Quotient rule y

( )
 dy dx
 
( )


( ) ( )
 
2

Logarithmic rule y

( )
 ln x dy
  
1 dx x y

( )
 log b x dy dx
  
1 1 x ln b

y
 x
3 y

25 y

7
4 x
4 y

8 x
2 
3 x

14

1 y x
3

2 
2 x
Exercise dy dx

3
2 x dy dx
 
7 x

5 dy dx dy dx

16

3 x x
2

1/ 2
 
2
3 x

3 
2

y
 
20 x

4 x
2 y
 e
2 x y

3ln x y

2( x

1)
2
Exercise dy

20 8 x dx dy

2 e
2 x dx dy

3 dx x dy dx

4( x

1)

f x

( x

1)
3 
( x
2  x
2 
5 f x

3( x

1)
2 
2( x
2  x x

2)
( )

(2 x

4)
99 f x
 x
 98
( ) 99(2 4) 2 f x
 e x ab

( )
 x ab

1 x
( ) e
 abe abx
( )

( e x a b
)

( )

( x a b

1 x
) a e ax a

1  ab e x a b a
( ) x

1
( )

( e
2
)
10

( ) 10( e
2
10
) ( b

2 )

y
 log (2
2 x

3) y
 x
2 log (8 )
4 y
 x
3 log
2 x
Exercise y
 
2 1
2 x

3 ln 2 y
 
16 x 1

2 1
8 x
2 ln 4 x ln 4 y
  x
2
3 log
2 x
 x
3
1 1 x ln 2 y
 log
3 x
2 x y
  
1
2 x

2 log
3 x

1 1 1
2 x x ln 3

Second derivative
• First derivative
– Rate of change
– Slope of the function
– Velocity dy dx
 
( )
• Second derivative
– Rate of change of the rate of change
– Rate of change of the slope of the function
– Acceleration
  2 d dy d y
 
  dx dx dx
2 f

( )

Example: square-root utility
( )
 ac
1
2 dU
 dc
2 d U dc
2


( )

1
2 ac
 1
2

( )
 
1
4 ac
 3
2

Square-root utility function
( )
 c
1
2 dU
 dc

( )

1
2 c
 1
2
2 d U dc
2
 
( )
 
1
4 c
 3
2

Example: logarithmic utility
( )
  ln x dU
 dx
2 d U dx
2


( )

 x

( )
 
 x
2

Logarithmic utility function

Second derivative and convexity
• Strictly convex
– Slope strictly increasing
– Second derivative positive: f”(x) > 0
• Strictly concave
– Slope strictly decreasing
– Second derivative negative: f”(x) < 0

Second derivative and convexity
• Convex
– Slope increasing
– Second derivative f”(x) 
0
• Concave
– Slope decreasing
– Second derivative f”(x) 
0

Concave and convex functions
Strictly concave
Strictly convex

Which do you prefer?
• $12 for sure
• 50-50 chance of $6 or $18
• Expected value of both is $12
.5($6) + .5($18) = $12
• Risk averse
– Prefer sure thing
– U(12) > .5U(6) + .5U(18)
– Concave utility
• Risk seeking
– Prefer gamble
– U(12) < .5U(6) + .5U(18)
– Convex utility

Risk-averse and risk-loving utility

y
Exercise: Concave or convex?
  x

6 x
2  x
3 y
   x

3 x
2 y x
30
25
20
Concave y”<0 Convex y”>0
1 2 3 4

Elasticities
• How sensitive is demand to a change in price?
• Does revenue increase or decrease when price is raised?
• Depends on price elasticity of demand
• % change in quantity / % change in price
 



Elasticities
0, percentage change in Q is dQ
Q
0, percentage change in P is dP
P
Price elasticity of demand
 
/
 dQ P
/ dP Q

Elasticities
• 
< -1 demand is price elastic
– quantity falls by more than 1% if price increases by 1%
• 
> -1 demand is price inelastic
– quantity falls by less than 1% if price increases by 1%
• Elasticities in different markets are comparable, since based on % change
P
Elastic
P
Inelastic
D
Q
D
Q

Elasticities with linear demand
Elastic
Inelastic q p
  dq p
  

  p
  p

2


, q


2 p p dp q q
     p

Arc elasticity for discrete changes
 arc

(
(
Q
P
Q

Q
A
A
P
B

B

Q

P
B
B
P
A
A
) / 2
) / 2

Alternative form for elasticity
Price elasticity of demand
 
/
 dQ P
/ dP Q d ln Q

1
 dQ Q d ln Q
 dQ
Q d ln P

1
 dP P d ln P
 dP
P
  dQ P
 d ln Q dP Q d ln P

Log-linear demand

Infant mortality and income/capita

Elasticity and other economic concepts
• Substitutes
– Products with good substitutes are more elastic
• Complements
– Products with many complements are less elastic
• Luxury goods tend to be more elastic
• Necessities tend to be less elastic

Do consumers or producers bear a tax?
• Depends on elasticity of demand and supply
– Party least able to substitute away pays more of tax
P
C

P
P

T
Q ( P
D C
)

Q P
S P
)

 dP
C
 dP
P
 dT dQ
D dP
C dQ P
D
 dQ
S
P dP Q dP Q
C P dP
P dT


D
  
S D

 dP
P
and
  dP
C dT dP
C


S
  
S
 dQ
S dP
P dQ P
D dP Q
C dT
D dP
P

Impact of quantity change on revenue
• Depends on elasticity of demand
– Producer wants to reduce quantity if it faces inelastic demand
R

PQ dR dP
 dQ dQ
Q P P

 1
 dP Q dQ P


P

1


1

• 
> -1 then revenue declines when produce more
• 
< -1 then revenue increases when produce more
• Monopolist allows produces in elastic portion of demand (

< -1)

2500
2000
1500
1000
500
P = 100 – Q; R = (100 – Q)Q

< -1

> -1
20 40 60 80 100

y
Exercise: Find elasticity
  
4
  dy x dx y
 
2
4
 
92
8 / 92 y

16 8 x
 2
,

1.5
    ln y
  x x

3
  
.5
x y


2 x
4
 x
 
3 / 2.5
y

50 x

2
, x

2
  
2
50

3 x x
50 x

2
 
2

Total, average, and marginal
• Total product
• Average product
• Marginal product
800
Q
Q
/

100

L
1/2
100

L

1/2

1/2
/ 50 L
Total product
600
Average product > marginal product since concave
400
200
60
L
10 20 30 40 50

Total, average, and marginal
• Total cost
C

Q
3 
15 Q
2 
93 Q

100
• Average cost
• Marginal cost
$ dC dQ

3 Q
2 
30 Q

93
Total cost
700
600
500
400
300
200
Average cost = marginal cost at minimum average cost
12
Q
2 4 6 8 10

300
250
200
150
100
50
Average cost = marginal cost at min
Competitive firm size
Marginal cost
Average cost
5 10 15