Econometrics questions

1

Differential Calculus II II. Higher Order Derivatives

A. Introduction The second-order derivative, written"( )f x , measures the slope and the rate of change of the first derivative, just as the first derivative measures the slope and the rate of change of the original or primitive function. The third-order derivative'"( )f x measures the slope and rate of change of the second-order derivative, etc. Higher-order derivatives are found by applying the rules of differentia- tion to lower-order derivatives. Common notation:

2nd order derivatives: 2

2 2

"( ) " d y

f x y D y dx

3rd order derivatives: 3

3 3

"'( ) "' d y

f x y D y dx

4th order derivatives: 4

(4) ( 4) 4 4

( ) d y

f x y D y dx

Examples:

1) Find the successive derivatives of the following function: 4 3 2 ( ) 2 5 3f x x x x= + +

3 2

2

(4)

(5)

'( ) 8 15 6

"( ) 24 30 6

"'( ) 48 30

( ) 48

( ) 0

f x x x x

f x x x

f x x

f x

f x

= + +

= + +

= +

=

=

For each of the following functions:

a) find the second-order derivative; and b) evaluate it at x = 2.

3 2

2

2

2

2

2

2) 7 5 12

a) 21 10

42 10

b) At = 2,

42(2) 10 94

y x x

dy x x

dx

d y x

dx

x

d y

dx

= + +

= +

= +

= + =

2

[ ]

2

2 2

22

2 4 4 4 3

2

2 3 3

5 3)

1 3

(1 3 )(5) (5 )( 3) a)

(1 3 )

(5 15 ) ( 15 ) 5

(1 3 ) (1 3 )

(1 3 ) (0) 5 2(1 3 )( 3) 30 90 30(1 3 ) 30

(1 3 ) (1 3 ) (1 3 ) (1 3 )

30 30 30 b) At = 2,

(1 3(2)) ( 5) 125

x y

x

dy x x

dx x

x x

x x

x xd y x x

dx x x x x

d y x

dx

= −

− − − =

−

− − − = =

− −

− − − − − − = = = =

− − − −

= = = = − − − −

6

25

Economic Application: If AR = f(Q), determine TR, MR and MR’.

2

2

( ) ( ); '( ) ( )

' "( ) '( ) '( ) "( ) 2 '( )

dTR TR Q AR Q Q f Q MR Q f Q f Q

dQ

d TR MR Q f Q f Q f Q Q f Q f Q

dQ

= ⋅ = ⋅ = = ⋅ +

= = ⋅ + + = ⋅ +

Economic Interpretation in the case of a linear demand curve (not required for question 12):

1) ( ) ( ) is the demand curve

2) '( ) is the slope of the demand curve

3) Since '( ) < 0 and "( )

AR Q f Q

f Q

f Q f Q

=

= 0, MR AR<

4) ' is the slope of the MR curve

5) Since "( ) 0 (the 2nd derivative of a linear function will always equal 0),

' 2 '( ) (the slope of MR is twice as steep as that of the demand curve)

6)

MR

f Q

MR f Q

= =

At 0, ( ) (0) ( the intercept for both MR and AR is equal).Q MR AR f Q f y= = = =

3

f(Q)

Q Q Q/2

f(Q)

Q

f(Q)

MR D = AR

Q

B. Concavity & Convexity

In chapter 2, it was established that a function is strictly convex over an interval if a secant line connecting any two points in that interval lies wholly above the graph of that function. A function is strictly concave over an interval if a secant line connecting any two points in that interval lies wholly below the graph of that function. We can now use our knowledge on 2nd derivatives to provide an alternative set of definitions for establishing concavity and convexity. Strictly Concave: A function f (x) is strictly concave over an interval if "( ) 0f x < for all values of x in that interval.

Strictly Convex: A function f (x) is strictly convex over an interval if "( ) 0f x > for all values of x in that interval.

Concave: A function f (x) is concave over an interval if "( ) 0f x ≤ for all values of x in that interval. Convex :A function f (x) is convex over an interval if "( ) 0f x ≥ for all values of x in that interval.

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III. Applications of the Derivative in Economics A. Differentiation and Marginal Analysis

Differentiation is concerned with finding the 'rate of change', 'slope', or 'gradient of a line or curve'.

a. Marginal Function

The marginal function is the 'slope of the change in the curve', which is equal to the rate of change of the variable being measured on the y axis with respect to that being measured on x axis. In other words, the first derivative of the function under consideration.

b. Marginal Cost

Marginal cost is the 'slope of total cost curve', which is equal to the 'rate of change of total cost with respect to a change in total output'.

dTC MC

dQ =

c. Marginal Revenue

Marginal revenue is the 'slope of the total revenue curve', which is equal to the 'rate of change of total revenue with respect to a change in total output'.

dTR MR

dQ =

d. Marginal Demand

Marginal demand is the 'slope of the demand curve', which is equal to the 'rate of change of quantity demanded of a good with respect to a change in the price of the good'

1) ( ) ( ) is the demand curve

2) '( ) (slope of the demand curve) is marginal demand

AR Q f Q

f Q

=

5

e. Marginal Propensity to Consume

The marginal propensity to consume is the 'slope of the consumption function', which is equal to the 'rate of change of consumption with respect to a change in disposable income'.

dC MPC

dY =

f. Marginal Utility

Marginal utility of a good is the 'rate of change of total utility with respect to a change in the quantity of the good consumed.

( , )

x

dU x y MU

dx =

( , ) y

dU x y MU

dy =

g. Marginal Product

Marginal product of a good is the 'rate of change of total product with respect to a change in quantity of the good produced.

dTP MP

dQ =

B. Cost, Revenue and Profit Functions 1. Given the Total Cost Function:

21( ) 10 5 4

TC Q Q Q= + +

a. Find Average Cost :

( ) 10 1( ) 5

4 TC Q

AC Q Q Q Q

= = + +

b. Find Marginal Cost

( ) 1( ) 5

2 dTC Q

MC Q Q dQ

= = +

2. Given the Demand Function: 8.175P =

a. Find Total Revenue

( ) 8.175TR Q PQ Q= =

b. Find Average Revenue

( ) ( ) 8.175

TR Q AR Q

Q = =

6

c. Find Marginal Revenue:

( ) 8.175 dTR

MR Q dQ

= =

3. Given the Profit Function

( ) ( )TR Q TC Qπ = −

a. Find the Profit Maximizing Level of Output:

Method 1: Taking the derivative of the profit function with respect to Q and solving for Q

2

( ) ( )

18.175 10 5 4

18.175 5 2

13.175 2

6.35

TR Q TC Q

Q Q Q

d Q

dQ

Q

Q

π

π

π

= −

 = − + +  

= − +

=

=

Method 2: Setting MR = MC and solving for Q

( ) ( )

18.175 5 2

13.175 2

6.35

MR Q MC Q

Q

Q

Q

=

= +

=

=

b. What is the Profit at this Level of Output:

[ ]

2

2

( ) ( )

( )

1(6.35)(8.17) 10 5(6.35) (6.35) 4

1(6.35)(8.17) 10 5(6.35) (6.35) 4

151.88 10 31.75 (40.323) 4

51.88 10 31.75 10.08 0

TR Q TC Q

PQ TC Q

π

π

= −

= −

 = − + +  

 = − + +  

 = − + +  

= − + + ≈

7

c. What might we conclude from this result?

Since profit = 0, it would be reasonable to conclude that the industry is characterized as perfectly competitive. The fact that the firm’s demand curve is perfectly horizontal and equal to its marginal revenue curve confirms this conclusion.

0 2 4 6 8 10 12 14 16 18 20 0

2

4

6

8

10

12

14

16

18

20

Quantity

P ri ce

P = MC = MR = min AC

Perfectly Competitive Firm

D = P = MR = AR

AC

MC

4. Downward Sloping Demand Function and Marginal Revenue Now consider the very same firm, but in this case, its demand curve is downward sloping as represented by the following demand function:

27.5 2P Q= −

a. Find the Total Revenue Function

2

( )

(27.5 2 )

27.5 2

TR Q PQ

Q Q

Q Q

=

= −

= −

b. Find Average Revenue

( ) ( ) 27.5 2

TR Q AR Q Q

Q = = −

8

c. Find Marginal Revenue

( ) ( ) 27.5 4

TR Q MR Q Q

dQ = = −

5. Given the Profit Function

( ) ( )TR Q TC Qπ = −

a. Find the Profit Maximizing Level of Output:

Method 1: Taking the derivative of the profit function with respect to Q and solving for Q

2 2

2 2

2

( ) ( )

127.5 2 10 5 4

127.5 2 10 5 4

2.25 22.5 10

4.5 22.5

4.5 22.5

5

TR Q TC Q

Q Q Q Q

Q Q Q Q

Q Q

d Q

dQ

Q

Q

π

π

π

π

π

= −

 = − − + +  

= − − − −

= − + −

= − +

=

=

Method 2: Setting MR = MC and solving for Q

( ) ( )

127.5 4 5 2

22.5 4.5

5

MR Q MC Q

Q Q

Q

Q

=

− = +

= =

b. Find the Price at this Level of Output:

27.5 2(5) 17.5P = − =

c. What is the Profit at this Level of Output:

2

( ) ( )

( )

1(17.5)(5) 10 5(5) (5) 4

187.5 35 (25) 4

46.25

TR Q TC Q

PQ TC Q

π

π

= − = −

 = − + +  

 = − +  

=

9

d. What might we conclude from this result?

Since profit > 0, it would be reasonable to conclude that the industry is a monopoly or an oligopoly. The fact that quantity (5) produced by the firm corresponds to only one price on the demand curve confirms this conclusion.

0 2 4 6 8 10 12 14 16 0

3

6

9

12

15

18

21

24

27

30

Quantity

P ri ce

MC

AC

5

17.5

D = AR MR

Monopoly

C. Demand function and elasticity The linear demand function for a product is given by:

36 6Q P= −

Where Q is quantity demand and P is price of the product

1. Calculate the slope of the demand curve (i.e. the marginal demand)

6 dQ

dP = −

10

2. Find the price elasticity of demand at P = 6 and P = 3. (Recall ( / )( /dQ dp P Q∈= )

( 6) dQ P P

dP Q Q ∈= ⋅ = − ⋅

At P = 6:

36 6(6) 0Q = − =

( 6)(5) ( 6) (perfectly elastic)

0

dQ P P

dP Q Q

− ∈= ⋅ = − ⋅ = → ∞

At P = 3: 36 6(3) 18Q = − =

( 6)(3)

( 6) 1 (unit elastic) 18

dQ P P

dP Q Q

− ∈= ⋅ = − ⋅ = = −

D. The Production Function A firm's production function is given by:

21000 (0.2)Q L L= −

where Q = total output and L = number of workers

1. Find the expression for the marginal production of labor (MPL)

1000 0.4L dQ

M P L dL

= = −

2. Find the value of MPL when

L = 1 L = 10 L = 100 L = 1000

If L = 1, 1000 0.4 1000 0.4(1) 999.6LM P L= − = − = If L = 10, 1000 0.4 1000 0.4(10) 996LM P L= − = − = If L = 100, 1000 0.4 1000 0.4(100) 960LM P L= − = − = If L = 1000, 1000 0.4 1000 0.4(1000) 600LM P L= − = − =

3. Does the law of diminishing marginal productivity apply to this particular function?

Yes, as L increases from 1 to 1000, MPL decreases from nearly 999.6 to 600 units.

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E. The Consumption function A non-linear consumption function is given by:

(0.9) 10000C Y= +

where C = consumption and Y = income

1. Calculate the marginal propensity to consume (mpc) and marginal propensity to save (mps)

.9 dC

mpc dY

= =

mps = 1- mpc = 1 - 0.9 = 0.1.

2. Use the fact that Y = C + S to derive the expression for the savings function. Substituting (0.9) 10000C Y= + into Y = C + S:

(0.9) 10000Y Y S= + +

Solving for S: .1 10000S Y= −

3. From your answer to question 2, use differential calculus to find the expression for the marginal

propensity to save (mps)

0.1 dS

mps dY

= =