Lectures 6

Section 11.6 from Fundamental methods of Mathematical Economics, McGraw Hill 2005, 4th Edition. by A. C. Chiang & Kevin Wainwright is covered. Please read this section from book. The brief summary of this section and some examples are solved here for better understanding.

Economics Applications

Multi product firms-Perfect Competition, so prices taken as given

Read detail of theory from book.

Example 1: A two-product competitive firm faces the following cost and revenue functions:

퐶 = 2푄 + 2푄

푅 = 12푄 + 18푄

(a) Will the production of the two goods be technically related? (b) Formulate the total-profit function 휋 in terms of 푄 and 푄 . (c) What will be optimal levels of 푄 and 푄 . (d) What is the value of 휋 ? What does this imply economically?

Solution:

(a) Two goods are technically related if is a function of both 푄 and 푄 . Similarly, is a function of

both 푄 and 푄 . 휕퐶휕푄 = 4푄

휕퐶휕푄 = 4푄

The production of the two goods is not technically related because the marginal cost of one commodity will be independent of the output of the other.

(b) The total profit function is

휋 = 12푄 + 18푄 − 2푄 − 2푄 (c) The first order partial derivatives of profit function are set equal to zero for optimal values

= 12− 4푄 = 0

= 18− 4푄 = 0

this yields 푄 = 3 , 푄 = 4.5. To ensure that these values represent a maximum profit, let us check the second order condition. The Hessian is

|퐻| = −4 00 −4

The leading principal minors are

|퐻 | = −4 < 0, |퐻 | = |퐻| = 16 > 0

The second-order condition for maximum is satisfied. The signs of the leading principal minors do not depend on where they are evaluated. Thus the maximum in this problem is a unique absolute maximum.

(d) 휋12 = 0 implies that the profit-maximizing output level of one commodity is independent

of the output of the other (see first-order condition). The firm can operate as if it has two plants, each optimizing the output of a different product.

Example 2: A two-product competitive firm faces the following cost and revenue functions:

퐶 = 2푄 + 2푄

푅 = 12푄 + 18푄

(a) Will the production of the two goods be technically related? (b) Formulate the total-profit function 휋 in terms of 푄 and 푄 . (c) What will be optimal levels of 푄 and 푄 . (d) What is the value of 휋 ? What does this imply economically?

Solution: See solution from book.

Multi product firms-Monopoly, so Prices dependent on Quantity

Read detail of theory from book.

Example 3: A two-product firm faces the following demand and cost functions:

푄 = 40− 2푃 − 푃

푄 = 35− 푃 − 푃

퐶 = 푄 + 2푄 + 10

(a) Write out the total-revenue function R in terms of 푄 and 푄 . (b) Formulate the total-profit function 휋 in terms of 푄 and 푄 . (c) Find the output levels that satisfy the first-order condition for the maximum profit. (d) Check the second-order sufficient condition.

Solution:

(a) We have given 푄 = 40− 2푃 − 푃 푄 = 35− 푃 − 푃

Rearranging both equations for quantity to get 푄 − 40 = −2푃 − 푃

푄 − 35 = − 푃 − 푃

Solving these two equations for 푃 ,푃 , we have

푃 = 5− 푄 + 푄 , 푃 = 30 + 푄 − 2푄

The total revenue function is 푅 = 푃 푄 + 푃 푄

푅 = 5푄 + 30푄 + 2푄 푄 −푄 − 2푄

(b) The total profit function is

휋 = 푅 − 퐶 = 푅 = 5푄 + 30푄 + 2푄 푄 − 2푄 − 4푄 − 10

(c) The first order partial derivatives of profit function are set equal to zero for optimal values = 5− 4푄 + 2푄 = 0

= 30 + 2푄 − 8푄 = 0

Solving equations simultaneously results in 푄 = 푄 = .

(d) Now we will check second order sufficient condition for maximum profit

휋 = −4 휋 = 2 휋 = 2 휋 = −8

|퐻| = −4 22 −8

|퐻 | = −4 < 0 | 퐻 | = 28 > 0

The second-order condition for maximum is satisfied.

Example 4: A two-product firm faces the following demand and cost functions:

푄 = 40− 2푃 + 푃

푄 = 15 + 푃 − 푃

퐶 = 푄 + 푄 푄 + 푄

(a) Write out the total-revenue function R in terms of 푄 and 푄 . (b) Formulate the total-profit function 휋 in terms of 푄 and 푄 . (c) Find the output levels that satisfy the first-order condition for the maximum profit. (d) Check the second-order sufficient condition. What is the maximum profit?

Solution: See solution from book.

Price discrimination

Read detail of theory from book.

Example 5: A monopolist producing a single output has two types of customers. If it produces 푄 units for customers of type 1, then these customers are willing to pay a price of

50− 5푄 dollars per unit. If it produces 푄 units for customers of type 2, then these customers are willing to pay a price of 100 − 10푄 dollars per unit. The monopolist’s cost of manufacturing 푄 units of output is 90 + 20푄 dollars. In order to maximize profits, how much should the monopolist produce for each market?

Solution: The average revenue functions of monopolistic firm are

푃 = 50 − 5푄 ,푃 = 100− 10푄

and that total cost function is

퐶 = 90 + 20푄 where 푄 = 푄 + 푄

The profit function is 휋 = 푅 (푄 ) + 푅 (푄 )− 퐶(푄)

휋 = 50푄 − 5푄 + 100푄 − 10푄 − 90− 20(푄 + 푄 )

휋 = 50− 10푄 − 20 = 30 − 10푄

휋 = 100 − 20푄 − 20 = 80− 20푄

Setting 휋 , 휋 to zero, we obtain

30 − 10푄 = 0,푄 = 3

80 − 20푄 = 0,푄 = 4

The profit maximizing output levels are 푄 = 3,푄 = 4.

To ensure that these values represent a maximum profit, let us check the second order condition. The Hessian is

|퐻| = −10 00 −20

The leading principal minors are

|퐻 | = −10 < 0, |퐻 | = |퐻| = 200 > 0

The second-order condition for maximum is satisfied. Example 6: A monopolistic firm sells a single product in three separate markets. Suppose the monopolistic firm has the specific average revenue functions

푃 = 63 − 4푄 ,푃 = 105− 5푄 ,푃 = 75− 6푄

and that total cost function is

퐶 = 20 + 15푄 + 푄 where 푄 = 푄 + 푄 + 푄

(a) Find the marginal cost and marginal revenue functions. (b) Find the equilibrium quantities and equilibrium prices. (c) Verify that the second-order sufficient condition is met. (d) On the basis of the equilibrium price and quantity, calculate the point elasticity demands

|휀 |, |휀 |, |휀 |.

|휀 | =푑푄푑푃

푃푄 , 푖 = 1,2,3

Which market has the highest and the lowest demand elasticities?

Solution:

(a) The average revenue functions of our monopolist firm are 푃 = 63 − 4푄 so that 푅 = 푃 푄 = 63푄 − 4푄

푃 = 105 − 5푄 so that 푅 = 푃 푄 = 105푄 − 5푄

푃 = 75 − 6푄 so that 푅 = 푃 푄 = 75푄 − 6푄

And the total cost function is

퐶 = 20 + 15푄 + 푄 where 푄 = 푄 + 푄 + 푄

The marginal revenue functions are

휕푅휕푄 = 푅′ = 63− 8푄

휕푅휕푄 = 푅′ = 105− 10푄

휕푅휕푄 = 푅′ = 75 − 12푄

and The marginal cost function is

퐶 ′(푄) = 15 + 2푄 = 15 + 2푄 + 2푄 + 2푄 (b) The profit function is 휋 = 푅 (푄 ) + 푅 (푄 ) + 푅 (푄 )− 퐶(푄)

휋 = 63푄 − 4푄 + 105푄 − 5푄 + 75푄 − 6푄 − 20 − 15푄 −푄

The first partial derivative of 휋 with respect to 푄 is

휋 = 63− 8푄 − 15− 2푄 = 48− 8푄 − 2(푄 + 푄 + 푄 ) 휋 = −10푄 − 2푄 − 2푄 + 48

The first partial derivative of 휋 with respect to 푄 is

휋 = 105− 10푄 − 15− 2푄 = 90− 10푄 − 2(푄 + 푄 + 푄 ) 휋 = −2푄 − 12푄 − 2푄 + 90

The first partial derivative of 휋 with respect to 푄 is

휋 = 75 − 12푄 − 15 − 2푄 휋 = −2푄 − 2푄 − 14푄 + 60

Setting 휋 , 휋 , 휋 to zero, we obtain

10푄 + 2푄 + 2푄 = 48, 2푄 + 12푄 + 2푄 = 90 푎푛푑 2푄 + 2푄 + 14푄 = 60

The solution of above linear system yields the following equilibrium quantities

푄 = 28897 ,푄 = 6

5197 ,푄 = 2

9197

and equilibrium prices are

푃 = 63 − 4 28897 = 51

3697

푃 = 105 − 5 6 = 72

푃 = 75 − 6 29197 = 57

3697

(c) Now we will calculate second order derivatives of profit function

휋 = −10푄 − 2푄 − 2푄 + 48,휋 = −10,휋 = −2,휋 = −2

휋 = −2푄 − 12푄 − 2푄 + 90,휋 = −12,휋 = −2 휋 = −2푄 − 2푄 − 14푄 + 60,휋 = −14

The Hessian matrix is

퐻 =−10 −2 −2−2 −12 −2−2 −2 −14

The leading principal minors of H are

|퐻 | = −10 < 0 | 퐻 | = −10 −2−2 −12 = 116 > 0, | 퐻 | = |퐻| =

−10 −2 −2−2 −12 −2−2 −2 −14

= −1552 < 0

The second order condition is met.

(d) Now the point elasticities are |휀 |, |휀 |, |휀 |.

|휀 | =푑푄푑푃

푃푄

푃 = 63 − 4푄 so that = −4

푃 = 105 − 5푄 so that = −5

푃 = 75 − 6푄 so that = −6

Now

|휀 | =푑푄푑푃

푃푄 = −

14

396 =

138

|휀 | =푑푄푑푃

푃푄 =

43

|휀 | =푑푄푑푃

푃푄 =

32

The highest is |휀 | and lowest is |휀 |

Example 7:

A monopolistic firm sells a single product in three separate markets. Suppose the monopolistic firm has the specific average revenue functions

푃 = 63 − 4푄 ,푃 = 105− 5푄 ,푃 = 75− 6푄

and that total cost function is

퐶 = 20 + 15푄

(a) Find the marginal cost and marginal revenue functions. (b) Find the equilibrium quantities and equilibrium prices. (c) Verify that the second-order sufficient condition is met. (d) On the basis of the equilibrium price and quantity, calculate the point elasticity demands

|휀 |, |휀 |, |휀 |. Which market has the highest and the lowest demand elasticities?

Solution: See solution from book.