Question

Exercise 14.5

Phillips Industries manufactures a certain product that can be sold directly to retail outlets or to the Superior Company for further processing and eventual sale as a completely different product. The demand function for each of these markets is

Retail Outlets: P1=120−Q1Retail Outlets: P1=120−Q1

Superior Company: P2=40−Q2Superior Company: P2=40−Q2

where P1P1 and P2P2 are the prices charged and Q1Q1 and Q2Q2 are the quantities sold in the respective markets. Phillips’ total cost function for the manufacture of this product is

TC=10+8(Q1+Q2)TC=10+8(Q1+Q2)

What is Phillips’ total profit function?

119Q1−Q12+30Q2−Q22−10119Q1−Q12+30Q2−Q22−10

112Q1−Q12+32Q2−Q22−10112Q1−Q12+32Q2−Q22−10

120Q1−Q12+40Q2−Q22−10120Q1−Q12+40Q2−Q22−10

112Q1−Q12+32Q2−Q22112Q1−Q12+32Q2−Q22

The profit-maximizing levels of price and output for the retail outlets market are

per unit and

units respectively.The profit-maximizing levels of price and output for the Superior company are

per unit and

units respectively.At these levels of output, the marginal revenue in the retail outlets market is

and the marginal revenue in the Superior Company market is

.What are the total profits if Phillips is effectively able to charge different prices in the two markets?

.If Phillips is required by law to charge the same per unit in each market, the profit-maximizing level of price and output are

per unit and

units, respectively. Total profits in this condition are

.

Answer 1

What is Phillips’ total profit function?

Revenue in case of Retail outlets, TR1

TR1=P1*Q1=(120-Q1)*Q1=120Q1-Q1²

Revenue in case of Superior Company, TR2

TR2=P2*Q2=(40-Q2)*Q2=40Q2-Q2²

Total Revenue=TR=TR1+TR2=120-Q1²+40Q2-Q2²

Profit=TR-TC

Profit=(120Q1-Q1²+40Q2-Q2²)-(10+8Q1+8Q2)

Profit=(120-8)Q1+(40-8)Q2-Q1^2-Q2^2-10

Profit=(120-8)Q1+(40-8)Q2-Q1^2-Q2^2-10

Profit=112Q1-Q1^2+32Q2-Q2^2-10

Option (b) is correct

The profit-maximizing levels of price and output for the retail outlets market are per--------- unit and -------units respectively.

Retail Outlets:

P1=120−Q1

Total Revenue=TR1=P1*Q1=(120−Q1)*Q1=120Q1-Q1²

Marginal Revenue=MR1=d(TR1)/dQ1=120-2Q1

TC=10+8(Q1+Q2)

Marginal Cost=MC1=dTC/dQ1=8

Set MR1=MC1 for profit maximization

120-2Q1=8

112=2Q1

Q1=56

P1=120-Q1=120-56=64

Retail Outlets Market Price=64

Output=56

The profit-maximizing levels of price and output for the Superior company are ---------per unit and --------units respectively.

Superior Company:

P2=40−Q2

Total Revenue=TR2=P2*Q2=(40−Q2)*Q2=40Q2-Q2²

Marginal Revenue=MR2=d(TR2)/dQ2=40-2Q2

TC=10+8(Q1+Q2)

Marginal Cost=MC2=dTC/dQ2=8

Set MR2=MC2 for profit maximization

40-2Q2=8

32=2Q2

Q2=16

P1=40-Q2=40-16=24

Superior Company Market Price=24

Output=16

At these levels of output, the marginal revenue in the retail outlets market is ------------- and the marginal revenue in the Superior Company market is --------

Marginal Revenue for retail outlets=MR1=120-2Q1=120-2*56=8

Marginal Revenue for superior company=MR2=40-2Q2=40-2*16=8

What are the total profits if Phillips is effectively able to charge different prices in the two markets?

Profit=112Q1-Q1²+32Q2-Q2²-10 (derived in first part)

Profit=112*56-56²+32*16-16²-10=3382

If Phillips is required by law to charge the same per unit in each market, the profit-maximizing level of price and output are--------- per unit and ---------units respectively.

In this case,

Q=Q1+Q2

And P1=P2=P

TC=10+8(Q1+Q2)=10+8Q

Marginal Cost MC=dTR/dQ=8

Given that in case of Retail Outlets

P1=120−Q1

On rearranging, we get

Q1=120-P1

In case of Superior Company:

P2=40−Q2,

Q2=40-P2

Q=Q1+Q2

Q=(120-P1)+(40-P2)

Set P1=P2=P

Q=(120-P)+(40-P)

Q=160-2P

Or

2P=160-Q

P=80-0.5Q

Total Revenue, TR=P*Q=(80-0.5Q)*Q=80Q-0.5Q²

Marginal Revenue=MR=dTR/dQ=80-Q

Set MR=MC

80-Q=8

Q=72 units

P=80-0.5*72=44 per unit

Total profits in this condition are

TC=10+8Q=10+8*72=586

TR=P*Q=44*72=3168

Profit=TR-TC=3168-586=2582