Question

2. A firm has a demand and cost function given by: D: P=100-Q, C: P=10+2q1+4q2

a) Demonstrate whether this firm has economies to scale.

b) Demonstrate whether this firm has economies of scope.

c) Assume that the firm is the only firm in the market, derive graphically and numerically the optimal quantity, price and the profits/losses.

d) What is the Lerner Index? Provide a numerical value and explain the relevance of the index/how it is used

Answer 1

A) economies of scale is cost advantage due to increase in scale. Average cost decreases when scale or production Increases.

C=10+2Q1+4Q2

AC=10/Q1+2+4Q2/Q1

DERIVATIVE of AC with respect to Q1

∆AC/∆Q1=-10/Q1^2-4Q2/q1^2=-(10+4Q2)/Q1^2

So negitive DERIVATIVE shows increasing Q1 ,leads to decrease in average cost ,so firm has economies of scale.

B) economies of scope is cost advantage due to production of variety of product together instead separately.

If Q1 and Q2 separately produced then

C(Q1)=10+2Q1

C(Q2)=10+4Q2

C(Q1+Q2)=20+2Q1+4Q2

Together,C(Q1+Q2)=10+2Q1+4Q2

So cost production of q1 and q2 together less than cost production of q1 and q2 separately.

So firm has economies of scope.

C)p=100-Q

P=100-q1-q2

MR1=100-2q1-q2

MC1=2

MR1=MC1

100-2q1-q2=2

Q1=49-0.5q2

MR2=100-2q2-q1

MC2=4

100-2q2-q1=4

Q2=48-0.5q1

Putting q2 into q1

Q1=49-0.5(48-0.5q1)=49-24+0.25q1

0.75q1=25

Q1=25/0.75=100/3=33.33

Q2=48-0.5*100/3=48-50/3=(144-50)/3=94/3=31.33

Q=100/3+94/3=194/3

P=100-194/3=(300-194)/3=106/3=35.33

Profit=TR -TC=64.66*35.33-10-2*33.33-4*31.33=2284.43-10-66.66-125.32=2082.45

D) Lerner index={p-MC}/p=

Average MC=(2+4)/2=6/2=3

Lerner index=(35.33-3)/35.33=32.33/35.33=0.915

Lerner index shows the market power of firm . it's value remain between 0 to 1. Zero means no market power and 1 means complete market power.

It can also be used for calculating Elasticity of demand at equilibrium price .

Lerner index=-1/e