Question

Which of the following cost functions exhibits economies of scope when three (3) units of good one and two (2) units of good two are produced?

1) C = 50 − 5Q1Q2 + 0.5Q12 + Q22.

2) C = 15 + 5Q1Q2 + 2Q1 + 4Q2.

3)C = 5 + Q1Q2 + Q12Q22.

4)C = 10 + 4Q1Q2 + Q12 + Q22.

Please tell me the # of the answer.and explain thanks.

Answer 1

Option '1' is correct .

Economies of scope is defined as the savings done on producing two products together instead of producing it differently.

So, economies of scale is defined when,

C(q_{​​​​​​1}​​​​​,q_​​2​​​​​ )< C(q_{​​​​​1}​​​​​ ,0) +C(0, q_{​​​​​​2}​​​ )

Here,q1=3,q2=2.

For option '1',

C(3,2)= 50 - 5*3*2 +0.5*3²+2² = 50-30+4.5+4= 28.5

C(3,0) = 50-5*3*0+0.5*3² + 0²= 50-0+4.5+0= 54.5

C(0,2) = 50-5*0*2 + 0.5*0² + 2² = 50-0+0+4 = 54

Clearly, C(3,2)<C(3,0)+C(0,2) as , 28.5< 54.5+ 54 => 28.5<108.5

So,this cost function exibit economies of scope.

Similarly we can check for other options.

For option 2,

C(3,2) = 15+5*3*2+2*3+4*2= 15+30+6+8= 59

C(3,0)= 15+5*3*0 + 2*3+4*0 = 15+0+6+0= 21

C(0,2)= 15+5*0*2+2*0+4*2=15+0+0+8=23

C(3,0)+C(0,2)=21+23= 44

Clearly ,C(3,2)>C(3,0)+C(0,2) as 59>44 ,so this cost function do not exibit economies of scope.

For option '3'

C(3,2)= 5+3*2+3²*2²= 5+6+9*4=5+6+36=47

C(3,0)= 5+3*0+ 3²*0²=5+0+0 =5

C(0,2) = 5+0*2+0²*2²= 5+0+0=5

C(3,0)+C(0,2)= 5+5=10

Clearly,C(3,2)>C(3,0)+C(0,2) as 47>10,so this cost function do not exhibit economies of scope.

For option 4,

C(3,2) = 10+4*3*2+3²+2²=10+23+9+4= 46

C(3,0)=10+4*3*0+3²+0²= 10+0+9+0=19

C(0,2)= 10+4*0*2+0²+2²= 10+0+0+4=14

C(3,0)+C(0,2)= 19+14=33

Clearly C(3,2)>C(3,0)+C(0,2) as 46>33, so this cost function also does not exhibit economies of scale.

Therefore,Only Option '1' exhibits economies of scale.