Question

Exercise 2. Two plants

A firm has two plants, both with increasing marginal costs. Plant 1 is more efficient than plant 2. Specifically, the cost function in plant 1 is c₁(y₁)=2y₁², while the cost function in plant 2 is c₂(y₂)=3y₂². Output produced in plant 1, y₁, is identical to output produced in plant 2, y₂. For any overall output level Y=y₁+y₂the firm wants to minimize costs.

1. What is the marginal cost in plant 1? Find ∂c₁/∂y₁.

2. What is the marginal cost in plant 2? Find ∂c₂/∂y₂.

3. The firm needs to produce a total of Y=10 units. If it decides to produce all of them in plant 1 (y₁=100 and y₂=0), what is the total cost?

4. What is the optimal production level, y₁^*and y₂^*, that would minimize cost? What is the total cost now?

Answer 1

a. Total Cost of plani 1, c1(y1) = 2y12

Marginal cost in plant 1, MC1 = ∂c1/∂y1 = ∂(2y12)/∂y1 = 4y1

b. Total Cost of plani 2, c2(y2) = 3y22

Marginal cost in plant 2, MC2 = ∂c2/∂y2 = ∂(3y22)/∂y2 = 6y2

c. if it produces all of output in firm 1 and total output is Y = y1 = 10

(In the question, it is mentioned that y1=100 and y2=0, but as Y = y1+y2 = 10, then  y1 cannot be greater than 10. So, I have considered y1 = 10 in this case)

Then, the total cost of production is, c1(y1) = 2y12 = 2 * (10)2 = 2 * 100 = 200

d. Optimal production condition in 2 plant model for cost minimization is,

MC1 = MC2

or, 4y1 = 6y2

or, y1 = (3/2)y2

Now,  y1 + y2 = 10

or, (3/2)y2 + y2 = 10

or, (5/2)y2 = 10

or, y2* = 4

y1* = (3/2)y2* = (3/2)*4 = 6

Optimal production level for cost minimization is, y1* = 6 and y2* = 4

Total Cost of plant 1, c1(y1) = 2y12 = 2*62 = 2*36 = 72 unit

Total Cost of plant 2, c2(y2) = 3y22 = 3*42 = 3*16 = 48 unit

So, total cost is, TC(Y) = c1(y1) + c2(y2) = 72 + 48 = 120 unit