Suppose a firm produces widgets on two assembly lines and operates with the foll

Question

Suppose a firm produces widgets on two assembly lines and operates with the following Total Cost function: TC = 3x*2 + 6y*2 – xy
Where x represents the number of widgets produced on one assembly line and y the production on the second. The firm is under contract to produce 20 widgets.

(a) Use the Lagrangian method to calculate the least-cost allocation of production to the two assembly lines.

(b) What is the firm’s total cost equal to if it fulfils its contract?

(c) Give an economic interpretation to the value of the Lagrangian multiplier.

(d) Should this firm accept an offer of $60 to produce an additional widget? How about an offer of $80? Explain.

Explanation / Answer

TCx= 6x-y
TCy= 12y-x
x+y=20
fx=fy=1

So we have
6x-y=

12y-x=

6x-y= 12y-x

7x=13y

x+y=20

7x= 13(20-x)

20x= 260

x=13

y=7

TC=507 +294- 91= 710

The economic interpretation of the Lagrangian multiplier is the marginal cost of production.

No because MC= 6*13-7=71 and it exceeds 60.

Yes because marginal revenue (80) is greater than marginal cost (71).

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