Question

A firm produces plastic water bottles according to the production function ? = ?(?,?) = ? 2?, where ? is the number of water bottles produced, ? is the quantity of labor employed and ? is the quantity of capital. The associated marginal product functions are ??? = 2?? and ??? = ? 2 . The cost per unit of labor and capital are ? = $4 and ? = $2, respectively.

1) If this firm wants to produce a given level of output, does its cost minimization problem have an interior solution or a corner solution? Justify your answer. (5 points)

2) Suppose that the firm wants to produce ? = 1000 units of output. What is the cost minimizing combination of labor and capital to produce this output target? (20 points)

3) Suppose that the price of output is $1 per unit and that in order to start production the firm must rent a workshop at a cost of $950, in addition to the cost of any capital and labor used.

a) If the firm has not yet rented the workshop, would it be better to produce 1000 units that are sold at this price or would it be better to not to enter this market? (3 points)

b) If the firm has already rented the workshop and cannot recover that money even if they shut down production, would it be better to produce 1000 units that are sold at this price or to leave the market? (3 points)

c) If your answers to parts a and b are different explain why. (4 points)

Answer 1

1. The cost minimization problem will have interior solution as isoquants produced by this production function will be negatively sloping and convex to the origin.

2.To obtain the answer first find cost function.This can be obtained by multiplying factors of production with their respective payoffs and then adding them together.After finding cost function take the first derivative with respect to both factors and set equal to zero..By doing so and solving the resulting equations cost minimizing amount of labour and capital can be obtained.

3. Find out the total cost for producing the specified output.Renting a workshop becomes additional cost.If by adding this the firm is able to cover atleast the variable cost of production from the receipts, the firm continues in the market.If not the best option will be not to enter the market.